A general theory of the seismic traveltime tomography
© V.S. Geyko
The main problems of seismic tomography are as follows.
- Inverse seismic problems in 3-D are generally non-linear. Their reduction to tomographic problems and the analysis of uniqueness and stability represents a complicated mathematical task.
- Problems of computer tomography are extremely numerically unstable. Use of cumbersome regularising algorithms on high-powered computers does not always provide positive result because of smoothness (or, opposite, oscillation) of the obtained solutions and its strong dependence on the choiced initial approach.
- Observed data are always noisy and incomplete. In this paper we consider two alternative ways: the linearization and a new Taylor approximation of non-linearity of the 3-D inverse problem for the wave equation and the eikonal equation. As the kernel of this approach appear transforming the original travel-times (wave) field to the CMP (central middle point) format.
Compared to the traditional linearization Taylor approximation for travel-time of refraction's has the following advantages.
- It ensures the considerable gain of the non-linearity approach accuracy.
- It is valid with fewer constraints imposed on a velocity function.
- It is not need assignment of a reference velocity as starting approach.
- It reduces to the problem, which is correct by Tikhonov, instead of an essential unstable one.
- It caused the significant reduction of dimension of the numerical inverse problem as it provides scanning of a traveltime field and inversion in place of entirely inversion.
The method proposed here was tested analytically. In this paper we consider explicitly the techniques of processing and inversion of body wave data from earthquakes. The developed method extends easily on case of inversion of reflected travel-times corresponding with subhorizontal discontinuities. It can be used to interpret 3-D data of seismology and 2-D seismic refraction/wide-angle reflection (deep seismic sounding) data as well as for (with corresponding modification) profile and spatial CMP data in seismic reflection and refraction wave exploration.