Analysis of Hydrologic Time Series Reconstruction Uncertainty Due to Inverse Model Inadequacy Using the Laguerre Expansion Method

In subsurface inverse problems, it is common to assume a perfect process model that, in the absence of measurement error, would exactly produce the data. Model bias has been recognized as important, but is resistant to systematic analysis. To address this problem, we introduce a new technique based on expansion in series of Laguerre functions, which maps inverse problems with a convolution structure into matrix inverse problems with triangular Toeplitz structure. Exploiting this form, we develop analytic lower bounds on the reconstruction error. We also use this as the foundation for a Monte Carlo study in which a reconstruction of a time series of hydraulic head values is attempted using remote measurements transmitted through an imperfectly characterized domain. Qualitative properties of the reconstruction error are related to qualitative properties of the oversimplified inverse model, and the expected square reconstruction error due to model inadequacy is compared with that due to measurement error.