The traveltime equation, tau-rho mapping and inversion of common midpoint data

Seismic traveltime data can be described in terms of instantaneous slope (dT/dX) and intercept time (tau ). With this parameterization, the most general form of the traveltime equation is developed and found to be valid for reflections and refractions, all source-receiver offsets, and all commonly used experimental geometries. For common source-receiver experiments, the instantaneous slope of the traveltime data represents the true horizontal ray parameter p, and the intercept time is the sum along the ray of thickness-vertical slowness products referenced to the fixed surface station. For common midpoint (CMP) experiments, we show that the instantaneous slope is the average slowness of upgoing and downgoing rays at the surface, and the intercept time is again the sum of thickness-vertical slowness products, but referenced to the common midpoint. This inherent averaging immediately explains the superiority of the CMP experiment in the presence of dip. In addition to providing a general traveltime equation, we find that tau -p parameterization reorganizes traveltime data in a way which greatly facilitates interpretation. For reflections, the tau -p trajectories are sums of ellipses and all turning rays (whether refracted or postcritically reflected) combine to form a well defined and easily identified trajectory. We present a method for deriving a velocity-depth function from this trajectory which is applicable to both refracted and reflected arrivals.