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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.

We study a model of continental collision in which one of the continents acts as a rigid die indenting the other plate which flows as an incompressible viscoplastic medium. We consider two extreme cases of plane deformation: (1) plane strain which corresponds to an infinitely thick lithospheric plate, and (2) plane stress corresponding to a very thin plate. Deformation of the lithosphere, a thick plate, should be intermediate between those extremes. We found that the flow in the plane strain case is quite similar to that obtained by slip line, theory. The plane stress results are quite different, since in this case most of the plate shortening is taken up by the thickening of the lithosphere. We also explored the role of boundary conditions on the flow, in particular, the role of the side walls containing the flow of the lithosphere. In the case of a free lateral boundary the main feature is a flow of matter toward this free wall and a S‐like pattern for the horizontal stress field. For a rigid wall, on the other hand, the plane strain and the plane stress results are quite different. In the first case, there is a large return flow on the sides of the punch, the material being extruded along the only free surface available. In the plane stress case, the return flow disappears, and the material displaced by the penetration of the die tends to thicken the plate. The role of a nonlinear constitutive relation is studied for power law creep. As the power of the flow limit increases, the flow retains its general features, but the deformation localizes creating sharper contrasts between high and low strain rate areas; in plane stress, the effect of nonlinearity is to increase the contrasts in vertical motion. Available data for Asia are discussed in the light of the new results.

SUMMARY An asymptotic linearized iterative elastic inversion method is proposed to invert 2‐D Earth parameters from multicomponent data and is tested numerically. The forward problem is solved by a combination of the Born approximation and ray theoretical methods. We express the perturbed seismogram in terms of perturbations of P‐ and S‐wave impedances and density. The inversion method is based on generalized least squares. We introduce a special form of the ρ2 norm with a weighting function that corrects for geometrical spreading and obliquity of the reflectors. The Hessian for this norm could be estimated in a closed form that is asymptotically valid at high frequencies. We propose a quasi‐Newtonian iterative method for the solution of the inverse problem. The first iteration of this inversion method resembles the operator proposed by Beylkin (1985) and Beylkin & Burridge (1990) for the asymptotic inversion of seismic data. Our method is more general than theirs because it can handle arbitrary discrete distributions of sources and receivers. Elastic inversion is generally ill‐posed because the problem is overdetermined but undersampled. We study the resolution of the asymptotic inversion method for general sets of sources and receivers. We show that simultaneous inversion for both P‐ and S‐wave impedance is generally ill‐conditioned if data for a single scattering mode are available. In particular, it seems that only one parameter can be reliably resolved from marine data. Simultaneous inversion for a finite set of parameters can be resolved only for multicomponent elastic data containing both P‐wave and S‐wave information. Inversion tests using synthetic data calculated by finite‐differences demonstrates that it is possible to invert simultaneously for P and S impedances.

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