Mathematics in Computational Science and Engineering. Группа авторов

Figure 2.1 This is a vertical cross-section of a 3-D model. The model represents an anomalous zone. Electrical resistivity of the anomalous zone is perturbed. Reciprocity may be achieved by interchanging the current and potential dipoles.

Equations (2.3) and (2.7) are compared here. Upon comparison of these two equations, it is observed that small changes in electrical potential field distribution is related to small changes in electrical resistivity of the medium. Also, in equation (2.7) behaves as current source for changes in potential fields.

      Now, let us consider the “generalized Green’s identity” of Lanczos [27], which is an integral over volume τ

      (2.8)

      If the adjoint problem D and its boundary conditions are chosen properly, then F(u^*,V) will vanish and the Green’s identity will reduce to the bilinear identity

      (2.9)

      The adjoint system is

(2.10)

      Here, ~ denotes Hermitian and * denotes complex conjugate. If

      (2.11)

      then we can write

      (2.12)

      In order to get expressions for V^T (Du)^*, we need to write complete differentials

      (2.13)

      (2.14)

      (2.15)

      Similar expressions for V^T(D^Tu^*) can be written for the y and z variables. Thus, the expression for V^T(D^Tu^*) will be

      (2.16)

      (2.17)

      (2.18)

      (2.19)

      then the adjoint system of equation D^Tu^* = γ^* will be written as

      (2.20)

      and

      (2.21)

      The above analysis will also yield the expression for F(u^*,V), which is

      (2.22)

      Rewriting the boundary terms

      (2.23)

      Substituting u₁, u₂, u₃, and u₄ values yields

      (2.24)

Rewriting the boundary term again, it gives

      (2.25)

      Changing to the surface integral, it becomes

      (2.26)

As the extended Green’s theorem is valid regardless of the field substituted for u and V, we replace V by δV in the bilinear identity. If the sides and bottom surfaces are chosen far enough from the anomalous region, then δρ and δJ will be zero. At the top surface, J and J′ are parallel to the surface, and J·ds = 0. Therefore, the boundary term vanishes and equation (2.10) becomes

      (2.27)

      Substituting equation (2.7) and (u)^* = γ^* in the above expression, one obtains

      (2.28)

      This expression can be rewritten using expressions for δV^T, γ^*, δD, V, and u^*

      (2.29)

      or

      (2.30)

      Considering a unit point source I^’_s = δ(x - x₀) δ(y - y₀) δ(z) at the observation point, the above equation becomes

(2.31)

Using equation (2.31), small changes in the potential field distribution due to small changes in electrical resistivity of a 3-D model may be expressed in a different format. Assuming the 3-D model is discretized into small individual blocks, equation (2.31) may be rewritten further in algebraic notation representing power loss or dissipated in the blocks. For a given electrode geometry (