# Persistent scatterer interferometric synthetic aperture radar for crustal deformation analysis, with application to Volcán Alcedo, Galápagos

## Abstract

[1] While conventional interferometric synthetic aperture radar (InSAR) is a very effective technique for measuring crustal deformation, almost any interferogram includes large areas where the signals decorrelate and no measurement is possible. Persistent scatterer (PS) InSAR overcomes the decorrelation problem by identifying resolution elements whose echo is dominated by a single scatterer in a series of interferograms. Existing PS methods have been very successful in analysis of urban areas, where stable angular structures produce efficient reflectors that dominate background scattering. However, man-made structures are absent from most of the Earth's surface. Furthermore, existing methods identify PS pixels based on the similarity of their phase history to an assumed model for how deformation varies with time, whereas characterizing the temporal pattern of deformation is commonly one of the aims of any deformation study. We describe here a method for PS analysis, StaMPS, that uses spatial correlation of interferogram phase to find pixels with low-phase variance in all terrains, with or without buildings. Prior knowledge of temporal variations in the deformation rate is not required for their identification. We apply StaMPS to Volcán Alcedo, where conventional InSAR fails because of dense vegetation on the upper volcano flanks that causes most pixels to decorrelate with time. We detect two sources of deformation. The first we model as a contracting pipe-like body, which we interpret to be a crystallizing magma chamber. The second is downward and lateral motion on the inner slopes of the caldera, which we interpret as landsliding.

## 1. Introduction

[2] Volcán Alcedo is one of six volcanoes located on Isla Isabela in the Galápagos Archipelago (Figure 1). Alcedo is unusual in that it is the only active Galápagos volcano known to have erupted rhyolite as well as basalt [*Geist et al.*, 1994]. The last known eruption occurred in late 1993 from the south caldera wall [*Green*, 1994]. Deformation of the caldera was detected by *Amelung et al.* [2000], who carried out interferometric analysis of synthetic aperture radar (SAR) data acquired over Alcedo between 1992 and 1999. However, they found the correlation to be too low to determine the deformation signal on the volcano flanks and were therefore unable to draw any conclusions about the source of the deformation. No measurements of surface displacement have been made on Alcedo by any other means, and displacements inferred from SAR data are therefore all we can currently use to constrain the movement of subsurface magma and volatiles. Conventional interferometric SAR (InSAR) fails on the upper flanks of Alcedo because a significant amount of vegetation is present and this leads to temporal decorrelation for most pixels in the image.

[3] When a SAR image is formed, even at the highest possible resolution, the value for each pixel remains the coherent sum of the returns from many scatterers on the ground. If these scatterers move with respect to each other between satellite passes, as is expected to be the case when many scatterers are vegetation, the phase of the return will vary in a random manner which leads to decorrelation. If, however, a pixel is dominated by one stable scatterer that is brighter than the background scatterers, the variance in the phase of the echo due to relative movement of the background scatterers will be reduced, and may be small enough to enable extraction of the underlying deformation signal (Figure 2). *Hooper et al.* [2004] identified this type of pixel as a persistent scatterer (PS) pixel. Physically, these stable scatterers might be a tree trunk or a single large rock or facet amongst the vegetation.

[4] Methods to identify and isolate these PS pixels in interferograms have been developed by several groups [e.g., *Ferretti et al.*, 2001; *Crosetto et al.*, 2003; *Lyons and Sandwell*, 2003; *Werner et al.*, 2003; *Kampes*, 2005]. All of these methods use a functional model of how deformation varies with time to identify PS pixels, and have been very successful in identifying PS pixels in urban areas undergoing primarily steady state or periodic deformation. In these algorithms an initial set of PS pixels with a high signal-to-noise ratio (SNR) is identified by analyzing the amplitude variation, pixel by pixel, in a series of interferograms. Approaches include analysis of each pixel alone [e.g., *Ferretti et al.*, 2001] and comparison with surrounding pixels [e.g., *Adam et al.*, 2005]. Once an initial set of amplitude-stable pixels has been identified, each candidate pixel is tested for phase stability by examining its phase differences with nearby candidates. Only a pixel whose phase history is similar to the assumed model of deformation is deemed stable and not merely the result of random chance. In this manner a network of reference PS pixels is identified that is then used to find additional PS pixels by further phase analysis of all (or a subset of) the remaining pixels.

[5] This approach can fail for two reasons, both related to the fact that the data are “wrapped” or modulo 2*π*. For reliable “unwrapping” or estimation of integer ambiguities, unmodeled phase must be small; *Colesanti et al.* [2003b] estimate that it must be less than 0.6 rad.

[6] First, it can fail if the distance between neighboring PS pixels is too large, such that the contribution to the unmodeled phase from the difference in delay along the raypaths through the atmosphere exceeds the limit for reliable unwrapping. For common atmospheric conditions, the PS pixel density should exceed 3 to 4 per km^{2} [*Colesanti et al.*, 2003b]. The initial selection using amplitude variation finds most bright PS pixels, such as those from man-made structures, and therefore works well in urban areas where the density of structures is high. However, in most natural terrains, including the majority of volcanoes, bright scatterers are rare and the density of reference PS pixels is generally too low to form a closely spaced reference network. This is the case for Long Valley volcanic caldera [*Hooper et al.*, 2004] and the central San Andreas Fault zone [*Johanson and Bürgmann*, 2001].

[7] The second limitation is that an approximate model for the temporal variation in deformation is needed to isolate the deformation signal from atmospheric, topographic and other phase errors. As the time dependence of deformation is not usually known a priori, it is usually assumed to be approximately constant in rate, or periodic in nature. If PS pixels can be identified, deviations from the initial parametric model may be estimated from the phase residuals [*Ferretti et al.*, 2000; *Colesanti et al.*, 2003b; *Kampes*, 2005]. Pixels can only be identified, however, when the deviations from the model are sufficiently small that unwrapping is reliable, as in the San Francisco Bay Area [*Ferretti et al.*, 2004; *Hilley et al.*, 2004], where deformation is due primarily to steady strain accumulation on various fault systems. In cases where deformation varies too much from steady state, such as many volcanoes and landslides, as well as certain tectonic settings e.g., those dominated by postseismic deformation, a reliable network of reference PS pixels will not result. A method is required that produces a time series of deformation, with no prior assumptions about its temporal nature.

[8] *Hooper et al.* [2004] introduced a PS method to extract the deformation signal from SAR data acquired over Long Valley caldera, an area that contains few man-made structures and that deformed at variable rates during the time interval analyzed. *Hole et al.* [2006] also applied a variation of this method to the Taupo volcanic zone, New Zealand. In this paper, we report on significant improvements to the method which increase the accuracy of the estimated displacements and also make it applicable in areas with widely varying deformation gradients. We first describe in detail our method, StaMPS (Stanford Method for PS), for identifying PS pixels and estimating their displacements. We then apply StaMPS to SAR data acquired over Volcán Alcedo and model the source of the deformation seen in the resulting PS interferograms.

[9] There are four parts to StaMPS, each discussed in detail in 2., 3., 4.–5:

[10] 1. For interferogram formation (section 2), there are aspects of interferogram formation for PS processing that differ to conventional interferogram formation. We summarize all the steps involved and describe the differences in detail. We also discuss the error terms associated with the processing.

[11] 2. For phase stability estimation (section 3), we make an initial selection of candidate pixels based on analysis of amplitude and then use phase analysis to estimate the phase stability of these pixels in an iterative process. Finally, there is an optional step to estimate the phase stability of those pixels that were initially rejected based on their amplitude characteristics.

[12] 3. For PS selection (section 4), we estimate for each pixel the probability it is a PS pixel based on a combination of amplitude and estimated phase stability. We then use the estimated probabilities to select PS pixels, rejecting those that appear to be persistent only in certain interferograms and those that appear to be dominated by scatterers in adjacent PS pixels.

[13] 4. For displacement estimation (section 5), once selected, we isolate the signal due to deformation in the PS pixels. This involves unwrapping the phase values and subtracting estimates of various nuisance terms.

## 2. Interferogram Formation

[14] For PS systems relying on a functional temporal model to select PS pixels, typically at least 25 interferograms are required to obtain reliable results [*Colesanti et al.*, 2003a]. Using StaMPS, however, fewer interferograms are required. We find that 12 interferograms are usually sufficient to identify a network of PS pixels and, in one case at least, have even been able to identify PS pixels using just four interferograms. The limiting factor is the accuracy in estimation of the look angle error (equation (18)), which is aided by good digital elevation model (DEM) accuracy and high SNR. The accuracy of the estimated look angle error and hence the estimated deformation signal improves as the number of interferograms increases, so it is desirable to use as many images as possible.

[15] It is possible to carry out PS analysis jointly on data acquired by sensors with different carrier frequencies, for example, data acquired by the ERS and ENVISAT satellites [e.g., *Adam et al.*, 2005; *Arnaud et al.*, 2004; *Arrigoni et al.*, 2004]. However, the number of PS pixels is reduced as only pixels dominated by the most point-like scatterers remain correlated at different frequencies. Because PS pixels in nonurban terrains tend to be less point-like in their scattering characteristics, we only consider here interferometry between images acquired by sensors with the same carrier frequency to maximize the number of PS pixels identified.

[16] There are several aspects of interferogram formation for StaMPS that differ from conventional InSAR processing, which we describe below, together with a discussion of the error terms that arise in interferometric processing.

### 2.1. Decorrelation and Choice of Master Image

*N*single-look interferograms from

*N*+ 1 images acquired at different times, all with respect to one master image. We choose as the master, the image that minimizes the sum decorrelation, i.e., maximizes the sum correlation, of all the interferograms. The correlation is a product of four terms, dependent on time interval (

*T*), perpendicular baseline (

*B*

_{⊥}, see Figure 3) difference in Doppler centroid (

*F*

_{DC}) and thermal noise [

*Zebker and Villasenor*, 1992]. A simple model for the total correlation,

*ρ*

_{total}, is

*ρ*denotes correlation and superscript

*c*denotes the critical parameter values, i.e., the value beyond which an interferogram exhibits almost complete decorrelation. The critical values are dependent on the data set, but typical values for data acquired by the ERS satellites in arid regions are

*T*

^{c}= 5 years,

*B*

_{⊥}

^{c}= 1100 m and

*F*

_{DC}

^{c}= 1380 Hz. We choose the master that maximizes Σ

_{i=1}

^{N}

*ρ*

_{total}, assuming a constant value for

*ρ*

_{thermal}.

[18] We do not apply any spectral filtering in range or azimuth, which would increase the correlation, as this would coarsen the resolution. Generally, the finer the resolution, the fewer scatterers will be contained within each resolution cell, and the greater the chance of the cell being dominated by one scatterer. The trade-off is that except for truly point-like PS pixels, phase values will include decorrelation noise related to perpendicular baseline and Doppler separation.

[19] Our algorithm for PS identification does not actually require that all interferograms are formed with respect to one master, only that all are coregistered to one master. Although we have not implemented this option in our code, combinations of interferograms using multiple masters may be chosen to increase the sum correlation.

### 2.2. Coregistration

[20] Some interferograms have values of temporal separation, perpendicular baseline and Doppler separation that are larger than usual for conventional InSAR. This fact leads to high decorrelation and a corresponding low coherence which make standard coregistration routines, based on cross correlation of amplitude, fail. To avoid this problem, we have developed a coregistration algorithm that uses an amplitude based algorithm to estimate offsets in position between pairs of images with good correlation. The function that maps the master image to each other image is then estimated by weighted least squares inversion (Appendix A).

[21] Once the mapping functions are estimated, we resample each image to the master coordinate system, using a 12 point raised cosine interpolation kernel. Then we form a raw interferogram by differencing the phase of each image to the phase of the master.

### 2.3. Geometric Phase Correction

[22] Raw interferograms contain a geometric phase term which is due to the master and slave images being acquired from different points in space. As in conventional InSAR, we correct for this geometric phase in two steps. First, we flatten the interferograms, which involves correcting the phase of each pixel as if the scattering surface were lying on a reference ellipsoid. Next, we estimate the phase due to the deviation of the real surface from the reference ellipsoid by transforming a DEM into the radar coordinate system. Two error terms arise in this processing, look angle error and squint angle error. For pixels with many distributed scatterers, which is the usual situation for conventional interferometry, the look angle error is due almost entirely to error in the DEM, and is commonly referred to as DEM error. For PS pixels, however, there is also a contribution due to the difference in range between the position of the dominant scatterer and the center of the ground patch that is resolved by the pixel; hence we prefer the more general term. Squint angle error is commonly avoided in conventional InSAR by processing both images to a common squint angle. However, this coarsens azimuthal resolution, which, as discussed in section 2.1, we wish to make as fine as possible to identify PS pixels.

#### 2.3.1. Look Angle Error

_{θ}, is proportional to the change in range, Δ

*r*, between the master and slave geometry,

*λ*is the radar wavelength. The minus sign comes from the definition of the phase measured at the sensor being phase delay. From the geometry (Figure 3),

*r*is the range in the master geometry,

*B*is the baseline distance between the slave and master sensor position, θ is the look angle in the master geometry and

*ω*is the angle between the baseline vector and the horizontal. Differentiating and using equation (2) gives

*r*≪

*r*so this simplifies to

*B*cos(θ −

*ω*), which is included in an orbit error term (equation (10)), we find then that the error, Δϕ

_{θ}, in our estimate of

_{θ}depends only on Δθ, the error in our knowledge of θ and for small Δθ

*B*

_{⊥}(θ) is the perpendicular component of the baseline. From the geometry, Δθ is obviously dependent on the accuracy of the estimated height above the reference surface (Figure 3) but is also dependent on any difference in the position of the pixel phase center to that assumed in the flattening step. Specifically,

*h*is the error in height,

*ξ*is the horizontal distance of the phase center from the middle of the pixel in range direction, and θ

_{i}is the incidence angle. Thus, even if the DEM were 100% accurate, there could still be an error in θ due to both the lateral offset of the phase center itself and any variation in height due to this offset. In the case that we consider here, where carrier frequency does not vary between passes, the DEM error is indistinguishable from the error due to phase center uncertainty. With varying carrier frequency, separation of the two errors would be possible [

*Colesanti et al.*, 2003b;

*Kampes*, 2005], yielding more accurate positioning of the scatterer; however, the above mentioned limitation of using varying carrier frequencies, that only the most point-like pixels remain persistent, would apply. In any case, the positional accuracy of several meters that is already obtained is sufficient for most deformation studies.

[26] As the look angle error term is estimated later in the PS processing, the accuracy of the DEM is not usually important, although higher accuracy reduces the ambiguity in the estimation step, and thus is important in the case where only a few (<12) interferograms are available.

#### 2.3.2. Squint Angle Error

*η*, cancels when the squint angle is the same for both the master and slave acquisition. When the squint angles differ, an error term

*v*is the sensor velocity (see

*Kampes*[2005] for a derivation). In the data sets we have analyzed until now,

*F*

_{DC}is generally small and Δϕ

_{ϑ}is expected to be less than 0.2 rad. Furthermore,

*F*

_{DC}is somewhat correlated with time. As deformation is also correlated with time, there is the potential for the phase due to deformation, which is often much larger, leaking into any estimate for Δϕ

_{ϑ}. We therefore do not attempt to estimate this term and, instead, treat it as noise.

### 2.4. Geocoding

[28] Finally, we estimate the position of every pixel in a geocoded reference frame using the orbital parameters and the DEM. Theoretically, our estimates of position could be improved using various parameters estimated during processing, but for the purposes of most deformation modeling, positional accuracy to within several meters is usually sufficient.

## 3. Phase Stability Estimation

[29] Initially, we select a subset of pixels based on analysis of their amplitudes, rejecting those least likely to be PS pixels. We then estimate the phase stability of each of these pixels through phase analysis, which we successively refine in a series of iterations. Finally, we include an optional step to estimate the phase stability of those pixels that were not included in the initial amplitude-based selection.

### 3.1. Amplitude Analysis

[30] Although it is the phase stability of a pixel that defines a PS pixel, there is a statistical relationship between amplitude stability and phase stability, that makes consideration of amplitude useful for reducing the initial number of pixels for phase analysis. Later, we also include amplitude characteristics in estimating the probability of a pixel being a PS pixel, as discussed in section 4.

*Ferretti et al.*[2001] as

*Ferretti et al.*[2001] threshold on to select a subset of pixels where most are PS pixels. We use differently, to select a subset of pixels that includes almost all of the PS pixels in the data set. The threshold value we use is consequently higher, typically in the region of 0.4, which leads to most of the selected pixels not being PS pixels (see Appendix B). There is an optional later step, described in section 3.3, in which pixels above this threshold are also analyzed for phase stability.

### 3.2. Phase Analysis

*ψ*

_{x,i}, of the

*x*th pixel in the

*i*th “flattened” and topographically corrected interferogram can be written as the wrapped sum of 5 terms,

_{D,x,i}is the phase change due to movement of the pixel in the satellite line-of-sight (LOS) direction, Δϕ

_{θ,x,i}is the residual phase due to look angle error, ϕ

_{A,x,i}is the phase due to the difference in atmospheric delay between passes, Δϕ

_{S,x,i}is the residual phase due to satellite orbit inaccuracies, ϕ

_{N,x,i}is a noise term due to variability in scattering, thermal noise, coregistration errors and uncertainty in the position of the phase center in azimuth, and

*W*{·} is the wrapping operator. The pixels we seek as PS are those where ∣ϕ

_{N,x,i}∣ is small enough that it does not completely obscure the signal. As the phase is wrapped, this term must certainly be less than

*π*, but in order to correctly estimate the integer ambiguity in the number of wraps when estimating the spatially uncorrelated look angle error, Δθ

_{x}

^{u}in equation (18), it must be even smaller in practice.

[33] Variation in the first four terms of equation (10) dominates the noise term, making it difficult to identify which scatterers are persistent directly from the wrapped phase. Hence we estimate these four terms and subtract them, giving an estimate for _{n,x,i} which we assess statistically. We then reestimate the first four terms, using the statistics of _{n,x,i} to weight the contribution of each pixel to the reestimaton, and subtract the reestimated values from equation (10) to give a new estimate of _{n,x,i}. We iterate around this loop, refining our estimates of _{n,x,i} each time, until the values converge, which generally occurs after only a few iterations.

[34] The deformation signal we are interested in is that due to deformation of the Earth's surface, which is correlated spatially. There could also be signal associated with the isolated movement of individual bright scatterers, but we are not interested here in these movements and they can be considered noise. Variation in atmospheric delay between passes is due mainly to variation in the total electron content (TEC) of the ionosphere and water vapor content of the troposphere. Both of these quantities are correlated spatially [*Hanssen*, 2001]. Orbital errors are also correlated spatially in azimuth, and interferometric processing leads to spatial correlation of the residual orbit error term also in range. Finally, DEM error tends to be partly spatially correlated and this maps into the look angle error. Hence estimating the spatially correlated part of *ψ*_{x,i} provides an estimate for the first three terms plus part of the fourth term in equation (10). *Hooper et al.* [2004] achieved this by calculating the mean of surrounding pixels that were judged most likely to be PS pixels. The number of pixels included in the mean was reduced with each iteration as confidence in ruling out pixels as PS increased. This method is, in effect, a crude low-pass filter implemented in the spatial domain and relies on knowing the length scale of the spatial correlation. While this method worked fine in Long Valley, where the limiting factor on distance of spatial correlation is the atmospheric term, in areas with steeper deformation gradients the deformation term can become the limiting factor, altering the length scale of correlation. A better approach is to apply a band-pass-filtering method that adapts to any phase gradient present in the data.

_{N,x,i}to estimate SNR. Figure 4 shows the relationship between signal, noise, amplitude and phase noise for a single pixel in a single image. If we assume that the amplitude of the signal,

*g*

_{x}, remains constant and that real and imaginary components of the noise,

*n*

_{R,x,i}and

*n*

_{I,x,i}, respectively, are characterized by a single Gaussian distribution with zero mean and standard deviation

*σ*

_{n,x}, then SNR =

*g*

_{x}

^{2}/2

*σ*

_{n,x}

^{2}. Our estimate of

_{x}is then simply

*g*

_{x,i}=

*A*

_{x,i}cosϕ

_{N,x,i}−

*n*

_{∥,x,i}where

*n*

_{∥,x,i}is the signal parallel component of noise, the mean of which is zero. Our estimate for

_{n,x}

^{2}is given by

*g*

_{x}with

_{x}gives

[37] We also experimented with other weight selection methods, for example, weighting each pixel based on the probability that it is persistent. Empirically, we find that setting the amplitude to 1/[*P*(*x* ∈ *PS*)]^{2}, where *P*(*x* ∈ *PS*) is the probability that the pixel is a PS pixel (see Appendix C), can give better results than weighting with the SNR, in the sense that more PS pixels are found.

*Z*(

*u*,

*v*) is the smoothed intensity of the 2-D FFT [

*Goldstein and Werner*, 1998]. We smooth the intensity by convolution with a 7 × 7 pixel Gaussian window. We combine the adaptive phase filter response,

*H*(

*u*,

*v*), with a narrow low-pass filter response,

*L*(

*x*,

*y*), to form the new filter response,

*L*(

*u*,

*v*) is a fifth-order Butterworth filter, with a typical cutoff wavelength of 800 m,

*α*and

*β*are adjustable weighting parameters, typical values being 1 and 0.3, respectively, and (

*u*,

*v*) is the median value of

*H*(

*u*,

*v*).

_{x,i}, is a wrapped estimate of the spatially correlated parts of each of the terms on the right-hand side of equation (10), so subtracting

_{x,i}from

*ψ*

_{x,i}and rewrapping gives

^{u}denotes the spatially uncorrelated part of ϕ. We expect ϕ

_{D,x,i}

^{u}, ϕ

_{A,x,i}

^{u}and Δϕ

_{S,x,i}

^{u}to be small, as most of their power lies at longer wavelengths, so replacing their sum with

*δ*

_{x,i}in equation (16) gives

_{θ,x,i}and Δθ

_{x}. As long as Δϕ

_{θ,x,i}

^{u}contains approximately the same frequency components for all

*i*, it follows that the same approximate relationship holds between Δϕ

_{θ,x,i}

^{u}and Δθ

_{x}

^{u}. Substituting for Δϕ

_{θ,x,i}

^{u}in equation (17) gives

[41] Because *B*_{⊥,x,i} is not expected to correlate with ϕ_{N,x,i}^{u} or *δ*_{x,i}, we can estimate Δ_{x}^{u} in a least squares sense. The contribution from the master image to ϕ_{N,x,i}^{u} +*δ*_{x,i} will be present in every interferogram, causing a constant offset, ϕ_{x}^{m,u}, that we must also resolve in our least squares inversion. Since the phase is wrapped, the inversion is not linear. We implement the inversion as a rough search of parameter space, followed by a linear inversion to estimate the best fitting model in the region of the rough estimate. We typically limit the rough search to values of Δ^{u} equivalent to ±10 m of height error and in increments such that the range of Δ_{θ,x}^{u} increases by *π*/4 (Figure 5). The inversion could also be implemented using the least squares ambiguity decorrelation (LAMBDA) method, initially developed for fast GPS double-difference integer ambiguity estimation [*Teunissen*, 1995] and adapted for InSAR data by *Kampes* [2005]. However, we find that as implemented, the inversion is not a time-limiting step.

_{x}

^{u}we derive Δ

_{θ,x,i}

^{u}which we subtract from equation (17) to give

*δ*′

_{x,i}=

*δ*

_{x,i}+ Δϕ

_{θ,x,i}

^{u}− Δ

_{θ,x,i}

^{u}.

*N*is the number of interferograms. Assuming that ϕ

_{N,x,i}≈ ϕ

_{n,x,i}

^{u}and

*δ*

_{x,i}≈ 0,

*γ*

_{x}is a measure of the phase noise level and an indicator of whether the pixel is a PS pixel. The measure

*γ*

_{x}is similar to a measure of coherence [

*Bamler and Just*, 1993], although in time rather than in space as is generally implied by the term in reference to SAR interferograms. The measure differs from coherence in that amplitude is assumed to be constant at all times, as we wish to give every image equal weight. A pixel that is bright in one image and dark with random phase in all other images would have high coherence if amplitude were included whereas in reality it would not be a good PS pixel.

[44] The value of *γ*_{x} also depends on the accuracy of _{x,i}. In the case where all the phase values for the surrounding grid, to which the 2-D FFT is applied, are completely decorrelated, _{x,i} will be random. The measure *γ*_{x} will then not be a true representation of the variation in residual phase for a pixel, and the pixel will likely not be selected.

### 3.3. Pixels With High-Amplitude Dispersion

[45] Some pixels with a high value of , which are rejected by our initial thresholding (section 3.1), will have stable phase, although the proportion will be small (see Appendix B). Once the low- pixels have been analyzed, we include an optional step to analyze the high- pixels, using the filtered phase of the low- pixels to calculate _{x,i}, and proceeding from equation (16) to calculate *γ*_{x}.

## 4. PS Selection

[46] Once we have converged on estimates for the phase stability of each pixel, we select those most likely to be PS pixels, with a threshold determined by the fraction of false positives we deem acceptable. We also seek to reject pixels that persist only in a subset of the interferograms and those that are dominated by scatterers in adjacent PS pixels.

*γ*

_{x}. When it ceases to decrease, the solution has converged and the algorithm stops iterating. We then select pixels based on the probability they are PS pixels. Appendix C describes PS probability estimation based on

*γ*

_{x}alone. As there is a correlation between amplitude variance and phase stability (see section 3.1), we can calculate the probability more accurately still by considering the amplitude dispersion of the pixels,

_{,x}, as well as

*γ*

_{x}. We bin the pixels by

_{,x}, ensuring there are at least 10

^{4}pixels in every bin, and then bin by

*γ*

_{x}in increments of 0.01, resulting in the probability distribution,

*p*(

*γ*

_{x},

_{,x}). For each bin in

_{,x}, we estimate

*α*, the proportion of the pixels that are PS pixels, as described in Appendix C. If only pixels with

*γ*

_{x}above a given threshold value,

*γ*

^{thresh}, are selected, the number of those pixels that are non-PS pixels is given by (1 −

*α*)

*p*

_{B}(

*γ*

_{x})

*dγ*

_{x}, where

*p*

_{B}(

*γ*

_{x}) is the probability distribution for pixels with random phase. We find

*γ*

^{thresh}(

_{,x}) for each bin in

_{,x}such that the fraction of pixels selected that are non-PS pixels (false positive identifications) is acceptable for our particular application, i.e.,

*q*is the specified acceptable fraction of false positives. As discussed in Appendix B, we generally expect

*γ*

_{x}, which is a measure of phase stability, to decrease with increasing

_{,x}. This implies that as

_{,x}increases,

*p*(

*γ*

_{x},

_{,x}) will skew to lower values of

*γ*

_{x}. The net effect on

*γ*

^{thresh}(

_{,x}) is to increase with increasing . Empirically we find that the relationship is approximately linear, i.e.,

*γ*

^{thresh}= κ where κ is a constant. We find the best fitting κ by least squares inversion and select pixels for which

*γ*

_{x}> κ

_{,x}as our PS pixels.

### 4.1. Partially Persistent Scatterers

[48] The stability of a pixel may change during the time interval spanned by the data set, for example, if a dominant scatterer is added or removed. Another issue arises when the scattering characteristics of a dominant scatterer are altered, but the scatterer remains dominant. The pixel would then exhibit two intervals of phase stability, although one of the intervals would be decorrelated with respect to the master image. In both cases the pixel may still be identified if its mean variance is small enough, although in some images the phase will be dominated by noise. Furthermore, if the phase characteristics of whole areas of an image change between two passes, e.g., if a field is plowed or there is a new lava flow, the signal of all PS pixels selected in that area will be dominated by noise in some of the images.

[49] It would be possible to keep these pixels only for those images where they provide useful deformation signal, and reject them for all other images. However, the three-dimensional phase unwrapping algorithms developed by *Hooper and Zebker* [2007] require pixels that persist throughout the time spanned by the images. We expect to relax this requirement in future phase unwrapping algorithms but, for now, we avoid picking these pixels; we estimate the variance of *γ*_{x} for each pixel using the bootstrap percentile method [*Efron and Tibshirani*, 1986] with 1000 iterations, and drop pixels with a standard deviation over a defined threshold, a typical value being 0.1.

### 4.2. Multiple Pixel PS

[50] A scatterer that is bright can dominate pixels other than the pixel corresponding to its physical location. The error in look angle and squint angle due to the offset of the pixel from the physical location usually results in these pixels not being selected as PS pixels. However, for pixels immediately adjacent to the PS pixel the error may be sufficiently small that the pixel appears stable. To avoid picking these pixels, we assume that adjacent pixels selected as PS pixels are due to the same dominant scatterer. As we expect the pixel that corresponds to the physical location to have the highest SNR, for groups of adjacent stable pixels we select as the PS pixel only the pixel with the highest value of *γ*_{x}.

## 5. Displacement Estimation

[51] Once we have selected the most likely PS pixels, we discard all other pixels and return to the original wrapped interferogram phase, *ψ*_{x,i} (equation (10)). In order to retrieve the phase due to deformation, ϕ_{D,x,i}, the phase must be unwrapped and other nuisance terms estimated.

### 5.1. Phase Unwrapping

*π*. For the spatially correlated part of the signal, this will be true as long as the spatial sampling of the signal by the PS pixels is sufficiently dense. Even when the sampling density is high, the contribution to the absolute difference in phase between neighboring PS pixels can still be greater than

*π*due to the spatially uncorrelated part of the signal. The most significant contribution to this part is the spatially uncorrelated part of the look angle error, Δϕ

_{θ}

^{u}, for which we already have an estimate. We also have an estimate for the contribution of the master to the spatially uncorrelated part of the signal,

_{x}

^{m,u}. We therefore subtract our estimates for these two terms before unwrapping, yielding

_{θ,x,i}

^{c}is the spatially correlated part of Δϕ

_{θ,x,i}, and Δϕ

_{N,x,i}is the residual spatially uncorrelated noise term, ϕ

_{N,x,i}−

_{x}

^{m,u}.

*Hooper et al.*[2004]. However, as we have three dimensions of phase data, two in space and one in time, more reliable results can be obtained using one of the three-dimensional unwrapping algorithms described by

*Hooper and Zebker*[2007]. Whichever algorithm is chosen, the results of unwrapping can be summarized as

_{x,i}is the unwrapped value of

*W*{

*ψ*

_{x,i}− Δ

_{θ,x,i}

^{u}−

_{x}

^{m,u}} and

*k*

_{x,i}is the remaining unknown integer ambiguity. The value of

*k*

_{x,i}includes an integer ambiguity associated with each interferogram as a whole, which is not estimated. If the unwrapping is sufficiently accurate,

*k*

_{x,i}will be the same integer for most

*x*in a given interferogram

*i*.

### 5.2. Spatially Correlated Nuisance Terms

[54] After unwrapping, several terms remain in equation (23) that mask ϕ_{D,x,i}. The spatially uncorrelated part of these nuisance terms can be modeled as noise in any subsequent deformation modeling, but the spatially correlated parts can bias the results, so we seek to estimate and subtract them. We separate the spatially correlated part of the nuisance terms into the part that is correlated in time and the part that is expected not to be. The former consists of the master contribution to (ϕ_{A,x,i} + Δϕ_{S,x,i}), which is present in every interferogram, and the latter consists of the slave contribution to (ϕ_{A,x,i} + Δϕ_{S,x,i}) and Δϕ_{θ,x,i}^{c}. We estimate both parts separately using a combination of temporal and spatial filtering as described below.

*k*

_{x,i}

*π*in equation (23), which is not estimated, absolute values of

_{x,i}are essentially decorrelated in time and we are not able to apply a temporal filter directly. However,

*k*

_{x,i}is identical for most neighboring PS pixels; thus calculating the phase differences between neighboring PS pixels cancels the 2

*k*

_{x,i}

*π*term in most cases, and the resulting phase can be filtered. We first form a spatial network connecting all PS pixels using Delaunay triangulation. In each interferogram we difference

_{x,i}between pairs of PS pixels around each triangle, in a clockwise direction, giving (from equation (23))

*x*

_{2}and

*x*

_{1}. For each PS pair, we low-pass filter the differenced phase in time by convolution with a Gaussian function yielding

^{T}{·} is the low-pass filter operator, and superscript

*m*indicates the master contribution to these terms. The width of the Gaussian is chosen to be less than the time over which deformation rate is expected to vary, in order to preserve the deformation phase. Evaluating

_{i}} at the master time, when ϕ

_{D,x}= 0 for all

*x*, gives an estimate for

_{A}

^{m}

_{S}

^{m}). The estimate, (

_{A,x}

^{m}+ Δ

_{S,x}

^{m}) with respect to an arbitrary reference PS pixel is retrieved by least squares inversion.

_{i}. We achieve this by subtracting the low-pass-filtered signal,

_{i}}, giving (from equations (25) and (24))

*s*indicates the slave contribution to these terms. For each interferogram the high-pass-filtered signal for each PS pixel, with respect to an arbitrary reference PS pixel, is retrieved from equation (26) by least squares inversion,

_{A,x,i}

^{s}+ Δ

_{S,x,i}

^{s}+ Δ

_{θ,x,i}

^{c}). Rearranging equation (23) and substituting in the terms estimated in this section gives

[57] Typically, we consider the resultant phase relative to some small region of the image, which cancels the whole interferogram contribution to the 2*k*_{x,i}*π* term, and leaves phase due only to deformation, spatially uncorrelated noise and unwrapping errors.

[58] In practice, it is possible that the spatially correlated nuisance terms may also be correlated temporally, in which case they will not be correctly estimated by the procedure described above. For the atmospheric term, this may be the case in regions where tropospheric moisture content varies seasonally, as in coastal areas where fog is more prevalent at certain times of the year. Further processing is required if this is suspected to be the case. For the orbit error term, we have found with data from the Radarsat-1 satellite that values can be large and, apparently by chance, correlated in time. This is not the case for the Volcán Alcedo data we analyze in section 6, but when this does occur, phase ramps can be removed from interferograms where the orbit error term is visible before estimating the other spatially correlated terms. Extra care should then be taken in interpreting the deformation signal, as any overall tilt signal will be removed by this procedure, and residual nonlinear contributions to the orbit error term will remain [*Kohlhase et al.*, 2003].

[59] Care should also be taken if the number, or distribution, of interferograms is such that they sample the signal in time at a rate that is too low to capture the detailed variation in deformation rate. Some of the deformation signal may then leak into the estimate for the spatially correlated nuisance terms, which may be visible as a recurring spatial pattern in the estimate of these terms for each interferogram.

## 6. Application to Volcán Alcedo

[60] We applied StaMPS to data acquired by ERS-1 and ERS-2 satellites over Volcán Alcedo between June 1992 and January 2001 (see Tables 1 and 2). Starting from the raw SAR data, the total CPU time we required was approximately four days, using a PC with a 1.5 GHz CPU and 1 GB of RAM. We processed 15 descending track images giving 14 interferograms, each with February 2000 as the reference master image (Figure 6). We also processed 14 ascending track images giving 13 interferograms, each with January 2000 as the reference master image (Figure 7). We used a DEM determined by the Shuttle Radar Topography Mission (SRTM), with a posting of three seconds of arc. The specific parameters we used in the processing are detailed in Table 3. The broad distribution of PS pixels is similar in both the descending and ascending cases, although the positions of individual PS pixels are not necessarily identical (Figures 8 and 9) . In the ascending data, the distribution of perpendicular baselines, *B*_{⊥}, falls into two distinct clusters separated by 707 m. Coregistration between the two clusters is challenging, leading to more error in ϕ_{N,x,i} from misregistration than in the descending case. The Doppler centroid separation is also generally larger in the ascending case, leading to a greater noise contribution from background scatterers. Note also that more long-wavelength atmospheric signal is present in the ascending data than the descending data, presumably related to the difference in acquisition time which is 10.45 p.m., local time, for the ascending orbit as opposed to 10.23 a.m. for the descending orbit.

^{a}

Orbit | Date | Sensor | B_{⊥}, m |
f_{DC}, Hz |
---|---|---|---|---|

4794 | 1992-06-15 | ERS-1 | 616 | −301.24 |

13020 | 1997-10-16 | ERS-2 | 176 | −677.10 |

18531 | 1998-11-05 | ERS-2 | 533 | −574.81 |

19032 | 1998-12-10 | ERS-2 | −917 | −633.35 |

19533 | 1999-01-14 | ERS-2 | −484 | −574.48 |

20034 | 1999-02-18 | ERS-2 | 976 | −447.77 |

20535 | 1999-03-25 | ERS-2 | −407 | −472.20 |

25044 | 2000-02-03 | ERS-2 | 0 | −551.03 |

25545 | 2000-03-09 | ERS-2 | −198 | −421.53 |

26046 | 2000-04-13 | ERS-2 | −274 | −684.07 |

26547 | 2000-05-18 | ERS-2 | −308 | −594.90 |

27048 | 2000-06-22 | ERS-2 | −453 | −746.82 |

27549 | 2000-07-27 | ERS-2 | −39 | −667.34 |

28551 | 2000-10-05 | ERS-2 | −337 | −883.46 |

29052 | 2000-11-09 | ERS-2 | −99 | −908.65 |

- a
*B*_{⊥}is the perpendicular baseline relative to the master acquisition on 3 February 2000, and*f*_{DC}is the absolute Doppler centroid.

^{a}

Orbit | Date | Sensor | B_{⊥}, m |
f_{DC}, Hz |
---|---|---|---|---|

17951 | 1998-09-26 | ERS-2 | −650 | 29.46 |

18452 | 1998-10-31 | ERS-2 | −707 | −92.33 |

19955 | 1999-02-13 | ERS-2 | −854 | −71.67 |

20456 | 1999-03-20 | ERS-2 | −707 | −25.25 |

24965 | 2000-01-29 | ERS-2 | 0 | 25.39 |

25466 | 2000-03-04 | ERS-2 | 298 | −184.15 |

25967 | 2000-04-08 | ERS-2 | 387 | −193.15 |

26468 | 2000-05-13 | ERS-2 | −900 | 592.97 |

26969 | 2000-06-17 | ERS-2 | −733 | 500.57 |

27470 | 2000-07-22 | ERS-2 | 33 | 274.85 |

28472 | 2000-09-30 | ERS-2 | 122 | 517.95 |

28973 | 2000-11-04 | ERS-2 | 370 | 143.69 |

29474 | 2000-12-09 | ERS-2 | 57 | 446.37 |

29975 | 2001-01-13 | ERS-2 | −726 | −176.83 |

- a
*B*_{⊥}is the perpendicular baseline relative to the master acquisition on 29 January 2000 and*f*_{DC}is the absolute Doppler centroid.

Parameter | Value |
---|---|

DEM | SRTM 3 arc sec |

Maximum DEM error | 10 m |

Band-pass phase filter grid cell size | 40 m |

Band-pass phase filter grid size | 64 × 64 |

Band-pass phase filter low-pass cutoff | 800 m |

Band-pass phase filter α |
1 |

Band-pass phase filter β |
0.3 |

Acceptable fraction of false positive PS pixels | 1% |

Partial PS pixels rejected? | no |

Spatially correlated filtering time window | 180 days |

Spatially correlated filtering minimum wavelength | 50 m |

Unwrapping algorithm | stepwise 3-D |

Unwrapping grid cell size | 100 m |

Unwrapping Gaussian width | 8σ |

[61] An event causing asymmetric deformation occurred within the caldera between June 1992 and October 1997 which is visible in the interferogram of *Amelung et al.* [2000]. Although this event is clear in the wrapped phase of the PS pixels in an interferogram covering this time interval (Figure 10), the spatial sampling of the PS pixels is not high enough everywhere to unwrap the signal associated with this event. The spatial pattern of the wrapped phase is, however, consistent with trapdoor faulting, similar to that observed on Sierra Negra [*Amelung et al.*, 2000; *Chadwick et al.*, 2006], another volcano located on Isla Isabela. In the case of Alcedo, the fault appears to be located in the southwest of the caldera, striking approximately northwest-southeast. For all subsequent acquisitions, which cover the time interval between October 1997 and January 2001, we are able to extract and unwrap the deformation signal (Figures 8 and 9).

### 6.1. Modeling

[62] Our results show a dominant deflation signal for the entire 1997 to 2001 time interval, largely confined within the caldera. The rate of deflation appears to be approximately constant over this time interval, and can be observed by the change in maximum LOS displacement in the descending data (Figure 11a). There is also a discontinuity in the displacement rates following the trend of a break in topography on the west side of the caldera. The rate of deformation to the west of this discontinuity also appears be approximately constant as observed by the change in LOS displacement between the west side and east side of the caldera in the ascending data (Figure 11b). Both modes of deformation appear to be constant in rate, and we calculate the mean LOS velocities for each PS pixel in both the descending and ascending data. For the descending data we are able to refer the velocities to the mean signal at the coasts (Figure 12), which we assume to be unaffected by the local deformation detected here and moving at the tectonic plate velocity. For the ascending data, unwrapping between the caldera and the coast is unreliable due to the higher phase noise; therefore we analyze only the data within the caldera and refer the velocities to the east side of the caldera (Figure 13).

[63] PS pixels on the west side of the caldera that seem most affected by the second mode of deformation are located on the inner slope of the caldera (see topography in Figure 13). Relative to the velocities expected from the deflationary signal alone, in the ascending data, PS pixels on the slopes are moving away from the satellite and, in the descending data, they are moving toward the satellite. This implies a horizontal component of deformation. Given that there is a sharp change in velocities moving from the slopes to the caldera floor, we interpret this mode of deformation as landsliding.

[64] The similarity between Alcedo and Sierra Negra, in terms of evidence of trapdoor faulting, might lead us to assume similar geometries for their shallow magma chambers. Deformation on Sierra Negra between September 1998 and March 1999 can be fit well by an inflating sill-like body situated entirely within the caldera boundaries and about 2 km below the surface [*Amelung et al.*, 2000; *Jónsson*, 2002; *Yun et al.*, 2006]. However, for Alcedo, the offset in the position of maximum LOS velocity on the caldera floor between ascending and descending geometries (Figures 12 and 13) indicates that there is significant horizontal displacement, more consistent with a three-dimensional source than a sill-like body. The asymmetry of the deformation pattern suggests that the source is not radially symmetrical, so we model it with a contracting, finite ellipsoid [*Yang et al.*, 1988]. We use only the velocities of PS pixels located on the caldera floor to avoid bias in our results from the second mode of deformation. The caldera is essentially flat and we approximate it with a half-space. We assume a shear modulus of 30 GPa, with no change in rheology to the depth of the deformation source, and also assume that the volcano deforms as a Poisson solid.

[65] Using both descending and ascending LOS velocities, Markov chain Monte Carlo sampling enables us to find the posterior probability distribution of the model parameters [*Mosegaard and Tarantola*, 1995]. We assume zero velocity at the coasts in the descending data. For the ascending data we estimate an additional velocity offset for the east side of the caldera. We also assume that the probability density of the velocities approximates a multivariate Gaussian distribution. Several randomly chosen marginal distributions of the data, estimated using the percentile bootstrap method of *Efron and Tibshirani* [1986], suggest this to be a reasonable assumption (Figure 14).

[66] We reduce the data to a manageable size for computing purposes by resampling both descending and ascending data sets to a 90 m grid. We combine velocities for PS pixels within the same grid cell by taking the weighted mean, using 1/_{rate,x} as the weight for each pixel, where _{rate,x}^{2} is the estimated variance of the velocity distribution for pixel *x*. We estimate this distribution for each PS pixel using the percentile bootstrap method [*Efron and Tibshirani*, 1986] to recalculate velocity 1000 times. The position we assign to each grid cell is the weighted mean position of all PS pixels within the cell. The variance-covariance matrix for each of our reduced data sets also follows from the percentile bootstrap method.

**is the data vector,**

*x***is the mean vector,**

*μ**n*is the number of data,

**Σ**is the variance-covariance matrix and superscript

*T*denotes the transpose. In order to regularize

**Σ**, we increase the diagonal variance terms by 10%. This broadens the marginal probability density function for each grid cell by 10%, which we expect to broaden the posterior model distribution by a negligible amount.

### 6.2. Results

[68] The maximum likelihood model is shown in Figure 15, together with the LOS velocities predicted by this model, along with the residual difference between these and the data. Although only velocities for PS pixels on the floor of the caldera are used in the inversion, the predicted velocities are shown for all PS pixels. Residual velocities for PS pixels located on the inner slopes of the caldera are assumed to be due to landsliding. The root-mean-square residual value for the PS pixels used in the inversion is 1.9 mm yr^{−1}.

[69] We plot marginal probability densities from Markov chain Monte Carlo sampling, for all the model parameters except position, in Figure 16. The depth of the source is well constrained, lying between 2.21 and 2.26 km below sea level at 95% confidence, with the best fit at 2.24 km. This is based on our assumption of constant shear modulus, however. If shear modulus actually increases with depth, as is usually the case, we might expect the depth range to be somewhat deeper. Although the semimajor axis of the source ellipsoid is well constrained, between 2.5 and 2.7 km at 95% confidence, the aspect ratio is less well constrained. The maximum likelihood model (shown in Figure 15) is very prolate, with an aspect ratio of 25.5, but an aspect ratio of 5.7 would also fit the data at 95% confidence. The volume decrease is well constrained, between 1.25 and 1.30 × 10^{6} m^{3} at 95% confidence, with a clear trade-off with both the depth and semiminor axis of the ellipsoid source. The strike of the ellipsoid lies between 127° and 128° and the dip is subhorizontal, dipping upward between 1.3 and 4.7° at 95% confidence.

[70] In order to determine the motion due to landsliding, we resolve the residual velocities, after subtraction of the deflationary source, into eastward and subvertical components. The subvertical component includes a small component of any northward motion. For the PS pixels lying within the circles marked on the residual plots in Figure 15, the eastward component is between 6.5 and 6.7 mm/year and the subvertical component is between 2.3 and 2.7 mm/year downward. This corresponds to an eastward plunge of between 19.4° and 21.9° for the motion.

## 7. Geophysical Interpretation

[71] Although the data are reasonably well fit by an ellipsoidal source, in all likelihood, the geometry is more complex. However, it appears that the source of the deflation is not equidimensional, and that the longest axis runs subparallel with the long axis of Isla Isabela in the region of Alcedo. If we assume that the source represents a preexisting magma chamber, the shape and orientation of the chamber imply that the axis of least compressive stress runs approximately southwest-northeast. The bathymetry of the region (Figure 17) shows a raised platform on all sides of Alcedo except to the northeast, where the depth drops rapidly from 500 m to 2500 m within 10 km. Thus we might expect the least resistance to intrusion to be oriented toward the northeast.

[72] There was no known eruption during this time interval, so the most likely causes of the deflation are contraction due to crystallization and cooling, and the loss of volatiles from the source area. Crystallization of magma emplaced at 3 km beneath the Krafla central volcano in Iceland was estimated to result in a reduction in volume of ∼9% [*Sigmundsson et al.*, 1997]. If we assume a similar value for magma emplaced beneath Alcedo, between 0.6 and 10% of the magma would need to crystallize each year, to give the model distribution of volume reduction at 95% confidence. Note that the probability distribution for percentage of crystallization is less well constrained than that for volume reduction, as it also depends on the total volume which is not so well constrained. Cooling of the solidified magma could also contribute further to the volume reduction, with up to 2% contraction being the estimated value for solidified Krafla magma. Loss of volatiles from the magma chamber would reduce the pressure, also leading to a reduction in volume, but the pressure decrease required to produce the surface displacements would be between 23 and 370 MPa yr^{−1} at 95% confidence. This range is too high to be plausibly produced by the loss of volatiles alone.

[73] The local eastward slope dip for the PS pixels circled in Figure 15 is 15.7°. Our estimate of the dip of the landsliding motion is somewhat steeper, which suggests that there is additional vertical settling. Given that the displacements we measure are those of the dominant scatterers within each pixel, which are most likely the larger boulders in this case, additional settling is possible.

## 8. Conclusions

[74] We have developed a method to extract the deformation signal from a series of SAR images for pixels whose scattering characteristics vary little with time and look angle (PS pixels). The method works in terrains with or without buildings and no assumptions about variations in displacement rate are required. We refer this method as the Stanford method for persistent scatterers (StaMPS).

[75] Using StaMPS we extract the deformation signal from SAR data acquired over Volcán Alcedo between 1997 and 2001. The signal we find implies deflation of a subhorizontal, prolate ellipsoidal source, extended in the direction of the long axis of Isla Isabela, which is consistent with crystallization of a pipe-like magma body at around 2.2 km depth. We also detect displacements on the inner slopes of the west side of the caldera that are consistent with landsliding.

## Acknowledgments

[87] We thank K. Feigl and an anonymous reviewer for thorough comments, which improved the manuscript. ERS data were provided by the European Space Agency. SRTM topographic data were provided by the Jet Propulsion Laboratory (JPL). Focused SAR images were produced using the ROI_PAC software package developed by JPL. Some interferometric processing was performed using the Doris software package developed by the Delft Institute for Earth-Oriented Space Research (DEOS), Delft University of Technology. The GMT public domain software [*Wessel and Smith*, 1998] was used to produce one figure. This research was supported by NSF grant EAR-0511035.

## Appendix A:: Coregistration Algorithm

*m*, we define functions

*f*

_{x}

^{m}(

*x*

^{m},

*y*

^{m}) and

*f*

_{y}

^{m}(

*x*

^{m},

*y*

^{m}) that map range position

*x*

^{m}→

*x*

^{0}and azimuth position

*y*

^{m}→

*y*

^{0}, respectively, where superscript 0 denotes the master image. For any point

*k*in image

*m*,

*f*

_{x}

^{m}(

*x*

_{k}

^{m},

*y*

_{k}

^{m}) is equal to the range offset of the point between image

*m*and the master image, Δ

*x*

_{m,k}

^{0}, and

*f*

_{y}

^{m}(

*x*

_{k}

^{m},

*y*

_{k}

^{m}) is equal to the azimuth offset, Δ

*y*

_{m,k}

^{0}. We approximate

*f*

_{x}

^{m}(

*x*

^{m},

*y*

^{m}) as a polynomial function in

*x*

^{m}and

*y*

^{m},

*a*

_{pq}

^{m}represents the coefficient that is

*pth*order in

*x*

^{m}and

*qth*order in

*y*

^{m}. Typically we use terms up to second order. We approximate

*f*

_{y}

^{m}(

*x*

^{m},

*y*

^{m}) similarly.

_{m,k}

^{0}directly at many points, using amplitude cross correlation between the two images. For PS analysis, however, we cannot rely on high correlation between master and slave images. Instead, we estimate offsets, Δ

_{m,k}

^{n}, between all pairs of images,

*m*and

*n*, that we expect to be highly correlated. We estimate the offsets using amplitude cross correlation centered on point

*k*, with a typical oversample factor of 32 and a typical window size of 64 pixels. The offsets between two slave images depend on offsets relative to the master image as

*x*

_{k}

^{n}=

*x*

_{k}

^{m}+ Δ

*x*

_{m,k}

^{n}and

*y*

_{k}

^{n}=

*y*

_{k}

^{m}+Δ

*y*

_{m,k}

^{n}. We then estimate the coefficients of function

*f*

_{x}

^{m}(

*x*

^{m},

*y*

^{m}) for all images simultaneously, by solving the following linear system of equations,

*σ*

_{Δ}where

*σ*

_{Δ}is the standard deviation of the estimate Δ. We estimate

*σ*

_{Δ}from the coherence of the cross correlation,

*γ*, using the formula derived by

*Bamler*[2000], which states

*N*is the number of samples in the estimation window and

*η*is the oversampling factor of the data. We find the coefficients of

*f*

_{y}

^{m}(

*x*

^{m},

*y*

^{m}) similarly.

*m*into the master coordinate system, the functions we require are actually the inverse functions,

*g*

_{x}

^{m}(

*x*

^{0},

*y*

^{0}) and

*g*

_{y}

^{m}(

*x*

^{0},

*y*

^{0}) that map position

*x*

^{0}→

*x*

^{m}and

*y*

^{0}→

*y*

^{m}, respectively. We obtain

*g*

_{x}

^{m}(

*x*

^{0},

*y*

^{0}) for each image by synthesizing a grid of values of

*x*

^{m}and

*y*

^{m}, estimating Δ

_{m,k}

^{0}and Δ

_{m,k}

^{0}from

*f*

_{x}

^{m}(

*x*

^{m},

*y*

^{m}) and

*f*

_{y}

^{m}(

*x*

^{m},

*y*

^{m}), and solving for the coefficients of

*g*

_{x}

^{m}(

*x*

^{0},

*y*

^{0}),

*b*

_{pq}

^{m}, by inverting the following system of linear equations,

*x*

_{k}

^{0}=

*x*

_{k}

^{m}+ Δ

*x*

_{m,k}

^{0}and

*y*

_{k}

^{0}=

*y*

_{k}

^{m}+ Δ

*y*

_{m,k}

^{0}. We find

*g*

_{y}

^{m}(

*x*

^{0},

*y*

^{0}) similarly.

## Appendix B:: Amplitude Dispersion Thresholding

[79] *Ferretti et al.* [2001] show that for a constant signal and high signal-to-noise ratio (SNR), ≈ *σ*_{ϕ}, where *σ*_{ϕ} is the phase standard deviation. They plot the relationship using simulation, which we have repeated in Figure B1. The model consists of a constant signal of amplitude 1, with additive noise selected from a complex circular Gaussian distribution with a characteristic standard deviation *σ*_{n} for both the real and imaginary components. Figure B1 shows that given 34 images, is a reasonable proxy for *σ*_{ϕ}, for low values of *σ*_{ϕ}.

[80] However, Figure B1 might lead one to conclude that is a better proxy for phase stability than it really is, for two reasons. First, the error bars show the variability of for a given value of *σ*_{n}, but it is the variability of *σ*_{n} for a given value of that is required. Although this variability depends on the chosen distribution of *σ*_{n}, which is arbitrary here, we can get an idea of the variability of *σ*_{n} for any given value of from Figure B1. For instance, given this simulated distribution of *σ*_{n}, = 0.3 indicates a range for *σ*_{n} of about 0.28 to 0.4 at 68% confidence. Second, although a given value of *σ*_{n} implies a particular value of *σ*_{ϕ}, the variability in phase for a finite sample is better represented by the standard deviation estimated from each sample, _{ϕ}, which varies about *σ*_{ϕ}. In this example, a range for *σ*_{n} of 0.28 to 0.4 implies a range for _{ϕ} of about 0.25 to 0.54, at 68% confidence. Thus, even for a relatively small value of = 0.3, the variability in phase stability is rather large. An alternate way to view the relationship between and phase stability is by plotting the same data in a scatter plot as in Figure B1b, where the variability of _{ϕ} for any given value of is immediately apparent. It is also apparent that although tends to the theoretical limit for the Rayleigh distribution of ≃ 0.523, as SNR tends to zero [*Ferretti et al.*, 2001], estimated values of from finite samples can be somewhat greater.

[81] The actual relationship between and _{ϕ} for any given data set depends on the distribution of SNR within the data set. We can estimate a model distribution of SNR for any given data set from , and then use this to build a simulated distribution of _{ϕ} versus . First, we estimate SNR distributions for the data set by applying the model of signal plus circular Gaussian noise. We simulate a distribution of for a range of SNR values, and solve for the weighted sum of individual SNR distributions that best fits the distribution of in the data, using a nonnegative least squares algorithm to determine the weightings. Two examples are shown in Figure B2, the first for 15 images acquired over Volcán Alcedo and the second for 24 images acquired over Mount St. Helens in Washington State. Initially, we tried fitting the data for all values of but found for higher values that the model always systematically underfit the data, so we only attempted to fit values of < 0.5. The systematic misfit of the data implies that the model does not exactly reflect reality. This was also the conclusion of *Kampes* [2005], who analyzed data acquired over Berlin and found the mean value of to be 0.56. As the maximum theoretical mean value of is 0.523, it is clearly impossible to explain this distribution as the sum of distributions for different SNRs. The underlying assumption of this model is that any pixel can be characterized by fixed SNR in time. As we expect the SNR for some pixels to vary with time, it is not surprising that the model is unable to fit the data distribution exactly. However, for lower values of the model is adequate. Given our model distributions of SNR, we then simulate the distribution of _{ϕ} versus , as shown in Figure B3. If we took _{ϕ} ≤ 0.6 as indicative of phase stability [*Colesanti et al.*, 2003b], simulated values of _{ϕ} for the Mount St. Helens data set would indicate that there were no stable scatterers at all, according to this definition of phase stability. For the Alcedo data set, the simulated values indicate that even choosing pixels with < 0.25 does not necessarily imply good phase stability.

[82] The phase analysis steps of PS methods that rely on a functional model of temporal deformation to identify PS pixels require an initial selection of pixels as PS candidates. These candidates should provide spatial coverage of at least 3 to 4 candidates/km^{2} [*Colesanti et al.*, 2003b] and most of the candidates must be stable [*Kampes*, 2005]. Figure B4 shows, for the Alcedo data set, the percentage of area that meets the minimum candidate density requirement and the percentage of candidates that have good phase stability, for any chosen threshold. If using to pick the candidates, the density requirement pushes the threshold value up and the stability requirement pushes it down; hence there is no value that satisfies both requirements for the Alcedo data set.

[83] The phase analysis step in StaMPS does not require that most PS candidates are in fact PS pixels, so we are free to set as high as we like. Our criterion for phase stability is also more relaxed than _{ϕ} ≤ 0.6, as we expect PS pixels in rural areas to have lower SNR than those in urban areas. Our only requirement is that the signal be distinguishable from the noise, and on this basis we find PS pixels even in the Mount St. Helens data set. In theory, we could run the phase analysis step with no thresholding at all, but in practice computational times are greatly improved by thresholding. We choose a threshold value of that reduces the data volume by about an order of magnitude and includes most low simulated _{ϕ} values. This is typically in the region of 0.4.

## Appendix C:: PS Probability

*γ*

_{x}and the probability that pixel

*x*is a PS pixel. By binning and normalizing the values of

*γ*

_{x}we can estimate the probability density of

*γ*

_{x},

*p*(

*γ*

_{x}). We treat our population of pixels as the union of two populations, population

*A*containing only PS pixels and population

*B*containing only non-PS pixels;

*p*(

*γ*

_{x}) is, then, a weighted sum of the probability density for the PS pixels,

*p*

_{A}(

*γ*

_{x}), and the probability density for the non-PS pixels,

*p*

_{B}(

*γ*

_{x}) (see Figure C1), i.e.,

*α*≤ 1. In order to derive

*p*

_{B}(

*γ*

_{x}), we simulate 10

^{6}pseudopixels with random phase, i.e.,

*W*{

*ψ*

_{x,i}−

_{x,i}} = exp(

*R*), where

*R*is a random variable in the interval [−

*π*,

*π*], and follow the steps described above to arrive at a value of

*γ*

_{x}for each pseudopixel. We bin these values and normalize the distribution to obtain an estimate for

*p*

_{B}(

*γ*

_{x}). For low values of

*γ*

_{x}, i.e., ≤0.3,

*p*

_{A}(

*γ*

_{x}) ≈ 0 which implies

*α*. For pixel

*x*, the probability that it is a PS pixel is

[86] In practice, due to the presence of noise, the function *p*_{B}(*γ*_{x})/*p*(*γ*_{x}) may not be monotonically decreasing as expected, so we smooth it first by convolution with a 7 point Gaussian window.