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Explicit expressions and numerical calculations for the Fréchet and second derivatives in 2.5D Helmholtz equation inversion
Author(s) -
Bing Zhou
Publication year - 1999
Publication title -
geophysical prospecting
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.735
H-Index - 79
eISSN - 1365-2478
pISSN - 0016-8025
DOI - 10.1046/j.1365-2478.1999.00139.x
Subject(s) - helmholtz free energy , helmholtz equation , inversion (geology) , mathematical analysis , parametrization (atmospheric modeling) , inverse problem , synthetic data , electrical resistivity tomography , seismic inversion , electrical resistivity and conductivity , finite element method , acoustic wave equation , mathematics , geology , geometry , physics , acoustic wave , algorithm , acoustics , optics , thermodynamics , paleontology , structural basin , radiative transfer , quantum mechanics , azimuth , boundary value problem
In order to perform resistivity imaging, seismic waveform tomography or sensitivity analysis of geophysical data, the Fréchet derivatives, and even the second derivatives of the data with respect to the model parameters, may be required. We develop a practical method to compute the relevant derivatives for 2.5D resistivity and 2.5D frequency‐domain acoustic velocity inversion. Both geophysical inversions entail the solution of a 2.5D Helmholtz equation. First, using differential calculus and the Green's functions of the 2.5D Helmholtz equation, we strictly formulate the explicit expressions for the Fréchet and second derivatives, then apply the finite‐element method to approximate the Green's functions of an arbitrary medium. Finally, we calculate the derivatives using the expressions and the numerical solutions of the Green's functions. Two model parametrization approaches, constant‐point and constant‐block, are suggested and the computational efficiencies are compared. Numerical examples of the derivatives for various electrode arrays in cross‐hole resistivity imaging and for cross‐hole seismic surveying are demonstrated. Two synthetic experiments of resistivity and acoustic velocity imaging are used to illustrate the method.