Harmonic and biharmonic biases in potential field inversion.

We found that minimum ℓ₂-norm and smoothness-constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic, respectively. In the case of a discrete distribution, the minimum ℓ₂-norm and smoothness-constrained solutions become biased toward being harmonic or biharmonic, respectively. As a result, the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle, which forces the solution maxima and minima to lie on any boundary of the discretized region. The above findings were illustrated with 2D numerical examples. The harmonic or biharmonic bias is brought forth when the strengths of the minimum ℓ₂-norm or the smoothness constraint are enhanced (relative to all other constraints) by approximating the continuous case (a large number of discretizing cells relative to the number of independent observations) and/or by using a regularizing parameter. We discovered that, by inspecting the rearranged normal equations, it is possible to qualify three different possibilities of designing nonharmonic estimators. Then we found that all three possibilities have in fact already been implemented in the literature, reinforcing, in this way, the validity of the theoretical demonstration. [ABSTRACT FROM AUTHOR]