To produce a unique and stable solution in potential-field interpretation, an inversion method must introduce particular constraints. These constraints will inevitably restrict the type of geological setting where the method may be applied.
We present a nonmathematical overview of most stabilizing constraints used in inversion methods. Our purpose is to demonstrate that the inversion results are valuable only if the mathematical stabilizing constraints are translated from the geological setting. We identify five basic types of constraints: (1) lower and upper bounds of parameter estimates; (2) proximity of a parameter estimate to a specified value; (3) proximity between pairs of parameter estimates; (4) concentration of the anomalous source about a geometrical element such as an axis; and (5) source compactness. In practice, if used in isolation, constraints (1), (2), (4), and (5) will not produce geologically meaningful results, regardless of the geological setting of the interpretation area. Constraint (3) may produce geologically meaningful results if the anomalous source has a spatially smooth attribute such as the physical property. We illustrate that constraints 1–4, if used in isolation, cannot delineate the geometry of a simulated sill intruded into a sedimentary basin.
The basic constraints may (and should) be combined in inversion to produce geologically meaningful results. We present two examples of effective constraint combination: (1) proximity to a specific value and mass concentration about an axis (used to delineate the thickness variation of a sill intruded in a sedimentary basin) and (2) inequality, proximity of a parameter estimate to a specified value, and proximity between pairs of parameter estimates (used to map a discontinuous basement relief).
Usually, the stabilizing constraints are too restrictive to hold at all points of a given geological environment. In this case, we use different constraints in different subareas. Each constraint is based on its compatibility with the actual geology of the subarea.