# Calibration of imperfect mathematical models by multiple sources of data with measurement bias

Model calibration involves using experimental or field data to estimate the unknown parameters of a mathematical model. This task is complicated by discrepancy between the model and reality, and by possible bias in the data. We consider model calibration in the presence of both model discrepancy and measurement bias using multiple sources of data. Model discrepancy is often estimated using a Gaussian stochastic process (GaSP), but it has been observed in many studies that the calibrated mathematical model can be far from the reality. Here we show that modeling the discrepancy function via a GaSP often leads to an inconsistent estimation of the calibration parameters even if one has an infinite number of repeated experiments and infinite number of observations in a fixed input domain in each experiment. We introduce the scaled Gaussian stochastic process (S-GaSP) to model the discrepancy function. Unlike the GaSP, the S-GaSP utilizes a non-increasing scaling function which assigns more probability mass on the smaller L_2 loss between the mathematical model and reality, preventing the calibrated mathematical model from deviating too much from reality. We apply our technique to the calibration of a geophysical model of Kīlauea Volcano, Hawai`i, using multiple radar satellite interferograms. We compare the use of models calibrated using multiple data sets simultaneously with results obtained using stacks (averages). We derive distributions for the maximum likelihood estimator and Bayesian inference, both implemented in the "RobustCalibration" package available on CRAN. Analysis of both simulated and real data confirm that our approach can identify the measurement bias and model discrepancy using multiple sources of data, and provide better estimates of model parameters.

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