LESSONS LEARNED FROM PSIC4: IMPROVING PSI RESULTS FOR A CONSTRAINED TEST SITE

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1 LESSONS LEARNED FROM PSIC4: IMPROVING PSI RESULTS FOR A CONSTRAINED TEST SITE Smi Smiei Esfhny, Freek J vn Leijen, Petr Mrinkovic, Gini Ketelr, nd Rmon F Hnssen Delft Institute of Erth Oservtion nd Spce Systems, Delft University of Technology, Kluyverweg, 2629 HS Delft, The Netherlnds Emil: SSmieiesfhny@studenttudelftnl ABSTRACT In the PSIC4 experiment, eight different processing pproches for time series nlysis of InSAR dt were pplied to dt set contining n unknown deformtion signl Results of the experiment showed different sptil point density for different processing pproches The results of most of the groups tht prticipted in the project suffer from low sptil density of persistent sctterers in deforming res This lck of PS points in the re of interest rises the hypothesis of type-i errors (flsely rejected PS) due to imperfections in the mthemticl model of PS processing We investigte the contriution of three different sources of type-i errors: tmospheric phse screen (APS) nd oritl errors, non-liner deformtion mechnisms, nd zimuthl su-pixel position of the sctterers We investigte these type-i errors nd present significnt improvements in their identifiction y optimiztion of the mthemticl model, leding to n improved density of persistent sctterers Key words: Persistent Sctterer; deformtion; InSAR results At TU Delft, the results of the PSIC4 study hve een used to evlute lgorithm performnce nd ssess which pproches re most importnt to retrieve relile deformtion prmeters t high sptil density This includes n nlysis of type-i errors (flsely rejected PS) due to imperfections in the functionl model (eg, imperfection due to highly non-liner deformtion, su-pixel position of the sctterer), or in the stochstic model (eg imperfection due to tmospheric errors, orit errors, geometricl decorreltion) In this pper, we investigte the contriution of three different sources of type I errors (functionlly nd stochsticlly) We optimized the mthemticl model to cope with these three sources of errors in order to decrese the flse rejection rte of sctterers The improvements re sed on the Guss-Mrkov model (which is used in the integer lest-squre nd ootstrpping technique), which llows for the ddition of extr prmeters in the functionl model nd for the doption of the stochstic model without n increse of the computtionl effort INTRODUCTION In the PSIC4 experiment (Rcoules et l, 2006), eight different processing pproches for time series nlysis of InSAR dt were pplied to dt set contining n unknown deformtion signl The experiment proved to e highly successful, s it clerly indicted tht different pproches resulted in different results, in terms of sptil point density, temporl deformtion ehvior, phse miguity resolution, nd precision nd reliility Results showed trde-off etween one or more of the ove chrcteristics, nd high sensitivity to chosen threshold vlues As PSIC4 ws lind experiment, where no -priori informtion on the loction, mgnitude, temporl ehvior, nd other chrcteristics of the deformtion ws ville, it rises the question to wht extent -priori informtion cn nd should e used in PSI processing Informtion, or ssumptions, either on temporl or sptil ehvior pper to e very importnt in the processing 2 MATHEMATICAL FRAMEWORK The sic concept of our PS processing pproch is sed on the mthemticl model which cn e formulted s: E{y} = A + B () D{y} = Q y, where E{} is the expecttion opertor, D{} the dispersion, y the vector of oservtions, A nd B re design mtrices for the integer nd rel vlued prmeter vectors nd respectively, nd Q y is the covrince mtrix This mthemticl model comprises of the functionl nd the stochstic model The functionl model descries the reltion etween the oservtions nd the unknowns, wheres the stochstic model represents the stochstic properties of the oservtions The functionl model for single mster stck, where the mster is indicted y zero nd N slve cquisitions re

2 ville, hs the form ψ 0 2π ψ 0N S E{ 2π H } = D DP 4π λ β 0 α (t 0 ) α P (t 0 ) 4π λ β 0N α (t 0N ) α P (t 0N ) + 0 0N S H D D P covrince mtrix The covrince mtrix of the pseudooservtions Q y contins vrinces which ound the solution spce of the unknowns Using lest-squre estimtion with the functionl model, Eq (2), nd the stochstic model, Eq (3), ech phse doule difference etween two PS cndidtes is unwrpped in time The testing criterion for ccepting sctterer s PS is the -posteriori vrince fctor ˆσ 2 = et Q ψ e, (4) r where e is the vector of residuls etween the unwrpped phse nd the deformtion model nd r is the redundncy in the functionl model A PS cndidte is ccepted s, (2) PS when ˆσ 2 is smller thn 0 3 SOURCES OF TYPE I ERRORS where ψ re the phse oservtions, S is the tmospheric dely of the mster cquisition, H the height, D p re deformtion prmeters with p = P, λ is the rdr wvelength, β is the height-to-phse conversion fctor, α p descries deformtion model s function of temporl seline t nd () denotes pseudo-oservle needed to solve for the rnk deficiency of the system (Kmpes nd Hnssen, 2004) The rnk deficiency is cused y the fct tht for ech oserved phse n miguity needs to e estimted, together with the prmeters of interest As result, the numer of unknowns exceeds the numer of oservtions With the introduction of pseudo-oservles, the mthemticl model is regulrized Without -priori knowledge of the topogrphy nd deformtion in the re, the pseudo-oservtions re set to zero To reduce the influence of tmospheric delys nd orit errors on the estimtion process, differentil phses etween two neighoring PS cndidtes re used s oservtions After unwrpping these rcs in time, sptil testing nd unwrpping lgorithm is pplied to otin estimtes with respect to single reference Additionl prmeters cn esily e dded to the functionl model, eg su-pixel position in zimuth direction to ccount for Doppler vritions in the cquisitions (Kmpes, 2005) The second prt of the mthemticl model is the stochstic model, represented y the covrince mtrix [ ] [ ] ψ Qψ 0 D{ y } =, (3) 0 Q y where y represents the vector of pseudo-oservtions The covrince mtrix of the phse oservtions Q ψ is otined y vrince component estimtion (VCE) (Teunissen nd Amiri-Simkooei, 2006; Kmpes, 2005) The VCE technique estimtes the covrince mtrix directly from the dt Hence, the covrince mtrix used in the estimtion process is not dependent on -priori ssumptions on the qulity of the dt After estimtion nd sutrction of certin signls, eg the tmospheric dely, the VCE lgorithm is pplied gin to updte the Any imperfection in the functionl model (eg, imperfection due to non-liner deformtion, su-pixel position of the sctterer) results in lrger devitions etween oservtions nd model As consequence, these lrge residues will result in lrger ˆσ 2 nd type-i errors, so the sctterer will e discrded for further nlysis In order to optimize the mthemticl model to hndle these kinds of deficiencies, n extended model with dditionl prmeters should e used Using extended functionl models, the numer of detected PS cn e enlrged, leit t the expense of decresed rnk of the estimtion prolem This decresed rnk increses the chnce on undetected unwrpping errors, denoted s type-ii errors Therefore, extended deformtion models should only e pplied when necessry On the other hnd, imperfections in the stochstic model (Q ψ ) cn lso cuse type I errors Using too optimistic stochstic model results in lrger ˆσ 2, leding to the flse rejection of PS For exmple, imperfections in the stochstic model cn e due to tmospheric errors, orit errors, geometricl decorreltion or su-pixel position of the sctterer In generl, there re two pproches to cope with these kinds of imperfections: ) dpt the covrince mtrix (dding dditionl vlues in entries of Q ψ ), or 2) remove the corresponding errors from the oservtions (filtering) In this study we will investigte three sources of type-i errors: ) APS nd orit errors 2) Non-liner deformtion model nd 3) Azimuthl su-pixel position of the sctterer We optimize the functionl nd the stochstic models in order to cope with these three sources of flse rejections 4 APS INTERPOLATION AND ORBIT ERRORS In the stndrd PS-InSAR methodology, the residul signl of ll differentil rcs in the initil network is filtered

3 Figure ) PSI results from stndrd PS processing using only kriging interpoltion for the mining re in Grdnne, Frnce (red dshed rectngle) ) PSI result otined y pplying dpted APS estimtion with trend nd strtifiction Liner deformtion rtes estimted from the unwrpped time series re shown to extrct the tmospheric signl per cquisition, see eg Ferretti et l (200) Bsed on the correltion of the tmosphere in spce nd its decorreltion in time, comined smoothing nd interpoltion opertion using kriging is used to derive the APS (ie, the tmosphere signl for the whole scene) After sutrction of the APS from the originl differentil phse, the estimtion procedure cn e repeted for ll pixels This pproch is optiml for interpoltion of the turulence signl of the tmosphere However filtered residuls not only contin the turulence signl ut lso orit errors nd verticl strtifiction signl of the tmosphere, see Hnssen (200) Although differentil phses etween two neighoring PS cndidtes re used s oservtions in the mthemticl model to reduce orit error nd strtifiction, these effects cn e source of error in the finl estimtion nd cuse PS to e flsely rejected especilly in res with low signl to noise rtio (eg non-urn res with high non-liner deformtion mechnism) Bsed on this hypothesis we optimized our APS interpoltion pproch Insted of only using kriging interpoltion on filtered residuls, we used three steps For ech cquisition, detrend the dt y fitting first order plne to the unwrpped phse nd sutrct the estimted trend from the originl phse 2 Estimte verticl tmospheric strtifiction sed on the liner correltion etween topogrphy (DEM) nd strtifiction signl nd sutrct it from the detrended phse 3 Interpolte turulence using kriging on residuls (s in stndrd pproch) Consequently, the finl estimted APS is summtion of trend signl, strtifiction, nd turulence Fig shows the detected PS nd their displcement rtes otined from stndrd processing using only kriging interpoltion Fig visulizes the results fter pplying detrending nd estimtion of strtifiction The numer of detected PS in deforming re (red dsh rectngle) incresed in this cse from 749 pplying the stndrd pproch to 889 (8% improvement) using detrending nd to 9 (9% improvement) using detrending nd strtifiction estimtion Also some PS re detected close to the center of susidence region (white circle in Fig ) 5 NON-LINEAR DEFORMATION MODEL Becuse the -posteriori vrince fctor ˆσ 2 is dependent on the devition etween the oservtions nd the ssumed deformtion model (usully liner model), lrge residues due to higher order deformtion mechnism cn result in type I errors In other words, persistent sctterers with more complex displcement histories will not e detected vn Leijen nd Hnssen (2007,) presented two strtegies to increse the numer of detected persistent sctterers using dptive deformtion models The first strtegy is sed on lterntive hypothesis testing during the PSI processing A sequentil scheme is used to pply nd test extended deformtion models per phse doule difference, intending to find model tht sufficiently fits to the dt to void type-i errors The dption of the deformtion model is sed on hypothesis testing The lgorithm is initilized with the selection of set of these models Then, ech phse doule difference etween two PS cndidtes is unwrpped in time pplying the sequentil scheme of lterntive hypothesis testing until deformtion model fits to the dt well enough A liner model is good null hypothesis ecuse of the mximum redundncy in the estimtion process Apply-

4 c d Figure 2 ) PSI result using liner deformtion model for the mining re in Grdnne, Frnce (red dsh rectngle) ) Results otined y pplying the sequentil hypothesis testing scheme using liner nd rekpoint model The rekpoint ws set -priori t 9 My 995 c) Result otined y pplying the sequentil hypothesis testing scheme using liner nd qudrtic model d) Result otined using integrtion of the sequentil hypothesis testing scheme nd itertive deformtion modeling using kriging Liner deformtion rtes estimted from the unwrpped time series re shown Figure 3 ) PSI result from stndrd PS processing using dpted functionl model with zimuthl su-pixel position for mining re in Grdnne, Frnce (red dsh rectngle) ) PSI result otined y pplying dopted stochstic model for zimuthl su-pixel position Liner deformtion rtes estimted from the unwrpped time series re shown

5 [] [%] All(%48) Higher order Deformtion(%25) Azimuthl Supixel Position(%2) Adopted APS (%7) Figure 4 ) PSI result from dpted mthemticl model to optimized APS interpoltion, non-liner deformtion, nd zimuthl su-pixel position for mining re in Grdnne, Frnce (red dsh rectngle) ) Contriution of different dpttions in the PS processing Note tht the sum of the three contriution (44%) is not equl to the totl improvement (48%) due to the synergy effect ing the sequentil scheme of hypothesis testing, certin deformtion model is ccepted when ˆσ 2 is smller thn 0 Otherwise, the next model is tested until the complete set of models is evluted The second method is sed on n itertive scheme of deformtion modeling After stndrd PSI processing under the null hypothesis, tht is, pplying liner model, deformtion model is estimted sed on interpoltion (eg, kriging) from the PS results The modeled deformtion is then sutrcted from the originl interferometric phse nd the PSI processing is repeted (gin using liner model) Becuse the deformtion models re estimted per epoch using the displcement time series, possile non-liner deformtion is modeled s well As result, points which were previously rejected s PS due to too lrge devitions from the model my now e ccepted Herey the density of PS improves Oviously, this procedure cn e repeted itertively Note tht similr procedure is often followed in stndrd PSI processing to remove tmospheric delys nd, sed on uxiliry dt, heights The results of stndrd processing using liner deformtion model re shown in Fig 2 Fig 2 shows the result fter pplying the sequentil testing scheme using liner nd rekpoint model The rekpoint model is chrcterized y two susequent liner models seprted y n event or rekpoint In this study, the rekpoint is defined -priori on 9 My 995, sed on the strt of the mining ctivities With this model PS re detected in the center of the susidence region To enle visuliztion, the figure shows liner deformtion rtes estimted through the unwrpped time series, even when the rekpoint model ws used for the unwrpping The numer of detected PS in the deforming re (red dshed rectngle) incresed in this cse from 749 using the liner model to 2069 (8% improvement) Fig 2c shows the result of the sequentil testing scheme using liner nd qudrtic deformtion model In this cse, the numer of detected PS in the re of interest incresed 749 to 26 (20% improvement) Applying itertive deformtion modeling using kriging interpoltion in comintion with sequentil hypothesis testing (with liner, rekpoint, nd qudrtic deformtion model) gve the results which is presented in Fig 2d The numer of detected PS incresed from 749 to 255 (23% improvement), confirming the expected increse in PS density Importntly, this dptive pproch enled the detection of PS in the center of susidence region 6 AZIMUTHAL SUB-PIXEL POSITION The su-pixel position of PS point in zimuth direction induces n dditionl phse component in the PSI oservtion, especilly when there is lrge difference etween the Doppler centroid frequency of the mster nd slve imge This term is only importnt for point sctterers, since for distriuted sctterers the effective phse center is pproximtely t center for ll pixels The zimuthl su-pixel position cn e estimted s n dditionl prmeter in the mthemticl model sed on the liner reltionship with the Doppler difference etween the mster nd slve imge (Kmpes, 2005; Mrinkovic et l, 2008) ψ 0n = 4π 0n fdc ξ x, (5) v where v is the stellite velocity, fdc 0n is the Doppler centeroid difference etween mster nd nth cquisition, nd ξ x is the su-pixel position of the PS in zimuth detection However, dding this dditionl prmeter to the functionl model leds to decresed rnk of the estimtion

6 prolem Moreover, the zimuth su-pixel position cnnot e estimted with high precision in the cse of smll vrition of the Doppler frequencies This suggests treting this prmeter stochsticlly rther thn functionlly Tht is, insted of estimting this prmeter s new unknown, introduce it in the stochstic model Tht is, dd extr vlues to the digonl elements of the Q ψ This extr vlue is scled reltive to the Doppler difference of ech interferogrm So oservtions with higher Doppler difference hve higher vrince nd so lower weight in the estimtion, leding to smller ˆσ 2 nd so cceptnce of the PS However, oth of these dptions (ie dopt functionl model or stochstic model) should e pplied only when it is necessry Estimtion of the zimuthl su-pixel position s n extr prmeter for ll pixels cuses tht pixels with smll zimuthl su-pixel position re unwrpped with n unnecessry complex model, incresing the chnce of unwrpping errors (type-ii errors) On the other hnd, dding this term to the stochstic model for ll pixels results in underestimtion of ˆσ 2, leding to the flse detection of PS (gin type-ii errors) So oth dptions should only e used for pixels with lrge zimuthl su-pixel position In order to detect these pixels, we nlyzed the residuls (devition etween the oservtions nd the model) fter temporl unwrpping Looking t the correltion etween residuls nd Doppler differences, we cn detect the pixels which show high correltion with Doppler differences Fig 5 shows the histogrm of this correltion in smll re in the Grdnne re It is cler tht these pixels cn e clssified in two groups: ) norml distriution prt which contins pixels with smll zimuthl su-pixel position nd lso distriuted sctterers with zero men correltion with Doppler differences (red norml ell shpe), nd 2) pixels outside the norml ell shpe which show correltion with Doppler differences (red circle) The second group contins pixels which we detected for dption of the mthemticl model for zimuthl su-pixel position 83% of these detected points re rejected in the stndrd PS pproch without tking the zimuthl su-pixel position into ccount For these rejected PS, we dpt the mthemticl model (oth functionl nd stochstic prt) nd iterted the PS estimtion Fig 3 shows the result fter itertion of the PS processing using the dopted functionl model with zimuthl su-pixel position The density of PS did in this cse not improve significntly (numer of PS incresed from 749 to 785 (2% improvement)) However, dding the su-pixel position term in the stochstic model results in incresed numer of PS from 749 to 972 in deforming res (Fig 3), tht is 2% improvement in density of PS in the re of interest These results show tht it is etter to del with zimuthl supixel position stochsticlly rther thn functionlly in this dtset Figure 5 Histogrm of correltion etween residuls nd Doppler differences 7 FINAL RESULTS After evluting the three mentioned sources of type-i errors in the Grdnne re, we comined ll proposed pproches together Tht is, we pplied the following dption Adpted APS interpoltion with trend nd strtifiction effect 2 Sequentil hypothesis testing scheme using liner, rekpoint, nd qudertic models 3 Itertive deformtion modeling using kriging 4 Adpted stochstic model for pixels with zimuthl su-pixel position Fig 4 shows the results The density of PS in the deforming re incresed significntly from 749 to 258, tht is 48% improvement in PS detection cpility Fig 4 presents the contriution of the different dptions in PS density in the re of interest Note tht the sum of the three contriution (44%) is not equl to the totl improvement (48%) due to the synergy effect mong these optimiztions Tht is, integrtion of ll these dptions cting together improves the results more thn pplying the indivisul dptions Improving the PS density in the deforming re results in etter estimtion of the finl velocity mp of the re Figs 6 nd show interpolted velocity mps using kriging interpoltion for the stndrd PS pproch nd the dopted pproch respectively These results show tht the dpted lgorithm cn etter ctch the deformtion signl in this mining re

7 Figure 6 ) nd ) Interpolted velocity mps using stndrd PS pproch nd dopted pproch respectively 8 CONCLUSIONS The results of the PSIC4 study hve een used to evlute lgorithm performnce nd ssess which pproches re most importnt to retrieve relile deformtion prmeters t high sptil density This includes n nlysis of type-i errors (flsely rejected PS) due to imperfections in the functionl model (eg, imperfection due to non-liner deformtion, su-pixel position of the sctterer), or in the stochstic model (eg imperfection due to tmospheric errors, orit errors, geometricl decorreltion) We investigte the contriution of three different sources of type I errors: tmosphere phse screen (APS) nd oritl error, non-liner deformtion mechnism, nd zimuthl su-pixel position of the sctterers nd optimized the mthemticl model to del with these errors The results show significnt improvements in the identifiction of these type I errors, leding to improved PS density in the deforming re Appliction of the proposed optimiztions shows 48% increse in the numer of detected PS in the deforming re This improvement cn significntly improve the finl velocity mp which is one of the min products of the PSI technique from n ppliction point of view REFERENCES Ferretti, A, C Prti, nd F Rocc (200, Jnury) Permnent sctterers in SAR interferometry IEEE Trnsctions on Geoscience nd Remote Sensing 39(), 8 20 Hnssen, R F (200) Rdr Interferometry: Dt Interprettion nd Error Anlysis Dordrecht: Kluwer Acdemic Pulishers Kmpes, B M (2005, Septemer) Displcement Prmeter Estimtion using Permnent Sctterer Interferometry Ph D thesis, Delft University of Technology, Delft, the Netherlnds Kmpes, B M nd R F Hnssen (2004, Novemer) Amiguity resolution for permnent sctterer interferometry IEEE Trnsctions on Geoscience nd Remote Sensing 42(), Mrinkovic, P, G Ketelr, F vn Leijen, nd R Hnssen (2008) InSAR qulity control: Anlysis of five yers of corner reflector time series In Fifth Interntionl Workshop on ERS/Envist SAR Interferometry, FRINGE07, Frscti, Itly, 26 Nov-30 Nov 2007, pp 8 pp Rcoules, D, B Bourgine, M de Michele, G L Coznnet, C Luc, C Bremmer, H Veldkmp, D Trgheim, L Bteson, M Crosetto, M Agudo, nd M Engdhl (2006, June) Psic4: Persistent sctterer interferometry independent vlidtion nd intercomprison of results Technicl report, BRGM BRGM/RP FR Teunissen, P J G nd A R Amiri-Simkooei (2006) Lest-squres vrince component estimtion Journl of Geodesy xx, sumitted vn Leijen, F J nd R F Hnssen (2007) Persistent sctterer density improvement using dptive deformtion models In Interntionl Geoscience nd Remote Sensing Symposium, Brcelon, Spin, July 2007, pp 4 pp vn Leijen, F J nd R F Hnssen (2007) Persistent sctterer interferometry using dptive deformtion models In ESA ENVISAT Symposium, Montreux, Switzerlnd, April 2007, pp pp