Theory of the Magnetotelluric Method

Transcription

1 Geophys. J. R. asir. SOC. (1966) 11, Theory of the Magnetotelluric Method for a Spherical Conductor * 4 S. P. Srivastava* (Received 1964 September 24. Revised 1965 August 19) Summary Impedance relations for various conductivity distributions are derived considering the Earth as a non-uniform spherical conductor. The impedance relations are so presented that they can be used for any number of layers inside the Earth, each representing a different conductivity distribution. It is shown theoretically that for small periods the impedance relations derived for a non-uniform spherical conductor are the same as those for a semi-infinite conductor with plane boundaries. Calculations of impedance and phase values made for plane and spherical canductors for periods from 1 to 108s show that the Earth's curvature is important only for periods longer than a day. However, for such long periods at the conductivities of 0.01 and ohm-' metre-', there are effectively no induced currents, and the difference obtained between curves for plane and spherical conductors is attributed to the difference in the coordinate systems used. It is concluded that for these conductivities and periods less than one day, the effect of the Earth's curvature can be neglected when determining conductivity distribution by the magnetotelluric method. 1. Introduction Two methods are generally used to calculate the electrical conductivity inside the Earth using the natural electromagnetic variations recorded on the surface of the Earth. The first, the magnetic potential method, which utilizes only magnetic variations, is based on the assumption that the magnetic field between the Earth and the ionosphere can be represented by a potential function. Such an assumption is justified as the conductivity above the Earth's surface is effectively zero. The second, the magnetotelluric method, which utilizes electric and magnetic variations, does not use the magnetic potential directly. It assumes that the sources of the magnetic variations lie outside the Earth and the air above the Earth's surface is effectively non-conductive. The basic ideas underlying the magnetotelluric method were presented by Cagniard (1953) where he assumed that the magnetic field observed at the surface of the Earth can be considered to be uniform over an infinite horizontal plane. However, it has been shown by Wait (1962) and Price (1962) that such an assumption is not justified and a more generalized form * Present address: Bedford Institute of Oceanography, Dartmouth, Nova Scotia, Canada. ** Canadian contribution No. 101 to the International Upper Mantle Project. 313

2 374 S. P. Srivastava of the method should be used. Price (1962) has shown that the methods, magnetic potential and magnetotelluric, are basically the same and the difference between them arises in the practical application of the theory, when choosing which components of the electromagnetic field to measure at the Earth s surface. The basic assumption of a potential, in the magnetic potential method, enables one to separate the total magnetic field observed on the surface of the Earth into two components, external and internal. The internal part is then attributed to electric currents induced in the Earth by the external part. Using the magnetic variations recorded all over the Earth, the amplitudes and phases of the ratios of the internal to the external parts for various harmonics are calculated. By comparing these ratios with the computed ratios obtained on the assumption of a particular conductivity distribution inside the Earth, one can obtain an approximate conductivity distribution if there is close agreement between the two. Several workers have used such a method to determine the conductivity distribution deep inside the Earth (Chapman & Whitehead 1922, Chapman & Price 1930, Lahiri & Price 1939). On the other hand the magnetotelluric method has been used until now only for the determination of the resistivity distribution at shallow depths (e.g. Srivastava & Jacobs 1954, Niblett & Sayn-Wittgenstein 1960) although an attempt has been made to use it for deep probing (Tikhonov & Lipskaya 1952). Again the results are questionable because of the assumption of a plane wave. The two methods are essentially based on different systems of coordinates. The Earth has been considered as a sphere in the potential theory method and as a semi-infinite conductor with a plane boundary in the magnetotelluric method. Attempts have been made to justify such an assumption in the magnetotelluric method (Scholte & Veldkamp 1955, Wait 1962) by considering the effect of the sphericity of the Earth on the magnetotelluric data, at short periods (of the order of 103s). It has been concluded from these studies that the effect is negligible. However, no attempt has been made so far to show a similar effect on the magnetotelluric data at longer periods. To do so it is essential to develop the magnetotelluric method for a spherical conductor. It has been pointed out by Price (1962) that the magnetotelluric method can be used to determine the resistivity distribution both in shallow and deep regions inside the Earth provided one knows the value(s) of the horizontal wavelength(s) of the source involved. This means recording electromagnetic variations at several places. However, it is possible to interpret the resistivity curves obtained from the recordings made at one place only with the help of master curves (Srivastava 1965). In order to apply a similar method of interpretation for deep probing, the theory of the magnetotelluric method is extended here to a spherical conductor. A comparison between the apparent resistivity curves obtained for spherical and plane conductors is made to study the effect of the sphericity of the Earth on the magnetotelluric method for deep probing. 2. Mathematical theory, spherical conductor (i) General solution. The theory of the magnetotelluric method will be first developed for a spherical conductor when the conductivity can have any distribution with depth. Subsequently it will be applied for a specific conductivity distribution. It will be sufficient for the present purpose to treat the Earth as a sphere of radius a and conductivity o = o(r) where r is the distance from the centre. For the sake

3 Theory of the magnetotellmic method for a spherical coadoctor 375 of simplicity the permeability p will be considered as unity in the following derivations. We assume that the varying magnetic field of arbitrary distribution in the region r > a induces electric currents in the conductor. We then have to find the tangential components of the electric and magnetic fields at the surface r = a. As a great deal of work on the problem of electromagnetic induction in uniform and non-uniform spherical conductors has been done in the past (e.g. Lamb 1883, Price 1930, Price 1932, Lahiri & Price 1939) the derivations will not be given in great detail. Assuming that the electric displacement current can be ignored, it is found that the electric and magnetic fields inside the Earth for the harmonic of degree n and order m can be given as and 1 H, = --f( r). n(n + 1 ). Sr(8, q5) r E, = 0 Ee = - iof(r).- ~ dq5 d sr(4 4) sin8 d E, = iw. f(r)-s;(8, q5) do where f(r) satisfies the equation r2-+2r-+(k2(r). d2f(r) dr2 dr r2-n(n+ l))f(r) = o and SF(& q5) is the usual surface harmonic ~ (COS 8) multiplied by sin or cos mq5 (Lahiri & Price 1939). Taking the ratio of orthogonal components of E and H from (1) and (2), we get Similarly If is quite clear from (3) and (4) that the impedance value at any point inside a spherical conductor depends only on the radial function f(r). (ii) a(r) = constant in diferent spherical shells. Let us divide the spherical conductor into a number of spherical shells in each of which the conductivity is

4 376 S. P. Srivastava constant. The radial functions for a spherical shell whose conductivity, a(r), remains constant within the limits rl to r2 is given by the differential equation where d2f(r) dr2 df(4 dr r r- + (k2r2 - n(n + l))f(r) = 0 k2 = -4naio. (5) Since k2 is constant, the solution of (5) is where j, and qn are spherical Bessel functions of first and second kind, of order n. Hence the impedance at a distance r inside the conductor will be given by E,= -_ Be io Akr. j,(kr) + B. kr. q,(kr) k. A(d/dkr)(kr. j,(kr)) + B. (d/dkr)(kr. q,,(kr))* In equation (7) B should be equal to zero because f(r) remains finite as r tends to zero. Hence for the last shell near the centre of the sphere the impedance value will be E+ = io kr. j,(kr) - -_ (9) Be k (d/dkr)(kr.j,,(kr))* (8) z -_- io AP(l,l)+BQ(l, 1) 1- kl AR(1,1)+ BS(1,l) Now let us find the impedance at a distance r = r2 from the centre of the sphere where r2 < rl but a(r) remains the same. Therefore z --_ io AP(2,l) + BQ(2,l) 2 - kl * AR(2, l)+bq(2, 1)

5 Theory of the magnetotelluric method for a spherical conductor 377 By eliminating A and B between (15) and (16) we get Now if we extend the distance rl to the surface of the sphere (Fig. la) then rl becomes equal to a and Z, becomes the impedance measured on the surface of the Earth (Z,,), Equation (17) gives a relation between the impedances which are measured at the boundaries of a layer or a shell. Surface FIG. l(a). Diagram showing the different resistivity shells inside a spherical conductor. Hence, using (17) we can write a general impedance relation iw z Q(n,n).P(n+l,n)-P(n,n).Q(n+l,n) ' = -- k, 'S(n, n). P(n+ 1, n)-r(n, n). Q(n+ 1, n) +Z,,+,.(k,,/io)[Q(n, n).r(n+l, n)-p(n, n).s(n+l,n)] +Z,+l.(k,,/ia)[S(n, n). R(n+l, n)-r(n,n).s(n+l,n)]. If there are n+l layers then Z,,+l will be given by (9). Substituting the values of Z,, Z3, Z,... Z,,, in (17) one can calculate the value of Z,. (iii) a(r) = ao(r)-'. In part (ii) of this section, impedance relations were derived for the case where the conductivity had constant values in a series of concentric shells. In this section similar expressions are derived when the conductivity is a function of depth. Lahiri & Price (1939) obtained the solution of (2a) for a conductivity distribution given by a(r) = ao(r)-'. Taking the ratio of the orthogonal components of E and H (18)

6 378 S. P. Srivasteva using (3), (4) and the solution of (2a) given by Lahiri & Price, the impedance relations for the various cases can be summarized in the following way: /l>2 1=2 1<2 where S = [i- (A; - n(n + l)]f, and z=- 2Ao (1-2)' v=- 2n ' A$ = -4xiw0,. Proceeding in the same way as in part (ii) of this section the surface impedance for a spherical conductor with concentric shells, each representing a different conductivity distribution can be written in the form Z,= -iw. where and Q(L1) - P(5 1)-p(1, 1). Q(5 1) S(1, 1). P(2, l)-r(l, 1). Q(2, 1),.l-+A, +(z2/i4[q(1, 1). R(2, l)-p(l, 1). S(2, 111 +(Z,/id[S(1, 1) * R(2,1)-W, 1) f S(2, 111 (24) The impedance relation given by (24) is a general expression for any distribution of conductivity and holds good for any value of 1 from 1 = 0 to infinity except when 1 = 2. When I = 2, P(a, b) will become (rap+*, where Sb = [i-(&-n(n+ 1))lf (31) and similar expressions for the other quantities in (281 (29) and (30). Equations (17) and (24) are exactly the same when 1 = 0.

7 Theory of tbe magnetotelluric metbod for a spherical conductor 3. Spherical conductor treated as a semi-infinite conductor with a plane dace In the previous section impedance relations for a spherical conductor with various conductivity distributions were derived. In order to apply these relations to a restricted portion of the spherical conductor, it is necessary to consider the case when the size of the spherical conductor is increased indefinitely, i.e. the spherical conductor could be treated as a semi-infinite conductor with plane surface. The impedance relations for such a case can be found from equation (5) when a tends to infinity. The solution of (5) is given by (7). From the theory of Bessel functions we know that the first derivative term in (5) can be neglected under the conditions that Re kr % 1 and lkri2 is somewhat greater than n(n+ 1) (Sommerfeld 1949) as has been pointed out by Wait (1962). The condition )kri2 > n(n+ 1) implies that 4a2r2a T<- n(n+ 1) When the above condition is satisfied, (5) can be written as 379 f (r) = 0. (33) If one attempts to treat the Earth as a semi-infinite conductor, it has been pointed out by Price (1962) that the longest wavelength (2a/v) that could be taken for the inducing field would correspond to the circumference of the Earth. For local fields one can assume that v = n/u, approximately, but this is not permissible for small n as was pointed out by Price (personal communication). The direction of r in the spherical coordinate system corresponds to z, in the Cartesian coordinate system. Hence (33) reduces to d2 --f(z) dz2 = (4aioa + v2)f(z) where n(n+ 1) N n2. Equation (34) is exactly the same as obtained by Price (1962, equation 11). This shows that for those values of T which satisfy (32) the effect of the Earth s curvature can be neglected. Such a property will be shown numerically in subsequent sections. In order to see the effect at periods greater than those given by (32) it is necessary to compute the impedance values for a spherical conductor and for a semi-infinite conductor with a plane surface for a large range of T values. The impedance value for a semi-infinite conductor with a plane surface, where conductivity remains constant within layers (Fig. lb) can be written in the form (34) a2h coth- a, where 2, is the surface impedance value; hl, h2, h3... are the thicknesses of the first, second, third... layers and a,,, a quantity given by (Srivastava 1965). a: = 411io0, + v2

8 380 S. P. Srivastava Surface pn I hn W FIG. l(b). Diagram showing the different resistivity layers inside a conductor with plane boundaries. 4. Computations of p4 and 8 The quantities which are usually used in the magnetotelluric method are the apparent resistivity p4, and the phase angle 8, between the orthogonal components of the electric and magnetic fields. Expressing the surface impedance relation (equations 17, 19,24 and 35) in the form iw X+iY =--- (I k, U+iV (37) where X, U are the real parts and Y, V the imaginary parts of the numerator and denominator of the final equation, we could then write p4 and 8 as X2+Y2 PdPl = U(Y-X)-V(X+Y) 8 = tan-' (39) U(X+ Y)+ V(Y -X) where p4 = 2T(ZOIZ. To study the effect on magnetotelluric method of the Earth's curvature at periods longer than those given by (32), it is necessary to calculate pa and 8 for various conductivity distributions inside the Earth. A simple illustrative case is that in which the conductivity remains constant with depth. For this simple example the impedance value for a spherical conductor is given by where jn(x) = (-I)Y(-$)"(~)

9 and Theory of the magnetotelluric method for a spherical conductor 38 1 (43) and for a plane conductor by io z = -- a1 where a1 is given by (36). Calculations have been made for pjpl and 0 (equations 38, 39) using (41) and (44) for a range of values of n from 1 to 5, and for a range of values of T from 1 to 10 s. The resistivity (l/a) has been taken as 100 ohm metres. The results are shown in Figs. 2 and 3. In Fig. 2 the values of pjpl are plotted against T for various values of n. A comparison between the values of pjp, for plane and spherical conductors, Fig. 2, reveals that the two differ considerably only at periods longer than 5 x lo5 s. This would then show that the Earth s curvature is important only for periods longer than 5 x lo5 s, a value which agrees well with (32). 1 I I I lo lo7 lo8 lo r(s) FIG. 2. Apparent resistivity/resistivity of the top layer versus period for the model shown. A similar effect can be seen if one compares the phase angles, Fig. 3. Fig. 2 is plotted on a log versus log scale so that any change in the value of p will result in horizontal shift of the curves to the right or left depending upon whether the resistivity has been decreased or increased compared to the one assumed for the present model. Only values of n of 5 or below are considered here, as fields of most geomagnetic fluctuations used in deep magnetotelluric probing would contain spherical harmonics of an order less than or equal to 5. The example chosen in Figs. 2 and 3 is a very simplified one as resistivity is assumed to be constant and thus does not correspond to the resistivity distribution which has generally been obtained by various investigators in the past. In order to see the effect when the resistivity suddenly increases at a certain depth, a two

10 382 S. P. Srivastava layer model has been considered. For this case the impedance value for a spherical conductor is given by (I i~ Q(1, 1). P(2, l)-p(l, 1). Q(5 1) = -- k1. S(1, 1). P(2,l)-R(l, 1). Q(2, 1) + [Q(L 1). R(2, U-PU, 1). S(2, l)lz2.(kl/iw) +[S(l, 1). R(2, 1)-R(1, 1). S(2, l)]z2. (k1/i0) where (45) and P, Q, R, S are given by (11, 12, 13, 14), and for a plane conductor by 1 a2 where a,,, is given by (36) f Plane 90 Spherical ' > T 105 (s) eo 15 FIG. 3. Phase angle between E and H versus period for the model shown in Fig : P. Calculations have been made for pjpl and 8 (38, 39) using (45) and (46) for a range of values of n from 1 to 5, and for a range of values of T from 1 to 10" s. The resistivity for the first'layer is taken as 10ohmmetres and for the second as lo3 ohm metres at a depth of 25 km below the surface. The results are shown in Figs. 4 and 5. A comparison between the values of pjpl for plane and spherical conductors, Fig. 4, reveals that the two differ to some extent in the same manner as shown in Fig. 2. An effect similar to that shown in Fig. 2 of the Earth's curvature on pjpl values at periods longer than 5 x lo5 s is obtained here. That is, the Earth's curvature is important only for periods longer than 5 x lo5 s. In order to see effects similar to those shown in Figs. 2 and 4 of the &rth curvature for cases where the resistivity discontinuities lie in deep regions within the Earth, it is necessary to consider a model similar to that shown in Fig. 4 with a resistivity discontinuity at a greater depth. Such a model is shown in Fig. 6 where 30 0

11 Theory of the mngnetotelloric metbod for a spherical conductor 383 FIG. 4. Apparent resistivity/resktivity of the top layer versus period for the model shown FIG. 5. Phase angle between E and H versus period for the model shown in Fig. 4.

12 384 S. P. Srivastava the resistivity discontinuity lies at 300 km below the surface of the Earth. A comparison between the apparent resistivity curves for plane and spherical conductor reveals that they start deviating after 4 x lo4 s and the deviation becomes maximum after 5 x lo5 s.. ---_ Plane -3 -Spherical ' ' ' I I I 1 lo ' 10 6 T(S) FIG. 6. Apparent resistivity/resistivity of the top layer versus period for the model shown. 5. Discussion and conclusions Although the above calculations have been made only for very simple distributions of conductivity, they are sufficient to show the effect of the Earth's curvature on both the amplitude ratio and the phase difference between the horizontal components of E and H in the magnetotelluric method. It is shown by Figs. 2-6 that the Earth's curvature is important only for periods longer than 5 x lo5 s even though the resistivity models considered in each case differ fairly widely from each other. Such a difference between the apparent resistivity curves for conductors with plane and spherical surfaces could be due either to the effect of the curvature of the conductor or to some other reason. To find wbether the difference is due to the curvature, the contribution to the surface field from the induced currents should be considered. Calculating the ratio of the amplitude of the induced magnetic field to that of the inducing field (Price 1950), it was found that, for the harmonics considered here, the contribution to the surface field at periods longer than 5 x lo5 s is negligibly small. This suggests that (taking the real resistivity or conductivity to be of the order of that assumed in the calculations), any useful magnetotelluric data would not be affected by the Earth's sphericity. The real conductivity is, in fact, probably much higher at km below the surface (Lahiri 8c Price 1939) than that assumed, and this would further reduce the effect of sphericity. To illustrate this a three layer model with a d'ecrease in resistivity at 650 km was considered (Figs. 7 and 8). The curves for the semi-infinite conductor with plane surface are calculated for various values of v using the relation v =,/(n(n+ l))/u instead of

13 Theory of the magnetotelluric method for a spherical conductor 385 the relation v = n/a which was used in the computation of curves shown in Figs It can be seen from Figs. 7 and 8 that the curves for conductors with plane and spherical surfaces do not differ significantly (less than 10%) for periods less than 5 x lo6 s. A difference of 30% is obtained for lower spherical harmonics at T = 18' s which gradually decreases with the increase in the order of spherical harmonic. It thus appears that the difference between the curves in Figs. 2 4 is due to the lo r( ~6371 km '2 =6346km r3=5m ~vn FIG. 7. Apparent resistivity/resistivity of the top layer versus period for the model shown r(s) 1 FIG. 8. Phase angle between E and H versus period for the model shown in Fig. 7.

14 386 S. P. Srivastava difference in coordinate systems used in the two cases and to the assumption that n/a can be equated to v in the case of a semi-infinite conductor with plane surface. The approximation that n/a can be equated to v is only true for large values of n as can be seen to some extent from Figs A comparison between the curves for plane and spherical conductors indicates that the difference between them decreases with the increase in the order of spherical harmonic. This clearly shows that the approximation n/a = v is only true for large values of n, for small values of n we should substitute n(n+ l)/a* = vz in equation (36). The difference between the curves for plane and spherical conductors will then be much smaller in comparison to the one obtained in Figs. 2-6 as shown in Figs. 7 and 8. It can be concluded from the above studies that for the conductivity distribution considered here, it is immaterial whether we treat the Earth as a conductor with a plane surface or a spherical surface. It is possible that for other conductivity distributions the difference between pa and 0 curves for plane and spherical conductors becomes significant. For such instances it would be necessary to consider the Earth as a spherical conductor. The interpretation of magnetotelluric curves in terms of the conductivity distribution is more difficult when the Earth is considered as a spherical conductor. The method of interpretation for such cases becomes more or less the same as that in the magnetic potential method (Price 1962). The only advantage of using the magnetotelluric method in such instances would be to use the electric field as a check on the conductivity distribution deduced from the magnetic potential method. An interesting property which can be seen in Figs. 2-8 is the dependence of apparent resistivity and phase values on n at periods longer than 5 x lo4 s, as has also been shown by Rikitake (1951), Price (1962) and Srivastava (1965). The distribution of resistivity at greater depths (of the order of 500 km) which can be obtained by the method described here is of course dependent on the fact that the measured surface field components are significantly affected by the induced currents flowing at that depth. During their investigation of the resistivity distribution inside the Earth, Lahiri & Price (1939) made some calculations of the variation of the amplitudes of the induced currents with depth. According to them for a conductivity distribution of the form 0 = 4 x 10-'4(r/a)37 in emu, where a is the radius of the Earth, the current density for the second harmonic will be a maximum at a depth 0.12~ (800 km) and will be negligible after 0.h (1 274 km) when diurnal variations are considered. This places a restriction on the use of the magnetic potential or magnetotelluric methods for determining the conductivity distribution in deep regions within the Earth. The greatest difficulty which can be forseen will be to record electric fields of long periods (lo4 to i06 s) as fields of such periods are generally very weak. However, thjs difficulty may be overcome if more sensitive and stable equipment is designed for recording Earth currents. Acknowledgments I would like to thank Professor J. A. Jacobs for his valuable suggestions. I am indebted to Dr T. Watanabe for suggesting this problem to me and his valuable comments during the development of this method. I would also like to thank Professor A. T. Price for critically reviewing this paper and for his valuable suggestions.

15 Theory of the magnetotelluric method of a spherical conductor 387 I gratefully acknowledge the assistance of the Computing Centre of the University of British Columbia with the computation. The work was supported by the Defence Research Board of Canada. Department of Geophysics, University of British Columbia, Vancouver, Canada August. References Cagniard, L., Geophysics, 18, 605. Chapman, S. & Price, A. T., Phil. Trans. R. SOC., A, 229, 427. Chapman, S. & Whitehead, T. T., Trans. Camb. phii. SOC., 22, 463. Lahiri, B. N. & Price, A. T., Phil. Trans. R. SOC., A, 237, 509. Lamb, H., Phil. Trans. R. SOC., A, 174, 526. Niblett, E. R. & Sayn-Wittgenstein, C., Geophysics, 25, 998. Price, A. T., Proc. Lond. math. SOC., (2), 31, 217. Price, A. T., Proc. Lond. math. SOC., (2), 33, 233. Price, A. T., Q. Jl Mech. appl. Math.,3,385. Price, A. T., J. geophys. Rex, 67, Rikitake, T., Bull. Earthq. Res. Inst. Tokyo Univ., 29, 271. Scholte, J. G. & Veldkamp, J., J. atmos. terr. Phys., 6, 33. Sommerfeld, A., Partial Differential Equations. Academic Press, New York. Srivastava, S. P. & Jacobs, J. A., J. Geomagn. Geoelect., Kyoto, 15, 280. Srivastava, S. P., J. geophys. Res., 70, 945. Tikhonov, A. N. & Lipskaya, N. V., Dokl. Akad. Nauk SSSR, 87, 547. Wait, J. R., J. Res. natn. Bur. Stand., D, 66, 509.