An algorithm for computation of resistivity soundings over inhomogeneous layers

Geophysics ◽  
1985 ◽  
Vol 50 (7) ◽  
pp. 1166-1172 ◽  
Author(s):  
S. P. Dasgupta
Keyword(s):  
Half Space ◽  
Kernel Function ◽  
Transition Layer ◽  
Recurrence Relations ◽  
The Other ◽  
Dc Resistivity ◽  
Resistivity Sounding ◽  
Transition Layers ◽  
Homogeneous Form ◽  
Layered Half Space

Calculation of dc resistivity sounding curves for a multilayer earth with transition layers has been treated by several authors since Mallick and Roy (1968). However, derivation of the kernel function for such problems has remained difficult for more than three‐layer models for want of a proper algorithm. The problem was first solved by Pekeris (1942) in the case of uniformly resistive layers. Other forms of recurrence relations for the kernel function of a half‐space containing such homogeneous layers were forwarded by Flathe (1955), Kunetz (1966), and Koefoed (1968). Patella (1977) considered the kernel function for a half‐space which contained a series of alternate layers, one having a linearly varying conductivity with depth while the other was homogeneously conductive. Koefoed (1979a) considered the case of a half‐space containing a transition layer situated anywhere among a series of homogeneous layers and possessing a resistivity that changed linearly with depth. In this article a very general form of algorithm is developed for generating the kernel function for a layered half‐space containing any number of transition layers having an arbitrary resistivity distribution [Formula: see text] in such ith layer. This new general form is very similar to the homogeneous form derived by Pekeris (1942).

On: “The interpretation of direct current resistivity measurements” by D. W. Oldenburg (GEOPHYSICS, April 1978, p. 610–625)

Geophysics ◽  
1982 ◽  
Vol 47 (2) ◽  
pp. 264-265 ◽  
Author(s):  
D. Guptasarma
Keyword(s):  
Characteristic Function ◽  
Half Space ◽  
Direct Current ◽  
Layered Medium ◽  
Inverse Theory ◽  
Dc Resistivity ◽  
Vertical Variation ◽  
Resistivity Sounding ◽  
Resistivity Measurements

Oldenburg made an excellent example of the application of linearized inverse theory to invert dc resistivity sounding data to fit a continuous vertical variation of resistivity. In the Introduction he mentioned that the Frechet kernels for resistivity are the same as the depth investigation characteristic function (DIC) used by Roy and Apparao (1971). In the second part of the paper, he showed that it is so for a uniformly conducting half space. He mentioned that the electrostatic analog which was used (by Roy and Apparao) becomes quite complex when a layered medium is introduced, and that the extension to a continuous ρ(z) would be a difficult task (p. 623).

Multipolar elastic fields in a layered half space

1968 ◽  
Vol 58 (5) ◽  
pp. 1519-1572 ◽  
Author(s):  
Ari Ben-Menahem ◽  
Sarva Jit Singh
Keyword(s):  
Half Space ◽  
Elastic Half Space ◽  
Strike Slip ◽  
The Other ◽  
Source Depth ◽  
Elastic Fields ◽  
New Type ◽  
Boussinesq Solution ◽  
Horizontal Thrust ◽  
Layered Half Space

Abstract Hansen's expansion is used to derive integral expressions for the displacement field due to a localized buried source of the mth order in a layered half space. The dipolar case (m ≦ 2) is worked out in detail for arbitrary source-depth in the layer and in the substratum. A new type of representation of the source is used which gives the final results in a concise form. Explicit expressions for the displacements at the free surface are obtained for a center of explosion, a vertical strike-slip fault and a vertical dip-slip fault. The results for a horizontal thrust are found to be the same as for a vertical dip-slip fault. The relations between the Galerkin vector and the biharmonic eigenvectors are clarified. It is shown that the Galerkin-Boussinesq solution for the elastic half space cannot be extended to structures of higher complexity, except for a few simple sources. On the other hand, the Hansen Solution is valid for a wide class of sources and structures. Both dynamic and static regimes are considered.

Parameterization of Eddy Fluxes near Oceanic Boundaries

2008 ◽  
Vol 21 (12) ◽  
pp. 2770-2789 ◽  
Author(s):  
Raffaele Ferrari ◽  
James C. McWilliams ◽  
Vittorio M. Canuto ◽  
Mikhail Dubovikov
Keyword(s):  
Boundary Layer ◽  
Transition Layer ◽  
General Circulation ◽  
Mesoscale Eddies ◽  
Turbulent Boundary Layers ◽  
Mesoscale Variability ◽  
Transition Layers ◽  
Boundary Layer Turbulence ◽  
Dynamical Coupling ◽  
Eddy Fluxes

Abstract In the stably stratified interior of the ocean, mesoscale eddies transport materials by quasi-adiabatic isopycnal stirring. Resolving or parameterizing these effects is important for modeling the oceanic general circulation and climate. Near the bottom and near the surface, however, microscale boundary layer turbulence overcomes the adiabatic, isopycnal constraints for the mesoscale transport. In this paper a formalism is presented for representing this transition from adiabatic, isopycnally oriented mesoscale fluxes in the interior to the diabatic, along-boundary mesoscale fluxes near the boundaries. A simple parameterization form is proposed that illustrates its consequences in an idealized flow. The transition is not confined to the turbulent boundary layers, but extends into the partially diabatic transition layers on their interiorward edge. A transition layer occurs because of the mesoscale variability in the boundary layer and the associated mesoscale–microscale dynamical coupling.

Seismic Waves in a Laterally Inhomogeneous Layered Medium, Part I: Theory

1997 ◽  
Vol 64 (1) ◽  
pp. 50-58 ◽  
Author(s):  
Ruichong Zhang ◽  
Liyang Zhang ◽  
Masanobu Shinozuka
Keyword(s):  
Wave Field ◽  
Half Space ◽  
Seismic Waves ◽  
Scattered Wave ◽  
Inhomogeneous Layer ◽  
Dislocation Source ◽  
Seismic Dislocation ◽  
Body Forces ◽  
The Mean ◽  
Layered Half Space

Seismic waves in a layered half-space with lateral inhomogeneities, generated by a buried seismic dislocation source, are investigated in these two consecutive papers. In the first paper, the problem is formulated and a corresponding approach to solve the problem is provided. Specifically, the elastic parameters in the laterally inhomogeneous layer, such as P and S wave speeds and density, are separated by the mean and the deviation parts. The mean part is constant while the deviation part, which is much smaller compared to the mean part, is a function of lateral coordinates. Using the first-order perturbation approach, it is shown that the total wave field may be obtained as a superposition of the mean wave field and the scattered wave field. The mean wave field is obtainable as a response solution for a perfectly layered half-space (without lateral inhomogeneities) subjected to a buried seismic dislocation source. The scattered wave field is obtained as a response solution for the same layered half-space as used in the mean wave field, but is subjected to the equivalent fictitious distributed body forces that mathematically replace the lateral inhomogeneities. These fictitious body forces have the same effects as the existence of lateral inhomogeneities and can be evaluated as a function of the inhomogeneity parameters and the mean wave fleld. The explicit expressions for the responses in both the mean and the scattered wave fields are derived with the aid of the integral transform approach and wave propagation analysis.

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