ON THE USE OF ANISOTROPIC COVARIANCE MODELS IN ESTIMATINGATMOSPHERIC DINSAR CONTRIBUTIONS

A. Refice, A. Belmonte, F. Bovenga, and G. Pasquariello

ISSIA-CNR, Via Amendola 122/d, 70126 Bari, Italy; E-mail: refice@ba.issia.cnr.it

ABSTRACT

When studying geophysical processes through Differen-tial SAR Interferometry (DInSAR), it is often necessaryto estimate and subtract signals due to atmospheric in-homogeneities. To this end, stochastic models are oftenused to describe atmospheric phase delays in DInSARdata. As a first approximation, these can be modelledas isotropic, though this is a simplification, because SARinterferograms often exhibit anisotropic atmospheric sig-nals. In view of this, it is increasingly advocated the useof anisotropic models for atmospheric phase estimation.However, anisotropic models lead to increased computa-tional complexity in estimating the correlation functionparameters with respect to the isotropic case. Moreover,performances can degrade when use is made of interfero-grams with only a few sparse points usable for computa-tions, such as in the case of persistent scatterers interfer-ometry (PSI) applications, especially when this estima-tion has to be done in an automated way on several tens ofinterferograms. In the present work we critically analysethe main aspects connected with the use of anisotropicmodels for DInSAR atmospheric delays, and we evalu-ate the advantage given by anisotropic modeling of at-mospheric phase in the case of sparse grids of points.Through analysis of APS simulated data, we observe thata slight increase in the performances of reconstruction ap-proaches can be obtained when sufficient sampling den-sities are available; based on these results, some recom-mendations for operational atmospheric phase estimationprocedures are proposed.

Key words: InSAR; Atmospheric Phase Screen; vari-ogram; anisotropy.

1. INTRODUCTION

Anisotropy is often considered important in modeling2D stochastic fields such as atmospheric phase contribu-tions in SAR interferograms. In fact, InSAR atmosphericphase screen (APS) fields published in the literature of-ten show evident anisotropy (see e.g. [1], or works suchas [2, 3, 4, 5, 6]).

The concept of anisotropy is usually quantified by mean-ing that estimates of spatial variance, such as those ob-tained through the variogram function, exhibit signifi-cant differences when estimated along different direc-tions. Anisotropy can be easily accounted for in theo-retical models of spatial variance. The most commonand simple form of anisotropic behavior is that referredto as geometric anisotropy, which can be described as avariability of some of the model parameters with the es-timation direction. Extending a given model of spatialvariance to a form which includes geometric anisotropyconsists basically in adding a pair of parameters to theisotropic model, controlling the amount and direction ofthe anisotropy.

Modeling of APS in SAR interferograms is important inquantitative applications, such as monitoring of surfacedisplacements, in which the APS represents a nuisance.The most advanced of such applications are those allow-ing the monitoring of temporal trends of displacementsthrough analysis of stable targets in series of SAR in-terferograms. In such applications, known as persistentscatterers interferometry (PSI), APS contributions are es-timated by filtering over selected pixels containing stableradar targets, and then a continuous APS field is interpo-lated over the entire investigated area, to remove it fromthe data and so to allow more PS targets to be recognized.This interpolation is usually performed through advancedmethods such as kriging, which benefit from the defini-tion of analytical models for the spatial variance of thefield to be interpolated (a concept known as model-basedgeostatistics [7]).

Kriging of APS fields in PSI applications is usually per-formed by assuming a certain analytical stochastic modelfor the fields, then estimating the model parameters fromthe APS data through model fitting, and finally by per-forming the interpolation of the APS field over the entiregrid using the analytical model with the fitted parameters.

Using mode accurate models to describe spatial variationof APS fields seems thus desirable to improve the per-formances of PSI systems. In this paper we review someof the concepts involved in the anisotropic modeling ofAPS fields, focusing specifically on their use in PSI ap-plications. We support our considerations by showing ex-amples of real and simulated interferograms.

_____________________________________________________ Proc. ‘Fringe 2009 Workshop’, Frascati, Italy, 30 November – 4 December 2009 (ESA SP-677, March 2010)

2. VARIOGRAM ANISOTROPY

Given a regionalized variablez (r), and defined the(isotropic) variogram modelV (h), as a function of the(scalar) lagh throughV (h) = E (z(r)− z(r+ h))2,the corresponding anisotropic version can be constructedby letting h become a vector valueh, and allowing fora generic elliptical deformation and rotation of the axesalong whichh is measured, according to the followingformulae:

V ′ (h) = V(

S ·R · h)

(1)

where

S =

[√δ 0

0 1/√δ

]

;R =

[

cos θ sin θ− sin θ cos θ

]

. (2)

The parametersδ and θ quantify the anisotropy of thefield, i.e., as appears from (2), model the affine trans-formation –an axis deformation followed by a rotation–necessary to bring theV ′ variogram to the isotropic formV .

Other forms of anisotropic variogram models can be con-ceived with more general forms of parameter variabilitywith estimation direction, such as, for instance, multi-plicative or additive models, i.e. products or sums of one-dimensional models valid along orthogonal directions.This behaviour is often indicated as zonal anisotropy, todistinguish it from the geometric anisotropy paradigmstated above. However, using such models in practicalapplications requires particular care, since they may vi-olate basic assumptions such as (second-order) stationar-ity [8] or pose problems in the kriging process itself [9].In this work, we restrict to models extended through geo-metric anisotropy as in (1) and (2).

3. MODELING ATMOSPHERIC PHASESCREEN INSAR SIGNALS

Hanssen [1] describes thoroughly the various compo-nents of the atmospheric phase contributions which canbe expected in a SAR interferogram. Among these, themost important for the study of ground movements is thecomponent due to spatial inhomogeneities in the watervapour content of the air. Hanssen’s review models theAPS water vapour statistics through multi-fractal func-tions, relying on the Kolmogorov turbulence theory. Thetheory predicts a power-law behaviour for the spatial vari-ation of atmospheric components, with different expo-nents for different scales.

The multi-fractal paradigm has theoretical appeal, andis receiving attention as a viable tool for APS model-ing [10], also in view of the fact that it solves some prob-lems connected with pure power-law functions especiallyconcerning finite asymptotic values at large lags [8]. Onthe other hand, if we focus on the reconstruction (i.e. in-terpolation from scattered data) problem, for which the

kriging approach constitutes a classic solution, then long-range behavior of covariance model functions is less im-portant than its asymptotic values near the origin [11].

For these reasons, in practical applications APS datais often modeled through a number of “well-behaved”model functions - such as the Matern [12], Gauss (a spe-cial case of the Matern model) or Bessel functions [13].The Matern model, with the variogram and the corre-sponding correlation function, can be written as follows:

V (h) = N + S ·[

1− 21−α

Γ(α)·(

2√αh

L

)α

· Kα

(

2√αh

L

)]

;

(3a)

C(h) = C(0)− V (h); (3b)

whereKα is the modified Bessel function of the secondkind of orderα. N , S,L andα are the model parameters.The Gaussian model is obtained forα = 2, and is thusbased on only three parameters,N , S, andL. Anotheruseful model is the Bessel:

V (h) = N + S ·[

1− 2αΓ (α+ 1)

(

h

L

)−α

· Jα(

h

L

)]

,

(4)

whereJα is the Bessel function of the first kind of orderα.

The above models can be extended to the anisotropicform using (1) and (2), As mentioned, in the anisotropicversions two additional parameters,δ andθ, are neededto specify completely the statistics of the field.

4. RECONSTRUCTION OF 2D APS FIELDSFROM SPARSE POINT SAMPLES

As mentioned in the introduction, real APS fields oftenexhibit evidently anisotropic characteristics, observationwhich has led to consider the use of anisotropic structurefunctions as more suitable descriptive tools than isotropicmodels [13]. As an example, we show in fig. 1(a) and(e) two ERS-1/2 tandem differential interferograms. To-pographic contributions have been removed using DEMdata from the SRTM mission. The SAR data are acquiredover a flat region in southern Italy, thus residual contribu-tions from different stratification of atmospheric layers,which could lead to components proportional to topog-raphy, are likely to be negligible. The short revisit time(1 day) also ensures that displacements between the twoacquisitions can be ruled out. It can therefore be assumedthat the whole interferometric signal is that relative to tro-pospheric inhomogeneities. Data have been multilookedwith a factor of 2×10 (range×azimuth), and the differ-ential phase has been unwrapped through a state-of-the-art minimum cost flow algorithm (as implemented in the

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Figure 1. ERS-1/2 unwrapped tandem differential interferograms and corresponding variograms estimated on differentnumbers of points. (a);(e) interferograms acquired on July 23-24, 1995, and March 24-25, 1996, respectively; color isproportional to absolute phase; multilook factor is 2×10 (range×azimuth). (b);(f) directional variograms of the inter-ferogram in (a);(e), respectively; (c)-(d) 2-dimensional variogram of the interferogram in (a), estimated over pixels withcoherence above 0.85 and 0.87, respectively. (g)-(h) same as (c)-(d), but for the interferogram in (e), with thresholds of0.9 and 0.93, respectively.

SNAPHU software [14]).

In fig. 1, directional variograms, estimated by restrictingthe classical scalar estimator to discrete direction inter-vals, as well as omnidirectional variograms, are shown inpanels (b) and (f), referring to the interferograms in (a)and (e), respectively. Two-dimensional variograms arealso shown: panels (c)-(d) refer to the interferogram in(a), panels (g)-(h) to that in (e). The 2D variograms havebeen obtained from the two-dimensional version of theclassical estimator:

2V (h) =E[

(z(r+ h)− z(r))2]

=

∑

(r′−r)∈Nh[z(r′)− z(r)]

2

|Nh|,

(5)

whereNh = N(hx,hy) = {((x, y), (x′, y′)) |(x′ − x) ≃hx, (y

′ − y) ≃ hy}, the≃ sign meaning equalilty withina certain tolerance interval, i.e. in practice distance vec-tors are binned in discrete intervals.|Nh| is the numberof elements inNh. The variograms in fig. 1 have beenestimated from a subset of the pixels in the raster inter-ferogram, selected according to their interferometric co-herence (γ) value. In detail, panel (c) has been derivedusing points with coherence above 0.85, correspondingto an average density of about 6.5%, while in panel (d)points withγ > 0.87 have been used, corresponding toabout 0.5% of the pixels. Analogously, the variogram inpanel (g) has been estimated on points withγ > 0.9, cor-responding to 11.8% of the pixels, while that in panel (h)from pixels withγ > 0.93 (∼ 4.5%).

These examples show how decreasing point densitycan decrease variogram estimation performance. Forthe interferogram of fig. 1(a), which has a pronouncedanisotropy, decreasing the point sampling by more than

an order of magnitude does not decrease significantly theestimation accuracy, since the general shape of the vari-ogram is preserved, while in the case of the interferogramin fig. 1(d) the estimate with less points appears muchharder to interpret, with spurious peaks which may hin-der model fitting.

It can obviously be expected that estimation procedureswill perform worse when fitting a given variogram modelobtained from less sample points. Estimating parametersfor anisotropic versions of a certain model exhibits highercomplexity with respect to the corresponding procedureusing the isotropic model version (essentially given bythe two additional model parameters). This may balancethe fact that an anisotropic model could describe betterthe APS field than an isotropic model does, so that it isnot a trivial point to compare, given the same point den-sity, estimation procedures to fit experimental variogramsto anisotropic and isotropic model versions. This issuewill eventually reflect on the quality of APS fields inter-polated over the entire interferogram grid, so that betterparameters estimation should in principle lead to betterreconstruction quality.

In the following, we investigate on these points, propos-ing some observations about the performance of estima-tion and reconstruction algorithms, based on simulations.

5. SIMULATIONS

Using simulated data allows to keep control of all thevariables in complex problems. We generated series ofrandom surfaces with assigned statistical characteristics;we performed model fitting from samples of these sur-faces taken at random with decreasing density, and then

surface reconstruction based on the fitted models throughkriging.

5.1. Surface generation

Series of random surface were generated with assignedcovariance functions. Conditional simulation of surfaceswith given covariance models can be achieved throughalgorithmic tools such as Cholesky or eigendecomposi-tion [15], assuming a Gaussian distribution. Other sim-ulation methods for fractal surfaces rely on spectral fil-tering, based on the Wiener-Khintchine theorem. In thepresent work, series of surfaces of size 100×100 pix-els were generated through spectral methods, as sug-gested in [1], with varying degrees of anisotropy; in de-tail, the results described in the following refer to seriesof isotropic surfaces, anisotropic surfaces withδ = 2 andrandom orientationsθ uniformly distributed in the inter-val [−π, π), and another series of surfaces withδ = 10and again random orientation angles. In view of the con-siderations in sect. 3 about model selection, a referenceset of parameters has been estimated using variogramsobtained from all samples of the simulated fields, andthen through LMS fit. To investigate the robustness of fitmethods to the number of samples available, variogramsparameters have then been fitted to experimental vari-ograms obtained from reduced sets of surface samplesselected at random positions, with sampling spatial den-sities ranging from 10% to 0.5% of the surface pixels.

Note that, for ERS sampling conditions at full resolution,a density of points of 1% corresponds to about 100 pixelsper km2. In the case of PSI applications, such a den-sity of PS per km2 corresponds to conditions typical ofurban environments [16], whereas on scarcely urbanisedareas, much lower values are typically observed [17].Higher values of PS densities, up to several percents,can be expected on urban areas from higher resolutionSAR data such as in the case of TerraSAR-X [18] orCOSMO/SkyMed. This is a huge increase in spatial den-sity, considering that for a full-resolution TSX or CSKstripmap image with ground spacing at 3×3 m2, 1% sam-pling corresponds to about 1000 pixels per km2.

Variogram model fitting in general can be a hard problem.Experienced geostatisticians use a series of practical pro-cedures and tools to infer the best statistical model whichfits experimental data. When faced with data exhibitinganisotropy, such empirical methods are an aid to infersuitable values for the anisotropy parameters and thus fi-nally aid the model fitting procedure. However, withinthe specific field of PSI applications, APS raster field in-terpolation has to be performed tipically for each acqui-sition of a time series of several tens of images. There-fore, the emphasis is posed over reconstruction methodswhich can work in an automated way, with as little oper-ator interventions as possible, to avoid excessive process-ing times. In this respect, automated parameter estima-tion methods should be viewed as the most useful. Thesimplest and best known of such methods is of course

least mean square (LMS) curve fitting. The method re-lies on the assumption that errors are Gaussian distributedaround the “exact” values and independent. Weights canbe used to guide the fit by increasing the importance ofsome points with respect to others. For the specific caseof variogram fitting, typical weighting schemes proposedin the literature are e.g. those inversely proportional tothe lag distance or its squared value, or proportional tothe bin population [19].

5.2. Reconstruction of simulated surfaces

In fig. 2, we show a pair of examples of model fit andreconstruction of simulated surfaces. It can be seen thatin the first example (panels a and b, original surface sim-ulated withδ = 10), the reconstruction at 5% sampling(panel a) is more faithful using the 2D model than theomnidirectional one, since in this case the high samplingdensity allows a good fit of the 2D variogram: the recon-structed surface better reproduces the anisotropic struc-ture of the original. In lower sampling conditions (2%,panel b) the surface reconstructed through the 2D modeldoes not differ too much from the one obtained by usingthe 1D model. In this case, then, using the more compli-cated 2D model seems unjustified. In the second exam-ple (panels c and d,δ = 2), which refers to a slightlyanisotropic original surface, with a 10% sampling rate(panel c) the reconstruction is good with both 1D and 2Dmodels, which give closely comparable results. If sam-pling is reduced to 3%, instead (panel d), the low numberof available points causes the 2D model to give unreliableresults, especially for the anisotropyδ andθ parameters,which results in a reconstruction which is much worsethan that obtained through the 1D model. In this case,then, robustness is more important than model accuracyfor the reconstruction purpose.

The degradation of model fit parameters as sampling den-sity gets lower can be appreciated by looking at the valuesin tab. 1. Here, the parameters for the surface analysed infig. 2(a) and (b) are shown, obtained through WLMS fitstarting from isotropic conditions, as sampling rate de-creases from 100% to 0.5%. It can be noticed the de-crease in accuracy, especially for the anisotropy param-eters, and the corresponding increase in RMSE, as sam-pling rate decreases.

This behaviour can be understood at least partially bymaking the following observations.

1. “Curse of dimensionality”: in practice, estimatingthe experimental variogram in a 2D domain requiresmore sampling points than using a 1D model, sincethe number of required bins increases exponentiallywhen adopting higher-dimensional models. Conse-quently, it is expected that, for a certain samplingdensity, estimates of model parameters from lower-dimensional models are more robust than those ob-tained from higher-dimensional estimates, in spiteof the fact that the surface a priori statistics could be

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(d)

Figure 2. Examples of model fit and reconstruction of simulated surfaces. Each panel shows original and reconstructedsurface through kriging using 1D and 2D model fit, 1D and 2d experimental variograms and theoretical Matern models,according to figure titles. (a) and (b): strongly anisotropic random surface (δ = 10) sampled at 5% and 2%, respectively;(c) and (d): slightly anisotropic random surface (δ = 2) sampled at 10% and 3%, respectively

better described by the latter.

2. By construction, the anisotropy parametersδ andθ are shape parameters, which are better estimated“away from the origin”, i.e. from variogram sam-ples with large distance values; the rest of the modelparameters, instead, which are the same involved inthe 1D form, have more influence close to the ori-gin [11]. Consequently, weighting schemes suitedfor estimating anisotropy parameters, assigning sig-nificant weights to samples at large distances, aredifferent from those suited to estimate the rest of theparameters, which should give more emphasis to theestimates close to the origin. This dichotomy canlead to errors in estimates of either type of parame-ters, regardless of the analytical model adopted.

The preceding observations explain what is observedby comparing original and reconstructed surfaces tak-ing means of the root-mean-square reconstruction error(RMSE).

Figs. 3, 4, and 5 reports results of some of these exper-iments by using the Matern, the Bessel, and the Gaussmodel, respectively. From the figures it can be seen that,in spite of the fact that the 2D variogram models in princi-ple represent better the simulated data, the RMSE valuesof the reconstructed surfaces stay slightly lower when us-ing 1D models than by adopting anisotropic models, aslong as low anisotropy surfaces are considered (δ = 1 orδ = 2). When considering stronger anisotropies, a slightadvantage can be seen in using the 2D models in somecases, although the error bars in all the plots overlap.

Note also the relatively low sensitivity of the reconstruc-tion performances to the particular model adopted (all thethree figures show similar treds).

In practice, then, stability of parameters estimation,which is higher for simpler 1D models, seems to over-come the advantage given by more flexible 2D models,which could in principle describe the data more accu-rately.

The above-mentioned results stem mainly from charac-teristics inherent to the (weighted) LMS fit procedure,which is affected by bin population, number of bin sam-ples, and model complexity. To overcome at least someof the problems connected with LMS fit, more sophis-ticated tools could be used, although at the cost of in-creased computational complexity and time. For exam-ple, methods such as the restricted maximum likelihood(REML) [20] allow to obtain model parameter estimateswithout explicitly recurring to binning of the data dis-tances.

6. CONCLUSIONS AND RECOMMENDATIONS

In this work, we have reported some observations relatedto the use of anisotropic models for atmospheric phase

0 2 4 6 8 100

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Figure 3. RMSE error in reconstruction of simulatedsurfaces vs. density of sampled points. Plots referto isotropic (top), anisotropic (middle), and stronglyanisotropic (bottom) surfaces. Reconstruction throughkriging via WLMS-fitted anisotropic Matern model co-variance function obtained starting from the “true” guessvalues (blue curves), isotropic guess values (red curves),and completely isotropic (1D) Matern model (greencurves).

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Figure 4. RMSE error in reconstruction of simulated sur-faces with Bessel model covariance function. Colors andplots as in fig. 3.

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Figure 5. RMSE error in reconstruction of simulated sur-faces with Gauss model covariance function. Colors andplots as in fig. 3.

Table 1. Estimated Matern parameters and surfaceRMSE at various sampling densities, for one simulatedsurface.

% N S L α δ θ RMSE100% 0.75 3.54 8.34 2.59 2.97 2.67 10−15

10% 0.49 3.87 8.05 2.64 3.38 −0.49 0.858% 0.53 3.93 8.63 2.27 2.68 −0.50 0.965% 0.63 3.69 8.25 10.00 3.55 2.75 1.083% 0.22 4.21 7.68 2.97 3.35 −0.41 1.322% 0.39 4.28 8.73 3.58 1.66 −0.31 1.531% 0.078 3.54 7.76 10.00 1.47 −0.64 1.92

0.5% 10−6 3.90 4.21 10.00 2.67 −3.14 1.99

screen modeling and reconstruction, with reference totheir use in persistent scatterers interferometry applica-tions.

We have shown that two-dimensional variogram modelscan better represent anisotropic atmospheric phase screenfields, provided sufficient densities of sampling points areavailable. This is hardly the case with medium resolutionsensors such as ERS or ENVISAT, while it may becomea viable solution with high-resolution sensors such asTerraSAR-X or COSMO/SkyMed which promise muchhigher stable target densities.

However, when dealing with sampling densities up to afew percents of the available pixels, LMS fit procedurescan perform poorly when used with complex 2D mod-els, due both to the increased size of the parameters statespace, which would require higher numbers of experi-mental samples, and the increased computational com-plexity connected with estimating parameters of a 2Dmodel, especially in view of the different importance ofsamples close and away to the origin in the estimationof parameters connected to the model 1D shape and itsanisotropy degree.

Consequently, when using the LMS-fitted models to re-construct the entire raster APS field from sparse samples,the potential advantage given by the higher flexibility of2D models can be reduced by the increased complexityjust described, and cause results to be substantially sim-ilar to those obtained by using simpler one-dimensionalvariogram models.

More sophisticated estimation methods than conventionalLMS fit, e.g. REML, could perhaps be used to ensure thatthe use of anisotropic models give a tangible advantagewith respect to 1D models.

Future work will concentrate on implementing someof the above-mentioned solutions within PSI processingchains, in order to improve the APS estimation and re-moval step, also by better constraining expectations aboutphase noise reduction.

ACKNOWLEDGEMENTS

ERS SAR data were provided by ESA through CAT-1Project n. 5367, entitled “Subsidence monitoring in Dau-nia and Capitanata (Puglia Region, Italy) through multi-temporal point-target DInSAR techniques”.

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