Many types of seismic data indicate that there are substantial three-dimensional variatons in velocity and Q in the lower as well as the upper mantle. A number of studies have established a qualitative link between these variations and fluctuations in body wave amplitudes. Quantitative analyses are rare since they require a method for treating wave propagation in three-dimensional structures. The three-dimensional slowness method has been investigated as possible tool for carrying out such calculations. Some elements of the theory behind the method have been clarified in such a way as to make the method easier to apply. Some new developments of the theory have been made as well. It has been shown that there are two possible choices for ray paths in the slowness calculations. It is recommended that ray paths related to generalized rays in the limit of vanishing layer thickness be used rather than the geometric paths in the medium. The slowness integral has been cast in such a form as to have an explicit integration over horizontal takeoff angle as well as vertical takeoff angle (classical ray parameter).

The integration over horizontal angle is evaluated in exactly the same way as the integration over the vertical angle. The latter integration has become familiar as the slowness representation of the response of a one-dimensional medium. It has been shown that the integral over horizontal takeoff azimuth reduces to the convolution integral with 1/t1/2 for the case of a one-dimensional medium. We present an alternate geometric interpretation of the slowness method rather than the disk ray interpretation. We consider several different possible applications of the method.