Wave-Equation Velocity Analysis

Velocity sensitive (wide propagation angle) seismic data do not comply with the r.m.s. small propagation angle approximation. A hyperbolic velocity spectrum and Dix’s equation cannot use wide-angle arrivals to estimate interval velocity accurately. The linear moveout method measures interval velocity exactly in stratified media. Snell midpoint coordinates are constructed to image the data before velocity estimation. Energy focuses at the arrival coordinates of a fixed reference Snell wavefront as required by the linear moveout method. The image of a common-midpoint seismic gather in Snell midpoint coordinates, for a nonvertical reference Snell wave, defines the wave-equation velocity spectrum. Two important properties of the velocity spectrum are locality, where energy is a local function of velocity; and linearity, that is invertible using linear transformations. Velocity sensitivity of wave-equation extrapolation operators increases with the angle of propagation. In Snell midpoint coordinates, angles are measured relative to an arbitrary slanted reference Snell wave. At this particular angle wave-equation operators are exact, independent of velocity. Approximations of the wave equation in Snell midpoint coordinates satisfactorily image wide-angle energy. To compute the velocity spectrum, we use the 15-degree finite-difference wave equation in the frequency domain. This equation is insensitive to the downward continuation velocity. This formation resolves multivalued, wide velocity spectrum data using in-homogeneous, offset and depth dependent, downward continuation velocity. Stepout filtering concurrent with downward continuation eliminates wide-angle propagation energy not modeled by the 15-degree wave equation.