Crosswell data frequently appear impossible to explain without anisotropy, and depth imaging often finds it impossible to explain both moveouts and well-ties with a single isotropic velocity. In this paper, I propose an algorithm using group velocities that can change with angle. The computational difficulty is only slightly greater, and the flexibility will be available when necessary.
Previously, I found explicit methods of extrapolating traveltime tables convenient, but such tables take too much memory to be stored for a large 3D dataset. Recomputation of traveltime tables is likely to be prohibitively expensive. Even sampled raypaths also can require too much memory to be stored. Instead I optimize the coefficients of smooth basis functions according to Fermat's principle, to minimize the ray traveltime. Only a few coefficients need be stored to describe each raypath.
The first obvious applications of this method are to crosswell data and to surface diving-wave tomography for near surface velocities. In later work, reflection points will be included. Because less work has been done on large 3D datasets for diving wave tomography and because there is less understanding of the non-uniqueness, I address diving waves first. Diving waves are generalizations of surface refractions. Conventional refraction analysis describes the near surface as layers of slabs, whose velocities change very smoothly, if at all, in the interior, and very sharply at the boundaries between slabs. The first arrival times are assumed to increase linearly with offset for a given refraction. An interpreter must distinguish individual refractions on the first arrival. Unfortunately, first arrivals rarely appear to be composed of a series of distinct linear segments.
Diving waves allow the interpreter to model all direct arrivals, without identifying individual refractions. Velocities are allowed to vary arbitrarily but are expected to increase overall with depth. When sources and receivers on the surface are relatively close (hundreds of meters), the wave energy that arrives first is confined to the very near surface. When the separation reaches several kilometers, then the earliest arrivals contain energy that has passed a kilometer or so in depth. For example, if the velocity at the surface of the earth were 2 km/s, and this velocity increased 1 km/s with every 1 km of depth (reaching 4 km/s at 2 km depth), then a source and receiver at an offset of 4.47 km would produce a circular arc that reached a maximum depth of 1 km. An offset of 6.93 km would reach 2 km depth.
Recent papers have used tomographic methods to reconstruct near surface velocities from surface measurements of first arrival times. Simmons, Bernitsas, and Backus were able to reconstruct impressive images with simple assumptions of semicircular raypaths. Zhu, Sixta, and Angstman allowed more flexible raypaths for improved 2D images. Stefani 1993 prepared an excellent 2D case study which demonstrated the accuracy of estimated velocities for depth conversion. Bell, Lara, and Gray 1994 showed that targeted tomography was appropriate in certain areas. Zhang and McMechan 1994 and Qin, Cai, and Schuster, Cai and Qin (1994); Qin et al. (1993) solved very general isotropic 3D geometries. J.A. Hole et al used explicit traveltime extrapolations for good inversions of 3D diving traveltimes with limited surface coverage. Many other papers consider diving waves without tomography Laski (1978); Levin (1994). The literature on the inversion of refracted waves also contains many good ideas which could be generalized for diving waves Clayton and McMechan (1981); Hagedoorn (1964); Hawkins (1961); Landa et al. (1994); Palmer (1981); Zanzi and Carlini (1991).
Diving wave tomography has only recently incorporated anisotropy, which appears more in crosswell tomographic applications Michelena et al. (1993); Michelena (1994); Pratt and McGaughey (1991); Saito (1991); Vassiliou et al. (1994). I will follow many of the suggestions of Grechka and McMechan for diving wave tomography. They use Chebyshev polynomials to describe raypaths and have already performed SVD analysis to show the non-uniqueness introduced by anisotropy into diving-wave tomography.
In this paper, a continuous anisotropic velocity model is parameterized with a minimal number of discrete parameters. Most of the cited publications prefer to work with discrete bins and may even discretize the physical modeling. Instead, I assume a continuous velocity model with a finite number of coefficients to scale continuous basis functions.
Most cited diving wave papers construct sampled rays from shooting methods or from traveltime tables. Shooting methods can be fast, but are efficient only if one sorts through the data in a particular order. Rays must be stored in their sampled form, which may take too much memory for 3D datasets. Traveltime tables cannot be saved even for smallish datasets. This paper constructs rays as a scaled sum of smooth basis functions. Only a small number of coefficients need to be stored to describe an entire continuous raypath. A generic Gauss-Newton algorithm optimizes raypaths and velocities without any features specific to this application.