Adjoint method and its application in mantle convection

Abstract

The adjoint method for data assimilation is a gradient-based inversion technique which is especially useful for inverting nonlinear dynamic systems. Recently, its application for simulating geophysical problems has been proved promising. In this paper, we try to introduce the adjoint method in a systematic way, from the theoretical basis to its practical implementation in numerical models of mantle convection, with several examples to help the understanding. The adjoint operator of a temporally evolving system can be derived based on the perturbation theory, where a mismatch in the model output against observation is attributed to an error in the input, with their relation approximated as a first-order derivative (gradient) of the least-squared mismatch with respect to the input. For a nonlinear system, iterative processing is inevitable; the efficiency and convergence rate depend on the amount of a-priori information about the input (e.g. the initial condition). Seismic tomography, which describes the present-day mantle structures, has seen steady progress on both regional and global scales, and allows inverting mantle convection to the past. By either assimilation or comparison with geological observations, especially the dynamic topography inferred from stratigraphy, adjoint calculation of mantle convection can constrain unknown mantle dynamic properties while recovering the initial condition. Enormous new insights about the earth’s dynamic mechanisms, therefore, can emerge. A specific example is the inversion of the Farallon flat subduction under North America during the Late Cretaceous time.