Mathematical Inverse Problem of Magnetic Field from Exponentially Varying Conductive Ground

Transcription

1 Applied Mathematical Sciences, Vol. 6, 212, no. 113, Mathematical Inverse Problem of Magnetic Field from Exponentially Varying Conductive Ground Warin Sripanya Faculty of Science and Technology, Nakhon Pathom Rajabhat University Nakhon Pathom 73, Thailand Suabsagun Yooyuanyong Department of Mathematics, Faculty of Science Silpakorn University, Nakhon Pathom 73, Thailand Abstract We derive an analytical solution of the steady state magnetic field due to a direct current source on a multilayered earth with a layer having exponentially varying conductivity. Our variation in conductivity is realistic and can be generalized to all cases of exponential profiles. The Hankel transform is introduced to our problem and analytical result is obtained. Our solution is achieved by solving a boundary value problem in the wave number domain and then transforming the solution back to the spatial domain. The effects of magnetic fields are plotted and compared to show the behavior in response to different ground structures at many depths while some parameters are approximately given. The curves of magnetic fields show some significance to the variation of conductivity. An inverse problem via the use of the Levenberg-Marquardt optimization technique is introduced for finding the conductivity parameters of the ground. The optimal result of our model is close to the true value with percentage errors of our two conductivity parameters less than.9% and 1.7%. Mathematics Subject Classification: 86A25 Keywords: inverse problem, magnetic field, direct current, exponentially varying conductivity Corresponding author.

2 564 W. Sripanya and S. Yooyuanyong 1 Introduction Many authors have investigated the nature of the resistivity response resulting from a direct current source on a heterogeneous ground whose electrical conductivity varies exponentially with depth. Paul and Banerjee [9] studied the problem of computing apparent resistivity for a half-space model. Similar results, but for a layered earth, were reported by Banerjee et al. [2], Kim and Lee [7], Stoyer and Wait [12]. Throughout these investigations, the electrical conductivity was assumed, for simplicity, to be exponentially dependent upon depth, denoted by σ (z) = α exp (βz), where α and β are the parameters that define the conductivity profile. Unfortunately, these works are not sufficiently general about the variation of conductivity to be used for many applications. This approach is realistic only for a very short range of depth. Fortuitously, Yooyuanyong [14] introduced a new approach for enhanced investigation of a varying conductive ground structure. The general form of the variation of conductivity was developed and defined by σ (z) = σ + (σ 1 σ ) exp ( βz), where β, σ and σ 1 are positive real number parameters that describe the conductivity profile. In this approach, the electrical conductivity varies exponentially and tends to a basement value σ as z tends to infinity. Chumchob [6] used this conductivity variation to study the problem of computing electromagnetic response for a multilayered earth structure. In this paper, the electrical exploration method based on the measurement of static magnetic fields associated with noninductive current flow (Sripanya and Yooyuanyong [11]) is applied to investigate the structure of earth. We develop an analytical solution of the steady state magnetic field for the problem of a horizontally stratified layered earth with a layer having exponentially varying conductivity presented by Yooyuanyong [14]. The Hankel transform is introduced to our problem and analytical result is obtained. The inversion process, using the Levenberg-Marquardt algorithm, is conducted to estimate the conductivity parameters of the ground. 2 Model and Basic Equations In our geometric model, a point source of direct current I is located at the interface between two half-spaces. The half-space above the interface (z < ) is the region of air with conductivity approximately equal to zero, whereas the half-space below the interface (z > ) is an n-layered horizontally stratified earth with depths to the layers h 1, h 2,..., h n 1 (the lowermost layer extending to infinity) measured from the ground surface, where n 2 is an integer. Each layer has conductivity as a function of depth, i.e., σ k (z) for layer 1 k n.

3 Mathematical inverse problem of magnetic field Magnetic Field Resulting from a DC Source The general steady state Maxwell s equations in the frequency domain (Chen and Oldenburg [4, 5]) can be used to determine the magnetic field for this problem, namely E =, H = σe, (1) where E is the vector electric field, H is the vector magnetic field and σ is the conductivity of the medium. Since the problem is axisymmetric and H has only the azimuthal component in cylindrical coordinates, for simplicity, we use H to represent the azimuthal component in the following derivations, and we now have 2 H z 2 + σ z ( ) 1 H σ z + 2 H r r H r 1 H =. (2) r2 The Hankel transform (Ali and Kalla [1]) is introduced and defined by and H (λ, z) = H (r, z) = λrh (r, z) J 1 (λr) dr (3) H (λ, z) J 1 (λr) dλ, (4) where J 1 is the Bessel function of the first kind of order one. Taking the transformation on both sides of equation (2), we obtain 2 H z 2 + σ z ( ) 1 H σ z λ2 H =. (5) 2.2 Boundary Conditions The magnetic field in each layer can be obtained by taking the inverse Hankel transform to the solution of equation (5), which satisfies the following boundary conditions: 1. The azimuthal component of the magnetic field needs to be continuous on each of the boundary planes in the earth, i.e., lim z h k H k = lim z h + k H k+1. (6) 2. The radial component of the electric field needs to be continuous on each of the boundary planes in the earth, i.e., lim z h k 1 H k σ k z = lim z h + k 1 H k+1. (7) σ k+1 z

4 5642 W. Sripanya and S. Yooyuanyong 3. The vertical component of the current density must be zero at the free surface except in an infinitesimal neighborhood around a current source (Kim and Lee [7]), i.e., 1 r r (rh 1) = I z= 2π δ (r) r. (8) 4. The azimuthal component of the magnetic field tends to zero as z tends to infinity, i.e., H n =. (9) lim z 3 Response of a Layer Having Exponentially Varying Conductivity For an exponentially varying conductive layer k, where 1 k n and n 2, the variation of conductivity is denoted by σ k (z) = c k + (a k c k ) e b kz, (1) where a k, b k and c k are positive real numbers. Hence, the equation for the magnetic field in each layer can be simplified by substituting equation (1) into (5) and we obtain 2 Hk + b k (σ k c k ) z 2 σ k The solution to the above equation is H k (λ, z) = ψ k ( Ak e ζ 1,k 2 F 1 (α 1,k, β 1,k ; γ 1,k ; ξ k ) H k z λ2 Hk =. (11) + B k e ζ 2,k 2 F 1 (α 2,k, β 2,k ; γ 2,k ; ξ k ) ), (12) where ϱ k α i,k = 3 2 λ + ( 1) i, β i,k = 3 b k 2b k 2 + λ + ( 1) i ϱ k, b k 2b k γ i,k = 1 + ( 1) i ϱ k, ϱ k = b 2k b + 4λ2, ξ k = c ke b kz, k c k a k ψ k = ( ( a k + c k e b k z 1 )) ( ) 2 z, ζi,k = ( 1) i ϱ k b k 2 (13) and 2 F 1 is the ordinary hypergeometric function. The unknown coefficients A k and B k are arbitrary constants, which can be determined by using the

5 Mathematical inverse problem of magnetic field 5643 propagator matrix technique applied to the above boundary conditions (see Sripanya and Yooyuanyong [11]). Thus, the magnetic field in layer k is H k (r, z) = ψ k ( Ak e ζ 1,k 2 F 1 (α 1,k, β 1,k ; γ 1,k ; ξ k ) + B k e ζ 2,k 2 F 1 (α 2,k, β 2,k ; γ 2,k ; ξ k ) ) J 1 (λr) dλ. (14) In the case of an exponentially varying conductive half-space, the magnetic field can be written as H (r, z) = I 2π 2F 1 (α, β; γ; ξ) ψ e ζ J 1 (λr) dλ, (15) 2F 1 (α, β; γ; ξ ) ψ where all the variables are given as above without subscript k while i = 1, and the subscript refers to z =. 4 Numerical Experiments In our numerical experiments, we calculate the magnetic fields due to a direct current source on two types of electrically conductive earth structures. Both of the example ground models are the heterogeneous conductive half-spaces. The first model has exponentially increasing conductivity with depth, whereas the second model has exponentially decreasing conductivity. The values of the model parameters are given in Table 1. Chave s algorithm [3] is used for numerically calculating the inverse Hankel transform of the magnetic field solutions. The special functions are computed by using the Numerical Recipes source codes (Press et al. [1]). The results from both models are plotted to show the behavior of magnetic fields against source-receiver spacing r at different depths z =,.5, 1,..., 1 metres as shown in Figure 1. Table 1: Model parameters used in our sample tests. Parameters Model a ( S m 1) b ( m 1) c ( S m 1)

6 5644 W. Sripanya and S. Yooyuanyong.16 Model 1.14 Magnetic Field Intensity (A/m) Source Receiver Spacing (m).16 Model 2.14 Magnetic Field Intensity (A/m) Source Receiver Spacing (m) Figure 1: Behavior of magnetic fields from synthesis models in our sample tests. 5 Inversion Process In our inverse model example, we simulate the reflection of magnetic radiation data from our forward model of practical interest. The magnetic fields are generated by the forward problem of the first model in our sample tests. Random errors up to 3% are superimposed on the scaled magnetic fields to simulate the set of real data. The iterative procedure using the Levenberg-Marquardt method (Press et al. [1]) is applied to estimate the model parameters of conductivity variation. The model parameter a is a conductivity of the earth s surface, which can be assumed to be known from the measurement. We start the iterative process to find the values of the conductivity parameters with initial guess values b =.1 m 1 and c =.1 S m 1. The inversion method leads to the optimal values of the parameters b and c in which their percentage errors are less than only 1.7% and.9%, respectively, after using 138 iterations. The graphs of the true and estimated conductivity models are plotted as shown in Figure 2.

7 Mathematical inverse problem of magnetic field Conductivity (S/m) True Model Estimated Model Initial Guess Model Depth (m) Figure 2: Graphs of conductivity σ against depth z for our inverse model example. 6 Discussions and Conclusions Analytical solution of the steady state magnetic field for exponentially conductive ground profile is derived. Our variation in conductivity is realistic and can be generalized to all cases of exponential profiles. The effects of magnetic fields are plotted and compared to show the behavior in response to different ground structures at many depths (see Figure 1). The magnetic curves of the model having exponentially increasing conductivity with depth (Model 1) are quite different from the model having exponentially decreasing conductivity (Model 2). The magnitude of magnetic field from the first model is much higher than the magnetic field from the second model at the same depth and source-receiver spacing. We observe that if the conductive ground has a high electrical conductivity, it will lead to the large sized magnitude of magnetic field. This means that the differences of curves for our models are depended on the variation of conductivity. An inverse problem via the use of an optimization technique is introduced for finding the conductivity parameters of the ground. The model of a simple case for the ground structure is used to investigate the conductivity profile. The iterative procedure using the Levenberg-Marquardt method is applied to estimate the model parameters of conductivity variation. The optimal result of our model converges to the true value with percentage errors of b and c less than only 1.7% and.9%, respectively, after using 138 iterations. The graphs of the true and estimated conductivity models are plotted as shown in Figure 2. We see that the graph of the estimated model is close to the true model. The inversion method leads to very good result and has high speed of convergence. This illustrates the advantage in using Levenberg-Marquardt method which gives the result much better than using another method of inversion (e.g.,

8 5646 W. Sripanya and S. Yooyuanyong Oldenburg [8], Vozoff and Jupp [13]). References [1] I. Ali and S. Kalla, A generalized Hankel transform and its use for solving certain partial differential equations, J. Austral. Math. Soc. Ser. B, 41 (1999), [2] B. Banerjee, B.J. Sengupta and B.P. Pal, Apparent resistivity of a multilayered earth with a layer having exponentiality varying conductivity, Geophys. Prosp., 28 (198), [3] A.D. Chave, Numerical integration of related Hankel transforms by quadrature and continued fraction expansion, Geophysics, 48 (1983), [4] J. Chen and D.W. Oldenburg, Magnetic and electrical fields of direct currents in a layered earth, Expl. Geophys., 35 (24), [5] J. Chen and D.W. Oldenburg, A new formula to compute apparent resistivities from marine magnetometric resistivity data, Geophysics, 71 (26), [6] N. Chumchob, Mathematical modelling of electromagnetic response on a multilayered earth with a layer having exponentially varying conductivity, Thai J. Math., 2 (24), [7] H.-S. Kim and K. Lee, Response of a multilayered earth with layers having exponentially varying resistivities, Geophysics, 61 (1996), [8] D.W. Oldenburg, The interpretation of direct current resistivity measurements, Geophysics, 43 (1978), [9] M.K. Paul and B. Banerjee, Electrical potentials due to a point source upon models of continuously varying conductivity, Pure Appl. Geophys., 8 (197), [1] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran 77: The Art of Scientific Computing, Fortran Numerical Recipes Vol. 1, 2nd ed., Cambridge Univ. Press, [11] W. Sripanya and S. Yooyuanyong, Mathematical inverse problem of magnetic field from heterogeneous earth structures, Far East J. Math. Sci. (to appear).

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