The Complex Gradient Method of Interpreting the Magnetic Anomalies due to Long Horizontal Cylinders

1980 ◽  
Vol 11 (1-2) ◽  
pp. 34-37 ◽  
Author(s):  
D. Atchuta Rao ◽  
H.V. Ram Babu
Keyword(s):  
Gradient Method ◽  
Magnetic Anomalies ◽  
Horizontal Cylinders ◽  
Complex Gradient

To: Erratum in GEOPHYSICS, v. 46, page 1572

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 973-973 ◽  
Keyword(s):  
Gradient Method ◽  
Magnetic Anomalies ◽  
Complex Gradient

D. Atchuta Rao, H. V. Ram Babu, and P. V. Sanker Narayam, authors of “Interpretation of magnetic anomalies due to dikes: The complex gradient method,” have forwarded us a change in their paper (November 1981 GEOPHYSICS, v. 46, p. 1572).

Magnetization vector imaging for borehole magnetic data based on magnitude magnetic anomaly

Geophysics ◽  
2013 ◽  
Vol 78 (6) ◽  
pp. D429-D444 ◽  
Author(s):  
Shuang Liu ◽  
Xiangyun Hu ◽  
Tianyou Liu ◽  
Jie Feng ◽  
Wenli Gao ◽  
...  
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Magnetization Vector ◽  
Real Data ◽  
Magnetic Anomalies ◽  
Magnetic Data ◽  
Preconditioned Conjugate Gradient ◽  
Magnetization Direction ◽  
Magnetization Intensity

Remanent magnetization and self-demagnetization change the magnitude and direction of the magnetization vector, which complicates the interpretation of magnetic data. To deal with this problem, we evaluated a method for inverting the distributions of 2D magnetization vector or effective susceptibility using 3C borehole magnetic data. The basis for this method is the fact that 2D magnitude magnetic anomalies are not sensitive to the magnetization direction. We calculated magnitude anomalies from the measured borehole magnetic data in a spatial domain. The vector distributions of magnetization were inverted methodically in two steps. The distributions of magnetization magnitude were initially solved based on magnitude magnetic anomalies using the preconditioned conjugate gradient method. The preconditioner determined by the distances between the cells and the borehole observation points greatly improved the quality of the magnetization magnitude imaging. With the calculated magnetization magnitude, the distributions of magnetization direction were computed by fitting the component anomalies secondly using the conjugate gradient method. The two-step approach made full use of the amplitude and phase anomalies of the borehole magnetic data. We studied the influence of remanence and demagnetization based on the recovered magnetization intensity and direction distributions. Finally, we tested our method using synthetic and real data from scenarios that involved high susceptibility and complicated remanence, and all tests returned favorable results.

A computer program for interpreting vertical magnetic anomalies of spheres and horizontal cylinders

1973 ◽  
Vol 110 (1) ◽  
pp. 2056-2065 ◽  
Author(s):  
B. S. R. Rao ◽  
I. V. Radhakrishna Murthy ◽  
C. Visweswara Rao
Keyword(s):  
Computer Program ◽  
Magnetic Anomalies ◽  
Horizontal Cylinders

scholarly journals Convergence of an Online Split-Complex Gradient Algorithm for Complex-Valued Neural Networks

2010 ◽  
Vol 2010 ◽  
pp. 1-27
Author(s):  
Huisheng Zhang ◽  
Dongpo Xu ◽  
Zhiping Wang
Keyword(s):  
Neural Networks ◽  
Fixed Point ◽  
Adaptive Learning ◽  
Gradient Method ◽  
Error Function ◽  
Gradient Algorithm ◽  
Training Procedure ◽  
Weight Sequence ◽  
Complex Valued ◽  
Complex Gradient

The online gradient method has been widely used in training neural networks. We consider in this paper an online split-complex gradient algorithm for complex-valued neural networks. We choose an adaptive learning rate during the training procedure. Under certain conditions, by firstly showing the monotonicity of the error function, it is proved that the gradient of the error function tends to zero and the weight sequence tends to a fixed point. A numerical example is given to support the theoretical findings.

On: “Nomogram for the direct interpretation of magnetic anomalies due to long horizontal cylinders,” by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, (GEOPHYSICS, 51, 2156–2159, November 1986.)

Geophysics ◽  
1987 ◽  
Vol 52 (10) ◽  
pp. 1442-1443
Author(s):  
Ronald Green
Keyword(s):  
Magnetic Anomaly ◽  
Magnetic Anomalies ◽  
Horizontal Cylinder ◽  
Direct Interpretation ◽  
Horizontal Cylinders

In the article by T. K. S. Prakasa Rao, M. Subrahmanyam, and A. Srikrishna Murty, the trivial problem of interpreting the magnetic anomaly over a horizontal cylinder was examined and a set of nomograms to assist with the interpretation was presented. Prakasa Rao et al. begin their discussion with equation (1) from Gay (1965).

Nomogram for the direct interpretation of magnetic anomalies due to long horizontal cylinders

Geophysics ◽  
1986 ◽  
Vol 51 (11) ◽  
pp. 2156-2159 ◽  
Author(s):  
T. K. S. Prakasa Rao ◽  
M. Subrahmanyam ◽  
A. Srikrishna Murthy
Keyword(s):  
Direct Method ◽  
Linear Equations ◽  
Arbitrary Point ◽  
Magnetic Anomalies ◽  
Horizontal Cylinder ◽  
Empirical Method ◽  
Master Curves ◽  
Hilbert Transforms ◽  
Direct Interpretation ◽  
Horizontal Cylinders

One of the widely used geometrical configurations for magnetic interpretation is the long horizontal circular cylinder. Gay (1965) provides a set of master curves for the interpretation of magnetic anomalies of these bodies. Rao et al. (1973) formulates functions of the anomaly at several distances from an arbitrary point, and the linear equations thus formed are solved for coefficients related to the parameters of the causative body. Prakasa Rao and Murthy (1976) propose an empirical method for rapid interpretation. Atchuta Rao and Ram Babu (1980), Mohan et al. (1982), and Sampath Kumar and Prakasa Rao (1984) describe methods based on Hilbert transforms. Radhakrishna Murthy et al. (1980) propose a method based on two components of the anomalous magnetic field. With the exception of the direct method of Prakasa Rao and Murthy (1976), the other methods mentioned involve reduction of field curves and then matching with master curves, solving linear equations, performing Hilbert transformations, and computation of derivatives, respectively. Hence they are not suitable for direct and rapid interpretation. This note contains a simple nomogram for the magnetic effect due to an arbitrarily magnetized horizontal cylinder.

Interpretation of magnetic anomalies due to dikes: The complex gradient method

Geophysics ◽  
1981 ◽  
Vol 46 (11) ◽  
pp. 1572-1578 ◽  
Author(s):  
D. Atchuta Rao ◽  
H. V. Ram Babu ◽  
P. V. Sanker Narayan
Keyword(s):  
Magnetic Anomaly ◽  
Magnetic Anomalies ◽  
Half Width ◽  
Amplitude Function ◽  
Maximum Value ◽  
Magnetic Inclination ◽  
Vector Quantity ◽  
Characteristic Points ◽  
Depth Of Burial ◽  
Complex Gradient

A method to interpret the magnetic anomaly due to a dipping dike using the resultant of the horizontal and vertical gradients of the anomaly is suggested. The resultant of both the gradients is a vector quantity and is defined as the “complex gradient.” A few characteristic points defined on the amplitude and phase plots of the complex gradient are used to solve for the parameters of the dike. For a dike uniformly magnetized in the earth’s magnetic field, the amplitude plot is independent of [Formula: see text], the index parameter, which depends upon the strike and dip of the dike and the magnetic inclination of the area. The phase plot of the complex gradient is an antisymmetric curve with an offset value equal to [Formula: see text]. For a dike whose half‐width is greater than its depth of burial, two maxima at equal distances on either side of a minimum value appear on the amplitude plot. For a dike whose half‐width is equal to or less than its depth of burial, the amplitude plot is a bell‐shaped symmetric curve with its maximum appearing directly over the origin. In the case of a thin dike, the amplitude function falls off to half its maximum value at the same point on the abscissa where the phase function reaches, i.e., [Formula: see text]. A combined analysis of the amplitude and phase plots of the complex gradient yields all the parameters of the dike. The method is applicable for the magnetic anomaly in either the total, vertical, or horizontal field. A field example is included to show the applicability of the method.

Magnetic interpretation using a least-squares, depth-shape curves method

Geophysics ◽  
2005 ◽  
Vol 70 (3) ◽  
pp. L23-L30 ◽  
Author(s):  
El-Sayed M. Abdelrahman ◽  
Khalid S. Essa
Keyword(s):  
Least Squares ◽  
Magnetic Anomaly ◽  
Least Squares Method ◽  
Random Noise ◽  
Synthetic Data ◽  
Theoretical Data ◽  
Magnetic Anomalies ◽  
Shape Factors ◽  
Characteristic Points ◽  
Horizontal Cylinders

We have developed a least-squares approach to depth determination from residual magnetic anomalies caused by simple geologic structures. By normalizing the residual magnetic anomaly using three characteristic points and their corresponding distances on the anomaly profile, the problem of determining depth from residual magnetic anomalies has been transformed into finding a solution to a nonlinear equation of the form z = f(z). Formulas have been derived for spheres, horizontal cylinders, thin dikes, and contacts. The method is applied to synthetic data with and without random noise. We have also developed a method using depth-shape curves to simultaneously define the shape and depth of a buried structure from a residual magnetic anomaly profile. The method is based on determining the depth from the normalized residual anomaly for each shape factor using the least-squares method mentioned above. The computed depths are plotted against the shape factors on a graph. The solution for the shape and depth of the buried structure is read at the common intersection of the depth-shape curves. The depth-shape curves method was successfully tested on theoretical data with and without random noise and applied to a known field example from Ontario.

STANDARD CURVES FOR MAGNETIC ANOMALIES OVER LONG HORIZONTAL CYLINDERS

Geophysics ◽  
1965 ◽  
Vol 30 (5) ◽  
pp. 818-828 ◽  
Author(s):  
S. Parker Gay
Keyword(s):  
Circular Cylinder ◽  
Magnetic Anomalies ◽  
Curve Matching ◽  
Profile Curve ◽  
Complete Family ◽  
Family Of Curves ◽  
Isolated Points ◽  
Horizontal Cylinders ◽  
Magnetizing Field ◽  
Horizontal Circular Cylinder

The magnetic anomalies in Z, H, and [Formula: see text] for the long horizontal circular cylinder are shown to belong to a single mathematical family of curves for all values of strike and all values of inclination of the magnetizing field, a characteristic that was previously shown to hold for long tabular bodies, or dikes (Gay, 1963). The complete family of standard curves has been constructed and is incorporated into an interpretational scheme based on superposition with observed magnetic profiles. Comparison of cylinder anomalies with dike anomalies shows only slight differences in the two types of curves, which would be very difficult, if not impossible, to detect using interpretational methods based on a few isolated points of a profile curve, such as half‐width, distance between maximum and minimum, etc. Curve‐matching, or superposition, appears to be mandatory for reliable quantitative interpretations.

Relationship of magnetic anomalies due to subsurface features and the interpretation of sloping contacts

Geophysics ◽  
1980 ◽  
Vol 45 (1) ◽  
pp. 32-36 ◽  
Author(s):  
D. Atchuta Rao ◽  
H. V. Ram Babu ◽  
P. V. Sanker Narayan
Keyword(s):  
Magnetic Anomaly ◽  
Magnetic Anomalies ◽  
Horizontal Cylinder ◽  
Horizontal Derivative ◽  
Horizontal Cylinders ◽  
Relationship Of

A study of the magnetic anomalies produced by sloping geologic contacts, thin dikes, and horizontal cylinders has revealed that a single relationship exists among the magnetic anomalies created by them. The magnetic anomaly due to a horizontal cylinder, the first horizontal derivative of the magnetic anomaly due to a thin dike, and the second horizontal derivative of the magnetic anomaly due to a sloping contact are found to be identical in shape. Gay (1963, 1965) presented standard curves to interpret the magnetic anomalies over long tabular bodies (1963) and long horizontal cylinders (1965). It is shown here that the same curves can also be used to interpret the total, vertical and horizontal magnetic anomalies due to sloping geologic contacts.

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