We present a direct (noniterative) inversion algorithm for the determination of the conductivity profile of a layered earth from the measurements of the apparent resistivity with the Schlumberger array. The necessary conditions for the existence of a one‐dimensional (1-D) continuous conductivity profile are determined, and the uniqueness of the solution is proved for complete and precise data. Examples are given in which the conductivity profile is determined from analytical data, incomplete and imprecise artificial data, and raw field data.
It is shown that the density and the two Lamé profiles of a layered elastic medium can be uniquely determined from the impulse responses due to obliquely incident SH plane waves at two precritical angles of incidence and a normally incident P plane wave. A direct (noniterative) inversion algorithm is developed which construct the three elastic profiles of the layered medium from the reflection data. The equation of motion for an obliquely incident SH plane wave is transformed to the Schrödinger equation whose potential is related to the density profile, shear modulus profile, and the angle of incidence. The Gelfand–Levitan theory is used to recover the potential of the Schrödinger equation from the impulse response. When the impulse response is available at two angles of incidence, the shear modulus and density profiles can be separately obtained as a function of depth. It is further shown that the incompressibility profile can be obtained from the impulse response due to a normally incident P plane wave and the density and shear modulus profiles.
An inverse acoustic scattering theory and algorithm is presented for the reconstruction of a two-dimensional inhomogeneous acoustic medium from surface measurements. The measurements of the surface pressure due to a harmonically oscillating surface point source at two arbitrary frequencies allows the separate reconstruction of the density and velocity of the subsurface. This is a first step towards solving the inverse problem of exploration geophyiscs.
IN A REFRACTIVE MEDIUMThe density and compressibility profiles of a layered fluid are obtained from the reflection coefficient due to plane waves at two precritical angles of incidence and all the frequencies. The inverse scattering problem for a layered fluid, at oblique incidence, is transformed to an equivalent inverse scattering problem for a layered refractive index profile, at normal incidence. The latter inverse scattering problem is transformed to an inverse scattering problem in quantum mechanics whose solution is obtained by the Gelfand-Levitan theory.An ideal fluid occupies the region -CC < z < 33. The halfspace z < 0 is homogeneous with known density pe and known compressibility ho, whereas the half-space z > 0 is inhomogeneous with unknown density p(z) and unknown compressibility A(z). A plane pressure wave is incident from the homogeneous region with 0 as the angle of incidence, which is assumed to be less than the critical angle. The reflected pressure wave has a reflection coefficient Srs(o, 0) that is a function of both the frequency w and the angle of incidence 8. Required is the density profile p(z), z > 0, and the compressibility profile A(z), z > 0, from the reflection coefficient SiZ(w, tl,), 1 = 1, 2 and --OO < 0 < so.
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