We present here a new InSAR persistent scatterer selection method using maximum likelihood estimation to identify persistent scattering pixels, which results in a denser network of reliable phase measurements than do existing methods. We analyze the phase of each pixel in a series of interferograms and estimate the relative strength of any slowly fluctuating component of the radar echo from a dominant scatterer to the background scattering within a pixel. We find a fairly dense network of scatterers with stable phase characteristics in areas where conventional InSAR fails due to decorrelation. We examined data over two vegetated regions in the San Francisco Bay Area. The average phases of these pixels clearly show the slip along the Hayward fault, and set upper bounds on any slip along the Bay Area segment of the San Andreas fault.
 Conventional InSAR has been an effective technique for measuring deformation in regions exhibiting good interferometric coherence. Even so, most interferograms still exhibit areas of low correlation in which it is difficult to extract useful phase observations, limiting the interpretation of the deformation signal [e.g., Jónsson et al., 1999, 2002]. The presence of vegetation, snow or weathering often alters the scattering properties of the surface over time and results in loss of InSAR coherence. Geometric, or spatial, decorrelation also inhibits us from using all available data by rendering few image pairs feasible for analysis, as orbit tracks separated in space by more than the critical baseline produce no useful phase measurements.
Ferretti et al.  first suggested identifying stable natural reflectors, or persistent (“permanent”) scatterers (PS), to exploit those pixels in an interferogram time series which exhibit a significant degree of coherence over time. Colesanti et al.  enhanced this approach and various groups [e.g., Adam et al., 2003; Werner et al., 2003] have since refined the method to increase the reliability of the identification. All of these methods initially select PS pixels from observations of the amplitude scintillations of each pixel – low scintillation indices often correspond to low phase variation [Ferretti et al., 2001]. These algorithms work well in urban settings, where there are many bright reflectors associated with man-made structures, but fail to identify a reliable and dense network of PS pixels in natural terrain.
Hooper et al.  proposed a coherence-based method to identify PS pixels in natural terrain for an experiment examining volcanic deformation at Long Valley caldera in California. In his method the deformation and atmospheric phases are modeled as spatially correlated, thus a low pass filter applied to the network permits subtraction from the observations. The method is rather empirical, however, and relies heavily on statistics obtained from simulation. One major advantage of Hooper's approach is that it does not require assuming a time-dependent form of the deformation signal, as needed in the Ferretti et al.  method, beyond its spatial correlation function.
 We present here a new method for selection of the PS pixels using maximum likelihood estimation techniques. We adopt the basic description of InSAR phase as the sum of specific terms defined by Hooper et al. , and similarly do not assume a deformation model during the selection process. The denser PS network suffices for estimation of deformation.
2. Signal Model
 The signal model follows from letting the signal from the dominant scatterers fluctuate slowly over time, with additional contributions from a scintillating background. Thus we use a Gaussian signal in Gaussian noise model to derive pixel phase. We parameterize the phase distribution by the relative brightness of a dominant scatterer, if any. Resolution elements with a dominant scattering element form persistent scatterers. The model is a function of the “signal to noise ratio” (which we denote γ) of the dominant scatterer in a pixel to the echoes from all other scatterers and any other noises. Note that this signal to noise ratio is not the usual SNR associated with radar signals, hence the γ-notation. We determine the theoretical phase variation for a pixel as a function of γ, and compare it with the observed interferometric phase corrected for atmosphere, topography, deformation and other correlated errors.
 Let z1 = s + n1 and z2 = s + n2 represent the return signals from a resolution element in a single look interferogram at two time instances. The signal s represents the return from the stronger scatterers in a resolution element, has circular Gaussian statistics and is independent of the noise terms n1 and n2, also circular Gaussian terms, representing the return from the weaker scatterers in the resolution cell. They are uncorrelated with each other. Just and Bamler  showed that the PDF of interferometric phase about a constant mean phase using this model is
where βϕ = ∣ρ∣ cos ϕ and the interferometric correlation ρ depends on γ through
 In the derivation of equations (1) and (2) we assume that noise variance is the same at the time of acquisition of each scene [Bamler and Just, 1993]. This PDF has been examined in the more general context of second order speckle statistics [Goodman, 1975]. Similar distributions for multilook interferograms are given by Lee et al. .
 Example plots of the theoretical phase distributions for different values of γ are shown in Figure 1, where we also plot γ as a function of the standard deviation of phase.
3. Maximum Likelihood Estimation
 We express the observed radar interferogram phase ϕ as the sum of phase contributions from several physical effects [Hooper et al., 2004]:
where ϕdef represents the phase change due to surface deformation between observations, ϕαrepresents the phase change due to atmospheric propagation [Zebker et al., 1997], ϕorb is the phase error from uncertainty in orbit parameters, and ϕɛ results from errors in the digital elevation model (DEM) used to remove the effects of topography. Hooper et al.  showed that these phase terms are spatially correlated over short distances, and as such they can be reliably estimated by spatial filtering and removed from each observation. Then, we are left with only the noise term, n, representing the combined phase contribution from the scatterers in the resolution cell. We compare the observed phases with the theoretical phase distributions to obtain the maximum likelihood estimate of γ.
 The value of γ maximizing P(γ∣n1, …., nN), where n1, n2, …., nN are the residual phases at a pixel in N interferograms, γ is again the brightness of the dominant scatterer, and P(A∣B) is the conditional probability of event A given event B. Under Bayes' rule we rewrite the probability function as follows:
 The term in the denominator of the right hand side of equation (4) is independent of γ, thus we need only to maximize the numerator over all values of γ. As we have no prior knowledge of γ, we assume that all values are equally likely, that is, P(γ) is constant for all values of γ. We also assume that the observed interferometric phase values are independent and identically distributed.
 Under these conditions it suffices to maximize the product P(n1∣γ) …. P(nN∣γ) over all possible values of γ. In other words, we can restate equation (4) as maximizing the product
 We compare the maximum likelihood estimate γ for each of the candidate pixels with a pre-determined threshold value (γthr) — those that exceed the threshold form the set of candidate PS pixels.
 The set of candidate PS pixels depends on our selection of the proper γ - threshold. If the threshold is placed too high, many potential PS pixels are not included and the resulting PS network is sparse. If the threshold is set too low, too many PS candidates are included and the network contains many points that are not truly persistent. Our threshold selection is currently rather arbitrary, because we cannot easily calculate the scattering statistics for persistent pixels amid all possible background terrains. If we could set a limit on the natural variation in the phase of each pixel, we could readily select the threshold parameter γthr from the dependence of phase standard deviation on γ (Figure 1) and use one of the common radar detection criteria [e.g., see Skolnik, 2001]. To use the constant false alarm rate criterion, for example, we would use the standard deviation of phase as a function of the parameter γ. If we believe that we should see a natural variation of 1 radian across all interferograms, then the threshold value is about 3. But at present we do not know if a variation of 1 radian is physically reasonable.
 Therefore, we choose to determine the threshold by determining the equivalent value of γ yielding at least that set of PS pixels obtained from accepted PS selection methods. No processed interferograms are available in the public domain for the method of Ferretti et al.  for comparison. Hence, we used the StaMPS method [Hooper, 2006] to determine the threshold. We examined data acquired over two different areas and chose threshold values yielding at least the set of PS pixels found using StaMPS. The threshold in both these cases was 1.85.
4. Application to Data
 We applied our PS pixel selection method to ERS data acquired over the San Francisco bay area in two different regions: along the San Andreas fault near the SFO airport region and along the Hayward fault near the Oakland-Alameda area. These areas consist of a combination of urban and natural terrain clearly traversed by active faults. Conventional InSAR studies combined with GPS observations [Burgmann et al., 2000] measure the fault creep deformation signal well in the urban regions along the Hayward Fault, with inconclusive measurements along the San Andreas fault. Previous PS studies [Burgmann et al., 2006] conducted in this area also pick out coherent scatterers in the urban areas well, but fail in identifying a reliable network of scatterers in the natural terrain on the more rural sides of the faults.
 We processed 19 descending scenes acquired by ERS-1 and ERS-2 between 1992 and 2000. We selected a scene from September 1995 as a master scene, based on minimization of the perpendicular baseline and the temporal baseline, and generated 18 interferograms. The maximum perpendicular baseline was 260 m.
 Using the maximum likelihood selection method, we identify coherent pixels on the non-urbanized sides of each fault, including where the Ferretti and StaMPS methods fail to locate any PS pixels (see Table 1). Figure 2 shows the observed average deformation rates in mm per year and the locations of PS pixels for the San Andreas Fault and Hayward Fault region. Creep of 3 mm/yr shows clearly along the Hayward fault, but no evidence of creep greater than possibly 1–2 mm/yr appears along this segment of the San Andreas fault. Notably, in both these regions our methods picked PS in regions where other methods have failed to identify any coherent scatterers.
Table 1. Comparison Between Different PS Selection Methods (Number of Pixels) for Two Test Regions in the San Francisco Bay Area
 Once we identify the proper set of PS pixels, we obtain a deformation time-series by unwrapping the measured phases in three dimensions, two spatial dimensions and one in time. We used the step wise-3D unwrapping method described by Hooper and Zebker . As with any phase unwrapping method, this algorithm can generate unwrapping errors (Figure 3), therefore results regarding the range change rates in Figure 2 need to be analyzed with caution. Figure 3 shows the comparison of the range change values obtained by unwrapping the phase of the selected PS points with GPS data obtained from the SOPAC GPS archive. The comparison clearly shows discrepancies of about 28 mm (2π) and 56 mm (4π) with GPS data, and the PS data appear to suppress motion seen in the GPS results. Thus the phase unwrapping over-smoothes the data. Yet the unwrapped solution is plausible in the sense that each interferogram in the time series seems to have a reasonable unwrapped solution. Since, we do not assume a deformation model in our method, it is necessary to use a true 3D unwrapping algorithm to estimate the deformation effectively. This remains a significant topic for future work before the denser PS network can be used reliably.
 Examination of the PS pixels outside of the urban areas in both regions of study shows that there are areas with spatially correlated phases distinct from the average background velocities. While these could be areas of coherent motion such as landslides, subsidence, or other localized movement, they can just as easily be the product of phase unwrapping errors. At present we are unable to validate the absolute accuracy of the phase unwrapping and this appears to be the limiting problem in the application of PS methods to natural terrains.
 Maximum likelihood methods for the identification of candidate PS pixels find many more PS points than published algorithms. When applied to ERS-1 and ERS-2 data acquired over the San Francisco bay area, our PS selection method identified 62% (San Andreas Fault region) and 35% (Hayward Fault region) more persistent pixels than the Hooper et al.  algorithm, which in turn finds more PS pixels than the original Ferretti et al.  method. We find 97% of the pixels identified by Hooper plus many more, notably in the vegetated areas in which both previous methods fail to find sufficient persistent scatterers to form a usable network to infer deformation.
 Thus the major advantage of the new method is that it allows us to find persistent scattering pixels in vegetated and non-urban terrain, as shown here in the hills west of the San Andreas fault near the SFO airport and the hills to the east of Hayward fault. Very few coherent scatterers are identified here using the methods of Ferretti et al.  and Hooper et al. , greatly limiting our ability to measure deformation accurately. The PS pixels identified by the maximum likelihood method are mostly a superset of those identified by published methods.
 Significant work remains in the development of useful 3-dimensional phase unwrapping algorithms. Phase unwrapping error is still the major source of uncertainty for PS-InSAR in vegetated areas. The increased density of PS points provided by the maximum likelihood selection method promises to make the unwrapping problem more tractable. Therefore, we hope to extract arbitrary deformation patterns, for now subject to limitations of three dimensional phase unwrapping, in regions where other conventional and persistent scatterer InSAR methods have to date failed.
 The authors would like to thank Andrew Hooper for helpful discussions on StaMPS and PS analysis, and George Hilley for providing us with the Permanent Scatterers ™ data set for the Bay Area to compare our results with. This work was supported by the USGS NEHRP Program and also by the National Science Foundation under grant 0511035.