TY - JOUR

T1 - Explicit expressions and numerical calculations for the Frechet and second derivatives in 2.5D Helmholtz equation inversion

AU - Bing, Zhou

AU - Greenhalgh, S. A.

PY - 1999/7

Y1 - 1999/7

N2 - In order to perform resistivity imaging, seismic waveform tomography or sensitivity analysis of geophysical data, the Frechet 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 Frechet 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.

AB - In order to perform resistivity imaging, seismic waveform tomography or sensitivity analysis of geophysical data, the Frechet 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 Frechet 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.

UR - http://www.scopus.com/inward/record.url?scp=0032873096&partnerID=8YFLogxK

U2 - 10.1046/j.1365-2478.1999.00139.x

DO - 10.1046/j.1365-2478.1999.00139.x

M3 - Article

AN - SCOPUS:0032873096

SN - 0016-8025

VL - 47

SP - 443

EP - 468

JO - Geophysical Prospecting

JF - Geophysical Prospecting

IS - 4

ER -