# Seismic tomography - book

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 3 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

Seismic traveltime tomography is a popular method for seismic velocity analysis (e.g., Lines, 1991^{[1]}; Stewart, 1991^{[2]}). Its principles are sketched in Figure 6. Traveltime tomography can provide velocity estimates from borehole or surface-reflection data. The goal of tomography is to image the internal material properties of a body by observing the wavefields that have passed through it. The tomographic method can be used to estimate seismic velocities from traveltime information and can be viewed as a three-step procedure, as is detailed next.

What is the first step in tomography? The first step is *data gathering*: Seismic traveltimes for various source/receiver positions are measured (Figure 7). This procedure is time-consuming because often a geophysicist must pick a multitude of events from the seismic data manually to determine the required traveltimes. For well-behaved seismic data, computer tracking programs can make the picks automatically, thereby greatly decreasing the effort expended by the geophysicist. However, even with good data, the traveltimes resulting from automatic picking programs have to be inspected visually for quality control.

What is the second step in tomography? The second step is *modeling*: Seismic raytracing methods are used in accordance with a velocity model. This model is employed to find expressions for the traveltimes, which are the so-called traveltime equations. The subsurface is divided into cells of some convenient geometric shape to which a given initial velocity configuration is assigned.

That initial velocity model also can allow for velocity gradients in both horizontal and vertical directions. Ray-tracing methods obeying Snell’s law are used so that the raypaths are curved correctly. It turns out that use of the slowness (or reciprocal velocity) simplifies the equations that need to be solved because then the traveltime in each cell is equal to the product of distance traveled in the cell times slowness in the cell. A traveltime equation expresses the total traveltime for a raypath as the sum of such products for the cells traversed.

The system of traveltime equations for all the raypaths is *linear in slowness* for the special case of straight raypaths. More generally, the raypaths curve according to Snell’s law, and then the distance values are functions of slowness. In such cases, the system of traveltime equations becomes nonlinear.

What is the third step in tomography? The third step consists of a nonlinear *iterative-improvement method*. In each iteration, the set of picked traveltimes obtained from the seismic data is matched to the computed traveltimes obtained from the traveltime equations. The traveltime equations require a velocity or slowness model. This velocity model starts with an initial best guess. For each iteration, the error (for example, the mean squared error) between the two sets of traveltimes is used to update the previous velocity model. That is, in each iteration, the slowness vector is adjusted so that the traveltimes given by the model agree more closely with the picked traveltimes. The iteration terminates when the agreement is deemed satisfactory. The result is the desired final velocity model.

What modeling is involved in tomography? Transmission tomography involves modeling rays that are transmitted without reflection from a known source position to a known receiver position. Transmission tomography can be used either in cross-borehole profiling or in vertical seismic profiling. In the cross-borehole case, the sources are in one borehole and the receivers in another. In the VSP case, the sources are on the earth’s surface and the receivers are in the borehole.

What is reflection tomography? Reflection tomography involves two-way transmission. The seismic waves from a surface source propagate to an interface from which they are reflected. Then the waves propagate back to receivers on a convenient recording surface. Because of the problem of defining the reflector position, reflection tomography is more difficult to apply than transmission tomography is. In reflection tomography, the ray-traced traveltimes of the model are matched to the picked traveltimes. The result of this iterative-improvement method is an estimate of the interval velocities. Such velocity estimates are required to produce reliable images with seismic migration methods. In this sense, tomography and migration are complementary processes.

How is migration involved in tomography? To implement tomography successfully, a good depth image depicting the horizons (reflectors) is needed. A reliable depth image can be obtained by migration. The ideal output of tomography is a reliable velocity model. To carry out migration, a reliable velocity model is needed, and we know that such a model can be obtained with tomography. And as we already stated, the ideal output of any migration method is a reliable depth image depicting the layer interfaces (reflectors). Thus, around and around it goes, and consequently, the processes of tomography and migration can be used iteratively to produce both reliable velocity models and reliable depth images.

## References

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Velocity analysis | Coherence |

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Digital Imaging | Sampling |

## Also in this chapter

- The seismic reflection method
- Seismic interpretation
- Porosity
- Absorption loss and transmission loss
- The wave equation
- Wave velocity
- Velocity analysis
- Coherence
- Appendix C: Exercises