Profiling the robustness, efficiency and limits of the forward-adjoint method for 3D mantle convection modelling

Knowledge of Earth’s past mantle structure is inherently unknown. This lack of knowledge presents problems in many areas of Earth science, including in mantle circulation modelling (MCM). As a mathematical model of mantle convection, MCM’s require boundary and initial conditions. While boundary conditions are readily available from sources such as plate reconstructions for the upper surface, and as free slip at the core-mantle boundary (CMB), the initial condition is not known. MCM’s have historically ‘created’ an initial condition using long ‘spin up’ processes using the oldest available plate reconstruction period available. Whilst these do yield good results when models are run to present day, it is difficult to infer with confidence results from early in a model’s history. Techniques to overcome this problem are now being studied in geodynamics, such as by assimilating the known internal structure (e.g. from seismic tomography) of Earth at present day backwards in time. One such method is to use an iterative process known as the forward-adjoint method, which, while an efficient means of solving this inverse problem still strains all but the most cutting edge computational systems. In this study we endeavour to profile the effectiveness of this method using synthetic test cases as our known data source. We conclude that savings in terms of computational expense for forwardadjoint models can be achieved by streamlining the time-stepping of the calculation, as well as determining the most efficient method of updating initial conditions in the iterative scheme. Furthermore we observe that in the models presented, there exists an upper limit on the time interval over which solutions will practically converge, although this limit is likely to be linked to Rayleigh number.