research article modeling and processing l-band ground based radar data...

Hindawi Publishing CorporationJournal of Electrical and Computer EngineeringVolume 2013 Article ID 804615 6 pageshttpdxdoiorg1011552013804615

Research ArticleModeling and Processing L-Band Ground Based Radar Data forLandslides Early Warning

A R Laganagrave M T Bevacqua and T Isernia

Dipartimento di Ingegneria dellrsquoInformazione delle Infrastrutture e dellrsquoEnergia SostenibileUniversita Mediterranea di Reggio Calabria Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT) Italy

Correspondence should be addressed to T Isernia tommasoiserniaunircit

Received 26 May 2013 Accepted 19 June 2013

Academic Editor Sandra Costanzo

Copyright copy 2013 A R Lagana et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

L-band radars have been proposed as possible way for monitoring landslides In this paper we examine and solve the principaldifficulties arising in modeling and processing radar data evidencing differences with more usual SAR imaging Numericalexamples in support of the proposed processing procedure are finally provided

1 Introduction

Amongst the different possible ways to have a continuousmonitoring of areas which may be subject to landslides L-band radars offer the possibility to penetrate foliage while stillbeing able to get some understanding of the evolution of thescenario by means of differential imaging techniques

Differently from a large body of the literature whereimaging is performed by means of satellite-based radars(so that the movement of the satellite allows to rely on asynthetic aperture) the research activity considered in thefollowing concern the exploitation of a fixed ground basedradar where the only eventually available movement of thesensor is achieved by a mechanical or electronic scanning ofthe antenna pattern

Such a circumstance implies a number of interestingdifferences which are discussed in the following with respectto more usual radar imaging modeling and processing tech-niques

In fact a more detailed and difficult model is requiredfor data simulation (which is useful to ldquotunerdquo imaging proce-dure) Moreover data processing requires more sophisticatedtechniques with respect to satellite-based imaging On theother side location on ground of the sensor allows a verysimple deployment on those areas which are judged to have arisk of landslides Also the assumptions on ldquocoherencerdquo (seebelow) which are needed for differential imaging are more

easily verifiedwith respect to differential interferometric SARtechniques

For the sake of simplicity of explanation most conceptsregarding simulation and processing are explained with ref-erence to a simple ldquo2D geometryrdquo that is in a case where bothfields and the scenario are invariant along one dimension andthe field is directed along such a direction Such a simplifyingassumption is then removed in Section 4

In the following Section 2 is concerned with the problemof accurately simulating scenarios of interest and attentionis paid to the need of developing models which take intoaccount the fact that the radar antenna is supposed to bein the near zone of the scenario under test Then Section 3presents the basic idea for monitoring possible deformationsof the soil on the basis of a differential imaging techniqueIn particular the need of solving such kind of inverse sourceproblem is underlined and a suitable processing technique isintroduced and discussed

Finally the extension to the 3D case is discussed inSection 4 and results of processing (on simulated data) arepresented in Section 5

Conclusions and possible developments follow

2 Modeling Radar Data

In order to provide whatever form of imaging technique itis mandatory to get a deep understanding of the physical

2 Journal of Electrical and Computer Engineering

mechanism underlying the process and to get a properaccurate mathematical modeling of this latter In fact thiswill allow a better comprehension of the phenomena at hand(possibly suggesting proper imaging techniques) and it willallow an accurate simulation of radar data which is of courseessential to properly tune imaging techniques

The basic physical mechanism is the electromagneticscattering [1] In general case backscattering is due to botha kind of reflection at the interface as well as from thesecondary field which is generated within the volume ofinterest

By taking into account the values of permittivity andconductivity at the frequencies of our actual interest thepredominant mechanism is ldquosurface scatteringrdquo In fact asit can be deduced from Figure 1 the penetration depthconcerning the four kinds of soils described in [2] at thefrequency range of interest is expected to be too small to beable to consider the mechanism of ldquovolume scatteringrdquo

As no exact closed form but only approximate solutionsexist for the problem of scattering from a irregular surface[1] to derive a mathematical approximate expression of thescattered field it is assumed a surface with gentle undulations[1] whose average horizontal dimension is large comparedwith the wavelength This is a general case because if onewants to consider a rough surface it is only necessary to adda term of roughness according to ldquoClapps modelsrdquo [3]

It is also assumed that the total field at any point on thesurface can be computed as if the incident wave is impingingupon a infinite plane tangent to the point according to thebasic assumption of the Kirch off method [4ndash6] or ldquofacetsmodelrdquo [1] A mathematical statement of the scattered fieldformulated by Stratton and Chu [4 5] and modified by Silver[6] is as follows

Es = 119870ns times int [n times E minus 120578119904ns times (n timesH)] 119890

119895119896119904rsdotns

119889119878 (1)

where

(i) a time factor of the form 119890

119895119908119905 is understood

(ii) 119870 = minus119895119896119904119890

119895119896119904119877041205871198770

(iii) ns is the unit vector in the scattered direction

(iv) n is the unit vector normal to interface

(v) r is the vector that scans the points on the surface

(vi) 120578119904and 119896119904are respectively the intrinsic impedance and

wave number of the medium in which the scatteredfield is evaluated

(vii) 1198770is the range from the center of the illuminated area

to the point of observation

(viii) E and H are the total electric and magnetic fields onthe interface

Equation (1) allows computeing the scattered field in farzone [7] as superposition of fields scattered by each point onthe surface On the other side in the case of our interest thereceiving antenna is in the near zone of the scenario at hand

0

02

04

06

08

1

12

14

16

18

2

300 400 500 600 700 800 900 1000 1100 1200 1300Frequency (MHz)

Pene

trat

ion

dept

h (m

)

Field 1Field 2

Field 3Field 4

Figure 1 Penetration depth for soils described in [2]

(which acts as a source) so that (1) is noted as adequate toaccurately modeling the radar data

So (1) has to be reformulated considering that someparameters like the incident angle or the distance betweenthe sensor and the scenario change for each point on thesurface Accordingly it is not possible to use the usualsimplification for far-field region that is to approximate thedistance factor in the denominator by 119877

0and exploit a first-

order approximation for the argument of the exponentialAnother basic difference with respect to SAR techniques

concerns the kind of data which is collectedObviously one needs some kind of data diversity to get an

image after processing and if a 2D image is the final goal theraw datamust be a function of two coordinates at least whichis commonly achieved in SAR systems by a translation of theplatform and registering delays of the backscattered signalsIn our scenario the ldquoplatformrdquo is fixed and we would like toexploitmonochromatic signals which can allow a very simpledata processing (see Section 3)

Hence the needed diversity is achieved by a suitable 2Dmechanical or electronic scanning of the antenna so that ourdata will be a 2D function of (120599

119901 120593119901) where such a couple

denotes the direction where the radar antenna is pointing toBy assuming a scenario (and incident fields) which is

invariant along one direction and assuming an incident fieldparallel to such a direction the overall model becomes scalarand the field collected at the radar position can be written as

119864119904(120599119901) =

minus1198951198961

4120587

int [(119860119899119904119909minus119861119899119904119911)]

times

119890

11989521198961|119903minus119903119905119909|

1003816100381610038161003816

119903 minus 119903119905119909

1003816100381610038161003816

ℎ (120599119901 120599119886) 119889119878

(2)

where

119860 = (1 + 119877perp) (minus119899119909) + (n sdot ni) (1 minus 119877perp) (minus119899119904119909)

119861 = (1 + 119877perp) (119899119911) + (n sdot ni) (1 minus 119877perp) (119899119904119911)

(3)

Journal of Electrical and Computer Engineering 3

where

(i) 120579119901is the observation angle of the measuring sensor

(ii) 1198961is the wave number of the medium

(iii) 119877perp is the Fresnel reflection coefficient(iv) h is the array factor of the sensor used formonitoring(v) ni is the unit vector that identifies the incident

direction(vi) 120579119886is the observation angle that scans the elemen-

tary areas on the surface (direction where the radarantenna is pointing to)

In (2) one can identify a factor depending on the antennaa factor depending on the round-trip and a factor dependingon the scenario (shape of the interface and its electromagneticcharacteristics)

A numerical simulator based on (2) has been developedsuccessfully and tested in some canonical scenarios (flat andsinusoidally varying surfaces)

3 2D Case Data Processing

31 The Basic Idea In order to describe the strategy it isconvenient to rewrite the relation between the scattered elec-tromagnetic field and the surface features already describedin (2) in a different form In particular the scattered electro-magnetic field at the radar position when the radar antennais pointing toward 120579

119901can be expressed by the following

equation

119864119904(120579119901) = int119864

119904 loc (120579119886 120579119901) 119889120579119886

= int 119869ind (120579119886)119890

minus119895119896119877(120579119886)

radic119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

= int119891 [119877perp (120579119886)] 119890minus1198952119896Δ119877(120579

119886)

times

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

(4)

where

(i) 119864119904 loc(120579119886) is the contribution to the overall received

field due to each elementary area on the surface(ii) R is the generic distance between antenna and surface

in particular 119877(120579119886) = 119877(120579

119886) + Δ119877(120579

119886) where 119877(120579

119886)

is a ldquoreference distancerdquo and Δ119877(120579119886) is the amplitude

corresponding to the surface deformation(iii) ℎarray is the array factor (in transmission and receiving

case)(iv) 119891(119877perp)119890

minus1198952119896Δ119877(120579119886) is a function called 120574(120579

119886) that

depends on surface characteristics that is local reflec-tion coefficient roughness and local height of thesurface

TXRX

X

R

ΔR

Figure 2 Reference scenario

(v) 119890minus1198952119896119877(120579119886)119877(120579119886) is a term that depends on the round-

trip path of the electromagnetic field

Figure 2 shows the graphic representation of the referencescenario

So (4) can be rewritten as follows

119864119904(120579119901) = int 120574 (120579

119886) 119860 (120579

119886 120579119901) 119889120579119886 (5)

The processing used herein in order to locate possiblesurface deformations is based on differential interferometry[8] which works as followsThe phase of the reflectivity func-tion contains information about the signal round-trip pathTherefore a surface deformation causes a different length ofround-trip path and consequently a phase displacement ofthe signal Then if 120574

1(120579119886) and 120574

2(120579119886) are reflectivity functions

to two times t1and t2 one can write

1205741(120579119886) = 119860

11205931(120579119886) 119890

minus1198951198962Δ1198771(120579119886)

1205742(120579119886) = 119860

21205932(120579119886) 119890

minus1198951198962Δ1198772(120579119886)

(6)

where1198601and119860

2are themaxim intensities of the two signals

the complex signals 1205931and 120593

2take into account possible

variations on the factor 119891[119877perp(120579119886)] and the exponentialterms contain information about surface displacementsThen ldquobeatingrdquo the two signals one achieves

1205741(120579119886) 1205742(120579119886)

lowast

= 11986011198602

1003816100381610038161003816

1205931(120579119886) 120593

2(120579119886)

1003816100381610038161003816

times 119890

119895(arg1205931(120579119886)minusarg120593

2(120579119886))

119890

minus1198951198962(Δ1198771(120579119886)minusΔ1198772(120579119886))

(7)

Then as long as the phase of120593(sdot)has not changed amongstthe different instant times (eg under a ldquohigh correlationrdquoassumptions) the phase of expression (7) gives back

2119896 (Δ1198771(120579119886) minus Δ119877

2(120579119886)) (8)

which is obviously related to displacements However a lastdifficulty has to be tackled In fact the phase factor has anambiguity of 2120587 (wrapped phase) So a (possibly effective)phase unwrapping procedure is needed

32 Extracting the Reflectivity Function Traditional syntheticaperture radar (SAR) processing procedures heavily rely onthe convolutional nature of the equation to be inverted [9]


In fact such a circumstance is exploited to develop computa-tionally effective codes based on fast Fourier transforms

In our case due to the expression of 119860(120579119886 120579119901) and 120574(120579

119886)

(a different choice would have been possibly by incorporating119890

minus1198952119896119877(120579119886)

119877(120579119886) in the reflectivity but then a much more

cumbersome unknown 120574(120579119886) has to be retrieved) one cannot

rely any more on convolutions so that different processingstrategies have to be devised

As in any integral equationwith a smooth kernel problem(5) is ill posed (According toHadamard definition a problemis ill posed when its solution is not unique or does not existor does not depend continuously on the data)

As a consequence only an approximate version of the truereflectivity function can be found by looking for some kind ofregularized solution

Amongst the different possibilities the well-known trun-cated singular-value decomposition (TSVD) [10] can be usedAs a suitable alternative a Tikhonov regularization [11] can beexploited which is the suggested solution in case one has todeal with a very large number of unknowns

33 The Phase Unwrapping Problem As previously de-scribed the phase factor in (8) has an ambiguity of 2120587(wrapped phase) [12 13] The mathematical statement of thephase unwrapping problem can be expressed by

120593unwrapped = 120593wrap + 2119873120587 (9)

and where the integer unknown N has to be determined foreach point of the scenario

Two-dimensional phase unwrapping is a classical prob-lem encountered in several fields such as electromagnetictheory optics and DInSAR data processing For this reasonthere is a large amount of research [14 15] and solutionapproaches to both 1D and 2D phase unwrapping problemRoughly speaking solution approaches span from model-based procedures [14] to ldquototal least squaresrdquo [15 16] Com-pressive sensing-based procedures have also been recentlyproposed [17]

4 Extension to the 3D Case

The approach previously described for the 2D geometry canbe extended to the 3D case In particular it is possibleto rewrite an equivalent approach in terms of open-circuitvoltage [7] by

119881

119875

oc (120579119901) = int

surface119881

119875

oc loc (120579119886 120579119901) 119889120579119886

= int

surface119864

119875

s loc (120579119886) hPrx (120579119886 120579119901) 119889120579119886

= int

surface

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

2hPR (120579119886 120579119901)

times 120574

119875

(120579119886) hPT (120579

119886 120579119901) 119889120579119886

= int

surface119860

119875

(120579119886 120579119901) 120574

119875

(120579119886) 119889120579119886

(10)

where

(i) 119881119875oc loc is the open-circuit voltage induced by 119864119875s loc

(ii) hPR is the array factor in transmission case

(iii) hPT is the array factor in receiving case

(iv) 120579119886= (120599119886 120593119886) corresponds to the polar coordinates of

the points on the surface

(v) 120579119901= (120599119901 120593119901) corresponds to the polar coordinates of

the observation point of the measuring sensor(vi) P is a superscript that identifies the polarization

channel

In fact in 3D case one can consider four different types oftransmission-reception setup [18] Notably for three of thesefour different ldquopolarization channelsrdquo a different reflectivityfunction could be defined but the signal processing requiredto get the unknown deformation would be exactly the sameFar from being a problem such a circumstance could beexploited in order to improve accuracy in the deformationreconstruction andor avoid possible ambiguities in the phaseunwrapping step On the other side such an approach wouldneed antennas able to work in both the two independentpolarizations

5 2D Case Data Processing NumericalProcessor and Some Examples

Using contents as above numerical simulators (for the 1D and2D cases) have been developed first Then the different stepsof the processors have also been developed in a MATLABenvironment [19]

After validating the numerical simulator by examiningresults furnished by the codes in simple cases (flat surfacesa single small deformation and a sinusoidally varyingdeformation) the processing chain has been tested

The first example is aimed at displaying the reconstruc-tion of a simple deformation that is a rigid shiftTheworkingfrequency is 2GHz The reference surface is the undulatingsurface shown in Figure 3 Assuming that at the time t

2this

surface is rigidly shifted of 1205824 the reconstruction is shownin Figure 4 The latter is obtained by the multiplication in(7) The unwrapping procedure used herein is the standardprocedure of the MATLAB library In correspondence to thenulls of the amplitudes of the functions 120574

1(120579119886) and 120574

2(120579119886)

where there is a loss of information about the phase apreprocessing based on technique described in [15] hasalso been used As it can be seen in Figure 4 the surfacedeformation is correctly indentified

A second example considers a surface deformation form-ing by the overlapping between the undulating surface inFigure 3 and the deformation shown in Figure 5Theworkingfrequency is 2GHz

The corresponding reconstruction is shown in Figure 6As it can be seen also in this case the deformation is correctlyidentified

As it can be seen one is able to retrieve deformations assmall as a quarter of a wavelength In all tests the simulated


minus15 minus10 minus5 0 5 10 15

0

05

1

15

2

25

3

35

4

45

5

Surface (m)

Am

plitu

de o

f the

und

ulat

ing

surfa

ce (m

)

Figure 3 Undulating reference surface

minus10 minus5 0 5 1002

021

022

023

024

025

026

027

028

029

ReconstructedReal

Am

plitu

de o

f the

def

orm

atio

n (120582

)

Surface (m)

Figure 4 Surface deformation reconstructed (blue line) realsurface deformation (red line)

data used for processing was affected by a simulated errorwith SNR = 20 dB

Similar kinds of results are achieved in the preliminarytest we are performing on the corresponding 3D processor

6 Conclusions

In this paper approaches commonly used in SAR for imagingof deformations have been generalized to the case of ground-based nonsynthetic radars operating in the L band

Such a case is indeed of interest in the applications as itallows a very simple deployment of low-cost radar systems[20] and it allows to monitor regions which could not beeasily accessible from satellites or airborne-based instru-mentations Moreover the use of relatively low frequenciesallows penetration of foliage Last but not least different fromwell-known DInSAR techniques which requires multiple


0

01

02

03

04

05

06

07

08

09

Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

minus01

Figure 5 Example of deformation

minus01

0

01

02

03

04

05

06

07

08

09

minus10 minus5 0 5 10Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

ReconstructedReal


passages in slightly different orbits the sensor is herein fixedwhich allows a much more easy fulfillment of the coherenceproperty recalled after (7)

On the other side proper modeling and processing atthese frequencies has required the development of both newsimulators as well as the development of the new processingtechniqueswhich have been introduced discussed and testedabove

Conflict of Interests

All authors state that the research herein described is notinfluenced by secondary interests (such as financial gain) andthat they have no conflict of interests concerning all termsused in this paper


Acknowledgment

This work has been carried out under the framework of PON01 01503 National Italian Project ldquoLandslides EarlyWarningrdquofinanced by the Italian Ministry of University and Research

References

[1] F T Ulaby R K Moore and A K Fung Microwave RemoteSensing vol III chapter 11-12 Addison-Wesley Boston MassUSA

[2] N R Peplinski F T Ulaby and M C Dobson ldquoDielectricproperties of soils in the 03ndash13-GHz rangerdquo IEEE Transactionson Geoscience and Remote Sensing vol 33 no 3 pp 803ndash8071995

[3] R E Clapp ldquoA theoretical and experimental study of radarground returnrdquo Tech Rep 1024MIT Radiation Lab April 1946

[4] J A Stratton Electromagnetic Theory McGraw-Hill New YorkNY USA 1941

[5] J A Stratton and L J Chu ldquoDiffraction theory of electromag-netic wavesrdquo Physical Review vol 56 no 1 pp 99ndash107 1939

[6] S SilverMicrowave Antenna Theory and Design vol 12 ofMITRadiation Lab Series McGraw-Hill New York NY USA 1947

[7] C Balanis AntennaTheory Analysis and Design John Wiley ampSons New York NY USA 2005

[8] D Massonnet and K L Feigl ldquoRadar interferometry andits application to changes in the earthrsquos surfacerdquo Reviews ofGeophysics vol 36 no 4 pp 441ndash500 1998

[9] G Franceschetti and G Schirinzi ldquoSAR processor based ontwo-dimensional FFT codesrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 26 no 2 pp 356ndash366 1990

[10] M Bertero and P Boccacci Introduction to Inverse Problems inImaging IOP Bristol UK 1998

[11] V A TikhonovMethodes de Resolution de Problemes mal PosesEditions de Moscou Traduction Francaise Paris France 1976

[12] O M Bucci and T Isernia ldquoElectromagnetic inverse scatteringretrievable information and measurement strategiesrdquo RadioScience vol 32 no 6 pp 2123ndash2137 1997

[13] M Costantini ldquoA novel phase unwrapping method based onnetwork programmingrdquo IEEE Transactions on Geoscience andRemote Sensing vol 36 no 3 pp 813ndash821 1998

[14] Z-P Liang ldquoA model-based method for phase unwrappingrdquoIEEE Transactions on Medical Imaging vol 15 no 6 pp 893ndash897 1996

[15] G Fornaro G Franceschetti R Lanari and E Sansosti ldquoRobustphase-unwrapping techniques a comparisonrdquo Journal of theOptical Society of America A Optics and Image Science andVision vol 13 no 12 pp 2355ndash2366 1996

[16] G Fornaro A Pauciullo and E Sansosti ldquoPhase difference-based multichannel phase unwrappingrdquo IEEE Transactions onImage Processing vol 14 no 7 pp 960ndash972 2005

[17] M Hosseini and O Michailovich ldquoDerivative compressivesampling with application to phase unwrappingrdquo in Proceedingsof European Signal Processing Conference (EUSIPCO rsquo09) pp115ndash119 Glasgow UK August 2009

[18] W-M Boerner H Mott E Luneburg et al ldquoPolarimetry inradar remote sensing basic and applied conceptsrdquo in Principlesand Applications of Imaging Radar F M Henderson and A JLewis Eds vol 2 of Manual of Remote Sensing Chapter 5 pp271ndash357 John Willey amp Sons New York NY USA 3rd edition1998

[19] MATLAB Userrsquos Guide 2009 httpwwwmathworksit[20] S Costanzo F Spadafora A Borgia O H Moreno A

Costanzo and G Di Massa ldquoHigh resolution software definedradar system for target detectionrdquo in Advances in IntelligentSystems and Computing vol 206 pp 997ndash1005 2013

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mechanism underlying the process and to get a properaccurate mathematical modeling of this latter In fact thiswill allow a better comprehension of the phenomena at hand(possibly suggesting proper imaging techniques) and it willallow an accurate simulation of radar data which is of courseessential to properly tune imaging techniques

The basic physical mechanism is the electromagneticscattering [1] In general case backscattering is due to botha kind of reflection at the interface as well as from thesecondary field which is generated within the volume ofinterest

By taking into account the values of permittivity andconductivity at the frequencies of our actual interest thepredominant mechanism is ldquosurface scatteringrdquo In fact asit can be deduced from Figure 1 the penetration depthconcerning the four kinds of soils described in [2] at thefrequency range of interest is expected to be too small to beable to consider the mechanism of ldquovolume scatteringrdquo

As no exact closed form but only approximate solutionsexist for the problem of scattering from a irregular surface[1] to derive a mathematical approximate expression of thescattered field it is assumed a surface with gentle undulations[1] whose average horizontal dimension is large comparedwith the wavelength This is a general case because if onewants to consider a rough surface it is only necessary to adda term of roughness according to ldquoClapps modelsrdquo [3]

It is also assumed that the total field at any point on thesurface can be computed as if the incident wave is impingingupon a infinite plane tangent to the point according to thebasic assumption of the Kirch off method [4ndash6] or ldquofacetsmodelrdquo [1] A mathematical statement of the scattered fieldformulated by Stratton and Chu [4 5] and modified by Silver[6] is as follows

Es = 119870ns times int [n times E minus 120578119904ns times (n timesH)] 119890

119895119896119904rsdotns

119889119878 (1)

where

(i) a time factor of the form 119890

119895119908119905 is understood

(ii) 119870 = minus119895119896119904119890

119895119896119904119877041205871198770

(iii) ns is the unit vector in the scattered direction

(iv) n is the unit vector normal to interface

(v) r is the vector that scans the points on the surface

(vi) 120578119904and 119896119904are respectively the intrinsic impedance and

wave number of the medium in which the scatteredfield is evaluated

(vii) 1198770is the range from the center of the illuminated area

to the point of observation

(viii) E and H are the total electric and magnetic fields onthe interface

Equation (1) allows computeing the scattered field in farzone [7] as superposition of fields scattered by each point onthe surface On the other side in the case of our interest thereceiving antenna is in the near zone of the scenario at hand

0

02

04

06

08

1

12

14

16

18

2

300 400 500 600 700 800 900 1000 1100 1200 1300Frequency (MHz)

Pene

trat

ion

dept

h (m

)

Field 1Field 2

Field 3Field 4

Figure 1 Penetration depth for soils described in [2]

(which acts as a source) so that (1) is noted as adequate toaccurately modeling the radar data

So (1) has to be reformulated considering that someparameters like the incident angle or the distance betweenthe sensor and the scenario change for each point on thesurface Accordingly it is not possible to use the usualsimplification for far-field region that is to approximate thedistance factor in the denominator by 119877

0and exploit a first-

order approximation for the argument of the exponentialAnother basic difference with respect to SAR techniques

concerns the kind of data which is collectedObviously one needs some kind of data diversity to get an

image after processing and if a 2D image is the final goal theraw datamust be a function of two coordinates at least whichis commonly achieved in SAR systems by a translation of theplatform and registering delays of the backscattered signalsIn our scenario the ldquoplatformrdquo is fixed and we would like toexploitmonochromatic signals which can allow a very simpledata processing (see Section 3)

Hence the needed diversity is achieved by a suitable 2Dmechanical or electronic scanning of the antenna so that ourdata will be a 2D function of (120599

119901 120593119901) where such a couple

denotes the direction where the radar antenna is pointing toBy assuming a scenario (and incident fields) which is

invariant along one direction and assuming an incident fieldparallel to such a direction the overall model becomes scalarand the field collected at the radar position can be written as

119864119904(120599119901) =

minus1198951198961

4120587

int [(119860119899119904119909minus119861119899119904119911)]

times

119890

11989521198961|119903minus119903119905119909|

1003816100381610038161003816

119903 minus 119903119905119909

1003816100381610038161003816

ℎ (120599119901 120599119886) 119889119878

(2)

where

119860 = (1 + 119877perp) (minus119899119909) + (n sdot ni) (1 minus 119877perp) (minus119899119904119909)

119861 = (1 + 119877perp) (119899119911) + (n sdot ni) (1 minus 119877perp) (119899119904119911)

(3)


where











equation

119864119904(120579119901) = int119864

119904 loc (120579119886 120579119901) 119889120579119886

= int 119869ind (120579119886)119890

minus119895119896119877(120579119886)

radic119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

= int119891 [119877perp (120579119886)] 119890minus1198952119896Δ119877(120579

119886)

times

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

(4)

where



in particular 119877(120579119886) = 119877(120579

119886) + Δ119877(120579

119886) where 119877(120579

119886)



case)(iv) 119891(119877perp)119890


119886) that


TXRX

X

R

ΔR






119864119904(120579119901) = int 120574 (120579

119886) 119860 (120579

119886 120579119901) 119889120579119886 (5)


1(120579119886) and 120574



1205741(120579119886) = 119860

11205931(120579119886) 119890

minus1198951198962Δ1198771(120579119886)

1205742(120579119886) = 119860

21205932(120579119886) 119890

minus1198951198962Δ1198772(120579119886)

(6)

where1198601and119860





1205741(120579119886) 1205742(120579119886)

lowast

= 11986011198602

1003816100381610038161003816

1205931(120579119886) 120593

2(120579119886)

1003816100381610038161003816

times 119890

119895(arg1205931(120579119886)minusarg120593

2(120579119886))

119890

minus1198951198962(Δ1198771(120579119886)minusΔ1198772(120579119886))

(7)


2119896 (Δ1198771(120579119886) minus Δ119877

2(120579119886)) (8)






119886)


minus1198952119896119877(120579119886)








120593unwrapped = 120593wrap + 2119873120587 (9)





119881

119875

oc (120579119901) = int

surface119881

119875

oc loc (120579119886 120579119901) 119889120579119886

= int

surface119864

119875

s loc (120579119886) hPrx (120579119886 120579119901) 119889120579119886

= int

surface

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

2hPR (120579119886 120579119901)

times 120574

119875

(120579119886) hPT (120579

119886 120579119901) 119889120579119886

= int

surface119860

119875

(120579119886 120579119901) 120574

119875

(120579119886) 119889120579119886

(10)

where








channel






2this


1(120579119886) and 120574

2(120579119886)







0

05

1

15

2

25

3

35

4

45

5

Surface (m)

Am

plitu

de o

f the

und

ulat

ing

surfa

ce (m

)



021

022

023

024

025

026

027

028

029

ReconstructedReal

Am

plitu

de o

f the

def

orm

atio

n (120582

)

Surface (m)




6 Conclusions




0

01

02

03

04

05

06

07

08

09

Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

minus01


minus01

0

01

02

03

04

05

06

07

08

09


Am

plitu

de o

f the

def

orm

atio

n (120582

)

ReconstructedReal







Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where











equation

119864119904(120579119901) = int119864

119904 loc (120579119886 120579119901) 119889120579119886

= int 119869ind (120579119886)119890

minus119895119896119877(120579119886)

radic119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

= int119891 [119877perp (120579119886)] 119890minus1198952119896Δ119877(120579

119886)

times

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

ℎarray (120579119886 120579119901) 119889120579119886

(4)

where



in particular 119877(120579119886) = 119877(120579

119886) + Δ119877(120579

119886) where 119877(120579

119886)



case)(iv) 119891(119877perp)119890


119886) that


TXRX

X

R

ΔR






119864119904(120579119901) = int 120574 (120579

119886) 119860 (120579

119886 120579119901) 119889120579119886 (5)


1(120579119886) and 120574



1205741(120579119886) = 119860

11205931(120579119886) 119890

minus1198951198962Δ1198771(120579119886)

1205742(120579119886) = 119860

21205932(120579119886) 119890

minus1198951198962Δ1198772(120579119886)

(6)

where1198601and119860





1205741(120579119886) 1205742(120579119886)

lowast

= 11986011198602

1003816100381610038161003816

1205931(120579119886) 120593

2(120579119886)

1003816100381610038161003816

times 119890

119895(arg1205931(120579119886)minusarg120593

2(120579119886))

119890

minus1198951198962(Δ1198771(120579119886)minusΔ1198772(120579119886))

(7)


2119896 (Δ1198771(120579119886) minus Δ119877

2(120579119886)) (8)






119886)


minus1198952119896119877(120579119886)








120593unwrapped = 120593wrap + 2119873120587 (9)





119881

119875

oc (120579119901) = int

surface119881

119875

oc loc (120579119886 120579119901) 119889120579119886

= int

surface119864

119875

s loc (120579119886) hPrx (120579119886 120579119901) 119889120579119886

= int

surface

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

2hPR (120579119886 120579119901)

times 120574

119875

(120579119886) hPT (120579

119886 120579119901) 119889120579119886

= int

surface119860

119875

(120579119886 120579119901) 120574

119875

(120579119886) 119889120579119886

(10)

where








channel






2this


1(120579119886) and 120574

2(120579119886)







0

05

1

15

2

25

3

35

4

45

5

Surface (m)

Am

plitu

de o

f the

und

ulat

ing

surfa

ce (m

)



021

022

023

024

025

026

027

028

029

ReconstructedReal

Am

plitu

de o

f the

def

orm

atio

n (120582

)

Surface (m)




6 Conclusions




0

01

02

03

04

05

06

07

08

09

Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

minus01


minus01

0

01

02

03

04

05

06

07

08

09


Am

plitu

de o

f the

def

orm

atio

n (120582

)

ReconstructedReal







Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119886)


minus1198952119896119877(120579119886)








120593unwrapped = 120593wrap + 2119873120587 (9)





119881

119875

oc (120579119901) = int

surface119881

119875

oc loc (120579119886 120579119901) 119889120579119886

= int

surface119864

119875

s loc (120579119886) hPrx (120579119886 120579119901) 119889120579119886

= int

surface

119890

minus1198952119896119877(120579119886)

119877 (120579119886)

2hPR (120579119886 120579119901)

times 120574

119875

(120579119886) hPT (120579

119886 120579119901) 119889120579119886

= int

surface119860

119875

(120579119886 120579119901) 120574

119875

(120579119886) 119889120579119886

(10)

where








channel






2this


1(120579119886) and 120574

2(120579119886)







0

05

1

15

2

25

3

35

4

45

5

Surface (m)

Am

plitu

de o

f the

und

ulat

ing

surfa

ce (m

)



021

022

023

024

025

026

027

028

029

ReconstructedReal

Am

plitu

de o

f the

def

orm

atio

n (120582

)

Surface (m)




6 Conclusions




0

01

02

03

04

05

06

07

08

09

Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

minus01


minus01

0

01

02

03

04

05

06

07

08

09


Am

plitu

de o

f the

def

orm

atio

n (120582

)

ReconstructedReal







Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

05

1

15

2

25

3

35

4

45

5

Surface (m)

Am

plitu

de o

f the

und

ulat

ing

surfa

ce (m

)



021

022

023

024

025

026

027

028

029

ReconstructedReal

Am

plitu

de o

f the

def

orm

atio

n (120582

)

Surface (m)




6 Conclusions




0

01

02

03

04

05

06

07

08

09

Surface (m)

Am

plitu

de o

f the

def

orm

atio

n (120582

)

minus01


minus01

0

01

02

03

04

05

06

07

08

09


Am

plitu

de o

f the

def

orm

atio

n (120582

)

ReconstructedReal







Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgment


References























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article modeling and processing l-band ground based radar data...

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