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.
I. Ravve and Z. Koren, "Hyperbolic Velocity Model," International Journal of Geosciences, Vol. 4 No. 4, 2013, pp. 724-745. doi: 10.4236/ijg.2013.44067.
[1] M. Muskat, “A Note on Propagation of Seismic Waves,” Geophysics, Vol. 2, No. 4, 1937, pp. 319-328. doi:10.1190/1.1438098 [2] M. M. Slotnick, “On Seismic Computations, with Applications, Part I,” Geophysics, Vol. 1, No. 1, 1936, pp. 9-22. doi:10.1190/1.1437084 [3] M. M. Slotnick, “Lessons in Seismic Computing,” Society of Exploration Geophysicists, Oklahoma, 1959. [4] M. M. Slotnick, “On Seismic Computations, with Applications, Part II,” Geophysics, Vol. 1, No. 3, 1936, pp. 299-305. doi:10.1190/1.1437111 [5] M. Al-Chalabi, “Instantaneous Slowness versus Depth Functions,” Geophysics, Vol. 62, No. 1, 1997, pp. 270-273. doi:10.1190/1.1444127 [6] C. H. Chapman and H. Keers, “Application of the Maslov Seismogram Method in Three Dimensions,” Studia Geophysica et Geodaetica, Vol. 46, No. 4, 2002, pp. 615-649. doi:10.1023/A:1021104820892 [7] C. E. Houston, “Seismic Paths, Assuming a Parabolic Increase of Velocity with Depth,” Geophysics, Vol. 4, No. 4, 1939, pp. 232-236. doi:10.1190/1.1440500 [8] M. Al-Chalabi, “Parameter Non-Uniqueness in Velocity versus Depth Functions,” Geophysics, Vol. 62, No. 3, 1997, pp. 970-979. doi:10.1190/1.1444203 [9] L. Y. Faust, “Seismic Velocity as a Function of Depth and Geologic Time,” Geophysics, Vol. 16, No. 2, 1951, pp. 192-206. doi:10.1190/1.1437658 [10] L. Y. Faust, “A Velocity Function Including Lithologic Variation,” Geophysics, Vol. 18, No. 2, 1953, pp. 271-288. doi:10.1190/1.1437869 [11] I. Ravve and Z. Koren, “Exponential Asymptotically Bounded Velocity Model, Part I: Effective Models and Velocity Transformations,” Geophysics, Vol. 71, No. 3, 2006, pp. T53-T65. doi:10.1190/1.2196033 [12] I. Ravve and Z. Koren, “Exponential Asymptotically Bounded Velocity Model, Part II: Ray Tracing,” Geophysics, Vol. 71, No. 3, 2006, pp. T67-T85. doi:10.1190/1.2194897 [13] I. Ravve and Z. Koren, “Conic Velocity Model,” Geophysics, Vol. 72, No. 3, 2007, pp. U31-U46. doi:10.1190/1.2710205 [14] Z. Koren and I. Ravve, “Constrained Dix Inversion,” Geophysics, Vol. 71, No. 6, 2006, pp. R113-R130. doi:10.1190/1.2348763 [15] E. Robein, “Velocities, Time-Imaging and Depth-Imaging in Reflection Seismics: Principles and Methods,” EAGE Publications, Houten, the Netherlands, 2003. [16] H. Kaufman, “Velocity Functions in Seismic Prospecting,” Geophysics, Vol. 18, No. 2, 1953, pp. 289-297. doi:10.1190/1.1437871 [17] R. M. Corless, G. H. Gonnet, D. G. Hare, D. J. Jeffrey and D. D. Knuth, “On the Lambert W Function,” Advances in Computational Mathematics, Vol. 5, No. 1, 1996, pp. 329-359. doi:10.1007/BF02124750