spaceABSTRACT Conventional crosswell direct-arrival traveltime to-mography solves for velocity in a 2-D slice of the subsur-face joining two wells. Many 3-D aspects of real crosswell surveys, including well deviations and out-of-well-plane structure, are ignored in 2-D models. We present a 3-D approach to crosswell tomography that is capable of handling severe well deviations and multiple-profile datasets. Three-dimensional pixelized models would be even more seriously underdetermined than the

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... ixelized models that have been used in 2-D tomography. We, therefore, employ a thinly layered, vertically discontinu-ous 3-D velocity model that greatly reduces the number of model parameters. The layers are separated by 2-D in-terfaces represented as 2-D Chebyshev polynomials that are determined using a priori structural information and remain fixed in the traveltime inversion. The velocity in each layer is also represented as a 2-D Chebyshev polynomial. Unlike pixelized models that provide lim-ited vertical resolution and may be overparameterized horizontally, this 3-D model provides vertical resolution comparable to the scale of wireline logs, and reduces the degrees of freedom in the horizontal parameterization to the expected in-line and out-of-well-plane horizontal resolution available in crosswell traveltime data. Ray tracing for the nonlinear traveltime inversion is performed in three dimensions. The 3-D tomography spaceproblem is regularized using penalty constraints with a continuation strategy that allows us to extrapolate the velocity field to a 3-D region containing the 2-D cross-well profile. Although this velocity field cannot be ex-pected to be accurate throughout the 3-D region, it is at least as accurate as 2-D tomograms near the well plane of each 2-D crosswell profile. Futhermore, multiple-profile crosswell data can be inverted simultaneously to re-solve better the 3-D distribution of velocity near the profiles. Our velocity parameterization is quite different from pixelized models, so resolution properties will be differ-ent. Using wave-modeled synthetic data, we find that near horizontal raypaths have the largest mismatch be-tween raytraced traveltimes and traveltimes estimated from the data. In conventional tomography, horizontal raypaths are essential for high vertical resolution. With our model, however, the highest resolution and most accurate inversions are achieved by excluding raypaths that travel nearly parallel to the geologic layering. We perform this exclusion in both a static and model-based manner. We apply our 3-D method to a multiple-profile crosswell survey at the Cymric oil field in California, an area of very steep structural dips and significant well tra-jectory deviations. Results of this multipleprofile 3-D tomography correlate very well with the independently-processed single profile results, with the advantage of an improved tie at the common well. space spaceINTRODUCTION Crosswell direct arrival traveltime tomography for a single profile between two wells is inherently a 2-D problem. Al-though the subsurface of the Earth is three dimensional, the energy recorded in a crosswell dataset that influences direct arrival traveltimes has traveled mostly through a quasi-2-D region close to a 2-D surface joining the wells. Because the spacevelocity in this quasi-2-D region is all that we can hope to determine from the traveltime data, approaches to crosswell tomography have generally attempted to determine a velocity profile defined on a planar 2-D slice. Many 3-D aspects of the geometry either have to be ignored in a planar 2-D model or included with great effort. Most wells are not perfectly straight and vertical. Each well separately may not even stay within a single plane, and it is not uncommon for two wells together to space Manuscript far from coplanar. In these situations, the quasi-2-D region of the subsurface influencing the crosswell traveltimes is not close to any 2-D plane. Even if the two wells are coplanar, energy can travel along paths out of that plane if there is out-of-plane dip in the structure or out-ofplane velocity variation near the wells. Multiple-profile crosswell tomography can be handled as a suite of separate single-profile 2-D problems, each providing the velocity field on a 2-D slice of the subsurface. Incorporating several separate 2-D profiles into one consistent 3-D model is not easy, however, particularly when there are misties between profiles at wells involved in more than one profile. The 2-D profiles can be extended beyond the well planes using inter-polation, extrapolation, or least-squares fitting to construct a 3-D model that is approximately consistent with all of the 2-D profiles. In most multiple-profile situations, these 2-D profiles are not dense enough to get an accurate 3-D model. Crosswell tomography in realistic geometries thus faces a serious difficulty. Because of the 3-D aspects of the problem mentioned above, we need to have a 3-D velocity field to trace rays accurately, but the traveltime data determine only one (or several) 2-D slices of the velocity field. We present here a strategy for overcoming this difficulty. Instead of a two-stage process of determining 2-D slices and then extrapolating them to obtain a 3-D model, we will deter-mine the 2-D profiles and their extrapolation to 3-D together simultaneously, obtaining directly a 3-D velocity model. Much of this 3-D velocity model will be inaccurate because there are insufficient data to determine it accurately. But the parts in the quasi-2-D regions between the wells will be recovered as accu-rately as they would be using a 2-D approach. Two-dimensional tomograms can be extracted from this 3-D model: the rays tell us where the 2-D surfaces are, and these surfaces can be pro-jected onto nearby planes for display purposes. Our strategy is able to handle well deviations and out-of-plane dip, and works effectively for both single-profile and multiple-profile tomography problems. The four aspects of our strategy that we discuss in detail in this paper are the following. First, we parameterize our velocity model in a manner appro-priate for this combined tomography and extrapolation, using a priori structural information to help determine ‡