ATMOSPHERIC ERROR, PHASE TREND AND DECORRELATION NOISE IN TERRASAR-X DIFFERENTIAL INTERFEROGRAMS Steffen Knospe () () Institute of Geotechnical Engineering and Mine Surveying, Clausthal University of Technology, Clausthal-Zellerfeld, Germany, Email: steffen.knospe@tu-clausthal.de ABSTRACT The potential of TerraSAR-X data for deformation measurements benefits from high spatial and temporal resolution and even allow detecting higher deformation gradients compared to other sensors. However, decorrelation noise from vegetation and atmospheric effects are of greater importance because of the shorter wavelength. The extent of a TerraSAR-X scene is small compared to common scale atmospheric features and therefore a phase trend appears that has to be addressed carefully. The error content of TerraSAR-X interferograms can be modeled with a stochastic approach as stationary autocorrelated component superimposed with a trend and white noise. These components were analyzed in a large stack of short-term interferograms; summary statistics of covariance parameters are given.. INTRODUCTION The high spatial resolution of TerraSAR-X data allow detecting higher deformation gradients compared to C-band and L-band sensors. However, the shorter revisit time of days compensates only partly the higher temporal decorrelation in vegetated areas. Atmospheric delay features take a greater portion of X-bands shorter wavelength. Therefore a correct error estimation and treatment of atmospheric effects especially are of greater importance. Covariance analysis allows estimating error variances, correlation lengths and special effects like anisotropies []. In a geostatistical covariance analysis it is possible to separate several phase contributions. However, it is not trivial to distinguish between deformation and atmospheric signals in single interferograms if their statistics are similar. The phase anomalies from tropospheric delay and moderate deformation can have similar spatial pattern and comparable amplitudes. With a-priori knowledge about the location of deformation features, the spatial statistics of tropospheric delay at high coherent pixel in non-deforming areas can be analyzed. While several methods of estimating the tropospheric phase delay contribution have been described already, correct error treatment or removal of this phase contribution remains challenging. A stochastic model based on the theory of Random Functions was used to describe spatial auto-covariance structures. The analysis includes calculation of experimental semi-variograms of the dinsar phase and fitting of variogram-models with Matérn- and Besselfamily types of functions that are suitable according Kolmogorov s elementary turbulence theory for tropospheric processes. A covariance model is needed to derive full covariance matrices which are necessary e.g. for reliable estimation of uncertainties and correct error treatment in geophysical inversion studies []. Over the last two years a large stack of TerraSAR-X scenes was collected in the frame of a research project supported by the German Federal Ministry of Economics (BMWi, under DLR-Grant No. 50EE080). TerraSAR-X data were provided by DLR in the frame of General AO project GEO095. Starting with the first available scene from 008-0-5 we attempt to collect all day revisits. Main objective of the still running project is deformation monitoring with TerraSAR-X interferometry in the Emscher Region, North Rhine- Westphalia, Germany, with specific interest on tunneling of main sewers along the river Emscher and its tributaries []. From the large number of analyzed interferograms trends and statistics of covariance model parameters are derived that are representative for the error inventory of TerraSAR-X interferograms in the investigation area and perhaps for that climate region in general. With these it is possible to simulate common error structures for TerraSAR-X deformation measurements and to use Geostatistical Prediction to estimate error free and noise filtered deformation signals.. SPATIAL AUTOCORRELATION A stochastic model of 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 [3]. Smooth variation of phase-values at small distances expresses either smooth deformation or a smoothly varying amount of water vapor and wet particles in the troposphere and can be described with spatial autocorrelation in a probabilistic framework as regionalized variable. Proc. Fringe 009 Workshop, Frascati, Italy, 30 November 4 December 009 (ESA SP-677, March 00)
According to the theory of Random Functions (RF) after Matheron, a regionalized variable is a realization of a RF or stochastic process [4]. If we assume nd order intrinsic stationarity for the spatial varying tropospheric phase delay, the expectation of the increments is a linear function of lag distance h and its variance is given with the semi-variogram function: ( ) E ( ( ) ( )) d γ h = Z x Z x+ h, h R () 3. TERRASAR-X DATA STACK The investigation area is the central part of the Ruhr Area (between the rivers Lippe and Ruhr in the North and the South respectively) covering the cities of Essen, Gelsenkirchen, Bochum, Bottrop, etc., Figure. The TerraSAR-X data stack (orbit 063 descending, strip 008R; single polarization HH, scientific mode orbit data) with 37 stripmap scenes analysed for this study starts with the first available scene from 008-0-5. An estimate of the theoretical semi-variogram of a homogeneous isotropic RF is the experimental semivariogram [4]: ˆ γ n n c ( ) ( h) = z( xi) z( xi + h) c i=, h H () c In practice we replace the experimental semi-variogram with a semi-variogram function that is assumed to be the true variogram of the RF. One wishes to choose a flexible model-function out of the class of negative definite functions consistent with elementary Kolmogorov turbulence [5], and to parameterize it to fit the experimental semi-variogram. It has already been shown in [6] that Matérn-family models [7] can describe the well-known ⅔-law of Kolmogorov s turbulence. In addition, the Matérnfamily has some nice properties to meet physical (adjustable smoothness for short distances) and statistical (saturating variance for long distances) needs. Therefore it is a suitable model candidate and can be described with: d dh dh γ ( h) = c K ( d ) d Γ ( d ) a a (3) where K is the modified Bessel-function of the second kind, Γ is the Gamma-function [8], d defines smoothness (with d = 0.5 we get the exponential function), a is the range (correlation-length) parameter, and c is the sill (variance) parameter. The Besselfamily: Figure.: TerraSAR X SLC 008-0-5; orbit063 descending, strip008r, projected to UTM3 N; investigation area for this study in red. We have attempted to collect all day revisits. Two scenes were excluded from the analysis because of snow cover and strong decorrelation. Short-term interferograms ( day or temporal neighbouring scenes respectively, Figure ) were calculated with the GAMMA software using a SRTM-X digital elevation model for differential interferometry. d h h ( d ) γ ( h) = c Γ ( d + ) J d a a (4) where J is the Bessel-function of the first kind [8] allows to describe periodic structures. Figure.: Baseline plot, TerraSAR X data stack; orbit_063 descending, strip_008r.
Orbit state vector data comes with the SAR data delivery (precise orbit, scientific mode). Perpendicular baselines are far below a critical spatial baseline and up to 330 m for the interferometric stack with 008-0-5 as master, Figure. We did a x multi-looking resulting in 3.3 m (ground range) x 3.7 m (azimuth) pixel spacing to keep high spatial resolution suitable for investigation of small scale phenomena. 4. ERROR ANALYSIS In the following section phase variability is analyzed as composition of deterministic trend, spatial autocorrelated signal and noise. All signal parts but deformation are called error for simplification. The investigation area for error analysis is a section of the central part of the scene with < 7 x km² (Figure ). accuracy of state vector data for TerraSAR-X it was attained that only a minor component of the trend arises from a baseline error. Trend surface analysis is based on unwrapped interferograms and as we do not need a high spatial resolution for it, the unwrapping is facilitated with strong multi-looking. From investigations of ERS or Envisat-ASAR interferograms [], [9] we know about large atmospheric structures (stratification part of tropospheric delay, weather fronts, etc.). Because of the small extent of TerraSAR-X stripmap data, longer wavelengths of these structures appear as a phase trend here. Be aware that if several datasets (dinsar, weather modelling, GPS, etc.) are to be combined in further analysis and data modelling, a common trend should be removed, which means that the detrended data sets are not trend-free. Estimated bilinear trend surfaces for the short-term interferograms of the data stack are shown in Figure 3. Trend reduction is a sensitive operation. The removal of higher order polynomial trend could introduce signal deformation to the analyzed deformation. Trend surface analysis is proposed to be done in the deformation free areas only. Trend estimation from weather models also could be used. 4.. Covariance analysis Figure.3: Bilinear trend surfaces of short-term interferograms in RDC, π/ rad per colour cycle (7.7 mm of apparent LOS deformation). 4.. Trend analysis Usually we take a phase trend as to be from orbit residuals and strictly remove it if large scale deformation features are of no concern. From previous studies with baseline re-estimation based on the fringe rate in interferograms and the knowledge about the After removing phase trend, a structure that is bounded within the scene (in the even smaller investigation area respectively) can be described with a stochastic approach as Random Field. Estimation and modelling of spatial signal characteristics of this autocorrelated signal component under stationarity assumptions (compare section ) are presented here. To increase performance we randomly select about 0000 pixel to create representative data-sets for the variogram analysis. For the selection we prefer high coherent pixel (>.85 and >.9 for most of winter interferogramsrespectively). The estimation of the spatial autocorrelated error component is not affected by this selection criterion. Deformation areas and areas with suspicious topographic errors were masked out in the selection process after detection in a summation stacking of all day interferograms, Figure 4. The omnidirectional experimental variogram measures spatial variability for autocorrelated data. It was calculated according to equation () for all short-term interferograms with distance bins of 500 m (lag size) and a maximum distance of km (about half the maximum distance of the analyzed area). The resulting experimental variograms are very different in both shape and variance (Figures 5 and 6). 4.3. Covariance modelling To use the information about autocorrelated error characteristics in further data modelling, e.g. simulation studies, we need to replace the experimental variogram
with a variogram function. This model we assume to be the real variogram function of the Random Field from which our data are just one sample and with which we can simulate arbitrary representations. Finding the parameters of a variogram model that best fit the experimental variograms is challenging because of physical and statistical constraints on the valid parameter space (negative definiteness, oscillation). A sequence of Simulated Annealing (to explore the parameter space) and nonlinear derivative-based optimization schemes were used to find the best-fitting parameters of the functional models for the variograms. Figure.4: Summation stacking, mean deformation rate, 0 cm a - apparent deformation per colour cycle. Variogram function fitting for the short-term interferograms of our data stack is presented in Figure 6. Isotropic Bessel- (5) and Matérn-family (6) functions were fitted; necessary parameters are explained in Figure 5. With d = 0.5 family members are specified as the exponential model and the so called hole-effect model for (5) and (6) respectively. Figure.5: Two examples for experimental omnidirectional variograms with explanation of the variogram-function parameter (red). Statistics of derived parameters are given in Table and show the strong variation of structures in the example data stack; with strong oscillation terms dominant in summer interferograms. Small spatial structures are not resolved in the 500m lag size and contribute to the noise part of the phase variability. Figure.6a: Experimental omnidirectional variograms with fitted model functions (solid blue line); semi-variogram values in [mm²] and lag distance in [m].
Figure.6b: Experimental omnidirectional variograms with fitted model functions (solid blue line); semi-variogram values in [mm²] and lag distance in [m].
7936 435989 930 35337 75503 364 90965 0950 540760 98058 38837 9840 589069 554370 008439 803 6347 79058 963397 698760 3876 54463 353458 995875 89434 69589 57978 09983 45406 77668 70589 65339 508393 498 4045 58864 6334 0809 830450 4.4. Noise White noise is measured as Nugget effect in the variogram analysis. However, the analysis does not represent the full error content because of the applied point selection criteria (high coherent pixel only); it does not cover the error contribution of temporal decorrelation. The preferential solution for dinsar to reduce noise with stronger multi-looking is not applicable for deformation analysis of small-scale phenomena. The tunnelling for the Marbach Sewer (Emscher tributary) runs through its former river floodplain where vegetation is dominant. In the highly urbanized area less decorrelation was expected. However, because of high spatial resolution, there are decorrelated, vegetation affected pixel between houses and infrastructure. Therefore, vegetation is still an issue for the unwrapping of high-resolution TerraSAR-X interferograms in urbanized areas. The noise from temporal decorrelation is a mayor error source. removal of a phase trend. For the example shown in Figure 7 the reduction is from 5.8 mm² to.05 mm² for the sill and from 5 km to 550 m for the correlation length (effective range) not covering the large scale drift, i.e. the strong increase of variance for long distances in the experimental variogram. The small investigation area and detrending explains the reduced variance of the atmospheric phase component compared to other studies, like [] and [9]. Variogram 3.5.5 0.5 504739 60960 090603_09053 566903 58034 5799 405040 0 0 000 000 3000 4000 5000 6000 7000 8000 9000 0000 000 Lag Distance Variogram 0 9 8 7 6 5 4 3 090603_09053.trend 64593 654030 6734 0 0 000 000 3000 4000 5000 6000 7000 8000 9000 0000 000 Lag Distance Table.: Summary statistics of the variogram model parameter for the stack of short-term interferograms mean min max STD nugget [mm²].54 0.75 7.05.4 sill_bessel [mm²] 4.7.4 9..5 sill_matern [mm²] 6.6.35 8. 4.8 range [m] range_eff_bessel range * 0 9083 5900 00 99 range_eff_matern range * 3 4504 700 9870 60 exponent_bessel 0.5 exponent_matern 0.5 4.5. Simulation The auto-covariance function C(h) is defined under more strict nd order stationarity (not all variograms have a covariance counterpart) with a variance C(0): ( h) C( 0) C( h) γ =. (5) The autocorrelation is C(h) divided by C(0). Simulation of zero-mean Gaussian Random Fields is straightforward based on the variance-covariance-matrix with Cholesky decomposition [0]. It is derived from the covariance function for a given sample point configuration; other techniques (e.g. the Circular embedding method []) are based on the variogram directly. 5. DISCUSSION In TerraSAR-X interferograms it is possible to reduce the atmospheric error content considerably with the Figure.7: Experimental omnidirectional variograms with fitted model functions (solid blue line), variogram values in [mm²], lag distance in [m]; comparison with not detrended data (right panel), corresponding interferograms (090603_09053, not detrended in the bottom panel). Removing the phase trend estimated from the fringe rate will also remove large scale deformation features. To avoid this, trend analysis is to be done in the known deformation free areas or by the introduction of independent data e.g. from weather models.
Geostatistical Simulation and Prediction is straightforward using the derived error model with autocovariance and noise term superimposed with a trend. The covariance model is needed to derive full covariance matrices for estimation of uncertainties in further data modelling like geophysical inversion studies [] and for the use as weights to advance adjustment approaches for Small Baseline methods, presented in e.g. []. ACKNOWLEDGEMENTS This research was approved within the German Aerospace Centre (DLR) call Preparation of Use of TerraSAR-X Data and is supported by the German Federal Ministry of Economics (BMWi) under the DLR-Grant No. 50EE080. The TerraSAR-X data we used were provided by the DLR in the frame of the General AO project GEO095. We acknowledge the Science Coordinator at the DLR and at the German Remote Sensing Data Centre (DFD) for providing the TerraSAR-X and the SRTM-X DEM. REFERENCES [] IEEE Paper in press: Steffen H.-G. Knospe and Sigurjón Jónsson, Covariance Estimation for dinsar Surface-Deformation Measurements in Presence of Anisotropic Atmospheric Noise, IEEE Transactions on Geoscience and Remote Sensing, vol. 48 (4), 009. [] Knospe, S. & Busch, W. (009). Monitoring a tunneling in an urbanized area with TerraSAR-X interferometry - surface deformation measurements and atmospheric error treatment. IEEE International Geoscience & Remote Sensing Symposium, Session TU.0: TerraSAR-X: Scientific Results, Cape Town, July -7, 009. [3] Gandin, L. S. (965). Objective analysis of meteorological fields, Israel Program for Scientific Translation, Jerusalem. [4] Wackernagel, H. (003). Multivariate Geostatistics, 3 rd ed., Berlin, Heidelberg: Springer. [5] Tatarski, V. I. (96). Wave Propagation in a Turbulent Medium, New York: McGraw-Hill. [6] Treuhaft, R. N. & Lanyi, G. E. (987). The effect of the dynamic wet troposphere on radio interferometric measurements. Radio Science, vol., pp. 5-65. [7] Guttorb, P. & Gneiting, T. (006). Miscellanea. Studies in the history of probability and statistics. XLIX. On the Matérn correlation family. Biometrika, vol. 93(4), pp. 989-995. [8] Abramowitz, M. & Stegun, I. A. (993). Handbook of Mathematical Functions, John Wiley & Sons. [9] Hanssen, R. (00). Radar Interferometry: Data Interpretation and Error Analysis, Dordrecht: Kluwer Academic Publisher. [0] Schlather, M. (999). Simulation and analysis of random fields. R news, vol. (), pp. 8-0. [] Dietrich, C. R. & Newsam, G. N. (993). A fast and exact method for multidimensional Gaussian stochastic simulations. Water Resour. Res., vol. 9, pp. 86-869. [] Ge, N., Knospe, S. & Busch, W. (009). Deriving High-Resolution Non-Linear Deformation Time Series from TerraSAR-X Interferograms with the Method of Least Squares. Proceedings of the Fringe workshop 009, ESA SP-677, this volume.