A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 605
Edgelist phase unwrapping algorithm for timeseries InSAR analysis
A Piyush Shanker and Howard Zebker
Departments of Electrical Engineering and Geophysics Stanford University 350 Serra Mall Packard 334Stanford California 94305 USA
Corresponding author shankerstanfordedu
Received August 31 2009 accepted December 8 2009posted January 5 2010 (Doc ID 116242) published February 26 2010
We present here a new integer programming formulation for phase unwrapping of multidimensional dataPhase unwrapping is a key problem in many coherent imaging systems including time series synthetic aper-ture radar interferometry (InSAR) with two spatial and one temporal data dimensions The minimum costflow (MCF) [IEEE Trans Geosci Remote Sens 36 813 (1998)] phase unwrapping algorithm describes a globalcost minimization problem involving flow between phase residues computed over closed loops Here we replaceclosed loops by reliable edges as the basic construct thus leading to the name ldquoedgelistrdquo Our algorithm hasseveral advantages over current methodsmdashit simplifies the representation of multidimensional phase unwrap-ping it incorporates data from external sources such as GPS where available to better constrain the un-wrapped solution and it treats regularly sampled or sparsely sampled data alike It thus is particularly ap-plicable to time series InSAR where data are often irregularly spaced in time and individual interferogramscan be corrupted with large decorrelated regions We show that similar to the MCF network problem theedgelist formulation also exhibits total unimodularity which enables us to solve the integer program problemby using efficient linear programming tools We apply our method to a persistent scatterer-InSAR data setfrom the creeping section of the Central San Andreas Fault and find that the average creep rate of 22 mmYris constant within 3 mmYr over 1992ndash2004 but varies systematically with ground location with a slightlyhigher rate in 1992ndash1998 than in 1999ndash2003 copy 2010 Optical Society of America
INTRODUCTIONhase unwrapping as used in synthetic aperture radar in-erferometry (InSAR) geodesy is the reconstruction of ab-olute phase from measured phase known only modulo 2n a finite grid of points Many methods have been devel-ped for unwrapping SAR interferograms [1ndash11] how-ver phase unwrapping is a key step in many other co-erent imaging techniques as well Most of these methodsocus primarily on regularly sampled two-dimensionalata (2D) sets Interferograms especially in time seriesnalysis are often irregularly sampled in both space andime so that existing algorithms do not always properlynwrap the data In this paper we develop a method toddress the most general form of the phase unwrappingroblemmdashunwrapping sparsely distributed multidimen-ional wrapped phase data Since we focus on applyingur new technique to InSAR phase data we restrict ouriscussion to conventional 2D (single interferogram) In-AR data and multitemporal InSAR [persistent scatterer
PS) and short baseline methods] 3D data setsThe performance of any phase unwrapping algorithm
or differential InSAR or time series InSAR applicationsepends on our ability to estimate the phase gradient be-ween two adjacent samples in our data set The basic as-umption in all phase unwrapping problems is that thenderlying continuous unwrapped phase function isell sampled in every dimension to enable us to recon-
truct it from wrapped phase measurements except atfinite relatively small number of discontinuities The
1084-752910030605-8$1500 copy 2
xistence of discontinuities in a data set produces path-ependent inconsistencies or residues [3] The branch cutlgorithm [3] and its derivatives are very popular ap-roaches to phase unwrapping because of their ease ofmplementation Hooper and Zebker [10] successfully ex-ended the idea of the shortest branch cut to the 3D phasenwrapping problem Least squares [2] and fast-Fourier-ransform-based [11] phase unwrapping algorithms areomputationally efficient but tend to distribute unwrap-ing errors globally instead of restricting errors to a smallet of points Ghiglia and Romero [5] suggested general Lporm objective function for phase unwrapping problemsnd argued that L0 and L1 norm solutions would produceewer errors than the traditional least squares solution
Costantini [1] developed the first network program-ing formulation to solve the regularly sampled 2D
hase unwrapping problem by using an L1 norm mini-um cost flow (MCF) approach Costantini and Rosen
12] adapted the algorithm to solve the irregularlyampled 2D phase unwrapping problems by usingelaunay triangulation [13] The performance of theetwork-programming-based unwrapping algorithms forD data sets was further improved by the development ofn iterative L0 norm approximation algorithm [9] and itspplication in combination with statistical cost functions14]
In this paper we propose a new minimum L1 norm for-ulation that is more flexible than the original MCF for-ulation allowing us to impose additional constraints on
010 Optical Society of America
tmmSrat4s
2LsitsuusDrwa
aauad
Hraocef
s
Ouifil
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c
MtoeM
teEts1gp
(tttlat
oeTi
Wats
FMl
606 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he unwrapped solutions based on available a priori infor-ation It also can be easily extended to analyze multidi-ensional data sets We describe our new algorithm inection 2 In Section 3 we explain how we can incorpo-ate additional geodetic observations as constraints Wepply the method to a time series of data acquired overhe creeping section of the San Andreas Fault in Section Finally in Section 5 we discuss implementation drawome conclusions and suggest possible improvements
PHASE UNWRAPPING FORMULATIONet V represent the set of points V=N on which theet of measured wrapped phase values i where 1 N is defined Let E define the set of edgesE=M on V that constitutes the unwrapping grid suchhat for every edge i jE i j Thus Gordf V E repre-ents the directed graph a formal representation of thenwrapping grid which will be used to estimate the set ofnwrapped phase values i where i 1 N Forimplicity we assume that the graph G represents theelaunay triangulation of the set of points V and E rep-
esents the set of the edges of the triangulation Therapped and unwrapped phase values at each point in Vre related by
i = i + 2 middot middot ni where i 1 N 1
nd ni denotes the integer number of cycles that must bedded to each point of the wrapped function to obtain thenwrapped function These variables can be interpreteds node potentials [15] For every edge i j in E weefine a new variable Kij such that
nj minus ni + Kij = i minus j
2 where ij E 2
ere [middot] represents the nearest integer function and Kijepresents the integer flow along the directed edge i jnd is equivalent to variables K1 and K2 in [1] As in theriginal MCF formulation we associate a nonnegativeonvex cost function fKij with the integral flow on everydge i jE We can then state our MCF problem asollows
Minimize
forallijE
fKij 3
ubject to
nj minus ni + Kij = i minus j
2 for all ij E 4
ni integer i 1 N 5
Kij integer ij E 6
ur formulation differs from that of [1] in that the basicnit in our algorithm is an edge on the unwrapping grid
nstead of a closed loop hence the name ldquoedgelistrdquo Thisormulation also reduces to a variation of the convex costnteger dual network flow problem [15] Lagrangian re-axation and cost scaling algorithms can be applied to
olve the general convex MCF problem [Eqs (3)ndash(6)] [15]ollowing Costantini [1] we restrict our discussion toinimum L1 norm solutions for ease in implementation
sing linear program (LP) solversOther salient features of the edgelist formulation in-
lude the following
bull The edgelist formulation reduces to the originalCF formulation if every edge of the Delaunay tessella-
ion is included as a constraint (see Fig 1) The constraintf our phase unwrapping formulation when applied to thedges of a loop produces a loop constraint in the originalCF formulationbull The edgelist formulation does not distinguish be-
ween 2D and 3D data where the third dimension is gen-rally the time dimension in a series of interferogramsach phase measurement is treated as a distinct vertex of
he graph G Consequently more variables and con-traints are needed to completely define a problem Tableprovides a comparison of the resources for when each al-orithm is applied to a regularly sampled 2D unwrappingroblembull The constraint matrix of the edgelist formulation
see Figs 1 and 2 for examples) is a total unimodular ma-rix (TUM see Appendix A) and the right-hand side ofhe constraints in Eq (4) is an array of integers Similaro the MCF and other TUM integer programming prob-ems the edgelist formulation can also be exactly solveds a LP when the associated objective functions minimizehe L1 norm [16]
bull The edgelist formulation can readily incorporatether geodetic measurements such as GPS (Fig 3) or lev-ling data as additional constraints without affecting theUM property of the constraint matrix This is discussed
n detail in Section 3
e alter the general edgelist formulation [Eqs (3)ndash(6)] tollow minimum L1 norm solutions using LP solvers byransforming the L1 problem into a LP Define two newets of nonnegative variables Pij and Qij such that
ig 1 Comparison of the edgelist formulation and the originalCF formulation We obtain the constraints of the MCF formu-
ation by summing up constraints of the edgelist formulation
Tpbvoi
s
TAa[o
Livswsbtn(i
Ievettz
putfshw
l
DNNN
Fistv
FblcieesGns
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 607
Kij = Pij minus Qij for all ij E 7
he resulting constraint matrix is also a TUM (See Ap-endix A) Hence the integer program [Eqs (3)ndash(6)] cane solved exactly by using its LP relaxation where all theariables are allowed to be real valued The solution tour phase unwrapping problem is thus obtained by solv-ng the LP
Minimize
forallijE
Cij middot Pij + Qij 8
ubject to constraints
nj minus ni + Pij minus Qij = i minus j
2 for all ij E 9
PijQij 0 and real for all ij E 10
Table 1 Parameters Needed To Represent a Regu-ar Grid 2D Unwrapping Problem of Size pAtildeqPixels by Using Edgelist and MCF Formulations
Parameter MCF Edgelist
imensions of image pq pqo of variables 4pqminus2pminus2q 5pqminus2pminus2qo of constraints pqminuspminusq+1 2pqminuspminusqo of sparse constraintmatrix entries
8pqminus8pminus8q+8 8pqminus4pminus4q
ig 2 Example 33 grid with pixel index and the correspond-ng wrapped phase values shown in i i format The corre-ponding edgelist constraints are also shown The optimal solu-ion corresponds to variable n5 being set to one and all flowariables (P and Q ) being set to zero
ij ij
ni real i 1 N 11
he node potentials and flows are solved simultaneouslys a result the final unwrapping step involves simpleddition of the node potentials to the wrapped valuesEq (1)] The functional form of Eq (8) ensures that flown an edge has the same cost irrespective of its direction
We illustrate the solution to our formulation by using aP solver on the example 33 square grid of Fig 2 The
ndex of the pixel and the corresponding wrapped phasealue are also shown in the i i format where i repre-ents the pixel index and i represents the correspondingrapped phase value A typical LP solver starts with a ba-
ic feasible solution and minimizes the objective functiony altering the solution in the direction of highest nega-ive gradient or the edge of the polytope with highestegative gradient The LP edgelist formulation [Eqs8)ndash(11)] always has a basic feasible solution correspond-ng to all ni=0
Pij = i minus j
2 0 Qij = i minus j
2 0
n case of the example in Fig 2 and uniform costs for alldges the basic feasible initial solution corresponds to allariables set to zero except P25 P45 Q56 and Q58 whichqual one The LP solver alters the solution in the direc-ion of the maximum negative gradient and the final op-imal solution corresponds to all variables being set toero except n5 which equals one
It is evident from Eqs (8)ndash(11) that the solution is de-endent on the unwrapping grid G used to define thenwrapping problem We used Delaunay triangulationso obtain simple nonoverlapping planar surface elementsor the reliable points in each interferogram [12] Ithould be noted that the edgelist formulation is capable ofandling nonplanar unwrapping grids for 2D data sets asell whereas the MCF formulation cannot We use
ig 3 We illustrate the ability to incorporate GPS observationsy using a regularly sampled 2D data grid and two GPS stationsabeled ldquoPrdquo and ldquoQrdquo With the original MCF formulation (left) aonstraint (as shown in the image) corresponding to any path ofntegration (dotted lines) joining the GPS stations along thedges of the grid can be used to constrain the solution In thedgelist formulation (right) a constraint similar to Eq (4) corre-ponding to a new edge (dotted curve) that directly connects thePS stations can be used to constrain the solution It should beoted that the TUM property of the constraint matrix is con-erved in case of the edgelist formulation only
Dctcr[wfilfttttbec
3MOGtTcwrast[btmd
dia
wcgetatselae
wtTmdtBcs
4TWsapCtSdmn
aambtpmtfgirwmg
ipaa
pttminusd3[
znAzFd
608 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
elaunay triangulations for ease in implementation Inase of 3D PS-InSAR and SBAS (Satellite-Based Augmen-ation System) data sets these triangulations are repli-ated in space for each individual interferogram and formectangular facets in time following Hooper and Zebker10] The edgelist algorithm allows us to evaluate the un-rapped solution on customized unwrapping grids by de-ning the set of edges E appropriately The optimal so-
ution of Eqs (8)ndash(11) also depends on the values of costunctions Cij for the edges These cost functions allow uso incorporate quality measures to control the flow acrosshe edges and consititute a feature that is common to bothhe original MCF and the edgelist formulations We usedhe deformation mode statistical cost functions developedy Chen and Zebker [17] here In Section 4 we show anxample where these cost functions are modified to in-lude a priori information about the area being analyzed
INCORPORATING OTHER GEODETICEASUREMENTS AS CONSTRAINTS
ften complementary geodetic measurements such asPS networks or leveling surveys are available in addi-
ion to frequent SAR acquisitions over regions of interesthis information can be used to direct edges in branch-ut-based algorithms or to adjust cost functions in a net-ork programming method for unwrapping interferomet-
ic phase constraining the solution to reflect thesedditional data In branch cut algorithms incorporatinguch changes involves defining and implementing a mul-icriteria or Pareto optimal spanning tree problem1819] In the original MCF formulation constraintsased on additional observations can be defined but athe cost of violating the TUM property of the constraintatrix (Appendix B) Violation of the TUM property ren-
ers the problem unsolvable exactly by LP solversIn the case of the edgelist formulation if alternate geo-
etic measurements are available at points p and q wentroduce a new edge between the points (see Fig 2) and
new constraint in the formulation
np minus nq = Npq minus p minus q
2 12
here Npq is the expected number of unwrapped phaseycles between p and q as determined by using additionaleodetic observations The new constraint adds additionalntries to the original node incidence matrix but retainshe TUM structure (Appendix A) If points p and q werelready connected in the original unwrapping grid Ghe corresponding constraint for the edge in the uncon-trained formulation [Eq (9)] can be replaced by the newquation [Eq (12)] Independent geodetic estimates ofine of sight (LOS) displacement for L vertices in V willllow us to construct CL 2= L middot Lminus1 2 additionalquations to constrain our unwrapped solutions
Simpler constraints of the form
0 P u for all ij E 13
ij
0 Qij u for all ij E 14
here u is a positive integer are further applied to reducehe solution space These constraints do not affect theUM property The primary advantage of the edgelist for-ulation is that it provides us with controls over every
ata point ni and every edge Pij Qij as opposed to con-rol over the edges alone in case of the MCF formulationoth of these properties can be suitably exploited to solvehallenging unwrapping problems as shown in the nextection
CASE STUDY CREEPING SECTION OFHE CENTRAL SAN ANDREAS FAULTe applied our edgelist unwrapping algorithm to a per-
istent scatterer InSAR (PS-InSAR) data set covering anrea of 40 km40 km around the Monarch and Austineaks (Fig 4) in the Central San Andreas Fault regiononventional InSAR stacking in earlier studies charac-
erized the spatial variation in slip deficit on the Centralan Andreas Fault [20] However the presence of largeecorrelated areas close to the fault severely compro-ised the ability to reliably estimate the deformation justorth of the fault (see Fig 2 from [20])We processed 21 SAR scenes (Track 27 Frame 2781)
cquired by ERS-1 and ERS-2 satellites between 1992nd 2004 We selected a scene from March 1997 as theaster scene from minimization of the perpendicular
aseline and the temporal baseline and generated 20 in-erferograms To optimize the accuracy of the correlatedhase estimates in the sparse PS network we limited theaximum perpendicular baseline to 400 m We applied
he maximum likelihood PS selection algorithm [21] andound a sparse network of 2067 PS points per interfero-ram Although the maximum likelihood PS algorithmdentified more PS points than other public domain algo-ithms the PS density 1 kmminus2 is still very low comparedith the suggested threshold of 4 PSkm2 [22] recom-ended for conventional PS-InSAR phase unwrapping al-
orithmsFigure 5(a) shows the average LOS displacement rate
n millimeters per year and Fig 5(b) shows the averagerofile computed as a function of distance from the faults computed by our algorithm [Eqs (8)ndash(11) (13) and (14)bove] To form this estimate we also did the following
1 Neglected elevation variation in the atmospherichase screen as the topography does not change by morehan a few hundred feet in this area Thus we computehe unwrapped solution on a 3D grid similar to the QuasiL norm algorithm of Hooper and Zebker [10] This isifferent from the two-step approach as in the stepwiseD algorithm [10] and the space-time MCF algorithm23]
2 Introduced an interferogram consisting purely ofero interferometric phase values to represent the combi-ation of the master scene with itself into the time seriesll the node potential variables ni for the vertices on theero interferogram were constrained to be zero (Seeig 6) This establishes a zero reference frame with noiscontinuities across the fault
eeoPgm
tr
allctfm
On
Fafa
FtC
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 609
3 Chose the PS point with the highest temporal coher-nce as the master PS point (see Fig 6) The phase of pix-ls in all the interferograms were referenced to the phasef the master PS point The node potential for the masterS point is also constrained to be zero in all the interfero-rams Thus all the unwrapped phase values are esti-ated with reference to the master PS point4 Forced the solution to concentrate phase changes in
he immediate vicinity of the fault trace by identifying aegion of 1 km width along the active fault trace in which
ig 4 (Color online) Map showing the location of the 40 km4he Central San Andreas Fault and Calaveras Fault are aanyon creepmeters are also shown
ll cost functions Cij associated with edges of the De-aunay triangulation and all time edges of the points thatie within this zone were decreased (See Fig 6) This in-orporates our knowledge that most of the creep is nearhe surface and that there is less deformation furtherrom the fault This is a good assumption for this fault butay not pertain to other parts of the world
ne of the main advantages of our method is that we doot require a temporal model for estimating deformation
rea (square) being analyzed by using PS-InSAR Fault traces forown in black Locations of the Melendy Ranch and Slack
ig 5 Left average LOS displacement rate image estimated by using the edgelist phase unwrapping algorithm The fault trace and therea used for computing the average profile are also indicated Right average LOS displacement rate (open circles red online) as aunction of distance from the fault Assuming that all the displacement was purely due to strike slip motion across the fault we estimateslip rate of 22 mmyr
0 km also sh
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
tmmSrat4s
2LsitsuusDrwa
aauad
Hraocef
s
Ouifil
sFmu
c
MtoeM
teEts1gp
(tttlat
oeTi
Wats
FMl
606 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he unwrapped solutions based on available a priori infor-ation It also can be easily extended to analyze multidi-ensional data sets We describe our new algorithm inection 2 In Section 3 we explain how we can incorpo-ate additional geodetic observations as constraints Wepply the method to a time series of data acquired overhe creeping section of the San Andreas Fault in Section Finally in Section 5 we discuss implementation drawome conclusions and suggest possible improvements
PHASE UNWRAPPING FORMULATIONet V represent the set of points V=N on which theet of measured wrapped phase values i where 1 N is defined Let E define the set of edgesE=M on V that constitutes the unwrapping grid suchhat for every edge i jE i j Thus Gordf V E repre-ents the directed graph a formal representation of thenwrapping grid which will be used to estimate the set ofnwrapped phase values i where i 1 N Forimplicity we assume that the graph G represents theelaunay triangulation of the set of points V and E rep-
esents the set of the edges of the triangulation Therapped and unwrapped phase values at each point in Vre related by
i = i + 2 middot middot ni where i 1 N 1
nd ni denotes the integer number of cycles that must bedded to each point of the wrapped function to obtain thenwrapped function These variables can be interpreteds node potentials [15] For every edge i j in E weefine a new variable Kij such that
nj minus ni + Kij = i minus j
2 where ij E 2
ere [middot] represents the nearest integer function and Kijepresents the integer flow along the directed edge i jnd is equivalent to variables K1 and K2 in [1] As in theriginal MCF formulation we associate a nonnegativeonvex cost function fKij with the integral flow on everydge i jE We can then state our MCF problem asollows
Minimize
forallijE
fKij 3
ubject to
nj minus ni + Kij = i minus j
2 for all ij E 4
ni integer i 1 N 5
Kij integer ij E 6
ur formulation differs from that of [1] in that the basicnit in our algorithm is an edge on the unwrapping grid
nstead of a closed loop hence the name ldquoedgelistrdquo Thisormulation also reduces to a variation of the convex costnteger dual network flow problem [15] Lagrangian re-axation and cost scaling algorithms can be applied to
olve the general convex MCF problem [Eqs (3)ndash(6)] [15]ollowing Costantini [1] we restrict our discussion toinimum L1 norm solutions for ease in implementation
sing linear program (LP) solversOther salient features of the edgelist formulation in-
lude the following
bull The edgelist formulation reduces to the originalCF formulation if every edge of the Delaunay tessella-
ion is included as a constraint (see Fig 1) The constraintf our phase unwrapping formulation when applied to thedges of a loop produces a loop constraint in the originalCF formulationbull The edgelist formulation does not distinguish be-
ween 2D and 3D data where the third dimension is gen-rally the time dimension in a series of interferogramsach phase measurement is treated as a distinct vertex of
he graph G Consequently more variables and con-traints are needed to completely define a problem Tableprovides a comparison of the resources for when each al-orithm is applied to a regularly sampled 2D unwrappingroblembull The constraint matrix of the edgelist formulation
see Figs 1 and 2 for examples) is a total unimodular ma-rix (TUM see Appendix A) and the right-hand side ofhe constraints in Eq (4) is an array of integers Similaro the MCF and other TUM integer programming prob-ems the edgelist formulation can also be exactly solveds a LP when the associated objective functions minimizehe L1 norm [16]
bull The edgelist formulation can readily incorporatether geodetic measurements such as GPS (Fig 3) or lev-ling data as additional constraints without affecting theUM property of the constraint matrix This is discussed
n detail in Section 3
e alter the general edgelist formulation [Eqs (3)ndash(6)] tollow minimum L1 norm solutions using LP solvers byransforming the L1 problem into a LP Define two newets of nonnegative variables Pij and Qij such that
ig 1 Comparison of the edgelist formulation and the originalCF formulation We obtain the constraints of the MCF formu-
ation by summing up constraints of the edgelist formulation
Tpbvoi
s
TAa[o
Livswsbtn(i
Ievettz
putfshw
l
DNNN
Fistv
FblcieesGns
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 607
Kij = Pij minus Qij for all ij E 7
he resulting constraint matrix is also a TUM (See Ap-endix A) Hence the integer program [Eqs (3)ndash(6)] cane solved exactly by using its LP relaxation where all theariables are allowed to be real valued The solution tour phase unwrapping problem is thus obtained by solv-ng the LP
Minimize
forallijE
Cij middot Pij + Qij 8
ubject to constraints
nj minus ni + Pij minus Qij = i minus j
2 for all ij E 9
PijQij 0 and real for all ij E 10
Table 1 Parameters Needed To Represent a Regu-ar Grid 2D Unwrapping Problem of Size pAtildeqPixels by Using Edgelist and MCF Formulations
Parameter MCF Edgelist
imensions of image pq pqo of variables 4pqminus2pminus2q 5pqminus2pminus2qo of constraints pqminuspminusq+1 2pqminuspminusqo of sparse constraintmatrix entries
8pqminus8pminus8q+8 8pqminus4pminus4q
ig 2 Example 33 grid with pixel index and the correspond-ng wrapped phase values shown in i i format The corre-ponding edgelist constraints are also shown The optimal solu-ion corresponds to variable n5 being set to one and all flowariables (P and Q ) being set to zero
ij ij
ni real i 1 N 11
he node potentials and flows are solved simultaneouslys a result the final unwrapping step involves simpleddition of the node potentials to the wrapped valuesEq (1)] The functional form of Eq (8) ensures that flown an edge has the same cost irrespective of its direction
We illustrate the solution to our formulation by using aP solver on the example 33 square grid of Fig 2 The
ndex of the pixel and the corresponding wrapped phasealue are also shown in the i i format where i repre-ents the pixel index and i represents the correspondingrapped phase value A typical LP solver starts with a ba-
ic feasible solution and minimizes the objective functiony altering the solution in the direction of highest nega-ive gradient or the edge of the polytope with highestegative gradient The LP edgelist formulation [Eqs8)ndash(11)] always has a basic feasible solution correspond-ng to all ni=0
Pij = i minus j
2 0 Qij = i minus j
2 0
n case of the example in Fig 2 and uniform costs for alldges the basic feasible initial solution corresponds to allariables set to zero except P25 P45 Q56 and Q58 whichqual one The LP solver alters the solution in the direc-ion of the maximum negative gradient and the final op-imal solution corresponds to all variables being set toero except n5 which equals one
It is evident from Eqs (8)ndash(11) that the solution is de-endent on the unwrapping grid G used to define thenwrapping problem We used Delaunay triangulationso obtain simple nonoverlapping planar surface elementsor the reliable points in each interferogram [12] Ithould be noted that the edgelist formulation is capable ofandling nonplanar unwrapping grids for 2D data sets asell whereas the MCF formulation cannot We use
ig 3 We illustrate the ability to incorporate GPS observationsy using a regularly sampled 2D data grid and two GPS stationsabeled ldquoPrdquo and ldquoQrdquo With the original MCF formulation (left) aonstraint (as shown in the image) corresponding to any path ofntegration (dotted lines) joining the GPS stations along thedges of the grid can be used to constrain the solution In thedgelist formulation (right) a constraint similar to Eq (4) corre-ponding to a new edge (dotted curve) that directly connects thePS stations can be used to constrain the solution It should beoted that the TUM property of the constraint matrix is con-erved in case of the edgelist formulation only
Dctcr[wfilfttttbec
3MOGtTcwrast[btmd
dia
wcgetatselae
wtTmdtBcs
4TWsapCtSdmn
aambtpmtfgirwmg
ipaa
pttminusd3[
znAzFd
608 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
elaunay triangulations for ease in implementation Inase of 3D PS-InSAR and SBAS (Satellite-Based Augmen-ation System) data sets these triangulations are repli-ated in space for each individual interferogram and formectangular facets in time following Hooper and Zebker10] The edgelist algorithm allows us to evaluate the un-rapped solution on customized unwrapping grids by de-ning the set of edges E appropriately The optimal so-
ution of Eqs (8)ndash(11) also depends on the values of costunctions Cij for the edges These cost functions allow uso incorporate quality measures to control the flow acrosshe edges and consititute a feature that is common to bothhe original MCF and the edgelist formulations We usedhe deformation mode statistical cost functions developedy Chen and Zebker [17] here In Section 4 we show anxample where these cost functions are modified to in-lude a priori information about the area being analyzed
INCORPORATING OTHER GEODETICEASUREMENTS AS CONSTRAINTS
ften complementary geodetic measurements such asPS networks or leveling surveys are available in addi-
ion to frequent SAR acquisitions over regions of interesthis information can be used to direct edges in branch-ut-based algorithms or to adjust cost functions in a net-ork programming method for unwrapping interferomet-
ic phase constraining the solution to reflect thesedditional data In branch cut algorithms incorporatinguch changes involves defining and implementing a mul-icriteria or Pareto optimal spanning tree problem1819] In the original MCF formulation constraintsased on additional observations can be defined but athe cost of violating the TUM property of the constraintatrix (Appendix B) Violation of the TUM property ren-
ers the problem unsolvable exactly by LP solversIn the case of the edgelist formulation if alternate geo-
etic measurements are available at points p and q wentroduce a new edge between the points (see Fig 2) and
new constraint in the formulation
np minus nq = Npq minus p minus q
2 12
here Npq is the expected number of unwrapped phaseycles between p and q as determined by using additionaleodetic observations The new constraint adds additionalntries to the original node incidence matrix but retainshe TUM structure (Appendix A) If points p and q werelready connected in the original unwrapping grid Ghe corresponding constraint for the edge in the uncon-trained formulation [Eq (9)] can be replaced by the newquation [Eq (12)] Independent geodetic estimates ofine of sight (LOS) displacement for L vertices in V willllow us to construct CL 2= L middot Lminus1 2 additionalquations to constrain our unwrapped solutions
Simpler constraints of the form
0 P u for all ij E 13
ij
0 Qij u for all ij E 14
here u is a positive integer are further applied to reducehe solution space These constraints do not affect theUM property The primary advantage of the edgelist for-ulation is that it provides us with controls over every
ata point ni and every edge Pij Qij as opposed to con-rol over the edges alone in case of the MCF formulationoth of these properties can be suitably exploited to solvehallenging unwrapping problems as shown in the nextection
CASE STUDY CREEPING SECTION OFHE CENTRAL SAN ANDREAS FAULTe applied our edgelist unwrapping algorithm to a per-
istent scatterer InSAR (PS-InSAR) data set covering anrea of 40 km40 km around the Monarch and Austineaks (Fig 4) in the Central San Andreas Fault regiononventional InSAR stacking in earlier studies charac-
erized the spatial variation in slip deficit on the Centralan Andreas Fault [20] However the presence of largeecorrelated areas close to the fault severely compro-ised the ability to reliably estimate the deformation justorth of the fault (see Fig 2 from [20])We processed 21 SAR scenes (Track 27 Frame 2781)
cquired by ERS-1 and ERS-2 satellites between 1992nd 2004 We selected a scene from March 1997 as theaster scene from minimization of the perpendicular
aseline and the temporal baseline and generated 20 in-erferograms To optimize the accuracy of the correlatedhase estimates in the sparse PS network we limited theaximum perpendicular baseline to 400 m We applied
he maximum likelihood PS selection algorithm [21] andound a sparse network of 2067 PS points per interfero-ram Although the maximum likelihood PS algorithmdentified more PS points than other public domain algo-ithms the PS density 1 kmminus2 is still very low comparedith the suggested threshold of 4 PSkm2 [22] recom-ended for conventional PS-InSAR phase unwrapping al-
orithmsFigure 5(a) shows the average LOS displacement rate
n millimeters per year and Fig 5(b) shows the averagerofile computed as a function of distance from the faults computed by our algorithm [Eqs (8)ndash(11) (13) and (14)bove] To form this estimate we also did the following
1 Neglected elevation variation in the atmospherichase screen as the topography does not change by morehan a few hundred feet in this area Thus we computehe unwrapped solution on a 3D grid similar to the QuasiL norm algorithm of Hooper and Zebker [10] This isifferent from the two-step approach as in the stepwiseD algorithm [10] and the space-time MCF algorithm23]
2 Introduced an interferogram consisting purely ofero interferometric phase values to represent the combi-ation of the master scene with itself into the time seriesll the node potential variables ni for the vertices on theero interferogram were constrained to be zero (Seeig 6) This establishes a zero reference frame with noiscontinuities across the fault
eeoPgm
tr
allctfm
On
Fafa
FtC
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 609
3 Chose the PS point with the highest temporal coher-nce as the master PS point (see Fig 6) The phase of pix-ls in all the interferograms were referenced to the phasef the master PS point The node potential for the masterS point is also constrained to be zero in all the interfero-rams Thus all the unwrapped phase values are esti-ated with reference to the master PS point4 Forced the solution to concentrate phase changes in
he immediate vicinity of the fault trace by identifying aegion of 1 km width along the active fault trace in which
ig 4 (Color online) Map showing the location of the 40 km4he Central San Andreas Fault and Calaveras Fault are aanyon creepmeters are also shown
ll cost functions Cij associated with edges of the De-aunay triangulation and all time edges of the points thatie within this zone were decreased (See Fig 6) This in-orporates our knowledge that most of the creep is nearhe surface and that there is less deformation furtherrom the fault This is a good assumption for this fault butay not pertain to other parts of the world
ne of the main advantages of our method is that we doot require a temporal model for estimating deformation
rea (square) being analyzed by using PS-InSAR Fault traces forown in black Locations of the Melendy Ranch and Slack
ig 5 Left average LOS displacement rate image estimated by using the edgelist phase unwrapping algorithm The fault trace and therea used for computing the average profile are also indicated Right average LOS displacement rate (open circles red online) as aunction of distance from the fault Assuming that all the displacement was purely due to strike slip motion across the fault we estimateslip rate of 22 mmyr
0 km also sh
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
Tpbvoi
s
TAa[o
Livswsbtn(i
Ievettz
putfshw
l
DNNN
Fistv
FblcieesGns
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 607
Kij = Pij minus Qij for all ij E 7
he resulting constraint matrix is also a TUM (See Ap-endix A) Hence the integer program [Eqs (3)ndash(6)] cane solved exactly by using its LP relaxation where all theariables are allowed to be real valued The solution tour phase unwrapping problem is thus obtained by solv-ng the LP
Minimize
forallijE
Cij middot Pij + Qij 8
ubject to constraints
nj minus ni + Pij minus Qij = i minus j
2 for all ij E 9
PijQij 0 and real for all ij E 10
Table 1 Parameters Needed To Represent a Regu-ar Grid 2D Unwrapping Problem of Size pAtildeqPixels by Using Edgelist and MCF Formulations
Parameter MCF Edgelist
imensions of image pq pqo of variables 4pqminus2pminus2q 5pqminus2pminus2qo of constraints pqminuspminusq+1 2pqminuspminusqo of sparse constraintmatrix entries
8pqminus8pminus8q+8 8pqminus4pminus4q
ig 2 Example 33 grid with pixel index and the correspond-ng wrapped phase values shown in i i format The corre-ponding edgelist constraints are also shown The optimal solu-ion corresponds to variable n5 being set to one and all flowariables (P and Q ) being set to zero
ij ij
ni real i 1 N 11
he node potentials and flows are solved simultaneouslys a result the final unwrapping step involves simpleddition of the node potentials to the wrapped valuesEq (1)] The functional form of Eq (8) ensures that flown an edge has the same cost irrespective of its direction
We illustrate the solution to our formulation by using aP solver on the example 33 square grid of Fig 2 The
ndex of the pixel and the corresponding wrapped phasealue are also shown in the i i format where i repre-ents the pixel index and i represents the correspondingrapped phase value A typical LP solver starts with a ba-
ic feasible solution and minimizes the objective functiony altering the solution in the direction of highest nega-ive gradient or the edge of the polytope with highestegative gradient The LP edgelist formulation [Eqs8)ndash(11)] always has a basic feasible solution correspond-ng to all ni=0
Pij = i minus j
2 0 Qij = i minus j
2 0
n case of the example in Fig 2 and uniform costs for alldges the basic feasible initial solution corresponds to allariables set to zero except P25 P45 Q56 and Q58 whichqual one The LP solver alters the solution in the direc-ion of the maximum negative gradient and the final op-imal solution corresponds to all variables being set toero except n5 which equals one
It is evident from Eqs (8)ndash(11) that the solution is de-endent on the unwrapping grid G used to define thenwrapping problem We used Delaunay triangulationso obtain simple nonoverlapping planar surface elementsor the reliable points in each interferogram [12] Ithould be noted that the edgelist formulation is capable ofandling nonplanar unwrapping grids for 2D data sets asell whereas the MCF formulation cannot We use
ig 3 We illustrate the ability to incorporate GPS observationsy using a regularly sampled 2D data grid and two GPS stationsabeled ldquoPrdquo and ldquoQrdquo With the original MCF formulation (left) aonstraint (as shown in the image) corresponding to any path ofntegration (dotted lines) joining the GPS stations along thedges of the grid can be used to constrain the solution In thedgelist formulation (right) a constraint similar to Eq (4) corre-ponding to a new edge (dotted curve) that directly connects thePS stations can be used to constrain the solution It should beoted that the TUM property of the constraint matrix is con-erved in case of the edgelist formulation only
Dctcr[wfilfttttbec
3MOGtTcwrast[btmd
dia
wcgetatselae
wtTmdtBcs
4TWsapCtSdmn
aambtpmtfgirwmg
ipaa
pttminusd3[
znAzFd
608 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
elaunay triangulations for ease in implementation Inase of 3D PS-InSAR and SBAS (Satellite-Based Augmen-ation System) data sets these triangulations are repli-ated in space for each individual interferogram and formectangular facets in time following Hooper and Zebker10] The edgelist algorithm allows us to evaluate the un-rapped solution on customized unwrapping grids by de-ning the set of edges E appropriately The optimal so-
ution of Eqs (8)ndash(11) also depends on the values of costunctions Cij for the edges These cost functions allow uso incorporate quality measures to control the flow acrosshe edges and consititute a feature that is common to bothhe original MCF and the edgelist formulations We usedhe deformation mode statistical cost functions developedy Chen and Zebker [17] here In Section 4 we show anxample where these cost functions are modified to in-lude a priori information about the area being analyzed
INCORPORATING OTHER GEODETICEASUREMENTS AS CONSTRAINTS
ften complementary geodetic measurements such asPS networks or leveling surveys are available in addi-
ion to frequent SAR acquisitions over regions of interesthis information can be used to direct edges in branch-ut-based algorithms or to adjust cost functions in a net-ork programming method for unwrapping interferomet-
ic phase constraining the solution to reflect thesedditional data In branch cut algorithms incorporatinguch changes involves defining and implementing a mul-icriteria or Pareto optimal spanning tree problem1819] In the original MCF formulation constraintsased on additional observations can be defined but athe cost of violating the TUM property of the constraintatrix (Appendix B) Violation of the TUM property ren-
ers the problem unsolvable exactly by LP solversIn the case of the edgelist formulation if alternate geo-
etic measurements are available at points p and q wentroduce a new edge between the points (see Fig 2) and
new constraint in the formulation
np minus nq = Npq minus p minus q
2 12
here Npq is the expected number of unwrapped phaseycles between p and q as determined by using additionaleodetic observations The new constraint adds additionalntries to the original node incidence matrix but retainshe TUM structure (Appendix A) If points p and q werelready connected in the original unwrapping grid Ghe corresponding constraint for the edge in the uncon-trained formulation [Eq (9)] can be replaced by the newquation [Eq (12)] Independent geodetic estimates ofine of sight (LOS) displacement for L vertices in V willllow us to construct CL 2= L middot Lminus1 2 additionalquations to constrain our unwrapped solutions
Simpler constraints of the form
0 P u for all ij E 13
ij
0 Qij u for all ij E 14
here u is a positive integer are further applied to reducehe solution space These constraints do not affect theUM property The primary advantage of the edgelist for-ulation is that it provides us with controls over every
ata point ni and every edge Pij Qij as opposed to con-rol over the edges alone in case of the MCF formulationoth of these properties can be suitably exploited to solvehallenging unwrapping problems as shown in the nextection
CASE STUDY CREEPING SECTION OFHE CENTRAL SAN ANDREAS FAULTe applied our edgelist unwrapping algorithm to a per-
istent scatterer InSAR (PS-InSAR) data set covering anrea of 40 km40 km around the Monarch and Austineaks (Fig 4) in the Central San Andreas Fault regiononventional InSAR stacking in earlier studies charac-
erized the spatial variation in slip deficit on the Centralan Andreas Fault [20] However the presence of largeecorrelated areas close to the fault severely compro-ised the ability to reliably estimate the deformation justorth of the fault (see Fig 2 from [20])We processed 21 SAR scenes (Track 27 Frame 2781)
cquired by ERS-1 and ERS-2 satellites between 1992nd 2004 We selected a scene from March 1997 as theaster scene from minimization of the perpendicular
aseline and the temporal baseline and generated 20 in-erferograms To optimize the accuracy of the correlatedhase estimates in the sparse PS network we limited theaximum perpendicular baseline to 400 m We applied
he maximum likelihood PS selection algorithm [21] andound a sparse network of 2067 PS points per interfero-ram Although the maximum likelihood PS algorithmdentified more PS points than other public domain algo-ithms the PS density 1 kmminus2 is still very low comparedith the suggested threshold of 4 PSkm2 [22] recom-ended for conventional PS-InSAR phase unwrapping al-
orithmsFigure 5(a) shows the average LOS displacement rate
n millimeters per year and Fig 5(b) shows the averagerofile computed as a function of distance from the faults computed by our algorithm [Eqs (8)ndash(11) (13) and (14)bove] To form this estimate we also did the following
1 Neglected elevation variation in the atmospherichase screen as the topography does not change by morehan a few hundred feet in this area Thus we computehe unwrapped solution on a 3D grid similar to the QuasiL norm algorithm of Hooper and Zebker [10] This isifferent from the two-step approach as in the stepwiseD algorithm [10] and the space-time MCF algorithm23]
2 Introduced an interferogram consisting purely ofero interferometric phase values to represent the combi-ation of the master scene with itself into the time seriesll the node potential variables ni for the vertices on theero interferogram were constrained to be zero (Seeig 6) This establishes a zero reference frame with noiscontinuities across the fault
eeoPgm
tr
allctfm
On
Fafa
FtC
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 609
3 Chose the PS point with the highest temporal coher-nce as the master PS point (see Fig 6) The phase of pix-ls in all the interferograms were referenced to the phasef the master PS point The node potential for the masterS point is also constrained to be zero in all the interfero-rams Thus all the unwrapped phase values are esti-ated with reference to the master PS point4 Forced the solution to concentrate phase changes in
he immediate vicinity of the fault trace by identifying aegion of 1 km width along the active fault trace in which
ig 4 (Color online) Map showing the location of the 40 km4he Central San Andreas Fault and Calaveras Fault are aanyon creepmeters are also shown
ll cost functions Cij associated with edges of the De-aunay triangulation and all time edges of the points thatie within this zone were decreased (See Fig 6) This in-orporates our knowledge that most of the creep is nearhe surface and that there is less deformation furtherrom the fault This is a good assumption for this fault butay not pertain to other parts of the world
ne of the main advantages of our method is that we doot require a temporal model for estimating deformation
rea (square) being analyzed by using PS-InSAR Fault traces forown in black Locations of the Melendy Ranch and Slack
ig 5 Left average LOS displacement rate image estimated by using the edgelist phase unwrapping algorithm The fault trace and therea used for computing the average profile are also indicated Right average LOS displacement rate (open circles red online) as aunction of distance from the fault Assuming that all the displacement was purely due to strike slip motion across the fault we estimateslip rate of 22 mmyr
0 km also sh
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
Dctcr[wfilfttttbec
3MOGtTcwrast[btmd
dia
wcgetatselae
wtTmdtBcs
4TWsapCtSdmn
aambtpmtfgirwmg
ipaa
pttminusd3[
znAzFd
608 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
elaunay triangulations for ease in implementation Inase of 3D PS-InSAR and SBAS (Satellite-Based Augmen-ation System) data sets these triangulations are repli-ated in space for each individual interferogram and formectangular facets in time following Hooper and Zebker10] The edgelist algorithm allows us to evaluate the un-rapped solution on customized unwrapping grids by de-ning the set of edges E appropriately The optimal so-
ution of Eqs (8)ndash(11) also depends on the values of costunctions Cij for the edges These cost functions allow uso incorporate quality measures to control the flow acrosshe edges and consititute a feature that is common to bothhe original MCF and the edgelist formulations We usedhe deformation mode statistical cost functions developedy Chen and Zebker [17] here In Section 4 we show anxample where these cost functions are modified to in-lude a priori information about the area being analyzed
INCORPORATING OTHER GEODETICEASUREMENTS AS CONSTRAINTS
ften complementary geodetic measurements such asPS networks or leveling surveys are available in addi-
ion to frequent SAR acquisitions over regions of interesthis information can be used to direct edges in branch-ut-based algorithms or to adjust cost functions in a net-ork programming method for unwrapping interferomet-
ic phase constraining the solution to reflect thesedditional data In branch cut algorithms incorporatinguch changes involves defining and implementing a mul-icriteria or Pareto optimal spanning tree problem1819] In the original MCF formulation constraintsased on additional observations can be defined but athe cost of violating the TUM property of the constraintatrix (Appendix B) Violation of the TUM property ren-
ers the problem unsolvable exactly by LP solversIn the case of the edgelist formulation if alternate geo-
etic measurements are available at points p and q wentroduce a new edge between the points (see Fig 2) and
new constraint in the formulation
np minus nq = Npq minus p minus q
2 12
here Npq is the expected number of unwrapped phaseycles between p and q as determined by using additionaleodetic observations The new constraint adds additionalntries to the original node incidence matrix but retainshe TUM structure (Appendix A) If points p and q werelready connected in the original unwrapping grid Ghe corresponding constraint for the edge in the uncon-trained formulation [Eq (9)] can be replaced by the newquation [Eq (12)] Independent geodetic estimates ofine of sight (LOS) displacement for L vertices in V willllow us to construct CL 2= L middot Lminus1 2 additionalquations to constrain our unwrapped solutions
Simpler constraints of the form
0 P u for all ij E 13
ij
0 Qij u for all ij E 14
here u is a positive integer are further applied to reducehe solution space These constraints do not affect theUM property The primary advantage of the edgelist for-ulation is that it provides us with controls over every
ata point ni and every edge Pij Qij as opposed to con-rol over the edges alone in case of the MCF formulationoth of these properties can be suitably exploited to solvehallenging unwrapping problems as shown in the nextection
CASE STUDY CREEPING SECTION OFHE CENTRAL SAN ANDREAS FAULTe applied our edgelist unwrapping algorithm to a per-
istent scatterer InSAR (PS-InSAR) data set covering anrea of 40 km40 km around the Monarch and Austineaks (Fig 4) in the Central San Andreas Fault regiononventional InSAR stacking in earlier studies charac-
erized the spatial variation in slip deficit on the Centralan Andreas Fault [20] However the presence of largeecorrelated areas close to the fault severely compro-ised the ability to reliably estimate the deformation justorth of the fault (see Fig 2 from [20])We processed 21 SAR scenes (Track 27 Frame 2781)
cquired by ERS-1 and ERS-2 satellites between 1992nd 2004 We selected a scene from March 1997 as theaster scene from minimization of the perpendicular
aseline and the temporal baseline and generated 20 in-erferograms To optimize the accuracy of the correlatedhase estimates in the sparse PS network we limited theaximum perpendicular baseline to 400 m We applied
he maximum likelihood PS selection algorithm [21] andound a sparse network of 2067 PS points per interfero-ram Although the maximum likelihood PS algorithmdentified more PS points than other public domain algo-ithms the PS density 1 kmminus2 is still very low comparedith the suggested threshold of 4 PSkm2 [22] recom-ended for conventional PS-InSAR phase unwrapping al-
orithmsFigure 5(a) shows the average LOS displacement rate
n millimeters per year and Fig 5(b) shows the averagerofile computed as a function of distance from the faults computed by our algorithm [Eqs (8)ndash(11) (13) and (14)bove] To form this estimate we also did the following
1 Neglected elevation variation in the atmospherichase screen as the topography does not change by morehan a few hundred feet in this area Thus we computehe unwrapped solution on a 3D grid similar to the QuasiL norm algorithm of Hooper and Zebker [10] This isifferent from the two-step approach as in the stepwiseD algorithm [10] and the space-time MCF algorithm23]
2 Introduced an interferogram consisting purely ofero interferometric phase values to represent the combi-ation of the master scene with itself into the time seriesll the node potential variables ni for the vertices on theero interferogram were constrained to be zero (Seeig 6) This establishes a zero reference frame with noiscontinuities across the fault
eeoPgm
tr
allctfm
On
Fafa
FtC
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 609
3 Chose the PS point with the highest temporal coher-nce as the master PS point (see Fig 6) The phase of pix-ls in all the interferograms were referenced to the phasef the master PS point The node potential for the masterS point is also constrained to be zero in all the interfero-rams Thus all the unwrapped phase values are esti-ated with reference to the master PS point4 Forced the solution to concentrate phase changes in
he immediate vicinity of the fault trace by identifying aegion of 1 km width along the active fault trace in which
ig 4 (Color online) Map showing the location of the 40 km4he Central San Andreas Fault and Calaveras Fault are aanyon creepmeters are also shown
ll cost functions Cij associated with edges of the De-aunay triangulation and all time edges of the points thatie within this zone were decreased (See Fig 6) This in-orporates our knowledge that most of the creep is nearhe surface and that there is less deformation furtherrom the fault This is a good assumption for this fault butay not pertain to other parts of the world
ne of the main advantages of our method is that we doot require a temporal model for estimating deformation
rea (square) being analyzed by using PS-InSAR Fault traces forown in black Locations of the Melendy Ranch and Slack
ig 5 Left average LOS displacement rate image estimated by using the edgelist phase unwrapping algorithm The fault trace and therea used for computing the average profile are also indicated Right average LOS displacement rate (open circles red online) as aunction of distance from the fault Assuming that all the displacement was purely due to strike slip motion across the fault we estimateslip rate of 22 mmyr
0 km also sh
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
eeoPgm
tr
allctfm
On
Fafa
FtC
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 609
3 Chose the PS point with the highest temporal coher-nce as the master PS point (see Fig 6) The phase of pix-ls in all the interferograms were referenced to the phasef the master PS point The node potential for the masterS point is also constrained to be zero in all the interfero-rams Thus all the unwrapped phase values are esti-ated with reference to the master PS point4 Forced the solution to concentrate phase changes in
he immediate vicinity of the fault trace by identifying aegion of 1 km width along the active fault trace in which
ig 4 (Color online) Map showing the location of the 40 km4he Central San Andreas Fault and Calaveras Fault are aanyon creepmeters are also shown
ll cost functions Cij associated with edges of the De-aunay triangulation and all time edges of the points thatie within this zone were decreased (See Fig 6) This in-orporates our knowledge that most of the creep is nearhe surface and that there is less deformation furtherrom the fault This is a good assumption for this fault butay not pertain to other parts of the world
ne of the main advantages of our method is that we doot require a temporal model for estimating deformation
rea (square) being analyzed by using PS-InSAR Fault traces forown in black Locations of the Melendy Ranch and Slack
ig 5 Left average LOS displacement rate image estimated by using the edgelist phase unwrapping algorithm The fault trace and therea used for computing the average profile are also indicated Right average LOS displacement rate (open circles red online) as aunction of distance from the fault Assuming that all the displacement was purely due to strike slip motion across the fault we estimateslip rate of 22 mmyr
0 km also sh
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
Tscpoodimpczbfimmi
tcstpwnbclpt2ttoicfittpfae
iga
tettboHtiSfsarvHs
Ftt
Fes
610 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
he approach described above models the points on oppo-ite sides of the fault as connected by loose (subsidizedosts) strings ie the unwrapped solution for a pixel de-ends more on the phase values of pixels on the same sidef the fault as itself and less on the phase of pixels on thether side of the fault In other words by quantitativelyecreasing the cost functions we allow the edges connect-ng pixels on opposite sides of the fault to accommodate
issing phase cycles more readily than edges connectingixels lying on the same side of the fault trace Phaseycle jumps are preferentially compensated in the bufferone As a consequence the unwrapped solutions in theuffer zone may not be entirely reliable owing to the arti-cial discounts applied on the cost functions But as weove away from the buffer zone the solutions should beore accurate Figure 5 shows the results obtained by us-
ng our new algorithmIt is useful to compare our solution to the results ob-
ained by other algorithms We have already noted theomparison with the Ryder and Burgmann [20] stack andee that the current solution produces estimates closer tohe fault We also reduced the data using the step-wise 3Dhase unwrapping algorithm (Fig 7) [10] In this analysise clearly see the presence of the fault but leakage of sig-al appears across the known fault trace This likely cane attributed to the inability to accommodate multipleycle jumps across the fault for the large temporal base-ine interferograms in the data set We overcome thisroblem by subsidizing the costs for edges cutting acrosshe fault In this area the fault creeps at roughly2 mmyr (approximately 7 mmyr in the radar LOS) sohat interferograms with temporal separation of morehan 4 yr exhibit multiple cycles across the fault In an L1r L2 norm formulation multiple cycle jumps are penal-zed more heavily than single-cycle jumps and requireareful and often impractical adjustment of the costunctions Also the San Andreas Fault fully bisects themage and forms a line discontinuity In other wordshere is no direct connection between the regions on ei-her side of the fault through an area that is not noisy inhase Hence it is not possible to unwrap around theault in this data set and correct for the phase jumpscross the fault by using conventional algorithms Thedges cutting across the fault are incorrectly unwrapped
ig 6 Left skeletal framework on which the PS-InSAR data setime series All the node potentials in areas marked by gray are she edges cutting across the fault is subsidized as a distance of c
n the temporal unwrapping stage of the stepwise 3D al-orithm [10] resulting in both overly smooth solutionsnd leakage of deformation signal across the faultFigure 8 shows the cumulative LOS displacements be-
ween regions that are located at a distance of 8 km onither side of the fault For reference the timeseries fromhe Melendy Ranch and Slack Canyon creepmeters (cour-esy USGS) are also included The creep rates estimatedy our method are consistent with the long-term trend ofbservations from the USGS creepmeters in the areaowever the PS-InSAR time series exhibits a nonlinear
rend that needs to be further investigatedmdashthis is seenn the Ryder and Burgmann [20] InSAR stack as well In-AR is more sensitive to the vertical component of the de-
ormation than to the lateral component because of theteep look angles of the instrument GPS stations in therea fail to characterize vertical deformation with compa-able accuracy and are often not used in modeling anyertical motion associated with creep across the faultence vertical deformation could be one source of this ob-
erved nonlinear trend that needs further investigation
rapped Each slice represents an interferogram in the PS-InSARero The plane of the fault trace is also shown Right the cost ofof edge from the fault as shown
ig 7 Average LOS displacement rate in millmeters per yearstimated by using the stepwise 3D unwrapping algorithm [10]hown on a geolocated latitudendashlongitude grid
is unwet to zenter
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
rtNrfatorcFi
5Waltldt
sssvttfmcwtaaisf
tca
kdisw
AWpv
a
aa
fo
T
wrap
FbrmEt
AWtl
ii
Wfgeotua
FmMah
A P Shanker and H Zebker Vol 27 No 3 March 2010J Opt Soc Am A 611
Most creep studies in the Central San Andreas Faultegion use a constant velocity to model the slip rate dis-ribution at depth [2024] However other studies such asadeau and McEvilly [25] suggest that the recurrence of
epeating microearthquakes affects the slip rate along theault significantly Comparing our results with [25] welso observe a similar pulselike variation in slip rate inhe time period 1992ndash1996 in Fig 8 If the results fromur phase unwrapping method are plausible the algo-ithm forms the basis for a new method to monitor thehanging strain field over the entire Central San Andreasault areamdashspaceborne PS-InSAR is capable of measur-
ng very small displacements over a large area
DISCUSSION AND CONCLUSIONSe propose a new phase unwrapping algorithm that is as
ccurate but more flexible than previously known formu-ations Our implementation exploits the TUM propertyo allow us to solve large-scale phase unwrapping prob-ems by using LP solvers Tests using simulated and realata sets demonstrate the validity of our new formula-ion
We implemented the edgelist algorithm by using theimplex modules of the CPLEX software [26] In the ab-ence of specialized constraints illustrated in the casetudy above the edgelist formulation is computationallyery inefficient compared with the original MCF formula-ion and requires more computer resources The quality ofhe solution however matches that of the original MCFormulation and permits a simplified representation ofultidimensional phase unwrapping problem and the in-
orporation of data from external sources such as GPShere available to better constrain the unwrapped solu-
ion Further research needs to be conducted on the scal-bility of the algorithm to unwrapping denser data setsnd using a denser set of GPS observations The possibil-ty of optimizing LP solvers to handle constraint matricesimilar to the edgelist formulation also needs to beurther explored Costantini et al [27] recently reported
ig 8 Cumulative displacement time series for the regionsarked out in Fig 5(a) The time series from the creepmeters atelendy ranch (solid curve) and Slack canyon (dotted curve) are
lso included for comparison The creepmeter measurementsave been projected onto the radar LOS direction for comparison
he independent development of similar edge-basedommercial unwrapping techniques for time series InSARnalysisThe edgelist method provides a way to apply a priori
nowledge to improve our ability to unwrap challengingata sets such as our case study above This makes it andeal candidate for improving the temporal unwrappingtep of the multitemporal PS-InSAR and SBAS phase un-rapping [2023]
PPENDIX Ae use the following properties of TUMs to prove that our
hase unwrapping problem can be solved by using a real-alued relaxation [16]
(a) The incidence matrix Gn of a directed graph G isTUM(b) If matrix A is a TUM and I is an identity matrix of
ppropriate dimensions then the following matrices arelso TUMs minusA AT A I A minusA [28](c) Let A be an mn integral TUM matrix Then the
ollowing polyhedrons are integral for any vectors b and uf integers
x RnA middot x b
he constraints expressed in Eq (9) can be rewritten as
GnI middot n
K = b A1
here n represents the node potential variables and Kepresents the flow variables From properties (a) and (b)bove A= Gn I is a TUM The equality (A1) can be ex-ressed as
A
minus A middot n
K b
minus b A2
rom property (c) we prove that our integer program cane solved by using a real-value relaxation as all the cor-esponding extreme points are integers A similar argu-ent can be provided to show that the LP relaxation inqs (8)ndash(11) also has a TUM constraint matrix and solves
he original integer program exactly
PPENDIX Be will use the following properties of matrices to prove
hat imposing additional constraints could render a prob-em with the original MCF formulation unsolvable
(a) A matrix with all elements in 0+1minus1 is a TUM ift contains no more than one +1 and no more than one minus1n each column
e provide the proof only for a regularly sampled 2D caseor ease of understanding It can be easily extended to theeneral case In case of the original MCF formulation [1]ach edge is traversed once in the clockwise direction andnce in the anticlockwise direction If we were to consis-ently traverse every loop in the same direction the col-mn of the constraint matrix corresponding to a flow vari-ble K would have exactly one +1 and one minus1 in the rows
i
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
capps
wtuMr
AWUtcss
R
1
1
1
11
1
1
1
1
1
2
2
2
2
2
2
2
2
2
612 J Opt Soc Am AVol 27 No 3 March 2010 A P Shanker and H Zebker
orresponding to the two loops of which it is a part Suchconstraint matrix satisfies (a) and hence is a TUM Sup-ose that there were two GPS measurements available atoints p and q Then the corresponding additional con-traint could be written as
P
Q
Kij = Npq minus P
Q i minus j
2 B1
here i j is an edge along any chosen path from point po point q and Npq is the expected number of cycles innwrapped phase Adding this constraint to the originalCF formulation would violate properties (a) and the
esulting constraint matrix would not be a TUM
CKNOWLEDGMENTSe thank Isabelle Ryder and Roland Burgmann fromniversity of California Berkeley for providing us with
he initial data for PS-InSAR analysis and for useful dis-ussions on the phase unwrapping model This work wasupported by the NASA Earth and Space Science Fellow-hip (NESSF)
EFERENCES1 M Costantini ldquoA novel phase unwrapping method based
on network programmingrdquo IEEE Trans Geosci RemoteSens 36 813ndash821 (1998)
2 B R Hunt ldquoMatrix formulation of the reconstruction ofphase values from phase differencesrdquo J Opt Soc Am 69393ndash399 (1979)
3 R M Goldstein H A Zebker and C L Werner ldquoSatelliteradar interferometry two-dimensional phase unwrappingrdquoRadio Sci 23 713ndash720 (1988)
4 N H Ching D Rosenfeld and M Braun ldquoTwo-dimensional phase unwrapping using a minimum spanningtree algorithmrdquo IEEE Trans Image Process 1 355ndash365(1992)
5 D C Ghiglia and L A Romero ldquoMinimum Lp-norm two-dimensional phase unwrappingrdquo J Opt Soc Am A 131999ndash2013 (1996)
6 M D Pritt ldquoPhase unwrapping by means of multigridtechniques for interferometric SARrdquo IEEE Trans GeosciRemote Sens 34 728ndash738 (1996)
7 T J Flynn ldquoTwo-dimensional phase unwrapping withminimum weighted discontinuityrdquo J Opt Soc Am A 142692ndash2701 (1997)
8 H A Zebker and Y Lu ldquoPhase unwrapping algorithms forradar interferometry residue-cut least squares andsynthesis algorithmsrdquo J Opt Soc Am A 15 586ndash598(1998)
9 C W Chen and H A Zebker ldquoNetwork approaches totwo-dimensional phase unwrapping intractability and twonew algorithmsrdquo J Opt Soc Am A 17 401ndash414 (2000)
0 A Hooper and H Zebker ldquoPhase unwrapping threedimensions with applications to InSAR time seriesrdquo JOpt Soc Am A 24 2737ndash2747 (2007)
1 D C Ghiglia and L A Romero ldquoRobust two-dimensional
weighted and unweighted phase unwrapping that uses fast
transforms and iterative methodsrdquo J Opt Soc Am A 11107ndash117 (1994)
2 M Costantini and P A Rosen ldquoA generalized phaseunwrapping approach for sparse datardquo in IEEE 1999International Geoscience and Remote Sensing Symposium1999 IGARSSrsquo99 (IEEE 1999) Vol 1 pp 267ndash269
3 C Barber and H Hudanpaa Qhull httpwwwqhull org4 C W Chen and H A Zebker ldquoTwo-dimensional phase
unwrapping with use of statistical models for cost functionsin nonlinear optimizationrdquo J Opt Soc Am A 18 338ndash351(2001)
5 R K Ahuja D S Hochbaum and J B Orlin ldquoSolving theconvex cost integer dual network flow problemrdquo in IntegerProgramming and Combinatorial Optimization GCornueacutejols R E Burkard and G J Woeginger eds Vol1610 of Lecture Notes in Computer Science (Springer1999) pp 31ndash44
6 A J Hoffman and J B Kruskal ldquoIntegral boundary pointsof convex polyhedralrdquo in Linear Inequalities and RelatedSystems H W Tucker and A W kuhn eds (PrincetonUniv Press 1965) pp 223ndash246
7 C W Chen and H A Zebker ldquoPhase unwrapping for largeSAR interferograms statistical segmentation andgeneralized network modelsrdquo IEEE Trans Geosci RemoteSens 40 1709ndash1719 (2002)
8 G Zhou and M Gen ldquoGenetic algorithm approach onmulti-criteria minimum spanning tree problemrdquo Eur JOper Res 114 141ndash152 (1997)
9 J D Knowles and D W Corne ldquoA comparison of encodingsand algorithms for multiobjective minimum spanning treeproblemsrdquo in Proceedings of the 2001 Congress onEvolutionary Computation 2001 (IEEE 2001) Vol 1 pp544ndash551
0 I Ryder and R Burgmann ldquoSpatial variations in slipdeficit on the Central San Andreas Fault from InSARrdquoGeophys J Int 175 837ndash852 (2008)
1 P Shanker and H Zebker ldquoPersistent scatterer selectionusing maximum likelihood estimationrdquo Geophys Res Lett34 L22301 (2007)
2 C Colesanti A Ferretti F Novali C Prati and F RoccaldquoSAR monitoring of progressive and seasonal grounddeformation using the permanent scatterers techniquerdquoIEEE Trans Geosci Remote Sens 41 1685ndash1701 (2003)
3 A Pepe and R Lanari ldquoOn the extension of minimum costflow algorithm for phase unwrapping of multitemporaldifferential SAR interferogramsrdquo IEEE Trans GeosciRemote Sens 44 2374ndash2383 (2006)
4 F Rolandone R Buumlrgmann D C Agnew I A JohansonD C Templeton M A drsquoAlessio S J Titus C DeMetsand B Tikoff ldquoAseismic slip and fault-normal strain alongthe creeping segment of the San Andreas faultrdquo GeophysRes Lett 35 L034437 (2007)
5 R M Nadeau and T V McEvilly ldquoPeriodic pulsing ofcharacteristic micro earthquakes on San Andreas FaultrdquoScience 303 220ndash222 (2004)
7 M Costantini M Costantini F Malvarosa and F MinatildquoA general formulation for robust integration of finitedifferences and phase unwrapping on sparsemultidimensional domainsrdquo presented at ESA FRINGE2009 Workshop Frascati Italy Nov 30ndashDec 4 2009
8 A F Veinott and G B Dantzig ldquoIntegral extreme pointsrdquoSIAM Rev 10 371ndash372 (1968)
