Sparse regularization of inverse gravimetry—case study: spatial and temporal mass variations in South America

Abstract

Sparse regularization has recently experienced high popularity in the inverse problems community. In this paper, we show that a sparse regularization technique can also be developed for linear geophysical tomography problems. For this purpose, we adapt a known matching pursuit algorithm. The main theoretical features (existence, stability, and convergence) of the new method are given. We also show further properties of some trial functions which we use. Moreover, the algorithm is applied to a static and a monthly varying gravitational field of South America which yields spatial and temporal variations in the mass distribution. The new approach represents essential progress in comparison to a corresponding wavelet method, which is not flexible enough for the use of heterogeneous data, and a respective spline method, where the resolution cannot exceed approximately 10 4 basis functions due to experienced numerical problems with the ill-conditioned and dense matrix. The novel sparse regularization technique does not require homogeneous data and is not limited in the number of basis functions due to its iterative algorithm.

Inverting GRACE gravity data for local climate effects

“…This method allows the inversion of noisy data with an increased spatial resolution by mixing different types of trial functions, e.g., functions with a global character as well as localized functions. Furthermore, in Fischer (2011), Fischer andMichel (2012), Fischer and Michel (2013) it was shown that the method is well-matched to invert heterogeneous data as well as large data sets.…”

“…For more information about the theoretical properties of algorithm 2.1 (RFMP), we refer to Fischer and Michel (2012). Code optimization and parallelization now allow a fast and competitive inversion of the data (for further details on the developed algorithm, see Fischer and Michel 2013).…”

“…Note that the choice of the dictionary strongly affects the convergence rate of the method (see Fischer 2011, Fischer andMichel 2012 for further details). Thus, it is advantageous to choose a wellmatched dictionary with respect to the structure of the solution.…”

“…For further numerical case studies on the mass transports in the Amazon area derived from GRACE data as well as the density distribution in the area of South America or the Himalayas out of the EGM2008, we refer to Fischer (2011), Fischer andMichel (2012), Fischer and Michel (2013). As discussed in Sect.…”

“…For this reason, a detailed and consecutive investigation of such phenomena is of great importance. We use here a novel technique called the Regularized Functional Matching Pursuit (RFMP see Fischer and Michel (2012)) to locate and analyze extreme weather events of the late summer months of six successive years starting with 2005 in the Amazon area, where the novel method allows for a much better localization and resolution of these effects than the usual methods or hydrological data. After a short introduction of the used data set and the RFMP, we concentrate on localizing particular extreme weather conditions in inverted GRACE potentials.…”