Multiple solutions to geophysical inverse problems via tracking flights across a deeply incised model landscape

Summary Certain geophysical inverse problems may be reduced to systems of sparse low-order polynomial equations. In simple systems of this type, the monomials are products of nodal values of a subsurface physical property, such as electrical conductivity, interior to a discretized modeling domain, whereas the coefficients are derived from the given boundary data. The latter are presumed to be over-specified by means of surface measurements made across a range of frequencies. Suppose an acceptable conductivity model is found, by any standard means, that fits the data. Then, provided other distinct, non-singular solutions exist, it is possible to find another one of them by charting a trajectory across model space using a path-tracking numerical method. A second path can then be tracked and, continuing in this manner, equivalent solutions to the inverse problem can be generated one by one. If the data are noisy, the equivalent models generate precisely the same numerical misfit. Regularization side constraints can be added which reduces the number of equivalent models and smooths the trajectories. This idea of tracking paths has been developed in the field of robotics where realizable configurations of kinematic mechanisms are sought; a similar approach is explored here, on small test systems, to explore its eventual applicability to practical geophysical inverse problems. In principle the method scales up to arbitrary size since the inverse of the forward matrix and the Jacobian can be expressed in polynomial form. The novel implications of a successful scaled-up implementation of this methodology is that multiple distinct inverse solutions can be obtained, each of which might entail a different geological interpretation. A surprising feature of the algorithm is that the forward problem becomes ill-posed while the inverse problem becomes well-posed.