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Extra resources for 100-Area Monthly Reports [for 1947] [declassified]
The reconstructed potential of figure 18 (top panel) is on the scale of the figure indistinguishable from the true potential. 5. 5 times as strong. This would lead to a near-perfect reconstruction of the potential. However, the reconstructed potential is given by the dashed line. 177 Figure 18. The original potential (solid lines) and the potential reconstructed with the Marchenko equation (dashed lines). 5 times as strong as the potential in the top panel. It can be seen that the left part of the potential (the side from which the waves are incident) the potential is reconstructed quite well, but that the part of the potential on the right is very poorly reconstructed.
99) For vector waves the Green’s function should be replaced by a second order tensor, but the principles are the same. Using this Green’s function the solution of (97) and (98) is given by u0 = GF and un = −GL1 un−1 (n ≥ 1) . (100) The solution can now be constructed by solving the above equation recursively and by inserting the result in (96): u= u0 Unperturbed wave − εGL1 u0 Single scattered wave + ε2 GL1 GL1 u0 + · · · (101) Double scattered wave In this formulation the total wavefield is written as a sum over the unperturbed wave, the waves that are scattered once by the inhomogeneity L1 , the waves that are scattered twice by the perturbation L1 and all the higher order scattered waves.
The kernel R(1) (x; x1 ) describes to what extent the estimated model is a blurred version of the true model. The higher order kernels such as R(2) (x; x1 , x2 ) can be interpreted as nonlinear resolution kernels that describe to what extent there is a spurious nonlinear mapping from the estimated model onto the true model in the inversion process. In the ideal case, the estimated model is equal to the true model: m(x) ˜ = m(x). This is the case when the linear resolution kernel R(1) (x; x1 ) is a delta function δ(x − x1 ) and when the nonlinear resolution kernels are equal to zero: R(n) (x; x1 , · · · , xn ) = 0 for n ≥ 2.
100-Area Monthly Reports [for 1947] [declassified]