PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 165　　　　　　　　　　　　　　　　　　　　　　　　　1969

ON THE INTERPRETATION OF MAGNETIC ANOMALIES CAUSED BY THE ACTION OF MAGNETIC BODIES OF FINITE DIMENSIONS

G. K. AVTEN’EV

(Presented by Professor D. S. Mikov)

The magnetic anomalies most simply interpreted are those caused by the action of magnetic bodies of great extent and considerable depth extent. Theoretically, these dimensions should be infinite. The problem becomes somewhat more complicated when the bodies have finite dimensions in depth, since the influence of the lower pole of the magnetized body makes itself felt. The limitation of the acting magnetic bodies in areal extent is especially strongly manifested. Moreover, this influence is qualitatively different from the influence of the limitation of the acting object in depth, which has led to the necessity of conditionally dividing magnetic anomalies into two-dimensional and three-dimensional ones and to the development of special techniques for interpreting magnetic anomalies from bodies of finite dimensions in areal extent.

As yet there are no general techniques for interpreting three-dimensional fields. Different authors approach the solution of this problem in different ways.

1. A number of authors propose finding the most suitable forms of three-dimensional bodies that could create the anomaly under consideration, and interpreting this anomaly in the usual way, for example, by characteristic points or palettes (A. A. Logachev, D. S. Mikov, I. A. Ivanov, Yu. P. Tafeev, and others). However, despite its simplicity, this approach is very limited in its possibilities by the number of regular geometric forms for which analytical relationships are known between the parameters of the acting object and the field value (sphere, ellipsoids of revolution, rods).

2. Others propose analytically transforming three-dimensional anomalies into two-dimensional ones and interpreting the already transformed anomalies [5, 8, 7, 6].

3. Integral methods for interpreting magnetic anomalies from bodies of arbitrary shape constitute a special case. When maps of the vertical and horizontal components of the magnetic field are available, these methods make it possible to determine the magnetic moments of the acting bodies, the coordinates of the center of magnetic masses, and the direction of magnetization (Zamorev, 1936; G. A. Gamburtsev, 1936, 1938; D. S. Mikov, 1954; K. F. Tyapkin, 1962).

4. However, in practice, in interpreting three-dimensional anomalies, methods developed for two-dimensional fields are most often used. The accuracy of such an interpretation is determined mainly by the degree of simi-

or differences between the anomaly under study and the anomaly of infinite strike; i.e., in the final analysis the criteria for these errors will be qualitatively determined by the criteria for assigning particular anomalies to one of three classes: one-dimensional, two-dimensional, or three-dimensional anomalies [10].

Such criteria exist [1, 10, etc.]. However, authors differ in their views on methods for finding these criteria. Some authors believe that the possibility of using simplified formulas and graphs can be established by comparing the characteristic parameters of elementary anomalies. Others (and they are the majority) propose establishing these criteria according to the shape of the anomaly being studied [1, 10]. A third group proposes transforming three-dimensional anomalies into two-dimensional ones [5, 8, 7, 6].

Below, in addition to the known methods, the possibility of quantitatively calculating the criteria under consideration is examined, and practical procedures are given for taking them into account for any point of a three-dimensional field.

Fig. 1

Thin Beds of Limited Strike

For vertical beds of finite strike and infinite strike down to depth, with vertical magnetization, for a profile directed along the strike of the bed with the origin at the center of the bed (Fig. 1), according to S. I. Blinstrubas [2, 3] we have

\[ Z_{yL}=\frac{2b\sigma}{h}\left[ \frac{y+L}{\sqrt{h^{2}+(y+L)^{2}}} - \frac{y-L}{\sqrt{h^{2}+(y-L)^{2}}} \right], \tag{1} \]

\[ Z_{y\infty}=\frac{4b\sigma}{h}, \tag{2} \]

where \(Z_{yL}\) is the value of the vertical component of the magnetic field on a line passing along the strike of the bed;

\(Z_{y\infty}\) is the value of the vertical component along a profile in the direction of strike for infinite strike of the bed;

\(\sigma\) is the linear density of magnetism.

Dividing (1) by (2), we obtain a formula by which the magnitude of the vertical component of the anomalous magnetic field from a three-dimensional body can be calculated in fractions of the field value for a two-dimensional body of the same cross section:

\[ \frac{Z_{yL}}{Z_{y\infty}} = \frac{1}{2} \left[ \frac{y+L}{\sqrt{h^{2}+(y+L)^{2}}} - \frac{y-L}{\sqrt{h^{2}+(y-L)^{2}}} \right]. \tag{3} \]

Using relation (3), one can find the value of the vertical component corrected for finite dimensions along strike:

\[ Z_{\mathrm{corr}}= \frac{Z_{\mathrm{obs}}}{\left(\dfrac{Z_{yL}}{Z_{y\infty}}\right)}. \tag{4} \]

Fig. 2. Graphs of correction coefficients for the central point of a sheet. The numbers indicate the values of \(h_2\) in fractions of \(h_1\).

Fig. 3. Graphs of correction coefficients for the central profile along the strike. The numbers in circles indicate the half-widths of the sheets along the strike.

The error itself due to the “non-two-dimensionality” of the anomaly for the central profile along the strike can be calculated by the formula:

\[ \Delta_y=\left(1-\frac{Z_{yL}}{Z_{y\infty}}\right)\cdot 100\%. \]

For beds of finite dimensions in depth, under the same conditions, we shall have:

\[ Z_{yL}=\frac{2b\sigma}{h_1}\left[ \frac{y+L}{\sqrt{h_1^2+(y+L)^2}}- \frac{y-L}{\sqrt{h_1^2+(y-L)^2}} \right]- \]

\[ -\frac{2b\sigma}{h_2}\left[ \frac{y+L}{\sqrt{h_2^2+(y+L)^2}}- \frac{y-L}{\sqrt{h_2^2+(y-L)^2}} \right], \tag{5} \]

\[ Z_{y\infty}=\frac{2b\sigma(h_2-h_1)}{h_1h_2}, \tag{6} \]

where \(h_1\) and \(h_2\) are the depths of the poles of the acting body. Hence, for any point of an arbitrarily chosen profile, with infinite dimensions of the bed in depth:

\[ \frac{Z_{xyL}}{Z_{xy\infty}}=\frac{1}{2}\left[ \frac{y+L}{\sqrt{h^2+x^2+(y+L)^2}}- \frac{y-L}{\sqrt{h^2+x^2+(y-L)^2}} \right] \tag{7} \]

and, analogously, for beds of finite dimensions in depth:

\[ \frac{Z_{xyL}}{Z_{xy}}= \frac{h_1(h_2^2+x^2)} {2(h_2-h_1)(h_1h_2-x^2)} \left[ \frac{y+L}{\sqrt{h_1^2+x^2+(y+L)^2}}- \right. \]

\[ \left. -\frac{y-L}{\sqrt{h_1^2+x^2+(y-L)^2}} \right] - \frac{h_2(h_1^2+x^2)} {2(h_2-h_1)(h_1h_2-x^2)} \times \]

\[ \times \left[ \frac{y+L}{\sqrt{h_2^2+x^2+(y+L)^2}}- \frac{y-L}{\sqrt{h_2^2+x^2+(y-L)^2}} \right]. \tag{8} \]

Thus, in the general form, the error due to the non-two-dimensionality of the anomalous field and the corrected value of the anomalous field for vertical beds of finite dimensions are expressed by the following relations:

\[ \Delta_{xy}=(1-K_i)\cdot 100\%, \tag{9} \]

\[ Z_{\text{corr}}=\frac{Z_{\text{obs}}}{K_i}, \tag{10} \]

where \(K_i=\dfrac{Z_L}{Z_\infty}\) is the correction coefficient, in fractions of \(Z_\infty\) or in percent, calculated by the corresponding formulas (3), (7), and (8).

Graphs of the correction coefficients \(K_i\) for various conditions are shown in Figs. 2, 3, 4.

Figure 5 shows an example of the practical use of the proposed method. The curve recalculated from curve 2 by formula (10) agrees well with theoretical curve 1, computed from the corresponding analytical expressions for the indicated conditions.

Cylindrical Bodies of Finite Strike

Using the formal relation between gravitational and magnetic fields and the expression \(\Delta g\) for cylindrical bodies of any profile across the strike [4], one can obtain the corresponding-

Fig. 4. Graphs of the correction coefficients \(K_i\) for vertical thin beds of great depth extent.

Fig. 5. Example of the practical use of correction coefficients for transforming a three-dimensional anomaly into a two-dimensional one: 1 — the \(Z_{\infty}\) curve for a vertical bed of infinite strike extent; 2 — \(Z_L\) for a bed with \(2L = 4h\); 3 — the correction curve by which curve 2 was recalculated. The points on curve 1 show the values recalculated from curve 2 by formula (10).

following expressions for the correction coefficients and errors for magnetic anomalies from cylindrical bodies of finite dimensions along the strike:

\[ Z=-\frac{\partial U}{\partial z}; \qquad U=\frac{1}{f\sigma}V_z,\qquad \text{where } V_z=\Delta q; \]

\(u\) is the magnetic potential of the cylindrical bodies,
\(z\) is the vertical coordinate,
\(\sigma\) is the excess density of the disturbing object,
\(f\) is the gravitational constant.

\[ V_{zL}=\frac{f\sigma h}{x^2+h^2} \left( \frac{y_2}{\sqrt{x^2+h^2+y_2^2}} - \frac{y_1}{\sqrt{x^2+h^2+y_1^2}} \right), \tag{11} \]

\[ V_{z\infty}=2f\sigma\,\frac{h}{x^2+h^2}, \tag{12} \]

where \(y_1\) and \(y_2\) are the horizontal coordinates of the ends of the disturbing cylinder along the central profile in the strike direction (Fig. 6).

Differentiating expressions (11) and (12) with respect to \(z\), dividing the resulting expressions by one another, and placing the origin of coordinates at the center of the cylinder, we obtain the general formula for the correction coefficient for any point of the profile:

Fig. 6

\[ \frac{Z_L}{Z_\infty} = \frac{(y+L)\left\{h^2(x^2+h^2)+(h^2-x^2)\left[x^2+h^2+(y+L)^2\right]\right\}} {2(h^2-x^2)\left[x^2+h^2+(y+L)^2\right]^{3/2}} - \]

\[ - \frac{(y-L)\left\{h^2(x^2+h^2)+(h^2-x^2)\left[x^2+h^2+(y-L)^2\right]\right\}} {2(h^2-x^2)\left[x^2+h^2+(y-L)^2\right]^{3/2}}. \tag{13} \]

The corrected values \(Z_a\) and the error due to the finite extent are calculated in the same way as for sheets, by the same formulas (10) and (9).

Expressions analogous to (8) and (13) can also be obtained for any other body shapes, for example, inclined sheets of arbitrary magnetization, of any thickness and any depth extent. The proposed method for transforming three-dimensional anomalies is very simple,

is visual and, provided that previously calculated plots of correction coefficients are available, does not require extensive calculations.

The transformed curve (or the entire field) can be interpreted by all the methods used in the interpretation of two-dimensional anomalies.

The method can be successfully used in the interpretation of detailed magnetic surveys where preliminary data are already available on the dimensions of bodies along strike, or where these dimensions can be approximately estimated from plans or graphs of the vertical component of the anomalous magnetic field. For example, for thin beds of finite dimensions in depth and for cylinders, this can be done from the half-maximum of the vertical component along a profile along strike (Fig. 3, curve 2).

REFERENCES

1. V. A. Andreev. On the conditions of applicability of the formulas of the two-dimensional problem in the interpretation of magnetic and gravitational anomalies. Tr. VIRG, issue III, 1950.

2. S. I. Blinstrubas. On the question of interpretation of magnetic anomalies. Scientific communication of the Institute of Geology and Geophysics, Academy of Sciences of the Latvian SSR, 2, 100—137, Vilnius, 1955.

3. S. I. Blinstrubas. A method for the quantitative interpretation of magnetic observations. Dissertation submitted for the degree of Doctor of Physical and Mathematical Sciences, Tbilisi State University, Tbilisi, 1962.

4. P. I. Lukavchenko. Gravity prospecting for oil and gas. Gostoptekhizdat, 1956.

5. A. K. Malovichko. On the transformation of three-dimensional anomalous fields into two-dimensional ones in solving problems from gravity and magnetic observations. Uch. zap. PGU, vol. XI, issue 1, Perm, 1959.

6. V. M. Novoselitskii. On the transformation of three-dimensional fields to a two-dimensional form in the analytic continuation of gravity anomalies. Uch. zap. PGU, vol. XXIV, issue 2, Perm, 1962.

7. G. A. Troshkov. On the question of the interpretation of magnetic and gravitational anomalies of three-dimensional bodies. Razvedka i okhrana nedr, No. 12, 1960.

8. K. F. Tyapkin. Interpretation of gravitational anomalies caused by geological objects finite along strike. Part I. Gosgeoltekhizdat, 1962.

9. G. A. Shchetnikov. On testing Tyapkin’s method on a theoretical field. Uch. zap. PGU, vol. XVIII, issue 4, Perm, 1961.

10. A. A. Yunkov. The belonging of \(V_{zx}\) anomalies to one-dimensional, two-dimensional, and three-dimensional bodies. Geophysical Prospecting, issue 9, 1962.