Nonlinear inversion of resistivity sounding data

The resistivity interpretation problem involves the estimation of resistivity as a function of depth from the apparent resistivity values measured in the field as a function of electrode separation. This is commonly done either by curve matching using master curves or by more formal linearized inversion methods. The problems with linearized inversion schemes are fairly well known; they require that the starting model be close to the true solution. In this paper, we report the results from the application of a nonlinear global optimization method known as simulated annealing (SA) in the direct interpretation of resistivity sounding data. This method does not require a good starting model but is computationally more expensive. We used the heat bath algorithm of simulated annealing in which the mean square error (difference between observed and synthetic data) is used as the energy function that we attempt to minimize. Samples are drawn from the Gibbs probability distribution while the control parameter the temperature is slowly lowered, finally resulting in models that are very close to the globally optimal solutions. This method is also described in the framework of Bayesian statistics in which the Gibbs distribution is identified as the a posteriori probability density function in model space. Computation of the true posterior distribution requires computation of the energy function at each point in model space. However, a fairly good estimate of the most significant portion(s) of the function can be obtained from simulated annealing run in a reasonable computation time. This can be achieved by making several repeat runs of SA, each time starting with a new random number seed so that the most significant portion of the model space is adequately sampled. Once the posterior density function is known, many measures of dispersion can be made. In particular, we compute a mean model and the a posteriori covariance matrix. We have applied this method successfully to synthetic and field data. The resulting correlation nd covariance matrices indicate how the model parameters affect one another and are very useful in relating geology to the resulting resisitivity values.