Mathematics in Computational Science and Engineering

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

(2.16)

(2.17)

(2.18)

(2.19)

then the adjoint system of equation D T u *= 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 u 1, u 2, u 3, and u 4values 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 uand V, we replace Vby δ Vin the bilinear identity. If the sides and bottom surfaces are chosen far enough from the anomalous region, then δρ and δ Jwill be zero. At the top surface, Jand 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 - x 0) δ(y - y 0) 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 ( Figure 2.1), equation (2.31)becomes

(2.32)

The current densities for each block can be computing using a forward resistivity modelling scheme, so that the equation (2.32)reduces to

(2.33)