Reply to comment by J. Trampert

Abstract

In response to the comment by Trampert (1990) on the paper ‘Comparison of iterative back-projection inversion and generalized inversion. . .’ by Ho-Liu, Montagner & Kanamori (1989; hereafter referred as HMK), we would like to reiterate the two main objectives of HMK. The first one was to compare the generalized least-squares inversion method without blocks by Tarantola & Valette (1982; hereafter referred to as TV) and a particular version of SIRT technique (Van der Sluis & Van der Vorst 1987) presented by Comer & Clayton (1985, unpublished manuscript; hereafter referred to as CC). We did not intend to provide a general relationship between TV and the general class of SIRT. We wanted to provide a way to choose the damping factor in CC using an approximate form of back projection algorithm (HMK, p. 22), and to assess the error and resolution. In the previous studies using CC, the damping factor was chosen more or less arbitrarily. The second purpose was to apply these two different tomographic techniques to the same data set to check the robustness of the inversion results. H. Trampert notes that two points need clarification. We will consider successively these two points. The first point concerns the damping factor p. Equation (1) from TV as reported by Trampert is the general algorithm solving a non-linear problem. We applied it to a linear problem (section 6 of HMK) without iteration (it is clearly pointed out in section 4 of HKM). We did not solve a non-linear problem. Comparing the two algorithms, we only showed that CC (actually slightly different from SIRT) cannot solve the problem by one iteration because CC’s expression is approximate. The correction term 6pcc = p k + l -pk of CC is always smaller than the exact term given by TV, 6 b =pl -po . Consequently, it is necessary to iterate when CC is used, in order to converge. That is the reason why we have kept an index k for 6pcc. To avoid confusion, we could‘ have used different indices for TV and CC. Trampert concludes that ‘As a consequence, their interpretation of p as the ratio between the data variance and the a priori model variance is incorrect.’ This statement is odd because our expression (17)