Think first, discretize later

A fundamental challenge in inverse problems is non-uniqueness: many models may fit a given data set exactly, or within observational uncertainty. A common remedy is the introduction of additional constraints encoding prior beliefs about the true model. Such explicit regularization mechanisms typically receive the most attention in inverse-problem research. However, it is well known that inversions may also be influenced by implicit sources of regularization. Discretization is perhaps the most prominent example, often introducing unintended and opaque prior assumptions.

 

Discretizations are commonly adopted for computational convenience and to avoid the theoretical complexities associated with models defined as functions. This practice, if done too early, can obscure fundamental questions concerning probabilities, regularity, and boundary behavior of the model. Although discretization may appear to eliminate these difficulties, it in fact makes choices for us that are often left unexamined.

 

In this contribution, we demonstrate that for linear(ised) problems in seismology, the undiscretized formulation can be treated rigorously using well-established theoretical tools. This perspective exposes hidden assumptions embedded in standard inversion workflows and allows prior choices to be made explicit and transparent. Although discretization is unavoidable in practice, we show that how and when it is introduced plays a crucial role, both for ensuring correct convergence and for computational efficiency.

 

Rather than attempting a fully rigorous solution of infinite-dimensional inverse problems—which can be expensive—we focus instead on probabilistic linear inference. Unlike classical inversion, linear inference targets specific properties of the model rather than a particular model realization. These quantities of interest are exactly representable in finite-dimensional spaces without discretizing the model itself. As a result, our framework delivers complete and mathematically consistent answers at reduced computational cost. We illustrate the proposed approach with synthetic inversion and inference examples in 1D.