Temporal decorrelation model for the bistatic SAR interferometry

Qilei Zhangand Wenge Chang

College of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China

Temporal decorrelation model for the bistatic SAR interferometry

Qilei Zhang*and Wenge Chang

College of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China

This paper develops a temporal decorrelation model for the bistatic synthetic aperture radar(BSAR)interferometry.The temporal baseline is one of the important decorrelation sources for the repeat-pass synthetic aperture radar(SAR)interferometry.The study of temporal decorrelation is challenging,especially for the bistatic confguration,since temporal decorrelation is related to the data acquisition geometry.To develop an appropriate theoretical model for BSAR interferometry,the existing models for monostatic SAR cases are extended,and the general BSAR geometry confguration is involved in the derivation.Therefore,the developed temporal decorrelation model can be seen as a general model. The validity of the theoretical model is supported by Monte Carlo simulations.Furthermore,the impacts of the system parameters and BSAR geometry confgurations on the temporal decorrelation model are discussed briefy.

temporal decorrelation,bistatic synthetic aperture radar(BSAR),interferometry,geometry confguration.

1.Introduction

The synthetic aperture radar(SAR)interferometry is an important technique for the earth observation and measurement[1,2].Nowadays,with the development of the bistatic SAR(BSAR)techniques,the interferometric application of BSAR has been focused by more and more institutes [3–5].

As a valuable parameter of the radar interferometry, the coherence indicates the extent of the similarity between consecutive radar images[6].Usually,the coherence performance of an SAR interferometer can be altered by several different factors,named as decorrelation sources.Among them,the temporal de-correlation plays an important role,especially,for the repeat-pass SAR interferometry[7].

The temporal de-correlation is due to physical changes of the observed scene over the time period between consecutive observations[8].Both natural and anthropogenic physical changes could possibly be involved[9].The natural changes,like vegetation growth,wind and precipitation,could be able to reduce the coherence,while the anthropogenic changes,such as construction,demolishing and irrigation,might completely destroy the coherence.For different interferometric applications,the temporal decorrelation plays different roles.In the interferometric height measurement,the temporal decorrelation might be a negative factor,since it increases the interferometric phase uncertainty.However,in the coherent change detection(CCD),the temporal decorrelation is the value to be measured,since it indicates the extent of the change.

In previous works,several useful models of the temporal decorrelation for the monostatic SAR interferometry have been developed[6,10–14].Most of these models are derived with the assumption that the temporal decorrelation mechanism is primarily due to random motions of element scatterers within the resolution cell[12].The validity of this assumption has been validated by real data experiments[6,14,15].An exponential model was derived by assuming the Gaussian-statistic motion in[6].Based on the assumption of the Brownian motion,F.Rocca extended the exponential model to account for different temporal baseline cases[12].Given the assumption of the Gaussianstatistic motion which varies along the vertical direction of vegetated land,M.Lavalle developed a temporal decorrelation model to evaluate the effect of wind in the forest canopy[14].

With the development of BSAR technologies and applications,the study of the temporal decorrelation for the BSAR interferometry is needed.However,this study is more challenging,since the data acquisition geometry needs to be taken into account.In BSAR interferometry, the phase variation due to the motion of element scatterers within the resolution cell depends on the data acquisition geometry.Moreover,the structure functionof the observedscene,such as vegetated land,varies with different acquisition geometries as well.Therefore,existing temporal models for the monostatic confguration is not valid any more. To deal with this problem,this paper develops a temporal decorrelation model for the BSAR interferometry.Even though the same approach as[14]is followed,this study is more advanced than[14],due to that the general bistatic confguration is considered here.

This paper is organized as follows.In Section 2,the fundamental theory of the temporal decorrelation for BSAR interferometry is developed based on the general BSAR imaging geometry.In Section 3,the temporal decorrelation model is derived by applying the random volume over ground(RVoG)model and by assuming that the random motion is characterized by independent Gaussian distributions.In Section 4,the validity of the developed model is verifed by performing Monte Carlo simulations.Finally,we present the sensitivity discussion of the developed model in Section 5,followed by some conclusions in Section 6.

2.Fundamental theory

In this section,given the principle of BSAR imaging,we present the fundamental theory of the temporal decorrelation for BSAR interferometry.Fig.1 illustrates the general imaging geometry of BSAR interferometry,where the volume scattering model is considered.As shown in Fig.1, the transmitter and the receiver illuminate a scene patch with exactly the same position at differenttimes.The complex image signals obtained at consecutive observations can be denoted as s1and s2.Let O represent the center of the resolution cell.For the resolution cell,the transmitter T is located at(αT,βT,RT)and the receiver R is located at(αR,βR,RR),where αTand αRare the azimuth angles,βTand βRare the elevation angles,RTand RRare the distance from O to T and the distance from O to R, respectively.

Fig.1 Imaging geometry of BSAR interferometry

Assuming that P with coordinate(x,y,z)is an arbitrary element scatterer within the resolution cell,and then the range sum T-P-R can be given by

The complex image signals s1and s2without the system thermal noise,measured at the position O,maybe represented as the sum of complex returns of all the element scatterers within the resolution cell[6].

where λ=c/fcis the radar wavelength,fcis the carrier frequency,c is the speed of light,W(x,y)is the BSAR point spread function(PSF),and f1(x,y,z)and f2(x,y,z) represent the complex bistatic scattering coeffcients of the element scatterer within the resolution cell for the primary acquisition and the repeat-pass acquisition,respectively. The temporal decorrelation resulted from the variation of the complex bistatic scatter can be evaluated by

Given that the dielectric constant of element scatterers is fxed and only the positions of element scatterers vary[14], the relationship between f1(x,y,z)and f2(x,y,z)can be expressed as

where ΔR(Δx,Δy,Δz)is the range variation caused by the position motion(Δx,Δy,Δz)of element scatterers. According to Fig.1 and(1),ΔR(Δx,Δy,Δz)can be calculated by

Therefore,the cross-correlation of the complex images pair can be given as

Then,assume that the imaged scene consists of the uniformly distributed and uncorrelated element scatterers[6]

where?·?indicates the ensemble average,and ρ(x,y,z) is the average complex bistatic scatter coeffcient,representing the structure function of the resolution cell.Furthermore,assume that the random motion of element scatterers is the dominant dynamic process that occurs within the resolution cell and the position changes can be characterized by independent probability distributions p(Δx), p(Δy)and p(Δz),we can get

Similarly,the auto-correlation of the complex images pair can be given as

By substituting(9)and(10)into(4),we can get the evaluation of temporal decorrelation effects.However,to get the explicit expression for the temporal decorrelation model,other assumptions have to be made.

3.Temporal decorrelation model

Observing(9)and(10)suggests that the temporal decorrelation depends on both the structure function and the random motion.A reasonable assumption for the imaged scene(especially for the vegetated land surface)is that the structure function and the random motion only depend on the initial vertical position of element scatterers[16,17]. This means that the structure function can be expressed as ρ(x,y,z)=σ0ρ(z)and the random motion can be characterized by p(Δx,z),p(Δy,z)and p(Δz,z),where σ0is a constant value,thus

where

Therefore,the expression of the temporal decorrelation can be rewritten as

We can fnd that the calculation of the temporal decorrelation rests with the structure function ρ(z)and the expressions of χ(z).

The structure function ρ(z)depends on the scene type, weather conditions,system confgurations,polarizations, the radar wavelength,etc.[17].The in-depth study of the structure function is not the aim of this paper.Here,a twolayer coherent scattering model of the structure function named as random volume over ground(RVoG)is applied. This model has been successfully adopted by many studies for interferometric and polarimetric radar[14,16,17].As shown in Fig.2,the RVoG model considers the scattering contribution from both a randomly oriented volume and an underlying dielectric surface located at zg[18].

Fig.2 RVoG model of the structure function

However,considering that the incident angle and the refected angle are different in BSAR confgurations,the structure function based on the RVoG model should be extended to

where zg≤z≤zg+hv,εvis the average bistatic scatter per unit length of the random volume layer and εgis that of the ground layer,respectively.In(15),κeis the one-way amplitude extinction coeffcient,zgis the height of the underlying ground surface,hvis the thickness of the randomvolume layer,cos θTand cos θRare the incident angle and the refected angle,respectively.

We consider that the random motion is characterized by independent Gaussian distributions and the variance of the random motion is dependent on the initial vertical position,i.e.,Δx～N(0,σ2Δx(z)),Δy～N(0,σ2Δy(z))and Δz～N(0,σ2Δz(z)).After some algebraic derivations,the result of(13)can be given as

where

To derive the explicit expression of the temporal decorrelation,here the variance of the random motion is assumed to be the linear function of the variable z as[14]. Assuming that the motion standard deviation of the element scatterers is σi,gat the bottom level zg,and σi,vat the top level zg+hv,then the motion variance with respect to the vertical position z can be evaluated by

where Δσi=σi,v?σi,gis the differential motion standard deviation,and i=Δx,Δy,Δz.

Substituting(15)and(16)into(14)yields

where

γTGstands for the temporal decorrelation effect caused by the random motion of element scatterers at the bottom level,γTVstands for the temporal decorrelation effect of the whole random volume layer,and μ represents the ground-to-volume scattering ratio.The parameters ξ1and ξ2are defned as

where

It can be seen from(18)that the temporal coherence value is between γTVand γTG,obtained for μ=0 and μ→∞,respectively.The former research showed that the value of the ground-to-volume scattering ratio μ is subject to the polarimetric character[14].Since the polarimetric diversity is not the emphasis of this study,the value of μ is assumed to be constant between consecutive observations.

The monostatic confguration can be seen as a particular case of the bistatic confguration.As shown in Fig.1,if we set αT=αR=π/2 and βT=βR=β,the bistatic imaging confguration will be degraded to the monostatic confguration,thus θT=θR=θ.Furthermore,assuming that the motion standard deviation of the element scatterers satisfes σΔy,g=σΔz,gand ΔσΔy=ΔσΔz,the derived temporal decorrelation model will be reduced to the expression presented in[14].If we consider the random volume layer only,i.e.,μ=0,the temporal decorrelation value satisfes γTemp=γTV.Then if the random motion is assumed to be uniform along the vertical direction,i.e. ΔσΔy=ΔσΔz=0,the temporal decorrelation model can be reduced to

The expression shown in(24)is in agreement with the temporal de-correlation model for monostatic SAR interferometry derived in[6].This means that the temporal decorrelation model derived here is a general model for BSAR interferometry.

4.Simulation validation

To validate the developed temporal decorrelation model, Monte Carlo simulations are performed in this section,for both monostatic and bistatic confgurations.The parameters used in the simulations are listed in Table 1.

The principle of the simulations here is similar with that in[6].However,several necessary modifcations need tobe made to meet the assumptions introduced above.Specifcally,the simulated temporal coherence can be obtained by performing the following steps.

Table 1 Simulation parameters

Step1It is assumed that a set of 10000 element scatterers is randomly located within a resolution cell to simulate the random volume layer,and then it is assumed that another set of 10 000 element scatterers is randomly located at the bottom level(z=zg)to simulate the ground layer.

It is worth noting that RTand RRcan be omitted in the phase calculation,since they are constant for all element scatterers both in the primary acquisition and the repeatpass acquisition.The PSF is chosen to be

where Rxand Ryare the resolutions along X-axis and Y-axis,and sinc(α)=sin(πα)/πα.According to the simulation experience,it seems that the PSF is irrelevant to the result.This is reasonable because that,as shown in(11) and(12),the impact of PSF will be eliminated in the calculation of the temporal decorrelation.

Step 3The frst image signal s1can be generated by coherent superposition based on(2).

Step 4According to(17),the variance of random motions can be calculated,and then Gaussian distributed random motions for all element scatterers can be generated.

Step 5Based on(3),the corresponding image signal s2can be generated by repeating Step 2 and Step 3.This time we just need to calculate the bistatic scattering coeffcient f2(x,y,z).Taking the element scatterer with the coordinate(xm,ym,zm)as an example again and assuming that the random motion is(Δxm,Δym,Δzm),the bistatic scattering coeffcient can be given as

Step 6After the generation of s1and s2,the temporal coherence between the two images is then calculated accordingto(4).This process(Step 1–Step 6)is repeated N times to get the ensemble average.

Meanwhile,the theoretical results of the temporal decorrelation could be calculated using the developed mathematical model and the listed parameters.At last,the comparison between the theoretical and the simulated results is presented to verify the validation of the developed bistatic temporal decorrelation model.

4.1Monostatic confguration

As listed in Table 1,we set αT= αR= 90°and βT=βR=45°in the simulation for the monostatic confguration.Here,the motion standard deviation is set to be σi,g=1 cm and Δσi=0:0.1:10 cm,i=Δx,Δy,Δz. Thus this simulation can be seen as a validation of the temporal decorrelation model presented in[14].The simulated and theoretical results can be obtained by applying processing approaches stated before.Fig.3 shows the comparison between the simulated and the theoretical results with different repetition times.

It can be observed from Fig.3 that,the agreement between the simulated and the calculated results becomes better when the repetition time N increases,as expected. The agreement is good enough when N reaches 1 000.The temporal coherence is decreasing with the increasing of the differential motion standard deviation.Furthermore,the temporal coherence value starts from 0.794.In this case, Δσi=0,ξ2=0,thus γTemp=γTV=γTG.This means that the temporal coherence value in this case is resulted from the random motion occurring at the bottom level.

Fig.3 Simulation validation of the developed temporal decorrelation model for the monostatic confguration with different repetition times

4.2Bistatic confguration

In this sub-section,another simulation is performed to verify the developed model for the bistatic confguration.As listed in Table 1,the geometric confguration parameters are set to be αT=50°,αR=165°,βT=60°,βR=45°. Therefore,the simulated bistatic confguration can be seen as a general confguration.The motion standard deviation is set to be σΔx,g=0.3 cm,σΔy,g= 0.5 cm, ΔσΔx=0.3×Δσ,ΔσΔy=0.5×Δσ,ΔσΔz=1×Δσ, Δσ=0:0.1:10 cm,and the repetition time is 1 000. Following the same approach as before,the simulated results can be generated.Then the comparison between the theoretical and the simulated results is illustrated in Fig.4.

Fig.4 Simulation validation for the bistatic confguration

It can be seen from Fig.4 that,for the bistatic confguration,the simulated results are in good agreement with the theoretical expectations.In Fig.4,the temporal coherence decreases when the differential motion standard deviation Δσ increases as well.The temporal coherence value starts from 0.863 due to decorrelation effects of the random motion occurring at the bottom level.

The agreement between the simulated results and the theoretical expectations verifes the validity of the developed temporal decorrelation model.This implies that the developed model offers a reasonable interpretation of the BSAR temporal decorrelation mechanism.Therefore,the developed model can be used to evaluate the temporal decorrelation effect for the BSAR interferometry with different confgurations and system parameters.

5.Sensitivity analysis

It can be seen from the derivation of the developed theoretical model that the BSAR temporal decorrelation is dependent on the system parameters and the BSAR geometry confguration.Therefore,the sensitivity of the derived temporal decorrelation model to these parametershas to be analyzed.

5.1Impact of system parameters

The system parameters include the radar carrier frequency (or wavelength),the extinction coeffcient,the thickness of random volume layers and the ground-to-volume scattering ratio.To analyze the impact of these parameters,both the BSAR geometry confguration and the random motion of element scatterers should be fxed.Without loss of generality,they are set to be the same as the ones used in the simulation,i.e.αT=50°,βT=60°,αR=165°,βT= 45°,σΔx,g=0.3 cm,σΔy,g=0.5 cm,σΔz,g=1 cm and ΔσΔx=0.3×Δσ,ΔσΔy=0.5×Δσ,ΔσΔz=1×Δσ, Δσ=[0,1 cm,2 cm].

The temporal decorrelation effects with different system parameters are plotted in Fig.5.Overall,the higher differential motion standard deviation Δσ,the lower temporal coherence value.Moreover,if Δσ=0,then γTemp= γTV=γTG.As given by(19)and verifed by Fig.5,the temporal coherence value is irrelevant to other three parameters except the radar carrier frequency in this case.

Fig.5(a)shows the impact of the carrier frequency on the temporal decorrelation.In the calculation,the extinction coeffcient is set to be κe=1 dB/m,the thickness of vegetation layers is set to be hv=10 m,and the groundto-volume scattering ratio is set to beμ=0.2.It canbe observed from Fig.5(a)that higher carrier frequency yields lower temporal coherence.This implies that the BSAR interferometry with lower carrier frequency benefts from lower temporal coherence loss.Fig.5(b)indicates the impact of extinction coeffcients on the temporal decorrela-tion.Here,system parameters are set to bef0=1.62GHz, hv=10 m,μ=0.2.In Fig.5(b),we can fnd that higher extinction coeffcients result in higher temporal coherence loss,as expected.Fig.5(c)illustrates the impact of the thickness of random volume layers on the temporal decorrelation.Similarly,the system parameters are fxed to be f0=1.62 GHz,κe=1 dB,μ=0.2.It seems that the temporal coherence is decreasing as the thickness of vegetable layers is increasing.Fig.5(d)shows the impact of the ground-to-volume scattering ratio on the temporal decorrelation.Here the system parameters are fxed to be f0=1.62 GHz,κe=1 dB,hv=10 m.The temporal coherence value increases as the ground-to-volume scattering ratio μ grows.

Fig.5 Impact of system parameters on the temporal decorrelation effect

5.2Impact of BSAR geometry confguration

From(18)to(23),we know that the geometry confguration is another important factor to impact the temporal decorrelation in BSAR confgurations.To study the impact of the geometry confguration,other parameters should be fxed.Here,they are set to be f0=1.62 GHz, κe=1 dB,hv=10 m,μ=0.2.Furthermore,it can be seen from the developed model that the impact of the transmitter’s position(αT,βT)and that of the receiver’s position(αR,βR)are symmetrical.Therefore,it is reasonable to assume that(αR,βR)is fxed while(αT,βT)varies.In the following calculations,the receiver’s position is set to be αR=165°,βR=45°,and the transmitter’s position is chosen as 0°≤ αT＜ 360°,20°≤ βT≤ 80°.Two sets of random motions are applied in the calculation.One of them is set to be σΔx,g=σΔy,g=σΔz,g=1 cm and ΔσΔx=ΔσΔy=ΔσΔz=0.1 cm,while the other oneis set to be σΔx,g=0.3 cm,σΔx,g=0.5 cm,σΔz,g=1 cm and ΔσΔx=0.3 cm,ΔσΔy=0.5 cm,ΔσΔz=1 cm. The calculated results are plotted in Fig.6.

Fig.6 Impact of BSAR geometry confguration on the temporal decorrelation effect

Fig.6 demonstrates that the bistatic acquisition geometry impacts the temporal decorrelation.For the same random motion,temporal decorrelation varies with the bistatic geometry confguration.However,comparing Fig.6(a)and Fig.6(b)suggests that the impact of the geometry confguration on the temporal decorrelation is operated through the structure function and random motions.

6.Conclusions

This paper develops a temporal decorrelation model for the interferometric application of BSAR.Based on a general BSAR geometry confguration,the theoretical model to describe the temporal decorrelation in this case is derived by extending existing monostatic models.The validity of the developed model is tested by Monte Carlo simulations. The good agreement between the theoretical and the simulated results verifes the validity of the derived model. Moreover,the sensitivity analysis of the derived model is presented as well.

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Biographies

Qilei Zhang was born in 1985.He received his B.S.degree in communication engineering and M.S.degree in information and communication engineering from the National University of Defense Technology,in 2007 and 2009,respectively, where he is currently working towards his Ph.D. degree.From January 2012 to July 2013,he was a visiting Ph.D.student with the University of Birmingham,UK.His research interests include bistatic synthetic aperture radar synchronization,imaging and interferometric applications.

E-mail:zhangqilei@nudt.edu.cn

Wenge Chang was born in 1965.He received his B.S.degree and Ph.D.degree from the National University of Defense Technology,in 1987 and 2001, respectively.From December 2007 to June 2008,he was an academic visitor with the University of Birmingham,UK.He is currently a professor with the National University of Defense Technology.His research interests include synthetic aperture radar and array signal processing.

E-mail:changwenge@nudt.edu.cn

10.1109/JSEE.2015.00011

Manuscript received April 28,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61101178;61271441).