A robust interactive estimation of the regularization parameter

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 mu in a regularized inverse problem. For each tentative value of mu, we perturb the observations with J sequences of pseudorandom noise and we track down the instability effect on the solutions. Then, we define a quantitative measure rho(mu) 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 rho(mu) cannot be used directly to estimate the best value of mu (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 (x, y, z) space, the pair (p) over cap (i) and (p) over cap (j) of the solutions most distant from each other and associated with the current rho(mu). From this comparison, the interpreter decides if the current mu produces stable solutions. Because the bisection method can be applied only to monotonic functions (or segments of monotonic functions) and because rho(mu) has a theoretical monotonic behavior that can be corrupted, in practice by a poor experiment design, the set of values of rho(mu) can be used as a quality control of the experiments in the proposed bisection method to estimate the best value of mu. 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.