Quasi-reversibility method for data assimilation in models of mantle dynamics

Abstract

S U M M A R YRapid progress in imaging deep Earth structures using seismic tomography and in studies of physical and chemical properties of mantle rocks facilitates research in assimilation of data related to mantle dynamics. In this paper, we present a new numerical approach for data assimilation, which allows for incorporating observations (at present) and unknown initial conditions (in the past) for mantle temperature and flow into a 3-D dynamic model in order to determine the initial conditions. The dynamic model is described by the backward heat, motion and continuity equations. The use of the quasi-reversibility (QRV) method implies the introduction into the backward heat equation of the additional term involving the product of a small regularization parameter and a higher order temperature derivative. The data assimilation in this case is based on a search of the best fit between the forecast model state and the observations by minimizing the regularization parameter. We apply the QRV data assimilation method to restore the evolution of (i) mantle plumes (a synthetic case study) and (ii) the lithospheric slab imaged by teleseismic body-wave tomography in the southeastern Carpathians. For both models the present temperature and mantle flow are assimilated to the geological past, and the prominent features of mantle structures are recovered. We then model the evolution of the mantle structures forward in time starting from the restored state to the present state and estimate the accuracy of the model predictions. The results of the QRV data assimilation are compared to that obtained by the variational (VAR) and backward advection data assimilation. Although the accuracy of the QRV data assimilation is lower than that of the VAR data assimilation, the QRV method does not require any additional smoothing of the input data or filtering of temperature noise as the VAR method does. Based on the results and the comparison of the methods, we consider the QRV method to be a highly promising approach to assimilation of data related to mantle dynamics.

Importance of initial buoyancy field on evolution of mantle thermal structure: Implications of surface boundary conditions

“…We underline that such predictions of the forward (i.e., future) evolution of the temperature anomalies beneath the Pacific plate cannot, of course, provide a direct geological reconstruction of their past evolution. Such a mapping requires the use of time-reversed convection modelling that employs data assimilation methods (e.g., Bunge et al, 2003;Ismail-Zadeh et al, 2007;Gli sovi c and Forte, 2014). We will instead focus on identifying a mechanism for generating intraplate volcanism that can be used as a theoretical basis for interpreting the broad spatial and temporal patterns of volcanism in the southern and eastern Pacific.…”

Linking mantle upwelling with the lithosphere descent and the Japan Sea evolution: a hypothesis

“…Using the quasi-reversibility method for data assimilation42 the set of the discrete equations governing the backward mantle convection flow are solved together with the initial and boundary conditions (see Supplementary Material) to restore the mantle evolution in the region of the Japan islands. In backward sense, the high-temperature patchy anomaly beneath the back-arc Japan Sea basin splits into two prominent anomalies showing two small-scale upwellings beneath the southwestern and northern part of the Japan Sea (Fig.…”

“…The equation of energy conservation (or the heat balance equation) with a small additional term responsible for heat relaxation is integrated backward in time using the quasi-reversibility method42. The additional term represents mathematically a higher order differential term and regularizes the ill-posed backward heat balance equation.…”

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

“…With an assimilated surface it is suggested that solutions will evolve towards a single solution, which corroborates the theorem of Serrin (1959) which states that two incompressible viscous bodies are equivalent given their boundary conditions are the same. Ismail-Zadeh et al (2007) meanwhile determine that the adjoint solution is limited by the characteristic thermal diffusion time, which for large features (> 100 km) suggests a limit over 300 Myr, and less than 5 Myr for fine features (< 10 km). This falls in line with what we have seen here, with the large scale features being predicted for all time scales (given sufficient iterations) and finer features not captured (such as in the upper mantle).…”

“…A third technique of recovering past mantle structures that is of note is the quasi-reversibility (QRV) method (Ismail-Zadeh et al 2007). This method is similar to the adjoint in that backwards heat, motion and continuity equations are solved, but here the backwards heat equation contains an additional term that involves a small regularisation parameter together with a higher order temperature derivative.…”