Multichannel appraisal deconvolution

The single channel scalar deconvolution method presented by Oldenburg has been extended to include N channels of data and vector models of the form ṁ(t) = (m1(t), m2(t), …, mα(t))T. The solution has its foundation in the linear inverse theory of Backus & Gilbert and is effected by computing a set of N filters, which, when convolved with the data, yield unique averages of one of the scalar functions of the model. Those averages are the summation of the scalar model convolved with a primary averaging function plus contamination from secondary averaging functions convolved with other model components. It is shown how a set of suitably selected weights can annihilate these secondary averaging functions and thereby greatly simplify the interpretation. The computations are efficiently carried out in the frequency domain and require the inversion of an N×N Hermitian matrix at each frequency. As a type example, we have shown how the time varying elements of a seismic moment tensor might be computed from a set of seismograms.