Three least‐squares minimization approaches to depth, shape, and amplitude coefficient determination from gravity data

Abstract

Three different least‐squares approaches are developed to determine, successively, the depth, shape (shape factor), and amplitude coefficient related to the radius and density contrast of a buried structure from the residual gravity anomaly. By defining the anomaly value g(max) at the origin on the profile, the problem of depth determination is transformed into the problem of solving a nonlinear equation, [Formula: see text]. Formulas are derived for spheres and cylinders. Knowing the depth and applying the least‐squares method, the shape factor and the amplitude coefficient are determined using two simple linear equations. In this way, the depth, shape, and amplitude coefficient are determined individually from all observed gravity data. A procedure is developed for automated interpretation of gravity anomalies attributable to simple geometrical causative sources. The method is applied to synthetic data with and without random errors. In all the cases examined, the maximum error in depth, shape, and amplitude coefficient is 3%, 1.5%, and 7%, respectively. Finally, the method is tested on a field example from the United States, and the depth and shape obtained by the present method are compared with those obtained from drilling and seismic information and with those published in the literature.

Interpretation of residual gravity anomaly caused by simple shaped bodies using very fast simulated annealing global optimization

“…15a. This anomaly has been interpreted by several authors (Shaw and Agarwal, 1990;Abdelrahman et al, 2001a;Salem et al, , 2004Tlas et al, 2005;Asfahani and Tlas, 2012;Mehanee, 2014) assuming a spherical structure. The observed anomaly is obtained by digitizing at 610 m interval from above mentioned published literature (Fig.…”

Global nonlinear optimization for the interpretation of source parameters from total gradient of gravity and magnetic anomalies caused by thin dyke

“…The general expression of a TG of gravity and magnetic anomaly V(x) for thin dyke at any point on the surface (Figure 1) is given by the equations [after Abdelrahman et al 2001a, b, Srivastava et al 2014]:…”

“…2001], Fourier transform [Odegard and Berg 1965, Bhattacharyya 1965, Sharma and Geldart 1968, Euler deconvolution [Thompson 1982], Mellin transform [Mohan et al 1986], Hilbert transforms [Mohan et al 1982], least squares minimization approaches [Gupta 1983, Silva 1989, McGrath and Hood 1973, Lines and Treitel 1984, Abdelrahman 1990, Abdelrahman et al 1991, Abdelrahman and El-Araby 1993, Abdelrahman and Sharafeldin 1995a, Werner deconvolution [Hartmann et al 1971, Jain 1976, Kilty 1983; Walsh Transformation [Shaw and Agarwal 1990], Continual least-squares methods [Abdelrahman and Sharafeldin 1995b, Abdelrahman et al 2001a, b, Essa 2012, Euler deconvolution method [Salem and Ravat 2003], Fair function minimization procedure andAsfahani 2011a, Asfahani andTlas 2012], DEXP method [Fedi 2007], deconvolution technique [Tlas and Asfahani 2011b]; Regularised inversion [Mehanee 2014, Mehanee andEssa 2015]; Simplex algorithm [Tlas and Asfahani 2015], simulated annealing methods [Gokturkler and Balkaya 2012], Very fast simulated annealing Acharya 2016, Biswas andSharma 2016a, b;Biswas 2015, b, Sharma and Biswas 2013a, particle swarm optimization [Singh and Biswas 2016] and Differential Evolution ] have been used to solve similar kind of no...…”

Depth and shape solutions from residual gravity anomalies due to simple geometric structures using a statistical approach

“…The general gravity anomaly expressions produced by a sphere, an infinitely long horizontal cylinder, and a semi-infinite vertical cylinder can be given as (Abdelrahman et al, 2001a):…”

“…The first category is the convolution methods, in which simple models are convolved with the same moving average filters or numerical horizontal derivative filters as applied to the observed gravity data (e.g. Abdelrahman and El-Araby, 1996;Abdelrahman et al, 2001aAbdelrahman et al, & 2006, and correlation factors between successive least-squares residual gravity anomalies (Abdelrahman and El-Araby, 1993). These methods can be applied to both residual and observed data.…”