Inversion using the conjugate gradient method

After tracing rays and representing the slowness field with spline coefficients, the object function we want to minimize in the least squares sense is  

$$J = ( \Delta {\bf t} - LS{\bf c}),$$ (41)

where S is the spline function shown in (), and ${\bf c}$ is a vector of spline coefficients. I used the conjugate gradient algorithm to solve the least squares problem  (). In using this algorithm, we do not need to construct the operator L in a matrix form, which is generally huge in size. Since summing slowness along the ray paths is the forward operation and putting the traveltime error along the ray paths is the adjoint of it, we only need to store the ray paths that correspond to nonzero components of the operator L.

After finding ${\bf c}$ for the measured traveltime error $\Delta {\bf t}$, the slowness error $\Delta {\bf w}$ is obtained from (). Figure  shows the slowness field updated by the least-squares inversion of $\Delta {\bf w}$.Comparing it to the true velocity model Figure , we can see that the low-frequency component of the slowness field was found very easily even in one iteration.

Figure  is a stacked image after planewave synthesis imaging that synthesized 31 different incidence angles, from -30 to +30 degrees, at the surface, using the updated slowness model (Figure ). Most of the reflectors are located closer to the real position than that of the images obtained using the initial reference model. And CSL gathers (Figure ) show that most of the reflectors are almost lined up horizontally.

 

synvel-iter1
Figure 16 Velocity model updated by adding $\Delta w$, which was calculated by inverting $\Delta t$, to the initial guess w. This corresponds to one iteration of the traveltime tomography inversion.

 

pws-syn-iter1
Figure 17 Stacked image after planewave synthesis imaging that synthesized 31 different incidence angles, from -30 to +30 degrees, at the surface. For imaging, the updated velocity model (Figure ) was used.

 

crp-iter1
Figure 18 Three CSL gathers at (a) 3 km, (b) 3.5 km, and (c) 4 km from the prestack image cube that was obtained by synthesizing plane waves at the surface using the updated velocity model (Figure ).