Impact of model inconsistencies on reconstructions of past mantle flow obtained using the adjoint method

Abstract

SUMMARY The adjoint method is a powerful technique to compute sensitivities (Fréchet derivatives) with respect to model parameters, allowing one to solve inverse problems where analytical solutions are not available or the cost to determine many times the associated forward problem is prohibitive. In Geodynamics it has been applied to the restoration problem of mantle convection—that is, to reconstruct past mantle flow states with dynamic models by finding optimal flow histories relative to the current model state—so that poorly known mantle flow parameters can be tested against observations gleaned from the geological record. By enabling us to construct time dependent earth models the adjoint method has the potential to link observations from seismology, geology, mineral physics and palaeomagnetism in a dynamically consistent way, greatly enhancing our understanding of the solid Earth system. Synthetic experiments demonstrate for the ideal case of no model error and no data error that the adjoint method restores mantle flow over timescales on the order of a transit time (≈100 Myr). But in reality unavoidable limitations enter the inverse problem in the form of poorly known model parameters and uncertain state estimations, which may result in systematic errors of the reconstructed flow history. Here we use high-resolution, 3-D spherical mantle circulation models to perform a systematic study of synthetic adjoint inversions, where we insert on purpose a mismatch between the model used to generate synthetic data and the model used for carrying out the inversion. By considering a mismatch in rheology, final state and history of surface velocities we find that mismatched model parameters do not inhibit misfit reduction: the adjoint method still produces a flow history that fits the estimated final state. However, the recovered initial state can be a poor approximation of the true initial state, where reconstructed and true flow histories diverge exponentially back in time and where for the more divergent cases the reconstructed initial state includes physically implausible structures, especially in and near the thermal boundary layers. Consequently, a complete reduction of the cost function may not be desirable when the goal is a best fit to the initial condition. When the estimated final state is a noisy low-pass version of the true final state choosing an appropriate misfit function can reduce the generation of artefacts in the initial state. While none of the model mismatches considered in this study, taken singularly, results in a complete failure of the recovered flow history, additional work is needed to assess their combined effects.