Constraining filtered Earth models using Backus Gilbert SOLA inferences

The quality of the solution to an inverse problem depends on several factors: data quality, data quantity, and modeling approximations. In seismology, observations are limited to a finite number of measurements, sometimes resulting in small datasets. This usually leads to an ill-posed inverse problem with many solutions. Classical inversion methods remove this non-uniqueness using regularizations and model discretization. These assumptions place strong constraints on the model space and may lead to systematically inaccurate models if they lack a good physical basis. Deterministic linear inferences, such as the Backus Gilbert (BG) SOLA method, can bypass the non-uniqueness issue by constraining properties of the unknown model, rather than the model itself. This approach also supports continuous model parameters with only mild regularizations. However, it requires a linear approximation of the data-model parameter relationship. The only regularization needed is a norm bound on the model space, which constrains the model space to a bounded subspace rather than a single point. Such prior information is less stringent than that needed for classic regularized inversions and reduces the risk of introducing subjective information into our problem. BG SOLA has mostly been used to estimate uniform local averages of Earth's properties using seismic data. In this contribution, we show how BG SOLA can be used to constrain other types of averages and gradients through different target kernels. We also demonstrate that certain properties can be better constrained than others, depending on the data used. An advantage of our methodology is that it provides uncertainty quantification and information about the resolution of the desired property. In addition, it can be used when the data simultaneously depend on multiple physical parameters. We apply this theory to a simple application where we aim to constrain averages and gradients of Earth’s 1D structure using normal mode data. Finally, we illustrate the advantages of our methodology by comparing our results with those obtained using regularized inversion methods.