Data and Model Uncertainty Estimation for Linear Inversion

Inverse theory concerns the problem of making inferences about physical systems from indirect noisy measurements. Information about the errors in the observations is essential to solve any inverse problem, otherwise it is impossible to say when a feature “fits the data.” In practice, however, one seldom has a direct estimate of the data errors. We exploit the trade-off between data prediction and model or data structure to determine both model-independent and model-based estimates of the noise characteristics from a single realization of the data. Noise estimates are then used to characterize the set of reasonable models that fit the data, for example, by intersecting prior model parameter constraints with the set of data fitting models. This prior information can also be used to set bounds on the bias. We illustrate our methods with synthetic examples of vertical seismic profiling and cross-well tomography.