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GEOPHYSICS
L. L. VANYAN, G. M. MOROZOVA
PALETTES FOR INTERPRETING THE ESTABLISHMENT OF A MAGNETIC FIELD
(Presented by Academician A. L. Yanshin, June 14, 1962)
Geophysical prospecting by the method of establishment of an electromagnetic field, the foundations of which were laid by A. N. Tikhonov \((^{1,2})\), is at present finding wide application, especially for elucidating the regional tectonics of oil-bearing areas. In most cases the establishment of a magnetic field is studied when a rectangular current pulse is switched on in a grounded supply circuit. The graph of the transient process is interpreted by selecting a suitable curve calculated for a model of a horizontally layered medium. Such calculations \((^{1-4})\) have clarified a number of important properties of the process of field establishment. However, the volume of these calculations is insufficient for interpreting the results of field work, taking into account the great variety of geoelectric conditions in different regions of the USSR. Proceeding from this, we carried out the calculation of three-layer and four-layer theoretical curves of magnetic-field establishment. The calculation was based on the relation between sounding of the upper layers of the Earth by means of a transient process (field establishment) and frequency sounding. This relation is expressed by the Fourier transform:
\[ \rho_{\tau}=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\rho_{\omega}\frac{e^{i\omega t}}{i\omega}\,d\omega \quad \text{for } t \ge 0, \]
where \(\rho_{\omega}\) is the complex apparent resistivity in the frequency-sounding method, and \(\rho_{\tau}\) is the apparent resistivity in the field-establishment method \((^5)\). Using the absence of singular points in the complex frequency function \(\rho_{\omega}\), the Fourier integral can be represented in the form
\[ \rho_{\tau}=\frac{1}{2\pi}\int_{0}^{\infty}\operatorname{Re}\rho_{\omega}\frac{\sin\omega t}{\omega}\,d\omega \quad (7). \]
Taking into account that at sufficiently high frequencies \(\omega>\Omega\), we write
\[ \rho_{\tau}=\frac{1}{2\pi}\int_{0}^{\infty}(\operatorname{Re}\rho_{\omega}-\rho_1)\frac{\sin\omega t}{\omega}\,d\omega +\frac{\rho_1}{2\pi}\int_{0}^{\infty}\frac{\sin\omega t}{\omega}\,d\omega = \]
\[ =\rho_1+\frac{1}{2\pi}\int_{0}^{\Omega}(\operatorname{Re}\rho_{\omega}-\rho_1)\frac{\sin\omega t}{\omega}\,d\omega \quad \text{for } t>0. \]
The first stage of calculating \(\rho_{\tau}\) is the computation of \(\rho_{\omega}\). It was carried out with the aid of the high-speed electronic computer of the Computing Center of the Siberian Branch of the Academy of Sciences of the USSR, as described in work \((^8)\). For computing the Fourier integral, a modified Filon method \((^6)\), also known as the trapezoidal method \((^7)\), was used. According to this method, the intermediate values of the complex frequency function \(\rho_{\omega}\), needed for computing the Fourier integral, are found by means of linear interpolation, taking which into account we have:
\[ \rho_{\tau}\approx \rho_1+\sum_{p}\frac{1}{2\pi}\int_{\omega_p}^{\omega_{p+1}} (a_{0p}+a_{1p}\omega)\frac{\sin\omega t}{\omega}\,d\omega, \]
where
\[ a_{1p}=\frac{\operatorname{Re}\rho_{\omega(p+1)}-\operatorname{Re}\rho_{\omega p}}{\omega_{p+1}-\omega_p}, \qquad a_{0p}=\frac{\omega_{p+1}\operatorname{Re}\rho_{\omega p}-\omega_p\operatorname{Re}\rho_{\omega(p+1)}}{\omega_{p+1}-\omega_p}. \]
The integral
under the summation sign is expressed in terms of the sine integral \(\operatorname{si}(\omega t)\) and \(\cos \omega t\):
\[ \int_{\omega_p}^{\omega_{p+1}} (a_{0p}+a_{1p}\omega)\,\frac{\sin \omega t}{\omega}\,d\omega = a_{0p}\bigl(\operatorname{si}\omega_{p+1}t-\operatorname{si}\omega_p t\bigr) - \]
\[ - a_{1p}\bigl(\cos \omega_{p+1}t-\cos \omega_p t\bigr). \]
In accordance with the last formula, a program was compiled for calculating the apparent resistivity \(\rho_\tau\) on a high-speed electronic computer.
Fig. 1. Template for the establishment of the magnetic field for \(\rho_2/\rho_1=1/8,\ \rho_3=\infty,\ h_2/h_1=2\). The numbers beside the curves are the ratios \(r/h_1\).
Using the method described above, theoretical curves of the establishment of the magnetic field were calculated for three-layer sections with parameters: \(h_2/h_1=1/8,\ 1/4,\ 1/2,\ 1,\ 2,\ 4,\ 8;\ \rho_2/\rho_1=1/16,\ 1/8,\ 1/4,\ 1/2,\ 2,\ 4,\ 8,\ 16;\ \rho_3=\infty\), and for four-layer sections with parameters: \(h_2/h_1=1/2,\ 2,\ 8;\ h_3/h_1=1/2,\ 2,\ 8;\ \rho_3/\rho_1=1/16,\ 1/4,\ 1,\ 4;\ \rho_2/\rho_1=\infty,\ 2,\ 1/2,\ 1/8;\ \rho_4=\infty\); where \(\rho_i\) and \(h_i\) are the resistivity and thickness of the layer with number \(i\). The apparent resistivity was calculated as a function of time for 6 fixed values of \(r\), the distance of the observation point from the source. The value of \(r\) was chosen so that it exceeded the depth of the reference horizon by a factor of 3–8. The theoretical curves are plotted on a double logarithmic scale, with \(\rho_\tau\) in fractions of \(\rho_1\) plotted along the vertical axis, and the dimensionless ratio \(\sqrt{\frac{1}{2}t\rho_1\mu_0/h_1^2}\) along the horizontal axis, where \(t\) is the time from the moment the current is switched on, and \(\mu_0\) is the magnetic permeability, which we assume everywhere to be equal to the magnetic permeability of vacuum.
We shall call the quantity \(\sqrt{\frac{1}{2}t\rho_1\mu}\) the establishment parameter of the electromagnetic field, \(\tau_1\). It is not difficult to verify that \(\tau_1\) has the dimension of length. Figure 1 shows theoretical curves of magnetic-field establishment for a three-layer section with \(\rho_2/\rho_1=1/8,\ h_2/h_1=2,\ \rho_3=\infty\). At small values of \(\tau_1/h_1\), the apparent resistivity approaches \(\rho_1\); as \(\tau_1/h_1\) increases, the value of \(\rho_\tau\) decreases, reflecting the low resistivity of the second layer. At maximum values of the establishment parameter, the graph has a maximum characterizing the influence of the nonconducting reference horizon. With the aid of the calculated templates of magnetic-field establishment, it is possible to interpret the results of field work, and also to determine the resolving power of the new method of electrical prospecting.
Institute of Geology and Geophysics
Siberian Branch of the Academy of Sciences of the USSR
Received
10 VI 1962
CITED LITERATURE
- A. N. Tikhonov, Izv. AN SSSR, Ser. Geogr. i Geofiz., No. 3 (1946).
- A. N. Tikhonov, O. A. Skugarevskaya, Izv. AN SSSR, Ser. Geogr. i Geofiz., No. 4 (1950).
- D. N. Chetaev, Tr. Geofiz. Inst. AN SSSR, issue 32 (159) (1956).
- S. M. Sheinman, Prikl. Geofiz., No. 3 (1947).
- L. L. Vanyan, Prikl. Geofiz., No. 25 (1960).
- K. Tranter, Integral Transforms in Mathematical Physics, 1956.
- V. V. Solodovnikov, Yu. I. Topcheev, G. V. Krutikova, Frequency Method for Constructing Transient Processes, 1955.