On the use of adjoints in the inversion of observed quasi-static deformation

Abstract

Summary An adjoint-based conjugate gradient algorithm provides an efficient means for imaging sources of deformation within the Earth, such as volume stresses associated with fluid flow in aquifers and reservoirs. For time intervals over which the overburden deforms elastically, one can calculate the gradient elements for a single model update using just two numerical simulations. The first is a forward run that is used to compute the residuals associated with the given iteration. The second simulation is to evaluate the application of the adjoint operator to the residuals. In this adjoint calculation, the residual displacements are applied as sources at the measurement locations, driving the deformation in the simulation. The volume stress on the source grid blocks, in response to the residual displacements, provide the gradient components. We apply this technique to satellite-based interferometric synthetic aperture radar (InSAR) line-of-sight displacements that were observed over an oil reservoir in California’s Central Valley. We find that the adjoint-based gradient estimates, requiring 18 CPU seconds, agree with conventional numerical calculations that take over 3700 CPU seconds to compute. Conjugate gradient algorithms utilizing the conventional approach and adjoint-based gradient computations give roughly the same reductions in misfit and similar final estimates of reservoir volume change.

Toward Retrieving Distributed Aquifer Hydraulic Parameters From Distributed Strain Sensing

“…Because of the combined effect of parameter crosstalk and data noise, the current solution may be solely a near‐optimal solution. A choice of other inversion methods (such as the adjoint‐based method; Vasco & Mali, 2020) and global optimization methods (Comola et al., 2016; Jones, et al., 1998), or a better regularization technique (Aster et al., 2018; Menke, 2018; Ren & Kalscheuer, 2020) may improve the solution.…”

Adjoint‐Based Inversion of Geodetic Data for Sources of Deformation and Strain

“…Thus, one does not solve Equation 1 directly for , as this is an under‐constrained problem. Typically, one resorts to a Green's function formulation in order to write the observed displacements directly in terms of the source parameters , effectively solving the forward problem (1) for (Menke, 2018; Vasco & Mali, 2021). We avoid the use of Green's functions, due to the difficulty of computing them for complicated three‐dimensional variations in elastic properties, and solve the inverse problem for directly in terms of a numerical simulation, that is, a numerical solution of the forward problem (1).…”

“…For example, if the sources are g E N grid blocks encompassing many nodes of the underlying finite-difference mesh, then E B is a map from the grid block to all of the associated nodes. Thus, the matrix (Menke, 2018;Vasco & Mali, 2021). We avoid the use of Green's functions, due to the difficulty of computing them for complicated 10.1029/2021JB021735…”

“…The use of adjoints for the much simpler source identification problem, which solves for an equivalent source, had been restricted to the dynamic case associated with the propagation of elastic waves (Kaderli et al., 2018). Recently, a Green's function formulation of an adjoint‐based inverse problem for the source of quasi‐static deformation was published (Vasco & Mali, 2021). In that work, the application of the Green's function to the distribution of sources was approximated by a numerical simulator.…”

“…Recently, a Green's function formulation of an adjoint-based inverse problem for the source of quasi-static deformation was published (Vasco & Mali, 2021). In that work, the application of the Green's function to the distribution of sources was approximated by a numerical simulator.…”