# Characterizing the impact of model error in hydrogeologic time series recovery inverse problems

## TL;DR

This paper investigates how systematic model errors affect hydrogeologic time series reconstruction, deriving bounds and diagnostics to improve understanding and handling of such errors in inverse problems.

## Contribution

It introduces a systematic analysis of model error impact in hydrogeologic inverse problems, including new bounds and diagnostic criteria for transfer function uncertainties.

## Key findings

- Derived upper and lower error bounds for model error effects.
- Monte Carlo simulations validate the bounds and reveal error behavior.
- Identified a diagnostic criterion for transfer function characterization.

## Abstract

Hydrogeologic models are commonly over-smoothed relative to reality, owing to the difficulty of obtaining accurate high-resolution information about the subsurface. When used in an inversion context, such models may introduce systematic biases which cannot be encapsulated by an unbiased "observation noise" term of the type assumed by standard regularization theory and typical Bayesian formulations. Despite its importance, model error is difficult to encapsulate systematically and is often neglected. Here, model error is considered for a hydrogeologically important class of inverse problems that includes interpretation of hydraulic transients and contaminant source history inference: reconstruction of a time series that has been convolved against a transfer function (i.e., impulse response) that is only approximately known. Using established harmonic theory along with two results established here regarding triangular Toeplitz matrices, upper and lower error bounds are derived for the effect of systematic model error on time series recovery for both well-determined and over-determined inverse problems. A Monte Carlo study of a realistic hydraulic reconstruction problem is presented, and the lower error bound is seen informative about expected behavior. A possible diagnostic criterion for blind transfer function characterization is also uncovered.