Optimally Uncertainty Estimation for Linear Inversion

In experimental sciences we often need to solve inverse problems. That is, we want to obtain information about the internal structure of a physical system from indirect noisy observations. 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. Here, we exploit the trade-off between data prediction and model or data structure to determine 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. By intersecting set of prior model parameter constraints with the set of data fitting models, we obtain a set of models that fit the data and are in agreement with prior constraints. This prior information can also be used to set bounds on the bias. We illustrate our methods with a synthetic example of vertical seismic profiling (VSP).