Buried Sphere Research Articles

A perfectly matched layer for the absorption of electromagnetic waves using differential forms in three-dimensional domains

As a significant contribution, we present a novel formulation of the complex frequency-shifted perfectly matched layer (CFS-PML) for differential forms, introducing the innovative concept of a directional incidence matrix. This paper explores the application of differential forms in simulating wave propagation problems in three-dimensional domains. To confine the unbounded domain, we utilize a discretization technique based on the CFS-PML. Additionally, we provide an illustrative example that involves a system comprising a GPR (Ground Penetrating Radar) horn antenna, air, lossy soil, and a buried sphere.

Static and dynamic features of granular material failure due to upward pulling of a buried sphere by a slowly increasing force.

A spherical intruder embedded in a confined granular column is extracted by pulling it upward by an attached string. As the tension of the string gradually increases, a failure event occurs at a certain pulling force, leading to rapid upward acceleration of the intruder. The threshold force and the dynamics of the failure event are experimentally investigated for different filling heights and column diameters, using Ottawa sand and glass beads. For the Ottawa sand, we find that the failure force can be fit by a model describing the weight of the granular material in a cone with the vertex at the bottom of the intruder and a vertex angle of 72°. The agreement between the model and experiments is good for heights less than the column (tube) diameter, but measured values deviate from the model for larger heights. We also report on experiments with glass beads that reveal unexpected effects for relatively small ratios of tube diameters to grain size. The dynamics of the intruder during the failure event is studied using high-speed video analysis. The granular drag force monotonically decays during the pullout for sufficiently large tube diameters. In narrow columns, a monotonic decay of drag force after failure is observed for low heights, whereas a secondary peak can be seen in sufficiently deep and narrow columns, indicating the existence of different mechanisms of failure. The normalized drag force declines with intruder displacement closely for all tube diameters within small displacements.

A Multiple-Reflection Solution to Electromagnetic Scattering by a Buried Sphere

Based on the image theory and addition theorem of spherical vector wave functions, a multiple-reflection solution is described for the electromagnetic scattering by a buried sphere. An iterative process is introduced to obtain the scattered field by enforcing the boundary conditions at the plane interface between free space and an isotropic medium and the spherical surface of a dielectric sphere buried in the isotropic medium. Numerical results for the scattered fields are provided and compared with those obtained by the commercial software FEKO. A good agreement is observed, which verifies the presented formulation.

Electromagnetic scattering by a buried sphere in a lossy medium of an inhomogeneous plane wave at arbitrary incidence: spectral-domain method.

A rigorous theoretical treatment to analyze the electromagnetic scattering of an inhomogeneous elliptically polarized plane wave by a sphere buried in a lossy half-space is presented. To consider the losses in the media an inhomogeneous plane wave is considered. The incident and the scattered electric field components are expanded in series of vectorial spherical harmonics using the Legendre functions generalized via hypergeometrical and gamma functions, with unknown expansion coefficients. The spectral-domain method to represent the scattered electric field is used in order to compute the scattered-reflected and scattered-transmitted fields, considering the reflection and transmission of each elementary plane wave by the interface. Finally, the unknown coefficients of the scattered field are computed by imposing the boundary condition on the spherical surface. In order to validate the model, a homemade code has been implemented. Comparisons with the simulations performed with a commercial software and the results in the literature are presented.

Acoustic scattering from elastic target buried in water-sand sediment

Acoustic scattering from objects buried in water-sand sediment is the foundation of target detection and identification. It is also a research hotspot in areas of acoustic scattering while the domestic research on scattered field from buried targets is not deep. This paper deduces an approximate Green's function of acoustic scattering from targets buried in water-sand sediment, which describes clearly the whole physical process during the propagation of scattered waves. Next, on basis of geometric acoustics, the corresponding Helmholtz-Kirchhoff formula of integration is presented. Complicated integration of the full wave number spectral representation of the Green's function is avoided by employing approximate formula derived from the method of ray acoustics. As a result of neglecting the influence from lateral waves, the Helmholtz-Kirchhoff integral given applies to supercritical incidence case. The function of COMSOL Multiphysics software is expanded by writing this formula of integration into it. By means of finite-element method, numerical calculation models for two-dimensional axisymmetric targets are established on the software platform. The proposed model built in free field is verified through comparing numerical results obtained with the Rayleigh method which has been validated in previous research achievements of acoustics. The target strength of buried elastic solid sphere is calculated under different conditions in order to analyze the change regularity of buried scattered field. We provide a summary about the law of target strength of the elastic sphere varying with frequency, buried depth, and the attenuation of sand. Finally, we conduct acoustic scattering experiments in free space and shallow buried conditions and process the data with the method of isolation and identification of resonance to separate eliastic echoes from reverberation echo and specular echo. Results from the experiment of free field show that components of the scattered wave should include Rayleigh waves and whispering gallery waves. The processed data of objects buried inside layered fluid media indicate that characteristics of resonance spectra can be used to identify and detect the target effectively while echo signal is not available for identification of target. The proposed technique is verified through the comparison of data from total scattered field between experiment and theoretical prediction. This study has important guiding significance for detecting and identifying targets embedded within layered acoustic media in practical applications.

Flow Past a Sphere Buried in a Porous Media: Concentration Boundary Layer Thickness

In this work, consideration is given to the problem of dissolution of a buried solid sphere in the liquid flowing uniformly through the packed bed around it. The differential equations describing fluid flow and mass transfer by advection and diffusion in the interstices of the bed are presented and the method for obtaining their numerical solution is indicated.From the surface concentration fields, given by the numerical solution, the concentration boundary layer thickness as a function of the relevant parameters were undertaken. Mathematical expressions that relate the dependence with the Peclet number and d/d1 ratio of an immersed active sphere are proposed to describe the approximate size of the concentration boundary layer thickness.

Spectral domain method for the electromagnetic scattering by a buried sphere

A rigorous method to analyze the electromagnetic scattering of an elliptically polarized plane wave by a sphere buried in a dielectric half-space, is presented. The electric field components of the incident and the scattered monochromatic plane waves are expanded in series of vectorial spherical harmonics, with unknown expansion coefficients. The scattered-reflected and scattered-transmitted fields are computed by exploiting the plane-wave spectrum of the scattered field, considering the reflection and transmission of each elementary plane wave by the interface. The boundary-condition imposition leads to a linear system that returns the unknown coefficients of the scattered field. To achieve a numerical solution, a code has been implemented, and a truncation criterion for the involved series has been proposed. Comparisons with the literature and simulations performed with a commercial software are presented. A generalization of the method to the case of a short pulse scattered by a buried sphere is presented, taking into account the dispersive properties of the involved media.

Estimation of depth of buried conductive sphere from electromagnetic induction anomaly data using linearization process

In this paper, we have developed a least-squares minimization method to estimate the depth of a buried conducting sphere using electromagnetic induction (EMI) data. This approach is basically based on the solving a set of algebraic linear equations to estimate the depth of sphere embedded in an insulating media. In electromagnetic induction method, the transmitter coil produces the incident magnetic and electric fields that obey the Maxwell’s equations. In the receiver coil, the received response is created in two modes—eddy-current mode (Vec) derived from the perfectly conductor placed in the shallow depth and another mode called current-channeling response (Vcc) which depends on the conductivity of the medium. As expected, these responses differ depending on the direction of the incident field related to the receiver coil’s axis. In our case, the transmitter coil’s axis is parallel to the ground surface, and only the eddy-current response is measured in the receiver coil. The validity of this new method is demonstrated through studying and analyzing synthetic EMI anomalies, using simulated data generated from a known model with different random error components and a known statistical distribution.

A gradient‐ratio method for the semi‐automatic interpretation of gravity map data sets

ABSTRACTThe application of semi‐automatic interpretation techniques to potential field data can be of significant assistance to a geophysicist. This paper generalizes the magnetic vertical contact model tilt‐depth method to gravity data using a vertical cylinder and buried sphere models. The method computes the ratio of the vertical to the total horizontal derivative of data and then identifies circular contours within it. Given the radius of the contour and the contour value itself, the depth to the source can be determined. The method is applied both to synthetic and gravity data from South Africa. The Matlab source code can be obtained from the author upon request.

The semiautomatic interpretation of gravity profile data

Potential field semiautomatic interpretation techniques can significantly decrease the interpretation workload of a geophysicist and are widely used. This paper generalises the tilt-depth method to gravity data using the horizontal cylinder and the buried sphere models. The method is compared to the Euler deconvolution and is applied to gravity data from South Africa. Matlab source code can be downloaded from the IAMG server at www.iamg.org.

Buried Sphere Detection Using a Synthetic Aperture Sonar

This paper presents observations of a buried sphere detected with a low-frequency (5-35-kHz) synthetic aperture sonar (SAS). These detections were made with good signal-to-noise ratios (SNRs) at both above and below the critical grazing angle. The raw data for the below-critical-grazing angle detection shows that the acoustic penetration is skewed by the 29deg offset of the ripple field relative to the sonar path. This observed skew is in agreement with T-matrix calculations carried out to model penetration into the bottom via ripple diffraction. Additionally, measured SNRs over different frequency bands are compared to predictions made using both first- and second-order perturbation theory for ripple diffraction. Both the data and the models indicate a peak detection region around 25 kHz for the environmental conditions present during the test. These results confirm that ripple diffraction can play a critical role in long range (subcritical angle) buried target detection.

A New Algorithm for Depth Determination from Total Magnetic Anomalies due to Spheres

We have developed an automatic method to determine the depth of a buried sphere from numerical second horizontal derivative anomalies obtained from total field magnetic data. The method is based on using a relationship between the depth and a combination of observations at symmetric points with respect to the coordinate of the projection of the center of the source in the plane of the measurement points with a free parameter (graticule spacing). The problem of depth determination has been transformed into the problem of finding a solution of a nonlinear equation of f(z) = 0. Procedures are also formulated to determine the magnetic moment and the effective angle of magnetization. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal. In all cases, the depth solutions are in good agreement with the actual ones.

The solution of nonlinear inverse problems and the Levenberg-Marquardt method

Although the Levenberg-Marquardt damped least-squares method is an extremely powerful tool for the iterative solution of nonlinear problems, its theoretical basis has not been described adequately in the literature. This is unfortunate, because Levenberg and Marquardt approached the solution of nonlinear problems in different ways and presented results that go far beyond the simple equation that characterizes the method. The idea of damping the solution was introduced by Levenberg, who also showed that it is possible to do that while at the same time reducing the value of a function that must be minimized iteratively. This result is not obvious, although it is taken for granted. Moreover, Levenberg derived a solution more general than the one currently used. Marquardt started with the current equation and showed that it interpolates between the ordinary least-squares-method and the steepest-descent method. In this tutorial, the two papers are combined into a unified presentation, which will help the reader gain a better understanding of what happens when solving nonlinear problems. Because the damped least-squares and steepest-descent methods are intimately related, the latter is also discussed, in particular in its relation to the gradient. When the inversion parameters have the same dimensions (and units), the direction of steepest descent is equal to the direction of minus the gradient. In other cases, it is necessary to introduce a metric (i.e., a definition of distance) in the parameter space to establish a relation between the two directions. Although neither Levenberg nor Marquardt discussed these matters, their results imply the introduction of a metric. Some of the concepts presented here are illustrated with the inversion of synthetic gravity data corresponding to a buried sphere of unknown radius and depth. Finally, the work done by early researchers that rediscovered the damped least-squares method is put into a historical context.

Least-squares Minimization Approaches to Interpret Total Magnetic Anomalies Due to Spheres

We have developed three different least-squares approaches to determine successively: the depth, magnetic angle, and amplitude coefficient of a buried sphere from a total magnetic anomaly. By defining the anomaly value at the origin and the nearest zero-anomaly distance from the origin on the profile, the problem of depth determination is transformed into the problem of finding a solution of a nonlinear equation of the form f(z)=0. Knowing the depth and applying the least-squares method, the magnetic angle and amplitude coefficient are determined using two simple linear equations. In this way, the depth, magnetic angle, and amplitude coefficient are determined individually from all observed total magnetic data. The method is applied to synthetic examples with and without random errors and tested on a field example from Senegal, West Africa. In all cases, the depth solutions are in good agreement with the actual ones.

Effects of soil electromagnetic properties on metal detectors

This paper presents an analysis, based on existing work in geophysics and nondestructive testing, of the effects of soil electromagnetic properties on the functioning of metal detectors widely used to detect buried landmines. The host soil is modeled as a half-space having real and frequency-independent electrical conductivity but frequency-dependent complex magnetic susceptibility. The analysis technique has been applied to three examples of soil of practical importance, namely, nonconducting soil with frequency-independent susceptibility, nonconducting soil with frequency-dependent susceptibility, and nonmagnetic soil with constant conductivity. Simplifications are made to clearly explain a number of previous field and experimental observations, for example, the greater influence of magnetic properties than of electrical conductivity on the performance of metal detectors. Results also show that soil magnetic properties affect continuous wave and pulsed-induction detectors differently. The effect that electrical conductivity and magnetic susceptibility of the host soil have on the signal produced by a target is investigated by computing the response of a buried small metallic sphere. Computations show that in some cases, which could represent practical landmine detection scenarios, the signal from the soil can dominate that due to the target, making it hard to detect the target. Further, it is shown that magnetic soil can alter a target's spectral response, which implies that, contrary to present practice, object identification techniques should take into account the electromagnetic parameters of the host medium.

Comparison of measured buried target backscatter levels with high-order predictions: Rippled interface effects

Controlled sonar measurements with buried spheres and cylinders have demonstrated diffraction by bottom ripple can significantly improve target detection performance with sonar operated at grazing angles shallower than the critical grazing angle. Furthermore, buried target scattering models based on low-order perturbation theory to handle sound transmission through bottom roughness confirm that a significant enhancement in detection performance can be expected. But comparisons between measurements and model predictions have not always produced good agreement. As the bottom ripple height increases, discrepancies have been noted in comparisons with the predicted backscatter levels from a buried sphere, suggesting a failure of low-order perturbation theory to properly account for high-amplitude roughness effects. Recently, a recursive algorithm to generate arbitrary-order perturbation corrections for transmission and reflection through an idealized sinusoidal ripple was formulated and incorporated into our existing buried target scattering codes. The resulting higher-order predictions indicate previous discrepancies were not necessarily due to the neglect of high-order corrections. Calculations show model inputs such as bottom attenuation can influence backscatter levels significantly and measurement of these parameters may need to be performed more carefully. We demonstrate this through a presentation of data and model comparisons. [Work supported by ONR and SERDP.]

Buried target signature extraction from ground‐penetrating radar signal: measurements and simulations

ABSTRACTGround‐penetrating radar (GPR) proves to be a very valuable tool in the field of humanitarian de‐mining, especially for the detection of plastic land‐mines. Recently, a monostatic stepped‐frequen‐cy continuous‐wave (SFCW) GPR, together with a conceptual model of the radar‐antenna‐soil system, has been developed for the characterization of the electromagnetic parameters of soil, i.e. dielectric permittivity (), magnetic permeability () and electric conductivity (). This approach is extended here to the extraction of the GPR signal and to modelling the signatures of buried targets. The equivalence principle is used to decompose the GPR signal into its soil and target‐in‐soil components, as well as to model the radar‐soil‐target system. It permits the soil contribution to be subtracted from the total GPR signal to provide the signature of the buried target. This signature is compared to simulations. For a proof of the concept, the GPR return signal from a buried metal sphere has been simulated using the Method of Moments and it shows good agreement with its measured counterpart. We also have extracted clean frequency‐ and time‐domain signatures of a PMN‐2 plastic mine embedded in a multilayered medium, subject to various water contents. The method is also applied to a B‐scan above a buried conducting cylinder. Finally, a study of the main sources of errors in the extraction of the signature of a buried target shows that mistakes in antenna height measurement lead to errors more important than those due to misestimating the relative dielectric permittivity of the soil.

Formal solution to the electromagnetic scattering by buried dielectric and metallic spheres

The canonical problem of electromagnetic scattering by a buried dielectric or metallic sphere is studied in this paper. Two expansions are presented, of which one is the expansion of the vector plane waves into the vector spherical waves. The other is its converse, the expansion of the vector spherical waves into the vector plane waves. The use of two expansions is to facilitate matching the boundary conditions at the planar ground surface and the sphere surface. A general full‐wave analytic solution is given in which not only the high‐order multipoles but also the multiple reflections between the sphere and the ground surface are considered. Oblique plane wave and vertical magnetic dipole are considered as sources, though the illumination can be of arbitrary form with the use of Weyl identity.

Mass transfer between flowing fluid and sphere buried in packed bed of inerts

AbstractThe equations describing fluid flow and mass transfer around a sphere buried in a packed bed are presented, with due consideration given to the processes of transverse and longitudinal dispersion. Numerical solution of the equations was undertaken to obtain point values of the Sherwood number as a function of the Peclet and Schmidt numbers over a wide range of values of the relevant parameters. A correlation is proposed that describes accurately the dependence found numerically between the values of the Sherwood number and the values of Peclet and Schmidt numbers. Experiments on the dissolution of solid spheres (benzoic acid and 2‐naphthol) buried in packed beds of sand, through which water was forced, at temperatures in the range 293 K to 373 K, gave values of the Sherwood number that were used to test the theoretical results obtained. Excellent agreement was found between theory and experiment, including the large number of available data for naphthalene–air, and this helps establish the proposed correlation as “general” for mass transfer between a buried sphere and the fluid (liquid or gas) flowing around it. © 2004 American Institute of Chemical Engineers AIChE J, 50: 65–74, 2004

Model/data comparisons of subcritical grazing angle detection of targets buried under a rippled sand interface

Subcritical grazing angle target detection results for a sphere buried in Coastal System Station’s Facility 383 pond are compared to predictions from two different models. Both models use perturbation theory to calculate the transmission across the rough interface due to ripple scattering. One is a simple sonar equation model that uses a first-order expression [Eq. (16), Jackson et al., IEEE J. Ocean. Eng. 27, 346–361 (2002)] to calculate the penetration, taking into account the measured ripple profile and the target strength estimate for the buried sphere. The other model uses the penetrating field to calculate target backscattering via a T-matrix formalism adapted to account for the specified sinusoidal roughness. These models have been combined with a reverberation model to predict detection SNR. The reverberation model uses a power law fit to the sand/water interface roughness spectrum determined from measured roughness profiles. [Work supported by ONR.]