Parameter Sensitivity Analysis of Plate Motion using the Adjoint Method and Automatic Differentiation

The adjoint method for the Stokes equations provides a versatile and highly efficient approach to investigate the underlying physics of geodynamic processes. Reuber et al. (2018) demonstrated that adjoint sensitivities can be used to develop scaling laws for processes like folding and subduction dynamics. The gradients derived using the adjoint method can also directly be used in inversions in geodynamic applications. However, previous implementations of the adjoint method have typically been highly problem-dependent and often limited to viscous rheologies. Extending it to other nonlinear rheologies typically required substantial additional work, which is likely one of the reasons that the method has not yet been widely adopted in solid Earth geosciences.

To overcome this problem, we use automatic differentiation (AD) to compute the gradients needed to develop an adjoint solver for the Stokes equations. The gradients are computed using the Julia package Enzyme.jl. The adjoint solver is designed to be problem-agnostic, where the gradients are automatically computed for any user-defined rheology, from a simple linear viscous model to a complex visco-elasto-viscoplastic composite rheology. This functionality is added to the JustRelax.jl thermo-mechanical solver, where we use the same pseudo-transient solver strategy to solve both the forward and adjoint problems. This approach ensures that the adjoint solver remains consistent and fully generic.

The method is applied to analyse horizontal plate motion around subduction zones. For different material parameters, it is possible to calculate sensitivity kernels that show, for each location in the numerical domain, how much these parameters influence the horizontal plate motion (e.g. Reuber et al (2020)). The scaling of sensitivities for different parameters is discussed to enable a quantitative comparison. This approach is then used to identify the most influential factors affecting plate motion.

 

Reuber, G. S., Popov, A. A., & Kaus, B. J. (2018). Deriving scaling laws in geodynamics using adjoint gradients. Tectonophysics, 746, 352-363.

Reuber, G. S., Holbach, L., Popov, A. A., Hanke, M., & Kaus, B. J. (2020). Inferring rheology and geometry of subsurface structures by adjoint-based inversion of principal stress directions. Geophysical Journal International, 223(2), 851-861.