N. Sundararajan

Interpretation of some two-dimensional magnetic bodies using Hilbert transforms.

Procedures are formulated using the Hilbert transform for interpreting vertical magnetic anomalies of (1) the sheets (finite and infinite depth extent), (2) the dike, and (3) the horizontal circular cylinder. The applicability of the method is tested on theoretical models. The method is also applied on the well‐known Kursk field anomaly of a sheet (infinite‐depth extent) and the field anomaly of a dike of Karimnagar, Andhra Pradesh, India.

3D gravity inversion of basement relief — A depth-dependent density approach

We present a 3D gravity inversion technique, based on the Marquardt algorithm, to analyze gravity anomalies attributable to basement interfaces above which the density contrast varies continuously with depth. The salient feature of this inversion is that the initial depth of the basement is not a required input. The proposed inversion simultaneously estimates the depth of the basement interface and the regional gravity background. Applicability and efficacy of the inversion is demonstrated with a synthetic model of a density interface. We analyze the synthetic gravity anomalies (1) solely because of the structure, (2) in the presence of a regional gravity background, and (3) in the presence of both random noise and regional gravity background. The inverted structure remains more or less the same, regardless of whether the regional background is simulated with a second-degree polynomial or a bilinear equation. The depth of the structure and estimated regional background deviate only modestly from the assumed ones in the presence of random noise and regional background. The analyses of two sets of real field data, one over the Chintalpudi subbasin, India, and another over the Pannonian basin, eastern Austria, yield geologically plausible models with the estimated depths that compare well with drilling data.

An analytical method to interpret self-potential anomalies caused by 2-D inclined sheets

The first‐order horizontal and vertical derivatives of the self‐potential (SP) anomalies caused by a 2-D inclined sheet of infinite horizontal extent are analysed to obtain the depth h, the half width a, the inclination α and the constant term containing the resistivity ρ and the current density I of the surrounding medium. The vertical derivative of the SP anomaly is obtained from the horizontal derivative via the Hilbert transform, which is also redefined to yield a modified version, a 270° phase shift of the original function. The point of intersection of these two Hilbert transforms corresponds to the origin. The amplitudes constitute a similar case. The practicability of the method is tested on a theoretical example as well as on field data from the Surda area of Rakha mines, Singhbhum belt, Bihar, India. The results agree well with those of other methods in use. Since the procedure is based on a simple mathematical expression involving the real roots of the derivatives, it can easily be automated.

Ridge-regression algorithm for gravity inversion of fault structures with variable density

We derive an analytical expression for gravity anomalies of an inclined fault with density contrast decreasing parabolically with depth. The effect of the regional background, particularly the interference from neighboring sources of a fault structure, is ascribed by a polynomial equation. We have developed an inversion technique employing the ridge-regression iterative algorithm to infer the shape parameters of the fault structure, in addition to the effect of regional background. We demonstrate the validity of the proposed technique by inverting a gravity anomaly of a theoretical model, both with and without adding a regional background. The technique is insensitive to the effect of regional background. Two density-depth models of the Godavari subbasin in India are used in our interpretation of the gravity anomalies of the Ahiri-Cherla master fault. The interpreted results of a parabolic density model are found to be more geologically reasonable in comparison with the constant density model. The variations of the misfit function of the theoretical and observed gravity anomalies, the damping factor, and the shape parameters of the fault against the iteration number indicate the reliability of the interpretation.

VES and VLF—an application to groundwater exploration, Khammam, India

Although the geophysical methods are being routinely used for exploration of groundwater, at times it becomes a challenge because of various factors such as geometry and depth of the aquifer and the yield of groundwater. Further, in the absence of surface manifestations of structures favorable for groundwater occurrence, instead of depending on one particular geophysical method, an integrated geophysical strategy plays an indispensable role not only in mapping and understanding the nature of aquifers but also ensures a better success rate of exploration.

An integrated geophysical approach for imaging subbasalt sedimentary basins: Case study of Jam River Basin, India

An integrated geophysical strategy comprising deep electrical resistivity and gravity data was devised to image subbasalt sedimentary basins. A 3D gravity inversion was used to determine the basement structure of the Permian sediments underlying the Cretaceous formation of the Jam River Basin in India. The thickness of the Cretaceous formation above the Permian sediments estimated from modeling 60 deep-electric-sounding data points agrees well with drilling information. The gravity effect of mass deficit between the Cretaceous and Permian formations was found using 3D forward modeling and subsequently removed from the Bouguer gravity anomaly along with the regional gravity field. The modified residual gravity field was then subjected to3D inversion to map the variations in depth of the basement beneath the Permian sediments. Inversion of gravity data resulted in two basement ridges, running almost east to west, dividing the basin into three independent depressions. It was found that the Katol and Kondhali...

2-D Hartley transforms

Two different versions of kernels associated with the 2-D Hartley transforms are investigated in relation to their Fourier counterparts. This newly emerging tool for digital signal processing is an alternate means of analyzing a given function in terms of sinusoids and is an offshoot of Fourier transform. Being a real‐valued function and fully equivalent to the Fourier transform, the Hartley transform is more efficient and economical than its progenitor. Hartley and Fourier pairs of complete orthogonal transforms comprise mathematical twins having definite physical significance. The direct and inverse Hartley transforms possess the same kernel, unlike the Fourier transform, and hence have the dual distinction of being both self reciprocal and having the convenient property of occupying the real domain. Some of the properties of the Hartley transform differ marginally from those of the Fourier transform.

Modeling and inversion of magnetic and VLF-EM data with an application to basement fractures: A case study from Raigarh, India

We present modeling of magnetic and very low frequency electromagnetic (VLF-EM) data to map the spatial distribution of basement fractures where uranium is reported in Sambalpur granitoids in the Raigarh district, Chhattisgarh, India. Radioactivity in the basement fractures is attributed to brannerite, U-Ti-Fe complex, and uranium adsorbed on ferruginous matter. The amplitude of the 3D analytical signal of the observed magnetic data indicates the trend of fracture zones. Further, the application of Euler 3D deconvolution to magnetic data provides the spatial locations and depth of the source. Fraser-filtered VLF-EM data and current density pseudosections indicate the presence of shallow and deep conductive zones along the fractures. Modeling of VLF-EM data yields the subsurface resistivity distribution of the order of less than 100 ohm-m of the fractures. The interpreted results of both magnetic and VLF-EM data agree well with the geologic section obtained from drilling.

On “The gravitational attraction of a right rectangular prism with density varying with depth following a cubic polynomial” (Juan García-Abdeslem, 2005, GEOPHYSICS, 71, 1793–1804)Discussion and Reply

oad, H 007 A.P F G a The author presents an interesting note that a cubic polynomial ould be more appropriate than a quadratic polynomial to simulate he density-depth variation of sedimentary rocks when the depth is onfined to 7 km. The author demonstrates this by simulating the ensity-depth data from the Green Canyon, offshore Louisiana, as resented by Li 2001 . In general, the density of sedimentary rocks increases with depth ith the larger increases occurring in the shallower depths. Thus, onuniform density-depth profiles play a significant role in modelng and inversion of gravity anomalies resulting from sedimentary asins. Many mathematical relations have been developed to assess he decrease in density contrast of sedimentary rocks with depth. hose described in the literature include the following: exponential ensity function by Cordell 1973 , Granser 1987 , Chai and Hinze 1988 , and Rao et al. 1993 ; hyperbolic density function by Litinky 1989 ; quadratic density function by Rao 1990 , and Gallardoelgado 2003 . The author, Juan Garcia-Abdeslem, used a cubic olynomial to simulate the density-depth variation of sedimentary ocks and adopted the density function to derive an analytical exression for the gravity anomaly of a 3D rectangular prism. Howevr, in hydrocarbon exploration, geophysical logs are generally conned to shallow depths from pilot or exploratory boreholes; in such ases, the cubic or quadratic polynomials may fail to simulate proper ensity-depth data of sedimentary rocks beyond the logging depth of oreholes. However, we feel that the parabolic density function Chakravarhi, 2003; Chakravarthi and Sundararajan, 2004 in this case — paricularly when the depth is beyond 7 km — could be a better choice or simulating the density-depth variation. We show here the paraolic density function in combination with quadratic and cubic polyomials. The dotted line shown in black Figure 1 is the observed density ontrast-depth variation of the Green Canyon, offshore Louisiana, btained by subtracting a reference density of 2670 kg/m3 Juan arcia-Abdeslem, 2005 from the observed density-depth data Li, 001 . The solid line shown in red represents the cubic polynomial t of the available density contrast-depth data up to nearly 7 km of he Green Canyon, offshore Louisiana, defined by Juan Garcia-Abeslem 2005 ,

On: “Interpretation of some two-dimensional magnetic bodies using Hilbert transforms” discussion and reply

We read Mohan et al.’s paper on interpretation of magnetic anomalies using Hilbert transforms and the discussion raised on it by Pauls (1985) and the authors’ reply (Mohan et al., 1985). While critically going through the authors’ reply (Mohan et al., 1985), we noticed a serious error in the subject paper related to the incompatibility of its equations (1) and (2).