Stability analysis and improvement of structural index estimation in Euler deconvolution

Abstract Euler deconvolution has been widely used in automatic aeromagnetic interpretations because it requires no prior knowledge of the source magnetization direction and assumes no particular interpretation model, provided the structural index defining the anomaly falloff rate related to the nature of the magnetic source, is determined in advance. Estimating the correct structural index and electing optimum criteria for selecting candidate solutions are two fundamental requirements for a successful application of this method. We present a new criterion for determining the structural index. The criterion is based on the correlation between the total-field anomaly and the estimates of an unknown base level. These estimates are obtained for each position of a moving data window along the observed profile and for several tentative values for the structural index. The tentative value for the structural index producing the smallest correlation is the best estimate of the correct structural index. We also propose a new criterion to select the best solutions from a set of previously computed candidate solutions, each one associated with a particular position of the moving data window. A current criterion is to select only those candidates producing a standard deviation for the vertical position of the source smaller than a threshold value. We propose that in addition to this criterion, only those candidates producing the best fit to the known quantities (combinations of anomaly and its gradients) be selected. The proposed modifications to Euler deconvolution can be implemented easily in an automated algorithm for locating the source position. The above results are grounded on a theoretical uniqueness and stability analysis, also presented in this paper, for the joint estimation of the source position, the base level, and the structural index in Euler deconvolution. This analysis also reveals that the vertical position and the structural index of the source cannot be estimated simultaneously because they are linearly dependent; the horizontal position and the structural index, on the other hand, are linearly independent. For a known structural index, estimates of both horizontal and vertical positions are unique and stable regardless of the value of the structural index. If this value is not too small, estimates of the base level for the total field are stable as well. The proposed modifications to Euler deconvolution were tested both on synthetic and real magnetic data. In the case of synthetic data, the proposed criterion always detected the correct structural index and good estimates of the source position were obtained, suggesting the present theoretical analysis may lead to a substantial enhancement in practical applications of Euler deconvolution. In the case of practical data (vertical component anomaly over an iron deposit in the Kursk district, Russia), the estimated structural index (corresponding to a vertical prism) was in accordance with the known geology of the deposit, and the estimates of the depth and horizontal position of the source compared favorably with results reported in the literature.