Prismatic Sources Research Articles

2D fast Fourier transform analytical solutions in all space for all gravity and magnetic components

ABSTRACTForward modelling of potential field data is an important part of optimization algorithms used to invert large datasets such as those involving rugged terrain or borehole data. Two‐dimensional fast Fourier transform modelling with a prism or a dipole is one of the most efficient methods compared to the forward modelling in the space domain. However, the exact solution of a prismatic source is limited to the case of a half‐space with the computation of data on a horizontal datum above the topography. Starting from the three‐dimensional Fourier forward modelling analytical formulation for a prism, an integration according to the wavenumber w is accomplished which allowed to find a two‐dimensional Fourier exact analytical formulation outside, at the interfaces of, and inside a prism for all potential field components. This new formulation requires the calculation of only four integrals. The gravity and magnetic fields are computed with this two‐dimensional fast Fourier transform formulation in the entire domain and compared with the analytical space domain and the three‐dimensional fast Fourier transform formulations. From the three‐dimensional calculated field, each component can be interpolated with the tri‐linear interpolation method along a borehole or on a drape surface simulating an airborne survey. Based on experiments demonstrated in this work, the two‐dimensional formulation in the Fourier domain gave accurate results with greater speed of execution in comparison to modelling in the space domain. The forward modelling method is tested on real gravity data from the north of Alberta (Canada).

A computational review of the line integral analytical formulation of the polyhedral gravity signal

ABSTRACTThe generally shaped polyhedron is a widely used model in gravity field modelling and interpretation. Its induced gravity signal – gravitational potential and its derivatives up to second order – has been studied extensively in the geophysical literature. The class of solutions with special interest is the one which leads to closed analytical expressions, as these offer an exact representation of the gravity signal of finite three‐dimensional distributions while being at the same time linked to a flexible means of geometric modelling. Of the several mathematical algorithms available, the line integral approach involves no approximations and is, however, connected with certain singularities, occurring for specific relative positions of the computation point with respect to the polyhedral source. The present contribution analyses the algorithmic details of the polyhedral line integral approach focusing on its geometric and computational aspects. Following the definitions of the individual coordinate systems and the two‐step application of the Gauss divergence theorem for each face and polygonal boundary of the source, the geometric insight of the algorithm is presented, which permits a deeper understanding of the significance of the involved numerical singular terms. An overview of the line integral analytical approach for the polyhedral gravity signal is presented with emphasis on its geometric and computational aspects. Matlab code and results for the two considered case studies, a prismatic source and asteroid Eros, are provided as electronic supplement.

Source Edge Detection of Potential Field Data Using Wavelet Decomposition

Edge detection of the sources of potential field anomaly is an important step in the interpretation of subsurface source geometries. The conventional methods based on calculation of horizontal or vertical derivatives identify the edges or center of sources by minima, maxima, or zero values in the transformed data. We present a wavelet source edge detector method (WSED) using wavelet multiresolution analysis to identify potential field sources boundaries. The two-dimensional wavelet decomposition is an effective method to understand the frequency components of the signal in different directions. We use a 2D-discrete wavelet transform using Haar wavelets in resolving lateral edges for source edge detection. We test the method on synthetic magnetic anomalies due to sources of complex geometries generated using prismatic sources. We investigated the robustness of the method on the magnetic data of the Bishop model and found the results useful in resolving the edges. We applied the method to gravity data of the north Delhi fold belt, India, to identify boundaries of different geological formations. Our results indicate distinct properties of the source edges in the wavelet domain, which is for the first time reported for the interpretation of the potential field anomalies.

Single-hole imaging using borehole gravity gradiometry

We study the feasibility of imaging density variations away from boreholes using gravity gradient measurements in a single borehole. The objective is to develop numerical interpretation tools and to understand the utility and limitations of borehole gravity gradiometry in anticipation of the development of such instruments for the exploration and production of minerals, oil, and gas. We develop an analytic solution for directly locating discrete off-hole mass anomalies and apply a generalized inverse approach to image anomalous density distribution. In the analytic approach, we solve for the distance and direction angles to a point source using the multiple components of the gravity gradient tensor from a set of observations in a single borehole; and then apply this approach to the scenarios of prismatic sources. In the generalized inverse approach, we invert multiple gradient tensor components to construct 3D density contrast distributions around the borehole. As few as two independent gravity gradient components in a single borehole can be used to locate and image isolated anomalous density bodies.

Improvements and Corrections to AT123D Code

Although based on exact analytical solutions, semi-analytical solute transport models can have significant numerical error in applications with high frequency oscillatory source terms and when parameter value combinations cause series solution approximations to converge slowly. Methods for correcting these numerical errors are presented and implemented in the AT123D code, which employs Green's functions to represent point, linear, and rectangular prismatic source zones. In order to increase its computational accuracy, a Romberg numerical integration scheme was added to AT123D with prespecified error criteria, variable time stepping, and partitioning of the integral to handle rapidly changing source terms. More rapidly converging series solution approximations for the Green's functions were also incorporated to improve both accuracy and computational efficiency for finite-depth aquifers. AT123D also has been modified to eliminate redundant calculations at points where approximate steady-state conditions have been reached to improve computational efficiency during numerical integration. These modifications help to decrease computer run times that can be excessive for three-dimensional problems with large numbers of computational points, small time steps, and/or long simulation time periods. Errors in the original AT123D code also were corrected in this modified version, AT123D-AT, in order to accurately simulate finite-duration (pulse) source releases.

Detection of potential fields source boundaries by enhanced horizontal derivative method

A high‐resolution method to image the horizontal boundaries of gravity and magnetic sources is presented (the enhanced horizontal derivative (EHD) method). The EHD is formed by taking the horizontal derivative of a sum of vertical derivatives of increasing order. The location of EHD maxima is used to outline the source boundaries. While for gravity anomalies the method can be applied immediately, magnetic anomalies should be previously reduced to the pole. We found that working on reduced‐to‐the‐pole magnetic anomalies leads to better results than those obtainable by working on magnetic anomalies in dipolar form, even when the magnetization direction parameters are not well estimated. This is confirmed also for other popular methods used to estimate the horizontal location of potential fields source boundaries.The EHD method is highly flexible, and different conditions of signal‐to‐noise ratios and depths‐to‐source can be treated by an appropriate selection of the terms of the summation. A strategy to perform high‐order vertical derivatives is also suggested. This involves both frequency‐ and space‐domain transformations and gives more stable results than the usual Fourier method.The high resolution of the EHD method is demonstrated on a number of synthetic gravity and magnetic fields due to isolated as well as to interfering deep‐seated prismatic sources. The resolving power of this method was tested also by comparing the results with those obtained by another high‐resolution method based on the analytic signal. The success of the EHD method in the definition of the source boundary is due to the fact that it conveys efficiently all the different boundary information contained in any single term of the sum.Application to a magnetic data set of a volcanic area in southern Italy helped to define the probable boundaries of a calderic collapse, marked by a number of magmatic intrusions. Previous interpretations of gravity and magnetic fields suggested a subcircular shape for this caldera, the boundaries of which are imaged with better detail using the EHD method.

Characteristics of Constrained Cavitating Bluff Body Wakes

The wake characteristics of constrained, cavitating two‐dimensional bluff bodies are studied over a large range of blockage. Prismatic sources with fixed separating points are chosen to reduce or eliminate Reynolds number effects. The effect of blockage on the geometrical characteristics of the wake such as the length and width of cavity are obtained. The frequency of cavity pressure pulsations, the pressure histograms, and the mean static pressure distributions along the wake axis are also determined at various cavitating conditions. Efforts are made to obtain the proper velocity and length scales to form the Strouhal numbers to absorb blockage effects. At very large blockages, the generation of small eddies in the shear layers accompanied by a reduction or elimination of Karman vortex street is noticed. Limited studies related to the wake characteristics of a circular source at a moderate blockage of 33% are also reported.

Anisotropy of the shock wave field from nonspherical sources

The authors investigate the spatial configuration of shock waves formed in air by sources of various shapes. Charts present triple-point trajectories relative to ideal reflecting surface for four prismatic sources; motion of Mach and incident shock waves for four prismatic sources; and overpressure on front of Mach and incident shock waves for four prismatic sources and for a cubical source. Diagrams illustrate the shapes of sources; pattern of interaction of shock waves from a cubicle source; shock wave front configurations produced in the near zone by four prismatic sources of identical mass; and shock wave configuration from a conical source in half space. The authors conclude that sources of non-spherical form with different angles between adjacent surfaces generate shock wave configurations in air which consist of waves originating from the source surfaces, Mach waves, and reflected waves. The parameters of these waves change with time by various laws.

Velocity Exponent for Erosion and Noise Due to Cavitation

The aim of the present study was to evaluate the effects of flow velocity on erosion and noise due to cavitation. To this end, tests were conducted in a two-dimensional water tunnel in which soft aluminum test specimens were mounted in the wake of a cavitating prismatic source. The prismatic source shape eliminated the Reynolds number effects on the flow characteristics. An excellent correlation appeared to be present between the erosion and noise due to cavitation in the range of velocities tested. At peak erosion conditions, the velocity exponents for erosion and intensity of noise were found to have an approximate value of 5.45. All the present tests were limited to a single cavitating source size for which the blockage ratio was fixed.

INVERSION OF AIRBORNE RADIOMETRIC DATA

The spectral representation describing the gamma ray intensity due to radioactive prismatic sources such as approximate areal distributions of outcropping rocks shows that such intensity fields result from the convolution of factors depending upon the height of the plane of observation, the geometry of the causative bodies, instrumental constants, and the concentrations of radioelements emitting gamma rays. Inverse filters, designed according to the Wiener least‐squares criteria, can deconvolve factors from the observed intensity field to provide direct mappings of the concentrations of radioactive elements.