Magneto-Telluric Fields for a Segmented Overburden. James R. WAIT and Kenneth P. SPIES

Transcription

1 J. Geomag. Geoelectr., 26, , 1974 Magneto-Telluric Fields for a Segmented Overburden James R. WAIT and Kenneth P. SPIES Cooperative Institute for Research in Environmental Sciences, NOAA, U.S. Department of Commerce, Boulder, Colorado, U.S.A. (Received August 13, 1974; Revised September 24, 1974) Adopting an idealized two-dimensional model, we consider the electromagnetic fields existing in a crustal layer that has any number of homogeneous segments with vertical interfaces. The magnetic field vector is everywhere parallel to these interfaces and to the earth's surface. Using a quasi-static approach, expressions for the electric fields at the earth's surface are derived and calculations for the actual surface impedance are then presented. It is shown that the vertical contacts will cause strong perturbations of the surface impedance calculated on the basis of a locally uniform slab model. This could be an important factor in carrying out the interpretation of magneto-telluric data. 1. Introduction In the magneto-telluric method of geophysical prospecting it is customary to represent the earth's crust as a system of parallel layers that are individually homogeneous. Thus the ratio of the observed tangential electric and magnetic fields are compared with calculated data for such a model. It has been appreciated for some time that lateral variations of the electric and geometrical characteristics will complicate the interpretation. In a now rather classic paper, D'ERCEVILLE and KUNETZ (1962) considered the effect of a vertical contact between two homogeneous media with contrasting conductivity. RANKIN (1962) used the same method to treat a vertical dike. In both these solutions, the magnetic vector was polarized parallel to the strike and the basement region was idealized as a perfect electric conductor or a perfect magnetic conductor. The extension of the single vertical contact to an infinite basement depth has been considered by WEAVER (1963) for the same assumptions. More recently, GEYER (1972) considered the effect of a dipping (i.e., non vertical) contact between the two homogeneous regions. Actually, Geyer treated both the H polarized and the E polarized situations. In the former case, the assumption that the magnetic field (taken to be parallel to the strike) is spatially constant is a physically reasonable approximation. However, this assumed constancy of the tangential magnetic field does not seem so convincing in the second case where the electric 449

2 450 J.R. WAIT and K.P. SPIES field is parallel to the strike. This comment also would apply to the finiteelement analysis of the same problem by REDDY and RANKIN (1973). When the geological structure becomes very complicated the single vertical contact or simple dike models do not seem to be adequate. The two-dimensional case where the region consists of blocks with differing conductivities separated by interfaces of general form has been treated by NEDOMA and PRAUS (1972) for a constant H-polarized surface field. They employed a numerical method based on a finite difference approximation. 2. Formulation We consider here a similar problem that really is a modest extension of the analytical methods discussed above. The segmented overburden model adopted is actually a generalization of RANKIN's (1962) dike geometry where we still take the H vector to be parallel to the strike. The two-dimensional situation is depicted in Fig. 1 where the overburden (AIR) (BASEMENT) Fig, la. Geometry of the general overburden of,m+ 1 segments. (AIR) (BASEMENT) Fig. 1b. Geometry for overburden with three segments. consists of any number of homogeneous segments all of constant depth h, but with variable width. Thus, for example, the conductivity and magnetic permeability of the m'th segment are 6m and pm, respectively. The lateral width of this segment is 4. We note that m ranges through values 0, 1, 2,... M and since the end segments are semi-infinite we have do = oo and dm = oo. In the model shown in Fig. la, the air region (i.e., z <0) and the basement region (i.e., z > h) are taken to be perfect insulators. Since all displacement currents are neglected, a condition of the solution is that the vertical current density within the segments is zero at both z=0 and z-h.

3 Magneto-Telluric Fields for a Segmented Overburden Quasi-Static Solution In terms of cartesian coordinate system (x, y, z), the agnetic vector has only ay component H while the electric vector has both E x and Ez components. For the m'th segment, we can write the total magnetic field as the sum of two parts as follows: Hm (x, Z)=Hm(Z)+(X, Pmz) (1) where the assumed functional dependences are indicated. In accordance with our assumptions Hm(x,0)= Ho and Hm(x, h) = 0 (2) where H0 is the constant value of the surface magnetic field that presumably is due to sources in the region z<0. The vanishing of the magnetic field at z = h is equivalent to the statement that the top of the basement is a perfect magnetic conductor. This idealization was used by D'ERCEVILLE and KUNETZ (1962) and claimed by RANKIN (1962) to be a valid approximation. Now we stipulate that Also we choose fl(z) where rm=16mm Hm(0)-R0, Pm(x, 0) = 0 &(h)=o, Pm(x, h)=0 to be a solution of the Helmholtz equation (3) (2/az2 a-rm)rm(z)= 0 (4) for an implied time factor exp (iwt). Thus, in view of(3), I(Z)-fl 'a sinh [7(h m-z)]isinh (7h) (s) is the appropriate form of the solution. Noting that both Hm and Hm satisfy the Helmholtz equation, from (1) that it follows [(a2/ax2+a2/az2) - rm]pm0 (6) An appropriate form of the solution, that satisfies (3), is Pm(x, z)fmi(x)sin n=1,2,3... where the x dependent coefficient satisfies Thus, for the m'th segment, we can write (n2rz/h) {(2/ax2)-[r+ (n2r/h)2]}f(x)=0 (8) fm,i(x)=am,n exp (-Km,nx)+Bm,n exp (Km,X) n (9) where Km= [(n7r/h)2+rm]1j2 is chosen such that Re. Km,n>0. Now, again dealing with the total fields, we have (7) Emz= (ipw/r)ahm/ax mm (10)

4 452 J.R. WAIT and K.P. SPIES Then, since ar /ax = 0, we see that Emz=(iw/r)(aPm/ax)mm (11) Thus, the required continuity of tangential electric fields at an interface, us that Pm-iaPm-iPm Ym-i ax apm Ym ax (at X = Xm-i) (12) Also, a statement of the required continuity of tangential magnetic fields at an interface is Hm-i = Hm (at x = xm-i) (13) The implication of the latter is that Pm and Hm are discontinuous across the vertical interfaces. This suggests that (for x=xm-1) we write tells (14) where Cm-1, is a coefficient to be determined. Application of (7), (9), (12) and (14) immediately equations leads to the 2M linear and (tm-ikmi,n rm-1 LAmi,n exp -(Kminxmi) -B m-i,n exp (Km-1,nxm-i)]-(PmKm,nlrm)[Am,n exp (Km,nxm-i) -B m, n exp (Km, nxm-i)=0 (15) Am-i, n exp (Kmi,nXmi)+Bmi,n exp (Kmi,nXmi) - A m, n exp (Km,nXmi) - Bm, n exp (Km, nxm-i) = Cm-1, n (16) where i =1, 2,..., M. In addition, we need to impose the conditions that the fields in the regions x < 0 and x > xmi are finite. This tells us that A0,n=BM,n=0 (17) The linear systems of Eqs. (15), (16) and (17) are adequate to solve for the unknown coefficients Am, and Bm,n in terms of the coefficient Cmi,n. The latter can be obtained explicitly as follows. First of all, we note that 1m=10 sinh rm(h-z)/sinh T'mh and thus, according to (14), we can write Cm-1,n=I2(Hm-Rm-i) Sin dz

5 Magneto-Telluric Fields for a Segmented Overburden Specialization to the Three-Media Case In the interests of saving space, we will not pursue the general case here. Instead, we specialize to the case of M=2 as illustrated in Fig. l b. The explicit forms for the relevant coefficients are now A0,n=0 A1,n= -C0, nmun(1+nvn)exp (Ln)+nC1,nNVn(1-MUn) (1+MUn)(1+NVn) exp (L)-(1-Nrun)(1-NVn) exp (-Ln) A2,n=[A1,n exp (-L)+B1,n (19) exp (L)- nc1 exp (VnLn) (20) B0,n=Al,n+B1,n+Co,n B1m= C1, xnvn(1 + MUn)- C0MUn(1- NV) exp (-Ln) (1+MUn)(1+NV) exp (L) n-(1-mun)(1- NVn) exp (-Ln) (21) B2,=0 C0,m= and Cln= where -i27rnh1(1-m-1) (r2n2+ihi)(ir2n2+ih0) -i2rnh1(jt'-1) (7r2n2+tHi)(7r2n2+1H2) Unr2n2+Z Ha ar2n2+ih21 1/2 Vn= Ln=(2r2n2+iH31)1/2X1, 7c2n2+iH22 r2n2+ih21 X1=x1/h, M=Ql/0, N=1/2, H1=(o1p1w)1/2h, H0=(ioiiow)2h 1/= Hl/(M)1/2 H2=(o'2p2w)12h/Hl/(N)1/2 M=Y21/Y20=Mi1/p0, and N=Y22/Y22=Np1/p2. 1/2 (22) (23) impedance To present the results in a meaningful form, we define an actual surface Zm as follows where Zm= Emx(X, 0)ipmO)C 1m(x, 0) Ymry (for m=0, 1, 2) (24) Sm=coth ll/2hm-1 N7cn[Amn exp [-(r2n2+ihm)1/2x] +Bm,n exp [(ir2n2+ih)'2x] (ZS) and X=x/h. Of course (25) would also apply to M>3. Here Sm can be described as the normalized surface impedance of the structure since the quantity ipmw/rm is the intrinsic or characteristic impedance of the m'th medium.

6 454 J.R. WAIT and K.P. SPIES Fig. 2a. Amplitude of the normalized surface impedance S as a function of the normalized depth H for a homogeneous overburden with a resistive basement. Fig. 2b. Phase of the normalized surface impedance S for a homogeneous overburden. If we had naively ignored the effect of the vertical contacts at the outset, the deduced value of Sm would be merely the coth term on the right hand side of (25). The corresponding value of Zm is then analogous to the input impedance of an open circuit lossy transmission line of length h and propagation constant Ym. It is of particular interest to know how Sm actually differs from coth i2hm as a function of the normalized lateral dimension X or x/h. A computer program was written to evaluate the indicated summations in (25). We illustrate here some typical results for the case where the magnetic permeability has the same value in each layer. Then, of course, M =M=a1/Q and N= N= 61f o 2. But first of all, we present plots of the amplitude and phase

7 Magneto-Telluric Fields for a Segmented Overburden 455 Fig. 3a. Amplitude of S as a function of distance for a three section overburden for M=N=0.1. Dashed lines indicate levels when vertical contact effects are ignored. Fig. 3b. Phase of S for three section overburden for M=N=0.1. Dashed lines indicate levels when vertical contact effects are ignored. of S= Goth i12h as a function of H, in Figs. 2a and 2b, respectively. As indicated, this corresponds strictly to the normalized surface impedance for the case where Q0=61=62=6. As it should, JS) approaches 1 and phase of S approaches zero as H tends to infinity. Actually, if H exceeds about 4 the function S is indistinguishable from 1. Thus the situation of interest corresponds to values of H0, H1 and H2 somewhat less than 10. In Figs. 3a and 3b we show the amplitude and phase of S, as a function of X=x /h, for the case M=N= 0.1 and for values of H1= (a1p0w)2h equal to 1, 2 and 5. Here we also choose X1= 1 corresponding to x1= h. The broken curves correspond to the simple approximation coth i2hm for the respective regions

8 456 J.R. WAIT and K.P. SPIES Fig. 4a. Amplitude of S as a function of distance for a three section overburden for M=N= 10. Dashed lines indicate levels when vertical contact effects are ignored. Fig. 4b. Phase of S for three section overburden for M=N=10. Dashed lines indicate levels when vertical contact effects are ignored. m=0, 1 and 2. (Note we drop the subscript on Sm here and in what follows.) The curves show clearly that the local surface impedance is only crudely approximated by the uniform slab model. In fact, SI is discontinuous at the vertical contacts. The latter is really a consequence of the fact that Ex is Note: Numerical results of S and phase of S in tabular form are available on request from the authors.

9 Magneto-Telluric Fields for a Segmented Overburden 457 Fig. 5a. Amplitude of S as a function of distance for a three section overburden for M=1 1/N=101/2. Dashed lines indicate levels when vertical contact effects are ignored. Fig. Sb. Phase of S for three section overburden for M=1/N=101/2. Dashed lines indicate levels when vertical contact effects are ignored. discontinuous if we enforce the continuity of the normal current flow at the vertical interfaces. The phase of S, however, is continuous. Similar conclusions apply to the curves in Figs. 4a and 4b where now M=N= 10 so the end sections are relatively poor conductors. Then, in Figs. 5a and Sb, we choose M=1/N=101/2 so that relative conductivities are decreasing from the left segment to the right segment. 5. Concluding Remarks Probably the most important conclusion from this work is that the local surface impedance, for such a segmented overburden model, must allow for the lateral changes of the structure. Of course, if the mid-section were chosen to

10 458 J.R. WAIT and K.P. SPIES be sufficiently wide, the perturbations would be localized to the regions near the vertical contacts. REFERENCES D'ERCEVILLE, I. and G. KUNETZ, Some observations regarding naturally occurring electromagnetic fields in applied geophysics, Geophysics, 27, , 1962, GEYER, R.G., The effect of dipping contact on the behavior of the electromagnetic field, Geophysics, 37, , NEDOMA, J. and 0. PRAUS, A contribution to the numerical treatment of the electromagnetic field (H polarization) in horizontally non-homogeneous models of the earth, Studia Geophysica et Geodaetica,16, 69-76, RANKIN, D., The magneto-telluric effect on a dike, Geophysics, 27, , REDDY, I.K. and D. RANKIN, Magnetotelluric response of a two dimensional sloping contact by the finite element method, Pure and Applied Geophysics, 105, , WEAVER, J.T., The electromagnetic field within a discontinuous conductor with reference to geomagnetic micropulsations near a coast-line, Can. J. Phys., 41, , 1963.