ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010) 884 ndash 898

DOI 103813AAA918347

Computational Model for Long-RangeNon-Linear Propagation over Urban Cities

T Leissing1) C Soize2) P Jean1) J Defrance1)

1) Universiteacute Paris-Est Centre Scientifique et Technique du Bacirctiment 24 rue Joseph Fourier 38400 Saint MartindrsquoHegraveres France thomasleissingcstbfr

2) Universiteacute Paris-Est Laboratoire Modeacutelisation et Simulation Multi Echelle MSME UMR8208 CNRS5 bd Descartes 77454 Marne-la-Vallee France christiansoizeuniv-paris-estfr

SummaryA computational model for long-range non-linear sound propagation over urban environments is described Firstthe probability model of the geometrical parameters of an urban environment are determined using InformationTheory and the Maximum Entropy Principle The propagation model is then presented it is based on the non-linear parabolic equation (NPE) and its extension to propagation in porous media in which the urban layer ofthe real system is represented by a porous ground layer The uncertainties introduced by the use of this simplifiedmodel and the presence of the variability of the real system are taken into account with a probabilistic model Ref-erence solutions are obtained thanks to the boundary element method (BEM) these experimental observationsare then used to identify the parameters of the probability model This inverse stochastic problem is solved usingan evolutionary algorithm which involves both the mean-square method and the maximum likelihood methodApplications and model validation are then presented for two different urban environment morphologies It isshown that the identification method provides an accurate and robust way for identifying the stochastic modelparameters independently of the variability of the real system Constructed confidence regions are in good agree-ment with the numerical observations

PACS no 4328-g 4350-x

1 Introduction

The long-term aim of this work is to develop a computa-tional model to simulate long-range non-linear wave prop-agation over urban cities We are interested in the effect ofthe surface irregularities (buildings) on the acoustic fieldabove the urban layer This problem could be studied withdeterministic numerical models where buildings geome-tries are explicitly given For example ray tracing methodsor models based on the Eulerrsquos equations allow the envi-ronmental context (buildings meteorological conditionssite topography ) to be taken into account and wouldtechnically be suited for this application However forlong-range propagation applications these methods sufferfrom their numerical complexity and the high computa-tional effort associated Moreover the suitability of thesemodels for the application under interest can be ques-tioned considering the high complexity of the real systemthe model approximations (eg using the uniform theoryof diffraction for ray-tracing methods) and the uncertain-ties on the model parameters (eg buildings geometries)the computational model could be improved introducing aprobabilistic model

Received 5 August 2009accepted 3 March 2010

In this work a different approach is proposed It con-sists in using a very simple model for the urban and atmo-spheric layers Propagation is modelled with a non-linearparabolic equation (NPE) whose fundamental principle isthe solution of a one-way non-linear wave equation over amoving window surrounding the wavefront Since the cal-culation domain is limited to a small area around the sig-nal computational cost is generally reduced compared toEulerrsquos equations methods For the propagation in the ur-ban layer a NPE for porous ground layers is used There-fore the simplified model is composed of two domains

1 the atmospheric layer where propagation is modelledby a NPE for air

2 the urban city layer where propagation is modelled bya NPE for porous ground surface

Equations to couple the two domains complete the sim-plified model of sound propagation over urban cities Us-ing this model as a predictive model for the real systemwill show very poor performances as the modelling cho-sen is far too simplistic to represent the real system com-plexity Hence a probabilistic approach of uncertainties isused to enhance the model capabilities Section 11 belowdescribes the scope and the range of applicability of sucha propagation model

884 copy S Hirzel Verlag middot EAA

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

11 Scope and range of applicability

The work presented in this paper was performed to studyhigh-amplitude wave propagation in complex media [1]The sound source of such a wave could for example be asonic bang or the wave produced by a ground-level explo-sion High-amplitude effects are hence considered and thefrequency range under interest is restricted to 0ndash100 Hz

It is not the intent here to develop an engineering modelfor the calculation of noise exposure in urban areas (suchas the Nord 2000 model see [2] or the Harmonoise modelsee [3 4]) but rather to present a method for the construc-tion of a stochastic model for propagation over irregularsurfaces such as an urban environment The propagationover the urban environment and not within is hence con-sidered in this paper

The urban environment is further characterised by a setof three random variables the heights and widths of thebuildings and the spacing between two consecutive build-ings (see section 3 for a description of these parameters)It is not the intent here to determine what the statisticaldependencies between these parameters are The probabil-ity distributions of these three random variables are de-termined using Information Theory [5] and the maximumentropy principle [6]

It was chosen to place the source above the urban citylayer (see section 4 for a complete description of the con-figuration) this could correspond to the case of the propa-gation of a sonic bang As a description of the sound fieldabove the city layer is sought the receiver is placed at thesame altitude as the source Note that nothing prevents oneto use the presented method for source and receiver loca-tions at any other location or for a wider frequency range

As explained in section 1 a NPE model is used as abasis for the development of the stochastic propagationmodel This choice was made regarding the application un-der interest in this work The NPE model is well adaptedfor long-range sound propagation applications and it canaccount for most features of weakly non-linear soundpropagation outdoors geometrical spreading weak non-linearities refraction effects (see for example [7 8]) sitetopography [9] ground impedance [10] and thermovis-cous effects [11] Different models such as the Fast FieldProgram (FFP see [12 13]) or the (linear frequency-domain) Parabolic Equation (PE see [14]) could as wellbe used The method for the construction of the stochas-tic model presented in this paper is general and is not re-stricted to the use of a specific propagation model

Section 2 explains and details the methodology to con-struct such a stochastic sound propagation model and theprobabilistic model of its parameters and outlines the con-tent of the paper

2 Construction of the computationalmodel principles and methodology

The objective is to develop a stochastic model for soundpropagation over urban cities using a NPE model origi-nally designed for sound propagation over porous ground

layers in which the urban city is taken into accountthrough independent random porous layer parametersnoted Γ Λ and Θ The probability distributions of theserandom variables depend on a parameter vector w (boldletters denote vectors throughout the paper) For exampleparameter w may contain the mean values and the stan-dard deviations associated with the random parameters ΓΛ and Θ The model output is the pressure at the receiverwhich is noted Pr (ω) A second propagation model inwhich the urban city is explicitly accounted for is usedto provide reference solutions The urban city geometryis characterised by a parameter vector u For example umay contain the mean values and standard deviations as-sociated with the heights and widths of the buildings (pa-rameter u is defined in section 3) For a given parameteru and several probability models an urban city realisationcan be generated and then used in the reference model toobtain the pressure P

expr at the receiver Figure 1 shows a

sketch and a diagram that detail the basic principle of eachmodel

To construct the stochastic sound propagation modelfor a given parameter u one has1 to construct the probability models of the urban city

geometrical parameters From one city to another onegeometrical parameters (eg building density mean el-evation ) can greatly vary The construction of aprobabilistic model of these parameters should hencebe done with measured data (for example data froma geographical information system) Moreover townplanning and buildings themselves complies with manyconstraints which introduce a statistical dependence be-tween the geometrical parameters The determinationof these dependencies being out of the scope of thiswork probability models are constructed assuming thatno information is available concerning the relations be-tween geometrical parameters The probability modelsare determined with the help of Information Theory[5] and the Maximum Entropy Principle [6] Once theprobability models of the geometrical parameters aredetermined different city realisations corresponding toa given parameter u can be generated

2 to use the city realisations generated in (1) and to per-form simulations with the reference model in orderto obtain statistical information on the model outputP

expr (ω)

3 to construct the probability models of the random pa-rameters ΓΛ andΘ These models are determined withthe help of Information Theory and the Maximum En-tropy Principle and depend on parameter w

4 using the outputs from the reference model Pexpr (ω)

(step 2) to identify parameter w corresponding to thegiven parameter u previously fixed This identifica-tion is done by solving an inverse stochastic problemthe ldquodistancerdquo between Pr (ω w) and P

expr (ω) is min-

imised so that the optimal parameter wopt is obtainedOnce wopt is determined the stochastic NPE model canbe used to study non-linear wave propagation over ur-ban cities

885

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 1 Sketches and diagrams detailing the basic principles of each model The reference linear propagation model is presented onthe left while the stochastic simplified NPE model is shown on the right

The paper is organised as follows Section 3 presents themethod to construct the probability models of an urbancity (step 1) The linear reference propagation model usedto obtain reference solutions (step 2) is detailed in section4 while the mean NPE model for sound propagation overurban cities and its parameters are presented in section 5In section 6 the probability models of the NPE model ran-dom variables are determined (step 3) The stochastic NPEmodel is then presented in section 7 Section 8 deals witha hybrid method based on the mean-square and the max-imum likelihood methods to solve the inverse stochasticproblem and to identify the parameter wopt (step 4) Fi-nally section 9 presents an application and a validationprocedure for the computational model of sound propaga-tion over urban cities Conclusions and perspectives aregiven in section 10

3 Prior probabilistic model of geometricalparameters of an urban city

A two dimensional cross-section of an urban city is con-sidered It is composed of n buildings of rectangular shapeparametrised by a set of three parameters For a given

building Bi its height and width are noted hi and wi andthe distance between two consecutive buildings Bi andBi+1 is noted di (see Figure 2) The prior probability modelof such an urban city is then introduced with the helpof the random variables H1 Hn W1 Wn D1 Dnminus1

Let Xi be the positive-valued real random variable rep-resenting either Hi either Wi or Di Let pXi

be the prob-ability density function of Xi and E be the mathematicalexpectation Let mXi

and δXi= σXi

mXibe its mean value

and its coefficient of variation in which σ2Xi

= m2 minusm2Xi

isthe variance and m2 the second-order moment One has

mXi= E Xi =

+infin

0xi pXi

dxi (1a)

m2 = E X2i =

+infin

0x2

i pXidxi (1b)

In order to construct the probability distributions of therandom variables H1 Hn W1 Wn D1 Dnminus1 Information Theory [5] and the Maximum En-tropy Principle [6] are used One then has to define theavailable information for these random variables which isthe following

886

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 2 Sketch of a representative urban city

Figure 3 Example of a city realisation with n = 20 and u = (mH mW mD δH δW δD) = (10 20 30 02 02 02)

For all i denoting by Xi either Hi either Wi or Di onehas1 Xi is a random variable with values in ] 0+infin [2 The mean value mXi

= E Xi is given and is equal tomX independent of i

3 The inverse Xminus1i of Xi is a second-order random vari-

ableThis condition is satisfied if

E log(Xi) = ci with |ci| lt +infin (2)

The constraint defined by equation (2) introduces an arbi-trary constant ci which does not have any physical mean-ing and which is then rewritten as a function of the coeffi-cient of variation of Xi which is δX independent of i Theuse of the Maximum Entropy Principle yields [15]

pH1HnW1WnD1Dnminus1

middot h1 hn w1 wn d1 dnminus1 (3)

=n

i=1

pHi(hi)

n

i=1

pWi(Wi)

nminus1

i=1

pDi(di)

with

pHi(hi) = pX (hi mH δH ) (4a)

pWi(wi) = pX (wi mW δW ) (4b)

pDi(di) = pX (di mD δD) (4c)

where

pX (x mX δX ) = ]0+infin[(x)1

mX

1

δ2X

(5)

1

Γ(1δ2X )

x

mX

1δ2X

minus1exp minus x

δ2XmX

In equation (5) ]0+infin[(x) = 1 if x gt 0 and 0 otherwiseand Γ(z) is the gamma function of argument z defined by

Γ(z) =+infin

0tzminus1eminust dt (6)

Since no available information concerning the statisti-cal dependence between the families of random variablesHii Wii and Dii is used the Maximum EntropyPrinciple yields independence of all the random variablesas a result as it can be seen in equation (3)

The following vector u of the parameters of the proba-bilistic model is introduced

u = mH mW mD δH δW δD (7)

Parameter u hence contains the mean values and the dis-persion parameters associated with the heights widths andspacings of the buildings It belongs to an admissible setU = (]0+infin[)6 Figure 3 shows an example of a realisa-tion with n = 20 and u = (10 20 30 02 02 02)

4 Reference model linear propagationover urban cities

This section presents the stochastic linear propagationmodel used to obtain reference solutions of the problemThe problem setting (source and receiver frequency range ) is first given in section 41 and the stochastic propa-gation model itself is then described in section 42

41 Setting the problem

Output Pexpr (ω) from the reference model is analysed in

the frequency domain on a frequency band defined byB =]0 ωmax] in which ωmax is such that B is a low fre-quency band Hence the buildings surfaces are assumed

887

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

to be acoustically rigid (no absorption and specular reflec-tions) The source and receiver are placed above the ur-ban city layer and the source is placed sufficiently far fromthe first building B1 to assume that the wave impinging onthis first building is a plane wave The receiver is placed atthe same altitude as the source and at the right of the lastbuilding Bn The quantity under interest is the normalisedspectrum Lexp(ω) at the receiver such that

Lexp(ω) = 10 log10P

expr (ω)

pfree(ω)

2

(8)

where pfree(ω) and Pexpr (ω) denote free field pressure and

the pressure in the presence of the urban city at the re-ceiver respectively Note that the reference model useddoes not account for building surfaces vibrations

42 Stochastic BEM solver for constructing a refer-ence solution

The model used to construct the reference solution is animplementation of the boundary element method [16 17](BEM) in which the city geometry is explicitly enteredinto the computational model For a fixed parameter uthe outputs Lexp(ω) are calculated using the Monte Carlomethod [18] with νexp independent realisations η1 ηνexp

of urban cities generated with the probability models pre-sented in section 3 The realisation of the experimental ob-servation for the urban city νp is Lexp(ω ηνp

) ω isin B

5 Mean non-linear parabolic propagationmodel for sound propagation over urbancities

This section describes the mean parabolic propagationmodel that will be later used associated with a probabilis-tic model of uncertainties to model sound propagationover urban environments Section 51 presents a generaloverview of propagation models based on the non-linearparabolic equation (NPE) section 52 formally defines thepropagation model and section 53 explains the choice ofthe propagation model parameters and details their alge-braic properties

51 General overview of NPE models

The non-linear parabolic equation (NPE) has first been de-veloped by McDonald and Kuperman in 1987 [19] (seealso [7 8]) and has been successfully used for underwateracoustics simulations [20 21] and blast wave propagationin air [22 23 24] The NPE model for a 2D domain withCartesian coordinates (x z) filed with air is

partt r(x z t) = minuspartx(c1r(x z t) + c0

β

2r(x z t)2)

minus c0

2part2

zr(x z t) dx (9)

where parti means partial derivation with respect to variablei x is the main propagation direction z is the transverse

propagation direction and t is the time variable The am-bient sound speed is c0 while c1 is the sound speed per-turbation in the moving window ie c1 = c(x z) minus c0 where c(x z) is the spatially-dependent sound speed Thedimensionless over-density variable is r(x z t) such thatr(x z t) = ρ (x z t)ρ0 with ρ (x z t) the acous-tic density perturbation and ρ0 the ambient medium den-sity In the following the (x z t) dependence is droppedie r(x z t) is written r For air the coefficient of non-linearity β is calculated with the help of the ratio of spe-cific heats γ ie β = (γ + 1)2 The first term on the righthand side of equation (9) simulates refraction and non-linear effects The second term accounts for propagationin the transverse direction The moving window operatorpartt is defined by

partt = partt + c0partx (10)

Note that in equation (9) the azimuthal spreading termc0r(2d) in which d is the distance from the source tothe point where the field is calculated has been droppedfrom the original NPE [19] There is no absorption fromair included in the model The NPE model derives fromEulerrsquos equations The assumptions used are

1 weak non-linearities ie ρ ρ0 12 weak sound speed perturbations ie c1 c0 3 propagation along a main direction

Various modifications and extensions to this original mo-del were made during the past two decades Spherical andcylindrical coordinate system versions [25] and high-angleformulation [26] have been developed Too and Lee [11]has added a complementary term in the NPE equation inorder to take into account thermoviscous effects Propa-gation in multiple media [20] and propagation through at-mospheric turbulences [27] were also successfully studiedusing this model

The NPE differential equation (9) is discretised with thefinite difference method The Crank-Nicolson method isused for linear terms yielding a tridiagonal system whichcan be solved with a Thomas algorithm (see for example[28]) For the non-linear term one has to use a specialisedalgorithm which is able to propagate discontinuities andwhich is stable In the numerical implementation used inthis work the flux corrected transport algorithm (FCT)[29 30] is used The principle of the FCT algorithm is tointroduce some artificial viscosity in the numerical schemeto limit the Gibb oscillations

In a previous work the original NPE model has beenextended to propagation within and over porous groundlayers [9 31] In this model the porous layer is assumedto be equivalent to a continuous fluid medium A wavecauses a vibration of air particles contained in the groundpores while the ground frame does not vibrate The modelderivation and the applications to propagation over porouslayers can be found in [10] In this work it is proposed touse this model to study long-range sound propagation overurban cities

888

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

889

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

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Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

11 Scope and range of applicability

The work presented in this paper was performed to studyhigh-amplitude wave propagation in complex media [1]The sound source of such a wave could for example be asonic bang or the wave produced by a ground-level explo-sion High-amplitude effects are hence considered and thefrequency range under interest is restricted to 0ndash100 Hz

It is not the intent here to develop an engineering modelfor the calculation of noise exposure in urban areas (suchas the Nord 2000 model see [2] or the Harmonoise modelsee [3 4]) but rather to present a method for the construc-tion of a stochastic model for propagation over irregularsurfaces such as an urban environment The propagationover the urban environment and not within is hence con-sidered in this paper

The urban environment is further characterised by a setof three random variables the heights and widths of thebuildings and the spacing between two consecutive build-ings (see section 3 for a description of these parameters)It is not the intent here to determine what the statisticaldependencies between these parameters are The probabil-ity distributions of these three random variables are de-termined using Information Theory [5] and the maximumentropy principle [6]

It was chosen to place the source above the urban citylayer (see section 4 for a complete description of the con-figuration) this could correspond to the case of the propa-gation of a sonic bang As a description of the sound fieldabove the city layer is sought the receiver is placed at thesame altitude as the source Note that nothing prevents oneto use the presented method for source and receiver loca-tions at any other location or for a wider frequency range

As explained in section 1 a NPE model is used as abasis for the development of the stochastic propagationmodel This choice was made regarding the application un-der interest in this work The NPE model is well adaptedfor long-range sound propagation applications and it canaccount for most features of weakly non-linear soundpropagation outdoors geometrical spreading weak non-linearities refraction effects (see for example [7 8]) sitetopography [9] ground impedance [10] and thermovis-cous effects [11] Different models such as the Fast FieldProgram (FFP see [12 13]) or the (linear frequency-domain) Parabolic Equation (PE see [14]) could as wellbe used The method for the construction of the stochas-tic model presented in this paper is general and is not re-stricted to the use of a specific propagation model

Section 2 explains and details the methodology to con-struct such a stochastic sound propagation model and theprobabilistic model of its parameters and outlines the con-tent of the paper

2 Construction of the computationalmodel principles and methodology

The objective is to develop a stochastic model for soundpropagation over urban cities using a NPE model origi-nally designed for sound propagation over porous ground

layers in which the urban city is taken into accountthrough independent random porous layer parametersnoted Γ Λ and Θ The probability distributions of theserandom variables depend on a parameter vector w (boldletters denote vectors throughout the paper) For exampleparameter w may contain the mean values and the stan-dard deviations associated with the random parameters ΓΛ and Θ The model output is the pressure at the receiverwhich is noted Pr (ω) A second propagation model inwhich the urban city is explicitly accounted for is usedto provide reference solutions The urban city geometryis characterised by a parameter vector u For example umay contain the mean values and standard deviations as-sociated with the heights and widths of the buildings (pa-rameter u is defined in section 3) For a given parameteru and several probability models an urban city realisationcan be generated and then used in the reference model toobtain the pressure P

expr at the receiver Figure 1 shows a

sketch and a diagram that detail the basic principle of eachmodel

To construct the stochastic sound propagation modelfor a given parameter u one has1 to construct the probability models of the urban city

geometrical parameters From one city to another onegeometrical parameters (eg building density mean el-evation ) can greatly vary The construction of aprobabilistic model of these parameters should hencebe done with measured data (for example data froma geographical information system) Moreover townplanning and buildings themselves complies with manyconstraints which introduce a statistical dependence be-tween the geometrical parameters The determinationof these dependencies being out of the scope of thiswork probability models are constructed assuming thatno information is available concerning the relations be-tween geometrical parameters The probability modelsare determined with the help of Information Theory[5] and the Maximum Entropy Principle [6] Once theprobability models of the geometrical parameters aredetermined different city realisations corresponding toa given parameter u can be generated

2 to use the city realisations generated in (1) and to per-form simulations with the reference model in orderto obtain statistical information on the model outputP

expr (ω)

3 to construct the probability models of the random pa-rameters ΓΛ andΘ These models are determined withthe help of Information Theory and the Maximum En-tropy Principle and depend on parameter w

4 using the outputs from the reference model Pexpr (ω)

(step 2) to identify parameter w corresponding to thegiven parameter u previously fixed This identifica-tion is done by solving an inverse stochastic problemthe ldquodistancerdquo between Pr (ω w) and P

expr (ω) is min-

imised so that the optimal parameter wopt is obtainedOnce wopt is determined the stochastic NPE model canbe used to study non-linear wave propagation over ur-ban cities

885

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 1 Sketches and diagrams detailing the basic principles of each model The reference linear propagation model is presented onthe left while the stochastic simplified NPE model is shown on the right

The paper is organised as follows Section 3 presents themethod to construct the probability models of an urbancity (step 1) The linear reference propagation model usedto obtain reference solutions (step 2) is detailed in section4 while the mean NPE model for sound propagation overurban cities and its parameters are presented in section 5In section 6 the probability models of the NPE model ran-dom variables are determined (step 3) The stochastic NPEmodel is then presented in section 7 Section 8 deals witha hybrid method based on the mean-square and the max-imum likelihood methods to solve the inverse stochasticproblem and to identify the parameter wopt (step 4) Fi-nally section 9 presents an application and a validationprocedure for the computational model of sound propaga-tion over urban cities Conclusions and perspectives aregiven in section 10

3 Prior probabilistic model of geometricalparameters of an urban city

A two dimensional cross-section of an urban city is con-sidered It is composed of n buildings of rectangular shapeparametrised by a set of three parameters For a given

building Bi its height and width are noted hi and wi andthe distance between two consecutive buildings Bi andBi+1 is noted di (see Figure 2) The prior probability modelof such an urban city is then introduced with the helpof the random variables H1 Hn W1 Wn D1 Dnminus1

Let Xi be the positive-valued real random variable rep-resenting either Hi either Wi or Di Let pXi

be the prob-ability density function of Xi and E be the mathematicalexpectation Let mXi

and δXi= σXi

mXibe its mean value

and its coefficient of variation in which σ2Xi

= m2 minusm2Xi

isthe variance and m2 the second-order moment One has

mXi= E Xi =

+infin

0xi pXi

dxi (1a)

m2 = E X2i =

+infin

0x2

i pXidxi (1b)

In order to construct the probability distributions of therandom variables H1 Hn W1 Wn D1 Dnminus1 Information Theory [5] and the Maximum En-tropy Principle [6] are used One then has to define theavailable information for these random variables which isthe following

886

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 2 Sketch of a representative urban city

Figure 3 Example of a city realisation with n = 20 and u = (mH mW mD δH δW δD) = (10 20 30 02 02 02)

For all i denoting by Xi either Hi either Wi or Di onehas1 Xi is a random variable with values in ] 0+infin [2 The mean value mXi

= E Xi is given and is equal tomX independent of i

3 The inverse Xminus1i of Xi is a second-order random vari-

ableThis condition is satisfied if

E log(Xi) = ci with |ci| lt +infin (2)

The constraint defined by equation (2) introduces an arbi-trary constant ci which does not have any physical mean-ing and which is then rewritten as a function of the coeffi-cient of variation of Xi which is δX independent of i Theuse of the Maximum Entropy Principle yields [15]

pH1HnW1WnD1Dnminus1

middot h1 hn w1 wn d1 dnminus1 (3)

=n

i=1

pHi(hi)

n

i=1

pWi(Wi)

nminus1

i=1

pDi(di)

with

pHi(hi) = pX (hi mH δH ) (4a)

pWi(wi) = pX (wi mW δW ) (4b)

pDi(di) = pX (di mD δD) (4c)

where

pX (x mX δX ) = ]0+infin[(x)1

mX

1

δ2X

(5)

1

Γ(1δ2X )

x

mX

1δ2X

minus1exp minus x

δ2XmX

In equation (5) ]0+infin[(x) = 1 if x gt 0 and 0 otherwiseand Γ(z) is the gamma function of argument z defined by

Γ(z) =+infin

0tzminus1eminust dt (6)

Since no available information concerning the statisti-cal dependence between the families of random variablesHii Wii and Dii is used the Maximum EntropyPrinciple yields independence of all the random variablesas a result as it can be seen in equation (3)

The following vector u of the parameters of the proba-bilistic model is introduced

u = mH mW mD δH δW δD (7)

Parameter u hence contains the mean values and the dis-persion parameters associated with the heights widths andspacings of the buildings It belongs to an admissible setU = (]0+infin[)6 Figure 3 shows an example of a realisa-tion with n = 20 and u = (10 20 30 02 02 02)

4 Reference model linear propagationover urban cities

This section presents the stochastic linear propagationmodel used to obtain reference solutions of the problemThe problem setting (source and receiver frequency range ) is first given in section 41 and the stochastic propa-gation model itself is then described in section 42

41 Setting the problem

Output Pexpr (ω) from the reference model is analysed in

the frequency domain on a frequency band defined byB =]0 ωmax] in which ωmax is such that B is a low fre-quency band Hence the buildings surfaces are assumed

887

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

to be acoustically rigid (no absorption and specular reflec-tions) The source and receiver are placed above the ur-ban city layer and the source is placed sufficiently far fromthe first building B1 to assume that the wave impinging onthis first building is a plane wave The receiver is placed atthe same altitude as the source and at the right of the lastbuilding Bn The quantity under interest is the normalisedspectrum Lexp(ω) at the receiver such that

Lexp(ω) = 10 log10P

expr (ω)

pfree(ω)

2

(8)

where pfree(ω) and Pexpr (ω) denote free field pressure and

the pressure in the presence of the urban city at the re-ceiver respectively Note that the reference model useddoes not account for building surfaces vibrations

42 Stochastic BEM solver for constructing a refer-ence solution

The model used to construct the reference solution is animplementation of the boundary element method [16 17](BEM) in which the city geometry is explicitly enteredinto the computational model For a fixed parameter uthe outputs Lexp(ω) are calculated using the Monte Carlomethod [18] with νexp independent realisations η1 ηνexp

of urban cities generated with the probability models pre-sented in section 3 The realisation of the experimental ob-servation for the urban city νp is Lexp(ω ηνp

) ω isin B

5 Mean non-linear parabolic propagationmodel for sound propagation over urbancities

This section describes the mean parabolic propagationmodel that will be later used associated with a probabilis-tic model of uncertainties to model sound propagationover urban environments Section 51 presents a generaloverview of propagation models based on the non-linearparabolic equation (NPE) section 52 formally defines thepropagation model and section 53 explains the choice ofthe propagation model parameters and details their alge-braic properties

51 General overview of NPE models

The non-linear parabolic equation (NPE) has first been de-veloped by McDonald and Kuperman in 1987 [19] (seealso [7 8]) and has been successfully used for underwateracoustics simulations [20 21] and blast wave propagationin air [22 23 24] The NPE model for a 2D domain withCartesian coordinates (x z) filed with air is

partt r(x z t) = minuspartx(c1r(x z t) + c0

β

2r(x z t)2)

minus c0

2part2

zr(x z t) dx (9)

where parti means partial derivation with respect to variablei x is the main propagation direction z is the transverse

propagation direction and t is the time variable The am-bient sound speed is c0 while c1 is the sound speed per-turbation in the moving window ie c1 = c(x z) minus c0 where c(x z) is the spatially-dependent sound speed Thedimensionless over-density variable is r(x z t) such thatr(x z t) = ρ (x z t)ρ0 with ρ (x z t) the acous-tic density perturbation and ρ0 the ambient medium den-sity In the following the (x z t) dependence is droppedie r(x z t) is written r For air the coefficient of non-linearity β is calculated with the help of the ratio of spe-cific heats γ ie β = (γ + 1)2 The first term on the righthand side of equation (9) simulates refraction and non-linear effects The second term accounts for propagationin the transverse direction The moving window operatorpartt is defined by

partt = partt + c0partx (10)

Note that in equation (9) the azimuthal spreading termc0r(2d) in which d is the distance from the source tothe point where the field is calculated has been droppedfrom the original NPE [19] There is no absorption fromair included in the model The NPE model derives fromEulerrsquos equations The assumptions used are

1 weak non-linearities ie ρ ρ0 12 weak sound speed perturbations ie c1 c0 3 propagation along a main direction

Various modifications and extensions to this original mo-del were made during the past two decades Spherical andcylindrical coordinate system versions [25] and high-angleformulation [26] have been developed Too and Lee [11]has added a complementary term in the NPE equation inorder to take into account thermoviscous effects Propa-gation in multiple media [20] and propagation through at-mospheric turbulences [27] were also successfully studiedusing this model

The NPE differential equation (9) is discretised with thefinite difference method The Crank-Nicolson method isused for linear terms yielding a tridiagonal system whichcan be solved with a Thomas algorithm (see for example[28]) For the non-linear term one has to use a specialisedalgorithm which is able to propagate discontinuities andwhich is stable In the numerical implementation used inthis work the flux corrected transport algorithm (FCT)[29 30] is used The principle of the FCT algorithm is tointroduce some artificial viscosity in the numerical schemeto limit the Gibb oscillations

In a previous work the original NPE model has beenextended to propagation within and over porous groundlayers [9 31] In this model the porous layer is assumedto be equivalent to a continuous fluid medium A wavecauses a vibration of air particles contained in the groundpores while the ground frame does not vibrate The modelderivation and the applications to propagation over porouslayers can be found in [10] In this work it is proposed touse this model to study long-range sound propagation overurban cities

888

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

889

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 1 Sketches and diagrams detailing the basic principles of each model The reference linear propagation model is presented onthe left while the stochastic simplified NPE model is shown on the right

The paper is organised as follows Section 3 presents themethod to construct the probability models of an urbancity (step 1) The linear reference propagation model usedto obtain reference solutions (step 2) is detailed in section4 while the mean NPE model for sound propagation overurban cities and its parameters are presented in section 5In section 6 the probability models of the NPE model ran-dom variables are determined (step 3) The stochastic NPEmodel is then presented in section 7 Section 8 deals witha hybrid method based on the mean-square and the max-imum likelihood methods to solve the inverse stochasticproblem and to identify the parameter wopt (step 4) Fi-nally section 9 presents an application and a validationprocedure for the computational model of sound propaga-tion over urban cities Conclusions and perspectives aregiven in section 10

3 Prior probabilistic model of geometricalparameters of an urban city

A two dimensional cross-section of an urban city is con-sidered It is composed of n buildings of rectangular shapeparametrised by a set of three parameters For a given

building Bi its height and width are noted hi and wi andthe distance between two consecutive buildings Bi andBi+1 is noted di (see Figure 2) The prior probability modelof such an urban city is then introduced with the helpof the random variables H1 Hn W1 Wn D1 Dnminus1

Let Xi be the positive-valued real random variable rep-resenting either Hi either Wi or Di Let pXi

be the prob-ability density function of Xi and E be the mathematicalexpectation Let mXi

and δXi= σXi

mXibe its mean value

and its coefficient of variation in which σ2Xi

= m2 minusm2Xi

isthe variance and m2 the second-order moment One has

mXi= E Xi =

+infin

0xi pXi

dxi (1a)

m2 = E X2i =

+infin

0x2

i pXidxi (1b)

In order to construct the probability distributions of therandom variables H1 Hn W1 Wn D1 Dnminus1 Information Theory [5] and the Maximum En-tropy Principle [6] are used One then has to define theavailable information for these random variables which isthe following

886

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 2 Sketch of a representative urban city

Figure 3 Example of a city realisation with n = 20 and u = (mH mW mD δH δW δD) = (10 20 30 02 02 02)

For all i denoting by Xi either Hi either Wi or Di onehas1 Xi is a random variable with values in ] 0+infin [2 The mean value mXi

= E Xi is given and is equal tomX independent of i

3 The inverse Xminus1i of Xi is a second-order random vari-

ableThis condition is satisfied if

E log(Xi) = ci with |ci| lt +infin (2)

The constraint defined by equation (2) introduces an arbi-trary constant ci which does not have any physical mean-ing and which is then rewritten as a function of the coeffi-cient of variation of Xi which is δX independent of i Theuse of the Maximum Entropy Principle yields [15]

pH1HnW1WnD1Dnminus1

middot h1 hn w1 wn d1 dnminus1 (3)

=n

i=1

pHi(hi)

n

i=1

pWi(Wi)

nminus1

i=1

pDi(di)

with

pHi(hi) = pX (hi mH δH ) (4a)

pWi(wi) = pX (wi mW δW ) (4b)

pDi(di) = pX (di mD δD) (4c)

where

pX (x mX δX ) = ]0+infin[(x)1

mX

1

δ2X

(5)

1

Γ(1δ2X )

x

mX

1δ2X

minus1exp minus x

δ2XmX

In equation (5) ]0+infin[(x) = 1 if x gt 0 and 0 otherwiseand Γ(z) is the gamma function of argument z defined by

Γ(z) =+infin

0tzminus1eminust dt (6)

Since no available information concerning the statisti-cal dependence between the families of random variablesHii Wii and Dii is used the Maximum EntropyPrinciple yields independence of all the random variablesas a result as it can be seen in equation (3)

The following vector u of the parameters of the proba-bilistic model is introduced

u = mH mW mD δH δW δD (7)

Parameter u hence contains the mean values and the dis-persion parameters associated with the heights widths andspacings of the buildings It belongs to an admissible setU = (]0+infin[)6 Figure 3 shows an example of a realisa-tion with n = 20 and u = (10 20 30 02 02 02)

4 Reference model linear propagationover urban cities

This section presents the stochastic linear propagationmodel used to obtain reference solutions of the problemThe problem setting (source and receiver frequency range ) is first given in section 41 and the stochastic propa-gation model itself is then described in section 42

41 Setting the problem

Output Pexpr (ω) from the reference model is analysed in

the frequency domain on a frequency band defined byB =]0 ωmax] in which ωmax is such that B is a low fre-quency band Hence the buildings surfaces are assumed

887

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

to be acoustically rigid (no absorption and specular reflec-tions) The source and receiver are placed above the ur-ban city layer and the source is placed sufficiently far fromthe first building B1 to assume that the wave impinging onthis first building is a plane wave The receiver is placed atthe same altitude as the source and at the right of the lastbuilding Bn The quantity under interest is the normalisedspectrum Lexp(ω) at the receiver such that

Lexp(ω) = 10 log10P

expr (ω)

pfree(ω)

2

(8)

where pfree(ω) and Pexpr (ω) denote free field pressure and

the pressure in the presence of the urban city at the re-ceiver respectively Note that the reference model useddoes not account for building surfaces vibrations

42 Stochastic BEM solver for constructing a refer-ence solution

The model used to construct the reference solution is animplementation of the boundary element method [16 17](BEM) in which the city geometry is explicitly enteredinto the computational model For a fixed parameter uthe outputs Lexp(ω) are calculated using the Monte Carlomethod [18] with νexp independent realisations η1 ηνexp

of urban cities generated with the probability models pre-sented in section 3 The realisation of the experimental ob-servation for the urban city νp is Lexp(ω ηνp

) ω isin B

5 Mean non-linear parabolic propagationmodel for sound propagation over urbancities

This section describes the mean parabolic propagationmodel that will be later used associated with a probabilis-tic model of uncertainties to model sound propagationover urban environments Section 51 presents a generaloverview of propagation models based on the non-linearparabolic equation (NPE) section 52 formally defines thepropagation model and section 53 explains the choice ofthe propagation model parameters and details their alge-braic properties

51 General overview of NPE models

The non-linear parabolic equation (NPE) has first been de-veloped by McDonald and Kuperman in 1987 [19] (seealso [7 8]) and has been successfully used for underwateracoustics simulations [20 21] and blast wave propagationin air [22 23 24] The NPE model for a 2D domain withCartesian coordinates (x z) filed with air is

partt r(x z t) = minuspartx(c1r(x z t) + c0

β

2r(x z t)2)

minus c0

2part2

zr(x z t) dx (9)

where parti means partial derivation with respect to variablei x is the main propagation direction z is the transverse

propagation direction and t is the time variable The am-bient sound speed is c0 while c1 is the sound speed per-turbation in the moving window ie c1 = c(x z) minus c0 where c(x z) is the spatially-dependent sound speed Thedimensionless over-density variable is r(x z t) such thatr(x z t) = ρ (x z t)ρ0 with ρ (x z t) the acous-tic density perturbation and ρ0 the ambient medium den-sity In the following the (x z t) dependence is droppedie r(x z t) is written r For air the coefficient of non-linearity β is calculated with the help of the ratio of spe-cific heats γ ie β = (γ + 1)2 The first term on the righthand side of equation (9) simulates refraction and non-linear effects The second term accounts for propagationin the transverse direction The moving window operatorpartt is defined by

partt = partt + c0partx (10)

Note that in equation (9) the azimuthal spreading termc0r(2d) in which d is the distance from the source tothe point where the field is calculated has been droppedfrom the original NPE [19] There is no absorption fromair included in the model The NPE model derives fromEulerrsquos equations The assumptions used are

1 weak non-linearities ie ρ ρ0 12 weak sound speed perturbations ie c1 c0 3 propagation along a main direction

Various modifications and extensions to this original mo-del were made during the past two decades Spherical andcylindrical coordinate system versions [25] and high-angleformulation [26] have been developed Too and Lee [11]has added a complementary term in the NPE equation inorder to take into account thermoviscous effects Propa-gation in multiple media [20] and propagation through at-mospheric turbulences [27] were also successfully studiedusing this model

The NPE differential equation (9) is discretised with thefinite difference method The Crank-Nicolson method isused for linear terms yielding a tridiagonal system whichcan be solved with a Thomas algorithm (see for example[28]) For the non-linear term one has to use a specialisedalgorithm which is able to propagate discontinuities andwhich is stable In the numerical implementation used inthis work the flux corrected transport algorithm (FCT)[29 30] is used The principle of the FCT algorithm is tointroduce some artificial viscosity in the numerical schemeto limit the Gibb oscillations

In a previous work the original NPE model has beenextended to propagation within and over porous groundlayers [9 31] In this model the porous layer is assumedto be equivalent to a continuous fluid medium A wavecauses a vibration of air particles contained in the groundpores while the ground frame does not vibrate The modelderivation and the applications to propagation over porouslayers can be found in [10] In this work it is proposed touse this model to study long-range sound propagation overurban cities

888

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

889

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 2 Sketch of a representative urban city

Figure 3 Example of a city realisation with n = 20 and u = (mH mW mD δH δW δD) = (10 20 30 02 02 02)

For all i denoting by Xi either Hi either Wi or Di onehas1 Xi is a random variable with values in ] 0+infin [2 The mean value mXi

= E Xi is given and is equal tomX independent of i

3 The inverse Xminus1i of Xi is a second-order random vari-

ableThis condition is satisfied if

E log(Xi) = ci with |ci| lt +infin (2)

The constraint defined by equation (2) introduces an arbi-trary constant ci which does not have any physical mean-ing and which is then rewritten as a function of the coeffi-cient of variation of Xi which is δX independent of i Theuse of the Maximum Entropy Principle yields [15]

pH1HnW1WnD1Dnminus1

middot h1 hn w1 wn d1 dnminus1 (3)

=n

i=1

pHi(hi)

n

i=1

pWi(Wi)

nminus1

i=1

pDi(di)

with

pHi(hi) = pX (hi mH δH ) (4a)

pWi(wi) = pX (wi mW δW ) (4b)

pDi(di) = pX (di mD δD) (4c)

where

pX (x mX δX ) = ]0+infin[(x)1

mX

1

δ2X

(5)

1

Γ(1δ2X )

x

mX

1δ2X

minus1exp minus x

δ2XmX

In equation (5) ]0+infin[(x) = 1 if x gt 0 and 0 otherwiseand Γ(z) is the gamma function of argument z defined by

Γ(z) =+infin

0tzminus1eminust dt (6)

Since no available information concerning the statisti-cal dependence between the families of random variablesHii Wii and Dii is used the Maximum EntropyPrinciple yields independence of all the random variablesas a result as it can be seen in equation (3)

The following vector u of the parameters of the proba-bilistic model is introduced

u = mH mW mD δH δW δD (7)

Parameter u hence contains the mean values and the dis-persion parameters associated with the heights widths andspacings of the buildings It belongs to an admissible setU = (]0+infin[)6 Figure 3 shows an example of a realisa-tion with n = 20 and u = (10 20 30 02 02 02)

4 Reference model linear propagationover urban cities

This section presents the stochastic linear propagationmodel used to obtain reference solutions of the problemThe problem setting (source and receiver frequency range ) is first given in section 41 and the stochastic propa-gation model itself is then described in section 42

41 Setting the problem

Output Pexpr (ω) from the reference model is analysed in

the frequency domain on a frequency band defined byB =]0 ωmax] in which ωmax is such that B is a low fre-quency band Hence the buildings surfaces are assumed

887

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

to be acoustically rigid (no absorption and specular reflec-tions) The source and receiver are placed above the ur-ban city layer and the source is placed sufficiently far fromthe first building B1 to assume that the wave impinging onthis first building is a plane wave The receiver is placed atthe same altitude as the source and at the right of the lastbuilding Bn The quantity under interest is the normalisedspectrum Lexp(ω) at the receiver such that

Lexp(ω) = 10 log10P

expr (ω)

pfree(ω)

2

(8)

where pfree(ω) and Pexpr (ω) denote free field pressure and

the pressure in the presence of the urban city at the re-ceiver respectively Note that the reference model useddoes not account for building surfaces vibrations

42 Stochastic BEM solver for constructing a refer-ence solution

The model used to construct the reference solution is animplementation of the boundary element method [16 17](BEM) in which the city geometry is explicitly enteredinto the computational model For a fixed parameter uthe outputs Lexp(ω) are calculated using the Monte Carlomethod [18] with νexp independent realisations η1 ηνexp

of urban cities generated with the probability models pre-sented in section 3 The realisation of the experimental ob-servation for the urban city νp is Lexp(ω ηνp

) ω isin B

5 Mean non-linear parabolic propagationmodel for sound propagation over urbancities

This section describes the mean parabolic propagationmodel that will be later used associated with a probabilis-tic model of uncertainties to model sound propagationover urban environments Section 51 presents a generaloverview of propagation models based on the non-linearparabolic equation (NPE) section 52 formally defines thepropagation model and section 53 explains the choice ofthe propagation model parameters and details their alge-braic properties

51 General overview of NPE models

The non-linear parabolic equation (NPE) has first been de-veloped by McDonald and Kuperman in 1987 [19] (seealso [7 8]) and has been successfully used for underwateracoustics simulations [20 21] and blast wave propagationin air [22 23 24] The NPE model for a 2D domain withCartesian coordinates (x z) filed with air is

partt r(x z t) = minuspartx(c1r(x z t) + c0

β

2r(x z t)2)

minus c0

2part2

zr(x z t) dx (9)

where parti means partial derivation with respect to variablei x is the main propagation direction z is the transverse

propagation direction and t is the time variable The am-bient sound speed is c0 while c1 is the sound speed per-turbation in the moving window ie c1 = c(x z) minus c0 where c(x z) is the spatially-dependent sound speed Thedimensionless over-density variable is r(x z t) such thatr(x z t) = ρ (x z t)ρ0 with ρ (x z t) the acous-tic density perturbation and ρ0 the ambient medium den-sity In the following the (x z t) dependence is droppedie r(x z t) is written r For air the coefficient of non-linearity β is calculated with the help of the ratio of spe-cific heats γ ie β = (γ + 1)2 The first term on the righthand side of equation (9) simulates refraction and non-linear effects The second term accounts for propagationin the transverse direction The moving window operatorpartt is defined by

partt = partt + c0partx (10)

Note that in equation (9) the azimuthal spreading termc0r(2d) in which d is the distance from the source tothe point where the field is calculated has been droppedfrom the original NPE [19] There is no absorption fromair included in the model The NPE model derives fromEulerrsquos equations The assumptions used are

1 weak non-linearities ie ρ ρ0 12 weak sound speed perturbations ie c1 c0 3 propagation along a main direction

Various modifications and extensions to this original mo-del were made during the past two decades Spherical andcylindrical coordinate system versions [25] and high-angleformulation [26] have been developed Too and Lee [11]has added a complementary term in the NPE equation inorder to take into account thermoviscous effects Propa-gation in multiple media [20] and propagation through at-mospheric turbulences [27] were also successfully studiedusing this model

The NPE differential equation (9) is discretised with thefinite difference method The Crank-Nicolson method isused for linear terms yielding a tridiagonal system whichcan be solved with a Thomas algorithm (see for example[28]) For the non-linear term one has to use a specialisedalgorithm which is able to propagate discontinuities andwhich is stable In the numerical implementation used inthis work the flux corrected transport algorithm (FCT)[29 30] is used The principle of the FCT algorithm is tointroduce some artificial viscosity in the numerical schemeto limit the Gibb oscillations

In a previous work the original NPE model has beenextended to propagation within and over porous groundlayers [9 31] In this model the porous layer is assumedto be equivalent to a continuous fluid medium A wavecauses a vibration of air particles contained in the groundpores while the ground frame does not vibrate The modelderivation and the applications to propagation over porouslayers can be found in [10] In this work it is proposed touse this model to study long-range sound propagation overurban cities

888

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

889

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

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Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

to be acoustically rigid (no absorption and specular reflec-tions) The source and receiver are placed above the ur-ban city layer and the source is placed sufficiently far fromthe first building B1 to assume that the wave impinging onthis first building is a plane wave The receiver is placed atthe same altitude as the source and at the right of the lastbuilding Bn The quantity under interest is the normalisedspectrum Lexp(ω) at the receiver such that

Lexp(ω) = 10 log10P

expr (ω)

pfree(ω)

2

(8)

where pfree(ω) and Pexpr (ω) denote free field pressure and

the pressure in the presence of the urban city at the re-ceiver respectively Note that the reference model useddoes not account for building surfaces vibrations

42 Stochastic BEM solver for constructing a refer-ence solution

The model used to construct the reference solution is animplementation of the boundary element method [16 17](BEM) in which the city geometry is explicitly enteredinto the computational model For a fixed parameter uthe outputs Lexp(ω) are calculated using the Monte Carlomethod [18] with νexp independent realisations η1 ηνexp

of urban cities generated with the probability models pre-sented in section 3 The realisation of the experimental ob-servation for the urban city νp is Lexp(ω ηνp

) ω isin B

5 Mean non-linear parabolic propagationmodel for sound propagation over urbancities

This section describes the mean parabolic propagationmodel that will be later used associated with a probabilis-tic model of uncertainties to model sound propagationover urban environments Section 51 presents a generaloverview of propagation models based on the non-linearparabolic equation (NPE) section 52 formally defines thepropagation model and section 53 explains the choice ofthe propagation model parameters and details their alge-braic properties

51 General overview of NPE models

The non-linear parabolic equation (NPE) has first been de-veloped by McDonald and Kuperman in 1987 [19] (seealso [7 8]) and has been successfully used for underwateracoustics simulations [20 21] and blast wave propagationin air [22 23 24] The NPE model for a 2D domain withCartesian coordinates (x z) filed with air is

partt r(x z t) = minuspartx(c1r(x z t) + c0

β

2r(x z t)2)

minus c0

2part2

zr(x z t) dx (9)

where parti means partial derivation with respect to variablei x is the main propagation direction z is the transverse

propagation direction and t is the time variable The am-bient sound speed is c0 while c1 is the sound speed per-turbation in the moving window ie c1 = c(x z) minus c0 where c(x z) is the spatially-dependent sound speed Thedimensionless over-density variable is r(x z t) such thatr(x z t) = ρ (x z t)ρ0 with ρ (x z t) the acous-tic density perturbation and ρ0 the ambient medium den-sity In the following the (x z t) dependence is droppedie r(x z t) is written r For air the coefficient of non-linearity β is calculated with the help of the ratio of spe-cific heats γ ie β = (γ + 1)2 The first term on the righthand side of equation (9) simulates refraction and non-linear effects The second term accounts for propagationin the transverse direction The moving window operatorpartt is defined by

partt = partt + c0partx (10)

Note that in equation (9) the azimuthal spreading termc0r(2d) in which d is the distance from the source tothe point where the field is calculated has been droppedfrom the original NPE [19] There is no absorption fromair included in the model The NPE model derives fromEulerrsquos equations The assumptions used are

1 weak non-linearities ie ρ ρ0 12 weak sound speed perturbations ie c1 c0 3 propagation along a main direction

Various modifications and extensions to this original mo-del were made during the past two decades Spherical andcylindrical coordinate system versions [25] and high-angleformulation [26] have been developed Too and Lee [11]has added a complementary term in the NPE equation inorder to take into account thermoviscous effects Propa-gation in multiple media [20] and propagation through at-mospheric turbulences [27] were also successfully studiedusing this model

The NPE differential equation (9) is discretised with thefinite difference method The Crank-Nicolson method isused for linear terms yielding a tridiagonal system whichcan be solved with a Thomas algorithm (see for example[28]) For the non-linear term one has to use a specialisedalgorithm which is able to propagate discontinuities andwhich is stable In the numerical implementation used inthis work the flux corrected transport algorithm (FCT)[29 30] is used The principle of the FCT algorithm is tointroduce some artificial viscosity in the numerical schemeto limit the Gibb oscillations

In a previous work the original NPE model has beenextended to propagation within and over porous groundlayers [9 31] In this model the porous layer is assumedto be equivalent to a continuous fluid medium A wavecauses a vibration of air particles contained in the groundpores while the ground frame does not vibrate The modelderivation and the applications to propagation over porouslayers can be found in [10] In this work it is proposed touse this model to study long-range sound propagation overurban cities

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Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

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ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

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Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 4 Sketch of the different domains with their respectiveboundaries of the NPE model for propagation in multiple media

52 NPE model for sound propagation in multiplemedia

In this section we summarise the NPE model for soundpropagation in multiple media proposed in [10] This prop-agation model is composed of three entities The two firstare non-linear parabolic equations for the air and the ur-ban layers and the third one is an interface condition tocouple the two domains Let Ωa and Ωu be two domainsoccupied by the atmosphere and the urban environmentand let Γa be the boundary at the top of the domain andΓu the boundary at the bottom The coupling interface be-tween the domains is noted Γ and the boundary to the leftand to the right of the domain Ωa cup Ωu are Γl and Γr re-spectively The mean acoustic perturbation field at pointsxa = (x z) isin Ωa and xu = (x z) isin Ωu are ra and rurespectively Figure 4 shows a sketch of the different do-mains with their boundaries

The NPE model for propagation in multiple media iswritten as

partt ra + partx c1ra + βc0

2r2a (11a)

+c0

2part2

zra dx = g(xa t) in Ωa

partt ru + partx c0 (micro minus 1)ru + β microc0

2r2u (11b)

+microc0

2part2

zru dx + αru = 0 in Ωu

partzra + partzra dx = γ partzru on Γ (11c)

ra = 0 ru = 0 on Γr (11d)

ra ru satisfy the Sommerfeld

radiation condition on Γl (11e)

partzru = 0 on Γu (11f)

ra satisfies the Sommerfeld

radiation condition on Γa (11g)

where g(xa t) is an external pressure field applied in the at-mospheric layer Equations (11a) and (11c) handle propa-gation in the atmospheric and urban layers Equation (11c)

is the interface condition between the domains and equa-tion (11d) is a standard boundary condition applied on theboundary Γr to the right of the moving window Equation(11e) states that the wave must not reflect at the left bound-ary of the domain and hence the acoustic field must satisfySommerfeld radiation condition on Γl It is supposed thatthe bottom of the domain is perfectly rigid hence partzru = 0on Γu (see equation (11f)) Since the propagation problemis infinite in the +z direction the acoustic field must satisfySommerfeld radiation condition on Γa (equation (11g))For the sake of brevity the explicit expressions of Sommer-feld radiation conditions are not given here In the presentwork the propagation domain is truncated in the +z direc-tion with the help of a perfectly matched layer (see [32]and [33])

Wave propagation in the urban layer (equation (11c))is characterised by micro which modifies the sound speed inthe urban layer so that cΩu

= microcΩa while α is the loss

rate in the layer The interface condition between domainsΩa and Ωu (equation (11c)) depends on parameters γ and On the boundary Γr a null pressure ra = 0 ru = 0 is

imposed meaning that no perturbation is introduced aheadof the wavefront The relative sound pressure level at thereceiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω)

pfree(xr zr ω)

2 (12)

where quantities with hats denote Fourier-transformedquantities and xr and zr are the receiver coordinates

53 Construction of the mean model and descrip-tion of the algebraic properties of its parame-ters

Below the mean model is derived from section 52 Onehas to define the parameters in the mean propagationmodel (see equations (11)) which have the capability torepresent the natural variability of the real system (the ur-ban environment) These defined parameters will be mod-elled by random variables as explained in section 2 Whena wave is reflecting on a plane surface its amplitude ischanged and a (possibly negative) delay is given to thereflected wave Parameters (micro α γ ) appearing in equa-tions (11c) and (11c) could be used because the propaga-tion model can recreate the behaviour of a porous groundlayer [10] and could thus be used ldquoas isrdquo to control thewave reflection However in order to reduce the numberof parameters in the mean model and consequently toreduce the stochastic model complexity another solutionis proposed The urban layer is now considered as semi-infinite and one writes that no waves are transmitted fromthe porous ground layer to the atmospheric layer This im-plies that the urban layer must behave like an atmosphericlayer which can be simulated in writing that micro = 1 andα = 0 in equation (11c) The time delay occurring dur-ing reflection is introduced by the use of a time-stretching

889

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

The time variable is then transformed according to

t minusrarr 1λ

t (13)

and the time derivative is changed accordingly

partt minusrarr λ partt (14)

The change of amplitude of the reflected wave is ac-counted for using the equation (11c) Parameter γ can beused to tune the amount of reflected and transmitted wavesThe time delay being already taken into account with thetime stretching there is no need to keep parameter inequation (11c) Parameter is set to = 0 One advantageof using such two parameters instead of four in the meanmodel is that these two parameters γ and λ do not notintroduce coupling effects between wave amplitude andtime delay This means that changing parameter λ doesnot change the reflected wave amplitude and changing pa-rameter γ does not change the time delay of the reflectedwave This property will later help designing the proba-bilistic model of these two parameters

Note that parameter λ cannot take the value 0 whichwould correspond to an infinite speed of sound Parameterγ is used to tune the amount of reflected wave in the atmo-spheric layer Setting γ = 0 yields partzra = 0 on couplinginterface Γ which is the condition for perfect reflectionSetting γ = 1 yields partzra = partzru on the coupling interfaceΓ which is the condition for perfect transmission param-eter γ thus belongs to [0 1] Introducing underlined quan-tities related to the mean model the NPE model for prop-agation in multiple media with the two above parametersis rewritten as

λ partt ra + partx c1ra + βc0

2ra

2

+c0

2part2

zra dx = g(xa t) in Ωa (15a)

λ partt ru + partx βc0

2ru

2

+c0

2part2

zru dx = 0 in Ωu (15b)

partzra = γ partzru on partΩ (15c)

ra = 0 ru = 0 on Γr (15d)

ra and rusatisfy the Sommerfeld

radiation condition on Γl (15e)

ru satisfies the Sommerfeld

radiation condition on Γu (15f)

ra satisfies the Sommerfeld

radiation condition on Γa (15g)

Furthermore introducing a third parameter θ in order tocontrol the output of the mean model the relative soundpressure level at the receiver is given by

L(ω) = 10 log10

ρ0c20ra(xr zr ω + θ)

pfree(xr zr ω + θ)

2 (16)

in which θ is a real and positive frequency shifting param-eter Its only intent is to help controlling the mean modeloutput The mean propagation model parameter s can nowbe defined as

s = (γ λ θ) (17)

Vector s belongs to the admissible set S = ([0 1]times]0+infin[times[0+infin[) The initial value of s is arbitrarily chosenas s0 = (γ 0 λ0 θ0) = (0 1 0) in which the superscript 0refers to the nominal value of s so that the nominal meanmodel simulates a wave reflecting on a plane which is anacoustically rigid surface

6 Construction of the probabilistic modelof random variables Γ Λ and Θ

Let Γ Λ and Θ be the random variables associated withthe mean model parameters γ λ and θ Information The-ory [5] and the Maximum Entropy Principle[6] are used toconstruct their probability distributions (see section 3)

61 Construction of the probability distribution ofrandom variable Γ

The available information for random variable Γ is the fol-lowing1 Γ is a random variable with values in [0 1] (see section

53)2 Its mean value mΓ = E Γ is given3 Its coefficient of variation δΓ is givenIt should be noted that the upper bound γ = 1 corre-sponds to perfect reflection at the interface and the lowerbound γ = 0 corresponds to perfect transmission Sincethe neighbourhoods of these two bounds can be reachedwith a non-zero probability it is not necessary to intro-duce an available information related to the behaviour ofthe probability distribution in the neighbourhood of thesetwo bounds With such an available information the Max-imum Entropy Principle yields

pΓ(γ) = [01](γ)eminusmicro0minusγ micro1minusγ2 micro2 (18)

where the constants micro0 micro1 and micro2 depend on mΓ and δΓand are the solutions of the equations

1

0γ eminusγ micro1minusγ2micro2 dγ minus mΓ

1

0eminusγ micro1minusγ2micro2 dγ = 0 (19a)

1

0γ2eminusγ micro1minusγ2micro2 dγ minus (m2

Γ + σ2Γ)

1

0eminusγ micro1minusγ2micro2 dγ = 0

(19b)1

0eminusγ micro1minusγ2micro2 dγ minus emicro0 = 0 (19c)

In the equations above integrals are numerically evaluatedwith the Monte Carlo method and the equations are solvedusing a non-linear Least-Squares method [34]

890

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

62 Construction of the probability distribution ofrandom variable Λ

Taking into account the algebraic properties given in sec-tion 53 random variable Λ is with values in ]0+infin[Since λ cannot take the value 0 which would correspond toan infinite speed of sound it is necessary to write that theprobability distribution goes sufficiently fast to zero whenλ goes to zero with superior values Such a property is sat-isfied in introducing the following condition

ElogΛ = c |c| lt +infin (20)

which implies that the inverse Λminus1 of Λ is a second-orderrandom variable Therefore the available information forrandom variable Λ is the following1 Λ is a random variable with values in ]0+infin[2 Its mean value mΛ = EΛ is given3 Equation (20) is satisfiedWith the constraints defined by the above available infor-mation the use of the Maximum Entropy Principle yields(see section 3)

pΛ(λ) = ]0+infin[(λ)1

mΛ

1

δ2Λ

1

Γ(1δ2Λ)

middot λ

mΛ

1δ2Λ

minus1

exp minus λ

δ2ΛmΛ

(21)

in which δΛ is the coefficient of variation of Λ

63 Construction of the probability distribution ofrandom variable Θ

The available information relative to random variable Θis the same as the available information defined in section62 for random variable Λ Consequently the probabilitydensity function of random variable Θ is written as

pΘ(θ) = ]0+infin[(θ)1

mΘ

1

δ2Θ

1

Γ(1δ2Θ)

θ

mΘ

1δ2Θ

minus1

exp minus θ

δ2ΘmΘ

(22)

in which mΘ and δΘ are the mean value and the coefficientof variation of random variable Θ

7 Stochastic non-linear propagation overurban cities

The following vectorw of the parameters of the probabilis-tic models of random variables Γ Λ and Θ is introduced

w = mΓ mΛ mΘ σΓ σΛ σΘ (23)

in which σΓ = mΓ δΓ σΛ = mΛ δΛ and σΘ = mΘ δΘ Pa-rameter w belongs to the admissible set W = (]0+infin[)6

The stochastic model for non-linear sound propagationover urban cities is defined by

Λ partt Ra + partx c1Ra + βc0

2R2

a

+c0

2part2

zRa dx = g in Ωa (24a)

Λ partt Ru + partx βc0

2R2

u +c0

2part2

zRu dx = 0 in Ωu (24b)

partzRa = Γ partzRu on Γ (24c)

Ra = 0 Ru = 0 on Γr (24d)

Ra and Ru satisfy Sommerfeld

radiation condition on Γl (24e)

Ru satisfies Sommerfeld

radiation condition on Γu (24f)

Ra satisfies Sommerfeld

radiation condition on Γa (24g)

The relative sound pressure level is calculated with

L(ωw) = 10 log10

ρ0c20Ra(xr zr ω + Θ)

pfree(xr zr ω + Θ)

2

(25)

In equations (24) and (25) the probability distributions ofrandom variables Γ Λ and Θ depend on w which in turndepends on u the parameter that describes the urban citygeometry Hence to complete the construction of the com-putational model and to obtain observations of L(ωw)one has to express parameter w as a function of parame-ter u (it should be noted that no explicit expression can beconstructed but the corresponding mapping will be numer-ically constructed)

8 Identification of parameter w of thestochastic model

81 Identification strategy

The identification of parameter w is performed by solv-ing an inverse stochastic problem Mean-square methods[35 36] and the maximum likelihood method [37] are gen-erally used to solve such a problem Both techniques havebeen tested (see below) for different values of parameter u(corresponding to different urban environments) The con-clusions of this comparative study are the followingi Since the mean-square methods introduce a mean-

square distance between the experimental data and therandom response of the stochastic model this type ofmethod is equivalent to a minimisation of the sum ofthe variance for the stochastic model response with thebias between the experimental mean value and the meanvalue of the random response If these two mean valuesare significantly different the bias can only be reducedin increasing the variance of the model In this casethe distance between the experimental mean value and

891

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

the mean value of the random response of the stochasticmodel is effectively reduced but in counterpart the con-fidence region of the random response increases (see forexample [38])

ii For the application analysed it has been seen that forcertain values of parameter u the maximum likelihoodmethod under-estimates the width of the confidence re-gion which means that an important number of experi-mental paths cross the upper and the lower envelopes ofthe confidence region This indicates that the maximumlikelihood method cannot lead to an accurate descrip-tion of the sound field for certain values of parameteru

To provide an accurate and robust identification methodfor all values of parameter u a hybrid method is used theidentification of the optimal parameter wopt is done in twosuccessive steps

1 The mean model is first updated with experimentaldata this step consists of finding the optimal meanmodel parameter sopt that minimises the norm betweenthe mean value of the reference model outputs and themean model output As only the mean value of theexperimental observations is used this first step onlyallows us to identify the parameter that optimises themean response of the stochastic model

2 Next to take into account the dispersion of the ex-perimental observations a multi-objective optimisa-tion problem is solved both the mean-square and themaximum likelihood methods are used concurrentlyThis multi-objective optimisation problem aims to findthe optimal parameter wopt that maximises the log-likelihood function between the experimental obser-vations and the stochastic model outputs while min-imising the areas where experimental observations donot belong to the confidence regions of the stochasticmodel This problem is solved using an evolutionary al-gorithm [39 40]

Step (1) allows us to obtain a rough approximation of themean values mΓ mΛ and mΘ which are the three first com-ponents of vector w and hence helps initialising the opti-misation problem in step (2) Section 82 presents the pro-cedure to update the mean model with experimental data(step 1) The mean square method and the maximum like-lihood method are presented in sections 83 and 84

82 Updating the mean model with experimentaldata

The observation from the mean model depending on thechoice of parameter s it is rewritten as Ls(ω) The per-formance level of the nominal model can be measured inestimating the norm

E Lexp minus LsB= (26)

ωisinB

E Lexp(ω) minus Ls(ω)2dω

12

and the nominal value of the mean model parameter s0 canbe updated in a vector sopt such that

sopt = arg minsisinS

E Lexp minus LsB

(27)

Hence for a fixed value of u = (mH mW mD δH δW δD)the parameter sopt that minimises the norm between themathematical expectation of the experimental observa-tions and the observation from the mean model can be de-termined

83 Identification of the stochastic model parameterwith the mean-square method

This section presents the mean-square method [35 36]with non-differentiable objective function [37] for theidentification of the parameter wopt of the stochastic mo-del The objective function is defined by writing that theηνexp experimental observations

Lexp(ω ηj) ω isin B j = 1 νexp

must belong to the confidence region of the stochasticmodel with a probability level Pc fixed in ]0 1[ Beforegiving the formal definition of the mean-square methodone must address the problem of the construction of con-fidence region

The confidence region is constructed by using the quan-tiles Let FL(ω) be the cumulative distribution function ofrandom variable L(ω) such that

FL(ω)( ) = P L(ω) le (28)

In equation (28) the right-hand side represents the proba-bility that the random variable L(ω) takes on a value lessthan or equal to For 0 lt p lt 1 the pth quantile (orfractile) of FL(ω) is defined as

ζ(p ω) = inf FL(ω)( ) ge p (29)

Then the upper and lower envelopes +(ω) and minus(ω) ofthe confidence region are given by

+(ω) = ζ1 + Pc

2 ω minus(ω) = ζ

1 minus Pc

2 ω

(30)

The estimation of +(ω) and minus(ω) is performed by usingthe sample quantile [41] Consider ν independent realisa-tions of the random variable L(ω) noted

1(ω) = L(ω η1) ν (ω) = L(ω ην)

and let 1(ω) lt lt ν (ω) be the ordered statistics asso-ciated with 1(ω) ν (ω) One has the following esti-mations for the upper and lower envelopes

+(ω) +j+ (ω) j+ = fix ν

1 + Pc

2 (31a)

minus(ω) minusjminus (ω) jminus = fix ν

1 minus Pc

2 (31b)

892

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

in which fix(x) is the integer part of the real number xNow introducing the dependence on parameter w the

formal definition of the mean-square method now followsLet +(w ω) and minus(w ω) be the upper and lower en-velopes of the confidence region of the stochastic modelThe functions +exp(ω) minusexp(ω) z+(w ω) and zminus(w ω)are such that

+exp(ω) = maxj

Lexp(ω ηj)

minusexp(ω) = minj

Lexp(ω ηj) (32)

z+(w ω) = +(w ω) minus +exp(ω) (33a)

middot 1 minus H +(w ω) minus +exp(ω)

zminus(w ω) = minus(w ω) minus minusexp(ω) (33b)

middot 1 minus H minus(w ω) minus minusexp(ω)

in which H(x) is the Heaviside function such thatH(x) = 1 if x ge 0 and H(x) = 0 otherwise Func-tions z+(w ω) and zminus(w ω) represent selected parts of

+(w ω) minus +exp(ω) and minus(w ω) minus minusexp(ω) wherethe experimental observations do not belong to the con-fidence region calculated with the stochastic model Thenon-differentiable objective function J (w) is then definedby

J (w) = z+(w )2B+ zminus(w )

2B

(34)

and the optimal parameter wopt is solution of the followingoptimisation problem

wopt = arg minwisinW

J (w) (35)

The mean-square method aims to minimise the areaswhere the experimental observations do not belong to theconfidence region of the stochastic model the only criteriato select the optimal parameterwopt is hence the amount ofinformation not covered by the stochastic model

84 Identification of the stochastic model parame-ter with the maximum likelihood method andstatistical reduction of information

This section deals with the maximum likelihood method toidentify the optimal parameter wopt Let ω1 ωm subB be a sampling of frequency band B and let w rarr L(w)be the log-likelihood function from W into R defined by

L(w) =νexp

j=1

log10 p Lexp(ω1 ηj) Lexp(ωm ηj) w

(36)

in which p(Lexp(ω1 ηj) Lexp(ωm ηj) w) is thejoint probability density function of random variablesL(ω1 w) L(ωm w) for the values Lexp(ω1 ηj)

Lexp(ωm ηj) The maximum likelihood method [35] con-sists in finding wopt as the solution of the following opti-misation problem

wopt = arg maxwisinW

L(w) (37)

The standard method requires a direct evaluation of thejoint probability density function appearing in equation(36) which involves a significant computational effort Inorder to decrease this effort the method of statistical reduc-tion of information introduced in [37] is used The princi-ple of the method is to proceed to a statistical reductionof information using a principal component analysis (seefor instance [42]) and then to use the maximum likelihoodmethod in the space of the uncorrelated random variablesrelated to the reduced statistical information

For all w fixed in W let be L(w) = L(ω1 w)

L(ωm w) Let m(w) = EL(w) be its mean value andlet [CL(w)] be its (m times m) covariance matrix defined by

CL(w) = E L(w) minusm(w) L(w) minusm(w)T

(38)

in which T superscript stands for transposition The fol-lowing eigenvalue problem is introduced

CL(w) x(w) = e(w)x(w) (39)

for which the first largest q le m positive eigenvalues aree1(w) ge e2(w) ge ge eq(w) and the associated eigen-vectors are x1(w) xq(w) in Rm The approximationLq(w) of L(w) is written as

Lq(w) = m(w) +q

α=1

eα (w) Yα (w) xα (w) (40)

in which Y1(w) Yq(w) are q real-valued random vari-ables such that for all α = 1 q

Yα (w) =1

eα (w)xα (w)T L(w) minusm(w) (41)

It can easily be proved that Y(w) = Y1(w) Yq(w) isa second-order random variable such that for all α and βin 1 q

E Yα (w) = 0 E Yα (w)Yβ (w) = δαβ (42)

which means that the centred random variables Y1(w) Yq(w) are uncorrelated The order q of the statisticalreduction is calculated in order to get an approximationwith a given accuracy which has to be chosen such that

maxwisinW

1 minusqα=1 eα (w)

tr CL(w)le (43)

From equation (41) one can deduce that random variablesY

expα (w ηj) associated with the experimental realisation ηj

are given for all α = 1 q and j = 1 νexp by

Yexpα (w ηj) =

1

eα (w)xα (w)T Lexp(ηj) minusm(w) (44)

893

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

Figure 5 Sketch for the reference model (numerical experiment)

Finally taking into account that the random variablesY1(w) Yq(w) are mutually independent one intro-duces the following approximation for the reduced log-likelihood function Lred

Lred(w) =νexp

j=1

q

α=1

log10 pYα (w) Yexpα (w ηj) w (45)

where pYα (w)(y w) is the probability density function ofthe real-valued random variable Yα (w) The optimal valuewopt of w is then given by

wopt = arg maxwisinW

Lred(w) (46)

This problem is solved by using a genetic algorithmbased on non-dominated sorting (NSGA-II) [39 40] Thismethod is used to solve the multi-objective optimisationproblem for which an initial population evolves over sev-eral generations of individuals For each generation a se-lection process selects the ldquobestrdquo parents from which chil-dren are generated

9 Application and numerical validation

It is proposed in this section to use the here developedstochastic parabolic propagation model to study soundpropagation over given urban environments Numericalsolutions from a BEM code are used as references

91 Summary of previous sections ndash Stochasticmodel validation procedure

In order to construct and validate the stochastic propaga-tion model one has to1 choose a parameter u = (mH mW mD δH δW δD) that

describes the urban city geometry studied With thisparameter the νexp urban environment realisations canbe generated with the probability distributions given insection 3 (see section 92)

2 using the νexp realisations of the probabilistic model ofthe given city and using the stochastic reference model(see section 4) produce the family (see section 92)Lexp(ω ηj) ω isin B

j

3 once the experimental realisations are obtained identifythe optimal parameter wopt of the stochastic model withthe help of the method described in section 8 (section93)

4 using the stochastic parabolic propagation model de-rived from section 7 construct the confidence regionassociated with the optimal parameter wopt determinedin (3) which finally allows the stochastic propagationmodel to be validated (section 94)

92 Choice of parameter u numerical experimentdescription and output calculation from the ref-erence model

As explained in section 4 the reference model is definedas a numerical experiment consisting of numerical calcu-lations using the Boundary Element Method (BEM) Thesource and receiver are positioned 2000 m from the cen-tral point of the urban environment at an angle of 5 andthe urban environment is composed of n = 20 buildingsThe ground and building surfaces are supposed acous-tically rigid and computations are performed on a fre-quency range B =]0 100] Hz with a frequency samplingΔf = 2 Hz yielding data vectors with 51 values Figure 5shows a sketch of the configuration

In this application two different city morphologies arechosen The corresponding parameters are u1 and u2 suchthat

u1 = (mH mW mD δH δW δD)

= (10 20 30 02 02 02)

and

u2 = (40 40 30 02 02 02)

Once u1 and u2 are fixed the buildings dimensions are gen-erated with the help of the probability distributions definedin section 3 The number of realisations of the probabilis-tic model of the real urban city is νexp = 500 inducinga convergence with a 10minus3 accuracy (at least) for the firstand second order moment of the random variable Lexp Acalculation with the nominal mean model described in sec-tion 5 is also performed for comparison and is initialisedwith

s0 = (γ 0 λ 0 θ 0) = (0 1 0) (47)

for which the interface behaves like a plane acousticallyrigid surface Figure 6 compares the reference model(made up of 500 outputs and the mean values estimatesmL

exp1

and mLexp2

) with the nominal model output Ls0

894

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

For the value u1 of the parameter describing the ur-ban environment a low dispersion on the observations andquasi-specular reflection behaviour are obtained The out-put from the nominal model and the mean value of thereference model output have a similar shape For the valueu2 of the parameter describing the urban environment lowdispersion on the low frequency range and high disper-sion in the range [65 90] Hz can be observed It can alsobe noted that the mean value estimate of the experimen-tal realisations L

exp2 (ω) is contained in a relatively narrow

region The maximum and minimum values never exceed6 dB and minus6 dB The shape of the experimental realisa-tions differs from the shape of the nominal model As itcan be seen in Figure 6 the output from the mean modelLs0 has misplaced interference dips for both urban envi-ronments parameters u1 and u2

93 Identification of the stochastic model optimalparameter wopt

In this section the procedure described in section 8 is usedto identify the optimal parameters wopt

1 and wopt2 of the

stochastic model corresponding to the parameters u1 andu2

931 Determination of the updated parameter of themean model

In the identification of the optimal parameter wopt of thestochastic model the first step is the determination of theupdated parameter sopt of the mean model (step 1 in sec-tion 82) solution of the minimisation problem defined inequation 27

The effects of parameters γ λ and θ on the reflectedwave being uncoupled the minimisation problem is re-duced to three one-dimensional searches The parame-ters are real and we use a parabolic interpolation and theBrentrsquos method [43] The optimal parameter found fromthis minimisation problem is

sopt1 = (γ opt

1 λ

opt1 θ

opt1 ) = (0366 1004 6377) for u1

and

sopt2 = (γ opt

2 λ

opt2 θ

opt2 ) = (0107 1814 21362) for u2

For u1 and u2 Figure 7 shows the experimental observa-tions L

exp1 (ω ηj) j

and Lexp2 (ω ηj) j

from the refer-ence model together with their respective mean value es-timates mL

exp1

and mLexp2

and the output from the updated

mean model with sopt1 and sopt

2 Figure 7 shows that the up-dated mean model yields an excellent prediction with re-spect to the reference model However it should be notedthat for u = u2 the reference model exhibits a high leveldispersion around the frequency 80 Hz These variabilitieswill be taken into account by the probabilistic model in thesection below

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativ

eto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 6 Comparisons of the reference model with the nomi-nal model (initial mean model) Reference model (made up of500 outputs (thin lines) and the mean values estimates mL

exp1

and

mLexp2

(thick white lines)) Nominal model output Ls0 (dashedline) The value of parameter u is u1 (top figure) and u2 (bottomfigure)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10SPLrelativ

eto

free

field

[dB]

u1 = (10 20 30 02 02 02)

101

102

minus25

minus20

minus15

minus10

minus5

0

5

10

Frequency [Hz]

SPLrelativeto

free

field

[dB]

u2 = (40 40 30 02 02 02)

Figure 7 Comparisons of the reference model with the updatedmean model Reference model (made up of 500 outputs (thinlines) and the mean values estimates mL

exp1

and mLexp2

(thick white

lines)) Updated mean model output Lsopt(dashed line) The

value of parameter u is u1 (top figure) and u2 (bottom figure)

932 Identification of the optimal parameter of thestochastic model with a genetic algorithm

The updated parameters of the mean model sopt1 and sopt

2are then used to define the reduced admissible set W red ofw which is used to initialise the genetic algorithm For theevaluation of the mean-square norm the 0 Hz point is re-moved from the calculation and the probability level usedfor the construction of the confidence region is Pc = 098For the statistical reduction of information performed inthe maximum likelihood method the order of decompo-sition q (see equation (39)) is q = 12 With this or-der of decomposition the accuracy in equation (43)is such that lt 10minus3 meaning that at least 999 ofthe information is contained within the random variablesY1(w) Y12(w)

Concerning the genetic algorithm the number of indi-viduals in the population is set to ni = 50 for each indi-vidual ν = 500 observations from the stochastic model aregenerated and the population evolved over 50 generationsAt each generation half the population is replaced withchildren

Figure 8 shows the values of the objective functionsminusLred(w) and J (w) of the 50 individuals at generation 1

895

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u1 = (10 20 30 02 02 02)

2 22 24 26 28 3

x 104

0

50

100

150

200

250

300

350

400

450

500

minusLred (w)

J(w

)

u2 = (40 40 30 02 02 02)

Figure 8 Values of the objective functions minusLred(w) and J (w) for the 50 individuals at generations 1 (circles) 10 (squares) and 50(final generation diamonds) The value of parameter u is u1 (left figure) and u2 (right figure) Note that some individuals are out of thefigure ranges

10 and 50 (the final generation) for both parameters u1 andu2 It can be seen that as the population evolves the valuesof the objective functions minusLred(w) and J (w) decreases

The chosen criteria to select the final individuals is themean-square norm individuals that show the lowest mean-square norm are selected first and within the remainingindividuals the individual that exhibit the highest log-likelihood is selected This way the least possible under-estimation of the experimental observations is achievedThe corresponding optimal parameters wopt

1 and wopt2 are

wopt1 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (48)

= (0213 0984 5136 0140 0023 0143)

wopt2 = (mopt

Γ moptΛ m

optΘ σ

optΓ σ

optΛ σ

optΘ ) (49)

= (0262 1776 20575 0132 0091 5036)

94 Solution of the stochastic propagation modelconstruction of confidence region and validation

Once the parameters wopt1 and wopt

2 are identified for u1and u2 one can solve the stochastic equations (24) and(25) using the Monte Carlo method with these parame-ters and construct the associated confidence region as ex-plained in section 83 Confidence regions are constructedwith a probability level Pc = 098 and with ν = 500 reali-sations of the stochastic model

Figure 9 shows the experimental observations

Lexp1 (ω ηj) j

and Lexp2 (ω ηj) j

their mean values estimates and the confidence region cal-culated using the stochastic model and the mean value ofthe stochastic model As one can see the experimental ob-servations belong to the constructed confidence regionsFor configuration corresponding to u2 the solutions calcu-lated using the stochastic model show a broad confidence

region in the very low frequency range whereas the refer-ence solutions exhibit a thin confidence region This meansthat the underlying deterministic model used is not robustin this very low frequency range with respect to statisticalfluctuations generated by the probabilistic model

10 Conclusions

In this work the development of a stochastic model forlong-range non-linear sound propagation over urban en-vironments has been initiated It provides an insight intothe construction method of stochastic models and demon-strates the feasibility of using such methods for study-ing sound propagation in complex environments Further-more it is shown that the amount of information providedby such prediction models can be improved by the use ofstochastic methods

The mean propagation model is in this paper based onthe non-linear parabolic equation and its extension forpropagation over porous ground layers in order to modelpropagation over urban environment This mean model ex-hibit low numerical cost but in counterpart induces modeluncertainties for simulation of sound propagation over ur-ban environments Indeed the high complexity of the urbanenvironment requires more advanced models The meanpropagation model is hence improved introducing a prob-abilistic model of uncertainties

The constructed stochastic model was shown to be ingood agreement with the reference model and can thusbe used to study non-linear wave propagation in complexenvironments Dissipation effects or refraction effects arenaturally present in the NPE model and could be incorpo-rated in the stochastic model Thanks to the low numericaleffort associated with this model large parametric stud-ies could be performed including main features of soundpropagation outdoors eg temperature and wind velocitygradients or site topography

896

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

Leissing et al Long-range propagation over urban cities ACTA ACUSTICA UNITED WITH ACUSTICAVol 96 (2010)

Figure 9 Comparisons of the reference model with the confidence region calculated with the stochastic model Reference model (madeup of 500 outputs (thin lines) and the mean values estimates mL

exp1

and mLexp2

(thick white lines)) The confidence region calculated withthe stochastic model is represented by the grey area delimited by thick black lines The value of parameter u is u1 (top figure) and u2(bottom figure)

References

[1] T Leissing Nonlinear wave propagation in complex me-dia Application to propagation over urban environementsPhD thesis Universiteacute Paris-Est 2009 httptelarchives-ouvertesfrtel-00455590en

[2] J Kragh B Plovsing S Storeheier G Taraldsen H GJonasson Nordic environmental noise prediction methodssummary report Denmark Delta av 171901 2002

[3] D van Maercke J Defrance Development of an analyt-ical model for outdoor sound propagation within the har-monoise project Acta Acustica united with Acustica 93(2007) 201ndash212

[4] J Defrance E Salomons I Noordhoek D Heimann BPlovsing G Watts H Jonasson X Zhang E Premat ISchmich F Aballea M Baulac F de Roo Outdoor soundpropagation reference model developed in the EuropeanHarmonoise project Acta Acustica united with Acustica93 (2007) 213ndash227

[5] C E Shannon A mathematical theory of communicationBell System Tech J 27 (1948) 379ndash423 and 623ndash659

[6] E T Jaynes Information theory and statistical mechanicsPhysical Review 106 108 (1957) 620ndash630 171ndash190

[7] B E McDonald P Caine M West A tutorial on the non-linear progressive wave equation (NPE) - Part 1 ApplAcoust 43 (1994) 159ndash167

[8] P Caine M West A tutorial on the nonlinear progressivewave equation (NPE) Part 2 Derivation of the three di-mensional cartesian version without use of perturbation ex-pansions Appl Acoust 45 (1995) 155ndash165

[9] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation in an inhomogeneous medium over a non-flat finite-impedance ground surface J Acoust Soc Am 123 (2008)3570 Acousticsrsquo08 Second ASA (Acoustical Society ofAmerica) - EAA (European Acoustics Association) jointinternational conference Paris France June 29ndashJuly 42008

[10] T Leissing J Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude sound propa-gation over porous ground layers J Acoust Soc Am 126(2009) 572ndash581

[11] G P J Too S T Lee Thermoviscous effects on transientand steady-state sound beams using nonlinear progressivewave equation models J Acoust Soc Am 97 (1995) 867ndash874

[12] F R DiNapoli R L Deavenport Theoretical and numer-ical Greenrsquos function field solution in a plane multilayeredmedium J Acoust Soc Am 67 (1980) 92ndash105

[13] R Raspet S W Lee E Kuester D C Chang W FRichards R Gilbert N Bong A fast-field program forsound propagation in a layered atmosphere above an im-pedance ground J Acoust Soc Am 77 (1985) 345ndash352

[14] K E Gilbert M J White Application of the parabolicequation to sound propagation in refracting atmosphere JAcoust Soc Am 85 (1989) 630ndash637

[15] C Soize Random matrix theory for modeling uncertain-ties in computational mechanics Comput Methods ApplMech Engrg 194 (2005) 1333ndash1366

[16] R D Ciskowski C A Brebbia Boundary element meth-ods in acoustics Elsevier Applied Science London 1991

[17] P Jean A variational approach for the study of outdoorsound propagation and application to railway noise JSound Vib 212 (1998) 275ndash294

[18] J M Hammersley D C Handscomb Monte carlo meth-ods Chapman and Hall 1964

[19] B E McDonald W A Kuperman Time domain formula-tion for pulse propagation including nonlinear behaviour ata caustic J Acoust Soc Am 81 (1987) 1406ndash1417

[20] J J Ambrosiano D R Plante B E McDonald W AKuperman Nonlinear propagation in an ocean acousticwaveguide J Acoust Soc Am 87 (1990) 1473ndash1481

[21] K Castor P Gerstoft P Roux W A Kuperman Long-range propagation of finite-amplitude acoustic waves in anocean acoustic waveguide J Acoust Soc Am (2004)2004ndash2010

897

ACTA ACUSTICA UNITED WITH ACUSTICA Leissing et al Long-range propagation over urban citiesVol 96 (2010)

[22] F van der Eerden E Veacutedy Propagation of shock wavesfrom source to receiver Noise Cont Eng J 53 (2005) 87ndash93

[23] K Attenborough A Cummings P Dutta P Schomer ESalomons E Standley O Umnova F van den Berg F vander Eerden P van der Weele E Veacutedy Blast sound absorb-ing surfaces Technical report ERDCCRREL September2004

[24] T Leissing Nonlinear outdoor sound propagation A nu-merical implementation and study using the nonlinear pro-gressive wave equation Masterrsquos thesis Chalmers Univer-sity of Technology Goumlteborg 2007

[25] G P Too J H Ginsberg Cylindrical and spherical coor-dinate versions of the NPE for transient and steady-statesound beams J Vib Acoust 114 (1992) 420ndash424

[26] B E McDonald High-angle formulation for the nonlinearprogressive wave equation model Wave Motion 31 (2000)165ndash171

[27] P Blanc-Benon L Dallois D Juveacute Long range soundpropagation in a turbulent atmosphere within the parabolicapproximation Acustica united with Acta Acustica 87(2001) 659ndash669

[28] W Press B Flannery S Teukolsky W Vetterling (eds)Numerical recipes in fortran Cambridge University PressCambridge 1996 Chapter 2 42ndash43

[29] J P Boris D L Book Flux-corrected transport III Mini-mal error FCT algorithms J Comp Phys 20 (1976) 397ndash431

[30] E Veacutedy Simulations of flows in porous media with aflux corrected transport algorithm Noise Cont Eng J 50(2002) 211ndash217

[31] T Leissing P Jean J Defrance C Soize Nonlinearparabolic equation model for finite-amplitude soundpropagation over porous ground layers 13th Long Range

Sound Propagation Symposium (LRSPS) Ecole Centralede Lyon Ecully France 16-17 October 2008 50ndash60

[32] J P Beacuterenger A perfectly matched layer for the absorptionof electromagnetic waves J Comp Phys 114 (1994) 185ndash200

[33] F Collino Perfectly matched absorbing layers for theparaxial equations J Comp Phys 131 (1997) 164ndash180

[34] J E Dennis Nonlinear least-squares state of the art in nu-merical analysis Academic Press 1977 269-312

[35] J C Spall Introduction to stochastic search and optimiza-tion John Wiley and Sons Hoboken New Jersey 2003

[36] E Walter L Pronzato Identification of parametric modelsfrom experimental data Springer 1997

[37] C Soize E Capiez-Lernout J F Durand C Fernandez LGagliardini Probabilistic model identification of uncertain-ties in computational models for dynamical systems andexperimental validation Comput Methods Appl MechEngrg 198 (2008) 150ndash163

[38] M H DeGroot Probability and statistics (2nd ed) Addi-son-Wesley London 1980

[39] K Deb A Pratap S Agarwal T Meyarivan A fast elitistmultiobjective genetic algorithm NSGA-II IEEE Transac-tions on Evolutionary Computation 6 (2002) 182ndash197

[40] N Srinivas K Deb Multiobjective optimization usingnondominated sorting in genetic algorithms EvolutionaryComputation 2 (1994) 221ndash248

[41] R J Serfling Approximation theorems of mathematicalstatistics John Wiley and Sons Hoboken New Jersey1980

[42] I T Jolliffe Principal component analysis Springer-Verlag New York 1986

[43] W Press B Flannery S Teukolsky W Vetterling (eds)Numerical recipes in fortran Cambridge University PressCambridge 1996 Chapter 10 395ndash398

898