z exp -3/2 powerlaw decay of Laplacian fields induced by disorder - Consequences for the inverse problem

Details z exp -3/2 powerlaw decay of Laplacian fields induced by disorder - Consequences for the inverse problem z exp -3/2 powerlaw decay of Laplacian fields induced by disorder - Consequences for the inverse problem

Dec 1992

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992georl..19.2377s&link_type=abstract

Geophysical Research Letters (ISSN 0094-8276), vol. 19, no. 24, p. 2377-2380.

Physics

Geophysics

Boundary Conditions, Geophysics, Laplace Equation, Dirac Equation, Gravitation, Green'S Functions, Magnetic Anomalies, Seismology

Scientific paper

We consider a Laplacian field in a semiinfinite medium bounded by a frontier delta-B, with imposed boundary values or sources at delta-B. When the boundary conditions are periodically distributed, it is well-known that the Laplacian field V decays exponentially due to screening of all multipoles. It is demonstrated that disorder is a singular perturbation in the sense that small random fluctuations around a periodic modulation, which are always present in nature, lead to long-range power-law decreasing fluctuations of the Laplacian field V. Power-law decay is due to the appearance of Fourier components of arbitrary large wavelengths in the boundary conditions. This result calls for a reexamination of various geophysical inverse problems (such as temperature, electrical, gravity, and magnetic anomalies), in which any possible small disorder on the source is traditionally neglected, thereby filtering out the very low wave-numbers. This singular effect of disorder must apply to more general partial differential operators as long as the Green function is long-range.

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