Navigating challenges: Revitalising the Direct Solution Method for 3D heterogeneous media in global seismology

Born in 1994 (Geller and Ohminato 1994) and grounded in the optimally accurate operator theory (Geller and Takeuchi 1994), the Direct Solution Method (DSM) has long been a reliable force in seismic wavefield computation for spherically symmetric planets, particularly excelling in high-frequency efficiency. However, as the nowaday seismology seeks greater realism in 3D heterogeneous media, DSM has been facing substantial hurdles in its extension.

This contribution first revisits the historical successes and failures of the DSM, spotlighting its efficiency in computing seismic wavefields in spherically symmetric planets. However, the shift towards more complex models exposes DSM's limitations due to its trial functions, prompting various attempts to adapt it to 3D heterogeneous media. Analysing these endeavours, we uncover pitfalls and missteps that have hindered the extension of DSM to 3D heterogeneous media. Out critical assessment identifies common challenges, offering insights into essential modifications needed to overcome them. This contribution acts as a guide, emphasising what to avoid and illuminating potential avenues to enhance the DSM's applicability in global seismology within 3D heterogeneous contexts.

We try to elucidate the shortcomings of previous efforts, providing valuable lessons for future endeavours. By steering researchers away from ineffective approaches, we aim to catalyse innovation, paving the way for the continued relevance and effectiveness of DSM in global seismology. This comprehensive exploration serves as a roadmap, directing attention towards necessary adjustments and innovations to sustain DSM's utility amidst the dynamic landscape of seismic wavefield computation.