VSP traveltime inversion for anisotropy in a buried layer Michael A. Slawinski, 1 * Michael P. Lamoureux, 2 Raphael A. Slawinski 3 and R. James Brown 3 1 Department of Earth Sciences, Memorial University of Newfoundland, Alexander Murray Building, St John's, NF A1B 3X5, 2 Department of Mathematics and Statistics, University of Calgary, Alberta T2N 1N4, and 3 Department of Geology and Geophysics, University of Calgary, Alberta T2N 1N4, Canada Received November 2001, revision accepted November 2002 ABSTRACT We present a method for calculating the anisotropy parameter of a buried layer by inverting the total traveltimes of direct arrivals travelling from a surface source to a well-bore receiver in a vertical seismic profiling (VSP) geometry. The method as- sumes two-dimensional media. The medium above the layer of interest (and separ- ated from it by a horizontal interface) can exhibit both anisotropy and inhomogeneity. Both the depth of the interface as well as the velocity field of the overburden are assumed to be known. We assume the layer of interest to be homogeneous and elliptically anisotropic, with the anisotropy described by a single parameter w. We solve the function describing the traveltime between source and receiver explicitly for w. The solution is expressed in terms of known quantities, such as the source and receiver locations, and in terms of quantities expressed as functions of the single argument x r , which is the horizontal coordinate of the refraction point on the interface. In view of Fermat's principle, the measured traveltime T possesses a stationary value or, considering direct arrivals, a minimum value, T  min xr tx r ; w  . This gives rise to a key result ± the condition that the actual anisotropy parameter  0  min xr  x r  . Owing to the explicit expression    x r  , this result allows a direct calculation of  0 in the layer of interest. We perform an error analysis and show this inverse method to be stable. In particular, for horizontally layered media, a traveltime error of one millisecond results in a typical error of about 20% in the anisotropy parameter. This is almost one order of magnitude less than the error inherent in the slowness method, which uses a similar set of experimental data. We conclude by detailing possible extensions to non-elliptical anisotropy and a non- planar interface. 1 INTRODUCTION An accurate description of the anisotropy of a medium pro- vides important lithological information and enhances seis- mic imaging of the subsurface. The importance of taking anisotropic effects into account has been emphasized in recent decades due to improved data-acquisition and data- processing methods. Neglecting even a seemingly modest amount of anisotropy can lead to significant degradation in processed seismic images (e.g. Winterstein 1986; Vestrum, Lawton and Schmid 1999). Various approaches for determin- ing anisotropy parameters have been formulated. For example, Gaiser (1990) showed how to estimate the aniso- tropy parameters of a transversely isotropic medium from vertical seismic profiling (VSP) using vertical and horizontal phase-slowness measurements. Miller and Spencer (1994) and Miller, Leaney and Borland (1994) presented methods for obtaining anisotropy parameters by inverting ß 2003 European Association of Geoscientists & Engineers 131 Geophysical Prospecting, 2003, 51, 131±139 *E-mail: mslawins@mun.ca