Stochastic regularization: Smoothness or similarity?

Abstract

Inversions of geophysical data often involve solving large-scale underdetermined systems of equations that require regularization, preferably through incorporation of a priori information. Since many natural phenomena exhibit complex random behavior, statistical properties offer important a priori constraints. Inversion constrained by model covariance functions, a form of stochastic regularization, is formally equivalent to imposing simultaneously the auxiliary constraints of (i) model correlation (smoothness) and (ii) similarity with a preferred model (damping). We show that a priori stochastic information defines uniquely the relative contributions of smoothing and damping, such that the higher the fractal dimension the greater the damping contribution. However, if the model discretization interval exceeds the characteristic scale length of the parameters to be resolved, stochastic regularization artificially reduces to only damping constraints.