Verification of correctness of using real part of complex root as Rayleigh-wave phase velocity with synthetic data |
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| Institution: | 1. Microseismic Inc., formerly Colorado School of Mines, United States;2. Halliburton, United States;3. Colorado School of Mines, United States;4. U.S. Geological Survey, United States;5. Seismic Science LLC, United States;6. ConocoPhillips, United States;1. School of Mechanical Engineering, University of Leeds, Leeds, United Kingdom;2. Leeds Academic Surgical Unit, St. James''s University Hospital, Leeds, United Kingdom;1. Hubei Subsurface Multi-scale Imaging Key Laboratory, China University of Geosciences, 388 Rumo Rd., Wuhan, Hubei 430074, China;2. Subsurface Imaging and Sensing Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, 388 Rumo Rd., Wuhan, Hubei 430074, China;1. Department of Earth and Space Sciences, Southern University of Science and Technology, Shenzhen 518055, China;2. College of Geo-Exploration Science and Technology, Jilin University, Changchun 130026, China;3. School of Geomatics and Prospecting Engineering, Jilin Jianzhu University, Changchun 130118, China;4. School of Geological Engineering and Geomatics, Chang''An University, Xi''an 710054, China;1. Subsurface Imaging and Sensing Laboratory, School of Earth Sciences, Zhejiang University, 38 Zheda Rd., Hangzhou, Zhejiang 310012, China;2. Subsurface Imaging and Sensing Laboratory, Institute of Geophysics and Geomatics, China University of Geosciences, 388 Rumo Rd., Wuhan, Hubei 430074, China |
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| Abstract: | High-frequency (≥ 2 Hz) Rayleigh-wave phase velocities have been utilized to determine shear-wave velocities in near-surface geophysics since the early 1980s. One of the key steps is to calculate theoretical dispersion curves of an earth model. When the earth model contains a low-velocity half-space, however, some roots of the dispersion equation turn out to be complex numbers, which makes phase velocities disappear at some frequencies. When encountering this situation, the common practice is to append an additional high velocity layer as the half-space to the model to make the roots real or use the real parts of complex roots as Rayleigh-wave phase velocities. The correctness of the first method has been verified. The correctness of the second method, however, remains to be unproved. We use synthetic data generated by numerical modeling of the wave equation to verify the correctness of the second method. In this paper, we firstly discuss the reasons that only complex numbers of the dispersion equation exist at some frequencies when an earth model contains a low velocity half-space. Then we discuss how the nearest offset affects a synthetic model and recommend an optimal nearest offset in generating synthetic data that are close to real-world situations. Several synthetic models are used to verify correctness of using real parts of complex roots as Rayleigh-wave phase velocities when an earth model contains a low velocity layer as the half-space. |
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