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(c) Computation of the Cost

The computation of J is not as automatic as it looks like. Indeed, the approach taken by Nishimura is as follows[25]: One computes at , , where K is a fixed number. The same number of time steps are used in the computation of J. In this case the far field can be computed for K time steps after the rise time of . However, the rise time of the scattered field from the temporary location of S in the process of NLP may not be identical with that of the observed far field. In the computation of J, however, one needs the values of for all the s where the observation is made. This is problematic if the computed rise time is earlier than the observed one, because the computed far fields for are not available. In this case one computes J assuming that stays constant for . With this method, however, the cost J may become constant if S is very far away from the origin, and NLP process may terminate. Hence one avoids this by introducing appropriate constraints to .

Notice that will include Dirac's delta terms when is aligned with the normal vector to S if the time shape function for is taken piecewise linear (see the term in (30)). This problem could be avoided, for example, by using spline () time shape functions. However, Nishimura used a simpler approach where one uses finite difference formulae to compute at from the computed values of at . One then uses linear interpolation of the discrete values of to compute .

N. Nishimura
Thu Feb 19 01:36:51 JST 1998