Electric field of a horizontal antenna above a homogeneous half‐space: Implications for GPR

Geophysics ◽  
2000 ◽  
Vol 65 (3) ◽  
pp. 823-835 ◽  
Author(s):  
Christoph E. Baumann ◽  
Edson E. S. Sampaio
Keyword(s):  
Electric Field ◽  
Half Space ◽  
Homogeneous Medium ◽  
First Integral ◽  
Analytic Solutions ◽  
Numerical Results ◽  
Branch Points ◽  
Order Zero ◽  
Homogeneous Half Space ◽  
Special Cases

Inverse modeling and interpretation of subsurface structures depend on accurate knowledge of the undisturbed field. This is especially true in the analysis of radargrams, in which it is difficult to resolve the upper homogeneous medium from the less shallow scatterers. The available forward models based on plane‐wave and ray approximation are not accurate enough for this task. To improve resolution capabilities, we determine the undisturbed field using exact expressions for the electric field of a sine‐shaped ground‐penetrating radar (GPR) signal antenna above a homogeneous half‐space. In the frequency domain it consists of the sum of two improper integrals with complex integrands. Each integrand contains a kernel multiplied by a Bessel function of the first kind and of order zero or one. In the general case these integrals do not have a solution in closed form, and their integrands are poorly convergent. Therefore, to solve the integrals we must use a special formalism involving integrals around branch points. When we assume that both the transmitter and the receiver are on the boundary of the half‐space, there exist analytic solutions for the first integral without further restrictions and for the second integral for two special cases: free space and half‐space, neglecting displacement currents. We check our corresponding numerical results against these analytic solutions. In the time domain we represent the electric field as a function of transmitter‐receiver offset and time. For a purely dielectric half‐space the backtransformation of the first integral is analytical under the assumed simplification, allowing us to check the numerical results obtained with a fast Fourier transform (FFT) algorithm. These results allowed us to design radargrams for five different models of a homogeneous earth, and they are fundamental for interpretation and further research of GPR modeling.

Constant Temperature at the Surface of an Initially Uniform Temperature, Variable Conductivity Half Space

1971 ◽  
Vol 93 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Leonard Y. Cooper
Keyword(s):  
Temperature Distribution ◽  
Half Space ◽  
Constant Temperature ◽  
Analytic Solutions ◽  
Numerical Results ◽  
Transient Temperature ◽  
Uniform Temperature ◽  
Transient Temperature Distribution ◽  
Temperature Variable ◽  
Variable Conductivity

The transient temperature distribution resulting from a constant and uniform temperature being imposed on the surface of an initially uniform temperature, variable conductivity half space is studied. Various solution expansion ideas are discussed. These are utilized in the solution of an example problem, and the resulting approximate analytic solutions representations are compared to exact numerical results. One of these approximations is found to be superior to the others, and, in fact, it is shown to yield useful results over a range of variables where the nonlinearities of the problem are significant.

On two-dimensional waves in an elastic half-space

1960 ◽  
Vol 56 (3) ◽  
pp. 269-285 ◽  
Author(s):  
J. W. Craggs
Keyword(s):  
Half Space ◽  
Elastic Waves ◽  
Analytic Solutions ◽  
Elastic Half Space ◽  
Numerical Results ◽  
Two Dimensional ◽  
Surface Traction ◽  
Concentrated Load ◽  
Dynamic Similarity

ABSTRACTTwo-dimensional elastic waves in a half-space 0 ≤ r < ∞, 0 ≤ θ ≤ π are examined under the assumption of dynamic similarity, so that the stresses depend only on r/t, θ. Analytic solutions are given for constant surface traction on θ = 0, 0 < r/t < V, where V is constant, the rest of the surface being unloaded, and for a concentrated load at r = 0.Numerical results are quoted for the particular case V → ∞, corresponding to a load on half the bounding plane.

The Elasto-Electric Field for a Rigid Conical Punch on a Transversely Isotropic Piezoelectric Half-Space

1999 ◽  
Vol 66 (3) ◽  
pp. 764-771 ◽  
Author(s):  
W.-Q. Chen ◽  
T. Shioya ◽  
H.-J. Ding
Keyword(s):  
Exact Solution ◽  
Electric Field ◽  
Potential Theory ◽  
Half Space ◽  
Piezoelectric Effect ◽  
Transversely Isotropic ◽  
Numerical Results ◽  
Theory Method ◽  
Elementary Functions ◽  
Potential Theory Method

This paper exactly analyzes the problem of a rigid conical punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to include the piezoelectric effect. By using the previous results of pure elasticity, exact solution is derived. It is found that all the elastoelectric variables are expressed in terms of elementary functions. Numerical results are finally performed.

Comments on the electromagnetic “smoke ring” concept

Geophysics ◽  
1998 ◽  
Vol 63 (6) ◽  
pp. 1908-1913 ◽  
Author(s):  
James E. Reid ◽  
James C. Macnae
Keyword(s):  
Electric Field ◽  
Half Space ◽  
Time Domain ◽  
Current System ◽  
One Dimensional ◽  
Homogeneous Half Space ◽  
Conductivity Structure ◽  
The Time Domain ◽  
Smoke Ring ◽  
Number Of Authors

The electromagnetic (EM) fields in a one‐dimensional (1-D) earth due to a dipole or loop transmitter have been studied by a number of authors, including Lewis and Lee (1978), Pridmore (1978), Nabighian (1979), and Hoversten and Morrison (1982). Nabighian (1979) aptly described the time‐domain‐induced current system in a homogeneous half‐space as resembling a “smoke ring” blown by the transmitter, which moves outwards and downwards and diminishes in amplitude with increasing time after the transmitter is turned off. In a homogeneous half‐space, the physical electric field maximum moves outward from the transmitter loop edge at an angle of approximately 30° with the surface. Hoversten and Morrison (1982) show how the direction of propagation of the time‐domain electric field maximum is affected by conductivity structure. In the case of a highly conductive overburden over a resistive basement, the electric field maximum travels essentially horizontally away from the transmitter, and is effectively trapped in the upper layer.

scholarly journals Bonding widths of Deposited Polymer Strands in Additive Manufacturing

Materials ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 871
Author(s):  
Cheng Luo ◽  
Manjarik Mrinal ◽  
Xiang Wang ◽  
Ye Hong
Keyword(s):  
Polymer Melt ◽  
Acrylonitrile Butadiene Styrene ◽  
High Viscosity ◽  
Numerical Results ◽  
Fused Filament Fabrication ◽  
Cross Sectional ◽  
Nozzle Height ◽  
Special Cases ◽  
Cross Sectional Profile ◽  
Sectional Profile

In this study, we explore the deformation of a polymer extrudate upon the deposition on a build platform, to determine the bonding widths between stacked strands in fused-filament fabrication. The considered polymer melt has an extremely high viscosity, which dominates in its deformation. Mainly considering the viscous effect, we derive analytical expressions of the flat width, compressed depth, bonding width and cross-sectional profile of the filament in four special cases, which have different combinations of extrusion speed, print speed and nozzle height. We further validate the derived relations, using our experimental results on acrylonitrile butadiene styrene (ABS), as well as existing experimental and numerical results on ABS and polylactic acid (PLA). Compared with existing theoretical and numerical results, our derived analytic relations are simple, which need less calculations. They can be used to quickly predict the geometries of the deposited strands, including the bonding widths.

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.

Inversion of slingram electromagnetic induction data using a Born approximation

Geophysics ◽  
2013 ◽  
Vol 78 (4) ◽  
pp. E201-E212 ◽  
Author(s):  
Jochen Kamm ◽  
Michael Becken ◽  
Laust B. Pedersen
Keyword(s):  
Half Space ◽  
Born Approximation ◽  
High Efficiency ◽  
Three Dimensions ◽  
Background Model ◽  
Desktop Computer ◽  
Near Surface ◽  
Homogeneous Half Space ◽  
Integral Equation Formulation ◽  
Forward Operator

We present an efficient approximate inversion scheme for near-surface loop-loop EM induction data (slingram) that can be applied to obtain 2D or 3D models on a normal desktop computer. Our approach is derived from a volume integral equation formulation with an arbitrarily conductive homogeneous half-space as a background model. The measurements are not required to fulfill the low induction number condition (low frequency and conductivity). The high efficiency of the method is achieved by invoking the Born approximation around a half-space background. The Born approximation renders the forward operator linear. The choice of a homogeneous half-space yields closed form expressions for the required electromagnetic normal fields. It also yields a translationally invariant forward operator, i.e., a highly redundant Jacobian. In connection with the application of a matrix-free conjugate gradient method, this allows for very low memory requirements during the inversion, even in three dimensions. As a consequence of the Born approximation, strong conductive deviations from the background model are underestimated. Highly resistive anomalies are in principle overestimated, but at the same time difficult to resolve with induction methods. In the case of extreme contrasts, our forward model may fail in simultaneously explaining all the data collected. We applied the method to EM34 data from a profile that has been extensively studied with other electromagnetic methods and compare the results. Then, we invert three conductivity maps from the same area in a 3D inversion.

A study of electric field components in shallow water and water half-space models in seabed logging

2016 ◽  
Author(s):  
Amir Rostami ◽  
Hassan Soleimani ◽  
Noorhana Yahya ◽  
Tadiwa Elisha Nyamasvisva ◽  
Muhammad Rauf
Keyword(s):  
Electric Field ◽  
Shallow Water ◽  
Half Space
Sign in / Sign up

Export Citation Format

Share Document