Non-Linear Reflection Tomography

Abstract

S U M M A R YReflection tomography, the determination of velocity distribution and reflector position from reflection travel-time data, is a very non-linear inverse problem. Unlike in transmission tomography, ray paths have to be iteratively updated, because travel-time variations cannot be computed by integration of slowness along the original unperturbed ray paths. From a study of parameter sensitivity we conclude that in seismic reflection experiments the vertical variation of slowness inside layers is poorly resolved from travel-time data. For this reason, in each layer the slowness was modelled with functions varying only in the horizontal direction. A B-spline representation is adopted for lateral velocity heterogeneity and interfaces. These splines are local and well adapted for tomography because the spline parameters have a geometrical interpretation and they may be explicitly used as unknowns in the inverse problem. For each iteration, and for every source-receiver pair, two-point ray tracing was performed by paraxial ray tracing, and the inverse problem was solved by iterative least-squares. A priori data, necessary to stabilize the inverse problem, were introduced by a penalty function approach. This method is equivalent to using a priori convariance matrices, but it has a simpler physical interpretation and is faster to use. Damping was used to improve the convergence. The method was first tested in the inversion of synthetic data. These synthetic examples illustrate the limitations of reflection tomography: non-linearity effects, poor vertical resolution of the velocity, and decrease of the resolution with the ratio of maximum offset to interface depth. Finally, we inverted a data set from the Paris Basin. The inversion method reduces the residual norm to 6 ms, which is less than the expected error on the data.

Seismic Traveltime Tomography of the Crust and Lithosphere

“…Undesirable effects of nonlocal support include poorly resolved portions of the model having a global influence and unrealistic velocity fluctuations between nodes (Shalev, 1993). In 2-D wide-angle traveltime tomography, cubic 11 spline interpolation has been employed by , while Farra & Madariaga (1988) and McCaughey & Singh (1997) have used cubic B-spline bases.…”

“…Červený et al (1984) parameterize a layered model with splines under tension for both interfaces and layer velocity fields. Cubic B-splines, which feature local control of interface geometry, have been used by Farra & Madariaga (1988) and McCaughey & Singh (1997). In 3-D, the use of smooth interfaces (see Fig.…”

“…24 is a specific application of paraxial ray approximation (Červený & Pšenčik, 1983;Červený et al, 1984;Červený, 1987;Farra & Madariaga, 1988;Červený, 2001), a method that is commonly used to solve the twopoint problem in reflection and refraction ray tracing. The method is based on using a ray-centered coordinate system, where a particular ray is taken to define one of the three coordinate axes.…”

“…Similar methods were used by Cassell (1982) and Langan et al (1985). Other applications of shooting methods in 2-D reflection and/or refraction traveltime inversion include those by Farra & Madariaga (1988), White (1989), , Williamson (1990), Zelt & Smith (1992) and McCaughey & Singh (1997). Examples of its use in 3-D reflection and/or refraction traveltime inversions are harder to find although several 3-D tomographic studies that combined refraction and local earthquake data (Benz & Smith, 1984;Ankeny et al, 1986;Sambridge, 1990) and some teleseismic tomography studies (Neele et al, 1993;VanDecar et al, 1995) have used shooting methods of ray tracing.…”

Fault zone structure determined through the analysis of earthquake arrival times

“…Farra and Madariaga (1988) used cubic B-splines in modelling seismic reflection data, for the parameterization of both the velocity field and the position of discontinuities in a two-dimensional model. Thomson and Gubbins (1982), were first to use cubic cardinal splines to determine the 3-D velocity structure beneath the NORSAR seismic array using teleseismic observations.…”