Focusing geophysical inversion images

Abstract

A critical problem in inversion of geophysical data is developing a stable inverse problem solution that can si multaneously resolve complicated geological structures. The traditional way to obtain a stable solution is based on maximum smoothness criteria. This approach, however, provides smoothed unfocused images of real geoelectri cal structures. Recently, a new approach to reconstruc tion of images has been developed based on a total varia tional stabilizing functional. However, in geophysical ap plications it still produces distorted images. In this paper we develop a new technique to solve this problem which we call focusing inversion images. It is based on specially selected stabilizing functionals, called minimum gradi ent support (MGS) functionals, which minimize the area where strong model parameter variations and disconti nuity occur. We demonstrate that the MGS functional, in combination with the penalization function, helps to generate clearer and more focused images for geologi cal structures than conventional maximum smoothness or total variation functionals. The method has been suc cessfully tested on synthetic models and applied to real gravity data. TIKHONOV REGULARIZATION AND STABILIZING FUNCTIONALS Consider the inverse problem d= Am, (1) where A is the forward modeling operator; m = m(r), a scalar function describing geological model parameter distribution in some volume V in the earth (m EM, where M is a Hilbert space of models with L z norm); and d = d(r), a geophysical data set (d E D, where D is a Hilbert space of data).

Salt-body Inversion with Minimum Gradient Support and Sobolev Space Norm Regularizations

“…According to the conventional approach by Portniaguine and Zhdanov (1999), the functional J mgs can be approximated by simple L 2 weighting of the model m for the gradient evaluation. Although, this approximation turns the problem into conventional Tikhonov regularization, it pays the price with a loss in sensitivity of the functional.…”

“…First, we apply minimum gradient support (MGS), a non-linear regularizer that is often used in geophysical imaging with electromagnetic and gravity fields. Historically, Portniaguine and Zhdanov (1999) introduced MGS to produce sharper inversion results compared to the most commonly used in seismic exploration regularizations such as total-variation (TV) and Tikhonov. However, updating higher wavenumbers of the model at early iterations might cause FWI failure.…”

Two‐dimensional magnetotelluric inversion of blocky geoelectrical structures

“…It could make the image look unrealistically sharp. Following Portniaguine and Zhdanov [1999a], we impose the upper bound s + (r) and the lower bound s À (r) for the conductivity values s(r) determined from inversion. During the iterative process we force the model parameter values to fit within these bounds.…”

“…In these situations it can be useful to search for a stable solution within the class of inverse models with blocky geoelectrical structures. The mathematical technique for solving this problem was developed by Portniaguine and Zhdanov [1999a]. It is based on introducing a new type of stabilizing functional in inverse problem solution, a so-called minimum support functional.…”

“…[5] This technique was successfully applied to the inversion of gravity data [Last and Kubik, 1983;Portniaguine and Zhdanov, 1999a] to generate a focused and resolved inverse image of the density distribution in a geological cross section. This idea was also implemented in inversion of three-dimensional (3-D) controlled-source MT (CSMT) data over the structures with sharp geoelectrical boundaries [Portniaguine and Zhdanov, 1999b;Zhdanov and Hursan, 2000].…”