Analytical Relationship Between Magnetic Anomalies and the Shape of Terrain Relief

G. K. Avten'ev, L. Ya. Erofeev, A. T. Sysoev

(Presented by the scientific seminar of the Department of Geophysical Prospecting Methods)

The numerous existing analytical dependences between the shape of disturbing bodies and the character of the magnetic field have, in most cases, been derived for isolated localized disturbing bodies under the assumption that the observation surface is a horizontal or uniformly inclined plane and that the enclosing medium is practically nonmagnetic (V. V. Kolyubakin and V. M. Lanina, 1960; Handbook of Geophysics, vol. VI, 1969, etc.). In general form, however, the character of the influence of irregularities of nonmagnetic relief has been studied only weakly. These investigations have mainly been reduced to clarifying the character of the change in local magnetic anomalies from two-dimensional bodies of regular geometric shape at various inclinations of the observation line (B. D. Mikov, 1963). The influence of magnetic relief of arbitrary form on the character of the distribution of the magnetic field has practically not been studied. This circumstance greatly complicates deciphering the nature of areal magnetic fields observed in mountainous terrain, since magnetic relief exerts a substantial influence on the structure of the magnetic field. Therefore, determining the influence of magnetic relief on the character of the distribution of the magnetic field of the area under study is extremely necessary (G. K. Avten'ev, L. Ya. Erofeev, 1970).

Certain aspects of this question are reflected in some published works (I. I. Farmandzhiev, 1957; N. E. Popkov, V. I. Pochtarev, 1959; A. A. Logachev, 1959; B. D. Mikov, 1959—1963; G. I. Kruglyakova, 1960; N. K. Stupak, G. Ya. Golizdra, 1962; B. E. Khesin, 1965—1969; B. N. Mozhaev, L. N. Ivanova, 1969, etc.). However, a detailed and systematic study of the quantitative side of this influence is delayed by the absence of recurrent analytical relations for calculating magnetic fields from the basic forms of magnetic relief. Nevertheless, such relations can be constructed. Thus, for two-dimensional vertically magnetized relief of arbitrary shape, under the condition of approximating it by a broken line, according to A. A. Logachev (1962) and D. Milkovenu (1970), one may write

\[ Z(p)=2J\left[\sum_{i=1}^{n}\cos^{2}\theta_i\,(\omega_i-\omega_{i-1}) -\sum_{i=1}^{n}\sin\theta_i\cos\theta_i \ln \frac{z_i}{z_{i-1}}\right] \tag{1} \]

The conventional notation is shown in Fig. 1. It is not difficult to show that, in principle, formula (1) is also valid for the case when the point

Fig. 1

\(P(x,y)\) will be located in the immediate vicinity of the approximation line. The segments of the broken line approximating the daytime relief may be differentially small; then, analogously to Milkovenu (1970), the expression for the magnetic field of a vertically magnetized relief described by an arbitrary curve will be:

\[ Z(P)=2J\int_{l(x,y)}\frac{h_i}{x^2+h_i^2}\,dx, \tag{2} \]

where

\(l(x,y)\) is some analytic expression of the curve of the real daytime relief;

\(y=h_i\) is the depth to the upper pole of an elementary vertical thin bed of a unipolar line. In this case the problem is considered correct if, when the real relief is approximated by a curve line,

\[ h_i=h_j+l\ne 0. \]

Under real conditions,

\[ Z(x_i,y_i)_{\mathrm{rel.}}=2J\cdot\Delta x \sum_{i,j=1}^{n} \frac{h_i-h_j+b}{(x_i-x_j)^2+(h_i+h_j+b)^2}, \tag{2a} \]

where

\(x_i, h_i\) are the coordinates of the point at which the magnetic field is determined;

\(x_j, h_j\) are the coordinates of the elementary bodies participating in the creation of this field.

If \(\Delta x\) is taken as the displacement step and \(x_i=\Delta x\cdot i\), \(x_j=\Delta x\cdot j\), then we finally obtain

\[ Z(x_i,h_i)=2J\cdot\Delta x \sum \frac{h_i-h_j+b}{\Delta x^2(i-j)^2+(h_i-h_j+b)^2}. \tag{2б} \]

Here \(i\) runs through all values from \(\dfrac{a}{\Delta x}\) to \(N-\dfrac{a}{\Delta x}\), and \(j\) varies from 0 to

\[ n=\frac{2a}{\Delta x}. \]

Here \(2a\) is the adopted length of the interval of influence, \(N\) is the number of points on the profile, and \(b\) is a quantity that accounts for the thickness of the overburden and the height of the instrument measuring the magnetic field. In principle, the displacement step may also be not equal to the thickness of the disturbing elementary bodies. For practice, the working algorithm of formula (1) is more convenient. For analytical calculations, however, the more exact algorithm (2) may be used.

It is easy to see that relations (1), (2б), when the magnitude of the in-

the magnetization intensity of the rocks composing the terrain relief; they make it possible to calculate the values of the vertical component of the magnetic-field intensity from a two-dimensional magnetic relief of any form with any prescribed accuracy. The latter will ultimately be determined by the degree of closeness between the approximation lines and the actual relief, i.e., for formula (1), by the number of segments of the broken line describing the daytime relief. Moreover, the greatest errors will occur at the break points of the approximating line.

The magnetic field from an arbitrarily magnetized relief (the whole of it or its separate parts) can be calculated in two ways. Both are based on one and the same known method of coordinate transformation. The first, in our view the simplest, consists in the fact that, for inclined magnetization of the masses composing the relief, the recurrence (generality) of formula (1) is not violated if, in calculating the vertical component of the magnetic-field intensity from an arbitrarily magnetized relief, we use the formula for \(Z\) of a vertically magnetized relief (1), but measure the angles \(\omega_i\) and \(\Theta_i\) not from the horizon (Fig. 1), but from the line perpendicular to the magnetization vector \(J_s\) (Fig. 2). Secondly, for inclined magnetization one can also derive an independent expression, using the known relation between the components of magnetic-field intensity for vertical \((Z_v, H_v)\) and inclined (arbitrary) magnetization \(Z_n\):

Fig. 2.

\[ Z_n = Z_v \sin J_i - H_v \cos J_i . \tag{3} \]

Here \(J_i\) is the inclination of the magnetization vector. Thus, the general expression for calculating the magnetic field from a two-dimensional arbitrarily magnetized relief can be written in the following form:

\[ Z_n(x_j,y_j)=2J\left\{\sin J_i\left[ \sum_{i=1}^{n}\cos^2\Theta_i(\omega_i-\omega_{i-1}) -\sum_{i=1}^{n}\sin\Theta_i\cos\Theta_i\ln\frac{r_i}{r_{i-1}} \right]\right. \]

\[ \left. -\cos J_i\left[ \sum_{i=1}^{n}\cos^2\Theta_i(\omega_i-\omega_{i-1}) +\sum_{i=1}^{n}\cos\Theta_i\sin\Theta_i(\omega_i-\omega_{i-1}) \right]\right\}. \tag{4} \]

After a simple transformation, it can be obtained in final form:

\[ Z_H = 2J \sum_{i=1}^{n} \cos \theta_i \left[ \sum_{i=1}^{n} \cos\left(J_i-\theta_i\right)\left(\omega_i-\omega_{i-1}\right) - \sum_{i=1}^{n} \sin\left(J_i-\theta_i\right)\ln \frac{z_i}{z_{i-1}} \right]. \tag{5} \]

For manual calculation it is more convenient to use the first (grapho-analytical) method.

For implementation on a computer, it is advisable to write the derived relations in Cartesian coordinates. Thus, for vertical magnetization, expression (1) can be written in the following form (Fig. 3):

Fig. 3.

\[ Z(x_i,H_i)= 2J \sum_{h=0}^{n} \left[ \frac{(x_{j+1}-x_j)^2}{(H_{j+1}-H_j)^2+(x_{j+1}-x_j)^2} \cdot \operatorname{arc tg} \frac{ \dfrac{x_{j+1}-x_i}{H_i-H_{j+1}+b} - \dfrac{x_j-x_i}{H_i-H_j+b} }{ 1+ \dfrac{(x_{j+1}-x_i)(x_j-x_i)} {(H_i-H_{j+1}+b)(H_i-H_j+b)} } - \right. \]

\[ \left. - \frac{0.5\,(H_j-H_{j+1})(x_{j+1}-x_j)} {(H_{j+1}-H_j)^2+(x_{j+1}-x_j)^2} \cdot \ln \frac{(H_i-H_j+b)^2+(x_j-x_i)^2} {(H_i-H_{j+1}+b)^2+(x_{j+1}-x_i)^2} \right]. \tag{6} \]

Here \(H_{ij}\) is the absolute elevation (altitude) of the magnetic relief.

If the numerical matrix (array) is specified with step \(\Delta x\), and the values of the magnetic field of the relief are calculated with another step equal to \(\Delta x \cdot c_1\) (where \(c_1\) is some constant number), then

\[ i=\Delta x c_1 k. \]

Here

\(k\) is the number of the point on the numerical axis, varying from \(\dfrac{a}{\Delta x}\) to \(\dfrac{L-a}{\Delta x}\), then

\[ j=i-\frac{a}{\Delta x}+c_2 n, \]

where

\(2a\) is the integration interval (the length of the zone of influence);

\(n\) is the calculation number, \(n=0,1,2,3,\ldots \dfrac{2a}{\Delta x c_2}\);

\(c_2\) is the number of matrix steps included in the apparent thickness of the disturbing object (in calculations it is convenient to take \(c_1=c_2\));

\[ C_2=\frac{x_{j+1}-x_j}{\Delta x}; \tag{7} \]

\(N\) is the number of points in the numerical array, equal to \(\dfrac{L}{\Delta x}\), where

\(L\) is the length of the profile.

Substitution of the adopted values into expression (6) makes it possible to simplify the working formula somewhat. In principle, expression (6) can also be generalized to the case of arbitrary magnetization.

REFERENCES

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