Bridging the gap between SOLA and deterministic linear inferences in the context of seismic tomography

SUMMARY Seismic tomography is routinely used to image the Earth’s interior using seismic data. However, in practice, data limitations lead to discretized inversions or the use of regularizations, which complicates tomographic model interpretations. In contrast, Backus–Gilbert inference methods make it possible to infer properties of the true Earth, providing useful insights into the internal structure of our planet. Two related branches of inference methods have been developed–the Subtractive Optimally Localized Averages (SOLA) method and Deterministic Linear Inference (DLI) approaches—each with their own advantages and limitations. In this contribution, we show how the two branches can be combined to derive a new framework for inference, which we refer to as SOLA-DLI. SOLA-DLI retains the advantages of both branches: it enables us to interpret results through the target kernels, rather than the imperfect resolving kernels, while also using the resolving kernels to inform us on trade-offs between physical parameters. We therefore highlight the importance and benefits of a more careful consideration of the target kernels. This also allows us to build families of models, rather than just constraining properties, using these inference methods. We illustrate the advantages of SOLA-DLI using three case studies, assuming error-free data at present. In the first, we illustrate how properties such as different local averages and gradients can be obtained, including associated bounds on these properties and resolution information. Our second case study shows how resolution analysis and trade-offs between physical parameters can be analysed using SOLA-DLI, even when no data values or errors are available. Using our final case study, we demonstrate that SOLA-DLI can be utilized to obtain bounds on the coefficients of basis function expansions, which leads to discretized models with specific advantages compared to classical least-squares solutions. Future work will focus on including data errors in the same framework. This publication is accompanied by a SOLA-DLI software package that allows the interested reader to reproduce our results and to utilize the method for their own research.