A Multiscale Approach to Solving One Dimensional Inverse Problems

2] suggests that multiscale methods are a useful tool for purposes of computational efficiency. In this paper we explore a multiresolution approach We consider inverse problems described by that class to solving one dimensional inverse problems. The apof operators which are made sparse under the action proach we take is motivated by the work of Chou, of the wavelet transform [2]. Thus, the inversion proGolden, and Willsky [1] and Beylkin, Coifman, and cedure is carried out in the wavelet transform domain. Rokhlin [2]. Specifically, we consider inverse probAs with many inverse problems, difficulties associated lems described by that class of operators which are with ill-posedness and ill-conditioning are overcome made sparse under the action of the wavelet transvia regularization. Because the inversion is performed form [2]. Moreover, statistically-based inversion proin scale-space, we employ a statistically-based regularcedures utilizing multiscale a priori stochastic models ization technique utilizing multiscale a priori models are considered [1]. As a concrete example, we examine similar to those explored by Chou and Willsky in [1]. a deconvolution problem arising in wellbore induction The vehicle for our work is an inverse problem atismeasurement of conductivity. ing in the area of geological exploration. Specifically, we are interested in determining the conductivity profile about a wellbore based upon a suite of induction measurements each of which contains conductivity in