Transformation of nonlinear problems into linear ones applied to the magnetic field of a two‐dimensional prism

Abstract

I present a magnetic interpretation method which transforms into a linear problem the nonlinear problem of obtaining the geometric and position parameters of a two‐dimensional vertical, infinite prism. The magnetization, the only linear parameter, becomes nonlinear after the transformation. By assuming a few discrete values over a prescribed interval for the magnetization, I obtain several solutions for the geometric and position parameters. By storing only the extreme solutions, bounds for each parameter are produced. The method was applied to synthetic anomalies due to isolated and interfering sources for which robust alternatives performed better than the least‐squares method. The correlation between the magnetization and the prism width is the most important factor controlling ambiguity of parameters. The horizontal position is the least affected parameter, followed by the depth to the top of the prism. Application to a real anomaly confirmed the results from synthetic data, except for a greater uncertainty in the estimation of the horizontal position. The uncertainty results from the requirement in the present method that the observations be reduced to the pole; an imprecise knowledge of the magnetization direction distorts the position, which is highly correlated with the magnetization inclination. Because the estimation of the position, depth, and width is transformed into a linear problem, the method is simple, fast, and independent of the initial guess. The method might, therefore, be useful in automatic interpretation of basement relief. By producing bounds for each parameter estimate, an analysis of parameter precision and ambiguity is also possible.