Analytical solution for the electric potential in arbitrary anisotropic layered media applying the set of Hankel transforms of integer order

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1 Geophysical Prospecting, 2006, 54, Analytical solution for the electric potential in arbitrary anisotropic layered edia applying the set of Hankel transfors of integer order E. Pervago 1,A.Mousatov 1 and V. Shevnin 1 1 Instituto Mexicano del Petróleo, México Received Noveber 2004, revision accepted March 2006 ABSTRACT The analytical solution and algorith for siulating the electric potential in an arbitrarily anisotropic ultilayered ediu produced by a point DC source is here proposed. The solution is presented as a cobination of Hankel transfors of integer order and Fourier transfors based on the analytical recurrent equations obtained for the potential spectru. For the conversion of the potential spectru into the space doain, we have applied the algorith of the Fast Fourier Transfor for logarithically spaced points. A coparison of the odelling results with the power-series solution for two-layered anisotropic structures deonstrated the high accuracy and coputing-tie efficiency of the ethod proposed. The results of the apparent-resistivity calculation for both traditional pole-pole and tensor arrays above three-layered sequence with an aziuthally anisotropic second layer are presented. The nuerical siulations show that both arrays have the sae sensitivity to the anisotropy paraeters. This sensitivity depends significantly on the resistivity ratio between anisotropic and adjacent layers and increases for the odels with a conductive second layer. INTRODUCTION The geoelectrical study of heterogeneous anisotropic edia provides iportant additional inforation about geological structures and petrophysical rock properties. The various kinds of resistivity anisotropy in layered forations are due to the presence of fractured systes or cross-bedded sedients. The deterination of anisotropy paraeters in heterogeneous anisotropic edia is a coplex proble that requires fast and accurate ethods for siulating the electric potential in such an environent. The general ethod for solving the electroagnetic proble in an anisotropic layered ediu uses 2D Fourier transfors 2D FT in the boundary plane. Gurevich 1975 proposed this approach for siulating vertical electrical sounding on the surface of a half-space consisting of layers with arbitrarily orientated anisotropy axes. By applying 2D FTs, Le Masne and Vasseur 1981 coputed the electroagnetic field of sources on the surface of a hoogeneous conductive half-space with a horizontal anisotropy axis. The schee for the nuerical coputation of the electric potential fro a point source in a layered earth with arbitrary anisotropy was developed by Das 1995 and extended by Li 1996 for the case of arbitrary sources. The 2D FT approach was used by Li and Pedersen 1991 to deterine the electroagnetic response of a horizontal electric dipole on the surface of an aziuthally anisotropic ediu. The current density and agnetic field for a direct current source in a layered conductor with an arbitrary anisotropy was calculated by Yin and Weidelt Yin and Maurer 2001 considered electroagnetic induction in a layered earth with arbitrary anisotropy. E-ail: epervago@ip.x C 2006 European Association of Geoscientists & Engineers 651

2 652 E. Pervago, A. Mousatov and V. Shevnin Another approach to odelling an aziuthal resistivity sounding over a two-layered structure with arbitrarily orientated transverse anisotropy was applied by Pervago 1998 and Bolshakov et al In this case, the solution of the equation for an aziuthally anisotropic ediu was represented by a set of Hankel transfors of integer order. Such a solution corresponds to the angular haronic decoposition of the aziuthal resistivity diagras. The analysis of the haronic coefficients allows the heterogeneity and anisotropy effects to be separated Pervago et al Additionally, by using the decoposition characteristics, the sensitivity of electrical surface arrays and well-logging sondes to anisotropy was estiated by Bolshakov et al and Mousatov and Pervago Here, we propose an analytical solution and coputation technique for calculating the electric potential of a point DC source in a ediu coposed of layers with three-axial, arbitrarily orientated anisotropy. The solution obtained is expressed by a set of Hankel transfors of integer order. We have found recurrent expressions for the coefficients of the electric-potential spectru in each layer. In order to convert the potential spectru to the space doain, we applied the algorith of the Fast Fourier Bessel Transfor for logarithically spaced points FFTLog designed by Hailton For the solution and algorith verifications, the calculation results were copared with the data published by Bolshakov et al Using the technique developed, we presented the responses of the traditional aziuthal resistivity surveys and advanced tensor array Mousatov et al. 2000, 2002, 2003 above a three-layered anisotropic odel. ELECTRIC POTENTIAL IN AN ANISOTROPIC LAYERED MEDIUM The odel consists of parallel hoogeneous layers with three-axial, arbitrarily orientated anisotropy. We assue that the z-axis of the Cartesian coordinate syste is perpendicular to layer boundaries and is orientated downwards. The electrical conductivity of the th layer is defined by the tensor σ.inananisotropic hoogeneous region, the electric potential U x, y, z produced by a point source of direct current I with coordinates x A, x A, y A satisfies the divergence equation, divσ gradu = Iδx x A, y y A, z z A 1 The solution of an inhoogeneous equation is the su of a general solution for a hoogeneous equation and the source function. Applying the 2D Fourier transfor in the horizontal xy-plane to the hoogeneous equation corresponding to 1, we obtain the hoogeneous equation in the spectral doain with spatial frequencies, k x and k y : σ 2 + 2i k z 2 x σ xz + k yσ yz z k 2 x σ xx + k2 y σ yy + 2k xk y σxy U = 0 2 Here, the electric potential U x, y, z and its spectru Ũ k x, k y, z are the Fourier transfor pair, k x, k y, z = 1 Ux, y, ze ikxx+ky y dxdy, 3 2π U x, y, z = 1 2π Ũ k x, k y, ze ikxx+ky y dk x dk y 4 In order to find the solution of 2 in the for, Ũz = e vz,wefirst obtain the quadratic equation for the paraeter v, σ v2 + 2i k x σ xz + k yσ yz v k 2 x σ xx + k2 y σ yy + 2k xk y σ xy = 0, 5 which has the following roots: v 1, = 1 [ + k σ 2 x σ xx σ ] i k x σxz + k yσyz, v 2, = 1 [ k σ 2 x σ xx σ ] i k x σxz + k yσyz. σ xz σ xz k x k y σxy σ σ xz σ yz + k 2 y σ yy σ σyz k x k y σxy σ σ xz σ yz + k 2 y σ yy σ σyz 6

3 Electric potential in anisotropic layered edia 653 Since v 2, k x, k y = v 1, k x, k y where v is the coplex conjugate of v, the coon solution of 2 with coplex coefficients can be written as the su of refracted and reflected parts, U k x, k y, z = A k x, k y e vz + B k x, k y e v z, 7 This solution can also be written in the cylindrical coordinate syste r, ϕ, z as a series of Hankel Fourier Bessel transfors of integer order. Transforing to cylindrical coordinates by substituting, x = r cos ϕ, y = r sin ϕ, k x = k r sin k ϕ, k y = k r cos k ϕ, 4 can be rewritten in the for, U r,ϕ,z = 1 π k r k r sin k ϕ, k r cos k ϕ, ze ikr r sinkϕ ϕ dk r dk ϕ 8 2π π 0 Then, using the Jacobi Anger expansion for Bessel functions of the first kind of integer order J n e it sin p = J n te inp, n= 9 and introducing the odified spectru, Ũ k r, k ϕ, z = k r Ũ k r sin k ϕ, k r cos k ϕ, z, 10 we can rewrite 8 as series of the Hankel transfor, [ 1 π ] U r,ϕ,z = e inϕ J n k r, r k r, k ϕ, ze inkϕ dk ϕ dk r. 11 2π n= 0 Finally, this equation can be written in the copact for, U r,ϕ,z = C n r, zeinϕ. n= π The coplex haronic coefficient Cn r, z isdefined by the Fourier and Hankel transfors with spatial frequencies k r, k ϕ as [ 1 π ] C n r, z = J n k r, r k r, k ϕ, ze inkϕ dk ϕ dk r. 13 2π 0 π The application of the Hankel transfor allows us to avoid the singularity of the potential spectru of a point DC source at spatial frequency zero, if the 2D Fourier ethod is used. Additionally, in this case, the potential spectru is discrete and its haronic coefficients decrease rapidly when the haronic nuber increases. If the ediu is isotropic, the odified spectru does not depend on the spatial frequency k ϕ, i.e. U k r, k ϕ, z = U k r, z, and the haronic coefficients Cn r, z becoe equal to zero for all n 0 due to the property of the Fourier integral, 1 π k r, ze inkϕ dk ϕ ={ Uk r, z n = 0, 0, n 0. 2π π Thus, the potential U r, z isequal to the zeroth haronic coefficient C 0 r, z equation 12, and 11 is converted into the well-known solution in the for of the Hankel integral, U r, z = 0 J 0 k r r U k r, z dk r. 12

4 654 E. Pervago, A. Mousatov and V. Shevnin Cobining 7 and 10, the odified spectru function U k r, k ϕ, z intheth layer can be written as k r, k ϕ, z = Ak r, k ϕ e wkr z + B k r, k ϕ e wkr z, 14 where w k ϕ = α iβ, w k ϕ = α + iβ, σxx σ σ xz sin 2 k ϕ + σ σ xy σ xz σ yz sin 2k ϕ + σ yy σ σyz cos 2 k ϕ α k ϕ =, 17 σ β k ϕ = σ xz sin k ϕ + σyz cos k ϕ. σ 18 The vertical coponent of the current density vector j z, in the th layer can be expressed in ters of the conductivity tensor σ and the potential U as j z. x, y, z = σ xz x U x, y, z + σ yz y U x, y, z + σ z U x, y, z. 19 After a Fourier transforation of 19, the spectru j z, k x, k y, z has the following for: j z, k x, k y, z = σ xz k x + σ yz k y Ũ k x, k y, z + σ zũk x, k y, z. 20 Using the odified potential spectru 10 in cylindrical coordinates, we obtain the Hankel spectru for the vertical coponent of the current density tensor, j z, k r, k ϕ, z = k r σ xz sin k ϕ + σ yz ϕ cos k k r, k ϕ, z + σ k r, k ϕ, z. 21 z Substituting 14 for k r, k ϕ, z into 21, the spectru j z, k r, k ϕ, z can be written as j z, k r, k ϕ, z = k r c A k r, k ϕ e wkr z + d B k r, k ϕ e w kr z, 22 where c k ϕ = γ k ϕ + σ w k ϕ, d k ϕ = γ k ϕ σ w k ϕ, γ k ϕ = σ xz sin k ϕ + σ yz cos k ϕ. The unknown coefficients, k r, k ϕ, z = +1 k r, k ϕ, z, j z, k r, k ϕ, z = j z,+1 k r, k ϕ, z. A k r, k ϕ and B k r, k ϕ, can be found using the boundary conditions, Using the conditions far fro the source point, i.e. U 0 k r, k ϕ, z 0, U N k r, k ϕ, z 0, z, z +, 28 29

5 Electric potential in anisotropic layered edia 655 as well as the M k r, k ϕ, z continuity and the j z,m k r, k ϕ, z discontinuity on the plane z A of a source placed in the Mth layer when ε tends to zero ε 0, we have M k r, k ϕ, z A + ε M k r, k ϕ, z A ε = 0 M k r, k ϕ, z A + ε 0 M k r, k ϕ, z A ε, 30 j z,m k r, k ϕ, z A + ε j z,m k r, k ϕ, z A ε = j 0 z,m k r, k ϕ, z A + ε j 0 z,m k r, k ϕ, z A ε. 31 The source potential spectru 0 M k r, k ϕ, z and the spectru of the vertical coponent of the source current density j 0 z,m k r, k ϕ, z are U 0 M k r, k ϕ, z = I 4πα M σ M e kr iβ Mz+α M z and j 0 z,m k r, k ϕ, z = Ik r ik r σ M sin k 4πα M σxz M ϕ + σ yz M cos k σ M ϕ + iβ M + α M signz e kr iβ Mz+α M z. 33 By substituting 32 and 33 into 30 and 31, we obtain the following conditions on the source plane: 32 U M k r, k ϕ, z A + ε U M k r, k ϕ, z A ε = 0, 34 j z,m k r, k ϕ, z A + ε j z,m k r, k ϕ, z A ε = Ik r 2π. 35 On solving this syste of linear equations, we obtain the analytical recurrent equations for the coefficients, A k r, k ϕ and B k r, k ϕ see Appendix. After deterining the quantities, A k r, k ϕ and B k r, k ϕ, we calculate the spectru of the electric potential 14 and then the coplex haronic coefficients using 13. Because the internal integral in 13 corresponds to the Fourier transfor, D n k r, z = 1 π k r, k ϕ, z e inkϕ dk ϕ, 36 2π π it is convenient to perfor its evaluation using the FFT ethod. The external integral of 12 represents the Hankel transfor of integer order, C n r, z = D n k r, zj n k r rdk r In geophysical applications, the calculation of the Hankel integral is presented with a specific proble as a liited nuber of points has to cover a relative distance range of Inthe case of linearly spaced points noral scale, the atrices of transfors are large To avoid this proble, we applied the algorith of the Fast Fourier Transfor in logarithically spaced points FFTLog, developed by Hailton Siilarly to the noral FFT, the FFTlog ethod gives the exact Fourier transfor of a discrete function that is periodic and uniforly spaced in logarithic space Talan The FFTLog calculates the fast Fourier and Hankel Fourier Bessel transfor of arbitrary order for a relative distance range of several orders with a liited nuber of points. The algorith proposed by Hailton 2000 converts the coputation of the Hankel transfor into two FFTs perfored sequentially, and it iproves the accuracy and speed of coputation. SIMULATION EXAMPLE Based on the analytical solution and the calculation algoriths presented above, we siulated apparent-resistivity curves for the vertical electrical soundings in edia coposed of transversely anisotropic layers. These edia are of interest for practical applications and correspond to geological structures fored by fractured rocks or intercalation of electrically contrasting thin beds. In this case, we assue that each layer is aziuthally anisotropic with the anisotropy axis perpendicular to the vertical

6 656 E. Pervago, A. Mousatov and V. Shevnin axis z.inthe local coordinate syste that coincides with the anisotropy axes, the conductivity tensor of the th layer is given by ση 0 0 ˆσ = 0 στ στ 38 If the Cartesian coordinate syste is used throughout the odel, the conductivity tensor σ can be written as σ = R T ˆσ R, 39 where cos β sin β 0 R = sin β cos β is the rotation atrix, 40 β denotes the rotation angle around the z-axis. All siulations were perfored for the relative distance of with a total of points for the electric potential spectra. To verify the technique described above, we calculated the haronics of the apparent resistivity using our approach and the analytic solution given by power series obtained for the special case of the two-layered aziuthally anisotropic odel Bolshakov et al In this odel, the anisotropy axes in each layer are arbitrarily orientated in the boundary plane and are described by the conductivity tensor 39. Each layer is characterized by the anisotropy coefficient λ and the ean resistivity ρ = σ τ σ η 1/2. The result of observations above an anisotropic ediu is usually given as the apparent resistivity ρ a, which can also be written as the su of haronics, ρ a r,ϕ,z = C ρ n r, zeinϕ, n= 41 where C ρ n r, z isthe coplex aplitude of the nth haronic. It is convenient to copare the coefficients C ρ n r, z ofthe apparent-resistivity haronics because the zeroth haronic aplitude is uch higher than the aplitude of any other haronic and a coparison of the apparent resistivities haronic sus will not be relevant Fig. 1. The C ρ n r, z graphs are plotted as a function of the noralized distance R/h where R is the array spacing and h is the upper layer thickness. These graphs deonstrate good agreeent between the two analytical solutions and indicate the high accuracy of siulation. The relative error in this case is less than 10 8 for all haronics shown. We siulated the apparent resistivity ρ a and apparent anisotropy coefficient λ a for a traditional aziuthal survey with a pole-pole array on the surface of three-layered structures for different odel paraeters and observation aziuths Fig. 2. In this odel, the second layer is transversely anisotropic with the anisotropy axis perpendicular to the vertical axis z aziuthally anisotropic layer. The apparent-anisotropy coefficient λ a was calculated as the ratio of the apparent resistivities easured perpendicular to and along the anisotropy axis. When the ean resistivity of the anisotropic second layer is higher than the resistivity of adjacent layers ρ 1 = ρ 3 = 1, ρ 2 = 20, the apparent-resistivity coefficient does not exceed the value 1.1, even for the true value λ 2 = 4 and thickness h 2 = 2h 1. For the conductive anisotropic layer ρ 1 = ρ 3 = 20, ρ 2 = 1, the apparent-anisotropy coefficient takes high enough values, even for a sall relative thickness of the second layer, h 2 = 0.2h 1.Inthis case, the behaviour of the ρ a and λ a curves shows the feasibility of thin anisotropic-layer detection by perforing easureents on various aziuths the traditional technology of an array rotation around a reference point.

7 Electric potential in anisotropic layered edia 657 Figure 1 Coparison of the coefficientsc ρ n of the apparent-resistivity haronics calculated for the two-layered aziuthally anisotropic structure by the power-series solid line and by the technique presented circles. The curve index is the haronic nuber. Model paraeters are A: ρ = 1, 100, λ = 1, 2; B: ρ = 100, 1, λ = 1, 2. R/hours is the noralized spacing. Figure 2 The apparent resistivity ρ a A B and the apparent-anisotropy coefficients λ a C D for a pole-pole array above the three-layered structures with an aziuthally anisotropic second layer. Model paraeters are A C: ρ = 1, 20, 1, h 1 = h; B D: ρ = 20, 1, 20, h 1 = h. 1:λ = 1, 2, 1, h 2 = 0.2 h; 2:λ = 1, 2, 1, h 2 = 2 h; 3:λ = 1, 4, 1, h 2 = 0.2 h; 4:λ = 1, 4, 1, h 2 = 2 h. AM/h is the noralized spacing.

8 658 E. Pervago, A. Mousatov and V. Shevnin Figure 3 Schee of the current and easuring electrode distribution for the tensor array, TEN FD. Figure 4 The apparent ean resistivity ρ a A B and apparent-anisotropy coefficients λ a C D for the tensor array, TEN FD., above threelayered structures with an aziuthally anisotropic second layer. The odel paraeters correspond to those given in Figure 2. AO/h is the noralized spacing. Additionally, for the sae odels, we calculated the response of the tensor arrays, proposed by Mousatov et al. 2000, 2002, 2003, for the anisotropy paraeter deterination in heterogeneous edia. This array TEN FD isbased on the easureents of the first U x = U 2 U 1, U y = U 3 U 4 and second U yy = 2U 0 U 3 U 4 differences of the electric potential fro a point source Fig. 3. The tensor array provides the deterination of all anisotropy paraeters ean resistivity, anisotropy coefficient and aziuth of anisotropy axis by carrying out the observation in one arbitrary direction only, and it does not require the array rotation. The apparent ean resistivities and apparent-anisotropy coefficients λ a obtained by the TEN FD Fig. 4 are siilar to the paraeters easured by the conventional aziuthal survey. CONCLUSIONS The analytical solution for edia coposed of arbitrarily anisotropic layers was obtained. Based on this solution, we developed an algorith for the siulation of the electric field produced by a point DC source in an anisotropic ultilayered ediu.

9 Electric potential in anisotropic layered edia 659 Coparison of the odelling results with the power series solution published for two-layered anisotropic structures deonstrated the high accuracy and coputing-tie efficiency of the ethod proposed. The analytical solution obtained is expressed as a su of discrete angular haronics with rapidly decreasing coefficients when the haronic nuber increases. These haronic coefficients are given by a set of Hankel transfors of integer order. Application of the Hankel transfor allows us to avoid the singularity of the electric-potential spectru at the spatial frequency zero that appears in the 2D Fourier transfor. To calculate the Hankel transfor we applied the fast Fourier-Bessel transfor for logarithically distributed points, thus providing high accuracy of calculation and significantly reducing the coputation tie. This calculation algorith can be used to odel the electric field for surface and borehole arrays, designed to deterine the anisotropy paraeters in heterogeneous edia with plane or cylindrical layers. ACKNOWLEDGEMENTS The authors are grateful to Instituto Mexicano del Petróleo, where this study was carried out within the fraework of the Yaciientos Naturalente Fracturados progra. We thank Dr Pedro Anguiano Rojas for his valuable help and support. REFERENCES Bolshakov D.K., Modin I.N., Pervago E.V. and Shevnin V.A Separation of anisotropy and inhoogeneity influence by the spectral analysis of aziuthal resistivity diagras. 3rd EEGS-ES Meeting, Aarhus, Denark, Expanded Abstracts, Bolshakov D.K., Modin I.N., Pervago E.V. and Shevnin V.A Modelling and interpretation of aziuthal resistivity sounding over twolayered odel with arbitrary-oriented anisotropy in each layer. 60th EAGE Conference, Leipzig, Gerany, Extended Abstracts, P110. Das U.C Direct current electric potential coputation in an inhoogeneous and arbitrary anisotropic layered earth. Geophysical Prospecting 43, Gurevich Y.M On the theory of vertical electrical sounding in anisotropic edia. Physics of the Earth 7, , in Russian. Hailton A.J.S Uncorrelated odes of the nonlinear power spectru. Monthly Notices of the Royal Astronoical Society 312, Le Masne D. and Vasseur G Electroagnetic field of sources at the surface of a hoogeneous conducting half-space horizontal anisotropy: application to fissured edia. Geophysical Prospecting 29, Li P A new solution for the electric potential in an arbitrary anisotropic and inhoogeneous layered earth. 66th SEG Meeting, Denver, USA, Expanded Abstracts, Li X. and Pedersen L.B The electroagnetic response of an aziuthally anisotropic half-space. Geophysics 56, Mousatov A. and Pervago E Aziuthally anisotropy estiation with electrical sondes applying haronic decoposition. 72nd SEG Meeting, Salt Lake City, USA, Expanded Abstracts, Mousatov A., Pervago E. and Shevnin V New approach to resistivity anisotropic easureents. 70th SEG Meeting, Calgary, Canada, Expanded Abstracts, Mousatov A., Pervago E. and Shevnin V Anisotropy deterination in heterogeneous edia by tensor easureents of the electric field. 72nd SEG Meeting, Salt Lake City, USA, Expanded Abstracts, Mousatov A., Pervago E. and Shevnin V Arrays for tensor easureents of the electric field. Proceedings of SAGEEP, San Antonio, USA, Pp Pervago E Anisotropy and inhoogeneity influence on the results of vertical electrical soundings. PhD Thesis, Moscow State University. Pervago E., Mousatov A. and Shevnin V Joint influence of resistivity anisotropy and inhoogeneity on exaple of a single dipping interface between isotropic overburden and anisotropic baseent. Proceedings of the SAGEEP, Denver, USA, ERP 7. Talan J.D Nuerical Fourier and Bessel transfors in logarithic variables. Journal of Coputational Physics 29 1, October, Yin C. and Maurer H.-M Electroagnetic induction in layered earth with arbitrary anisotropy. Geophysics 66, Yin C. and Weidelt P Geoelectrical fields in layered earth with arbitrary anisotropy. Geophysics 64, APPENDIX For coputing convenience 14 and 22 can be rewritten in the following for Fig. 5: for the layers below the source point z z A, = M,..., N: U + k r, k ϕ, z = b + a + ewkr z z + e w kr z z, A1

10 660 E. Pervago, A. Mousatov and V. Shevnin Figure 5 Schee for calculating the coefficients a, a+, b, b+,inananisotropic ultilayered ediu. J + z, k r, k ϕ, z = k r b + c a + ewkr z z + d e w kr z z, A2 and for the layers above the source point z z A, = 0,..., M: k r, k ϕ, z = b a e w kr z z 1 + e wkr z z 1, A3 J z, k r, k ϕ, z = k r b d a e w kr z z 1 + c e wkr z z 1, A4 where a + = A / B e 2Rewkr z, b + = B e w kr z, a = B / A e 2Rewkr z 1, b = A e wkr z 1. A5 A6 A7 A8 In these equations, z 1 and z are the top and botto of the th layer. Using expressions A1 A4, the syste of equations and 34,35 can be rewritten as follows: a 0 = 0, = 1..,M, a 1e w 1 kr h 1 + e w 1k r h 1 = b a + 1, b 1 A9 A10 b 1 b + M d 1 a 1e w 1 kr h 1 + c 1 e w 1k r h 1 = b d a + c, A11 a + M ew Mk r z A z M + e w M kr z A z M b M a M e w M kr z A z M 1 + e w Mk r z A z M 1 = 0, A12

11 Electric potential in anisotropic layered edia 661 b + M c M a + M ew Mk r z A z M + d M e w M kr z A z M b M d M a M e w M kr z A z M 1 + c M e w Mk r z A z M 1 = I 2π, A13 and = M,..., N 1, a = b + +1 b + b + a + +1 e w +1k r h +1 + e w +1 kr h +1, A14 c a + + d = b + +1 c +1 a + +1 e w +1k r h +1 + d +1 e w +1 kr h +1, A15 a + N = 0 A16 where h = z z 1 is the th layer thickness. Starting fro the known coefficients, a 0 and a+ N,wecan find the coefficients a+, and a, sequentially towards the Mth layer, which contains the source Fig. 5: a = c 1 c + a 1 d 1 c e 2Rew 1k r h 1 c 1 d + a 1 d 1 d e 2Rew 1k r h 1, = 1,...,M, A17 a + = d d +1 + a + +1 d c +1 e 2Rew +1k r h +1 c d +1 + a + +1 c, = N 1,...,M. A18 c +1 e 2Rew +1k r h +1 The coefficients, b M and b+ M,intheMth layer, which contains the source, can be found based on A12 and A13: b M = I 1 + a + e 2Rew Mk r z M z A 2π c M d M a + e w Mk r z A z M 1, A19 M a M e 2Rew Mk r h M 1 b + M = I 1 + a e 2Rcw Mk r z A z M 1 2π c M d M a + e w M k r z M z A. A20 M a M e 2Rcw Mk r h M 1 Finally, the coefficients, b + M and b M, are expressed through the coefficients previously obtained fro the layers above and below the layer containing the source: b = 1 + a b +1 +1, = M 1,...,0, A21 a e w k r h + e w k r h b + = 1 + a + b+ 1 1, = M + 1,...,N. A22 a + e wkr h + e w k r h