COVARIANCE ESTIMATION FOR DINSAR DEFORMATION MEASUREMENTS IN PRESENCE OF STRONG ATMOSPHERIC ANISOTROPY

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1 COVARIANCE ESTIMATION FOR DINSAR DEFORMATION MEASUREMENTS IN PRESENCE OF STRONG ATMOSPHERIC ANISOTROPY Steffen Knospe (1,*) and Sigurjón Jónsson (1) (1) ETH Zurich, Institute of Geophysics, Schafmattstr. 3, 893 Zurich (Switzerland), Phone: , knospe@erdw.ethz.ch (*) Steffen Knospe is supported by ESA within the postdoctoral fellowship programme ABSTRACT The dinsar phase signal is a superposition of different contributions. When studying geophysical processes, we are usually only interested in the surface deformation part of the signal, and therefore, to obtain high quality results, we would like to remove other phase components. Although several ways of estimating the tropospheric phase delay contribution have already been described in the literature, a correct error treatment or removal are still challenging. A stochastic model based on the theory of Random Functions has been found to be appropriate to describe atmospheric phase delay in dinsar images. However, these phase delays are usually modelled as being isotropic, which is a simplification, because InSAR images often show directional atmospheric anomalies. Here we analyse anisotropic structures based on the theory of Random Functions and show validation results using simulated and real data. We calculate experimental semi-variograms of dinsar image examples and fit anisotropic variogram models to them. With these variogram models we calculate covariance matrices for a given point raster, which we use to simulate multiple anisotropic phase delay realisations. We add these realisations as noise to simple forward model calculations of surface deformation. Then we invert for the source parameters of the deformation model and compare the results for different error scenarios, which shows the importance of including the anisotropic data covariance information. 1. INTRODUCTION The dinsar signal contains different phase contributions, such as surface deformation, tropospheric delay, topographic residual, and several errors from other sources. When studying geophysical processes, we are usually only interested in the surface deformation part of the signal, and therefore we would like to remove other phase components. However, it is not trivial to distinguish between deformation and atmospheric signals in single interferograms, because tropospheric delay signals and moderate deformation (e.g. inter- or post-seismic deformation) can have similar amplitudes and sometimes comparable spatial extent, i.e. similar structure of variability in the spatial domain. Several methods of estimating the tropospheric phase delay contribution have been described, but a removal of this phase contribution is difficult. One way is to use independent data from other sources, like GPS or MERIS. ENVISAT offers possible concurrent ASAR-MERIS data acquisitions, but some limitations come with the availability (cloud free, day-time) and the resolution of MERIS data. GPS phase delay estimations provide information only at a poor spatial resolution, due to the limited spatial density of continuously operating GPS stations in almost all places. A stochastic model describing a structure of variability (i.e. spatial autocorrelation) based on the theory of Random Functions has been found to be appropriate to describe tropospheric phase delay of electromagnetic waves propagating through the atmosphere. It is usually described based on elementary (Kolmogorov) turbulence theory [1] as a power-law dependency on spatial frequency in Very Long Baseline Interferometry (VLBI) [2], Global Positioning System (GPS) [3], and dinsar measurements [4]. Presuming good quality (i.e. high coherence and high SNR) dinsar observations the wet tropospheric delay usually dominates local phase variations within the area covered by an interferogram. Therefore atmospheric signal delay is an important limitation in many dinsar surface deformation applications. Tropospheric signals are usually modelled in the frequency domain as being isotropic, which is a simplification, as InSAR images often show directional atmospheric anomalies, i.e. an anisotropy in how phase values vary. In this paper we begin by introducing concepts and measures for spatial error structures. We then describe an example data stack of several ERS-1/2 tandem interferograms and calculate experimental semivariograms describing the spatial autocorrelation structure. Next we describe a way of estimating anisotropic variogram-models that are consistent with the theory of Random Functions and Kolmogorov s turbulence and also respect the specific data characteristics, including several aspects of anisotropy. We show examples on how to use this stochastic error model to weight dinsar observations in geophysical source parameter modelling. We then add simulated anisotropic phase delay realisations to simple forward model calculations of surface deformation from a magma intrusion. And finally, we show the importance of accounting for anisotropy by comparing different scenarios in source-parameter inversions based on dinsar surface deformation data. Proc. Envisat Symposium 27, Montreux, Switzerland April 27 (ESA SP-636, July 27)

2 2. SPATIAL AUTOCORRELATION AND VARIOGRAMS The self-similarity of phase-values at small distances, which we term spatial autocorrelation, is an expression of a similar amount of wet particles, i.e. atmospheric delay is a smoothly varying physical quantity. The socalled semi-variogram measures the spatial autocorrelation of a regionalized variable z(x); the variogram is also known as structure function [5]. If h is a classified separation distance, then the experimental semivariogram is: N( h) 1 2 γ ( h) = z x z x +. (1) 2N h ( ( ) ( )) ( ) i i h i= 1 Note that Eq.1 depends only on distance, but later we use a vectorized 2D variogram-map to characterize anisotropy. The auto-covariance function C(h) is defined under the less general 2 nd order stationarity (not all variograms have a covariance counterpart) with a variance C(): ( h) C( ) C( h) γ =. (2) The autocorrelation is C(h) divided by C(). Under the assumption of stationarity (at least intrinsic stationarity, i.e. 2 nd order stationarity of the distance increments), according to the theory of Random Functions after Matheron [6], there exists a theoretical variogram of a homogeneous isotropic Random Function Z(x) that has the same spatial characteristics as the experimental structure. With the Random Function this structure inherently obtains a physical meaning (a process from which the data are just one sample). Therefore we can replace the experimental semi-variogram with a theoretical semivariogram function (for convenience we use only variogram in the following) that has to be conditionally negative definite [7], which guarantees positive variances for any linear combination of sample values: n n n Var ωiz ( xi) = ωω i jγ ( xi x j) i= i= j= n with ω =. (2) i= i The replacement of an experimental semi-variogram with a theoretical variogram (model-function) is a critical topic in Geostatistics of which good function fit is only one criterion. While respecting its (assumed to be known) physical nature, we wish to choose out of the class of negative definite functions (that will be parameterized to fit the experimental semi-variogram) a flexible model consistent with elementary Kolmogorov turbulence [8]. A review of several studies where spatial statistics of water vapour fluctuations have been described with a power law in the frequency domain can be found in [4]. However as a power-law model with an exponent greater than zero is unbounded (not 2 nd order stationary with no finite variance), additional constraints are needed. To account for this and for smoothness, a power-law with varying exponent is necessary [1, 9]. The Wiener-Khintchin theorem defines the transition of the power spectral density to the autocorrelation function of a 2 nd order stationary process. It has already been shown [1] that a -family model [1] can describe the well-known 2/3 law of Kolmogorov s turbulence. In addition, it has some nice properties to meet physical (adjustable smoothness for short distances) and statistical (saturating variance for long distances) needs. Therefore we choose it here as a suitable model candidate. It can be described by [11]: d 1 2 dh 2 dh γ ( h) = c 1 K 2 ( d 1) d Γ( d ) a a (3) where K is the modified Bessel-function of the second kind, Γ is the Gamma-function [12], d defines smoothness, a is a range (correlation-length-) parameter, and c is the sill (variance-) parameter. Another suitable model is the Bessel-family model, which with d = is a 2D analogue of the cosine structure [13]: d h h ( d ) γ ( h) = c 1 2 Γ ( d + 1) J d. (4) a a This model choice is motivated because of its adjustable oscillation (J is the Bessel-function of the first kind [12]): To explore the spatial structure in terms of anisotropy we calculate 2D experimental variogrammaps and fit 2D variogram-models (alternative one can calculate several variograms for fixed directions respectively classes of directions ). As the variogrammodels are defined isotropic we need to transform the coordinate space to allow for anisotropic Random Functions. Without loss of generality we can vice versa transform isotropic variogram-functions to produce a whole class of ellipsoidal anisotropic variogramfunctions with different correlation length in different directions. This phenomenon is called Geometric Anisotropy. Another aspect of anisotropy is Zonal Anisotropy, i.e. a directional varying sill. To hold stationarity assumptions, in practice we use nested models to explain the different variability in different directions. Nested models are a superposition of several variograms that explain the case of superimposed

3 3. DATA EXAMPLE As our example data set we use a stack of ERS-1/2 tandem-scenes covering the area NE of Marseille (descending, track 65, frame 2727). ERS tandem interferograms are likely to have high coherence and SNR and are generally not affected by deformation due to the short one-day time interval. Therefore, it is an ideal data set to study anisotropic atmospheric structures as one might expect for this part of the world. InSAR processing software and the data stack were provided by Gamma RS. From this stack 8 tandem interferograms are calculated: , , , , , , , For demonstration purposes we choose the two interferograms with the most different spatial variability. We refer to these interferograms ( and ) as I1 and I2 in the following. The interferograms are shown in Fig. 1 with the atmospheric delay expressed in centimetres of apparent deformation. Figure 1. Interferograms I1 and I2 showing atmospheric phase delay, 4cm colour-cycle, UTM31 projection (45.7x33.6 km²). 3.1 Variograms We randomly sub-sample the images in Fig.1 to create two sets of about 45 data points. The omnidirectional experimental variograms in Fig.2 are different in shape and variance for the two interferograms. We fit a Bessel-model to the experimental variogram for I1 and a -model for I2 to get variogram-functions Vo1 and Vo2 (see Fig. 2), both containing a nugget-effect (describing white noise with correlation length zero) E E-6 7E-5 5E-6 6E-5 5E-5 Variogram in m² Variogram in m² Random Functions (from several uncorrelated processes). A nested structure is in line with our expectations, because there are several processes that can act on the atmospheric signal delay and its spatial properties. If one variogram gets a geometric anisotropy with major correlation length far beyond the limits of the investigation area, it only adds to the sill in the perpendicular direction (with short correlation length) [7]. For the purpose of function fitting we introduce additional predefined anisotropy parameters in Eq.3 and Eq.4: a rotation angle (orientation of the range-ellipses, i.e., the direction of maximum correlation length), and axis ratio (maximum to minimum range). 4E-5 3E-5 4E-6 3E-6 2E-6 2E-5 1E-6 1E Lag Distance in m Lag Distance in m Figure 2. Experimental omnidirectional variogram with variance (dashed line) and variogram models Vo1 and Vo2 (blue line) Variogram maps (showing variability from the centre of images into the different directions) can reveal anisotropic pattern (Fig. 3). For I1 we detect very strong anisotropy, but for I2 only some anisotropy is visible. This anisotropy is of course hidden (i.e. averaged) in the omnidirectional variograms shown in Fig. 2. Figure 3. Experimental variogram-maps for interferograms I1 and I2. The fitting procedure for 2D variograms is somewhat challenging because of (physical and statistical) constraints on function-parameters as well as the large number of parameters. First, from an eye-inspection we put reasonable bounds on the parameters. Then, we used a sequence of simulated annealing (to explore the parameter space) and nonlinear least squares optimisation to find the best-fitting parameters of the 2D function model for the experimental variogram maps. The best-fitting parameter values are given in Tab. 1. The 2D variogram-model (Vd1) for I1 shows that a Bessel-function describes well the strong oscillation in the NNW-SSE direction (Fig. 4). Figure 4. Variogram model Vd1 (left) and the residual (right) between the model and the experimental variogram for I1.

4 To calculate the variogram-model (Vd2) for I2 we used a -model without oscillations, see Fig.5. Figure 5. Variogram model Vd2 (left) and the residual (right) between the model and the experimental variogram for I2. Table 1. Variogram model parameters (rounded) I1 omnidirectional nugget [m²] type exponent range [km] anisotropic nugget [m²] type exponent max_range [km] direction [deg] min_range [km] nested anisotropic nugget [m²] Type Exponent max_range [km] direction [deg] min_range [km] Type Exponent max_range [km] direction [deg] min_range [km] I2.2 Bessel bessel bessel Bessel able to define correct covariances for any point configuration, because the functions are conditionally negative definite. Figure 6. Nested variogram models for interferograms I1 and I2. We prepared a point raster (32x32, 1 km spacing) for the study in Section 4 and created the corresponding covariance matrices (Co1, Co2 and Cd1, Cd2), which are guaranteed to be positive semi-definite and invertible. These covariance matrices are shown in Fig.7. Figure 7. Covariance matrices, top left and right: Co1 and Co2, bottom left and right: Cd1 and Cd2. To allow for Geometric as well as Zonal Anisotropies, we introduce a nested structure consisting of a nugget effect, a Bessel-model, and a -model. The resulting best-fit model variograms have somewhat smaller residuals. These nested 2D variogram models are presented in Fig. 6. However, for simplicity, we do not use these models in our simulations in the next Section. 3.2 Simulation of Atmospheric Delay Structures In the last Section we have examined the anisotropic spatial structures of the Random Fields, which explain interferograms I1 and I2 well. With the continuous variogram functions Vo1, Vo2 and Vd1, Vd2 we are Figure 8. Simulation examples, top So1 and So2, bottom Sd1 and Sd2.

5 We simulate zero-mean Gaussian Random Fields according to the corresponding covariance structures. For performance reasons, we use two different fast algorithms to produce 1 realisations based on each variogram: a direct method with Cholesky decomposition for Co1 and Cd1, and the Circular embedding method for Co2 and Cd2 [14]. Examples of the simulated atmospheric noise structures (So1, Sd1, So2 and Sd2, based on Co1, Cd1, Co2, and Cd2, respectively) are shown in Fig. 8. The simulations were carried out using the software package RandomFields [13] written in the R language [15]. 4. SOURCE PARAMETER MODELING To show the importance of considering anisotropic effects we investigate source parameter inversions with a simulated deformation signal affected by atmospheric noise. The deformation signal is calculated using a simple model of a Mogi point source embedded in an elastic half space. We fix source strength (volume change) to m 3 and the source location (depth 5 km), and calculate the expected vertical displacements (Fig. 9). We then add to this simulated deformation signal the different realizations of atmospheric noise described in Section 3.2, generating multiple noisy data sets. where Σ -1 is the inverse of the data covariance matrix. In the inversion calculations we use each of the noisy data sets and invert only for the volume change, which we can compare to the true value of m 3. Furthermore, for each data set we estimate the volume change three times, using different data weighting. In the first case we use no data weights, i.e. treating the data points as uncorrelated and equally weighted. The resulting scatter in solution values is shown in Figs (top panels). The second case uses the isotropic covariance matrices Co1 and Co2 (Fig.7) as data weights in the volumechange inversions and the third case the anisotropic covariance matrices Cd1 and Cd2. The resulting solution scatter about the true value for the second and third cases is also shown in Figs (middle and bottom panels, respectively). The 1 inversions provide an empirical standard deviation in each case that characterizes the quality and reliability of the inversion (STD values in Figs. 1-11). The Mean_D value listed in Figs represents the mean deviation from the true volume-change value. Figure 9. Vertical displacements predicted by a Mogi point source at 5 km depth in an elastic half-space. The linear forward problem can be described by: d = Gm (5) where d is a vector containing noisy data values, G is the kernel matrix linking the data and the model parameters, and m is a vector of model parameters (only volume change in this problem). The weighted leastsquares estimate of the model parameters is: T ( ) 1 ˆ = Σ Σ (6) 1 T 1 m G G G d Figure 1. Histograms, scatter of the estimated volume change using no data weights (top panels), isotropic weights (middle), and anisotropic weights (bottom). The added anisotropic noise had I1 statistics (left panels) and I2 statistics (right column). Mean_D and STD are explained in the text. The histograms in Fig.1 show the solution scatter for 1 realizations of anisotropic atmospheric noise. It is clear that including spatial autocorrelation in the data weighting (case 2) improves the quality of inversion results, even if we neglect the anisotropy (Fig. 1, middle panels). Furthermore, we dramatically improve the results by taking anisotropy into account (case 3, Fig. 1, bottom panels), and the improvement is greater the stronger the anisotropic effects are. We also investigated what happens if anisotropic data covariance weights are wrongly applied to data containing isotropic noise. The bottom panels of Fig. 11 show these results. It is surprising that the solution degradation in this case is at least as dramatic as the

6 improvement for the correctly applied anisotropic weights in Fig. 1. Figure 11. Histograms, same as Fig. 1, except here the added noise was isotropic. Therefore, it seems that the model-parameter estimation is sensitive to the inclusion of anisotropic weights and one needs to be careful in applying such weights in parameter inversions. 5. CONCLUSIONS InSAR data are sometimes influenced by strong atmospheric noise. To use such data in geophysical model-parameter estimations, it is beneficial to describe the noise characteristics well. Atmospheric noise is spatial autocorrelated and often shows anisotropy. We have shown that inversion results improve when we include information about the correlated noise in the data weighting. Furthermore, the results can improve strongly if we fully include anisotropic data weighting in cases where deformation signals are biased by significant anisotropic noise. However, more sensitivity tests need to be carried out to investigate how strongly anisotropic data weighting affects the model parameter estimation, both in presence of isotropic and anisotropic noise. 6. REFERENCES 1. Tatarski, V. I. (1961). Wave Propagation in a Turbulent Medium. McGraw-Hill, New York. 2. Treuhaft, R.N. & Lanyi, G.E. (1987). The effect of the dynamic wet troposphere on radio interferometric measurements. Radio Science 22, Tralli, D.M. & Lichten, S.M. (199). Stochastic Estimation of Tropospheric Path Delays in Global Positioning System Geodetic Measurements. Bulletin Géodésique 64, Williams, S., Bock, Y. & Fang, P. (1998). Integrated satellite interferometry: tropospheric noise, GPS estimates and implications for interferometric synthetic aperture products. J. Geophys. Res. 11, Gandin L. S. (1963). Objective analysis of meteorological fields (in Russian). Israel Program for Scientific Translation, 1965, Jerusalem. 6. Chilès, J.-P. & Delfiner, P. (1999). Geostatistics: modelling spatial uncertainty. Wiley series in probability and statistics. 7. Wackernagel, H. (1995). Multivariate Geostatistics. Springer, Berlin, Heidelberg. 8. Gneiting, T. (1999). Correlation functions for atmospheric data analysis. Q. J. R. Meteorol. Soc. 125, Hanssen, R.F. (21). Radar Interferometry. Data Interpretation and Error Analysis. Kluwer Academic Publisher Dordrecht. 1. Guttorb, P. & Gneiting, T. (26). Miscellanea Studies in the history of probability and statistics XLIX On the correlation family. Biometrika 93(4), Knospe, S. (22). Anwendungen Geostatistischer Verfahren in den Geowissenschaften Fallbeispiele unter Einbeziehung zusätzlicher Fachinformationen. Mensch & Buch Verlag, Berlin. 12. Abramowitz, M. & Stegun, I.A. (1993). Handbook of Mathematical Functions. John Wiley & Sons Inc. 13. Schlather, M. (1999). Simulation and analysis of random fields. R news 1(2), Dietrich, C. R. & Newsam, G.N. (1993). A fast and exact method for multidimensional Gaussian stochastic simulations. Water Resour. Res. 29, R Development Core Team (26). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, URL Okada, Y. (1992). Internal deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Am. 82,