### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Application to Long Valley, CA
- 4. Future Directions
- 5. Conclusions
- Appendix A:: Decorrelating Power of Orthogonal Wavelet Bases
- Acknowledgments
- References
- Supporting Information

[1] We present a new approach to extracting spatially and temporally continuous ground deformation fields from interferometric synthetic aperture radar (InSAR) data. We focus on unwrapped interferograms from a single viewing geometry, estimating ground deformation along the line-of-sight. Our approach is based on a wavelet decomposition in space and a general parametrization in time. We refer to this approach as MInTS (Multiscale InSAR Time Series). The wavelet decomposition efficiently deals with commonly seen spatial covariances in repeat-pass InSAR measurements, since the coefficients of the wavelets are essentially spatially uncorrelated. Our time-dependent parametrization is capable of capturing both recognized and unrecognized processes, and is not arbitrarily tied to the times of the SAR acquisitions. We estimate deformation in the wavelet-domain, using a cross-validated, regularized least squares inversion. We include a model-resolution-based regularization, in order to more heavily damp the model during periods of sparse SAR acquisitions, compared to during times of dense acquisitions. To illustrate the application of MInTS, we consider a catalog of 92 ERS and Envisat interferograms, spanning 16 years, in the Long Valley caldera, CA, region. MInTS analysis captures the ground deformation with high spatial density over the Long Valley region.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Application to Long Valley, CA
- 4. Future Directions
- 5. Conclusions
- Appendix A:: Decorrelating Power of Orthogonal Wavelet Bases
- Acknowledgments
- References
- Supporting Information

[2] Geodetic imaging aims to discover new crustal deformation processes, to monitor known sources of deformation, and to estimate the values and uncertainties of the parameters controlling these processes. Here, we focus on the use of repeat pass satellite interferometric synthetic aperture radar (InSAR) data. (For a review of InSAR techniques, we refer the reader to *Simons and Rosen* [2007] and references therein.) For many important geophysical targets, we already have deep archives of radar images from a given satellite and viewing geometry. More importantly, future radar missions should provide frequent image acquisitions with high interferometric correlation. In response to the increase in temporal density of radar acquisitions, we present a new approach to exploit InSAR archives to determine the spatiotemporal evolution of surface deformation.

[3] The simplest approach to utilizing multiple interferograms is to average them (often referred to as “stacking”, although in this paper we do not use the term “stack” to be synonymous with “average”). If the primary geophysical target is a single event that occurred quickly (i.e., anything taking less than the image acquisition interval) or is a gradual process occurring at constant rate, averaging is commonly used to increase the signal-to-noise ratio [e.g., *Peltzer et al.*, 2001; *Lyons and Sandwell*, 2003; *Gourmelen and Amelung*, 2005; *Pritchard and Simons*, 2006]. For example, averaging reduces the effects due to tropospheric delays, as these effects are typically uncorrelated on timescales of more than a day [e.g., *Hanssen*, 2001; *Emardson et al.*, 2003]. Averaging images can also reduce computational burden in parameter estimation schemes by reducing the amount of observations [e.g., *Pritchard and Simons*, 2006]. In the case of a single rapid event, the displacements are averaged, whereas for a constant rate process, it is common to average the velocities. We note that by averaging a set of interferograms, one is implicitly assuming a functional form for the deformation field (a step or a linear function) and simply estimating the appropriate constants [*Simons and Rosen*, 2007].

[4] Over the last decade, the community has made considerable progress in estimating time-dependent deformation from InSAR data. Approaches include Permanent or Persistent Scatterer (PS) techniques [e.g., *Ferretti et al.*, 2000, 2001; *Colesanti et al.*, 2003; *Wegmüller*, 2003; *Hooper et al.*, 2004; *Bürgmann et al.*, 2006] as well as interferogram time series techniques [e.g., *Lundgren et al.*, 2001, 2009; *Berardino et al.*, 2002; *Schmidt and Bürgmann*, 2003; *Lanari et al.*, 2004; *Hooper*, 2008]. Most of these techniques exploit long time series of over a decade of SAR observations. PS techniques are restricted to using only temporally coherent point scatterers, and require a parametrization of the time dependent deformation. In contrast, other time series techniques use all available pixels in all scene combinations where the interferometric baseline is well below a critical baseline [*Lundgren et al.*, 2001; *Berardino et al.*, 2002; *Schmidt and Bürgmann*, 2003; *Lanari et al.*, 2004]. As a result, in the latter techniques one can expand the area of usable pixels, and for well-correlated areas, potentially the time-dependent deformation of every pixel could be determined. For each usable pixel, interferometric pairs yield phase differences over available time intervals in the time series, with the shortest possible sampling interval being the satellite repeat period. In contrast to PS techniques, the phase difference measurements at a given pixel in all combinations of radar images are inverted to determine the time-dependent ground deformation. Typically, a piecewise linear time-dependence is sought, with the most common version of this latter technique being the Small Baseline Subset (SBAS) method [e.g., *Lundgren et al.*, 2001; *Berardino et al.*, 2002; *Doubre and Peltzer*, 2007]. Recently, *Hooper* [2008] proposed a hybrid approach to estimating time-dependent ground deformation, utilizing both PS and SBAS methodologies.

[5] While providing a solid foundation, these existing approaches have several important shortcomings. Principally, current methods are applied on a pixel-by-pixel basis, with pixel-to-pixel connection made by assuming a single master reference image. Pixel-by-pixel methods ignore the known spatial covariances in the observations, as well as expectations for “reasonable” behavior even in regions of low interferometric correlation. Current methods also use *ad-hoc* strategies to deal with interferograms containing regions of low interferometric correlation, or to deal with sets of interferograms that are not connected to other interferograms by common image acquisitions.

[6] We propose a multiscale InSAR time series (MInTS) approach to determine the spatiotemporal ground deformation from a catalog of interferograms of a region. For brevity, we refer to a catalog of interferograms as a “stack” throughout this paper. In our approach, we use a wavelet decomposition in space and a general parametrization in time. Spatial wavelet decompositions have been effectively applied to InSAR data in order to provide better estimates of orbital errors [*Shirzaei and Walter*, 2011]. Additionally, although not using wavelet based filters, *Lin et al.* [2010] used a multiscale decompositions in order to provide a more robust estimation of topographically correlated tropospheric path delays. The spatial wavelet decomposition we use in MInTS is chosen in order to efficiently deal with the spatial covariances in the InSAR phase difference measurements and to provide an efficient method for interpolating across regions of low interferometric correlation.

[7] Our approach to the temporal parametrization is highly influenced by common approaches used for post-processing continuous GPS data to extract velocities, co-seismic offsets, etc (e.g., QOCA and GLOBK) [*Herring et al.*, 1990; *Dong et al.*, 1998, 2002]. Determination of time-dependent ground deformation using InSAR data substantially differs from GPS time series techniques, as InSAR has a much higher density of samples in space, but much lower density of samples in time. Moreover, InSAR measures differences in ground position over time spans between repeat orbits, whereas GPS measures position through time relative to some reference. In this paper, we only focus on unwrapped interferograms from a single viewing geometry, thereby only resolving ground deformation along the satellite line-of-sight (LOS). In the following, we describe the theory and methodology of MInTS, we illustrate the application of MInTS to an example InSAR stack from Long Valley caldera, CA, and we discuss some future directions of MInTS development.

### 3. Application to Long Valley, CA

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Application to Long Valley, CA
- 4. Future Directions
- 5. Conclusions
- Appendix A:: Decorrelating Power of Orthogonal Wavelet Bases
- Acknowledgments
- References
- Supporting Information

[35] To illustrate the application of MInTS, we consider a stack of interferograms of the Long Valley, CA, volcanic region. Several uplift events in Long Valley have been observed using ground based geodesy over the last few decades [*Langbein et al.*, 1993, 1995; *Langbein*, 2003; *Feng and Newman*, 2009]. The largest was in late 1997, where 2-color electronic distance measurements (EDM) indicated short-lived rapid uplift of the resurgent dome in the Long Valley caldera [*Langbein*, 2003].

[36] We construct an interferometric stack using 24 ERS-2 and 15 Envisat SAR descending acquisitions, with a total of 92 interferograms (63 ERS-2 and 29 Envisat interferograms) spanning just over 16 years (Figure 6a). The interferograms are looked down eight times, equivalent to a resolution of about 196 and 160 meters in the range and azimuth directions, respectively. We unwrap the interferograms using the SNAPHU algorithm [*Chen and Zebker*, 2001]. The SNAPHU algorithm underestimates the regions of low interferometric correlation, for instance almost always unwrapping over Mono lake. Therefore, we calculate a common mask based on regions below a correlation coefficient of 0.25 in an interferogram formed from a one-day ERS pair (01–02 June 1996). We apply this common mask to all interferograms, in addition to those determined for each interferogram during unwrapping. We remove the best fit bi-linear ramp from each interferogram to account for long-wavelength signal we attribute to orbital errors.

[37] We follow the procedure outlined in section 2.6 to take the DWT of the interferogram stack. The time it takes to compute the DWT and calculate the weights of the wavelet coefficients of one interferogram depends on the interferogram size and the amount of interferometric decorrelation. For each of these interferograms, it takes a few minutes on a desktop computer with dual 3.2 GHz quad-core CPUs. Each interferogram is independently processed, and thus the process is inherently parallelizable. For each stack of interferograms, the DWT and the weights only need to be computed once.

[38] We invert the stack assuming a time-dependent deformation model composed of 42 evenly spaced B^{∫}-splines, and we regularize by damping the magnitude of the amplitudes of the B^{∫}-splines. Since the SAR acquisitions are not evenly spaced in time (Figure 6a), we apply a variable shape smoothing in order to more heavily damp the B^{∫}-spline amplitudes during periods of no SAR acquisitions, relative to the amplitudes during periods of dense acquisitions. Three parameters control the model regularization in the inversion: (1) the penalty parameter, *λ*, (2) the singular value truncation, *p*, of the pseudo-inverse used to determine *S*_{p}, and (3) the exponent in the variable smoothing, *a*. We use 12-fold cross-validation in order to determine the optimal *λ* at each scale and location separately. We assume a common *p* and *a* in all inversions, since cross-validation on these two parameters is not well defined as the cross-correlation objective functions do not have a clear minimum in terms of *p* or *a*. The choice of *a* has the smallest affect on the inversion results, and we use *a* = 1/2. Recall that each model parameter is regularized by a composite penalty parameter, and using a small *p* results in all of the model parameters being regularized almost uniformly (Figure 6b). As *p* approaches the number of model parameters, corresponding to model parameters during times that are densely sampled decreases (Figure 6b), and the model is largely non-regularized during those periods. We use *p* = 20, although the results we present here are consistent with those using other values of *p*. A more formal approach to selecting the total regularization is left to a future analysis.

[39] MInTS analysis of this example stack takes just under 12 hours on the aforementioned computer, which includes cross-validated inversion of 262,088 sets of wavelet coefficients. In Figure 7, we show the cumulative LOS ground displacement determined in the MInTS analysis over four time periods, roughly corresponding to prior to the 1997–1998 uplift event, during the event, during the period of sparse radar acquisitions (cf. Figure 6a), and during the last four years of the stack time span. The spatiotemporal deformation is presented in the radar range and azimuth coordinates, and for these descending orbits, increasing range and azimuth is roughly westward and southward, respectively. The deformation is along the satellite LOS direction. We use a convention wherein increasing *ρ* corresponds to ground deformation toward the satellite, which we loosely refer to as uplift. In Figure 8a, we show the time-dependent deformation in three regions, roughly corresponding to the region of maximum uplift during the 1997–1998 uplift event (labeled A in Figures 7 and 8), the Casa Diablo geothermal field (labeled B), and north of Mammoth mountain (labeled C).

[40] The results show a steady uplift of the resurgent dome during 1993 to about mid-1997, followed by an abrupt increase in uplift rate until about mid-1998. The maximum uplift is to the eastern edge of the resurgent dome, and the location of maximum uplift shifts slightly to the south when the uplift accelerated during 1997–1998 (Figure 7). Following the uplift event, there is a slight subsidence of the Long Valley caldera to the west of the resurgent dome. We do not resolve the uplift event that occurred between 2002 and 2003, which was of smaller magnitude than the 1997–1998 event [e.g., *Feng and Newman*, 2009]. That we do not capture the smaller uplift is not surprising, as there are no SAR acquisitions during 2001 to 2004, and the model parameters are highly damped during this time period due to the model-resolution-based variable shape smoothing (Figure 6). There is a slight subsidence over the resurgent dome during 2004–2008, and greater subsidence (≈1 cm/yr) of the Casa Diablo geothermal plant.

[41] The results we obtain here are broadly consistent with those obtained using SBAS [*Tizzani et al.*, 2007, 2009], although our results have higher spatial density. One notable difference is that we do not resolve the subsidence at the Casa Diablo geothermal field during the 1990's [*Howle et al.*, 2003; *Tizzani et al.*, 2007]. We do resolve that the Casa Diablo region is not uplifting as rapidly as in the rest of the caldera, but we do not resolve the subsidence over several kilometers that has been documented by leveling measurements [*Howle et al.*, 2003]. The early subsidence in the Casa Diablo region is not apparent in the interferograms we consider here (see auxiliary material), which indicates that the Casa Diablo deformation was essentially smoothed over in the InSAR processing.

[42] The inferred spatiotemporal deformation is broadly consistent with the EDM observations (Figure 7b). EDM measures changes in distance between two points on the ground, whereas the recovered deformation from the InSAR stack is of ground deformation along the satellite LOS. Due to the differing nature of these two data sets, we do not attempt a direct comparison between our results and the EDM measurements. Continuous GPS measures the ground deformation in three components through time, and we show the projection of these measurements onto the LOS at four continuous GPS sites in Figure 8c. The continuous GPS records start in late 2001, and resolve a relatively small magnitude uplift from 2002 to 2004, followed by relatively steady deformation [*Feng and Newman*, 2009]. The along LOS deformation we constrain from the MInTS analysis is noisier than the GPS time series (Figure 8c).

[43] In the present version of MInTS, we do not use either the EDM or GPS data to constrain the spatiotemporal deformation. The spatiotemporal deformation is determined such that the predicted interferograms best fit the observed interferograms in the InSAR stack. There are systematic residuals that are correlated with the resurgent dome and caldera boundary (see auxiliary material). There is a residual signal in the center of the resurgent dome in several of the interferograms that is similar to the observed phase differences, and probably indicates un-modeled ground deformation. That there is a slight amount of un-modeled ground deformation apparent in the residuals indicates this spatiotemporal deformation field estimated from MInTS analysis is slightly too smooth in time. On the other hand, some of the residuals appear to be correlated with topography, and may indicate topographically correlated path delays due to tropospheric effects [e.g., *Delacourt et al.*, 1998; *Cavalié et al.*, 2007; *Lin et al.*, 2010]. The topographically correlated residuals are most apparent in the Envisat interferograms, which capture a time during which there was less ground deformation than in the proceeding ten years.

### 4. Future Directions

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Application to Long Valley, CA
- 4. Future Directions
- 5. Conclusions
- Appendix A:: Decorrelating Power of Orthogonal Wavelet Bases
- Acknowledgments
- References
- Supporting Information

[44] MInTS is a new tool for exploring crustal deformation using temporally deep collections of InSAR observations. The framework of MInTS also lends itself to being extended, for instance to include better treatment of non-ground deformation processes during the inversion, to utilize more robust estimation algorithms, and to further constrain ground deformation by incorporating ground-based geodetic measurements. In this section, we outline several issues that need to be addressed in future revisions of MInTS.

[45] The method described here can be easily extended to include multiple viewing geometries (i.e., ascending and descending orbits, or other satellite platforms). With sufficient observations on multiple viewing geometries, there is the potential to recover the underlying 3D deformation. Additionally, the wavelet based approach may possibly be extended to operate on the original wrapped interferograms, thereby avoiding errors incurred in unwrapping [e.g., *Goldstein et al.*, 1988; *Chen and Zebker*, 2001; *Simons and Rosen*, 2007].

[46] In our representation of interferometric measurements, equation (10), we lumped all interferometric signal that is not associated with ground deformation into an un-described noise. Either correcting the interferograms for significant spatially coherent noise prior to MInTS analysis, or simultaneously estimating a model of such noise, may yield a better estimation of ground deformation [e.g., *Pritchard et al.*, 2006; *Cavelié et al.*, 2007; *Lin et al.*, 2010; *Shirzaei and Walter*, 2011]. Given even a purely horizontally stratified troposphere, one expects topographically correlated delays [e.g., *Delacourt et al.*, 1998; *Cavalié et al.*, 2007]. These so-called “tropostatic delays” are typically a large contribution to the interferometric measurement, and are relatively straightforward to model [e.g., *Cavelié et al.*, 2007; *Lin et al.*, 2010]. Excursions from horizontal stratification, due to atmospheric turbulence are often important [e.g., *Lin et al.*, 2010], and are more difficult to model. In the case of significant turbulent tropospheric signal, high resolution weather models may be used to estimate the spatial variation in wet delay [*Foster et al.*, 2006; *Puyssegur et al.*, 2007].

[47] The interferometric measurements for Long Valley show strong topographically correlated signal (see auxiliary material). *Lin et al.* [2010] developed a robust method for the estimation of topographically correlated signal using a multiscale decomposition filter-bank approach, and estimated the tropostatic signal in the ERS interferometric data for Long Valley. The multiscale filter-bank approach of *Lin et al.* [2010] is similar to as is used in MInTS, although their filter-bank is built with Gaussian filters and the estimation of the tropostatic signal is done in the spatial domain. In contrast, MInTS is based on a wavelet-based filter-bank, and the estimation is done in the wavelet domain. The multiscale approach of *Lin et al.* [2010] may be extended to the use of a wavelet-based filter-bank and the estimation of tropostatic signal in the wavelet domain. In which case, it would be trivial to incorporate the simultaneous estimation of tropostatic signal and ground-deformation into MInTS.

[48] In the regularized estimation, we use an *n*-fold cross-correlation to select the optimal penalty parameter. In cross-validation, it is essential that the data in the learning and testing sets are independent [*Picard and Cook*, 1984]. In the case of inverting the wavelet coefficients of InSAR interferograms, ensuring the independence of the learning and testing sets is only possible if the interferograms in the learning and testing sets share no common SAR acquisitions. Hence, to ensure that the interferograms in the learning and testing are formed from different radar acquisitions it is likely that interferograms would need to be held out of both the learning and testing sets. The particular interferograms that would need to be removed would depend on the *n* random learning/testing sets in the *n*-fold cross-validation. The random removal of data from the estimation scheme is problematic. For instance, particular folds in which a larger number of interferograms are dropped might have significantly less resolving power that other folds in the cross-validation, thereby complicating the selection of the optimal penalty parameter [e.g., *Picard and Cook*, 1984]. We have tested the cross-validation scheme using learning/testing sets formed on the interferograms and on the acquisitions, and found that in almost all instances the cross-validation returns the lowest tested *λ* (i.e., noisiest model) when the data are partitioned based on interferograms. When the learning/testing sets are formed on SAR acquisitions, the cross-validation is almost always stable, with a clear *λ* in which the estimated models from inverting all learning sets best fits the data in the associated testing sets. When the learning/testing sets are formed from the acquisitions, as long as the number of folds in the cross-validation is not comparable to the number of acquisitions, the dependence between the learning and testing sets is minimized. In MInTS, cross-validation is not required, and users can choose to invert for ground deformation with specified penalty parameters.

[49] The estimation of ground deformation in the present version of MInTS is entirely built upon least squares. The use of a purely least squares approach, relying on an L2 norm, is somewhat debatable since phase noise is not Gaussian. Additionally, interferometric noise contains other non-Gaussian processes, such as unwrapping errors [e.g., *Goldstein et al.*, 1988; *Chen and Zebker*, 2001], and tropospheric path delays [e.g., *Massonnet et al.*, 1993; *Goldstein*, 1995]. It may be more robust to use an L1 norm when dealing with interferometric data.

[50] Due to the often large contributions to interferometric measurements from processes other than ground deformation, it is difficult to constrain ground deformation during periods of low deformation rates. This can be readily seen in our demonstration of MInTS with the Long Valley InSAR stack (Figures 7–8), where although the deformation we estimated is broadly consistent with ground-based geodetic measurements, our estimation does not capture the small amplitude deformation after about 2002. It may be that the Envisat interferograms are dominated by topographically correlated signal, and simultaneous estimation of a tropostatic noise model may improve the estimates of the spatiotemporal ground deformation. However, our inability to capture the low signal deformation may also be due to the inherent sensitivity of interferometric measurements. Incorporation of ground based deformation measurements, primarily GPS, into the MInTS analysis has the potential to add strong constraints on the spatiotemporal ground deformation, as ground based geodetic measurements tend to include much more redundant time sampling than InSAR measurements. GPS and InSAR are largely complementary measurements, as GPS is a more continuous record of time-dependent ground deformation at specific locations, while InSAR provides a near continuous image of ground offsets. Incorporating the point-measurements of GPS into the multiscale wavelet decomposition of MInTS is not a straightforward task, and we save it for a future study.

### 5. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Methods
- 3. Application to Long Valley, CA
- 4. Future Directions
- 5. Conclusions
- Appendix A:: Decorrelating Power of Orthogonal Wavelet Bases
- Acknowledgments
- References
- Supporting Information

[51] We present MInTS, a new approach to extracting continuous spatiotemporal ground deformation from InSAR catalogs. We use a Meyer wavelet decomposition, which is chosen in order to efficiently deal with the spatial covariances in the InSAR phase difference measurements, as the Meyer wavelet transform essentially diagonalizes the interferogram spatial covariances. We base MInTS on a generalized model of time-dependent ground deformation, that can include linear rates, offsets, logarithmic or exponential trends, sinusoidal oscillations, and a collection of B-spline-based basis functions. We propose the B-spline-based representation of time-dependent ground deformation to capture unknown processes, while the other functions may be used to parameterize known processes (e.g., earthquakes, postseismic deformation, seasonal signals).

[52] The estimation of ground deformation is done in the wavelet-domain at each location and scale independently. We use a regularized least squares estimation, optionally using *n*-fold cross-validation in order to select the optimal regularization penalty parameter. We include an additional model-resolution-based regularization in the estimation, which may be used to more heavily damp the model during periods of sparse SAR acquisitions, compared to during periods of dense acquisitions. The MInTS approach provides an efficient method for interpolating across regions of low interferometric correlation, providing a constrained estimation of the continuous spatiotemporal ground deformation field. In this paper, we only focus on unwrapped interferograms from a single viewing geometry, so this approach only resolves ground deformation along the satellite line-of-sight. MInTS is developed as a toolbox in Matlab, and provides an extensible framework for estimation of deformation from catalogs of interferograms.

[53] We illustrate the application of MInTS to a catalog of 92 ERS and Envisat interferograms in the Long Valley, CA, region, spanning 16 years. The MInTS estimation of ground deformation resolves the uplift in the caldera prior to 1999, with accelerated uplift initiating in late 1998. In early 1999, the ground slightly subsides within the caldera, and then slightly inflates again prior to 2004. Following 2004, there is a larger subsidence signal over the western regions of the Long Valley caldera, with more pronounced subsidence localized at the Casa Diablo geothermal plant. The estimated deformation is broadly consistent with that detected by ground-based geodetic measurements and conventional InSAR time series techniques.