A robust interactive estimation of the regularization parameter

Abstract

We have developed a new and robust method (in the sense of it being applicable to a wide range of situations) to estimate the regularization parameter [Formula: see text] in a regularized inverse problem. For each tentative value of [Formula: see text], we perturb the observations with [Formula: see text] sequences of pseudorandom noise and we track down the instability effect on the solutions. Then, we define a quantitative measure [Formula: see text] of the solution instability consisting of the largest value among the Chebyshev norms of the vectors obtained by the differences between all pairs of the perturbed solutions. Despite being quantitative, [Formula: see text] cannot be used directly to estimate the best value of [Formula: see text] (the smallest value that stabilizes the solution) because, in practice, instability may depend on the particular and specific interests of the interpreter. Then, we determine that the interpreter, at each iteration of a bisection method, visually compares, in the ([Formula: see text], [Formula: see text], [Formula: see text]) space, the pair [Formula: see text] and [Formula: see text] of the solutions most distant from each other and associated with the current [Formula: see text]. From this comparison, the interpreter decides if the current [Formula: see text] produces stable solutions. Because the bisection method can be applied only to monotonic functions (or segments of monotonic functions) and because [Formula: see text] has a theoretical monotonic behavior that can be corrupted, in practice by a poor experiment design, the set of values of [Formula: see text] can be used as a quality control of the experiments in the proposed bisection method to estimate the best value of [Formula: see text]. Because the premises necessary to apply the proposed method are very weak, the method is robust in the sense of having broad applicability. We have determined part of this potential by applying the proposed method to gravity, seismic, and magnetotelluric synthetic data, using two different interpretation models and different types of pseudorandom noise.