On the application of Euler deconvolution to the analytic signal

Standard Euler deconvolution is applied to potential-field functions that are homogeneous and harmonic. Homogeneity is necessary to satisfy the Euler deconvolution equation itself, whereas harmonicity is required to compute the vertical derivative from data collected on a horizontal plane, according to potential-field theory. The analytic signal modulus of a potential field is a homogeneous function but is not a harmonic function. Hence, the vertical derivative of the analytic signal is incorrect when computed by the usual techniques for harmonic functions and so also is the consequent Euler deconvolution. We show that the resulting errors primarily affect the structural index and that the estimated values are always notably lower than the correct ones. The consequences of this error in the structural index are equally important whether the structural index is given as input (as in standard Euler deconvolution) or represents an unknown to be solved for. The analysis of a case history confirms serious errors in the estimation of structural index if the vertical derivative of the analytic signal is computed as for harmonic functions. We suggest computing the first vertical derivative of the analytic signal modulus, taking into account its nonharmonicity, by using a simple finite-difference algorithm. When the vertical derivative of the analytic signal is computed by finite differences, the depth to source and the structural index consistent with known source parameters are, in fact, obtained.