Premium
z −3/2 powerlaw decay of Laplacian fields induced by disorder: Consequences for the inverse problem
Author(s) -
Sornette Didier
Publication year - 1992
Publication title -
geophysical research letters
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.007
H-Index - 273
eISSN - 1944-8007
pISSN - 0094-8276
DOI - 10.1029/92gl01699
Subject(s) - physics , laplace operator , wavenumber , bounded function , inverse , perturbation (astronomy) , inverse problem , boundary (topology) , exponential decay , boundary value problem , magnetic field , mathematical analysis , mathematics , quantum mechanics , geometry
We consider a Laplacian field (▽ 2 V=0) in a semi‐infinite medium bounded by a frontier ∂B, with imposed boundary values or sources at ∂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 in this paper 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 powerlaw decreasing fluctuations of the Laplacian field V. Powerlaw 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 sources is traditionally neglected, thereby filtering out the very low wavenumbers. This singular effect of disorder must apply to more general partial differential operators as long as the Green function is long‐range.