Scattering, symmetry, and bias analysis of source‐position estimates in Euler deconvolution and its practical implications

Abstract

Despite being widely used, Euler deconvolution has received little attention in its theoretical aspects. We obtain analytical estimators [Formula: see text] and [Formula: see text] for the horizontal [Formula: see text] and the vertical [Formula: see text] source positions, which are the parameters to be estimated when the structural index is assigned a tentative value. By analyzing the estimators’ properties, we show two things. (1) Scattering of the alternative solutions associated with different data window positions has two components: data noise and the use of a wrong tentative value for the structural index, (2) The differences [Formula: see text] and [Formula: see text], as a function of the data window position, exhibit antisymmetric and symmetric behaviors, respectively, about [Formula: see text] when the magnetization and the geomagnetic field inclinations are 90°, 0°, or 45°. For intermediate inclinations there is a moderate departure from the symmetry properties. This analysis shows that the criterion for determining the structural index as the tentative value producing the smallest solution scattering is theoretically sound but occasionally fails in practice because data noise also contributes to solution scattering. In addition, the lower sensitivity of the averages of estimates of the horizontal position (as compared with the vertical position) occurs because [Formula: see text] displays an antisymmetric behavior with respect to the true horizontal position while the estimates of the vertical position display a symmetric behavior with respect to the same point.