Crosswell tomography using generalized traveltime inversion

Abstract

A nonlinear inversion method is presented for determining the velocity model from crosswell data. The method is a modification of Rytov inversion and is based on the phase of the wavefield. The modified Ryton norm is applied to the reconstruction of depth-dependent velocity models and we conclude that, in its behavior, it is similar to an error norm based on arrival times but it avoids the necessity of having to accurately determine the arrival times. Especially when many direct arrivals are present, for instance, if source and receivers are located in a low-velocity region, this is an important advantage. I n t r o d u c t i o n Imaging of crosswell reflection data can reveal the fine structure of the region between the wells. In principle, imaging methods for crosswell data are similar to the methods that are currently applied to surfaceseismic data. These methods only give accurate results if a reliable estimate of the background model is available. Determination of this background model is a nonlinear problem and it seems fair to say that, until now, the most successful and widely used methods are based on arrival times of the direct waves between sources and receivers (see, for instance, Luo and Schuster, 1990). This type of methods is highly dependent upon accurate measurements of arrival times. The main disadvantage of these methods, therefore, is the cumbersome step of accurately measuring the arrival times. This is especially the case when several ray paths exist between source and receiver due to the presence of low-velocity layers, giving rise to more than one direct arrival. In the present paper, a method is presented that is, in a certain sense, a generalization of these arrival-time methods, in which actual arrival times have been replaced by time windows that contain the direct waves of interest. In this way, the requirement of accurately having to measure arrival times has been replaced by the (less stringent and less subjective) requirement of having to define time windows around the main arrivals of interest. It is a modification of a method developed by Herman (1992) for the case of reflection seismology. F o r m u l a t i o n o f t h e f o r w a r d p r o b l e m We consider the crosswell geometry shown in figure 1, with sources and receivers placed in two vertical boreholes. For the tomographic problem under consideration, we only consider the direct waves from sources to receivers. We assume that the velocity model is smooth and, furthermore, not depending upon the y-coordinate perpendicular to the plane through the wells. For smooth velocity models, the direct waves can be computed accurately and efficiently with the aid of high-frequency asymptotics. We only consider scalar compressional waves. With the aid of high-frequency asymptotics, the direct wave P in the frequency domain can be expressed in terms of a number of different arrivals, where and denote the coordinates of the receiver and source, respectively, and where w denotes angular frequency. The multiple arrivals if present, are caused by folding of the wavefront which occurs if more than one ray path exists between source and receiver. They are given by In this equation, is the traveltime between source and receiver of the i-th arrival and satisfies the eikonal equation