Invariance Rules in the Regularized Inversion of Gravity and Magnetic Fields and their Derivatives

In potential field inversion problems we usually solve underdetermined systems and this leads to a very shallow solution, typically known as minimum length solution. This may be avoided introducing a depth weighting function in the objective function (Li and Oldenburg, 1996). In this paper we derive invariance rules for either the minimum norm minimization and for the regularized inversion with depth weighting and positivity constraints. For a given source class, corresponding to a specific structural index N, the invariance rule assures that the same solution is obtained inverting the magnetic (or gravity) field or any of its nth order vertical derivatives. Although we demonstrate mathematically this invariance rule for the minimum norm minimization only, it is shown to occur also for the regularized inversion with depth weighting and positivity constraints. In this case, a source-class invariant form of depth weighting is derived, referring to that of the magnetic field, in the magnetic case, and to the 1st derivative of the gravity field, in the gravity case. We also illustrate how the combined effect of regularization parameter and depth weighting influences the estimated source model depth in the regularized inversion with depth weighting and positivity constraints.