On the Determination of the Direction of Magnetization of Geological Bodies from Magnetic Anomalies

D. S. Mikov

The article gives formulas and an example of the practical determination of the direction of magnetization of two closely spaced spherical bodies from a theoretically computed anomaly.

The form of magnetic anomalies created by disturbing geological bodies, as is known, depends to a great extent not only on the shape of the bodies, but also on the direction of their magnetization. The latter is the vector sum of residual and induced magnetization and depends on the direction of the residual magnetization of the disturbing bodies, on the direction of the total vector of the Earth’s magnetic field in the area of investigation, on the spatial orientation of the bodies relative to the magnetic meridian, and on the relative shape of the disturbing bodies themselves.

As an example of the change in the form of anomalies with the direction of magnetization, we shall point out only that, for all two-dimensional bodies, when the direction of magnetization is changed by \(90^\circ\) (from vertical to horizontal, or in general from one inclined direction to another inclined direction), the vertical and horizontal projections of the anomalous magnetic field change so strongly that the curve \(Z\) takes the form of \(H\), while the curve \(H\) takes the form of \(Z\).

Therefore any change in the direction of magnetization of bodies, even by an amount of \(10\)--\(20^\circ\), will give a substantial change in the form of magnetic anomalies. If one does not determine and take into account the true direction of magnetization of the disturbing objects and assumes it, as is often done, to be always vertical, then large errors may be obtained in the interpretation of such anomalies.

All this shows that, for the correct geological interpretation of magnetic anomalies, it is necessary to determine the direction of magnetization of each disturbing object. In addition, we note that, for the purposes of determining the shape and parameters of bodies, it is more important to know the direction of magnetization than the magnitude of its intensity, since the latter changes only the intensity of the anomaly, whereas the direction of magnetization substantially changes both the intensity and the form of the anomalies produced. Therefore, without taking the direction of magnetization into account, it is in principle impossible to determine the shape of the disturbing body from \(Z\) and \(H\) separately.

The direction of magnetization of disturbing bodies can be determined by studying oriented samples of these bodies and by studying the anomalies produced by them.

The first method is the best known, but laborious; it requires the study of a large number of samples taken from different parts of the body. If, however, the disturbing object has not yet been exposed by exploratory workings, then it does not appear possible to apply it.

For determining the direction of magnetization of bodies from the anomalies they produce, several methods may be used. In some methods, combined magnetic and gravitational measurements are used, while in others only the properties of the magnetic anomalies themselves are used.

In the case of vertical magnetization of two-dimensional bodies, the projections of the magnetic field \(Z\) and \(H\), as is known, at all points of the anomaly will be respectively proportional (similar) to the gradients of the gravitational field of the same bodies, \(V_{\Delta}\) and \(V_{zs}\). For three-dimensional bodies of arbitrary shape, the same proportionality (similarity) of fields exists between the components of the magnetic field \(X\), \(Y\), and \(Z\) and the corresponding gradients of the gravitational field \(V_{xz}\), \(V_{yz}\), and \(V_{zz}\). Therefore, if such similarity in the fields is observed, the disturbing object may be considered vertically magnetized. In the case of inclined magnetization, however, the indicated similarities in the fields will be absent, and the direction of magnetization of the bodies in such cases can be determined from the following relations [1]:

for two-dimensional bodies

\[ \operatorname{tg}\gamma = \frac{ZV_{zs}-HV_{\Delta}}{HV_{zs}+ZV_{\Delta}}, \tag{1} \]

\[ \operatorname{tg}\gamma = \frac{\displaystyle\int_{-\infty}^{\infty} Zx\,dx} {\displaystyle\int_{-\infty}^{\infty} Hx\,dx}, \tag{2} \]

\[ \sin\gamma=\frac{2(-Z)}{(+Z)}-1, \]

\[ \cos\gamma=\frac{2(+H)}{(-H)}-1, \tag{3} \]

and for bodies of arbitrary shape

\[ \operatorname{tg}\gamma= \frac{\sqrt{M_x^2+M_y^2}}{M_z}, \qquad \operatorname{tg}A=\frac{M_y}{M_x}, \tag{4} \]

where \(Z\), \(H\), \(V_{\Delta}\), and \(V_{zs}\) are the projections of the anomalous magnetic field and the gradients of the gravitational field, measured at one and the same point of the anomaly;

\(\gamma\) is the angle of the body’s magnetization vector with the vertical;

\((+Z)\), \((-Z)\), \((+H)\), and \((-H)\) are the absolute values of the largest areas bounded by the positive and negative branches of the graphs of \(Z\) and \(H\) with the abscissa axis;

\(A\) is the azimuth of the plane of magnetization of the body, measured from the \(x\)-axis;

\(M_x\), \(M_y\), \(M_z\) are the projections of the total inclined magnetic moment of the body \(M\) onto the coordinate axes, determined from the anomaly by calculating the following integrals

\[ \left. \begin{aligned} M_x&=-0.159\int_{\sigma} Zx\,ds,\\[4pt] M_y&=-0.159\int_{\sigma} Zy\,ds,\\[4pt] M_z&=-0.159\int_{\sigma} Xx\,ds=-0.159\int_{\sigma} Yy\,ds,\\[4pt] M&=\sqrt{M_x^2+M_y^2+M_z^2}. \end{aligned} \right\} \tag{5} \]

The indicated methods for determining the direction of magnetization of two-dimensional bodies were tested by us on a large number of theoretical and practical examples. In this connection, formulas (3), although exact only for two-dimensional bodies with a circular cross section, can in practice be used in all cases when the relative elongation of the shape of the cross section of the disturbing body in some direction does not exceed a three- to fourfold ratio.

The practical possibility of determining the direction of magnetization of bodies of arbitrary shape by formulas (5) was checked by us on a special theoretical example. As the anomaly we took the total magnetic field of two closely spaced, parallelly magnetized, identical spheres with projections of the magnetic moments \(M_x = M_z = 1 \cdot 10^8\) CGS and \(M_y = 0.333 \cdot 10^8\) CGS, located at the same depth of \(100\) m, but at different positions in plan—in points \(C_1\) and \(C_2\) (Fig. 1). The computation of each projection of the field \(X\), \(Y\), and \(Z\) from each sphe-

Fig. 1. Plan of the projection \(X\) of the field of two spherical bodies magnetized obliquely, and computation of the integral
\(\int' Xx\,ds = -102058\). (To simplify the drawing, the numerical values of the field at the vertices of each small square of the plan are not written.) \(\Sigma X\) is the sum of the values of \(X\) along horizontal rows; \(x\) is the coordinate for a given row from the selected coordinate origin; \(Xx\) is the product \(\Sigma X\) by the coordinate \(x\) for the given row, and \(\Sigma' Xx\), i.e., the value of the integral \(\int' Xx\,ds\), is for the entire plan; \(ds\) on the plan \(= 1\ \text{cm}^2\); \(C_1\) and \(C_2\) are the centers of the spherical bodies. The values of \(X\) are in gammas.

of the sphere was carried out in sufficient detail using the formulas given by us in the book [1].

After computing the field projections from each sphere, by adding them we obtained composite plans of the \(X\), \(Y\), and \(Z\) components from both spherical bodies, magnetized, according to the adopted data, in a plane with azimuth

\[ A=\operatorname{arc\,tg}\frac{M_y}{M_x}=18^\circ 25' \]

and at an angle to the vertical

\[ \gamma=\operatorname{arc\,tg}\frac{\sqrt{M_x^2+M_y^2}}{M_z}=46^\circ 24'\ldots \tag{6} \]

The values of the integrals computed from the anomalies within the limits of the plans (Figs. 1, 2) were

\[ \int' Zx\,dS=-83360\cdot 10^{-6}; \]

\[ \int' Zy\,dS=-27431\cdot 10^{-6}; \]

\[ \int' Xx\,dS=-102058\cdot 10^{-6}. \]

Taking into account the scale of the plan \((1:2000)\), after conversion to the CGS system we obtained

\[ \int Zx\,dS=-83360\cdot 10^{-6}\cdot 2000^3=-668\cdot 10^6\ \text{CGS}, \]

\[ \int Zy\,dS=-27431\cdot 10^{-6}\cdot 2000^3=-219\cdot 10^6\ \text{CGS}, \]

\[ \int Xx\,dS=-102058\cdot 10^{-6}\cdot 2000^3=-815\cdot 10^6\ \text{CGS}, \]

\[ \int Yy\,dS=-95295\cdot 10^{-6}\cdot 2000^3=-765\cdot 10^6\ \text{CGS}. \]

From these values, according to formulas (5), the projections of the total magnetic moments of the two spherical bodies were computed:

\[ M_x=-0.159\int Zx\,dS=1.06\cdot 10^8\ \text{CGS}, \]

\[ M_y=-0.159\int Zy\,dS=0.35\cdot 10^8\ \text{CGS}, \]

\[ M_z=-0.159\int Xx\,dS=1.30\cdot 10^8\ \text{CGS}, \]

\[ M_z=-0.159\int Yy\,dS=1.22\cdot 10^8\ \text{CGS}, \]

\[ M_z\ \text{average}=1.26\cdot 10^8\ \text{CGS}, \]

and for one sphere these values will be

\[ M_x=0.53\cdot 10^8\ \text{CGS}, \]

\[ M_y=0.175\cdot 10^8\ \text{CGS}, \]

\[ M_z=0.63\cdot 10^8\ \text{CGS}. \]

Next, by calculation using formulas (6), the azimuth of the plane of magnetization and the angle of the magnetization vector with the vertical were obtained:

\[ \operatorname{tg} A=\frac{M_y}{M_x}=0.33\ \text{or}\ A=18^\circ 20', \]

\[ \operatorname{tg}\gamma=\frac{\sqrt{M_x^2+M_y^2}}{M_z}=0.89\ \text{or}\ \gamma=41^\circ 40' \]

instead of the values adopted in calculating the anomaly, \(A=18^\circ 25'\) and \(\gamma=46^\circ 24'\).

Thus, in determining the direction of magnetization, very good results were obtained, which indicates the possibility of broad practical use of this method on natural local anomalies. At the same time, let us note that the determination of the magneti-

Fig. 2. Plan of the projection of the \(Z\) field of two spherical bodies magnetized obliquely, and calculation of the integrals
\(\int Zx\,ds = -83360,\quad \int Zy\,ds = -27431.\)
(To simplify the drawing, the numerical field values at the vertices of each square of the plan are not written.) \(\Sigma Z\) denotes the sums over horizontal and vertical rows; \(x\) and \(y\) are the coordinates for the rows from the chosen origin of coordinates, and \(\Sigma Zx\) and \(\Sigma Zy\) are their products; \(\Sigma' Zx\) and \(\Sigma' Zy\) are the sums of products, i.e., \(\int Zx\,ds\) and \(\int Zy\,ds\) within the limits of the plan; \(ds\) on the plan \(= 1\ \text{cm}^2\); \(C_1\) and \(C_2\) are the centers of the spherical bodies. Values of \(Z\) are in gammas.

The direction of magnetization was obtained by us without taking the residual integrals into account, since neglecting them, as was indicated earlier (1), does not produce large errors.

The results obtained for determining the magnetic moments without taking the residual integrals into account were considerably worse. In comparison with those adopted for the calculations, they amounted, for \(M_x\) and \(M_y\), to only 53%, and for \(M_z\) to only 63%.

If even an approximate allowance is made for the residual integrals, the result can be substantially improved.

The calculation of the integrals from plans was in fact carried out by us not over infinite limits, but only within the limits of field values amounting to 5–10% of the maximum intensity of the anomaly. According to investigations, the value of the calculated integral for these limits will amount to only 50–60% of its total value over infinite limits. Therefore, after allowance for the correction for the residual inte-

graphs, the magnitude of the magnetic moment computed from the plan will also be close to that specified in computing the anomaly.

From the magnetic moment of the body \(M\) thus found, knowing the excess magnetization of the bodies \((I)\) and the specific weight of the body \((d)\), one can then compute the volume of the body \(v\) and the amount of ore mass \(P\) from the formulas

\[ v=\frac{M}{I}\quad \text{and}\quad P=vd. \]

Therefore, the procedure described here for determining the direction of magnetization of local bodies, their magnetic moments, and also the volumes and amounts of ore mass can be successfully applied in the interpretation of magnetic anomalies.

REFERENCES

1. Mikov D. S. Methods for the Interpretation of Magnetic Anomalies. TSU Press, 1962.