Beyond smooth inversion: the use of nullspace projection for the exploration of non-uniqueness in MT

Regularized inversion is the most commonly used interpretation method for magnetotellurics (MT) data. This leads to smooth models, which minimize the differences between the model response and the observed data. These models, however, do not represent the only possible solution. The inherent non-uniqueness of the problem and the errors in the data lead to the existence of equivalent models, that is, different models with very similar responses.Traditionally, equivalent models have been studied by using the singular value decomposition (SVD) of the Jacobian matrix (also known as the sensitivity matrix). The sensitivity matrix relates small changes in the model parameters with the associated changes in the response. By adding a linear combination of singular vectors of the nullspace of the sensitivity matrix to a model vector (in the parameter space), the resulting model has the same fit to the data as the original model. This approach has been used successfully to construct equivalent models in 1-D cases, represented by a small number of parameters. The application of this technique to 2-D or 3-D cases, however, is unpractical because of the large number of parameters involved. This makes the intuitive interpretation of the singular vectors in physical and geological terms impossible.In this paper, we present a hybrid algorithm, which seeks to unite the desirable features of the probabilistic and deterministic approaches to model appraisal; it generates a collection of models by a random modification of the segmented geoelectrical image's geometry. This represents a limited exploration of a subset of the parameter space characterized by similar geometry. We can then efficiently construct conservative models by projecting the changes introduced by the modification algorithm onto the nullspace of the original problem, thus maintaining the obtained fit to the data (in the linear approximation). In this way it is possible to test the structures obtained in 2-D MT inversions with respect to their resolution and inherent equivalencies. To illustrate the method, several numerical experiments with different structures and conditions are presented. In addition to these synthetic examples, a case study from SW Iberia is also shown.