Constant-velocity model

Explicitly computing the Hessian is possible when following a target-oriented strategy that exploits the Hessian sparsity and structure. We created a synthetic dataset, using equation 6, with a constant-reflectivity flat reflector (at $z=0.8\,\, km$) in a constant-velocity medium ($v=2.0 \,\, km\,s^{-1}$). Assuming a land acquisition geometry, where the shots and receivers were positioned every $20 \,\, m$ on the interval ${ \bf x}=[-0.8, 0.8 ] \,\, km$. Valenciano and Biondi (2004) discuss in detail the banded nature and sparsity of the Hessian matrix for the constant-velocity model.

Figure shows a $15\times15$ coefficient filter at constant depth as the x coordinate moves from the corner to the center of the acquisition. Figure a shows point 1, with coordinates ${\bf x}=(0.8,-0.6)$ (corner of the acquisition). Figure b shows point 2, with coordinates ${\bf x}=(0.8,-0.4)$. Figure c shows point 3, with coordinates ${\bf x}=(0.8,-0.2)$. Figure d shows point 4, with coordinates ${\bf x}=(0.2,0)$ (at the center of the acquisition).

 

hesian_phase_const
Figure 1 Hessian of the constant-velocity model, (a) point 1 ${\bf x}=(0.8,-0.6)$, (b) point 2 ${\bf x}=(0.8,-0.4)$, (c) point 3 ${\bf x}=(0.8,-0.2)$, and (d) point 4 ${\bf x}=(0.2,0)$.

Figure shows the envelope of the $15\times15$ coefficient filter shown in Figure . The energy of the ellipses become dimer away from the center, indicating that these points have lower illumination due to the acquisition geometry. To correct this effect we computed the least-squares inverse image, by the method described in the above section.

 

hesian_const
Figure 2 Envelope of the Hessian of the constant-velocity model, (a) point 1 ${\bf x}=(0.8,-0.6)$, (b) point 2 ${\bf x}=(0.8,-0.4)$, (c) point 3 ${\bf x}=(0.8,-0.2)$, and (d) point 4 ${\bf x}=(0.2,0)$.

Two different numbers of filter coefficients were used. Figures and show the inversion results for a filter of $11\times11$ coefficients, whereas Figures and show the inversion results for a filter of $15\times15$ coefficients. Figure shows a comparison of the best results of both filter sizes.

The panels in Figure show the least-squares inverse image for different numbers of iterations for a filter of $11\times11$ coefficients: a for 10 iterations, b for 20 iterations, c for 100 iterations, and d for migration. Notice how the image amplitudes become more even. Figure shows the comparison of the same least-squares inverse image results at the reflector depth. The image amplitude after 100 iterations is the best result. The conjugate gradient algorithm further balances the image amplitudes, which reduces the effects of the acquisition geometry and the bandlimited characteristic of the seismic data.

 

inv_const_11
Figure 3 Constant-velocity inversion using a filter size of $11\times11$ coefficients: (a) 10 iterations, (b) 20 iterations, (c) 100 iterations, and (d) migration.

 

inv_const_pp_11 Figure 4 Amplitudes extracted at reflector depth from Figure , filter size of $11\times11$ coefficients.

The panels in Figure show the least-squares inverse image for different number of iterations for a filter of $15\times15$ coefficients: a for 10 iterations, b for 20 iterations, c for 100 iterations, and d for migration. Notice again, how the image amplitudes become more even. Figure shows the comparison of the same least-squares inverse image results at the reflector depth. The image amplitude after 100 iterations is the best result. The conjugate gradient algorithm once again further balances the image amplitudes.

 

inv_const_15
Figure 5 Constant-velocity inversion using a filter size of $15\times15$ coefficients: (a) 10 iterations, (b) 20 iterations, (c) 100 iterations, and (d) migration.

 

inv_const_pp_15 Figure 6 Amplitudes extracted at reflector depth from Figure , filter size of $15\times15$ coefficients.

Figure compares the migration result to the best inversion results for filter sizes $11\times11$ coefficients, $15\times15$ coefficients. There is not much difference in the recovered amplitudes, thus a filter size of $11\times11$ should be sufficient.

 

inv_const_filter Figure 7 Comparison the migration result to the best inversion results for filter sizes $11\times11$ coefficients and $15\times15$ coefficients.