Toward Efficient InSAR Time Series Analysis

16 Pages·2017·25.06 MB·English

IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017 5637
Sequential Estimator: Toward Efficient
InSAR Time Series Analysis
Homa Ansari, Student Member, IEEE, Francesco De Zan, and Richard Bamler, Fellow, IEEE
Abstract—Wide-swath synthetic aperture radar (SAR) NOTATION
missions with short revisit times, such as Sentinel-1 and the n ∈N Number of the SLCs in the SAR data stack.
planned NISAR and Tandem-L, provide an unprecedented
l ∈N Number of samples in a statistically
wealth of interferometric SAR (InSAR) time series. However,
the processing of the emerging Big Data is challenging for homogeneousspatial neighborhood.
state-of-the-art InSAR analysis techniques. This contribution s ∈N Number of the SLCs included in the
introduces a novel approach, named Sequential Estimator, for mini-stack.
efficientestimationoftheinterferometricphasefromlongInSAR m ∈N Dimension of the low-rank signal subspace.
timeseries.Thealgorithmusesrecursiveestimationandanalysis (cid:2)∈C1×l Group of l statistically similar pixels
of the data covariance matrix via division of the data into small
batches, followed by the compression of the data batches. From surrounding a single pixel of interest, the
eachcompresseddatabatchartificialinterferogramsareformed, interferometric signal is assumed to be
resulting in a strong data reduction. Such interferograms are stationary within this ensemble.
used to link the “older” data batches with the most recent Z∈Cn×l Matrix of acquired SLCs, the rows correspond
acquisitions and thus to reconstruct the phase time series. This
to the images in the stack, the columns
schemeavoids thenecessityof reprocessingtheentiredatastack
atthefaceofeachnewacquisition.Theproposedestimatorintro- to the ensemble of pixels in (cid:2).
ducesnegligibledegradationcomparedtotheCramér–Raolower C∈Cn×n Complex coherence matrix of the stack;
bound under realistic coherence scenarios. The estimator may (cid:3) ∈Rn×n Coherence matrix of the interferograms defined
thereforebeadaptedforhigh-precisionnear-real-timeprocessing as (cid:3) =|C |.
ooffflIinnSeAtRo aanmdoancictoorminmgodgaeotedetthice ctooonlv.erTshioenpoefrfIonrSmAaRncefroomf tahne I(cid:2)∈Cn×n Comipjlex maijtrix of the interferograms.
Sequential Estimator is compared to state-of-the-art techniques φ ∈Cn×1 Sought phase series in phase-linking, estimated
via simulations and application to Sentinel-1 data. phase of each SLC inclusive of the systematic/
deterministic phase components.
Index Terms—Big Data, coherence estimation error, data
compression, differential interferometric synthetic aperture A Capital letter indicating a generic matrix.
radar(DInSAR),distributedscatterers,efficiency,erroranalysis, A Subscripted capital letter indicating the ith row
i
low-rankapproximation,maximum-likelihoodestimation(MLE). of the generic matrix A.
b Small bold letter indicating a generic vector.
ACRONYMS
(cid:2). Hat accentuation distinguishing an estimated
CCG Complex circular Gaussian.
CRLB Cramér–Rao lower bound. from an observed variable.
DS Distributed scatterer.
EVD Eigenvalue decomposition.
I. INTRODUCTION
MLE Maximum-likelihood estimation.
NRT Near-real-time. TIME-SERIES analysis of interferometric synthetic aper-
PCA Principal component analysis. ture radar (InSAR) has proved to be a high-precision
PS Persistent scatterer. geodetic approach for monitoring the crustal deformation of
RMS Root mean square. the earth. The retrieval of the geophysicalsignal from InSAR
SBAS Small-baseline subset approach. data stacks is limited by the atmospheric perturbation as
StBAS Short temporal-baseline subset. well as the temporal decorrelation of the SAR signal. In the
SLC Single-look complex. pursuit of overcoming these limitations, the exploitation of
InSAR stacks was initially limited to the Persistent Scatter-
Manuscriptreceived December13,2016;revisedApril12,2017;accepted
May 15, 2017. Date of publication September 1, 2017; date of current ers (PSs) [1], [2]. The PSs are phase-stable scatterers and
version September 25, 2017. This work was supported by the ‘Helmholtz donotundergoseveretemporaldecorrelation.ExploitingPSs,
AllianceRemoteSensingandEarthSystemDynamics´fundedbytheInitiative
the separation of the geophysicalsignal from the atmospheric
andNetworkingFundoftheHelmholtzAssociation.(Correspondingauthor:
HomaAnsari.) perturbations follows in a separate step [1], [2]. Although
H. Ansari and F. De Zan are with the German Aerospace Center, Oberp- precise in signal retrieval, the scarcity of PSs in nonurban
faffenhofen,Germany(e-mail:[email protected];[email protected]).
areashas led the InSARcommunitytoward relaxingthe strict
R.Bamler is withthe German Aerospace Center, Germany, andalso with
the Chair of Remote Sensing Technology, Technical University of Munich, limitonthephasestabilityandincludingtheareasaffectedby
80333Munich, Germany(e-mail: [email protected]). decorrelation, referred to as the distributed scatterers (DSs).
Color versions of one or more of the figures in this paper are available
The pioneering DS approach in minimizing the effect of
onlineathttp://ieeexplore.ieee.org.
Digital ObjectIdentifier 10.1109/TGRS.2017.2711037 decorrelation was to limit the analysis to subsets of moderate
0196-2892©2017IEEE.Translations andcontent miningarepermitted foracademic researchonly. Personaluseisalsopermitted,
butrepublication/redistribution requires IEEEpermission. Seehttp://www.ieee.org/publications_standards/publications/rights/index.html formoreinformation.
5638 IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017
to high-coherent interferograms, referred to as small-baseline having a performance deviant from the CRLB [9], [10] in
subsets (SBASs) [3], therefore excluding the decorrelated generic cases, the EVD is a suboptimum phase estimator.
interferometric pairs. Setting an a priori threshold on the Despite their various implementation details, all DS tech-
geometric and temporal baseline, or equivalently on the niques rely on the analysis of the complex coherence matrix
coherence, the data stack is divided into coherent subsets of of the data stack. Estimated from the time series, the com-
interferograms, each subset is then spatially unwrapped. The plex coherence is a Hermitian matrix containing all possible
unwrappedphasesofthedifferentsubsetsaretemporallyinte- interferograms as well as their respective coherence. It is
grated to estimate a phase series inclusive of the geophysical comprised of the total number of n(n−1)/2 unique entities,
signal of interest. withn beingthenumberofsingle-lookcomplex(SLC)images
Improving one step further, the imposed strict coherence in the data stack. The formation and exploitation of the full
limit for the subset selection was relaxed, allowing all inter- coherence matrix can, however, be computationally demand-
ferograms to be properly included in the retrieval of the ing, especially in largedata stacks. The computationalburden
geophysicalsignal [4], [5]. The effect of decorrelation in this increases at least cubically with the dimension of the data at
case is minimized by exploitation of the statistical properties hand.
of the interferometric stack in a maximum-likelihood esti- The current and future spaceborne SAR missions with
mation (MLE) scheme. Approximating the mean scattering systematicearthmonitoringobjectivetendtofollowthewide-
of DS by a PS-resembling mechanism, the MLE estimates swath design with revisit cycle as low as possible to allow
a wrapped phase series which pertains to the superposed for high temporal resolution monitoring of the earth. With
systematic phase components. Each element of the series missions such as ESA’s Sentinel-1 or, in the near future,
corresponds to an image in the stack and is relative NASA’s NISAR, the short revisit of 6–12 days gives birth
to an arbitrarily fixed master image. Borrowed from to unprecedented SAR data volumes. The interferometric
Guarnieri and Tebaldini [4], the retrieval of the wrapped processing of such emergingBig Data stacks with the current
phase is hereafter referred to as phase-linking. The estimated available DS algorithms would be infeasible, especially if
phase series bears the sought geophysical signature, although systematic high-precisionnear-real-time (NRT) monitoring of
superposed by an atmospheric-inducedsignal. The separation even small earth surface motion is the objective. The NRT
of the atmospheric contribution from the geophysical signal processing will open a new chapter in InSAR applications,
may be followed in a second estimation step as in the case converting the method form an offline analysis mode to a
of PSs [1], [2], [4]. high-precisionnearlyonlinemonitoringtoolwithapplications,
The MLE designs a temporal filter adaptive to the coher- e.g., in early warning systems.
ence, as the second-order statistic of the data. It provides TheInSARcommunityhasrecentlycommencedaddressing
an asymptotically optimum estimator with variance close to this new demand [14]–[17]. However, in the recent attempts,
the Cramér–Rao lower bound (CRLB) [4], [6]. If unknown, thefocusismostlyontheautomationoftheSARdataacquisi-
the coherence is substituted by its estimation. The MLE tionandoptimizationoftheprocessingpriortothetime series
is therefore bound to the performance of the coherence analysis. As of present, the main attempt has been on the
estimation. The well-known suboptimality of the coherence reductionoftheproductlatencythroughparallelizedcomputa-
estimation [7], [8] is the major limitation for MLE to reach tion and improvementof the computationalcapacity, e.g., via
its asymptotic behavior. In the case of poor performance of cloud computing. Being an inherently nonparallel problem,
the coherence estimator, the MLE may be shown to deviate the core time series analysis techniques have been left intact.
largely from the CRLB; hence, its optimality is compromised As briefly reported, Heu et al. [14] and Gonzalez et al. [15]
(see Section VI-C, [9], [10]). exploit SBAS-like techniques in retrieval of the geophysical
Building on the MLE, the estimation of the geophysical signalwhile Adam et al. [16] is tailored to the PSs. Although
signal has been further investigated by substitution of the sufficient for retrieval of large-scale deformation, the SBAS
ML optimization by robust M-estimators [11] or integer least approach fails in efficient processing of large data stacks.
squares [10]. The former aims at increasing the accuracy Indefiningtheefficiency,twodistinctcriteriaaredistinguished
of phase-linking by addressing special cases where the data here,namely,thealgorithmicefficiency,i.e.,theprocessingrun
characteristics deviate from the assumed stationary Complex time; and the estimation efficiency, i.e., the optimality of the
CircularGaussian(CCG)statisticsinMLE;thelatterprimarily estimatorwithrespecttotheCRLB.Thealgorithmicefficiency
focuses on the addition of an a posteriori quality indicator of ofSBASmaybeimprovedbyrestrictingthetemporalbaseline
the estimated phases. and exploiting a limited subset of interferograms. However,
A second category of phase-linking based on the exploita- such restrictions compromise the estimation efficiency and
tion of all interferometric pairs was first proposed in [12] consequentlydegradethe sensitivity to small surface motions.
and later in [13]. It primarily focuses on the extraction of Hereafter, efficiency implies both the algorithmic and estima-
the scattering mechanisms of the DS through eigenvalue tion optimality criteria.
decomposition (EVD) of the data covariance matrix. The idea of an efficient stacking technique was raised for
Compared to the MLE, it relaxes the single-dyadic the first time by De Zan and López-Dekker [18]. Propos-
(i.e., PS-resembling) approximation of the DS-scattering ing a technique tailored to a special coherence scenario,
mechanism and allows the retrieval of the decomposed phase termed long-term coherence, the authors succeeded in retain-
signature with multiple dyads of different power. However, ing performance close to the CRLB while avoiding the full
ANSARIetal.: SEQUENTIALESTIMATOR:TOWARDEFFICIENTINSARTIMESERIESANALYSIS 5639
exploitation of the entire data stack. The scheme, however, using an ensemble of l pixels in statistically homogeneous
fails in the absence of the long-term coherence. neighborhoods. The statistical homogeneity of the chosen
Inspired by De Zan and López-Dekker [18], this con- ensemble is assured via statistical similarity tests between the
tribution proposes a novel generic efficient stacking tech- centeringand neighboringpixels [5], [20].Sorting the chosen
nique,namedtheSequentialEstimator.Theproposedapproach ensemble in a spatiotemporal data matrix Z, the MLE of
allows efficient phase-linking in sequences by using isolated complex coherence matrix reads as
dtoataaccbeastcshtehseoefnttihree ttiimmee sseerriieess.aIttethacuhs apvroocidessstihnge nseeqcueessnictye Cˆ = (cid:3) ZZH (1)
T
and refrains from reprocessing the entire data archive at [[Z]]2([[Z]]2)
the face of each new SAR acquisition. Compressing the where [[Z]] gives the column-wise L2 norm of the matrix Z,
isolated data batches and generating artificial interferograms the power-2 and deviation operations are element wise. Cˆ is
with the compressed data, the Sequential Estimator retains a an n×n matrix thatencapsulatesthe availableinterferometric
performance close to the CRLB. The method competes with information with its argument containing the interferometric
expensivefull-stackphase-linkingtechniquesandevenoutper- phases and its modulus(cid:3)ˆ providingan estimate of the coher-
forms them in case the exploited estimation of coherence is ence of the correspondinginterferograms. Under the assump-
suboptimum. tionoftheCCGstatistics,theMLEofphaseisretrievedvia[4]
(cid:2) (cid:3)
theTofirtshtepbreosptoosaflofuorrkanogwenleedrigce,etfhfieciSenetquIennStAiaRl Essttaicmkaintogriins φˆML =argminφ ζˆMHL((cid:3)ˆ−1°Cˆ)ζˆML (2)
the realm of DS. The algorithm provides a recursive solution here ° is the Hadamard operation and ζˆ is a vector con-
ML
to the temporally nonparallelizable problem of phase-linking. taining the sought phase of each SLC with its elements set to
The proposedestimator may be combined with the optimized exp(jφ ). With this assumed model, the scattering behavior
i
wideareaprocessingtechniquesin[16]and[19],andtherefore of the DS neighborhoodis approximated by a PS-resembling
contribute to the state-of-the-art automatic InSAR process- mechanism. The MLE may be interpreted as a temporal filter
ing in the pursuit of high-precision NRT earth deformation that compresses the information of n(n−1)/2 interferograms
monitoring. to a phase series of size n.
In the following, a short introduction of the possible
optimized stacking schemes is provided in Section II. The B. Alternative Agile Phase-Linking Schemes: Partial
descriptionoftheSequentialEstimatorstartsfromSectionIII, Accessibility to the Stack
where the compression technique used for the SAR data The first requirement for an efficient stacking scheme is to
stack is elaborated; further algorithmic steps are described avoid reprocessing and rereading the entire data history when
in Section IV. Section V highlights the computational gain encountering a new acquisition. To fulfill this requirement,
of the scheme. Finally, the performance of the estimator is the intuitive solution is to follow the footsteps of SBAS with
assessedthroughsimulationsandapplicationtoSentinel-1data a slight modification: merely allowing the short temporal-
in Sections VI and VII, respectively. baseline interferogram in the phase retrieval. This approach
is hereafter referred to as the short temporal-BAseline sub-
II. CONVENTIONAL VERSUS AGILEPHASE-LINKING set (StBAS). Inspired by Rocca [21] and assuming an expo-
nential decrease of coherence in SAR stack, StBAS exploits
This section starts from laying down the formulation of
successive SLCs with temporal separation up to an a priori
the phase-linking which is used throughout this paper. The
time lag. The StBAS interferograms are combined via MLE
MLE phase-linking as well as possible agile, i.e., computa-
phase-linking. Compared to the conventional MLE, here a
tionallylight,stackingtechniquesareintroduced.Asitwillbe
banded matrix replaces the full coherence matrix. The band-
shownthroughsimulations(see Section VI-C),the introduced
widthofthematrixisfixedbytheaprioritimelag.Although
techniques are inefficient, in the sense that their optimality is
straightforward, there are three fundamental problems to this
compromisedbytheintroducedlimitondataaccessibility.The
approach as follows.
reviewofthesealternativeschemesis,however,worthwhileto
1) At each processing step, up to a certain lag of the
highlight the encountered challenges for an efficient stacking
previouslyacquireddatashallberereadandreprocessed,
method.
i.e.,theproposedstackingmethodstillundergoesredun-
Notethatthespatialanalysisisalwaysconsideredpixelwise
dant computations.
throughout this paper, although the statistics of each single
2) The choice of the a priori temporal baseline imposes a
pixel is inferred from an ensemble of pixels belonging to a
trade-offbetweentheestimationefficiencyandthecom-
statistically homogeneous region, hereafter referred to as the
(cid:2) neighborhood. putationalburden,i.e.,toavertexclusionofthecoherent
interferograms and the consequent performance degra-
dation of the estimator, a longer time lag is required.
A. Conventional MLE Phase-Linking: Full Accessibility
3) StBAS assumes that the interferograms with temporal
to the Stack
baselines of larger than the time lag bear no coherent
TheMLEschemeexploitsthecomplexcoherencematrixto signal. This assumption loses validity in explaining
estimate the phase series inclusive of the geophysical signal. seasonal decorrelation or long-term coherence observed
The complex coherence matrix of the data stack is estimated for C- and L-bands SAR [23], [24].
5640 IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017
To address some of the aforementionedproblems, an alter- basis T reads as
native approach may be taken that emphasizes the long-term, T : Zn×l → Z˜m×l (4)
andpossiblyweak,coherentsignalsratherthantheshort-term,
high-coherenceinterferometricpairs.Thishasbeenintroduced wheren isthedimensionofdataandm isthereduceddimen-
in [18], where the authors suggest using two subsets of the sion of its compressed version. Note that the compression is
SLCsandretrievingthelong-termcoherentsignalfromfiltered temporal and l as the size of the spatial neighborhood (cid:2) is
versionofthetwosubsets.Allowingeachsubsetas Zs ∈Cs×l intact. The transformation T may be defined by any arbitrary
with s (cid:3) n/2, its filtering follows from exploiting the set of orthonormal vectors, e.g., Fourier and wavelet bases.
MLE-estimated phase series φˆ via The efficiency of the compression is driven by the choice of
s
the basis. In the most efficient case, the basis is chosen to
(cid:4)s
vˆ (q)= 1 Z (p,q)exp(−jφˆ (p)). (3) capture the maximum variation of the data space, i.e., by a
s S s s subset of the most powerful eigenvectors, such that
p=1
Here, q is the spatial index from 1 to l, indicating that the T ={v1;...;vm} (5)
proposed filter is temporal and not spatial. Termed virtual
with the basis vectors derived from the EVD
image v is a filtered SLC representing the subset. Exploiting
s (cid:4)n
two virtual images at the beginning and end of the stack, Cˆn = λ v vH. (6)
p p p
a coherent interferogram may be formed which bears the
p=1
soughtlong-termcoherentsignalandprovidesanestimationof
the phase between the first and last SLC. The performanceof Here, Cˆn is the complex coherence matrix estimated by sub-
this approach is, however, compromised if no coherent signal stitution of Zn×l in (1), λi are the eigenvalues in descending
is presentamongthe chosen subsets. Thismethodis hereafter order and v are the corresponding eigenvectors. This setup
i
referred to as virtual image estimator. is the well-known principal component analysis (PCA) [25],
In pursuit of an efficient stacking technique, the shortcom- exploited in [12], [13], and [26]. PCA provides a spectral
ings of both StBAS and the virtual image estimator shall decompositionofthedataspace,suchthattheeigenvectorcor-
be conquered. An efficient estimator shall provide a generic respondingtothehighesteigenvaluerepresentstheunderlying
solution that properly exploits both the short- and the long- most coherent signal and vice versa.
termcoherence.Todoso,itmustbeabletoadaptivelyinclude The estimation of most coherent signal component v via
1
both weak and strong coherent signals, precisely as does generic PCA may be interpreted as [13]
(cid:5) (cid:6)
theTMheLSEe.quentialEstimatorisabletomeeta balancebetween v1 =argmaxv1 v1HCˆv1 (7)
the aforementioned processing regimes. In fact, it may be subject to vHv = 1. In this way, PCA approximates the
1 1
seen as a generalization of the virtual image estimator that complex coherence matrix with the single dyad v vH. Being
1 1
follows the same idea of filtering, or better compressing the a geometrical rather than a probabilistic approach, PCA fails
information, of the subsets, but without the emphasis on the to correctly incorporate the statistical properties of the data
long-termcoherence.Priortothedescriptionofthealgorithm, stack; nevertheless, it provides a fair approximation of the
the SAR data compression is explained in Section III. phase signature. The MLE, on the other hand, is a purely
probabilisticapproach.StartingfromtheCCGassumptionand
III. MULTIPASSSAR DATACOMPRESSION trusting the coherence as the true statistic of the data stack,
it formulates the correct metric for phase estimation as the
Datacompressionisaclassicapproachindealingwithhigh
Hadamard product of (cid:3)ˆ−1°|Cˆ| in (2) and allows the sought
data volumes. In the case of multipass SAR, the objective is
to compress a stack of coregisteredSAR data in the temporal dyad of ζˆζˆH, to merely explain the interferometric phases
direction, such that the size of the time series is reduced but rather than the amplitude variations. The power of PCA is in
the spatial size of each image is intact. Performed locally, thedecompositionoftheprobablemultiplecoherentscattering
the compression is adaptive to the time series at each reso- mechanisms rather than providing precise estimation to the
lution cell. The temporal compression is valid since despite phase-linking. Thus, in the interest of retaining estimation
the high dimensionality of the data, the prevailing scattering precision, the dyad providedby ML is preferred over the one
mechanismspansamuchlowerdimension(seeSectionIV-A). of the PCA.
The introduction of this low-rank signal subspace is the aim Followingthisrationale,aneworthonormalbasisissought,
of this section. where its first component is defined as the normalized
As a common compression technique, the linear trans- ML-estimated signal of (2), that is
formations are chosen here. The transforms are mathemati- exp (jφˆ )
callystraightforwardandcomputationallyefficient,hencewell vML = (cid:5)exp (jφˆML)(cid:5). (8)
suited to an efficient stacking technique. They provide a ML
mapping from the high-dimensional data space, defined by v replaces the first componentin (5), now the complemen-
ML
therowspace of Zn×l,to a lower-ranksubspacespanningthe tary componentsof this basis are desired. In order to form an
rowspaceof Z˜m×l.Thelinearmappingunderatransformation orthonormalbasis,thesecomponentsshallspantheorthogonal
ANSARIetal.: SEQUENTIALESTIMATOR:TOWARDEFFICIENTINSARTIMESERIESANALYSIS 5641
Fig.1. SchematicdepictionoftheSequentialEstimatorbythecoherencematrix.a)FullcoherencematrixofastackofSLCs:theSequentialEstimatordivides
thestackintoisolatedmini-stacks indicated bythetransparent boxesalongthediagonal. Ateachsequence, onemini-stackisprocessedandcompressed;the
mini-stacks are replaced by their compressed components in further sequences. The coherent signals among the mini-stacks are retrieved by generation of
artificial interferograms. b)Coherence matrix attheinitial sequence. c)Second sequence: theisolated dotonthediagonal indicates thecompressed SLCof
theunavailable firstmini-stack;thesquaredepictstheacquiredmini-stack;andthesparserectangles representthegeneratedartificial interferograms between
thecompressed andthe acquired SLCs.d)Thirdsequence: here, theartificial interferograms aregenerated withrespect tothecompressed SLCsofthe first
and the second mini-stacks. Estimated between the compressed SLCs and the mini-stack, the coherence of the artificial interferograms in (c) and (d) is an
implicationoftherelationbetweentheisolatedmini-stacks.Indicatedbythiscoherence, thequalityoftheartificialinterferogram isevaluated adaptivetothe
datacontentofthemini-stack. Exploitation oftheartificial interferograms whileincorporating theircoherence rendersthedataadaptability oftheSequential
Estimator.
complement of the subspace spanned by v . The projection Expanding the above matrix product, it can easily be shown
ML
matrix corresponding to v reads as that the above formulationis equivalentto the coherentfilter-
ML
ing of (3).
C =v vH (9)
ML ML ML The introduceddata reduction is the core method for infor-
⊥
and its orthogonal complement C as mation preservation in the Sequential Estimator, as it will be
ML
C⊥ ⊕C = I. (10) introduced in Section IV.
ML ML
Note that C and consequently C⊥ are projection matri- IV. SEQUENTIALESTIMATOR
ML ML
ces, and I is the identity matrix. The coherence matrix can The proposed Sequential Estimator pursues an efficient
therefore be propagated via the orthogonalcomplement as stacking scheme (see Section I for the defined criteria). The
C =C⊥ CnC⊥ . (11) algorithmic efficiency criterion is imposed by processing the
proj ML ML stack in isolated small batches, as schematically depicted
Cproj corresponds to the coherence matrix of the projected in Fig. 1(a). To meet the requirement for the estimation
data to the residual subspace, after elimination of the ML efficiency, the scheme retrieves the coherent signal among
component.Theeigendecompositionofthiscoherencematrix the isolated batches. As it will be discussed in Section VI-C,
providesthecomplementaryvectorsofthesoughtorthonormal neglectingevenlow-coherentinterferometricpairsinthestack
basis degrades the estimation performance.
(cid:4)n−1 Inspired by De Zan and López-Dekker[18], the Sequential
Cproj = λpvpvHp (12) Estimator is established based on the idea of retrieving the
p=1 coherentsignal withouthaving full access to the data archive.
and the transformation matrix of (5) is redefined by the The backbone of the method is using data compression and
retrieving the coherence via formation of artificial inter-
resulted components, that is
ferograms between the compressed and the newly acquired
T ={vML;v1;...;vm−1}. (13) data. The inclusion of the artificial interferogramsenablesthe
performance preservation.
Thedataiscompressedbyitstransformationtotherangespace
of the defined T Theestimatorisessentiallyarecursivealgorithmwithalink
totheproductsofthepriorstepsateachsequence.Forthesake
Z˜ = THZ. (14) of clarity, the recursion of the established method is shown
Asdesired,thetransformationprojectsthen-dimensionaldata schematically in Fig. 1 and algorithmically in Table I. The
in Z to the m-dimensional subspace represented by T, thus flowchartof the estimator at each sequence is given in Fig. 2.
compressing the data volume. Z˜ contains the m sorted com- Abriefsynopsisoftheschemeisprovidedbelow.Eachsingle
pressed SLCs in its rows, such that the first row corresponds module of Fig. 2 is further elaborated in the sections.
to the most coherent signal component. The first compressed With reference to Table I, the Sequential Estimator starts
SLC is given by with the acquisition of a small chunk of s SLCs of the data
stream, where s (cid:3) n. The data is collected and processed
z˜ = THZ
ML 1 in sequences, hence the name sequential. The data chunk is
= vH Z. (15) hereafterreferredtoasamini-stack.Themini-stackundergoes
ML
5642 IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017
TABLEI this subspace. Processing of the first mini-stack concludes by
HIGH-LEVELPSEUDOCODEDESCRIBINGTHERECURSIVENATUREOF archiving the compressed SLCs.
THESEQUENTIALESTIMATOR;EACHMENTIONEDBLOCKIS With reference to Fig. 2, at the subsequent mini-stacks,
COMPRISEDOFDIFFERENTMODULESEXPRESSEDINFIG.2;
NOTETHATONLYSOMEMODULESARE the same two-step phase estimation and data compression
RELEVANTTOTHEINITIALSEQUENCE is pursued, although with minor changes. Prior to the phase
estimation,the compressedSLCsareprependedtothe current
mini-stack.Exploitingthe augmenteddata, artificialinterfero-
gramsaregeneratedbetweentheacquiredandthecompressed
SLCs. They in fact substitute the lost coherent signal among
the isolated mini-stacks. The phase-linkingincludes the artifi-
cialinterferogramsandtheircorrespondingcoherence.Inorder
to link the estimated phases of different sequences, a datum
connection step follows. After the phase estimation block,
themini-stackiscompressedandarchivedsimilartotheinitial
sequence.
The different processing modules are elaborated in the
following.Thesectionsfollowaprocessingorderstartingfrom
datacompressionofthefirstmini-stackintheinitialsequence
andendingatthephaseestimationofitssubsequentsequence.
A. Signal Subspace Identification
This module aims at finding the low-rank subspace for
temporal compression of the mini-stack. The proposed ortho-
normal basis T of (13) is an optimum representative of this
subspace, in case a linear transformation is desired. This
section addresses the question of the optimum dimension of
this subspace, i.e., the choice of m in (13).
AsdiscussedinSectionIII,thecomplexcoherencesumma-
rizes the information content of the SAR stack. The modulus
of coherence matrix indicates the temporal decorrelation of
the SAR signal. The temporal decorrelation arises from the
position changesof the subresolutionscatterers [27]. Random
position changes impose a variation in the coherent mean of
thescatterers.Theyarethereforereflectedinboththemodulus
and the argumentof the complexcoherence. Systematic posi-
tionchangesofthesubresolutionscatterers,ontheotherhand,
only impose a systematic phase shift in the mean scattering
response. The latter effect is captured by a systematic phase
term on the complex coherence rather than imposing a tem-
poral decorrelation [27]. The systematic position changes are
attributed,amongother effects,to the surface motion,i.e., the
geophysical signal. Therefore, although the coherence matrix
isfullrank,itmostlyrepresentsthedecorrelationphenomenon.
The geophysical signal is of much lower rank and spans a
much lower dimensional subspace. MLE, in fact, estimates a
rank-1dyad as the first componentof this low-ranksubspace.
Assuming a single PS-resembling scattering, it suffices for
Fig. 2. Algorithmic flow of the Sequential Estimator at its kth sequence:
the sequential scheme to include only the ML components
each module isaccompanied byits final product; superscripts correspond to
the index of the sequence with i referring to the entire history prior to the of the subspace; by setting m to 1. Higher dimension of
currentsequence.Thespecifiedlettersinparenthesiscorrespondtothesection signal subspace is required in cases where the DS region
inwhichthemoduleiselaborated.
exhibits multiple dominant systematic position changes along
the elevation profile of the SAR 3-D resolution cell. The
a two-step process: signal estimation and data compression. precise retrieval of such multiple geophysical signals falls in
In the signal estimation step, the phase of each SLC in the therealmofdifferentialSARtomography[28],[29].Hereafter,
mini-stackisestimatedviaMLE.Inthedatacompressionstep, a single dominant deformation signal is assumed and m is
theSLCsarecompressedbyestimationoftheunderlyinglow- set to 1. Section VIII discusses the consequence of this
rank signal subspace and a further projection of the data to simplification.
ANSARIetal.: SEQUENTIALESTIMATOR:TOWARDEFFICIENTINSARTIMESERIESANALYSIS 5643
B. Mini-Stack Compression thisapproachisthattheestimatedphasesateachsequenceare
The acquired SLCs in the mini-stack are compressed by relative to their respective datum. The solution is to connect
projection of the data to the identified signal subspace fol- the defined datum of different sequences in order to provide
lowing (14). For the sake of clarity, the compressed data is a single fully connected phase series.
hereafter specified by accentuation with ˜. The datum connection can be achieved by performing
a phase-linking on the z˜i components. Recall that these
ML
C. Data Archiving components represent their isolated mini-stacks. The inter-
Z˜ is archived to be further exploited at the subsequent ferometric phase between them therefore implies the datum
separations. These phases are temporally integrated via a
mini-stacks.
separate ML phase-linking; i.e., by treating the compressed
D. Data Augmentation SLCsasanewstack,generatingtherespective(cid:3)˜ML andC˜ML,
and retrieving the phase of each sequence relative to a new
After the initial sequence,the data available to each further
arbitrary but unique datum via MLE
kth sequence is comprised of the following:
12)) ZZ˜ski ::tthheecaocmqupirreesdsesdSvLeCrssioonf othfethkethitmhimnii-nstia-sctka;ck prior φˆcal =argminφ{ζˆH((cid:3)˜M−1L °C˜ML)ζˆ}. (19)
to the current one (i =1,...,k−1). φˆ is a vector containing the calibration phases that con-
cal
The compressed components are prepended to the acquired nects the isolated mini-stacks. The datum connection for the
mini-stack, such that the augmented data Zˆk reads as ith sequence is thus carried out
(cid:7) (cid:8)
Zˆk = Z˜1;...;Z˜k−1;Zsk (16) φˆiUnified =φˆi +φˆcal(i). (20)
with Zˆk ∈C(s+k−1×l).Theaugmentationishereafterindicated Here, the superscripts identify the sequence, while φˆ (i)
cal
by accentuation withˆ. indicates the ith element of the calibration vector. Following
this calibration step, a uni-datum phase time series results.
E. Interferogram Generation
At each sequence of processing, the coherent signal
H. NRT Processing Capability of the Sequential Estimator
between the mini-stack and the unavailable data history
Uptothispoint,theproposedschemeawaitstheacquisition
is retrieved through generation of artificial interferograms,
of s imagesforformationof a mini-stack andonly then starts
formedbetweeneachacquiredSLCin Zk andthecompressed
s theprocessingateachsequence.Theschememaybeexpanded
SLCs, that is
(cid:9) (cid:10) (cid:9) (cid:10) within the mini-stack to accommodate the NRT processing
Iˆ(cid:2)k ij = Z˜i Zsk Hj (17) of each acquired SLC. In other words, upon acquisition of
where the subscript indicates the jth row of the accompanied eachSLCwithinthemini-stack,wemayallowtheaccustomed
matrix. processingschemevia replacingthe Zk in (16)bythe interim
s
The artificial interferograms are exploited jointly with data chunk of less than s SLCs. The size of the interim
the observed interferograms in the phase-linking. This joint data chunk increases from 1 to s − 1 SLCs and enables
exploitation prevents an expected performance loss for the phase estimation of each SLC upon its acquisition. Note
batch-processing schemes. that the compressed data prior to the interim data chunk is
still exploited in the phase-linking. In this fashion, the phase
F. Phase-Linking estimation is repeated within the mini-stack until s SLCs are
At each sequence k, the phase-linking is carried out using acquired.Thenumberofrepetitionsis,however,limitedbythe
the ML estimator of (2), that is size of the mini-stack, rendering the increased computational
burden bounded as opposed to the conventional full-stack
φˆk =argminφ{ζˆH((cid:3)ˆk−1°Cˆk)ζˆ}. (18) processing approaches.
Here, Cˆk and (cid:3)ˆk are estimated based on the augmented With reference to Fig. 2, only Blocks 1, 2, and 3 are
data Zˆk. Thus, they encapsulate both the observed and relevant for NRT processing within the mini-stack. The data
the artificial interferograms accompanied by their coherence, compression and archiving blocks are carried out only after
representing their statistics. the accumulation of the entire s SLCs of the mini-stack.
G. Datum Connection V. NOTEON THECOMPUTATIONALGAIN
The phase-linking formulated in (2) poses an under- InconventionalMLEphase-linking,thecomputationaltime
determinedproblem[4],[5].InmultipassInSAR,thisproblem complexity is driven by the number of SLCs involved. The
is tackled by constraining the phase of an arbitrary SLC, computational burden is primarily imposed by the iterative
i.e., by setting this SLC as a datum in the time series and ML optimization for phase estimation and secondarily by the
estimating the phase of the entire stack with respect to this regularizationand inversionofthe complexcoherencematrix.
datum. This solution is as well adopted in the Sequential Trivially, both are affected by the number of interferograms.
Estimator, i.e., at each sequence the phases are estimated Assumingastackofn SLCs,n(n−1)/2interferogramsare
relative to one SLC in the augmenteddata. The problem with generated and exploited in the phase-linking. Adopting the
5644 IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017
sequential approach, the size of the available SLCs in the TABLEII
last mini-stack reduces to (s2 − s + n)/s, the number of PERFORMANCEOFTHEVARIOUSPHASE-LINKINGSCHEMES
interferograms decreases accordingly. COMPAREDTOTHECRLB,USING300LOOKSIN
COHERENCEESTIMATION;REPORTEDISTHE
Besides the reduced computational burden, the need for RMSEOFTHEESTIMATEDPHASEFORTHE
updating and re-estimation of the entire phase history at the INTERFEROGRAMWITHLONGESTTEMPORAL
face of each single acquisition is prevented by the sequential BASELINE(600DAYS).THEPROPOSED
SEQUENTIALESTIMATORPROVIDESA
scheme. The scheme in factenablesefficientprocessing of an TRADE-OFFBETWEENTHE
inherently nonparallel problem in the temporal domain. The MLEANDStBAS
reduction of the data volume, from the entire stack to the
compressed SLCs is another gain of the algorithm regarding
the data storage capacity.
Asanexampleoftheefficiencyofthealgorithm,letustake
the expected seven-year life of Sentinel-1 mission. With the
six-day revisit of the mission, a rough number of 400 SLCs
maybeexpectedforaregionofinterest.Conventionalprocess-
ing of such stacks requires the generation and exploitation
of 79800 interferograms. Adopting the Sequential Estimator
with mini-stacksize of 20,the numberof SLCs reducesto 39
and the number of interferograms to 741 at the last mini-
stack. However, it shall be mentioned that the cumulative
number of interferograms of all mini-stacks prior to the last
sequence sums up to 8740. However, in contrast to the full- respectively, the initial and residual coherence, tp,q stands
stackschemes,theprocessingofthisnumberofinterferograms for the temporal baseline, and τ0 is the time constant of the
isspreadovertheacquisitiontimeandisnotimposedatonce. decorrelation process. Note that for γ∞ = 0 the two models
In terms of storage capacity, the 400 SLCs may be replaced are identical. The aforementioned scenarios manifest two
by 20 compressed SLCs. extremecasesforanyphase-linkingscheme.Intheexponential
Trivially, the processing gain of the Sequential Estimator decay,theinterferogramswithlargetemporalbaselinebearno
is affected by the mini-stack size. The choice of an optimum coherent signal while in the long-term coherence, a possibly
mini-stacksizerequiresfurtherinvestigation.Aseparateyearly weak, coherent coherent signal is present regardless of the
compression of the data might as well be considered to temporal separation of the SLCs.
enhance the processing/archiving reduction factors; the cor- Based on the temporal models, two coherence matrices
responding estimation efficiency shall, however, be studied. are simulated corresponding to the exponential decay and
long-term coherence. Given the simulated coherence matri-
VI. PERFORMANCE ASSESSMENT WITH SIMULATIONS ces, two stacks of 100 SLCs, each containing an ensemble
For validation and performance evaluation purposes, of 300 statistically homogeneous samples, are synthesized as
the Sequential Estimator is tested and compared to the con- follows.
ventional phase-linking algorithms using simulated data. 1) The CCG statistic and spatial stationarity are assumed
Reflectedinthecoherencematrix,themaximumachievable in the generation of the data stack.
precision is bound by signal decorrelation [6]. The impact 2) Thetopographic,atmospheric,anddeformationphaseis
of the decorrelation process on the performance of different set to zero.
approaches including the proposed estimator is investigated 3) The temporal sampling interval, similar to Sentinel-1,
here. is set to six days.
The decorrelation model parameters of the simulated stacks
are provided in Table II.
A. Simulation Scenarios
The simulated coherence matrix and its estimation using
The error source to be tackled by phase-linking is the
the synthesized stack are depicted in Fig. 3. Comparing the
temporal decorrelation. In order to investigate its impact, two
estimated and simulated coherence matrices, the well-known
models are considered here. The first is a purely exponential
error of coherence estimation is observable [7], [8]. The
decorrelation between acquisition pairs, that is
(cid:11) (cid:12) performance of coherence estimation is governed by the size
(cid:3)p,q =γ0exp −τtp,q (21) olefveelns[e7m],bl[e8]u.seItdiisn dtheegreasdteimdaftoiorncaoshewreenllceass tchloesceohtoereznecroe
0
and may be improvedby exploitinga larger (cid:2) neighborhood.
while the second reveals a residual coherence even for large
However,the increasein the size of (cid:2) shall be dealtcarefully
temporal baselines, that is [23]
(cid:11) (cid:12) as it can compromise the spatial stationarity of the ensemble.
(cid:3)p,q =(γ0−γ∞)exp −τtp,q +γ∞. (22) Inpractice, thepositive definitenessof the estimated coher-
0 ence is not guaranteed. In such cases, the coherence matrix
Intheseformulations,(cid:3)p,q is,asusual,thecoherencebetween is regularized via diagonal loading, i.e., the addition of
the pth and qth SLC of the stack, γ0 and γ∞ indicate, small fraction to the diagonal elements of the coherence
ANSARIetal.: SEQUENTIALESTIMATOR:TOWARDEFFICIENTINSARTIMESERIESANALYSIS 5645
Fig. 3. Coherence matrix of the two considered simulation models; (a) and (b) fast exponential decay and (c) and (d) long-term coherence model;
(a)and(c)simulated(true)coherence;(b)and(d)estimationofthecoherenceusingthesimulatedCCGensemblewith300looks;thecolorscaleisidentical
for all matrices; and the well-known coherence estimation error is observable and is more pronounced forlower coherence level. Increasing the number of
samples,theperformance ofcoherence estimation improves andtheestimated coherence asymptotically approaches itstruevalues.
Fig.4. RMSEofphase estimation asthe performance indicator ofthe Sequential Estimator compared totheMLE,EVD,StBAS,virtual image estimator,
and the CRLB; considering (a) Fast exponential decaying coherence and (b) Long-term coherence scenario. TheStBAS is the optimum solution in case of
fast coherence loss while MLEoutperforms it when a weak but long-term coherent signal is present. Evidently, the optimality ofthe mentioned estimators
dependsonthecoherencescenario.Inbothcases,theSequentialEstimatorretainsabalancedperformanceclosetotheCRLB,provingtobeagenericsolution
andadaptable tothecoherence scenario[notethedifferent scaleof(a)and(b)].
matrix (see [11]). This operation iteratively increases the C. Performance Assessment
negative eigenvalues of the coherence matrix, thus ensures The root–mean-squareerror (RMSE) of the estimated with
the positive definiteness of the regularized coherence and respect to the simulated phases is considered for the perfor-
consequently its inverse. mance assessment. This measure encapsulates both the bias
and precision of the corresponding estimator.
B. Comparison Scenarios
Using 1000 realizations, the RMSE of the two defined
The objective is to compare different phase-linking tech- simulation scenarios is calculated. Table II summarizes the
niques. The following estimators with their specified details simulation cases as well as the RMSE of the estimated phase
are considered. of the last SLC in the simulated stacks.
1) MLE: Using the iterative solution proposed in [4]. Fig.4(a)and(b)depictstheperformanceofdifferentestima-
2) EVD: Exploiting the dominant scattering mechanism torsfortheexponentialdecorrelationandlong-termcoherence
corresponding to the largest principal component. scenarios, respectively. Note that the CRLB is calculated
3) StBAS: phase-linking exploiting the short temporal- with the theoretical coherence given by (21) and (22), while
baseline interferograms with baseline of up to 60 days. the different estimators use the estimation of the coherence
It is equivalent to accessing up to lag-10 SLCs at each according to (1).
processing level. Comparing Fig. 4(a) with Fig. 4(b) reveals the influence
4) VirtualImageEstimator:Usingtwosubsetsof10SLCs, of the decorrelation mechanism on the performance of the
thefirstisfixedatthebeginningofthestack,thesecond estimators. Evena weak signal, with coherenceas low as 0.2,
coincides with the Sequential Estimator’s mini-stacks; improves the precision of ML phase estimation by a fac-
5) Proposed Sequential Estimator: Accessing isolated tor of ∼12. This observation emphasizes the importance of
mini-stacks of 10 SLCs and setting m =1; the inclusion of even low-coherent interferograms in phase-
6) Cramér–Rao Lower Bound: The theoretical, i.e., sim- liking. As explained in Section III, it is further noticeable
ulated coherence is exploited for the calculation of that the EVD asymptotically approximates the MLE in both
the CRLB. cases. The behavior of the virtual image estimator in the
5646 IEEETRANSACTIONSONGEOSCIENCEANDREMOTESENSING,VOL.55,NO.10,OCTOBER2017
two different cases is also worthy of notice. As expected, interferometricpairsinphase-linking.Retrievingthelong-term
thephaseestimationiscompromisedintheabsenceofacoher- coherent signal among the mini-stacks via artificial interfero-
entsignalbetweenthetwosubsets;i.e.,beyondthecorrelation gram,theSequentialEstimatormaintainsitsperformanceclose
length of the exponential signal in Fig. 4(a) (∼210 days). to the CRLB.
Inspecting Fig. 4(a), StBAS outperforms other approaches Coming back to the comparison of these two coherence
byhavingtheclosestperformancetotheCRLB.Theoretically, scenarios, different stacking strategies are observed to be
however,the MLE is expected to be the closest to the bound. suited to each case. The proposed Sequential Estimator is,
The odd behavior of MLE lies in the suboptimality of the however, shown to provide a balance between the two alter-
coherence estimation (see Fig. 3). It may in fact be shown native schemes as it retains performance close to the CRLB
that substituting the simulated, i.e., true, coherence, the MLE in both scenarios. It may therefore be proposed as a generic
attains its asymptotic performance to CRLB. The coherence approach, adaptive to the coherence pattern.
estimator performs poorer for coherence levels close to zero,
as is the case for the majority of interferogramsin the current VII. EXPERIMENTS WITH REALDATA
simulation. Exploiting such suboptimum coherence matrix, A time series of Sentinel-1 data is chosen for the first
the MLE is misled toward relying on the pure noise-bearing demonstrationoftheSequentialEstimator.Atestsiteispicked
interferogramsin its estimation of the phase series. However, in the southern volcanic islands of Italy known as Salina.
theStBASnaivelyignoresinterferogramswithtemporalbase- Fig. 5 is an optical view of the Salina Island revealing the
lines of larger than 60 days and merely exploits the high- variety of land cover in the scene, ranging from rocky areas
coherent interferograms for which the coherence estimation as probable PSs to sparse vegetation as possible DS regions.
is known to perform better. The latter is therefore immune Data sets in interferometric wide-swath mode are obtained
to the coherence error. Comparing the different estimators, for this test site. The acquisitions were taken from Decem-
all schemes based on the full exploitation of the coherence ber 2014 to April 2016 from a descending orbit, providing
matrix are compromised by the coherence error. This impact 38 SLCs. Fig. 6(a) and (b) shows the coherence of the
is also conveyed by an independent study in [10]. Coming observedinterferogramswiththeshortestandlongesttemporal
to the proposed Sequential Estimator, the performance loss baselines, revealingthe interferometricqualityof the data set.
is not as dramatic as the full-stack-exploiting schemes. The As apparent, the data stack undergoes severe decorrelation,
performance preservation may be explained by the exploited rendering the phase-linking a necessary but challenging task
data compression in the Sequential Estimator. Recall that the forthisdatastack.Fig.6(c)depictstheobservedinterferogram
noisecomponentsofthe data spaceare suppressedin the data pertaining to the longest temporal baseline of 564 days. This
compression. The artificial interferograms between the mini- interferogramisestimatedviaspatialadaptivefiltering.Infact,
stacks are therefore less affected by the noisy interferograms thephase-linkingschemesfurtherperformatemporalfiltering
andingeneralofhigherSNRcomparedtotherespectiveinitial that improves the estimation of this interferogram, as it will
SLCs. This intermediary filtering of the mini-stacks enhances be shown later in this section.
the performance of the Sequential Estimator compared to the The aim here is to estimate the wrapped phase series,
MLE or EVD. inclusive of the geophysical and atmospheric signals, via
The impact of coherence estimation on the performance of phase-linking. The separation of the atmosphere from the
MLE highlights the importance of the considered stochastic geophysical signal follows from a second processing step as
model in the phase-linking scheme. Although this issue is in the case of PSs [1], [2]. This step is common to all DS
beyond the scope of this contribution, we briefly discuss the schemes, thus of no interest for the current demonstration of
potential solutions and leave further investigations to future the Sequential Estimator.
works.Theimprovementofthecoherenceestimation[7],[30] Setting s = 10, the Sequential Estimator divides the data
or modification of the stochastic model is two possible ways set into four isolated mini-stacks; the last mini-stack contains
to approach the problem. Regarding the latter, the robust eight SLCs. The phase estimation is performedon full spatial
estimation schemes, e.g., the robust M-estimator of [11], resolution. The pointwise complex coherence matrices are,
may be adopted to modify the stochastic model relative to however, estimated based on an ensemble of pixels in the
a posteriori residuals. While the MLE relies solely on the homogenous(cid:2)regionsurroundingeachpixel.The(cid:2)isfound
inverse of the coherence matrix and is misled by its poor usingAnderson–Darlingstatisticalsimilaritytestontheampli-
estimation, the robust scheme adaptively down weights the tude data [31]. The false alarm rate, a.k.a. the p-value, of the
outlyingnoise-bearinginterferograms,hencebalancestherole test is setto 5%. Note thatthe detectionof (cid:2) is merelybased
of the coherence as the only weighting criterion in the MLE. on the first mini-stack. Such a chosen homogeneousregion is
Examining the long-term coherence scenario in Fig. 4(b) furtherutilizedforthefuturemini-stacks.Toimprovethespa-
reveals a contrasting behavior of the estimators compared to tial stationarity in the homogeneous region, the topographic-
the exponential decay. Having exploited the long-term coher- induced phase is simulated using the Shuttle Radar Topog-
ence, the full-stack-exploiting schemes outperform StBAS in raphy Mission digital elevation model and reduced from
this scenario. The performance of StBAS deviates from the the SLCs prior to the coherence estimation. At each mini-
CRLB. Thisdegradationisduetodiscardingthelow-coherent stack, the data is compressed by fixing m to 1; using z˜
ML
interferograms with γ∞ = 0.2. This observation corroborates compressed SLCs only. The data volume is thus compressed
theimportanceofinclusionofevenlow-coherent(butnonzero) by 90 percent.