END CORRECTIONS IN MAGNETIC PROFILE INTERPRETATION

Geophysics ◽  
1973 ◽  
Vol 38 (3) ◽  
pp. 507-512 ◽  
Author(s):  
R. T. Shuey ◽  
A. S. Pasquale
Keyword(s):  
Magnetic Anomalies ◽  
The Body ◽  
Two Dimensional ◽  
Total Field ◽  
Magnetic Profile ◽  
Arbitrary Magnetization ◽  
End Corrections

Simple expressions are presented for the vertical and total field magnetic anomalies due to a polygonal body of finite strike length and arbitrary magnetization. These formulas incorporate end corrections into the well‐known Talwani‐Heirtzler (1964) formulas for two‐dimensional polygonal bodies and reduce to the latter for large strike length. Because of their simplicity, the formulas with end corrections lend themselves to rapid use in digital interpretation. Analysis of the formulas shows that interpretation with end corrections gives a body which is deeper and has a larger magnetization and different shape than the body inferred without end corrections.

A comparative study of the relation figures of magnetic anomalies due to two‐dimensional dike and vertical step models

Geophysics ◽  
1982 ◽  
Vol 47 (6) ◽  
pp. 926-931 ◽  
Author(s):  
H. V. Ram Babu ◽  
A. S. Subrahmanyam ◽  
D. Atchuta Rao
Keyword(s):  
Comparative Study ◽  
Major Axis ◽  
Magnetic Anomalies ◽  
The Body ◽  
Two Dimensional ◽  
Depth Ratio ◽  
Characteristic Points ◽  
Vertical Step ◽  
Bottom To Top ◽  
Body Parameters

Magnetic anomalies in vertical and horizontal components, when plotted one against the other in polar form, result in a curve called the relation figure (Werner, 1953). In this paper, a comparative study of the relation figures of magnetic anomalies due to two‐dimensional (2-D) dike and vertical step models is made. The relation figures for these two models are found to be ellipses with different properties. The tangent at the origin to the ellipse is parallel to the major axis of the ellipse for the dike model, whereas it is perpendicular to the major axis for the vertical step. This property may be used to distinguish whether the source is a dike or a vertical step. For both of the models, the angle made by the axis of symmetry of the ellipse with the coordinate axis is equal to θ, the combined magnetic angle. The ratio between the lengths of the major and minor axes of the ellipse is directly related to the width‐to‐depth ratio of the dike or the bottom‐to‐top depth ratio of the vertical step. A few characteristic points defined on the ellipse are used to evaluate the body parameters. The major portion of the ellipse is obtained in the close vicinity of the source. Because of symmetry, the ellipse may be extrapolated easily outside the data length, and hence the effect of noise caused by adjacent objects is kept at a minimum.

Two‐dimensional transient electromagnetic responses

Geophysics ◽  
1985 ◽  
Vol 50 (12) ◽  
pp. 2849-2861 ◽  
Author(s):  
Jopie I. Adhidjaja ◽  
Gerald W. Hohmann ◽  
Michael L. Oristaglio
Keyword(s):  
Magnetic Field ◽  
Half Space ◽  
Time Domain ◽  
The Body ◽  
Two Dimensional ◽  
Total Field ◽  
Secondary Field ◽  
Field Solution ◽  
Transient Electromagnetic ◽  
Simple Boundary

The time‐domain electromagnetic (TEM) modeling method of Oristaglio and Hohmann is reformulated here in terms of the secondary field. This finite‐difference method gives a direct, explicit time‐domain solution for a two‐dimensional body in a conductive earth by advancing the field in time with DuFort‐Frankel time‐differencing. As a result, solving for the secondary field, defined as the difference between the total field and field of a half‐space, is not only more efficient but is also simpler and eliminates several problems inherent in the solution for the total field. For example, because the secondary field varies slowly both in space and time, it can be modeled on a coarse grid with large time steps. In addition, for a simple body the field is local; therefore, because the field can be assumed to satisfy a simple boundary condition in the earth computation is greatly simplified. Our tests show that for the same accuracy, the secondary‐field solution is roughly five times faster than the total‐field solution. We compute and analyze the magnetic field impulse response for a suite of models, most of which consist of a thin body embedded in a conductive half‐space—with or without overburden. The results indicate the conductive half‐space will both delay and attenuate the response of the body and even obscure it if the conductivity contrast is small. The results also suggest that the conductive host can alter the decay rate of the response of the body from its free‐space counterpart. Our results for multiple bodies illustrate the importance of early‐time measurements to obtain resolution, particularly for measurements of the horizontal magnetic field. The vertical magnetic field, however, can be used to infer the dip direction of a dipping body by studying the migration of the crossover. The results for models which include overburden show that the effect of a conductive overburden, in addition to the half‐space effect, is to delay the response of the body, because the primary current initially tends to concentrate and slowly diffuse through the overburden, and does not reach the body until later time. This effect also complicates the early‐times profiles, becoming more severe as the conductivity of the overburden is increased.

MAGNETIC ANOMALIES DUE TO PRISM‐SHAPED BODIES WITH ARBITRARY POLARIZATION

Geophysics ◽  
1964 ◽  
Vol 29 (4) ◽  
pp. 517-531 ◽  
Author(s):  
B. K. Bhattacharyya
Keyword(s):  
Polarization Vector ◽  
Magnetic Anomalies ◽  
Field Vector ◽  
The Body ◽  
Simple Extension ◽  
Second Derivative ◽  
Positive Anomaly ◽  
Total Field ◽  
Arbitrary Polarization ◽  
Dip Angle

A study is made of magnetic anomalies due to prism‐shaped bodies with arbitrary polarization. Expressions of the total field and its first and second derivatives are derived on the assumption of uniform magnetization through out the body. Formulas for all possible cases in connection with a rectangular prism with vertical sides can be obtained either directly from this paper or by simple extension of the formulas given here. Using the exact expressions given in this paper, the total field and its derivatives are evaluated conveniently and rapidly with the aid of a digital computer. The effect of the dip angle anti declination of the polarization vector on the size and shape of the magnetic anomaly is then studied for the case when the earth’s normal total field vector has a dip angle of 60° and declination of 0°. With an increase in the dip angle of the polarization vector, the negative anomaly occurring on the north of the causative body diminishes in magnitude, whereas the positive and second derivative anomalies increase to maximum values and then decrease. With an increase in declination, this latter trend is repeated with the positive anomaly but the negative and second‐derivative anomalies decrease systematically. Both the positive and second‐derivative anomalies become more and more symmetrical with respect to the prismatic body with increase in either the inclination or declination of the polarization vector.

MID-ATLANTIC RIDGE NEAR 45° NORTH: I. THE MEDIAN VALLEY

1966 ◽  
Vol 3 (3) ◽  
pp. 327-349 ◽  
Author(s):  
B. D. Loncarevic ◽  
C. S. Mason ◽  
D. H. Matthews
Keyword(s):  
Gravity Anomaly ◽  
Bouguer Anomaly ◽  
Magnetic Anomalies ◽  
Two Dimensional ◽  
Total Field ◽  
Dimensional Model ◽  
Free Air ◽  
Mid Atlantic Ridge ◽  
Strong Resemblance ◽  
Free Air Gravity

Detailed maps of bathymetry, free air gravity anomaly, and total field magnetic anomaly are presented for an area approximately 50 × 20 mi along the crest of the ridge. The median valley and the associated belt of large positive magnetic anomalies are continuous and display a striking lineation in direction 019°. The free air gravity anomaly shows a strong resemblance to topography. This correlation disappears when the Bouguer anomaly is calculated, indicating that the intrusive body immediately underlying the median valley is not significantly different in density from those bodies beneath the elongated sea mounts which overlook the valley. Small variations in the Bouguer anomaly indicate that there is an increase in density in a northwest direction across the survey area. Magnetic anomalies within the surveyed area can be simulated by a two-dimensional model in which steeply dipping contacts separate blocks of rock having different magnetizations. These blocks could be entirely within the volcanic layer extending to a depth of 5 km below sea level, but the central block, underlying the median valley, must be much more strongly magnetized than those adjacent to it. The mechanism by which the valley was formed remains obscure.

A LEAST‐SQUARES METHOD OF INTERPRETING MAGNETIC ANOMALIES CAUSED BY TWO‐DIMENSIONAL STRUCTURES

Geophysics ◽  
1969 ◽  
Vol 34 (1) ◽  
pp. 65-74 ◽  
Author(s):  
William W. Johnson
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Magnetic Anomalies ◽  
Descent Method ◽  
The Body ◽  
Steepest Descent Method ◽  
Two Dimensional ◽  
Gauss Method ◽  
Least Squares Estimates

The equations relating the magnetic anomalies to the shape and susceptibility of a body are nonlinear with respect to the coordinates describing the shape. Therefore, iterative procedures must be used to obtain least‐squares estimates of the body coordinates. One method in general use for obtaining nonlinear least‐squares estimates is the Gauss method. This method often fails when the initial values for the structures and susceptibilities do not adequately account for the magnetic anomalies. Another method known as the steepest descent method generally converges to a solution; however, a large number of iterations are required. A method suggested by Marquardt (1963) incorporates the best features of the previous methods. In this paper the Marquardt method is applied to the interpretation of magnetic anomalies. For this purpose the two‐dimensional formulas derived by Talwani and Heirtzler (1964) are used to relate the geometry of a body to the resulting magnetic anomalies. The procedure efficiently controls the amount of change made to an interpreted structure at each iteration, assuring rapid convergence to a solution which satisfies the observed data better in the least‐squares sense than does the initial solution. The method is applied to representative problems.

Magnetic anomalies of two‐dimensional bodies in a magnetic half‐space

Geophysics ◽  
1982 ◽  
Vol 47 (8) ◽  
pp. 1229-1234 ◽  
Author(s):  
Edson E. S. Sampaio
Keyword(s):  
Fourier Series ◽  
Country Rock ◽  
Magnetic Anomalies ◽  
The Body ◽  
The Self ◽  
Two Dimensional ◽  
Body Dimensions ◽  
Bipolar Coordinates ◽  
Susceptibility Contrast ◽  
Fourier Series Analysis

The presence of magnetization in country rock modifies the anomaly caused by magnetic bodies. Such a modification is distinct from the self‐demagnetization effect of the body, and the concept of susceptibility contrast is not adequate to explain it. We can achieve an exact understanding of the problem by solving the potential function in three media: air, magnetic country rock, and magnetized body. This paper sets up the solution of this problem when the magnetized body is a circular cylinder with an infinitely long horizontal axis, for both a horizontal and a vertical inducing ambient field. It expresses the solution of Laplace’s equation in bipolar coordinates for the potentials in the form of Fourier series. Analysis of the vertical, horizontal, and total magnetic anomalies shows that neglect of country rock magnetization reduces the apparent causative body dimensions.

scholarly journals Spinning rough disc moving in a rarefied medium

2010 ◽  
Vol 466 (2119) ◽  
pp. 2033-2055 ◽  
Author(s):  
Alexander Plakhov ◽  
Tatiana Tchemisova ◽  
Paulo Gouveia
Keyword(s):  
The Body ◽  
Two Dimensional ◽  
Zero Temperature ◽  
Optimal Mass Transport ◽  
Dense Gases ◽  
Magnus Effect ◽  
Inverse Effect ◽  
Transversal Force ◽  
The Moment ◽  
Complementary Mechanism

We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from non-elastic interaction of gas particles with the body. We propose another (complementary) mechanism of creating the transversal force owing to multiple collisions of particles in cavities of the body surface. We limit ourselves to the two-dimensional case of a rough disc moving through a zero-temperature medium on the plane, where reflections of the particles from the body are elastic and mutual interaction of the particles is neglected. We represent the force acting on the disc and the moment of this force as functionals depending on ‘shape of the roughness’, and determine the set of all admissible forces. The disc trajectory is determined for several simple cases. The study is made by means of billiard theory, Monge–Kantorovich optimal mass transport and by numerical methods.

A DIRECT COMPUTATION OF GRAVITY AND MAGNETIC ANOMALIES CAUSED BY 2- AND 3-DIMENSIONAL BODIES OF ARBITRARY SHAPE AND ARBITRARY MAGNETIC POLARIZATION BY EQUIVALENT‐POINT METHOD AND A SIMPLIFIED CUBIC SPLINE

Geophysics ◽  
1977 ◽  
Vol 42 (3) ◽  
pp. 610-622 ◽  
Author(s):  
Chao C. Ku
Keyword(s):  
Arbitrary Shape ◽  
Gaussian Quadrature ◽  
Magnetic Anomalies ◽  
Cubic Spline ◽  
The Body ◽  
Point Method ◽  
Magnetic Polarization ◽  
3 Dimensional ◽  
Equivalent Point ◽  
Gravity And Magnetic

A computational method, which combines the Gaussian quadrature formula for numerical integration and a cubic spline for interpolation in evaluating the limits of integration, is employed to compute directly the gravity and magnetic anomalies caused by 2-dimensional and 3-dimensional bodies of arbitrary shape and arbitrary magnetic polarization. The mathematics involved in this method is indeed old and well known. Furthermore, the physical concept of the Gaussian quadrature integration leads us back to the old concept of equivalent point masses or equivalent magnetic point dipoles: namely, the gravity or magnetic anomaly due to a body can be evaluated simply by a number of equivalent points which are distributed in the “Gaussian way” within the body. As an illustration, explicit formulas are given for dikes and prisms using 2 × 2 and 2 × 2 × 2 point Gaussian quadrature formulas. The basic limitation in the equivalent‐point method is that the distance between the point of observation and the equivalent points must be larger than the distance between the equivalent points within the body. By using a reasonable number of equivalent points or dividing the body into a number of smaller subbodies, the method might provide a useful alternative for computing in gravity and magnetic methods. The use of a simplified cubic spline enables us to compute the gravity and magnetic anomalies due to bodies of arbitrary shape and arbitrary magnetic polarization with ease and a certain degree of accuracy. This method also appears to be quite attractive for terrain corrections in gravity and possibly in magnetic surveys.

Finite Element Study of Oxygen Diffusion in the Intervertebral Disc

2000 ◽  
Author(s):  
E. Sélard ◽  
A. Shirazi-Adl ◽  
J. P. G. Urban
Keyword(s):  
Intervertebral Disc ◽  
Blood Supply ◽  
Oxygen Diffusion ◽  
Consumption Rate ◽  
The Body ◽  
Cellular Viability ◽  
Two Dimensional ◽  
Concentration Gradients ◽  
Exchange Area ◽  
The Matrix

Abstract The intervertebral disc consists of a water-rich extra-cellular matrix which is synthesized and maintained by its cells. The disc is the largest avascular tissue in the body with its cells lying as much as 8mm away from the blood supply. Nutrients, essential for maintaining cellular viability, diffuse through the matrix from blood supply under a concentration gradient arising from cellular demand. The oxygen concentration gradients in the intervertebral disc are investigated to examine the effects of exchange area and disc thickness on oxygen flux in the disc. The concentration gradients are computed using the two-dimensional Poisson’s equation and measured values for oxygen consumption rate and oxygen diffusion.

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