Green’s tensor formula

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... 

Now we will integrate identity (13.117) over the domain V and apply the Green s tensor formula for the Lamp s operator (F.16) to the left-hand side of the resulting formula ... 

The Green s tensor formulae are derived from expression (F.8). Indeed, let us specify an auxiliary tensor field G (r) ... 

Substituting equation (F.9) into the Gauss tensor formula (F.8), we write the Green s tensor formula in the final form ... 

If the vector field F is replaced by the tensor field Q, we arrive at another-Green s tensor formula ... 

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... 

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 ... 

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 Green s tensor can be treated as the solution of the vector wave equation with the right-hand side given, according to formula (13.86), by the product of the... 

Note that the differential sensitivities are vector functions, because they characterize the sensitivity of the vector wavefield to the square slowness variation. From the last formula we see that the Green s tensor G provides the sensitivity estimation of the vector wavefield. 

Green s tensor and vector formulae for Lame and Laplace operators... 

Finally, if the tensor field P in equation (F.IO) is replaced by the vector field B, we obtain the Green s vector formula ... 

In these formulae, = Sj + As is the square slowness model for which we calculate the variation of the forward modeling operator 6s is the corresponding variation of the square slowness which is obviously equal to the variation of the anomalous square slowness, 6s — 6As expressions G (r r o ) and G (r r u ) stand for the Green s function and tensor defined for the given square slowness and the function p(r,u>) and vector u(r,cu) represent the total acoustic and vector wavefields for the given square slowness s. ... 

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