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Application to seismic tomography

Following the earlier work by () and Maysami and Clapp (2009), I apply the methods discussed above to regularize the seismic velocity tomography problem. In this optimization problem, the data misfit term includes the physics and kinematics of the tomography problem, while the regularization (model-shaping) term imposes the structure present in the auxiliary field, which is chosen to be either the reflectivity or the resistivity field (see Figure 3). This problem can be stated as

\begin{displaymath}\begin{cases}\arg \min_{\Delta {\bf s}} &\quad {\Big \Arrowve...
...})  {\Big \Arrowvert}^2_2 \quad \thickapprox 0 \end{cases} ,\end{displaymath} (5)

where $ {\bf s}$ and $ {\bf r}$ are the slowness and auxiliary fields, respectively; $ \Delta {\bf t}$ and $ {\bf T}_L$ represents traveltime updates and the linearized tomography mapping operator; and $ \bf A$ is the model-shaping operator, which is picked as either the cross-gradient operator or the dip residual operator. In the latter case, the regularization operator is given by $ \mathbf{A({\bf r})} = \frac{\partial}{\partial x}+ p_{r} \frac{\partial}{\partial z}$.

Although the reflectivity field is in practice a function of velocity, we assume that it is an accurate representation. One may consider a more general case, where the regularization operator is also a function of model, which needs to be linearized around the current estimation.

Figure 4 shows the results of solving the tomography problem (Equation 5) for an optimal update in velocity. I start with a smooth velocity as the initial guess. I use either reflectivity or resistivity as auxiliary data and I choose either the cross-gradient function or the dip residual for the regularization operator. This leads to four different updates based on the choice of auxiliary data and data integration method. Note the different frequency contents of the two types of auxiliary data in Figure 3. Updated velocities with cross-gradient functions (Figure 4(a) and 4(b)) seem to have more contribution of structure (model-shaping term) than the physics (data misfit term) of the problem, as the structure of model is clearly visible in the results. The dip residual method, however, shows a better portion of physics in the results and less of the actual structure. This suggests that we may be able to benefit from combining these methods in a specific fashion to obtain a better velocity estimation.

vel-t vel-0 softdata1-0 softdata2-0
vel-t,vel-0,softdata1-0,softdata2-0
Figure 3.
Synthetic 2-D model used as an example for comparison of different data integration methods: (a) True velocity, (b) initial guess for velocity, (c) approximation of resistivity, and (d) true reflectivity. [ER]
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velx1-ds0 velx2-ds0 velp1-ds0 velp2-ds0
velx1-ds0,velx2-ds0,velp1-ds0,velp2-ds0
Figure 4.
Updated velocities for the model shown in Figure 3 obtained by solving the tomography problem using (a) resistivity and the cross-gradient function, (b) reflectivity and the cross-gradient function (c), resistivity and the dip residual, and (d) reflectivity and dip residual. [CR]
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next up previous [pdf]

Next: Conclusions Up: Maysami: Geophysical data integration Previous: Dip residual

2009-10-19