- journal_title：Geophysics
- Contributor：Carlos Torres-Verdín ; Vladimir L. Druskin ; Sheng Fang ; Leonid A. Knizhnerman ; Alberto Malinverno
- Publisher：Society of Exploration Geophysicists
- Date：2000-
- Format：text/html
- Language：en
- Identifier：10.1190/1.1444858
- journal_abbrev：Geophysics
- issn：0016-8033
- volume：65
- issue：6
- firstpage：1733
- section：ELECTRICAL AND ELECTROMAGNETIC METHODS
We develop a solution to the nonlinear inverse problem via a cascade sequence of auxiliary least-squares minimizations. The auxiliary minimizations are nonlinear inverse problems themselves, except that they are implemented with an approximate forward problem that is at least an order of magnitude faster to solve than the algorithm used to simulate the measurements. Any given auxiliary minimization in the cascade is fully self-contained and yields a solution of the unknown model parameters. This solution, in turn, is used to perform a numerical simulation of the data to quantify its agreement with the input measurements. If the difference between the measured data and the simulated data is above the estimated noise threshold, it is input as data to the subsequent auxiliary inverse problem in the cascade. Otherwise, the inversion is brought to a successful completion. We describe the theory and operating conditions under which this cascade inversion approach converges to a solution equivalent to that of a single inversion implemented with the original forward problem.
Depending on the choice of approximate forward problem and regardless of the computer algorithm employed to solve the inversion, the cascade sequence of auxiliary minimizations can be made many times more efficient than a single inversion performed with the original forward problem. In this paper, we consider a practical and flexible way to construct the approximate forward problem by using a subset of the finite-difference grid used to simulate the measurements numerically, i.e., a dual-grid construction approach.
Numerical examples of performance are described based on the inversion of cross- and single-well 2.5-D direct-current (dc) resistivity data. Our results suggest that the proposed dual-grid minimization technique may be able to improve greatly the performance of even the most effective multidimensional nonlinear inversion procedures currently in use.