Characterising the parameter space of a highly nonlinear inverse problem

In inverse problems, often there is no available analytical expression relating the physical quantities of interest and the available data. In these cases, one resorts to using a numerical model with a finite number of parameters, resulting in a discrete problem. Also, many discrete inverse problems involve a highly nonlinear mapping between the model parameters and the simulation of the data by the model. Algorithms exist for estimating the model parameters in nonlinear discrete inverse problems. However, one needs to investigate how these estimated models relate to the true structure of the studied system (i.e., the truth model). This is known as model appraisal and it is greatly affected by three sources of uncertainty: misleading search, non-uniqueness and errors. In this work, we aim at analysing the impact of the first two sources of uncertainty in model appraisal (misleading search and non-uniqueness) by characterising the parameter space of a highly nonlinear geophysical inverse problem. Our approach is to use an error-free synthetic data problem so that the truth model is known in advance and therefore the results can be compared against a known truth. The characterisation requires an exceptionally high quality sampling of the parameter space. A new Real-parameter Genetic Algorithm called Genetic Sampler (GS) is used for this purpose. The GS successfully characterises the parameter space by locating multiple unconnected optimal regions, including that of the truth model, in several instances of the inverse problem. It is argued that, given the characteristics of the parameter space, many search methods could misidentify a local optimum as the global optimum. Such a misleading search will, in this case, have terrible consequences, as all the local optima had no predictive value. Another important result is related to the issue of non-uniqueness. It seems that, for this model and in the absence of any error, there is a single global optimum, which constitutes a unique solution to this nonlinear inverse problem. Lastly, we investigated the effect of including more data in the inversion process. This resulted in the parameter space having fewer and less prominent locally optimal regions, which makes the search for the global optimum easier.