Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion
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A MATLAB^(R)-based computer code that analyses the traveltime distribution and performs quality analysis at the pre-inversion stage for 2D transmission experiments is presented. The core tools of this approach are the so-called mean traveltime curves. For any general recording geometry, the user may select any pair of subsets of contiguous sources and receivers. The portion of the domain swept by the implied rays defines a zone of analysis, and for each source (receiver) the outcoming (incoming) ray fan is named a source (receiver) gather. The empirical mean traveltime curves are constructed, for each zone, by assigning the average and the standard deviation of the traveltimes in the gathers to the positions of the sources (receivers). The theoretical expressions assume isotropic homogeneous velocity inside each zone. The empirical counterparts use the observed traveltimes and make no assumptions. Isotropic velocity in each zone is inferred by least-squares fitting of the empirical mean traveltime curves. The user may refine the analysis considering different zones (multi-zone analysis). Initially the whole domain is modelled as a single zone. The procedure compares empirical versus theoretical curves. In addition, residuals can be plotted using source-receiver positions as plane coordinates. The results are used to unravel the possible presence of anomalous gathers, heterogeneities, anisotropies, etc. Depending on the kind of anomalies, velocity estimation and mean time residuals are different in the source and receiver gather curves. This software helps to grasp a better understanding of the data variability before the inversion and provides to the geophysicist an approximate zonal isotropic model and a range of velocity variation that can be used in the inverse problem as a priori information (regularization term). Its use is described through tutorial examples. A guided user interface leads the user through the algorithm steps.