Automatic differentiation in geophysical inverse problems

Abstract

S U M M A R YAutomatic differentiation (AD) is the technique whereby output variables of a computer code evaluating any complicated function (e.g. the solution to a differential equation) can be differentiated with respect to the input variables. Often AD tools take the form of source to source translators and produce computer code without the need for deriving and hand coding of explicit mathematical formulae by the user. The power of AD lies in the fact that it combines the generality of finite difference techniques and the accuracy and efficiency of analytical derivatives, while at the same time eliminating 'human' coding errors. It also provides the possibility of accurate, efficient derivative calculation from complex 'forward' codes where no analytical derivatives are possible and finite difference techniques are too cumbersome. AD is already having a major impact in areas such as optimization, meteorology and oceanography. Similarly it has considerable potential for use in non-linear inverse problems in geophysics where linearization is desirable, or for sensitivity analysis of large numerical simulation codes, for example, wave propagation and geodynamic modelling. At present, however, AD tools appear to be little used in the geosciences. Here we report on experiments using a state of the art AD tool to perform source to source code translation in a range of geoscience problems. These include calculating derivatives for Gibbs free energy minimization, seismic receiver function inversion, and seismic ray tracing. Issues of accuracy and efficiency are discussed.

A Study of the Sensitivity of Response Spectral Amplitudes on Seismological Parameters Using Algorithmic Differentiation

“…This extremely powerful approach is extensively used in sensitivity analysis of complex models, such as those used in meteorology (Marotzke and Giering, 1999) or in the field of aerodynamics (Gauger et al, 2008). Sambridge et al (2007) was the very first application of AD in geophysics. To the best of our knowledge, AD has not been explored yet in the context of seismic-hazard analysis.…”

Reply to comment by D. Gibert and P. Sailhac on “Self‐potential signals associated with preferential groundwater flow pathways in sinkholes”

“…An example of these methods is the MAP method (maximum a posteriori estimate [see Tarantola and Valette , 1982]). The use of automatic differentiation [see Rath et al , 2006; Sambridge et al , 2007] could be a very efficient approach to determine the Jacobian of the forward model used in these optimization algorithms. The use of these approaches is the topic of future works where, in particular, we wish to perform a joint inversion of self‐potential and thermal data to obtain the three‐dimensional distribution of the seepage velocity in the ground.…”

Calibration‐constrained Monte Carlo analysis of highly parameterized models using subspace techniques

“…AD is a broad methodology for obtaining forward or reverse (adjoint) derivatives. AD exploits the fact that computer programs execute a sequence of arithmetic operations and/or functions, regardless of the complexity of the computer model, and that application of the chain rule of derivative calculus to these operations can be used to automatically compute derivatives [ Sambridge et al , 2007]. It should be noted, however, that the use of perturbation sensitivities does enable the SSMC method to be used with any model or sequence of models, without any specialized programming requirements.…”