forwardscatteringandvolterrarenormalizationfor acoustic ... · communicated by lianjie huang...

Commun. Comput. Phys.doi: 10.4208/cicp.050515.210116a

Vol. 20, No. 2, pp. 353-373August 2016

Forward Scattering and Volterra Renormalization for

Acoustic Wavefield Propagation in Vertically

Varying Media

Jie Yao1,∗, Anne-Cecile Lesage2, Fazle Hussain2 andDonald J. Kouri1

1 Department of Mechanical Engineering, Physics, Mathematics, University ofHouston, Houston, Texas 77204, USA.2 Department of Mechanical Engineering, Texas Tech University, Lubbock,Texas 79409, USA.

Communicated by Lianjie Huang

Received 5 May 2015; Accepted (in revised version) 21 January 2016

Abstract. We extend the full wavefield modeling with forward scattering theory andVolterra Renormalization to a vertically varying two-parameter (velocity and density)acoustic medium. The forward scattering series, derived by applying Born-Neumanniterative procedure to the Lippmann-Schwinger equation (LSE), is a well known toolfor modeling and imaging. However, it has limited convergence properties depend-ing on the strength of contrast between the actual and reference medium or the angleof incidence of a plane wave component. Here, we introduce the Volterra renormal-ization technique to the LSE. The renormalized LSE and related Neumann series areabsolutely convergent for any strength of perturbation and any incidence angle. Therenormalized LSE can further be separated into two sub-Volterra type integral equa-tions, which are then solved noniteratively. We apply the approach to velocity-only,density-only, and both velocity and density perturbations. We demonstrate that thisVolterra Renormalization modeling is a promising and efficient method. In addition,it can also provide insight for developing a scattering theory-based direct inversionmethod.

AMS subject classifications: 35R30, 45D05, 45G10, 45Q05, 65R20, 81Q15, 81U40, 86A15

Key words: Acoustic modeling, scattering theory, Volterra renormalization, velocity and densityvariation.

∗Corresponding author. Email addresses: [email protected] (J. Yao), [email protected] (A.-C.Lesage), [email protected] (F. Hussain), [email protected] (D. Kouri)

http://www.global-sci.com/ 353 c©2016 Global-Science Press

354 J. Yao et al. / Commun. Comput. Phys., 20 (2016), pp. 353-373

1 Introduction

Scattering theory is a powerful method for analyzing wave propagation in a givenmedium [1, 30]. It relates the propagation of the wave in a general medium with thepropagation of the wave in a reference medium and a perturbation operator which char-acterizes the difference between the actual and reference media. It is well known that therelation between the actual wave-field and the perturbation is nonlinear and can be rep-resented as an infinite series (scattering series or Born-Neumann series). The main useof the forward scattering series arose from the application of scattering theory to solvinginverse problems. Based on the early work of Jost and Kohn [6], Moses [14], Razavy [20]and Prosser [17], Weglein and co-workers developed the inverse scattering series (ISS)method [22,24–26,28,37]. The ISS has proven to be a good framework for a wide range ofseismic problems, e.g., multiple attenuation, imaging, direct nonlinear inversion. How-ever, the forward scattering using Born-Neumann series method suffers from the issueof convergence [18]. Convergence properties of the forward scattering series for differenttypes of acoustic medium have been extensively studied. Maston shows that the conver-gence occurs only for a ratio less than

√2 between the reference and the actual velocities

for a 1-D acoustic medium. Later Maston [12] and Nita [27] extended the study to a twodimensional vertically varying acoustic medium. They demonstrated that the forwardscattering series only converges for either limited velocity contrast or limited incidentangle. The same limited convergence condition was shown by Ramirez and Otens forthe acoustic multi-parameter case [19]. To overcome the limitation of convergence, var-ious methods have been proposed. Nita [16] showed that, using a Pade approximation,it is possible to extend the convergence properties of the forward scattering series to anyvelocity contrast in acoustic media. In addition, there also exist several partial summa-tion techniques where only those nonlinear components that contribute to specific events(e.g., primary, multiple) are retained. One is the De Wolf approximation (DWA) [2, 3],introduced to seismic problems by Wu and co-workers [31–34]. The DWA is a multipleforward scattering and single backscattering approximation after reordering and renor-malization of the Born series. Another nonlinear Born series partial summation is thatderived by Weglein and co-workers [22, 24, 28], who divided the full series into varioussubseries, where each subseries is responsible for a single task [28]. Based on that idea,Innanen also derived a nonlinear integral transform relationship between the primarydata and the earth model to model primary events [4, 5].

However, the convergence issue of the forward scattering problem can be overcomeby employing the Volterra renormalization technique to the LSE. The Volterra renormal-ization technique was first proposed by Kouri and co-workers for studying quantumscattering [21]. Later, it was applied to a 1-D acoustic medium for modeling wave prop-agation with normal plane wave incident [8,9]. In this paper, we extend this method to alayered medium, for which both velocity and density are allowed to vary. With the renor-malization, the Fredholm type of the LSE is transformed to a Volterra type. It has beenproved that solutions related to the renormalized LSE converge for any perturbation for

J. Yao et al. / Commun. Comput. Phys., 20 (2016), pp. 353-373 355

normal incident case [8].

Currently, the most widely used approaches for wave modeling involve finite differ-ence or finite element techniques. However, these methods require discretizing the waveequation in a finite region. To numerically model wave propagation in an unboundedregion, absorbing boundary conditions need to be applied to absorb energy at the artifi-cial boundaries of the discretization domain. However, our approach is a type of integralequation modeling method. As the causal Green’s function is employed, the solutionof the Lippmann-Schwinger integral equation satisfies both the wave equation and theSommerfeld radiation boundary condition. This is an important merit of integral equa-tion methods compared with partial differential equation formulations [23]. In addition,the domain of computation for our integral equation-based method is only the support-domain of the perturbation. However, in the partial differential equation formulationslike FD method, the domain of computation is the whole space contained by the artificialboundary, which can be very large for some cases.

The structure of this paper is the following. In Section 2, we briefly present the formal-ism of the forward scattering theory for a vertically varying acoustic media. In Section3, we introduce the Volterra renormalization technique for the LSE. In addition, we alsoemploy a wave separation to divide the whole renormalized LSE into two sub-Volterraintegral equations, and discuss the procedure to perform Volterra renormalized modeling(VRM). The constant density varying velocity and constant velocity varying density casesare studied in Sections 4, and 5, respectively. In Section 6, we extend the VRM methodfor a medium where both density and velocity change. Discussion of the VRM and con-clusions are presented in the final section. The VRM framework provides insights thatcan be applied to the study of wave modeling, and also to developing scattering theorybased inversion algorithms.

2 The Lippmann-Schwinger equation

We present a brief introduction of the scattering theory formulation. The acoustic waveequation with both velocity and density variations is

(∇· 1

ρ(x)∇+

ω2

c2(x)ρ(x)

)P(x,xs,ω)=δ(x−xs), (2.1)

where x is the observation point, xs is the source location, c(x) is the velocity model, ρ(x)is the density distribution, ω is the angular frequency, and δ is the Dirac delta function.

The reference medium is chosen as a homogeneous whole space satisfying the acous-tic wave equation,

(∇· 1

ρ0∇+

ω2

c20ρ0

)G0(x,xs,ω)=δ(x−xs), (2.2)


where ρ0,c0 are the reference density and velocity, respectively, and G0(x,xs,ω) is thecausal free space Green’s function. Then the Lippmann-Schwinger equation for the totalwave field is

P(x,xs,ω)=G0(x,xs,ω)+∫

dx′G0(x,x′,ω)V(x′,∇,ω)P(x′,xs,ω), (2.3)

or in operator form,

P=G0+G0VP, (2.4)

with the perturbation operator given by [37]

V(x,∇,ω)=( ω2

c20ρ0

− ω2

c2(x)ρ(x)

)+∇·

( 1

ρ0− 1

ρ(x)

)∇

=1

ρ0[k2

0α(x)+∇·β(x)∇], (2.5)

and the reference wavenumber k0 =ω/c0, the bulk modulus contrast α= 1−K0/K, thedensity contrast β=1−ρ0/ρ, and K= c2ρ is the bulk modulus.

For simplicity, considering the case of a 2D medium that varies only vertically, theperturbation is explicitly given by [28, 37]

V(z,∇,ω)=1

ρ0

[k2

0α(z)+β(z)∂2

∂x2+

∂

∂zβ(z)

∂

∂z

]. (2.6)

We can also define the velocity contrast γ= 1−c20/c2. Since velocity and density are

independent, the bulk modulus contrast can be expressed using these two parameters

α(z)=γ(z)+β(z)−γ(z)β(z) (2.7)

and the perturbation can also be represented as [19]

V(z,∇,ω)=1

ρ0

[k2

0(γ(z)+β(z)−γ(z)β(z))+β(z)∂2

∂x2+

∂

∂zβ(z)

∂

∂z

]. (2.8)

As discussed in [19], the selection of the parameters is very important for the interpreta-tion of each term and its role in modeling. Unlike in [19], we propose a novel and conciseform of the perturbation (for more details see appendix A)

V(z,∇,ω)=1

ρ0

[ω2

c20

γ+1

1−β

∂β

∂z

∂

∂z

]. (2.9)

From Eq. (2.9), we find that the transformed perturbation only contains two terms.One is responsible for a velocity perturbation, and the other is for a density perturbation.The effects of the velocity and density perturbation are isolated. The main advantage


of this form is that it can lead to a clear understanding of the effects of each parameter.In addition, it is also convenient for developing an inversion algorithm, which will bestudied in future.

The Green’s function representing a wave propagating with reference velocity c0 anddensity ρ0 in a 2D homogeneous medium is given by [13]

G0(x,xs,ω)=ρ0

2π

∫dkx

eikx(x−xs)+iq0|z−zs|

2iq0, (2.10)

where x=(x,z) and xs =(xs,zs) are the coordinates for the observation and source pointrespectively, kx is the horizontal wavenumber associated with the x coordinate, and q0 isthe vertical wavenumber satisfying the dispersion relation

q0= sgn(ω)√

k20−k2

x. (2.11)

For a non-evanescent plane wave component, we can define the angle of incidence θ by

sinθ= kx/k0. (2.12)

We first consider wave propagation with an incident plane wave, and the pressure wave-field from a point source can be constructed using all plane wave components. Takingthe Fourier transforms over the source and receiver horizontal coordinates for Eq. (2.3)

P(kg,z,ks,zs,ω)=1

2π

∫dxs

∫dxei(ks xs−kgx)P(x,xs,ω), (2.13)

we obtain

P(kg,z,ks,zs,ω)=P0(kg,z,ks,zs,ω)+∫

dz′ρ0

2iqgeiqg |z−z′|V(z′,∇,ω)P(kg,z′,ks,zs,ω), (2.14)

where the incident wave is

P0(kg,z,ks,zs,ω)=ρ0

2iqgeiqg |z−zs|δ(kg−ks). (2.15)

Eq. (2.14) represents a set of Fredholm type integral equations which correspond to dif-ferent incident and receiver angles. For a vertically varying medium, we always havethe relation kg = ks. Thus, the pressure wave for kg 6= ks is always zero. The plane wavepressure wavefield can be characterized using only one horizontal wavenumber. And inthe following section, we will use P(kg,z,zs,ω) in place of P(kg,z,ks,zs,ω).

The common way to solve the LSE (Eqs. (2.3) and (2.14)) is to formulate it as an infi-nite series, known as Born-Neumann series, or forward scattering series. It is achievedby repeatedly substituting the integral equation for the unknown total pressure wave Pwithin the integral term on the right hand side. However, the series converges only fora sufficiently small contrast between the actual and reference medium or for limited in-cident angles. This limitation prevents scattering theory from being an useful methodfor acoustic forward modeling. In the following section, we will introduce the VolterraRenormalization technique to transform the original Fredholm type LSE to a Volterratype, which is absolutely convergent.


3 Volterra renormalization of LSE

The absolute value of the exponential in the Green’s function implies that the wave prop-agates in two directions (upward and downward) in the space domain. We can separatethe wave propagation direction by removing the absolute value of the kernel. If we splitthe scattered wave into up-going and down-going parts over depth z, we obtain

P(kg,z,zs,ω)=ρ0

2iqgeiqg|z−zs|+

∫ z

−∞

dz′ρ0

2iqgeiqg(z−z′)V(z′,∇,ω)P(kg,z′,zs,ω)

+∫

∞

zdz′

ρ0

2iqgeiqg(z′−z)V(z′,∇,ω)P(kg,z′,zs,ω). (3.1)

Adding and subtracting the extra term∫ z−∞

dz′ ρ0

2iqgeiqg(z′−z)V(z′,∇,ω)P(kg,z′,zs,ω), we ob-

tain

P(kg,z,zs,ω)=ρ0

2iqgeiqg |z−zs|+

∫∞

−∞

dz′ρ0

2iqgeiqg(z

′−z)V(z′,∇,ω)P(kg,z′,zs,ω)

+∫ z

−∞

dz′ρ0

2iqg[eiqg(z−z′)−eiqg(z′−z)]V(z′,∇,ω)P(kg,z′,zs,ω). (3.2)

Similar to [8], we introduce the reflection coefficient [20] as

R(kg,ω)=∫

∞

−∞

dz′eiqgz′V(kg,z′,ω)P(kg,z′,zs,ω), (3.3)

and the Volterra Green’s function

G(kg,z,z′,ω)=ρ0

2iqg[eiqg(z−z′)−eiqg(z

′−z)]η(z−z′), (3.4)

where η(z) is Heaviside function (η(z)=0 for z<0, η(z)=1 for z≥0).Without loss of generality, we further assume the source is located at zs = 0 and the

observer points are all located at z>0, then Eq. (3.2) becomes

P(kg,z,ω)=P0(kg,z,ω)+Pr(kg,z,ω)+∫ z

−∞

dz′G(kg,z,z′,ω)V(z′,∇,ω)P(kg,z′,ω), (3.5)

where P0(kg,z,ω) is the incident wavefield, and Pr(kg,z,ω)=R(kg,ω) ρ0

2iqge−iqgzg is defined

as the receiver wavefield. Note that the renormalized integral equation is of the Volterratype, and the related Neumann series converges absolutely and uniformly, independentof the strength of the contrast [8,15]. We also stress that the renormalized LSE for the pres-sure wavefield is exact; no approximation has been made. The original causal Green’sfunction has the property that only contributions from times before the present time in-fluence the wave propagation and in the space domain the wave can propagate upwardand downward. The renormalized Volterra Green’s function has similar properties by


considering depth as analogous to time. Only contributions from points above the cur-rent position influence the wave propagation, which can be forward and backward intime. The renormalized Volterra Green’s function has additional properties: 1) it is realvalued; and 2) it is triangular in space. These properties will later prove to be very usefulfor developing efficient “one-way”-like modeling algorithms.

For wavefield modeling, the reflection coefficient R(kg,ω) is initially unknown, andso also the receiver wavefield. Hence, the renormalized integral equation cannot be di-rectly employed for forward modeling. However, this difficulty can be overcome byintroducing a wave separation technique to the renormalized LSE. We assume that thetotal wavefield can be separated into two sub-wavefields

P(kg,z,ω)=P1(kg,z,ω)+P2(kg,z,ω), (3.6)

and substitute this into the renormalized LSE (Eq. (3.5)). Actually there are many possiblechoices of wave separation. One separation approach is to require

[P1

P2

]=

[P0

Pr

]+

[G0V 0

0 G0V

][P1

P2

], (3.7)

and P1 and P2 are decoupled wave fields, and are expressed by two separate Volterraintegral equations. Explicitly, the first integral equation can be represented as

P1(kg,z,ω)=P0(kg,z,ω)+∫ z

−∞

dz′G0(kg,z,z′,ω)V(z′,∇,ω)P(kg,z′,ω). (3.8)

The Neumann series corresponding to Eq. (3.8) is

P1=P0+G0VP0+G0VG0VP0+···[G0V]nP0+··· , (3.9)

where

[G0V]nP0=∫ z

−∞

dz1G0(kg,z,z1,ω)V(z1,∇,ω)∫ z1

−∞

dz2G0(kg,z1,z2,ω)V(z2,∇,ω)···

×∫ zn−1

−∞

dznG0(kg,zn−1,zn,ω)V(kg,zn,ω)P0(zn,zs,ω), (3.10)

with z> z1> z2> ···zn > zs.We can see that each term of the series is a Volterra-type integral. Therefore, the whole

series also converges absolutely and uniformly.Although the iterated Volterra integral equation converges absolutely, the rate of con-

vergence depends on the strength of the perturbation. If the perturbation is relativelystrong, a large number of terms are required to be calculated. Due to the triangular na-ture of the renormalized Green’s function (G0(kg,z,z′,ω)=0, when z = z′), the pressurewave at a given depth can be explicitly represented in terms of the pressure at previ-ous depths. Hence, the Volterra integral equation can be calculated non-iteratively with


simple trapezoidal rule. In addition, since the Volterra renormalized Green’s function isreal valued, for a scalar perturbation we have the following relationship between the twosub-wavefields

P2(kg,z,ω)=R(kg,ω)P∗1 (kg,z,ω), (3.11)

where ∗ denotes the complex conjugate.It implies that arduous calculation for the second integral equation can be avoided

provided the reflection amplitude is known. Substituting Eq. (3.11) into Eq. (3.3) andafter certain simplification, we obtain

R(kg,ω)=

∫∞

−∞dz′eiqgz′VP1

1−∫

∞

−∞dz′eiqgz′VP∗

1

. (3.12)

Eq. (3.12) is a renormalized expression for the reflection coefficient. It provides a wayto calculate the reflection coefficient from sub-wavefield P1, given the information of theperturbation V. Thus after computing the sub-wavefield P1, the full pressure wave canbe constructed by summing two separated wavefields using Eq. (3.6). It seems that thecomputational effort of the integral in Eq. (3.8) is heavy, because it requires at each deptha sum over all the previous depth steps. However, due to the simple exponential form ofthe kernel, the calculations can be drastically reduced by introducing a memory variable.The pressure wave at a given depth can be calculated recursively with the results of lastdepth.

4 Velocity-only perturbation case

To have a better understanding of the VRM method, we now consider a special acousticcase with a velocity-only perturbation. The actual medium satisfies the acoustic waveequation in Eq. (2.1), with ρ(x)=ρ0. In this case, β(z)=0,α(z)=γ(z), the perturbation isgiven by

V(z,∇,ω)=1

ρ0k2

0α(z). (4.1)

We study the VRM with a simple three layer constant density acoustic model (asin Fig. 1 with ρ0 = ρ1 = 1g/cm3). Actually, the simplest model should be the two layermodel with a single interface. However, modeling with scattering theory assumes thatthe perturbation should have finite range (compact support). For non-compact supportperturbation, additional consideration is needed to calculate the reflection coefficient,and is addressed in Appendix B.

For this one parameter case, Nita [16] discussed the convergence properties for the fullforward scattering series. It has been shown that the series converges only for small per-turbations and limited angles of incidence. For plane waves arriving at pre-critical and


Figure 1: Three layer acoustic model.

critical angle, the series converges. For plane waves arriving at post-critical angles, theseries diverges. We want to know whether the VRM will produce stable and convergentresults for different incident angles. Actually, the normal incident case (kg =0) has beenpresented in [9], and its absolute convergent property is proved in [8]. Here, we show thebehavior of VRM for different incident angles. Fig. 2 shows the VRM results at three dif-ferent incident angles, corresponding to normal, pre-critical and post-critical angles. Theparameters of the model are z1=150m, z2=300m, c0=1500m/s, c1=3000m/s, ρ0=1g/cm3

and ω=200s−1. For this case, the critical incident angle is θc = sin−1(c0/c1)=30◦. FromFig. 2(a) to Fig. 2(d), both the sub-pressure wavefield P1 (corresponding to Eq. (3.8)) andthe total pressure for plane waves incident at angles less that critical angle produce “sta-ble” result. The pressure wave has different wavelengths and amplitudes for differentvelocity regions. When the wave is incident above the critical angle, the sub pressurewave P1 grows exponentially in the high velocity region (z1 < z < z2). Roughly speak-ing, this is because evanescent waves are generated at the interface for wave incidentsabove critical angle. Numerically, both growing and decaying evanescence waves areproduced. The growing evanescent parts will be dominant, and make the sub pressurewave and finally the total pressure wave unstable. However, from the results in Figs. 2(e)and 2(f), we find that the total pressure wave is still stable and produces a reasonableresult which exponentially decays in the high velocity region.

We can also construct the pressure wave for a point source and point receiver byperforming the inverse Fourier transform of Eq. (2.13) using all plane wave componentsobtained using the VRM

P(x,xs,ω)=1

2π

∫dxs

∫dxei(kgx−ksxs)P(kg,z,ks,zs,ω). (4.2)

The integration is not possible for the sub-pressure wavefield, since the result is growingexponentially for post-critical incidence. Fortunately, the solution for the total pressure


-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(a)

-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(b)

-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(c)

-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

deDepth(m)

(d)

-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(e)

-3-2-1 0 1 2 3

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(f)

Figure 2: The plane wave modeling result of the sub-pressure wave P1 (solid), total pressure wave P (dashed)at ω = 200 for model in Fig. 1 with parameter: z1 = 150m,z2 = 350m, c0 = 1500m/s, c1 = 3000m/s at threedifferent incident angles: (a-b) θ1=0◦; (c-d) θ1=20◦; (e-f) θ1=60◦ (left is the real part, right is the imaginarypart of the pressure wave).

wave is always stable, and allows for the construction of a point source response. Fig. 3shows the total pressure wave field for a point source located at (0m,0m) at the frequencygiven above. To further validate the VRM, we construct the time domain snapshots forthis constant density model. The time domain pressure wave field is generated by

P(x,xs,t)=∫

dωeiωtS(ω)P(x,xs,ω), (4.3)

where S(ω) is the frequency domain source signature. In this paper, the source is chosenas a Ricker wavelet with central frequency 15Hz. To validate the result, we also displaythe result of the finite difference (FD) method. The FD calculations are done with secondorder difference both in space and time. We have used dx=5m, and dz=5m for both ofthe methods. The time step for FD is chosen to be 0.001s, and the frequency step for VRMis dω = 2s−1. Fig. 4 shows the comparison of snapshots at different times calculated byVRM (Figs. 4(a), 4(c)) and FD (Figs. 4(b), 4(d)). We see that the results of VRM are almostidentical with the FD method.


-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(a)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

(b)

Figure 3: A point source modeling result of the total pressure wave P(x,z,xs,zs,ω) at ω=200s−1 for model inFig. 1 with parameter: z1=150m, z2=350m, c0 =1500m/s, c1 =3000m/s: (a) real part; (b) imaginary part.

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(a)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500D

epth

(m

)

(b)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(c)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(d)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(e)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(f)

Figure 4: The comparison of time domain snapshot using Volterra scattering (left) and FD method (right) forthe same model as Fig. 2 at different times: (a-b) t=0.1s; (c-d) t=0.2s; (e-f) t=0.3s.

The example described above is for the simple case of an acoustic medium with onlyvelocity variations and two interfaces. The result shows that the renormalized LSE con-verges for any incident angle, and demonstrates the effectiveness of VRM method for


acoustic wavefield modeling in this type of medium. However, the method also worksfor more complex cases with arbitrary contrast between actual and reference media. Thegeneralization of the method to a multi-dimensional constant density acoustic mediumwith lateral velocity variation will be the subject of future research.

5 Density-only perturbation case

In this section, we consider another special case–a medium which only involves densityvariation. If we choose the reference medium as the constant density, constant velocityone, for the transmitted wave, the reference wave field should have the same time behav-ior as the actual wave field. For a receiver above the perturbation, the actual wave fieldwill contain a direct wave which has propagated directly from the source, and reflectedwave which is generated due to the interaction with the perturbation. For this specialcase, α(x,z)=β(x,z), and the density perturbation is given by

V(z,∇,ω)=1

ρ0

[k2

0β(z)+β(z)∂2

∂x2+

∂

∂zβ(z)

∂

∂z

]. (5.1)

In [19], the total density perturbation (Eq. (5.1)) is decomposed as the isotropic part (thefirst term on the right hand side) and anisotropic part (the second and third terms, con-taining gradients). The isotropic part is analogous in form and behavior to the velocityperturbation in the previous section. Hence, considering only the isotropic par will pro-duce a change of arrival time as well as amplitude of the actual wavefield. We know thereshould have no arrival time change in the constant velocity case. Hence, the anisotropicterms perform the task of not only correcting amplitude, but also canceling the effects ofincorrect arrival time [19].

For a medium with only density variation, we find it is better to employ the velocityand density decomposition expression of the perturbation (Eq. (2.9)). In this case, thevelocity contrast γ(z)=0, and the perturbation reduces to

V(z,∇,ω)=1

ρ0

1

1−β

∂β

∂z

∂

∂z. (5.2)

Instead of separating isotropic and anisotropic parts, Eq. (5.2) shows that the densityperturbation contains only a single term with gradients of density contrast and gradientsof pressure wave. It is responsible for the amplitude change of the transmitted wave, andgenerates the reflected wave caused by the density change.

We next study the three layer acoustic model as in Fig. 1 but with constant velocityc0 = c1 =1500m/s. The parameters of the model are z1 =150m, z2 =300m, c0 =1500m/s,ρ0 = 1g/cm3, ρ1 = 1.5g/cm3. For the constant velocity case, there is no critical angle.Waves incident at a certain angle θ will continue to propagate along the same direction,and generate reflected waves at the interface with the same reflected angle. In Fig. 5, wealso display the VRM results at three different incident angles for frequency ω=200s−1.


-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(a)

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(b)

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(c)

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(d)

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(e)

-2-1.5

-1-0.5

0 0.5

1 1.5

2

0 100 200 300 400 500

Am

plitu

de

Depth(m)

(f)

Figure 5: The plane wave modeling results of the sub-pressure wave P1 (solid), total pressure wave P (dashed) at

ω=200 for model in Fig. 1 with parameter: z1=150m, z2=350m, c0=1500m/s, ρ0=1g/cm3, ρ1=1.5g/cm3

at three different incident angles: (a-b) θ1 =0◦; (c-d) θ1 =20◦; (e-f) θ1 =60◦ (left is the real part, right is theimaginary part of the pressure wave).

From Fig. 5, we find that although waves propagate at different wavenumbers for dif-ferent incident angles, they remain constant along each incident angle. Only amplitudechanges occur at different density regions. Snapshots at different times calculated byVRM (Figs. 6(a), 6(c), 6(e)) and FD (Figs. 6(b), 6(d), 6(f)) are displayed. The VRM stillproduces almost the same result as FD method, except for some artifacts due to the dis-cretization of the wavenumber.

6 Changing velocity and density medium

After studying the single parameter changing media in the previous two sections, wenext investigate how the VRM performs for an acoustic medium with varying velocityand density. In this section, we will study the modeling which involves the same struc-ture as shown in the previous section, but we will now allow both velocity and density


-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(a)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(b)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(c)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(d)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(e)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(f)

Figure 6: The comparison of time domain snapshot using Volterra scattering and FD method for the samemodel as Fig. 1 at different times:(a-b) t=0.1s ;(c-d) t=0.2s ;(e-f) t=0.3s.

change over the interface. The perturbation is expressed as

V(z,∇,ω)=1

ρ0

[k2

0γ+1

1−β

∂β

∂z

∂

∂z

]. (6.1)

The first term is analogous to the velocity perturbation in Section 4. It is responsible forchanging both the arrival time and amplitude. The last term is analogous to the densityperturbation in Section 5. It leads to the same effects as discussed in last section foramplitude correction.

The parameters of the model are z1 = 150m, z2 = 300m, c0 = 1500m/s, c1 = 3000m/s,ρ0=1g/cm3, ρ1=1.5g/cm3. From the analysis above, we know that the actual wavefieldshould have the same time behavior as the constant density one in Section 4. The onlydifference is in the amplitude of the wavefield, since the acoustic impedance (I = ρc) isdifferent for these two cases. We compare the amplitude difference between these twocases by display the amplitude of reflection coefficient |R(kg,ω)| at frequency ω=200s−1

in Fig. 7. As in Section 4, the critical angle is θc =sin−1(c0/c1)=30◦, and the correspond-


0

0.2

0.4

0.6

0.8

1

1.2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Am

plitu

de

Horizontal wavenumber

Figure 7: The norm of reflection coefficient at different incident angles for ω=200s−1: both changing velocityand density (solid); only changing velocity (dashed).

ing critical horizontal wavenumber kxc = k0sinθc = 0.67. From Fig. 7, for a wavenumberabove the critical value kxc and less than k0, the amplitude of reflection coefficient is al-most one. This is consistent with real situations, and most of the incident wave shouldbe reflected at these angles. For wave incident at less than the critical angle, the ampli-tude for variation in the two parameters is always larger than that if only velocity varies.Although in this example, the reflection amplitude is for a two interface model, we cananalyze it analogous to the single interface case. For θ<θc, the reflection amplitude at thefirst interface is given by [7, 37]

R(θ)=I1

√1−sin2 θ− I0

√1−(c2

1/c20)sin2θ

I1

√1−sin2 θ+ I0

√1−(c2

1/c20)sin2θ

=2cosθ

cosθ+(ρ0/ρ1)√(c2

0/c21)−sin2θ

−1. (6.2)

In the two parameter varying case, ρ0/ρ1 is less than one. From Eq. (6.2), we know thatthe reflection coefficient for this two parameter varying case should always be larger thanthe constant density case. This explains why the amplitude of the reflection coefficientfor the two parameter case should be larger in Fig. 7.

Fig. 8 shows snapshots at different times for the two parameter case calculated byVRM. We can see that the waveform in this case is identical with that shown in Fig. 4. Theamplitude of the waveform is larger than for the constant density case. Certain multiplyreflected events which are weak for the constant density case can be seen in this case.

7 Discussion and conclusion

As we mentioned above, there is an extensive literature using scattering theory methodfor forward modeling [4, 12, 27]. However, these studies are mainly limited to the Bornseries method and focus on determining the task performed by each term of the series.


-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(a)

-1000 -500 0 500 1000

Lateral (m)

0

100

200

300

400

500

Dep

th (

m)

(b)

Figure 8: The time domain snapshot using Volterra scattering for the two parameter varying model at differenttimes: (a) t=0.2s; (b) t=0.3s.

The full Born series converges for either limited velocity contrast or limited incident an-gle [28]. For large and spatially extended perturbations, renormalization and approxima-tion need to be employed [16,31,34]. For example, the Pade approximation was employedby Nita [15] to extend the convergence properties of the Born series for any velocity con-trast of single interface perturbation. Another one is the De Wolf approximation, whichis a multiple forward scattering and a single backscattering approximations after oper-ator splitting, reordering and renormalization of the Born series. Instead of employingthe series expansion of scattering theory, in this paper we present a different way of us-ing scattering theory for forward modeling of multi-parameter acoustic medium withVolterra renormalization. The VRM possesses several advantages. First, the issue of con-vergence is solved. Unlike the original LSE, the renormalized LSE is a Volterra type ofintegral equation, and converges absolutely and uniformly under iteration. Second, it iscomputationally efficient. For a vertical variation model, only a set of integral equationsrepresenting different plane wave angles are required to be calculated for each frequency.The Volterra LSE can be further separated into two uncoupled sub-equations. Due to thespecial properties of the Volterra Green’s function, only one integral equation is requiredto be solved, and can be solved recursively. And the full pressure wavefield can be con-structed for any source positions and signatures. For a 3-D model, only an extra Fouriercomponent is needed to obtain the full wavefield.

The study of forward modeling is closely linked to the development of inversion al-gorithms [8, 28, 29, 34]. In [37], Zhang and Weglein derived an inversion method for thismulti-parameter acoustic medium using inverse scattering sub-series. Since the renor-malized LSE and its related Neumann series converge absolutely, it is promising to de-velop an inversion algorithm based on them. Actually, some preliminary work has beendone for acoustic medium with only velocity variation [8–11, 35, 36]. These inversionmethods follow the same idea of the ISS method, by expanding the perturbation in ”or-ders of the data”. Then the high order perturbations can be expressed with respect to thelow order approximation and the reference Green’s function. When the perturbation islarge, the resulted inverse Born approximation can be very different from the true per-turbation, and the power series of the perturbation can be slowly convergent, or evennot convergent. In future, more robust nonlinear inversion algorithms will be studied toovercome this limitation.


Currently we assume that the velocity and density can only change vertically. Al-though simplistic, the model shows the power of the Volterra renormalization techniquefor acoustic wave modeling. In future, we will extend this work to an acoustic mediumwith lateral velocity and density variations.

Acknowledgments

Partial support of this research under R.A. Welch Foundation Grant E-0608 is gratefullyacknowledged.

A Simplification of perturbation

In this appendix, we provide detail derivation of another form for the perturbation. Asin Eq. (2.6), the perturbation is given by

V(z,∇,ω)=1

ρ0

[k2

0α(z)+β(z)∂2

∂x2+

∂

∂zβ(z)

∂

∂z

]. (A.1)

Substituting the explicit expression of β, we obtain

V(z,∇,ω)=1

ρ0

[k2

0α(z)+(

1− ρ0

ρ

) ∂2

∂x2+

∂

∂z

(1− ρ0

ρ

) ∂

∂z

]

=1

ρ0

[k2

0α(z)+( ∂2

∂x2+

∂2

∂z2

)+ρ0

(1

ρ

∂2

∂x2+

∂

∂z

1

ρ

∂

∂z

)]. (A.2)

With the expression of the wave equation, we have

1

ρ

∂2

∂x2+

∂

∂z

1

ρ

∂

∂z=− ω2

c2ρ. (A.3)

Then Eq. (A.2) can be simplified as

V(z,∇,ω)=1

ρ0

[k2

0α(z)+( ∂2

∂x2+

∂2

∂z2

)+

ω2ρ0

c2ρ

]. (A.4)

In addition, we also have

1

ρ

∂2

∂z2=

∂

∂z

1

ρ

∂

∂z−( ∂

∂z

1

ρ

) ∂

∂z. (A.5)


Then the Laplacian operator in Eq. (A.2) can be expressed as

∂2

∂x2+

∂2

∂z2=ρ

[1

ρ

∂2

∂x2+

1

ρ

∂2

∂z2

]

=ρ[1

ρ

∂2

∂x2+

∂

∂z

1

ρ

∂

∂z−( ∂

∂z

1

ρ

) ∂

∂z

]

=−ω2

c2−ρ

( ∂

∂z

1

ρ

) ∂

∂z. (A.6)

The perturbation becomes

V(z,∇,ω)=1

ρ0

[k2

0α(z)−ω2

c2−ρ

( ∂

∂z

1

ρ

) ∂

∂z+

ω2ρ0

c2ρ

]. (A.7)

Substituting the expression of α, we obtain

V(z,∇,ω)=1

ρ0

[ω2

c20

(1− ρ0c2

0

ρc2

)−ω2

c2−ρ

( ∂

∂z

1

ρ

) ∂

∂z+

ω2ρ0

c2ρ

]

=1

ρ0

[ω2

c20

(1− c2

0

c2

)−ρ

( ∂

∂z

1

ρ

) ∂

∂z

]

=1

ρ0

[ω2

c20

γ−ρ( ∂

∂z

1

ρ

) ∂

∂z

]. (A.8)

We also have

∂

∂z

1

ρ=− 1

ρ0

∂β

∂z. (A.9)

Then the final form of the perturbation is given as

V(z,∇,ω)=1

ρ0

[ω2

c20

γ+ρ

ρ0

∂β

∂z

∂

∂z

]

=1

ρ0

[ω2

c20

γ+1

1−β

∂β

∂z

∂

∂z

]. (A.10)

B Reflection coefficients for non-compact perturbation

In this paper, we only consider the case where the perturbation have finite range. Themain reason is to employ Eq. (3.12) to calculate reflection coefficient. For perturbationwith an infinite range, the integral in the equation is not well defined. However, thislimitation can be overcome if we assume the parameters approaches constant. At theconstant range, from Eqs. (3.6) and (3.11), the pressure wave can be represented as

P(kg,z,ω)=P1(kg,z,ω)+R(kg,ω)P∗1 (kg,z,ω). (B.1)


Since the sub-Volterra integral equation is an one-way propagator, P1(kg,z,ω) can be cal-culated with knowledge of perturbation only as above. As in [35, 36], we introduce thetransmission coefficient Tk(kg,ω). The pressure wave can also be expressed as

P(kg,z,ω)=T(kg,ω)P3(kg,z,ω), (B.2)

where P3(kg,z,ω) is another sub-pressure wave and is also in Volterra form (see [36] fordetail derivation). Consider pressure wave at two different depths z1 and z2, we have

[−P∗

1 (kg,z1,ω) P3(kg,z1,ω)−P∗

1 (kg,z2,ω) P3(kg,z2,ω)

][R(kg,ω)T(kg,ω)

]=

[P1(kg,z1,ω)P1(kg,z2,ω)

]. (B.3)

To avoid using Eq. (3.12), the reflection coefficients can be obtained from above linearalgebra equations.

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