The seismic reflection inverse problem
Inverse Problems, 25 (12):
123008+ (December 2009)DOI: 10.1088/0266-5611/25/12/123008
Abstract
The seismic reflection method seeks to extract maps of the Earth's
sedimentary crust from transient near-surface recording of echoes,
stimulated by explosions or other controlled sound sources positioned
near the surface. Reasonably accurate models of seismic energy propagation
take the form of hyperbolic systems of partial differential equations,
in which the coefficients represent the spatial distribution of various
mechanical characteristics of rock (density, stiffness, etc). Thus
the fundamental problem of reflection seismology is an inverse problem
in partial differential equations: to find the coefficients (or at
least some of their properties) of a linear hyperbolic system, given
the values of a family of solutions in some part of their domains.
The exploration geophysics community has developed various methods
for estimating the Earth's structure from seismic data and is also
well aware of the inverse point of view. This article reviews mathematical
developments in this subject over the last 25 years, to show how
the mathematics has both illuminated innovations of practitioners
and led to new directions in practice. Two themes naturally emerge:
the importance of single scattering dominance and compensation for
spectral incompleteness by spatial redundancy.