For more than half a century, geoscientists have sought new ways to solve inverse problems, which occur when observations only indirectly constrain some property of interest. In the case of geophysics this usually means using surface observations to quantify properties of the Earth hidden from us within its interior, or processes which occurred in the past. Both the target is not directly accessible, and measurements which constrain it are not completely under our control. This is a challenging situation, where the search for new efficient and practical methods of solution to inversion problems has received regular attention.
A convenient way to view inverse problems as a way of asking questions of data. A common class of question might be to ask `Which set of model parameters, within a chosent class, fits a subset of the data best?’ How one measures `best fit’ constitutes a fundamental component of the question being asked. Another example might be `Which probability distribution best describes a `state of knowledge’ about a set of representative parameters?’. As the question changes, naturally so does the solution, even if the data does not. This talk will examine this approach to inversion and explore some new forms of question that can be asked of data. In particular, cases will be examined where the same answer can arise from different style of questions, some of which are much easier to solve than others. A focus will be on optimal model generation in nonlinear cases using data questions based on mathematical ideas from the field of Optimal Transport.