A Parameter Choice Strategy for the Inversion of Multiple Observations

TL;DR

This paper introduces a general parameter choice strategy for multi-parameter inverse problems in geosciences, leveraging noisy measurements and inner product estimations to achieve near-optimal accuracy.

Contribution

It presents a novel method for selecting parameters in multi-parameter inverse problems that guarantees accuracy close to the best possible, based on measurement data and inner product estimates.

Findings

Method achieves accuracy within a constant factor of the noise level.

Applicable to problems with noisy measurements and discretization.

Provides a systematic approach for parameter selection in inverse problems.

Abstract

In many geoscientific applications, multiple noisy observations of different origin need to be combined to improve the reconstruction of a common underlying quantity. This naturally leads to multi-parameter models for which adequate strategies are required to choose a set of 'good' parameters. In this study, we present a fairly general method for choosing such a set of parameters, provided that discrete direct, but maybe noisy, measurements of the underlying quantity are included in the observation data, and the inner product of the reconstruction space can be accurately estimated by the inner product of the discretization space. Then the proposed parameter choice method gives an accuracy that only by an absolute constant multiplier differs from the noise level and the accuracy of the best approximant in the reconstruction and in the discretization spaces.