CROSSWELL TOMOGRAPHY APPLICATION

Figures 1-4 provide some numerical examples comparing and contrasting the results obtained using standard SVD resolution calculations with the LSQR Paige and Saunders (1982) resolution calculations described in an earlier paper Berryman (1994). We consider a $4\times4$ model using strictly crosswell data, so there are 16 source/receiver pairs as well as 16 cells in 2D. Model slowness values are shown in the upper block of each figure, while diagonal resolution values are shown in the lower block. The first two examples (Figures 1 and 2) show results for the actual model used to compute the traveltime data [see Berryman 1990 for a description of the code used to generate both the forward and inverse solutions]. The second two examples (Figures 3 and 4) show results obtained after 15 iterations of the reconstruction code of Berryman 1990. The LSQR resolution examples (Figures 2 and 4) were computed using ten iterations of the LSQR algorithm, so the maximum size of the resolved model vector space has dimension ten. To aid in the comparison, the SVD resolution examples use only the 10 eigenvectors associated with the 10 largest eigenvalues of the ray-path matrix. We find the results are in qualitatively good agreement. Better quantitative agreement is not anticipated because the 10-dimensional vector spaces spanned by these two approximations, although having large regions of overlap, will nevertheless almost always differ to some degree.

 

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Figure 1 Target model slowness (a) and resolution (b) for truncated SVD using 10 largest eigenvalues.

 

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Figure 2 Target model slowness (a) and resolution (b) for LSQR after 10 iteratons.

 

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Figure 3 Slowness (a) and resolution (b) for truncated SVD using 10 largest eigenvalues.

 

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Figure 4 Reconstructed slowness (a) and resolution (b) for LSQR after 10 iterations.