Use of Difference-Based Methods to Explore Statistical and Mathematical Model Discrepancy in Inverse Problems

Normalized differences of several adjacent observations, referred to as pseudo measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. In the presence of model misspecification, we also propose to use the information provided by pseudo measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.