Green’s tensors

Go and po represent a stationary Green s tensor and initial density matrix, respectively. The initial density matrix can be obtained by solving the system of N equations given by ... 

To obtain spectra, equation 21 is used, with the elements of the Green s tensor obtained frcnn equation 19 and 25, and the initial density matrix elements from equation 23. Enhanced spectra are calculated using the results from CDE, as shown in the previous section. The results are seen in Figure 3.5. 

The electromagnetic Green s tensors Ge, G// are introduced as the fields of an elementary electric source (Zhdanov, 1988 Felsen and Marcuvitz, 1994). They satisfy the Maxwell s equations... 

The EM Green s tensors exhibit symmetry and can be shown, using the Lorentz lemma, to satisfy the following reciprocity relations (Felsen and Marcuvitz, 1994) ... 

The last conditions show that by replacing source and receiver (i.e. the points r and r) and by going simultaneously to the reverse time —t, (therefore, by retaining the causality, because the condition t < t n ordinary time implies the condition —t > —t in reverse time), we obtain the equivalent electromagnetic field, described by the Green s tensors Ge (r, t r,t) and G (r, t r,t). 

Following Morse and Feshbach (1953) and Felsen and Marcuvitz (1994), we can also introduce the adjoint Green s tensors ... 

Note that we use the same symbols for Green s tensors in time and frequency domains to simplify the notations. One can eeisily recognize the corresponding tensor by checking for arguments t or u in the corresponding equations. 

Electromagnetic Green s tensors represent an important tool in the solution of the forward and inverse electromagnetic problems and in migration imaging. We will illustrate Green s tensor applications in the next Chapter. 

Wc can apply an approach, similar to the one used in the 2-D case, to derive the electromagnetic integral equations in three dimensions. Electromagnetic Green s tensors, introduced in the previous chapter, make it possible to determine the electromagnetic field of an arbitrary current distribution j (r) within a medium with background conductivity (Ti, ... 

Note that the arguments in the expressions for the Frechet differentials, F/. //(5,6ct), consist of two parts. The first part, d, is a conductivity distribution, at which we calculate the forward modeling ojicrator variation, the Green s tensors are... 

We can give a simple, but important physical interpretation of the expressions for sensitivities, (9.55), based on the reciprocity principle. Note that, according to definition (see Chapter 8), the Green s tensors (r r") and Gh (r r"), are the electric and magnetic fields at the receiver point, r, due to a unit electric dipole source at the point r" of the conductivity perturbation. Let us introduce a Cartesian system of coordinates x,y,z, and rewrite these tensors in matrix form ... 

The expression for the scattering tensor is obtained from equation (9.73) by approximating E (r) in the integral by its value at the peak point r = tj of the Green s tensor ... 

Following ideas of the extended Born approximation outlined above, we use the fact that the Green s tensor Ge (j" 1 r) exhibits either singularity or a peak at the point where r = r. Therefore, one can expect that the dominant contribution to the integral Ge [ActAE "] in equation (9.83) is from some vicinity of the point r = r. Assuming also that A (r) is slowly varying within domain D, one can write... 

Taking into account once again the fact that the Green s tensor (r r) exhibits either a singularity or a peak at the point where = r, one can calculate the Born approximation Gg [A5 (r) E (r)] using the formula... 

In order to develop the numerical analogs of the electromagnetic field integral representations, we have to discretize, also, the field components and the Green s tensors within the anomalous domain D of integration. We can treat all integral representations, considered in this chapter, as operators acting on the vector functions, x,... 

The 3N X 3N matrix is formed by the volume integrals over the dementary cells Dn of the components of the corresponding electric Green s tensor G, acting... 

The approximate anomalous conductivity in formula (10.33) is obtained as a scalar product of the auxiliary field E (r) with the complex conjugate background field at the point r, normalized by the magnitudes of the background field and the norm of the corresponding Green s tensor at the same point ... 

Note that each equation, (10.41) and (10.42), is bi-linear because of the product of two unknowns. Ad and E. However, if we specify one of the unknowns, the equations become linear. The iterative Born inversion is based on subsequently determining Ad from equation (10.42) for specified E, and then updating E from equation (10.41) for predetermined Ad, etc. Within the framework of this method the Green s tensors Ge and Gh, and the background field stay unchanged. 

In the distorted-Born iterative method, the background field and the Green s tensors are updated after each iteration according to the updated conductivity Ad. However, this approach is difficult to implement in practice, because it requires calculation of the Green s tensors for inhomogeneous media, which is an extremely time consuming problem in itself. 

In the last formulae a — db + Ad is a conductivity distribution, for which we calculate the forward modeling operator variation 6d is the corresponding variation of the conductivity a, which is obviously equal to the variation of the anomalous conductivity, 6d = 6Ad. Tensors Gg jf are electric and magnetic Green s tensors calculated for the given conductivity a. Vector E in expressions (10.54) and (10.55) represents the total electric field, E = E -t-E for the given conductivity d. 

We can extend the integral representations in the frequency domain, formulae (9.37) of Chapter 9, to the time domain. As a result, the anomalous electromagnetic field in the model can be expressed as an integral over the anomalous domain D of the product of the corresponding Green s tensors and excessive currents An (E -I- E ) ... 

The solution of the boundary value problem (11.69) and (11.70) for the concentration equation can be obtained with the aid of Green s tensor formula (F.IO) (see Appendix F). We assume that the volume V is bounded by the surface S, which... 

According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F . For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G" (r, t), which we call the Green s tensor of the vector wave equation. 

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