Fusing continuous and discrete data, on the example of merging seismic and gravity models in geophysics

In many application areas, we need to fuse continuous and discrete models of the same phenomena. For example, in geophysics, we have two main models for describing how the sound velocity changes with location and depth: a discrete gravity-based model, in which we have several layers with abrupt transition between layers, and a seismic model, in which the velocity continuously changes with the change in location and depth - and a transition is represented by a steeper change. Due to inevitable uncertainty, in two fused models, the same actual transition is placed at slightly different depths. If we simply fuse these models, the fused model will inherit both nearby transitions and therefore, will, misleadingly, correspond to two nearby transitions instead of one. It is therefore necessary, before fusing, to first get a fused (more accurate) location of the transition surface. In this paper, we show how to find such a location.