ON SUBSECTION OF A POSITIVELY SKEWED CURVE MEDIUM OF EM ON A HALF SPACE EARTH MODEL

The objective of this paper is to study the structure of the earth by constructing a half-space earth model with a positively skewed curve medium. The subsection of the ground is used mathematically to find the electric fields in each layer of the ground. Numerical solutions are computed to show graphically the behaviors of the electric field. The responses of electric field are obliged by a positively skewed curve medium ground profile. AMS Subject Classification: 86A25


Introduction
There are diverse approaches used in exploration geophysics.The electromagnetic method is the most commonly used because it is much less cost than others.Besides, it responses best to a good electrical conductors at a shallow depth to as deep as many kilometers from the earth surface depending on frequency.Lee and Ignetik [3] considered the forward problem of the transient electromagnetic response of a half-space with an exponentially varying conduc-tivity profiles.They indicated out that the conductivity variation of the ground may sometimes be reasonably approximated by an exponential variation since soil salinity profiles frequently show monotonically increasing or decreasing salt conductivity of the ground.Yooyuanyong and Siew [4] gave the mathematical model of electromagnetic response of a disk beneath an exponentially varying conductive overburden.
The goal of this paper is to present mathematical model and techniques for studying the structure of the earth's surface layer.We consider the ground having the conductivity which is given by σ(z) = (σ 0 + z)e −bz/2 , where σ 0 a is positive constant, b is constant.The conductivity ground profile in this paper is unlike to the model used by Yooyuanyong and Siew [4] but similar to the model used by Haarsa and Pothat [6] with different method to approach the solution of electric field.

Formulation and Solution of the Problem
Firstly, we consider a cylindrical polar coordinates (r, θ, z) which is introduced with z > 0 and taken vertically positive downward.The origin under the center of the horizontal circular loop is used.The azimuthally symmetry gives merely electric field E θ , and magnetic fields H r ,and H z components.These field quantities are found to satisfy Maxwell's equations Hohmann and Raiche [2] in the form of equations and where i = √ −1 is an imaginary number, J s = aI(ω)δ(r − a)δ(z + h)/r is the source current density, ω is the angular frequency, δ is the delta function, σ(z) is the electrical conductivity of medium depending on depth only, ε is the electric permittivity of medium, µ is the magnetic permeability of medium, and I(ω) is the current in a coil of a small radius a.The equations (1) -(3) can be solved to find the differential equation for electric fields as.
Taking Hankel transform which is defined as where J 1 is Bessel function of the first kind of order 1, and equation ( 4) becomes We next consider the ground having a positively skewed curve conductivity profile with depth denoted by σ(z) = (σ 0 + z)e −bz/2 , where σ 0 is a positive constant, b is constant.We consider a primary alternating source current carried by a coil of radius a, at z = −h above the surface of the earth z = 0.The electric field in air can be denoted by E air (λ, z, ω) and expressed as the sum of two components, where E p (λ, z, ω) is the primary field and E s (λ, z, ω) is the secondary field.Both electric fields can be obtained from equation (5).In air, σ air (z) ∼ = 0 and the electric field is given by which remains bounded as z → −∞, and ̥ is arbitrary constant to be determined.
In ground, we can obtain the partial differential equation for the electric filed as where E gro (r, z, ω) is the electric field in ground, k 2 g = iωµ g σ(z) + ω 2 µ g ε g , µ g is the magnetic permeability of ground, ε g is the electric permittivity of ground and σ(z) = (σ 0 + z)e −bz/2 is the conductivity of ground.Taking Hankel transform to equation (7), we obtain Using the exponential substitution ξ = e −bz/2 to equation (8).We obtain Equation ( 9) is a non-homogeneous differential equation, and it can be solved for a homogeneous equation by Cauchy-Euler equation method and a particular solution by the method of variation of parameters.Hence, the solution of equation ( 9) is where , and G(ξ; τ ) is the Green's function.We calculate the integration term of equation ( 10) by using method of subsection.Therefore, equation ( 10) becomes where C 1 , C 2 are arbitrary constants to be determined from the boundary conditions.Under the condition z → ∞, we have E gro (λ, z, ω) = 0. Hence, we require C 2 = 0. Therefore, equation ( 11) becomes We can find constants ̥ from equation ( 6) and C 1 from equation ( 12) by imposing the continuity of E and ∂ E ∂z at air-earth interface.That is, and By using inverse Hankel transform, we obtain the electric field in air as and the electric field in ground as 2 ) By partitioning on the ground, we can rewrite equation ( 14) in the matrix form, we now have where 2 , and E jgro and d 1 are the electric field and the width of each ground layer, respectively.The electric field on the ground surface can be determined from equations (13) or equation ( 14) by taking z = 0. Chave [1] is used for numerical calculating the inverse Hankel transform of the electric field solutions.

Conclusions
In this paper, we present the approach to study the structure of the earth's surface layer.The forward problem is produced by the integral equations.We formulated the problem to get the electric fields, which could be used to find  the electric fields on the ground surface.The Cauchy-Euler equation method and the method of variation of parameters have been used to attain a solution of a non-homogeneous differential equation.We regard a half-space model having a positively skewed curve conductivity profile which is denoted by σ(z) = (σ 0 + z)e −bz/2 , where σ 0 is a positive constant, b is constant.The electric field was expressed in term of mathematical expressions and plotted.The graphs are shown the behavior of the electric field against source-receiver spacing r.The values of b and σ 0 are fixed while the value of the width of each ground layer d 1 is varied.In the experiments, we set up the frequencies which depend on d 1 .The curves of the electric fields are similar to the shape of a positively skewed curve conductivity ground profile as shown in figures 1 − 3. The curves of both parts of electric field are oscillated and tend to zero as the source receiver spacing r increases which is caused by the ground medium profile.These can be used for studying the structure of the earth's surface layers and ground exploration.It is important to study the optimal values for all the parameters in the future work.More layered earth structures will be conducted as well as other mathematical approaches may be used to study the problems.

Figure 1 :
Figure 1: Graph of real and imaginary part of electric field E versus r for a half space earth model with a positively skewed curve medium by inputting σ 0 = 0.01 S m −1 , b 1 = 0.1 m −1 , d 1 = 1.5 m, frequency = 200 Hz.

Figure 2 :Figure 3 :
Figure 2: Graph of real and imaginary part of electric field E versus r for a half space earth model with a positively skewed curve medium by inputting σ 0 = 0.01 S m −1 , b 1 = 0.1 m −1 , d 1 = 3.5 m, frequency = 200 Hz.