We have developed and implemented a new method for estimating traveltimes and amplitudes in a general smooth two-dimensional (2-D) model. The basic idea of this wavefront (WF) construction approach is to use ray tracing to estimate a new WF from the old one. The WF is defined as a curve (in 2-D) of constant traveltime from the source. The ray direction and amplitude will then be a function of s, the distance along the front. To maintain a sufficiently small sampling distance along the WF, it is scanned at every time step and new rays are interpolated whenever the distance between two rays becomes larger than a predefined limit. As the wavefronts are constructed, the data (i.e. traveltimes, amplitude coefficients, etc.) are transferred to the receivers by interpolation within the ray cells.Advantages of the WF construction method are its flexibility, robustness, and accuracy. First, second, and later arrivals may be found at any point in the model. Any shape of the initial wavefront is possible.The drawbacks of the method are the same as for conventional ray tracing: large velocity contrasts, caustics and near-critical incidence angle of rays onto interfaces will give less accurate solutions. If the distance between two neighboring points on the new WF exceeds DS max' a new start position of a ray is

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A new 3D wavefield modelling approach based on dynamic ray tracing is presented. This approach is called wavefront construction, and it can be used in 3D models with constant or smoothly varying material properties (Sand P-velocity and density) separated by smooth interfaces. Wavefronts consisting of rays arranged in a triangular network are propagated stepwise through the model. At each time step, the differences in a number of parameters are checked between each pair of rays on the wavefront. New rays are interpolated whenever this difference between pairs of rays exceeds some predefined maximum value. A controlled sampling of the wavefront at all time steps is thus obtained. Receivers are given multiple-event values by interpolation when the wavefronts pass them. The strength of the wavefront construction method is that it is robust and efficient.

MSC: 35R30 86A15 53C21Keywords: Geometric inverse problems Riemannian manifold Shape operatorWe analyze the inverse problem, if a manifold and a Riemannian metric on it can be reconstructed from the sphere data. The sphere data consist of an open set U ⊂M and the pairs (t, Σ) where Σ ⊂ U is a smooth subset of a generalized metric sphere of radius t. This problem is an idealization of a seismic inverse problem, originally formulated by Dix [8], of reconstructing the wave speed inside a domain from boundary measurements associated with the single scattering of waves. In this problem, one considers a domain M with a varying and possibly anisotropic wave speed which we model as a Riemannian metric g. For our data, we assume that M contains a dense set of point diffractors and that in a subset U ⊂M , we can measure the wave fronts of the waves generated by these. The inverse problem we study is to recover the metric g in local coordinates anywhere on a set M ⊂M up to an isometry (i.e. we recover the isometry type of M ). To do this we show that the shape operators related to wave fronts produced by the point diffractors within M satisfy a certain system of differential equations which may be solved along geodesics of the metric. In this way, assuming that we know g as well as the shape operator of the wave fronts in the region U , we may recover g in certain coordinate systems (e.g. Riemannian normal coordinates centered at point diffractors). This generalizes the method of Dix to metrics which may depend on all spatial variables and be anisotropic. In particular, the novelty of this solution lies in the fact that it can be used to reconstruct the metric also in the presence of the caustics. (M.V. de Hoop), sean.holman@manchester.ac.uk (S.F. Holman), Einar.Iversen@norsar.com (E. Iversen), matti.lassas@helsinki.fi (M. Lassas), bjorn.ursin@ntnu.no (B. Ursin). http://dx.doi.org/10.1016/j.matpur.2014.09.003 0021-7824/© 2014 Elsevier Masson SAS. All rights reserved. M.V. de Hoop et al. / J. Math. Pures Appl. 103 (2015) 830-848 831 r é s u m éOn analyse un problème inverse, si une variété riemannienne peut être reconstruite à partir des données sphère. Les données sphère sont constituées d'un ensemble ouvert U ⊂M et les paires (t, Σ), où Σ ⊂ U set un sous-ensemble lisse d'une sphère métrique généralisée. Ce problème est une idéalisation d'un problème sismique inverse, à l'origine formulé par Dix [8], consistant à reconstruire la vitesse d'onde dans un domaine à partir des mesures aux frontières associées à la dispersion simple des ondes sismiques. On considère un domaine M avec une vitesse d'onde variable et éventuellement anisotrope modélisée par une métrique riemannienne g. On suppose que M contient une densité élevée de points diffractants et que dans un sous-ensemble U ⊂M , correspondant à un domaine contenant les instruments de mesure, on peut mesurer les fronts d'onde de la diffusion simple des ondes diffractées depuis les points diffractants. Le problème inverse étudié consiste à reconstruire la métrique g en coordon...

Seismic time migration is known for its ability to generate well-focused and interpretable images, based on a velocity field specified in the time domain. A fundamental requirement of this time-migration velocity field is that lateral variations are small. In the case of 3D time migration for symmetric elementary waves (e.g., primary PP reflections/diffractions, for which the incident and departing elementary waves at the reflection/diffraction point are pressure [P] waves), the time-migration velocity is a function depending on four variables: three coordinates specifying a trace point location in the time-migration domain and one angle, the so-called migration azimuth. Based on a time-migration velocity field available for a single azimuth, we have developed a method providing an image-ray transformation between the time-migration domain and the depth domain. The transformation is obtained by a process in which image rays and isotropic depth-domain velocity parameters for their propagation are esti-mated simultaneously. The depth-domain velocity field and image-ray transformation generated by the process have useful applications. The estimated velocity field can be used, for example, as an initial macrovelocity model for depth migration and tomographic inversion. The image-ray transformation provides a basis for time-to-depth conversion of a complete time-migrated seismic data set or horizons interpreted in the time-migration domain. This time-to-depth conversion can be performed without the need of an a priori known velocity model in the depth domain. Our approach has similarities as well as differences compared with a recently published method based on knowledge of time-migration velocity fields for at least three migration azimuths. We show that it is sufficient, as a minimum, to give as input a time-migration velocity field for one azimuth only. A practical consequence of this simplified input is that the image-ray transformation and its corresponding depth-domain velocity field can be generated more easily.