Characterizing the impact of model error in hydrologic time series recovery inverse problems. (January 2018)

• Record Type: Journal Article
• Authors: Hansen, Scott K.; He, Jiachuan; Vesselinov, Velimir V.
• Is Part Of: Advances in water resources. Volume 111(2018)
• Page Start: 372
• Page End: 380
• Publication Date: 2018-01
• Subjects: Hydrology -- Periodicals; Hydrodynamics -- Periodicals; Hydraulic engineering -- Periodicals; 551.48
• DOI: 10.1016/j.advwatres.2017.09.030
• Languages: English
• ISSNs: 0309-1708

Abstract

Highlights: A hydrologically important class of inverse problems is analyzed mathematically. A new way of symbolically representing model error via Fourier series is presented. Error bounds derived using only most significant terms in model series expansion. Extra data collection locations do not reduce expected model error when inverting. A case study showing problem severity, transfer function ID heuristic is shown. Abstract: Hydrologic models are commonly over-smoothed relative to reality, owing to computational limitations and to the difficulty of obtaining accurate high-resolution information. 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 an 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 model error on time series recovery for both well-determined and over-determined inverse problems. It is seen that use of additional measurement locations does not improve expected performance in the face of model error. 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.