DC telluric fields in three dimensions: A refined finite‐difference simulation using local integral forms

Geophysics ◽  
1983 ◽  
Vol 48 (3) ◽  
pp. 331-340 ◽  
Author(s):  
John F. Hermance
Keyword(s):  
Finite Difference ◽  
Thin Sheet ◽  
Circular Disk ◽  
Three Dimensions ◽  
Anomalous Field ◽  
Closed Surfaces ◽  
Integral Forms ◽  
Difference Form ◽  
Symmetric Structures ◽  
Improved Accuracy

A new finite‐difference form is developed for simulating the distortion of telluric fields by 3-D azimuthally symmetric structures. The technique involves a sequence of local integrations of the electric current density crossing closed surfaces surrounding each mesh node. The resulting expressions, which are accurate to second degree everywhere, correctly describe first‐order discontinuities in the electric field normal to electrical discontinuities in the interior of the model. Moreover, the new form (a nine‐point finite‐difference operator) accounts for cross‐derivative (e.g., [Formula: see text]) effects in the region about each node, which can lead to significantly improved accuracy near sharp, localized discontinuities where the anomalous field decays as [Formula: see text] or [Formula: see text] with distance. Numerical simulations are compared with analytical solutions for two simple models: (1) a circular disk‐shaped heterogeneity in a thin sheet; and (2) a sphere imbedded in a homogeneous, infinite medium. The comparison between the analytical and numerical results for both of these models indicates that an accuracy of better than a few percent is not exceptional.

Refined finite‐difference simulations using local integral forms: Application to telluric fields in two dimensions

Geophysics ◽  
1982 ◽  
Vol 47 (5) ◽  
pp. 825-831 ◽  
Author(s):  
John F. Hermance
Keyword(s):  
Finite Difference ◽  
Differential Forms ◽  
Integral Form ◽  
Closed Surface ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Difference Operators ◽  
Anomalous Field ◽  
Integral Forms ◽  
Point Operator

This paper describes a new finite‐difference form for simulating the behavior of telluric fields near electrical inhomogeneities. The technique involves a local integration of the electric current density crossing a closed surface surrounding a mesh node. To illustrate the concept, a two‐dimensional (2-D) model is considered, but it is readily possible to generalize to three dimensions. The resulting expressions, which are accurate to second degree everywhere, have the form of nine‐point finite‐difference operators, but they have a higher precision than those derived from the usual differential forms which result in five‐point operators. In particular, the new form accounts for cross‐derivative [Formula: see text] effects in the region about each node. Including this term can provide significant improvements in accuracy near sharp, localized discontinuities, where the anomalous field decays rapidly (as 1/r or [Formula: see text]) with distance. An analytical solution is compared to finite‐difference calculations using both the conventional five‐point differential form and the new nine‐point integral form developed here. The results suggest that, in some cases, one might expect at least a factor of three improvement when using the nine‐point operator instead of the five‐point operator. This is particularly true in the vicinity of localized structures where the curvilinear character of the distorted field is most pronounced and one would expect the cross‐derivative term to be large.

Synthetic reflection seismograms in three dimensions by a locked‐mode approximation

Geophysics ◽  
1989 ◽  
Vol 54 (3) ◽  
pp. 350-358 ◽  
Author(s):  
G. Nolet ◽  
R. Sleeman ◽  
V. Nijhof ◽  
B. L. N. Kennett
Keyword(s):  
Finite Difference ◽  
Born Approximation ◽  
Large Scale ◽  
Layered Medium ◽  
Three Dimensional ◽  
Simple Algorithm ◽  
Realistic Model ◽  
Three Dimensions ◽  
Acoustic Response ◽  
Many Sources

We present a simple algorithm for computing the acoustic response of a layered structure containing three‐dimensional (3-D) irregularities, using a locked‐mode approach and the Born approximation. The effects of anelasticity are incorporated by use of Rayleigh’s principle. The method is particularly attractive at somewhat larger offsets, but computations for near‐source offsets are stable as well, due to the introduction of anelastic damping. Calculations can be done on small minicomputers. The algorithm developed in this paper can be used to calculate the response of complicated models in three dimensions. It is more efficient than any other method whenever many sources are involved. The results are useful for modeling, as well as for generating test signals for data processing with realistic, model‐induced “noise.” Also, this approach provides an alternative to 2-D finite‐difference calculations that is efficient enough for application to large‐scale inverse problems. The method is illustrated by application to a simple 3-D structure in a layered medium.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

Numerical simulation of incompressible flows within simple boundaries: accuracy

1971 ◽  
Vol 49 (1) ◽  
pp. 75-112 ◽  
Author(s):  
Steven A. Orszag
Keyword(s):  
Numerical Simulation ◽  
Finite Difference ◽  
Spectral Methods ◽  
Degrees Of Freedom ◽  
Incompressible Flows ◽  
Finite Difference Methods ◽  
Fourier Method ◽  
Passive Scalar ◽  
Mathematical Basis ◽  
Improved Accuracy

Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here.

BOUNDARY LAYER CALCULATIONS ON CYLINDERS OF RANKINE-OVAL SECTIONS

1989 ◽  
Vol 13 (3) ◽  
pp. 65-68 ◽  
Author(s):  
M. ABDULHADI
Keyword(s):  
Boundary Layer ◽  
Finite Difference ◽  
Approximate Calculation ◽  
Separation Line ◽  
Boundary Layer Equations ◽  
Finite Difference Form ◽  
Difference Form ◽  
Long Cylinder

An approximate calculation of the boundary layer parameters around a long cylinder of Rankine-oval section is carried out. The calculations are undertaken as a step-by-step analysis using the finite-difference form of the integral formulations of the boundary layer equations. The separation line of the boundary layer for an oval of thickness = 0.2 is located in the rear portion of the oval in a plane making an angle of 137° with the direction of the flow.

Modeling near‐field GPR in three dimensions using the FDTD method

Geophysics ◽  
1997 ◽  
Vol 62 (4) ◽  
pp. 1114-1126 ◽  
Author(s):  
Roger L. Roberts ◽  
Jeffrey J. Daniels
Keyword(s):  
Electrical Properties ◽  
Finite Difference ◽  
Time Domain ◽  
Near Field ◽  
Characteristic Impedance ◽  
Fdtd Method ◽  
Three Dimensions ◽  
Field Pattern ◽  
Scattered Fields ◽  
Transmitting Antenna

Complexities associated with the theoretical solution of the near‐field interaction between the fields radiated from dipole antennas placed near a dielectric half‐space and electrical inhomogeneities within the dielectric can be overcome by using numerical techniques. The finite‐difference time‐domain (FDTD) technique implements finite‐difference approximations of Maxwell's equations in a discretized volume that permit accurate computation of the radiated field from a transmitting antenna, propagation through the air‐earth interface, scattering by subsurface targets and reception of the scattered fields by a receiving antenna. In this paper, we demonstrate the implementation of the FDTD technique for accurately modeling near‐field time‐domain ground‐penetrating radar (GPR). This is accomplished by incorporating many of the important GPR parameters directly into the FDTD model. These variables include: the shape of the GPR antenna, feed cables with a fixed characteristic impedance attached to the terminals of the antenna, the height of the antenna above the ground, the electrical properties of the ground, and the electrical properties and geometry of targets buried in the subsurface. FDTD data generated from a 3-D model are compared to experimental antenna impedance data, field pattern data, and measurements of scattering from buried pipes to verify the accuracy of the method.

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