Frequency domain least‐squares inversion of thick dike magnetic anomalies using Marquardt algorithm

Abstract

The elegance and ease with which the Fourier transformation method can be employed in the interpretation of potential field geophysical data (Dean, 1958; Gudmundsson, 1966) has resulted in the formulation of a number of frequency‐domain approaches for interpreting the magnetic effects of specific source geometries, e.g., Bhattacharyya (1966), Sengupta (1974), Sengupta and Das (1975), Bhattacharyya and Leu (1975, 1977), etc. In these techniques, the parameters of interest are estimated from the characteristics of the amplitude and phase spectra, a process that is often beset with several problems which adversely affect the estimation of parameters. These problems arise partly from the ill‐defined nature of characteristic points in the spectra, either due to spectral noise or due to inadequate digitization spacing, and partly from the fact that the errors in the estimation of certain parameters which are used for subsequent estimation of the other parameters usually cause added spectral distortions. Rao et al. (1978) advocated the use of end corrections to overcome these problems. An effective alternative would be to incorporate the least‐squares method in the interpretational process. This necessitates iteration owing to the nonlinear nature of equations relating the source parameters and the Fourier spectra of corresponding anomalies. While it is tempting to employ the iteration scheme formulated by Gauss in 1821 for this purpose, as has indeed been done in a number of space‐domain approaches (e.g., Hall, 1958; Bosum, 1968; Al‐Chalabi, 1970; Rao et al., 1973; Won, 1981), the convergence in this approach requires the initially assumed values of the parameters to be very close to the final solution. The steepest descent method, an alternative to this, has an extremely slow convergence. It is shown here, with the example of a two‐dimensional (2-D) structure, that Marquardt’s algorithm (Marquardt, 1963), which offers a compromise between these two iteration schemes and has been effectively employed by Johnson (1969) and Pedersen (1977) in their space‐domain interpretation schemes, provides an efficient approach to accomplish the iteration in the case of frequency‐domain magnetic interpretation.