GEOSTATISTICAL ANALYSIS AND MITIGATION OF ATMOSPHERIC PHASE SCREENS INKU-BAND TERRESTRIAL RADAR INTERFEROMETRY

Simone Baffelli1 Othmar Frey1,2 Irena Hajnsek1,3

(1) Chair of Earth Observation and Remote Sensing, ETH Zurich, Switzerland(2)Gamma Remote Sensing AG, Gumligen, Switzerland

(3)Microwaves and Radar Institute, German Aerospace Center - DLR, Oberpfaffenhofen, Germany

ABSTRACT

A geostatistical analysis of the atmospheric phase screen(APS) affecting Ku-Band terrestrial radar interferometric(TRI) observations of a fast-flowing alpine glacier is madeassuming a separable spatio-temporal covariance structure.Such a structure facilitates the mitigation of APS: the at-mospheric phase affecting individual interferograms can beextrapolated form a set of persistent scatterers (PS) usingregression-Kriging. After removing this estimate the residualAPS is only correlated in time; its effect on surface dis-placement estimation is mitigated with a generalized leastsquares (GLS) inversion employing an estimate of the tem-poral covariance of the APS. The applicability of a separablecovariance structure and the performance of the APS correc-tion method are assessed on a TRI timeseries of Bisgletscher,a glacier in the southwestern Swiss Alps.

1. INTRODUCTION

Terrestrial Radar Interferometry (TRI) can be used to monitordisplacements in natural terrain[1], [2] at day and night, re-gardless of weather. Typical applications are the observationof instable slopes[3], rock glaciers and glaciers[4]–[6]. Ter-restrial radar interferometers are very suitable for those appli-cations requiring zero-baseline differential interferometry asthey provide great flexibility in the acquisition geometry andtiming.Differential interferometry is a coherent method: the phase ofa target in a radar acquisition is proportional to its distance tothe radar; when acquisitions at different times are combinedin an interferogram, the phase difference is is proportional tothe displacement of the target. However, the phase is also af-fected by changes in the refraction index of the propagationmedium, which are usually due to variations in the spatialdistribution of atmospheric water vapor content. This signalis superimposed to the displacement phase and is of compara-ble magnitude, reducing the accuracy of estimated displace-ments. However APS and displacement have different spatio-temporal behaviors, permitting statistical inferences regard-ing their relative contributions, particularly if multiple inter-

ferograms are used for the estimation[7].

2. METHODS

2.1. Signal Model

The problem of estimating terrain displacements given a set ofM interferograms each consisting of P pixels can be writtenas[8]:

z =4π

λTv + εatm + εdecorr (1)

where z is a P M vector of unwrapped, referenced interfero-metric phases obtained from a vector y of P N SLC phasesordered by time of acquisition. The two vectors are relatedthrough the incidence matrix A[8]:

z = Ay. (2)

where A indicates that the incidence matrix is a block diago-nal matrix built of P identical M ×N blocks. v is a vector ofP S, S ≤ M velocities at S times and T is a block-diagonalmatrix with blocks of size M × S. Each block in T, cor-responding to a pixel in the interferogram stack consists ofthe same matrix: its i-th column is the vector of temporalbaselines for the M/S interferograms in z for which a con-stant displacement velocity vi is assumed for that pixel. As anexample, assuming a single velocity v for each pixel for theentire stack duration (S = M ) corresponds to interferogramstacking[9].Finally, εatm is the atmospheric phase (APS) and εdecorr rep-resents other sources of noise such as thermal noise and tem-poral decorrelation. Both these contributions are assumed tobe zero-mean Gaussian random vectors with covariance ma-trix Σ = Σatm + Σdecorr.The best linear unbiased (BLUE) estimate of the velocity vec-tor v from the interferogram stack z is the generalized leastsquares solution:

v =(TTΣ−1T

)−1TTΣ−1z. (3)

However the covariance matrix Σ is usually not known; evenif it were its large size would make the solution of (3) imprac-tical.

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2.2. Covariance Model and Inversion Strategy

Instead of directly computing (1), a two-step inversionscheme can be employed.To do so, εatm the APS affecting each SLC acquisition in yis decomposed in two terms: a stratified APS contributionεy,atm,strat and a turbulent APS εy,atm,turb. The formeris used to describe atmospheric layering[4], [10]; it is de-scribed with a multiple regression model using functions ofthe pixel’s locations. The turbulent APS is assumed to be azero mean, second order spatially and temporally stationaryGaussian random process with separable spatio-temporal co-variance structure[11]: its covariance can be written as theKronecker product of spatial Σatm,s and temporal covari-ance matrices Σatm,t. The covariance of the differential APSobserved in the interferogram vector is then [7]:

Σatm = A (Σatm,s ⊗ Σatm,t) AT

= Σatm,s ⊗(AΣatm,tA

T) (4)

In spaceborne InSAR Σatm,t = I is usually assume: suc-cessive passes are separated by several days or longer, whichensures uncorrelated atmospheric turbulence. This assump-tion does not necessarily apply to TRI observations, wherethe repeat time is much shorter. Possible correlations in timeare allowed through a non-diagonal Σatm,t.Spatio-temporal separability implies that spatial and tempo-ral covariance structures are independent of each other andare respectively functions of spatial and temporal separations:the spatial covariance structure cannot change as a function oftime[7]. The only way time can affect the spatial covarianceis by changing its overall magnitude as a function of the tem-poral baseline. Thus the APS in two interferograms with dif-ferent spatial baselines will have the same spatial correlationstructure; only their phase variance can differ.The decorrelation process Σdecorr is assumed to be spatiallyuncorrelated and can estimated using the procedure describedin [7].The above model is used to estimate the solution of equa-tion (1) in the following manner:

• Identify a set of persistent scatterers (PS) where v = 0.The selection of the PS usually proves to be unprob-lematic in the case of alpine glaciers: The locationsaffected by displacement are normally known a prioriand can be easily masked out; other deformation pro-cesses outside of that region happen at widely differenttimescales. With these PS a second interferogram stackis computed which is virtually free of deformation anddecorrelation contributions.

• Estimate Σatm,s and Σatm,t using variogram analy-sis on the PS stack. Σatm,s is obtained by averag-ing spatial variograms computed on individual interfer-ograms, assuming spatio-temporal separability. Simi-larly, Σatm,t is computed as the square average of un-

wrapped interferometric phase differences binned bytemporal baseline.

• Extrapolate the phases at the PS to the locations ofthe interferogram z using regression-Kriging (RK)[12]with a model of APS stratification and the estimatedspatial variogram. This gives a 2D prediction of thespatially correlated APS which can be removed fromthe interferogram. The residual interferogram vectorzres is then approximatively spatially uncorrelated.

• Compute the Generalized Least Squares (GLS) esti-mate (3) of v using zres instead of z. The GLS inversecan be computed pixel by pixel: the APS correctionusing RK makes zres approximately uncorrelated inspace.

3. DATA

The methods described above are tested with a dataset ac-quired using KAPRI (Ku-Band Advanced Polarimetric RadarInterferometer)[13], a polarimetric extension of the GPRI[14]real aperture TRI operating at 17.2 GHz.The data was acquired during the summer months of 2015 inthe Mattertal, Southwestern Swiss Alps, canton of Valais dur-ing a campaign to monitor Bisgletscher, a steep, fast flowing–up to 2 meters/day in its fastest parts– glacier near the vil-lage of Randa.In the course of the campaign a timeseries of SLC imagesspaced 2:30 minutes apart was acquired; from which 10 datesare chosen randomly as reference dates. The random sam-pling ensures that SLCs representing different atmosphericconditions are analyzed.Variogram analysis is performed on the set of all interfero-grams with a temporal baseline less than one hour from thereferences; while GLS inversion to derive velocity maps is ap-plied on all the interferometric pairs with a maximum tempo-ral baseline of 5 minutes. The threshold is chosen to minimizephase wraps and temporal decorrelationn. A model assuminga constant velocity over the course of 30 minutes (S = 15)was chosen for the GLS inversion.

4. RESULTS

Figure 1 is used to test the separability of the APS covarianceby computing experimental variograms using interferogramsfrom different dates and increasing temporal baselines; theseare encoded in the point’s color in the plot; the mean var-iogram is shown as a dashed line while the variance in thespatial semivariance across all interferograms is displayed asa gray ribbon. The blue line indicates a fitted exponential var-iogram model that is employed to extrapolate the APS ob-served at the PS using regression-Kriging (RK), which pre-dicts the APS considering the spatial statistics and the APS

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20

0 1000 2000 3000 4000 5000

Spatial distance [m]

Sem

ivar

ianc

e[ ra

d2 ]

1000

2000

3000

Temporal lag [s]

Fig. 1. Experimental spatial variogram; each dot shows thesemivariance for an interferogram at a given spatial separa-tion; the temporal lag is given by the dot’s color; the dashedblack line shows the variogram obtained by averaging thesemivariance over all interferograms while the gray surfacedisplays the standard deviation. The blue line is a fitted expo-nential variogram model.

stratification model simultaneously. The performance of RKis evaluated by cross-validation: the RK prediction of the APSis computed for each interferogram to derive the residual in-terferogram zres. A second set of PS located on stable areasin proximity of the glacier, where v = 0 is expected, is usedto extract the residual phases after RK, which are convertedinto line of sight velocities; they are shown in the form ofresidual velocity histograms in Figure 2.Finally, an example displacement map obtained using RK cor-rection and GLS inversion is displayed in Figure 2, overlaidto a topographic map.

5. DISCUSSION

Separability seems a reasonable assumption considering themoderate difference between variograms computed fromindividual interferograms, as shown by the gray ribbon ofFigure 1. Assuming the covariance to be separable, the spa-tial variograms must be the same up to a scaling factor andoffset for all temporal baselines.However, variations in variogram’s shape are still observedin the plot; they may relate to non-stationarity in the atmo-spheric turbulence caused by varying weather conditions and

stacked: APS+Coh stacked: no

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

0.0

2.5

5.0

7.5

10.0

Residual Velocity [m/day]N

orm

aliz

ed F

requ

ency

of O

ccur

ence

method

kriged

lm

unprocessed

Distribution of velocity on stable areas, binwidth: 0.025 m/day

Fig. 2. Cross-validation of APS correction and inversion per-formance: the estimated velocities on a set of stable locations,where no displacements are expected (v = 0) are extractedand their distribution plotted as an histogram. The columnnamed stacked: no on the right shows the solutions obtainedby applying APS correction methods (kriged: RK, lm: Strati-fied model only, unprocessed: no correction) to the interfero-grams and converting their phases into velocities; the columnAPS+Coh on the left display the distribution of estimates ob-tained with the same APS correction methods followed byGLS inversion.

amount of solar radiation at different dates and times of theday.A significant reduction in phase variance compared to theunprocessed interferograms (label method: unprocessed) isobserved when the RK APS predictions are subtracted (labelmethod: kriged). Removing only the stratified APS predictedby the multiple regression model (label method: lm) does notsubstantially reduce phase variance. This observation wastested quantitatively by computing the statistics of R2 overall runs of the multiple regression APS model, giving an aver-ageR2 of 0.2, suggesting that most of the phase variance is tobe attributed to turbulence of the APS and not to atmosphericstratification.When regression-Kriging is combined with GLS estima-tion of the velocities an additional decrease in varianceis achieved, as shown in the left column named stacked:APS+Coh of Figure 2.The estimated velocities are in agreement with the currentknowledge on the Bisgletscher, where time-lapse camera ob-

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IcefallTongue

Reference

Upper Icefall

Rock Below

Rock Left

Rock Right Wisslaub

0.000 0.222 0.444 0.667 0.889 1.111 1.333 1.556 1.778 2.000

LOS Displacement [m/day]

Fig. 3. Example displacement map obtained using spatialAPS correction with RK plus GLS inversion weighted bythe temporal covariance of the APS. The outline of the Bis-gletscher is shown in black. Areas of low coherence or lo-cated in radar shadows are masked. The times correspond tothe first and last acquisition used in the stack.

servations gave estimated flow velocities of up to 2 m/day inthe fastest parts.

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