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

Geophysics ◽  
1998 ◽  
Vol 63 (5) ◽  
pp. 1551-1555 ◽  
Author(s):  
N. Sundararajan ◽  
P. Srinivasa Rao ◽  
V. Sunitha
Keyword(s):  
Surrounding Medium ◽  
Mathematical Expression ◽  
Constant Term ◽  
Hilbert Transforms ◽  
The Hilbert Transform ◽  
Point Of Intersection ◽  
Self Potential ◽  
Derivatives Of ◽  
Sp Anomaly ◽  
Inclined Sheet

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.

scholarly journals A study of the self-potential field produced by a polarised inclined sheet in an electrically homogeneous and transversely anisotropic ground

2001 ◽  
Vol 34 (4) ◽  
pp. 1343
Author(s):  
Γ. Α. ΣΚΙΑΝΗΣ ◽  
Τ. Δ. ΠΑΠΑΔΟΠΟΥΛΟΣ ◽  
Δ. Α. ΒΑΪΟΠΟΥΛΟΣ
Keyword(s):  
Potential Field ◽  
The Self ◽  
Quantitative Interpretation ◽  
Potential Field Data ◽  
Synthetic Model ◽  
Transversely Anisotropic ◽  
Self Potential ◽  
Sp Anomaly ◽  
Interpretation Method ◽  
Inclined Sheet

In the present paper, the self-potential (sp) field is studied, which is produced by an inclined sheet (thin dyke) in an electrically homogeneous and transversely anisotropic ground. At first, the mathematical expression for the sp anomaly is deduced, by integration of the formula for the self-potential field produced by a point pole in a transversely anisotropic medium (Skianis & Herntmdez 1999). Then, the behavior of the sp curve is studied, for various angles of schistosity. The whole anomaly may be displaced along the horizontal axis and deformed in terms of amplitude and shape. Particular emphasis is given on the enhancement and suppression of the positive center of the self-potential, which depends on the values and orientations of the schistosity angle of the ground and the dip angle of the inclined sheet. These deformations of the sp anomaly, may introduce significant errors in the calculation of the parameters of the polarized body, if ground anisotropy is not taken into account. Therefore, new methodologies have to be developed, for a reliable quantitative interpretation of self-potential field data. In this paper, a direct interpretation method is proposed, which consists of two steps: In step one, the parameters of the inclined sheet are determined, assuming a homogeneous and isotropic ground. In this stage, any quantitative interpretation method, referred in the international bibliography, may be used. Secondly, the true parameters of the dyke are estimated, by a set of transformations in which the anisotropy coefficient and the schistosity angle are introduced. In order to apply this method, a priori information about ground anisotropy should be available, by dc geoelectrical and geological investigations. The efficiency of the method was tested on a synthetic model. In the first stage, the quantitative interpretation method of Murty & Haricharan 1985 was employed. In the second stage, the calculated parameters of the first step, served as input values of the transformations, and the real parameters of the inclined sheet were estimated. There was a good agreement between the parameter values of the synthetic model and the ones found by the proposed method. The results and conclusions of this paper, may be useful in detecting sulfide mineralization deposits or graphite.

A rapid graphical method for the interpretation of the self‐potential anomaly over a two‐dimensional inclined sheet of finite depth extent

Geophysics ◽  
1988 ◽  
Vol 53 (8) ◽  
pp. 1126-1128 ◽  
Author(s):  
H. V. Ram Babu ◽  
D. Atchuta Rao
Keyword(s):  
Graphical Method ◽  
Amplitude Ratio ◽  
Finite Depth ◽  
Ore Deposits ◽  
Characteristic Features ◽  
Zero Potential ◽  
Depth Extent ◽  
Self Potential ◽  
Sp Anomaly ◽  
Inclined Sheet

The inclined sheet is an important model for interpreting self‐potential (SP) anomalies over elongated ore deposits. Many techniques (Roy and Chowdhurry, 1959; Meiser, 1962; Paul, 1965; Atchuta Rao et al., 1982; Atchuta Rao and Ram Babu, 1983; Murty and Haricharan, 1985) have been proposed for interpreting SP anomalies over this model. We propose a simple graphical procedure for locating the upper and lower edges of an inclined sheet of infinite strike extent from its SP anomaly V(x) using a few characteristics points including [Formula: see text] [Formula: see text], and [Formula: see text] The amplitude ratio [Formula: see text], is shown to vary with θ, the dip of the sheet, making it possible to estimate θ. The two edges of the sheet are equidistant from the abscissa of [Formula: see text] the zero potential point. The sheet, when extrapolated onto the line of observation, meets the x‐axis at a point where [Formula: see text] From these characteristic features of V(x), the sheet can be located easily using the simple geometrical construction presented below.

scholarly journals Implications of Self-Potential Distribution for Groundwater Flow System in a Nonvolcanic Mountain Slope

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Tada-nori Goto ◽  
Kazuya Kondo ◽  
Rina Ito ◽  
Keisuke Esaki ◽  
Yasuo Oouchi ◽  
...  
Keyword(s):  
Groundwater Flow ◽  
General Trend ◽  
Flow System ◽  
Ground Surface ◽  
Mountain Slope ◽  
Infiltration Process ◽  
Groundwater Flow System ◽  
Self Potential ◽  
Sp Anomaly ◽  
Fft Analysis

Self-potential (SP) measurements were conducted at Mt. Tsukuba, Japan, which is a nonvolcanic mountain, to infer groundwater flow system in the mountain. Survey routes were set around the northern slope, and the reliability of observed SP anomaly was checked by using SP values along parallel survey routes; the error was almost within 10 mV. The FFT analysis of the spatial SP distribution allows us a separation of raw data into two components with shorter and longer wavelength. In the shorter (altitudinal) wavelength than ∼200 meters, several positive SP peaks of more than 100 mV in magnitude are present, which indicate shallow perched water discharges along the slope. In the regional SP pattern of longer wavelength, there are two major perturbations from the general trend reflecting the topographic effect. By comparing the SP and hydrological data, the perturbation around the foothill is interpreted to be caused by heterogeneous infiltration at the ground surface. The perturbation around the summit is also interpreted to be caused by heterogeneous infiltration process, based on a simplified numerical modeling of SP. As a result, the SP pattern is well explained by groundwater flow and infiltration processes. Thus, SP data is thought to be very useful for understanding of groundwater flow system on a mountain scale.

Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 780-786 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Hilbert Transform ◽  
Potential Field ◽  
Field Data ◽  
Three Dimensional ◽  
Analytic Signal ◽  
Three Dimensions ◽  
Hilbert Transforms ◽  
Potential Field Data ◽  
Automatic Interpretation ◽  
The Hilbert Transform

The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.

scholarly journals Generalized multidimensional Hilbert transforms in Clifford analysis

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Fred Brackx ◽  
Bram De Knock ◽  
Hennie De Schepper
Keyword(s):  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Clifford Analysis ◽  
Hilbert Transforms ◽  
Singular Operators ◽  
The Hilbert Transform

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces ofL2-functions.

Hilbert transforms and almost periodic functions

1960 ◽  
Vol 56 (4) ◽  
pp. 354-366 ◽  
Author(s):  
J. Cossar
Keyword(s):  
Hilbert Transform ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Hilbert Transforms ◽  
Cauchy Principal ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

The Hilbert transform, Hf, of a function f is defined by Hf = g, whereP denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists andIn the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

Use of the Hilbert transform to interpret self-potential anomalies due to two-dimensional inclined sheets

1990 ◽  
Vol 133 (1) ◽  
pp. 117-126 ◽  
Author(s):  
N. Sundararajan ◽  
I. Arun Kumar ◽  
N. L. Mohan ◽  
S. V. Seshagiri Rao
Keyword(s):  
Hilbert Transform ◽  
Two Dimensional ◽  
The Hilbert Transform ◽  
Self Potential

On: “Interpretation of magnetic bodies using Hilbert transforms” by N. L. Mohan, N. Sundararajan, and S. V. Seshagiri Rao (GEOPHYSICS, 47, 376–387, March, 1982).

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 168-168
Author(s):  
Rob Pauls
Keyword(s):  
Data Processing ◽  
Hilbert Transform ◽  
Geophysical Data ◽  
Hilbert Transforms ◽  
Magnetic Interpretation ◽  
The Hilbert Transform

The Fourier and Hilbert transform techniques have played extremely important roles in geophysical data processing. We were naturally very interested to find the paper by Mohan et al. which described a possible application of the Hilbert transform approach in magnetic interpretation. However, we found some very serious problems in the paper.

On: “Interpretation of two dimensional magnetic bodies using Hilbert transforms,” by N. L. Mohan, N. Sundararajan, and S. V. Seshagiri Rao (March 1982 GEOPHYSICS, 52, p. 376–387)

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 690-691
Author(s):  
B. N. P. Agarwal
Keyword(s):  
Magnetic Field ◽  
Hilbert Transform ◽  
Two Dimensional ◽  
Hilbert Transforms ◽  
The Hilbert Transform

While going through some of the publications (Mohan and Babu, 1995), I became interested in the work of Mohan et al. (1982) which proposed a technique for interpretation of magnetic field anomalies over different geometrical sources using the Hilbert transform (HT). Before I put forward my observations, it would be appropriate to look into some important properties of HT (Whalen, 1971, pages 63 and 69.)

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