Finite cube elements method for calculating gravity anomaly and structural index of solid and fractal bodies with defined boundaries

Abstract

S U M M A R YIn this paper, we present the finite cube elements method (FCEM); a novel numerical tool for calculating the gravity anomaly g and structural index SI of solid models with defined boundaries and variable density distributions, tilted or in normal position (e.g. blocks, faulted blocks, cylinders, spheres, hemispheres, triaxial ellipsoids). Extending the calculation to fractal objects, such as Menger sponges of different orders and bodies defined by polyhedrons, demonstrates the robustness of FCEM. In addition, approximating the cube element by a sphere of equal volume makes the calculation of gravitation and related derivatives much simpler. In gravity modelling of a sphere, cubes with edges of 100 m and 200 m achieve a good compromise between running time and overall error.Displaying the distribution of SI of the studied models on contour maps and profiles will have a strong impact on the forward and inverse modelling of potential field data, especially for Euler deconvolution.For Menger sponges, plots of gravity elements g and its derivatives show similar patterns independent of fractal order. Moreover, both the pattern and magnitude of SI are independent of fractal order, allowing the use of SI as a new invariant measure for fractal objects. However, SI pattern and magnitude strongly depend on the depth to the buried bodies as do other elementsIn this study, we also present a new type of plot; the structural index against distance variation diagrams from which we extract the three critical SI (CSI) values, one per axis. The inversion of gravity anomaly data at CSI values gives the optimal mean location of the buried body.

Towards a new potential field theory of fractal objects

“…The error in calculating gravity and magnetic fields, and their respective derivatives due to the variation of cube element edge length is reported for bodies of known closed formulas by Mostafa (2008 and. It was found that modeling objects with cube elements of edge length up to 100 m gives a reasonable error less than 3%.…”

“…The Finite Cube Elements Method (FCEM) developed by Mostafa (2008 and) is a numerical tool used for modeling potential field anomalies of solids or FOs. In addition, FCEM allows studying the distribution of the estimated SI of the modeled bodies on maps and profiles using Equation (1).…”

Modelling magnetic anomalies of solid and fractal bodies with defined boundaries using the finite cube elements method

“…1(a) shows a solid triaxial ellipsoid, (b) a third‐order MS and (c) a second‐order ELST. The construction of the triangle at the first three fractal orders is shown in Table 1, while that of the sponge is given in Mostafa (2008). The order of fractal object refers to the number of removals of successively smaller parts of the object.…”

“…Instead of the lengthy and limited analytical solution, and for considering realistic spatial distributions of magnetic properties under different field conditions, we propose a numerical calculation of magnetic anomalies of bodies with defined boundaries. This is done the same way gravitational anomalies are calculated using the finite cube elements method (FCEM) developed by Mostafa (2008). In addition to solid bodies, FCEM allows calculating magnetic anomalies and estimating SI of fractal objects which are widely used in many domains such as geology, geophysics and communication.…”

Mapping local singularities using magnetic data to investigate the volcanic rocks of the Qikou depression, Dagang oilfield, eastern China

“…Additionally, Mostafa (2008Mostafa ( , 2009) studied the characteristics of gravity and magnetic anomalies produced by self-similar fractal forward models. Cheng (2001) considered that geo-anomalies including geochemical and geophysical data exhibit properties of spatial or statistical self-similarity.…”