Electromagnetic Induction in an Irregular Layer Overlying the Earth. The 3rd Paper: A Conducting Layer Having a Plane Top and

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1 J. Geomag. Geoelectr., 39, 1-18, 1987 Electromagnetic Induction in an Irregular Layer Overlying the Earth. The 3rd Paper: A Conducting Layer Having a Plane Top and an Undulatory Bottom Naoto OHSHIMAN artment of Earth Sciences, College of Dep Humanities and Sciences, Nihon University, Sakurajosui, Setagaya-ku, Tokyo, Japan (Received March 19, 1986; Revised June 11, 1986) Electromagnetic induction by a uniform magnetic field in a conducting layer having a plane top and an undulatory bottom is studied. A source field parallel to the surface is assumed. The apparent resistivity (pa) vs period of magnetic variation (T) relation is obtained at various points over the top surface by means of conventional magneto-telluric technique. The phase vs Trelation is also obtained from the electric and magnetic fields studied. It turns out that the Pa-T relation and the phase- Trelation are the same as those of the two-layer model developed by Cagniard provided a small amplitude of the undulatory bottom is assumed. Also these relations are independent of the observation point. When the amplitude of undulation is large, however, the pa-t and the phase- Trelations are modified in such a way that an apparently low-resistive zone lies at the prescribed depth of undulatory boundary between the two layers of the original model, the conclusion being on the three-layer MT analysis due to Cagniard. 1. Introduction The magneto-tellurics (MT) formulated by CAGNIARD (1953) is one of the powerful means for estimating resistivity structure of the earth's crust. It is obvious, however, that the MT method leads us to an exact solution only in the case of an earth structure which is laterally uniform provided the inducing field is fairly uniform (e.g., RIKITAKE, 1966). Such a condition is not always fulfilled in an actual case. In that case, sounding curves are affected by the distortion of magneto-telluric field caused by inhomogeneity of the earth's crust (e.g., PARK, 1985). Therefore, we should evaluate correctly such an effect in order to apply the conventional MT analysis to forming a preliminary picture or a starting model of resistivity structure in two-dimensional (2-D) or three-dimensional (3-D) interpretation. RIKITAKE and OHSHIMAN (1985) investigated electromagnetic induction in a 2-D model consisting of a superficial thin layer over a non-conductor underlain by an undulating perfect conductor. Meanwhile, OHSHIMAN and RIKITAKE (1985) estimated the effect of topography on MT results basing on electromagnetic induction by a uniform magnetic field in a semi-infinite medium of finite conductivity having a 1

2 2N. OHSHIMAN sinusoidal undulation at the surface. They concluded that the pa-t relation was little affected by a surface undulation for which the wavelength and peak-to-peak amplitude are 10.0 and 2.0 km, respectively. HUGHES (1973) studied the effect of induced magnetic fields on geomagnetic micropulsation measurements by means of two 2-D models, one of which consisting of two layers which were separated by a boundary whose depth varied sinusoidally in the horizontal direction. He presented an analytical form of the transfer function (the ratio of vertical magnetic field to the horizontal one). Much attention was paid to the transfer function rather than the apparent resistivity in his paper. In this paper, the author investigates electromagnetic induction in a layer having a plane top and an undulatory bottom in order to bring out the above-mentioned distortion of MT results using a theory similar to that developed by RIKITAKE (1965, 1968). 2. Theory Let us think of a conducting layer of which the top surface is plain and bottom surface is undulatory as shown in Fig. 1 in which the coordinate system used in this paper is also shown. Everything is assumed to be uniform in the y direction perpendicular to the xz plane. In free spaces I and III, we can define magnetic potentials WI and WIII which should respectively satisfy the general solutions of which may be written as (1) (2) (3) Fig. 1. The coordinates and system of conductor. Regions I and III are isulating, while Region II is a conducting layer having a plane top and an undulatory bottom. The undulatory boundary between Regions II and III is given by z f(x). The plane boundary between Regions I and II is corresponding to the earth's surface.

3 Electromagnetic induction in an Irregular Layer3 where we assume that a time-dependent uniform field Ho parallel to the z=0 surface is applied to the conducting layer. It is clear that the first term on the righthand-side of (2) indicates the potential of inducing field and that the second term is that of the field produced by the electric currents induced in the conductor by the inducing field. It is assumed that the bottom undulation of the layer has a periodicity with a wavelength l and that f(x), which describes the shape of undulation, is symmetric with respect to x=0. This is the reason why the righthand-side of (2) and (3) involve only sine terms. We can define a vector potential in the conducting layer. In a two-dimensional problem considered here, it is obvious that the vector potential has a non-vanishing component only in they direction. Denoting the component of vector potential by Ay, we see that Ay satisfies (4) in which 6 and t denote the electrical conductivity and time, respectively. The magnetic permeability is assumed as unity in the electromagnetic unit. A typical solution of (4) which fits in the present problem is given as where From (2), (3) and (5), the magnetic field components in regions I, II and III are respectively obtained as (5) (6) (7) (8)

4 4N. OHSHIMAN (9) The boundary conditions at z=0 require that both the tangential and normal components of the magetic field are continuous there, so that we have (10) and At the lower boundary, which is defined by z=f(x), similar conditions should hold good. Although the boundary is undulatory, the continuity of the tangential and normal magnetic field components necessarily leads to the continuity of both the x and z field components. We therefore have (11) (12) (13) (14)

5 Electromagnetic Induction in an Irregular Layer5 where (15) Solving Bn and Cn from (10) and (11), we obtain (16) (17) Eliminating Bn and Cn from (13) and (14), we obtain

6 6N. OHSHIMAN (18) where (19) Equations (18) for N=1, 2,... provide a set of simultaneous equations solving which we can obtain A0, A1, A2,..., D1, D2,..., the infinite series involved being to be truncated at a suitable value of n in an actual calculation. 3. Magnetic and Electric Fields over the Top Surface Since An's can be calculated in the last section, the magnetic and electric fields can readily be obtained. Corresponding to the inducing magnetic field components which are given by (20)

7 Electromagnetic Induction in an Irregular Layer7 the induced magnetic field components at z=0 become (21) The y component of electric field is given by (22) so that we have (23) at z=0. Taking Eq. (11) into account, (23) is rewritten as (24) The ratio of electric field Ey to magnetic field Hx, which are perpendicular to each other, can then be obtained as (25) which is useful for estimating the apparent resistivity of the earth's crust by the magneto-telluric method. 4. Conducting Layer with a Bottom of Sinusoidal Undulation Let us assume that the time variation is purely periodic with an angular frequency (26) From (6), we obtain (27) where

8 8N. OHSHIMAN In the meantime, we assume that which indicates that the bottom of the conducting medium undulates in a sinusoidal way at a mean depth of L with an amplitude h and a wavelength l/q. Putting (28) into (15), we obtain (28) (29) If we specify l, q, h and L, the integrals involved in (29) can be calculated for according to the electrical structure considered. Assuming (30) dependence of the pa-t relation and the phase- T relation between Ey and Hx at the earth's surface on L, the mean depth of the layer-bottom and on h, the amplitude of the undulation are studied in this paper.

9 Electromagnetic Induction in an Irregular Layer9 Solving simultaneous Eq. (18) for n=0, 1,...,12, An and Dn are obtained in units of Ho as shown in Table 1 for T=0.143 and T=0.02 sec in the case of L=0.3 km and h=100 m, as examples. With the aid of An and Dn thus determined, the apparent resistivities at formula in MT, i.e., pa=0.2 T Ey/Hx 2, as shown in Fig. 2(a). Pa, T, Ey and Hx are measured in units of ohm-meter, second, mv/km and nt, respectively. Solid lines in Fig. 2(a) show the pa-t curves for which a parameter L is 0.3, 0.5,1.0, 2.0 and 5.0 km, respectively, h being assumed as 100 m. These results are exactly the same as those for a two-layer model due to CAGNIARD (1953). Such agreement between the model described in this paper and Cagniard's one is also seen in the phase-trelation. These Table 1. An, A*n, Dn and D*n in units of Ho for the two periods. Coefficients denoted by and * are real part and imaginary part of An and Dn, respectively. (a) T=0.143 sec. (b) T=0.02 sec.

10 10N. OHSHIMAN relation of each case in which parameter L amounts to 0.3, 0.5,1.0, 2.0 and 5.0 km, respectively, while h is assumed as 100 m. The dashed curve indicates the pa-t relation for a model for which h=1000 m and

11 Electromagnetic Induction in an Irregular Layer11 Fig. 3. (a) The pa-tcurves at various points on the earth's surface for a model for which h=1000 m and L=1.0 km. (b) The phase-t curves at various points on the earth's surface for a model for which h=1000 m and L=1.0 km.

12 12N. OHSHIMAN results indicate that the undulation model with a small h described here is approximately equivalent to the two-layer model of Cagniard. The pa-t relations are practically the same from observation point to observation point in the case of h=100 m, and the phase-t relations are almost the same, as will be seen in Fig. 4. On the other hand, the broken line in Fig. 2 shows the pa-t relation for a the curve obtained runs parallel to that of L=1.0 km and h=100 m although the value than that corresponding to the skin depth. Such an effect appears around T=0.1 sec where the skin depth is almost equal to the mean depth of the bottom of conducting layer. Therefore, the effect of the undulation of the layer-bottom on the pa-t relation results in an incorrect value of the resistivity of upper layer as long as we rely on a two-layer model of Cagniard. The phase-t relation of L=1.0 km and h=1000 m in Fig. 2(b) also shows the discrepancy between the undulatory model and Cagniard's one. It is clearly seen, therefore, that the distortion to the telluric field becomes more effective as the amplitude of the undulation increases. Fig. 4. Dependence of the apparent resistivity on truncation number n of An and Dn.

13 Electromagnetic Induction in an Irregular Layer13 The Pa-T relation and the phase- Trelation are different from observation point to observation point for the above model as are shown in Fig. 3. The dependence of the apparent resistivity on truncation number n of An and Dn in Eq. (18) is checked by varying n in the case of L=1.0 km, h=1000 m and T= sec. As shown in Fig. 4, estimated values of the apparent resistivity in the case of n=12 are considered to be good approximations to converged values. Cagniard's MT method will be described in the following section. One of the main purposes of MT analysis is to find the position and the shape of the boundary between the layers having different resistivities, and to determine the resistivity values of each layer. In order to see how well the layer undulation is brought out, the apparent resistivity, pa, and the phase difference between Ey and Ez at the frequency of 7 Hz are plotted against the position of observation point in Fig. 5. Sinusoidal curves are obtained for all cases although the curve for a small value of h, 100 m say, becomes almost flat. Fig. 5. The pa and phase vs x curves for various values of h on the earth's surface at a frequency 7 Hz. The mean depth L is 1.0 km.

14 14N. OHSHIMAN. MT Interpretation by Two- and 5Three-Layer Models In this section, conventional MT interpretations are applied to the results described in the previous section (see Fig. 3) using Cagniard's method. First of all, we apply the two-layer method due to Cagniard to the pa-t curve at each observation point shown in Fig. 3(a). Figure 6 shows the best-fitting pa-t curves at each observation point obtained by the two-layer method. The resistivity of the upper layer infinite. It turns out that complete fitting cannot be achieved by the two-layer method only. It is planned therefore, to deal with fitting only for longer periods. The two-layer Fig. 6. The best fitting pa-t curves to those in Fig. 3(a) using the two-layered model due to Cagniard.

15 Electromagnetic Induction in an Irregular Layer15 model thus obtained for each observation point, which explains the pa-t relation at each observation point shown in Fig. 3(a), is shown in Fig. 7. The solid curve in Fig. 7 shows the bottom shape of the conducting layer. As is clearly seen in the figure, the depths of boundary between the two layers estimated at each observation point are, systematically, larger than those of the original model. The above discussion makes it clear that a simple MT analysis leads to a resistivity structure which is considerably different from the original model when the bottom undulation is large. The shape of the bottom of conducting layer is approximately preserved even if two-layer MT analyses are applied although the estimated bottom happens to lie at a greater depth. The resistivity of top layer is Fig. 7. Schematic diagram of resistivity structure determined by the two-layered model due to Cagniard. The numerals indicate the resistivity in ohm-meter.

16 16 N. OHSHIMAN Fig. 8. The best fitting pa-t curves to those in Fig. 3(a) using the three-layered model due to Cagniard. clearly seen in the figure, a low-resistive layer (the second layer) always appears at a depth of undulatory interface assumed for the original model. The upper layer of three layer model has larger values of the resistivity than that of the original model phase- T relation of the original model shown in Fig. 3(b) fairly well, although the phase- T relations calculated from the analyzed model in Fig. 9 are not shown here. 6. Discussion and Conclusions Electromagnetic induction by a uniform inducing field in a conducting layer having a plane top and an undulatory bottom is studied in this paper using Fourier series expression for scalar and vector potentials of magnetic field. Much attention is drawn to the distortion of MT sounding curves due to undulatory bottom of the layer

17 Electromagnetic Induction in an Irregular Layer17 Fig. 9. Schematic diagram of resistivity structure determined by the three-layered model due to Cagniard. The numerals indicate the resistivity in ohm-meter. when a conventional MT analysis, which assumes a semi-infinite earth consisting of a number of horizontal strata of arbitrary thickness and resistivities, is applied to the model. When the undulation amplitude is appreciably smaller than the mean depth of the undulatory bottom, electromagnetic induction is solely controlled by the mean depth of the undulatory bottom, and the pa-t and phase- T relations for the model described in this paper are the same as those of the two-plane layer model due to Cagniard. In other words, therefore, it is said that we cannot find the undulation of the interface between two layers, of which the wavelength and amplitude are e.g km and around 100 m, respectively. On the other hand, if the amplitude is so large that it cannot be ignored relative to the mean depth of undulation, the distortion of MT sounding curves becomes considerable. Although the portion of pa-t relation where pa increases linearly as T becomes longer is mainly controlled by the depth of undulatory bottom immediately under each observation point, the distortion effect due to the undulation gives rise to a marked difference in a part of longer period of each pa- T relation in such a way that we obtain a systematically deeper interface when a two-layer MT model is adopted. The pa-t relation for periods shorter than the period corresponding to the skin depth considerably differs from observation point to observation point. Such distortion gives rise to a low resistive zone, which does not exist i the original induction model, when a three-layer MT analysis is utilized.

18 18N. OHSHIMAN The model studied in this paper consists of a conducting layer having a plane top and undulatory bottom and a semi-infinite non-conducting region having an undulatory top. In actual case of MT observation, we may well encounter an earth structure for which the resistivity of the second layer is much higher than that of upper layer, and the interface between two layers is not a plane. In such cases, much care should be taken for interpreting the results derived from a conventional MT analysis as demonstrated in the foregoing sections. The author is very grateful to Professor T. Rikitake for critically reading the manuscript. And many of the ideas presented here originated with him. The author is also thankful to Mr. H. Utada of the Earthquake Research Institute for critically reading the manuscript and giving valuable comments. Computations involved in this paper were made on a HITAC M280H system at the Computer Center, University of Tokyo. REFERENCES CAGNIARD, L., Basic theory of the magneto-telluric method of geophysical prospecting, Geophys., 18, ,1953. HUGHES, W. J., The effect of two periodic conductivity Anomalies on geomagnetic micropulsation measurements, Geophys. J. R. Astr. Soc., 31, , OHSHIMAN, N. and T. RIKITAKE, Electromagnetic induction in an irregular layer overlying the earth. The 2nd paper: A semi-infinite medium of finite conductivity having an undulatory surface, J. Geomag. Geoelectr., 37, ,1985. PARK, S. K., Distortion of magnetotelluric sounding curves by three-dimensional structure, Geophys., 50, , RIKITAKE, T., Electromagnetic induction in a semi-infinite conductor having an undulatory surface, Bull. Earthq. Res. Inst., Univ. Tokyo, 43, , RIKITAKE, T., Electromagnetism and the Earth's Interior, 308pp.,Elsevier, Amsterdam, RIKITAKE, T., Electromagnetic induction in uniform and non-uniform sheets underlain by an undulating perfect conductor, Bull. Earthq. Res. Inst., Univ. Tokyo, 46, , RIKITAKE, T, and N. OHSHIMAN, Electromagnetic induction in an irregular layer overlying the earth. The 1st paper: A thin layer over a non-conductor underlain by undulating perfect conductor, Proc. of the Institute of Natural Sciences, Nihon Univ., 20, 43-67,1985.