Interferometric synthetic aperture radar atmospheric correction: GPS topography-dependent turbulence model

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1 JOURNAL OF GOPHYSICAL RSARCH, VOL. 111,, doi: /2005jb003711, 2006 Interferometric synthetic aperture radar atmospheric correction: GPS topography-dependent turbulence model Zhenhong Li, 1 ric J. Fielding, 2 Paul Cross, 1 and Jan-Peter Muller 1 Received 2 March 2005; revised 3 November 2005; accepted 16 November 2005; published 10 February [1] Over the last two decades, repeat pass interferometric synthetic aperture radar (InSAR) has been widely used as a geodetic technique for measuring the arth s surface, including topography and deformation. Like other astronomical and space geodetic techniques, repeat pass InSAR is limited by the variable spatial and temporal distribution of atmospheric water vapor. In this paper, a topography-dependent turbulence model (GTTM for short) has been developed using GPS data only to produce zenith path delay difference maps for InSAR atmospheric correction. Application of the GTTM model to RS Tandem data over the Los Angeles Southern California Integrated GPS Network area has shown that use of the GTTM can reduce water vapor effects on interferograms from 10 mm down to 5 mm, which is of great interest to a wide community of geophysicists. The principal finding of this paper is that interpolation methods should be applied to zenith total delay (ZT) differences from different times instead of ZT values themselves for the purpose of InSAR atmospheric correction. This is crucial to reduce (if not completely remove) topographic effects on ZT values. Citation: Li, Z.,. J. Fielding, P. Cross, and J.-P. Muller (2006), Interferometric synthetic aperture radar atmospheric correction: GPS topography-dependent turbulence model, J. Geophys. Res., 111,, doi: /2005jb Introduction [2] It is well known that radio signals suffer from propagation delays when they travel through the atmosphere with the main uncertainties being due to water vapor in the troposphere. Moreover, the state of the atmosphere is never identical when two images are acquired at different times for repeat pass InSAR. Therefore any difference in path delays between these two acquisitions will result in additional shifts in phase signals. Massonnet et al. [1994] first identified atmospheric effects in repeat pass InSAR measurements when they studied the 1992 Landers earthquake. Zebker et al. [1997] suggested that a 20% spatial or temporal change in relative humidity could result in a cm error in deformation measurement retrievals, independent of baseline parameters, and possibly m of error in derived digital elevation models (M) for those interferometric pairs with unfavorable baseline geometries. A series of 26 RS tandem SAR interferograms was investigated to assess the heterogeneous effects of the atmosphere on the interferometric phase observations in the Netherlands [Hanssen, 1998]. This study showed that the RMS values of the atmospheric effects ranged from 0.5 to 3.6 rad, while the observed phase values ranged from 1 epartment of Geomatic ngineering, University College London, UK. 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA. Copyright 2006 by the American Geophysical Union /06/2005JB003711$ to 2.3 phase cycles (one cycle corresponds to 28 mm path delay) at a 95% significance level with a Gaussian distribution. The phase error, however, reached 4 cycles during thunderstorms. [3] Several techniques have been proposed for reducing atmospheric effects in interferograms, including stacking and calibration. Stacking involves temporal averaging of N independent interferograms p ffiffiffiffi to reduce the temporally uncorrelated noise by 1/ N [Zebker et al., 1997; Williams et al., 1998; Peltzer et al., 2001; Fialko, 2004]. Ferretti et al. [1999] developed a weighted averaging method to construct Ms taking into account the normal baseline value, the coherence level, and the phase distortion due to atmospheric effects. mardson et al. [2003] calculated the water vapor spatial variation using zenith atmospheric delays from GPS data from the Southern California Integrated GPS Network (SCIGN). Using these results, they showed the possibility of calculating the number and duration of interferograms required to achieve a desired sensitivity to deformation rate at a given length scale for a given orbit revisit time and image archive duration. Calibration involves spatial reduction (if not complete removal) of path delays using independent data such as those provided by continuous Global Positioning System (GPS) networks [Williams et al., 1998] and the NASA Moderate Resolution Imaging Spectroradiometer (MOIS) [Li et al., 2005b]. GPS measurements can be used to provide a highly accurate three-dimensional (3-) position, to derive water vapor products, and to map deformation. It therefore appears advantageous to integrate GPS with InSAR measurements as (1) they can be used to cross validate each other; (2) GPS water vapor products can be applied to reduce atmospheric effects on InSAR 1of12

2 measurements; and (3) GPS positioning results can be used as constraints to refine baselines in InSAR processing. [4] Integration of InSAR and GPS was first suggested by Bock and Williams [Bock and Williams, 1997]. Williams et al. [1998] used the Southern California Integrated GPS Network (SCIGN) to assess the possibility of reducing atmospheric effects on interferograms using GPS data. They demonstrated that the atmospheric effects appear to conform to a power law and a reduction in power law noise can be achieved by removing the long-wavelength effects (estimated from GPS zenith delays), leaving the higher-frequency, lower power components. Since current GPS networks are not optimal for InSAR purposes and GPS-derived zenith delays represent an average along the paths of 4 12 (or more) GPS satellites as they pass the station (which is different from the two-way slant range propagation in SAR images), it is regarded as being impossible to remove artifacts with smaller scales than the GPS station spacing [Hanssen, 2001]. Wadge et al. [2002] compared GPSderived zenith delays estimated from 14 continuous GPS (CGPS) and InSAR measurements over Mount tna, and found the equivalent delays for InSAR-GPS had an RMS value of 19 mm with a mean of +12 mm. With 16 GPS stations over Houston, Texas, Buckley et al. [2003] applied an atmospheric correction to a Tandem interferogram. Although the artifact reduction was marginal, they demonstrated utility in using GPS-derived zenith delays for the assessment of atmospheric effects on interferograms. To date, there have been few satisfactory results for the reduction of atmospheric effects on interferograms. This is usually believed to be due to two factors: (1) the limited spatial resolution of GPS stations and (2) the lack of an efficient spatial interpolator. In this study, an efficient spatial interpolator is developed and tested along with the most commonly used method, traditional inverse distance weighted interpolation method (IW) [Shepard, 1968]. 2. GPS Topography-ependent Turbulence Model 2.1. Spatial Structure Function (SSF) [5] For a random function x(r 0 ), where r 0 is a spatial coordinate, the spatial structure function for a vector L is defined as [Tatarskii, 1971; Treuhaft and Lanyi, 1987; Williams et al., 1998] x ðlþ ¼ ½xr ð 0 ; LÞ xr ð 0 ÞŠ 2 where the angle brackets denote an ensemble average; x(r 0, L) is the same random function at a point separated from r 0 by the vector L. For homogeneous, isotropic and ergodic random fields, the spatial structure function depends only on the separation (or distance) L = jlj and can be written as x ðlþ ¼ x ðlþ ¼ ½xr ð 0 ; LÞ xr ð 0 ÞŠ 2 The spatial structure function is often described as a power law process [Williams et al., 1998]: x ðlþ ¼ CL a ð1þ ð2þ ð3þ where C characterizes the roughness or scale of the process and a is the power index which expresses the rate at which the random function loses correlation with increasing distance. In the frequency domain, a power law process has a power spectrum P x f b ðf Þ ¼ P 0 where P 0 and f 0 are normalizing constants, f is the spatial or temporal frequency, and b is the spectral index (often 1 < b < 3). The spatial structure function with the power spectral form of equation (4) can be written as [Agnew, 1992; Hanssen, 2001] P 0 x ðlþ ¼ C x f b 0 f 0 L b 1 ¼ CL b 1 where C x = p b /[2 1 b G(b)cos( bp/2)]. The relationship between the power index a in equation (3) and the spectral index b can then be given by a ¼ b 1 When a = 2/3 (or b = 5/3), this is referred to as Kolmogorov turbulence [Tatarskii, 1971]. [6] On the basis of the spatial structure function, Treuhaft and Lanyi [1987] developed a statistical model (TL hereafter) of water vapor fluctuations to estimate wet tropospheric effects on very long baseline interferometry (VLBI). The TL model relied on two principal assumptions: First, the spatial structure of the fluctuations can be closely approximated by Kolmogorov turbulence theory; Second, temporal fluctuations are caused by spatial patterns which are moved over a site by the wind; that is, temporal fluctuations are caused by the frozen atmosphere being moved by the wind. For simplicity, the TL model also assumed that both the water vapor spatial structure and the wind vector were independent of height up to an effective scale height, and the frozen atmospheric water vapor moved across a flat arth. Treuhaft and Lanyi [1987] estimated a = 2/3 for a distance L of up to 3000 km, while a = 5/3 for L much smaller than 1 km, with a smooth transition between these two limits. Williams et al. [1998] suggested that tropospheric variations conform temporally and spatially to the TL statistical model. [7] For interferometric observations such as InSAR, its spatial structure function can be simulated by double differencing all possible observable combinations between time t n and t m [mardson et al., 2003]: x ðl; tþ ¼ f½x tn ðr 0 ; LÞ x tn ðr 0 ÞŠ ½x tm ðr 0 ; LÞ x tm ðr 0 ÞŠg 2 ð7þ where t = t n t m represents observation intervals. From equation (7), it is clear that a unique temporal change in the observations leads to x (L, t) equaling zero even when there is strong spatial variation at a given time (t n or t m ), indicating that atmospheric effects on interferograms are caused by spatiotemporal variations (instead of solely ð4þ ð5þ ð6þ 2of12

3 spatial or temporal variations) of the atmosphere (primarily water vapor). [8] Goldstein [1995] found that the spatial spectrum of atmospheric effects on an spaceborne imaging radar C-band (SIR-C) radar interferogram acquired over the Mojave esert in California followed a power law with a spectral index ( b) of 8/3 for spatial scales between 0.4 and 6 km. Hanssen [2001] studied atmospheric effects on eight Tandem interferograms acquired over Groningen in the Netherlands during 1995/1996 and found that the noise power spectrum typically has similar power law behavior, although the absolute power varied from one interferogram to another. The spectral indices ( b) varied between 5/3 and 8/3 for the different spatial scales: 5/3 for spatial scales larger than 2 km, 8/3 for intermediate scales between 0.5 and 2 km, and 2/3 for spatial scales smaller than 0.5 km. Hanssen [2001] argued that the 5/3 power law behavior represented scales larger than the thickness of the turbulent layer where an approximation of two-dimensional turbulence can be applied, while the intermediate scales were smaller than the thickness of the turbulent layer and the noise structure had a spectral index of 8/3. Hanssen [2001] suggested that the 2/3 power law behavior at the smallest spatial scales (<0.5 km) was unlikely to have an atmospheric origin, but more likely to result from decorrelation effects or interpolation errors Variance and Covariance [9] The variance (s 2 ) of a homogeneous, isotropic and ergodic random field can be written as a function of distance (L): s 2 ðlþ ¼ Varf½xr ð 0 ; LÞ xr ð 0 ÞŠg ð8þ where x represents zenith path delays or their time differences in this paper. [10] If the mean of [x(r 0, L) x(r 0 )] is zero, then s 2 ðlþ ¼ ½xr ð 0 ; LÞ xr ð 0 ÞŠ 2 ¼ L ð Þ ð9þ where the last equality is from equation (2). [11] Spatial covariance describes the relation between two points (or pixels). The covariance function of a 2- random field can be defined in terms of spatial covariance as CL ð Þ ¼ Cov½xr ð 0 Þ; xr ð 0 ; LÞŠ ¼ hxr ð 0 Þxr ð 0 ; LÞi hxr ð 0 ; LÞihxr ð 0 Þi ¼ hxr ð 0 Þxr ð 0 ; LÞi ðconstþ 2 ð10þ It should be noted that the mean of x(r 0 ) in equation (10) is assumed to be a constant, which is consistent with the assumption in equation (9). Thus, under such an assumption the covariance function can be written in terms of the variance (s 2 )as CL ð Þ ¼ Cð0Þ 1 2 s2 ðlþ ð11þ and in terms of the spatial structure function (SSF, (L)) as CL ð Þ ¼ Cð0Þ 1 2 L ð Þ ð12þ For interferometric observations such as interferograms, the variance (s 2 int ) can be written as a function of distance (L) and observation interval (t = t n t m ) (also see equation (7)): s 2 intðl; tþ ¼ Varf½x tn ðr 0 ; LÞ x tn ðr 0 ÞŠ ½x tm ðr 0 ; LÞ x tm ðr 0 ÞŠg ð13þ If the epochs t n and t m are sufficiently separated so that [x tn (r 0, L) x tn (r 0 )] and [x tm (r 0, L) x tm (r 0 )] are uncorrelated, then equation (13) reduces to s 2 intðl; tþ ¼ Var½x tn ðr 0 ; LÞ x tn ðr 0 ÞŠþVar½x tm ðr 0 ; LÞ x tm ðr 0 ÞŠ ¼ 2Var½x t ðr 0 ; LÞ x t ðr 0 ÞŠ ð14þ As in equation (9), if the mean of [x t (r 0, L) x t (r 0 )] is zero, then s 2 intðl; tþ ¼ 2 ½x t ðr 0 ; LÞ x t ðr 0 ÞŠ 2 ¼ 2ðLÞ ð15þ which is consistent with mardson et al. [2003]. It is clear that the only difference between equations (9) and (15) is a factor of 2. [12] Under the assumptions of equations (14) and (15), the mean of {[x tn (r 0, L) x tn (r 0 )] [x tm (r 0, L) x tm (r 0 )]} is zero, so equation (13) can also be written as s 2 intðl; tþ ¼ f½x tn ðr 0 ; LÞ x tn ðr 0 ÞŠ ½x tm ðr 0 ; LÞ x tm ðr 0 ÞŠg 2 ¼ int ðl; tþ ð16þ where the last equality is from equation (7). [13] From equations (9), (15), and (16), it is clear that int ðl; tþ ¼ 2ðLÞ s 2 intðl; tþ ¼ 2s 2 ðlþ ð17aþ ð17bþ The covariance function of interferometric observations can be written as C int ðl; tþ ¼ Covð½x tn ðr 0 Þ x tm ðr 0 ÞŠ; ½x tn ðr 0 ; LÞ x tm ðr 0 ; LÞŠÞ ¼ 2hxr ð 0 Þxr ð 0 ; LÞi 2ðconstÞ 2 ð18þ where the last equality is derived under two assumptions: (1) The epochs t n and t m are sufficiently separated so that [x tn (r 0, L) x tn (r 0 )] and [x tm (r 0, L) x tm (r 0 )] are uncorrelated as in equation (14) and (2) The mean of x(r 0 ) is assumed to be a constant as in equation (10). Hence, taking into account equations (17a) and (17b), equation (18) becomes C int ðl; t Þ ¼ C int ð0; tþ L ð Þ ¼ C int ð0; t Þ 1 2 intðl; tþ ¼ C int ð0; tþ 1 2 s2 intðl; tþ ð19þ 2.3. GPS Topography-ependent Turbulence Model (GTTM) [14] On the basis of the TL model, Jarlemark and mardson [1998] proposed a topography-independent tur- 3of12

4 bulence-based method to spatially interpolate path delays due to water vapor: ^ e ¼ X a i i ð20þ i where ^ e are the interpolated (estimated) values, i are the GPS-derived values, and a i are their corresponding weights. [15] Using equations (9) and (3), the covariance matrix C m,m between all the GPS-derived values can be calculated as well as the covariance matrix C m,e between the GPSderived values and the interpolated (estimated) values, and then all the GPS-derived values can be used to construct a best linear unbiased estimator (BLU) [e.g., Kay, 1993]. The optimal weights can then be written in vector form as a BLU ¼ C 1 m;m C m;e þ 1 Cm;e T C 1 m;m s s T C 1 m;m s C 1 m;m s ð21þ where s is a vector of ones whose length is the number of measured values. [16] Jarlemark and mardson [1998] reported that the turbulence model produced lower RMS errors when used to interpolate zenith wet delays (ZW) in different directions at different time instants as compared with a method that estimates horizontal gradients in the wet delay. mardson and Johansson [1998] demonstrated that it was possible to use the turbulence model to interpolate ZW to a particular location with an accuracy of 1 cm, even with a widely distributed (100 km spacing) permanent GPS network. It should be noted that in their study the elevation variation reached a maximum of only 214 meters, and thus this approach might not be applicable in mountain areas such as SCIGN which have an elevation variation of about 3000 m. [17] Using GPS data from 126 stations in SCIGN spanning the period from January 1998 to March 2000, mardson et al. [2003] found that the spatiotemporally averaged variance of water vapor depends not only on the distance between observations (L), but also on the height difference (H) as follows: s int ¼ c L g þ kh ð22þ where c = 2.8, g = 0.44, k = 0.5 for observation intervals of one day, s is in millimeters, and L and H are in kilometers. It should be noted that the estimate of g in equation (22) corresponds to half of a in equation (3), i.e., g a/2. [18] In this paper, equations (19) and (22) were used to estimate the covariance matrix of the turbulence model, and typical values of c, a, and k estimated by mardson et al. [2003] were adopted for the SCIGN area. This topographydependent turbulence model is designated as GPS topography-dependent turbulence model (GTTM) in this paper since only GPS data are required in this turbulence model. [19] It should be noted that equation (22) can only be used with interferometric observations since it was derived from simulated interferometric values [mardson et al., 2003]. In order to apply equation (22) to interpolate PWV or ZW in a 2- water vapor field, the relationship between s 2 int and s 2 has to be taken into account (see equation (17b)) Cross Validation of the GTTM [20] In order to estimate zenith total delays (ZT), the GPS data were analyzed separately for each UTC day using the GIPSY-OASIS II software package in two steps: First, ZTs were determined together with the site position and receiver clocks. Then the site position was fixed to the average of that day, and only the ZTs and receiver clocks were estimated. For more details, see Li et al. [2003]. [21] It should be noted that surface pressure is required to calculate zenith hydrostatic delays (ZH) and to separate ZW from ZT. Although ZH is typically around 2.3 m and ZW varies roughly from 0 to 30 cm between the poles and the equator and from a few centimeters to about 20 cm at midlatitudes, ZW is much more highly variable (both spatially and temporally) [lgered, 1993]. Therefore ZT difference is a good approximation of water vapor variation. In order to avoid additional uncertainties in surface pressure measurements, ZT values were used instead of ZW in this paper. [22] To assess the capability of GTTM, a cross-validation test was applied to the GTTM model as well as the traditional inverse distance weighted interpolation method (IW) [Shepard, 1968] with 288 samples over up to 27 CGPS stations during the period from 1995 to 1996; that is, the total number of cases used in this test was about = 7776 (actual total of 7164). Cross validation involved removing a sampled point from the data set and using all the other data to estimate the value at that point [Williams et al., 1998]. This procedure was repeated for all sampled points and the observed data compared with the predicted. [23] Figure 1a shows the differences between the GTTM and IW interpolated values and GPS estimates against station height, where GPS estimates mean the ZT values derived directly from GPS data, and considered as true values here. It is clear in Figure 1a that the GTTM interpolated values appeared to be in closer agreement with the GPS estimates than the IW interpolated values. However, the RMS difference between the GTTM interpolated values and the GPS estimates was 5.5 cm with a maximum difference of up to 30 cm. This means that these interpolated values could not be applied directly to correct InSAR measurements. On closer inspection of the relationship between large differences and station heights, it was found that large differences usually existed over GPS stations at a height greater than 1100 m, indicating that the uncertainty of the interpolated values appeared to be mainly due to the high correlation between integrated column water vapor and topography; even the GTTM model could not account for such a high degree of correlation. [24] In Figure 1b, it is clear that the trends of different ZT estimates for the Table Mountain (TABL) GPS station were similar though offset by large, but nearly constant, amounts. Similar trends could be observed over other GPS stations including CHIL, CMP9, HOLC, WLSN, and OAT2 (Figure 1a). The constancy of this offset implies that elevation effects could be reduced to a large extent when differencing ZT values from different times. Taking into account the fact that what matters to an interferogram is the change in ZT from scene to scene, rather than the absolute value of ZT itself, another cross validation test was performed using the differences between ZT values one day apart (Figure 2). All together, there were 3418 ZT differences in this test. It is shown that the GTTM model was slightly better than the IW method with a standard deviation of 6.3 mm for the GTTM (Figure 2a) against 7.2 mm for the 4of12

5 Figure 1. (a) Cross validation of GTTM and IW methods on ZT values. The left vertical axis represents ZT values in mm, while the right vertical axis shows station height in meters. (b) Trend of three ZT time series derived by different methods over the TABL GPS station with a height of 2228 m. It should be noted there was no arbitrary shift of these three ZT series for better visibility. IW (Figure 2b). Therefore, from equation (16) of Zebker et al. [1997], the uncertainty introduced by the GTTM model might lead to additional uncertainties of 6.8 mm for deformation estimates when using RS-1/2 data with incidence angles of 23, and even the uncertainty introduced by the IW could only result in additional uncertainties of 7.8 mm. This implies that both the GTTM and IW methods could be used to produce zenith path delay difference maps (ZPM) for InSAR atmospheric correction using ZT (or ZW) differences between different times. 3. A GPS and InSAR Integration Approach [25] On the basis of the two-pass method in the Jet Propulsion Laboratory/California Institute of Technology (JPL/Caltech) ROI_PAC software [Rosen et al., 2004], a GPS/InSAR integration approach was designed to produce differential interferograms with a water vapor correction (Figure 3), which is similar to the approach we used with MOIS water vapor data [Li et al., 2005b]. This integration involves the usual steps of image coregistration, interferogram formation, baseline estimation from the precise orbits, and interferogram flattening and removal of the topographic signal by use of a digital elevation model (M). At this point, the integration approach diverges from the usual interferometric processing sequence with the insertion of a zenith path delay difference map (ZPM), which aims to reduce water vapor variations in interferograms. The ZPM is mapped from the geographic coordinate system to the radar coordinate system (range and azimuth) and 5of12

6 (a) (b) Figure 2. Cross validation of (a) GTTM and (b) IW methods on ZT daily differences. subtracted from the interferogram. For longer time intervals, a model for ongoing deformation can also be subtracted in the same way [Peltzer et al., 2001] at this step. This corrected interferogram can be unwrapped and then used in baseline refinement. The wrapped, water-vapor-corrected interferogram is made by flattening the original interferogram using the refined baseline and precise M, and then the ZPM is subtracted from the differential interferogram. In order to obtain the unwrapped water vapor corrected interferogram, a new simulated interferogram is created using the refined baseline and topography and is subtracted from the unwrapped phase (including orbital ramp) with the water vapor model removed. [26] Prior to this study, several studies had been carried out to calibrate water vapor effects on InSAR using atmospheric delay models or independent data sources, including elacourt et al. [1998], Bonforte et al. [2001], and Wadge et al. [2002]. Among these studies, atmospheric effects were subtracted from (or compared with) the wrapped (or unwrapped) phase, but these water vapor corrections were not used to improve the InSAR processing. Buckley et al. [2003] suggested applying water vapor corrections to the unwrapped phase, and then using the corrected unwrapped phase to refine the baseline. Although the difference between the uncorrected and corrected interferograms was marginal, it was the first time water vapor correction was integrated with InSAR processing. However, this method might suffer from smearing of atmospheric artifacts during the filtering process, widely used in InSAR processing to reduce phase noise [e.g., Goldstein and Werner, 1998]. For instance, some topography-dependent water vapor signals, whose wavelength is relatively short but could be estimated using GPS data or other data sets, might be spread over a larger area when a nonlinear filter is applied before subtracting water vapor. It is believed that the water vapor correction approach proposed here has advantages over all previous approaches: (1) since the water vapor correction is performed directly on the unfiltered, wrapped interferogram followed by filtering, the performance of the filter does not have any unequal impacts on the water vapor correction; (2) reducing atmospheric effects on the wrapped interfero- Figure 3. Two-pass differential InSAR processing flowchart with water vapor correction. 6of12

7 Table 1. etails of Interferograms Utilized for This Study Track Frame ate 1 ate 2 t, days B?, a m s, b rad Ifm Jan Jan to Ifm Oct Oct to Ifm Apr Apr to a Perpendicular baseline at center of swath which varies along the track between the values shown. b Possible phase error due to the topographic uncertainty of SRTM M. grams may improve phase unwrapping; and (3) the corrected unwrapped phase is expected to improve the refined baseline. 4. Application to RS Tandem ata Over SCIGN [27] In order to evaluate the efficiency, utility, and potential of GTTM, three case studies (Table 1) were performed with two processing procedures: (1) the usual two-pass method and (2) the use of water vapor correction. RS-1/ RS-2 Tandem data acquired just 1 day apart were used, so there should be no significant deformation signals in the differential interferograms. The phase remaining in the Tandem interferograms after removing the known topographic and baseline effects should be almost entirely due to changes in the atmosphere between the two acquisitions. [28] The topographic phase contribution was removed using a 1-arc sec (30 m) M from the Shuttle Radar Topography Mission (SRTM) [Farr and Kobrick, 2000]. In order to assess the absolute accuracy of the SRTM M for the test area, it was compared with GPS-derived heights over 100 CGPS stations. It should be noted that the SRTM M distributed by the U.S. Geological Survey is referenced to the GM96 geoid, so that GPS-derived ellipsoid heights had to be converted to orthometric heights using geoid separations before comparisons [Zilkoski, 2001]. The mean difference of (SRTM GPS) is 3.4 m with a standard deviation of 7.9 m. The differences between these two data sets are mainly due to the different levels to which they refer: [29] 1. The SRTM M is based on heights from different penetration depths into vegetation and urban canopies; that is, it represents a height related to the average phase center of the radar return with a spatial resolution of 30 m. Many of the SCIGN GPS stations are located in urban or suburban parts of the Los Angeles metropolitan area so there are many buildings that strongly reflect the SRTM radar and map to heights well above ground level. [30] 2. The GPS-derived value represents the height over a GPS station that is usually around 1 m above the surface. [31] Taking into account the high accuracy of the GPSderived orthometric heights (<2 5 cm [Zilkoski, 2001]), it can be concluded that the accuracy of the SRTM M is within 7.9 m, which is consistent with Farr and Kobrick [2000]. From the elevation sensitivities in Table 1, it can be concluded that atmospheric effects should dominate over M errors in the differential interferograms (referred to as Ifms hereafter) with short baselines (e.g., Ifm 1 and 3). [32] For the analysis of the spatial variation of unwrapped phase (or water vapor signals), a 2- spatial structure function (2-SSF) was defined as [Hanssen, 2001] x ðr; aþ ¼ ½dðr 0 ; r; aþ dðr 0 ÞŠ 2 ð23þ where d is the unwrapped phase (or water vapor signals), r 0 is any random pixel location in the image, r is the distance from the denoted pixel, a is the azimuth from the denoted pixel, and the angle brackets indicate an ensemble average. The 2-SSF is similar to a variogram and gives the expectation of the squared difference between two pixels at a certain distance r and azimuth a in the image and reveals the spatial phase variation in the interferogram. From the definition of the 2-SSF in equation (23), the following conclusions can be made: [33] 1. The larger the SSF value, the larger the phase variation at the given distance and azimuth. [34] 2. The 2-SSF is symmetric about a point at the origin, and the center of the plot is usually selected as the origin. [35] 3. There are fewer measurements available for the edges and corners of the plot; consequently, some caution needs to be exercised when interpreting the borders [Hanssen, 2001]. [36] The unwrapped phase is employed to show interferograms in this paper. It should be noted that the unwrapped phase has been converted to range change in millimeters where a positive range change means apparent motion of the ground away from the satellite (or an increase in the delay of radar propagation due to the atmosphere) Interferogram: January 1996 [37] Figure 4a shows topography from the 1-arc sec SRTM M, and Figure 4b shows the unwrapped phase of the RS-1/RS-2 Tandem interferogram January 1996 (i.e., Ifm1). It is clear that atmospheric signals in Ifm1 appear to be highly correlated with topography. [38] Figure 5 shows the use of the GTTM and IW methods to correct Ifm1. After applying the GTTM water vapor correction to the original interferogram, it is clear that the Tandem interferogram was significantly improved. Most residual fringes were removed with the RMS decreasing from 1.30 rad (0.58 cm) (Figure 5a) to 0.87 rad (0.40 cm) (Figure 5c). On the other hand, when the IW was used, the RMS of the resultant interferogram decreased to only 1.08 rad (0.49 cm) (Figure 5e), indicating that the GTTM model works more efficiently than the IW. On closer inspection of the magnitude of the unwrapped phase over the Palos Verdes hills (indicated by a black rectangle), it was found that these signals were significantly reduced after applying the GTTM correction (Figure 5c), while the amount slightly increased after the IW correction (Figure 5e), providing strong supporting evidence for the conclusion that the GTTM model can reduce topography-dependent water vapor signals better than the IW method. [39] It should be noted that there is a larger area of unwrapped phase in the San Gabriel Mountains at the N corner of Figure 5c (and/or Figure 5e) than indicated in Figure 5a, implying that the reduction of the sharp phase 7of12

8 The masking of the more mountainous areas due to their low correlation also removes the areas where the topography-dependent correction would have the greatest effect. [43] From Figures 6b, 6d, and 6f, for 2-SSF for Ifm2 before and after correction, it is obvious that the phase variation decreased after both water vapor corrections, indicating that both the GTTM and IW methods can reduce topography-independent water vapor effects significantly. A further comparison between Figures 6d and 6f shows that the GTTM is slightly better at reducing the topography-independent water vapor effects than the IW method in this case. Figure 4. Correlation between topography and unwrapped phase. (a) SRTM M, elevations in meters. SGM, San Gabriel Mountains; SMM, Santa Monica Mountains; and PV, Palos Verdes. (b) Interferogram gradients due to the atmosphere in the mountains improved the filtering and phase unwrapping. [40] Comparing the figures for the square root of the 2- SSF of Ifm1, with and without water vapor correction (Figures 5b, 5d, and 5f), one can conclude that the phase variation decreased after both the GTTM and IW corrections. Comparison between Figures 5d and 5f shows that the GTTM model appeared better at reducing atmospheric effects on this interferogram than the IW method Interferogram: October 1995 [41] Figure 6 shows an RS Tandem interferogram from October 1995 with atmospheric signals that were uncorrelated (or poorly correlated) with topography (see M in Figure 4a). The RMS of the uncorrected unwrapped phase was 1.56 rad (0.70 cm) (Figure 6a). The larger RMS value than that of the first example (namely, Ifm1) could be due to the larger perpendicular baseline in the second interferogram (Table 1), but the phase shows a long-wavelength pattern across the whole scene. The areas of steep slopes in this interferogram have very low interferometric correlation due to the long baseline [Fielding et al., 2005] and are masked out (gray in Figure 6). [42] After applying the GTTM correction to Ifm2, the RMS of the unwrapped phase decreased to 1.26 rad (0.56 cm) (Figure 6c), while the RMS decreased to 1.35 rad (0.61 cm) after the IW correction (Figure 6e). It should be noted that the residuals were greater on the right hand side than on the left-hand side in Figures 6c and 6e after water vapor correction, which may be attributable to the sparseness of GPS stations (indicated by triangles) in the east. This interferogram has atmospheric effects with a large spatial scale, which are easier to measure with the GPS stations, so the water vapor correction is quite successful Interferogram: 5 6 April 1996 [44] Atmospheric ripples (probably due to inertial gravity waves in the northeastern part of the area covered) with a characteristic wavelength of 4 12 km were observed in the third example, an RS Tandem pair from April 1996 (Ifm3) (Figure 7a). The atmospheric ripples are still present in the water vapor corrected interferograms (Figures 7c and 7e). However, the RMS slightly decreased from 1.31 rad (0.59 cm) to 1.22 rad (0.55 cm) after the GTTM correction, and to 1.26 rad (0.57 cm) after the IW correction, indicating that both the GTTM and IW models can reduce atmospheric effects to some extent even under such conditions. Comparisons between the 2-SSF images (Figures 7b, 7d, and 7f) show that the phase variation decreased after both corrections, particularly in the S-NW direction. [45] Note that there were only three GPS stations in the N part of the interferogram (Figures 7c and 7e) and none were located in the area with atmospheric ripples. Water vapor effects were reduced in the western part after both corrections where there were more GPS stations. It is concluded that neither the GTTM nor IW method can remove atmospheric artifacts with a wavelength shorter than the spacing of GPS stations (except where the atmospheric variations are correlated with elevation and the topographic relief has a short wavelength for the GTTM model). Thus it appears that the distribution of GPS receivers still plays a key role in the integration of GPS and InSAR. In these cases, independent water vapor data sources, such as those provided by the NASA Moderate Resolution Imaging Spectroradiometer (MOIS) [Li et al., 2005b], might be a better option for InSAR atmospheric correction. 5. Conclusions [46] In this paper, a topography-dependent turbulence model (i.e., GTTM) has been developed using GPS data only. Cross validation tests on the GTTM and IW methods showed that (1) in order to produce zenith path delay difference maps (ZPM) for InSAR atmospheric correction, the GTTM and IW methods should be applied to ZT differences (instead of ZT values); this is crucial to reduce (if not completely remove) the component due to topographic effects; and (2) the GTTM model appeared to perform better than the IW, with a standard deviation of 6.3 mm for the GTTM (Figure 2a) against 7.2 mm for the IW (Figure 2b). [47] A GPS and InSAR integration approach was successfully incorporated into the JPL/Caltech ROI_PAC soft- 8of12

9 Figure 5. Interferogram (a) Original Ifm1; (b) 2-SSF for original Ifm1; (c) corrected Ifm1 using the GTTM; (d) 2-SSF for the GTTM-corrected Ifm1; (e) corrected Ifm1 using the IW; (f) 2-SSF for the IW-corrected Ifm1. Note solid black triangles in Figures 5c and 5e represent GPS stations used, and the grey in Figures 5b, 5d, and 5f implies that no valid pair of unwrapped phase existed at the given distance and azimuth. ware. This integration approach reduced atmospheric effects in interferograms, and appeared to improve InSAR processing such as phase unwrapping. The application of this integration approach to RS tandem data showed that the GTTM can reduce significantly not only topographydependent but also topography-independent atmospheric effects. The failure to reduce short-wavelength atmospheric ripples using the GTTM and IW methods indicated that the reduction of topography-independent short-wavelength water vapor effects is limited by the spatial distribution of GPS stations. However, it is evident that the long-wavelength water vapor variations can be removed through the use of GPS data and, particularly, some height-dependent effects can be reduced when using the GTTM. The extent to which the reduction of atmospheric water vapor artifacts can be done depends on two factors: [48] The first factor is the density of Continuous GPS (CGPS) stations across the region of interest. In southern California, the number of CGPS stations in the SCIGN, the arthscope Plate Boundary Observatory (PBO) and other western U.S. geodetic networks has greatly increased since the time frame covered by the RS Tandem mission. The average station spacing increased from 30 km on 1 May 1995 to 7 km on 1 May 2005 in the Los Angeles SCIGN region. Planned installation of arth- Scope PBO stations over western North America will continue to increase the CGPS station coverage in the near future. [49] The second factor concerns the atmospheric water vapor distribution over the region of interest at the time of SAR data acquisition. When water vapor varies significantly with short wavelengths, taking out or adding a GPS station from the CGPS network should make a significant difference to InSAR atmospheric water vapor correction in that local area. When water vapor only has low-frequency variations, particularly in flat areas, a well-distributed GPS network with 25 stations could have almost the same impact as one with 100 GPS stations. [50] It is worth mentioning that a subsequent validation study was performed using SA s NVISAT Advanced 9of12

10 Figure 6. Interferogram (a) Original Ifm2; (b) 2-SSF for original Ifm2; (c) corrected Ifm2 using the GTTM; (d) 2-SSF for the GTTM-corrected Ifm2; (e) corrected Ifm2 using the IW; (f) 2-SSF for the IW-corrected Ifm2. Solid black triangles represent GPS stations used and the grey in Figures 6b, 6d, and 6f implies that no valid pair of unwrapped phase existed at the given distance and azimuth. 10 of 12

11 Figure 7. Interferogram (a) Original Ifm3; (b) 2-SSF for original Ifm3; (c) corrected Ifm3 using the GTTM; (d) 2-SSF for the GTTM-corrected Ifm3; (e) corrected Ifm3 using the IW; (f) 2-SSF for the IW-corrected Ifm3. Solid black triangles represent GPS stations used. 11 of 12

12 Synthetic Aperture Radar (ASAR) data collected in 2004 and 2005 [Li et al., 2005a]. It was shown that even under cloudy conditions, the RMS differences between GPSderived and InSAR-derived range changes in the line-ofsight direction varied from 0.57 cm to 0.80 cm with a reduction of up to 0.55 cm after applying the GTTM water vapor correction model. [51] It should also be noted that the model parameters c, a, and k were fixed to the values estimated from the 126 GPS stations over SCIGN during the period from January 1998 to March 2000 [mardson et al., 2003]. A better reduction might be achieved if the model parameters were estimated from case to case, taking into account the large water vapor variations observed during spatial structure analysis, which will be an important issue in future work. [52] Acknowledgments. This work is supported by an Overseas Research Students Award (ORS) and a UCL Graduate School Research Scholarship to Z. Li at University College London. It is also associated with the NRC arth Observation Centre of xcellence: Centre for the Observation and Modeling of arthquakes and Tectonics (COMT). Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We are very grateful to Simon Williams, an anonymous reviewer, and Associate ditor Yehuda Bock for thoughtful and thorough reviews and to Tim J. Wright and Ant Sibthorpe for useful discussion. We thank JPL/Caltech for the use of the ROI_PAC software to generate our interferograms. The RS data were supplied under SA NVISAT data grant AO I = 853 (HAZARMAP), and the GPS data were obtained from Scripps Orbit and Permanent Array Centre (SOPAC, References Agnew,. C. (1992), The time-domain behavior of power-law noises, Geophys. Res. Lett., 19(4), Bock, Y., and S.. P. Williams (1997), Integrated satellite interferometry in southern California, os Trans. 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