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General principles of tomography

Both reflection tomography and transmission tomography start from the idea that we can use the following fitting goal to invert the unknown slowness model s from traveltime t :
\begin{eqnarray}
{\bf t= T(s)s }\end{eqnarray} (1)
The operator T(s) is usually constructed in terms of rays passing through the slowness model s, so it is model dependent and therefore non-linear. By doing a Taylor expansion and ignoring the second and higher order terms, we can linearize this problem:
\begin{eqnarray}
{\bf \Delta t= T_0 \Delta s }\end{eqnarray} (2)
Here ${\bf \Delta s}$ and ${\bf \Delta t}$ are the change of slowness and corresponding change of travel time, respectively. The tomography operator ${\bf T_0}$ now is model-independent. To use the fitting goal (2) for inversion, we need some prior information about the unknown slowness field to construct an initial model, then construct ${\bf T_0}$ by ray tracing through it. If the initial model does not seriously deviate from the true slowness field, we can use the operator ${\bf T_0}$ to invert the change of slowness ${\bf \Delta s}$ from the change of traveltime ${\bf \Delta t}$. After we obtain ${\bf \Delta s}$, we can update the initial model to obtain more accurate slowness field.


next up previous print clean
Next: Integrated tomography for slowness Up: Methodology Previous: Data Calibration
Stanford Exploration Project
7/8/2003