Electromagnetic Greens tensors

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... 

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, ... 

We now can use the Lorentz lemma to derive the reciprocity relations for the Green s electromagnetic tensor. Let us assume that the electric dipoles with moments a and b are located at points with the radius-vectors r and r",... 

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). 

Note that the differential sensitivities are vector functions, because they characterize the sensitivity of the vector electric and magnetic fields to the conductivity variation. From the last formulae we see that Green s electromagnetic tensors provide the sensitivity estimation of the electromagnetic field to the model conductivity. 

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,... 

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 ) ... 

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