Image and Model Fusion: Unexpected Counterintuitive Be- havior of Traditional Statistical Techniques and Resulting Need for Expert Knowledge

In many real-life situations, we have different types of data. For example, in geosciences, we have seismic data, gravity data, magnetic data, etc. Ideally, we should jointly process all this data, but often, such a joint processing is not yet practically possible. In such situations, it is desirable to “fuse” models (images) corresponding to different types of data: e.g., to fuse an image corresponding to seismic data and an image corresponding to gravity data. At first glance, if we assume that all the approximation errors are independent and normally distributed, then we get a reasonably standard statistical problem which can be solved by the traditional statistical techniques such as the Maximum Likelihood method. Surprisingly, it turns out that for this seemingly simple and natural problem, the traditional Maximum Likelihood approach leads to non-physical results. To make the fusion results physically meaningful, it is therefore necessary to take into account expert knowledge. Model (and Image) Fusion: Formulation of a Problem Need to combine data from different sources In many areas of science and engineering, we have different sources of data. For example, in geophysics, there are many sources of data for Earth models: • first-arrival passive seismic data (from actual earthquakes); see, e.g., [8];