Abstract
Asymptotically bounded velocity profiles describe the vertical velocity variations in compacted sediments in a more realistic way than unbounded velocity models, and allow presenting the subsurface by a smaller number of thicker layers. The first and the simplest asymptotically bounded model is the Hyperbolic velocity profile proposed by Muscatin 1937, and our paper is an extension of this early study. The Hyperbolic model has an advantage over other bounded models: The velocity increases with depth and approaches the limiting value with a more smooth and gradual rate. We derive the time-depth relationships, forward and backward transforms between the instantaneous velocity profile and the effective models (average, RMS and fourth order average velocities), study the trajectories for pre-critical and post-critical curved rays and derive the equations for traveltime, lateral propagation and arc length. We compare the ray paths obtained with the Hyperbolic model and with the other bounded velocity profiles.
Highlights
The Hyperbolic velocity model was first proposed by Muscat [1] and published in 1937
The first and the simplest asymptotically bounded model is the Hyperbolic velocity profile proposed by Muscat in 1937, and our paper is an extension of this early study
The Hyperbolic model has an advantage over other bounded models: The velocity increases with depth and approaches the limiting value with a more smooth and gradual rate
Summary
The Hyperbolic velocity model was first proposed by Muscat [1] and published in 1937. Since the model was not extensively studied and is unjustifiably ignored in the literature. The objective of this research is to extend the original study and to correct the inaccuracies. We show the place of the Hyperbolic model among the other asymptotically bounded models, analyze its basic relationships and attempt to develop a complete theory
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