Frechet derivatives provide the vital information for parametric resistivity inversion, but the calculation for a multidimensional problem is often computer intensive. This paper presents a new technique for fast calculation of the Frechet derivatives of resistivity measurements with respect to formation resistivity properties. The technique, referred to as the auxiliary source array method (ASAM), generalizes the reciprocity principle‐based methods in that for closely spaced receivers it may not be necessary to place a fictitious source at each receiver location. Rather, an auxiliary source array comprised of sparsely spaced fictitious sources can be constructed from which the field for any fictitious source location can be reconstructed. The ASAM was tested with a deviated‐well resistivity model for an array resistivity device that acquires 8 potential, 16 first potential difference, and 14 second potential difference data points at each depth level. The Frechet derivatives calculated by the ASAM agree well with those obtained through the parameter perturbation method. The tests showed that the calculation time of the ASAM has little dependence on the number of parameters for which the Frechet derivatives are to be calculated. The method can calculate the Frechet derivatives of 5 to 138 resistivity parameters with only 20% to 50% additional computer time. For the 138‐parameter model, the ASAM is about two orders of magnitude faster than the parameter perturbation method.
Gâteaux and Fréchet derivatives of the operator of geometrically nonlinear bending problem of sandwich plate
