.[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] When targeting small amplitude surface deformation, using repeat orbit Interferometric Synthetic Aperture Radar (InSAR) observations can be plagued by propagation delays, some of which correlate with topographic variations. These topographically-correlated delays result from temporal variations in vertical stratification of the troposphere. An approximate model assuming a linear relationship between topography and interferometric phase has been used to correct observations with success in a few studies. Here, we present a robust approach to estimating the transfer function, K, between topography and phase that is relatively insensitive to confounding processes (earthquake deformation, phase ramps from orbital errors, tidal loading, etc.). Our approach takes advantage of a multiscale perspective by using a band-pass decomposition of both topography and observed phase. This decomposition into several spatial scales allows us to determine the bands wherein correlation between topography and phase is significant and stable. When possible, our approach also takes advantage of any inherent redundancy provided by multiple interferograms constructed with common scenes. We define a unique set of component time intervals for a given suite of interferometric pairs. We estimate an internally consistent transfer function for each component time interval, which can then be recombined to correct any arbitrary interferometric pair. We demonstrate our approach on a synthetic example and on data from two locations: Long Valley Caldera, California, which experienced prolonged periods of surface deformation from pressurization of a deep magma chamber, and one coseismic interferogram from the 2007 Mw 7.8 Tocapilla earthquake in northern Chile. In both examples, the corrected interferograms show improvements in regions of high relief, independent of whether or not we pre-correct the data for a source model. We believe that most of the remaining signals are predominately due to heterogeneous water vapor distribution that requires more sophisticated correction methods than those described here.

S U M M A R YWe present a spherical wavelet-based multiscale approach for estimating a spatial velocity field on the sphere from a set of irregularly spaced geodetic displacement observations. Because the adopted spherical wavelets are analytically differentiable, spatial gradient tensor quantities such as dilatation rate, strain rate and rotation rate can be directly computed using the same coefficients. In a series of synthetic and real examples, we illustrate the benefit of the multiscale approach, in particular, the inherent ability of the method to localize a given deformation field in space and scale as well as to detect outliers in the set of observations. This approach has the added benefit of being able to locally match the smallest resolved process to the local spatial density of observations, thereby both maximizing the amount of derived information while also allowing the comparison of derived quantities at the same scale but in different regions. We also consider the vertical component of the velocity field in our synthetic and real examples, showing that in some cases the spatial gradients of the vertical velocity field may constitute a significant part of the deformation. This formulation may be easily applied either regionally or globally and is ideally suited as the spatial parametrization used in any automatic time-dependent geodetic transient detector.

Since the beginning, Mathematical Morphology has proposed to extract shapes from images as connected components of level sets. These methods have proved very efficient in shape recognition and shape analysis. In this paper, we present an improved method to select the most meaningful level lines (boundaries of level sets) from an image. This extraction can be based on statistical arguments, leading to a parameter free algorithm. It permits to roughly extract all pieces of level lines of an image, that coincide with pieces of edges. By this method, the number of encoded level lines is reduced by a factor 100, without any loss of shape contents. In contrast to edge detections algorithm or snakes methods, such a level lines selection method delivers accurate shape elements, without user parameter: no smoothing involved and selection parameters can be computed by Helmholtz Principle.

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