# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021031

## Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion

 1 Shandong University of Science and Technology, College of Earth Science and Engineering, Qingdao 266590, China 2 Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China 3 University of Chinese Academy of Sciences, Beijing 100049, China 4 Innovation Academy for Earth Science, Chinese Academy of Sciences, Beijing 100029, China

* Corresponding author: Y. F. Wang

Received  July 2020 Revised  February 2021 Published  May 2021

Fund Project: This work is sponsored by the National Key R & D Program of The Ministry of Science and Technology of China (Nos. 2018YFC1504203, 2018YFC0603500) and the key programs IGGCAS-2019031 & SZJJ-201901

This paper studies the S wave velocity modeling based on the Rayleigh wave dispersion curve inversion. We first discuss the forward simulation, and present a fast root-finding method with cubic-order of convergence speed to obtain the Rayleigh wave dispersion curve. With the Rayleigh wave dispersion curve as the observation data, and considering the prior geological anomalies structural information, we establish a sparse constraint regularization model, and propose an iterative solution method to solve for the S wave velocity. Experimental tests are performed both on the theoretical models and on the field data. It indicates from the experimental results that our new inversion scheme possesses the characteristics of easy calculation, high computational efficiency and high precision for model characterization.

Citation: Yan Cui, Yanfei Wang. Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion. Inverse Problems & Imaging, doi: 10.3934/ipi.2021031
##### References:
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##### References:
 [1] X. F. Chen, A systematic and efficient method of computing normal modes for multilayered half-space, Geophysical Journal International, 115 (1993), 391-409.  doi: 10.1111/j.1365-246X.1993.tb01194.x.  Google Scholar [2] X. F. Chen, Seismogram synthesis in multi-layered half-space Part Ⅰ, Theoretical formulations, Earthquake Research in China, 13 (1999), 149-174.   Google Scholar [3] H. A. Diao, P. J. Li and X. K. Yuan, Inverse elastic surface scattering with far-field data, Inverse Problems and Imaging, 13 (2019), 721-744.  doi: 10.3934/ipi.2019033.  Google Scholar [4] M. V. de Hoop, G. Nakamura and J. Zhai, Unique recovery of piecewise analytic density and stiffness tensors from the elastic-wave Dirichlet-to-Neumann map, SIAM Journal on Applied Mathematics, 79 (2019), 2359-2384.  doi: 10.1137/18M1232802.  Google Scholar [5] M. V. de Hoop, A. Iantchenko and R. D. van der Hilst and J. Zhai, Semiclassical inverse spectral problem for seismic surface waves in isotropic media Ⅰ: Love waves, Inverse Problems, 36 (2020), 075015, 27 pp. doi: 10.1088/1361-6420/ab8446.  Google Scholar [6] M. V. de Hoop, A. Iantchenko and R. D. van der Hilst and J. Zhai, Semiclassical inverse spectral problem for seismic surface waves in isotropic media Ⅱ: Rayleigh waves, Inverse Problems, 36 (2020), 075016, 31 pp. doi: 10.1088/1361-6420/ab8447.  Google Scholar [7] P. Gabriels, R. Snieder and G. Nolet, In situ measurements of shear-wave velocity in sediments with higher-mode Rayleigh waves, Geophysical Prospecting, 35 (1987b), 187-196.  doi: 10.1111/j.1365-2478.1987.tb00812.x.  Google Scholar [8] N. A. Haskell, The dispersion of surface waves on multilayered media, Bulletin of the Seismological Society of America, 43 (1953), 17-34.   Google Scholar [9] C. G. Lai, G. J. Rix, S. Foti and V. Roma, Simultaneous measurement and inversion of surface wave dispersion and attenuation curves, Soil Dynamics and Earthquake Engineering, 22 (2002), 923-930.  doi: 10.1016/S0267-7261(02)00116-1.  Google Scholar [10] G. Nakamura and G. Uhlmann, Inverse problems at the boundary for an elastic medium, SIAM Journal on Mathematical Analysis, 26 (1995), 263-279.  doi: 10.1137/S0036141093247494.  Google Scholar [11] F. Schwab and L. Knopoff, Surface-wave dispersion computations, Bulletin of the Seismological Society of America, 60 (1970), 321-344.   Google Scholar [12] K. T. Tran and D. R. Hiltunen, One-dimensional inversion of full waveforms using a genetic algorithm, Journal of Environmental & Engineering Geophysics, 17 (2012), 197-213.  doi: 10.2113/JEEG17.4.197.  Google Scholar [13] K. T. Tran and D. R. Hiltunen, One-dimensional inversion of full waveforms using a genetic algorithm, Journal of Geotechnical & Geoenvironmental Engineering, 138 (2012b), 1075-1090.   Google Scholar [14] Y. F. Wang and T.-Y. Xiao, Fast realization algorithms for determining regularization parameters in linear inverse problems, Inverse Problems, 17 (2001), 281-291.  doi: 10.1088/0266-5611/17/2/308.  Google Scholar [15] Y. F. Wang, I. E. Stepnova, V. N. Titarenko and A. G. Yagola, Inverse Problems in Geophysics and Solution Methods, Higher Education Press, Beijing, 2011a.   Google Scholar [16] Y. F. Wang, A. G. Yagola and C. C. Yang (eds.), Optimization and Regularization for Computational Inverse Problems and Applications, Springer, Berlin, 2011b. Google Scholar [17] J. H. Xia, R. D. Miller and C. B. Park, Estimation of near-surface shear-wave velocity by inversion of Rayleigh wave, Geophysics, 64 (1999), 691-700.  doi: 10.1190/1.1444578.  Google Scholar [18] J. H. Xia, High-Frequency Surface-wave Methods, China University of Geosciences Press, Wuhan, 2015.   Google Scholar [19] W. C. Yang, Theory and Methods of Geophysical Inversion, Geological Publishing House, Beijing, 1997. Google Scholar
Configuration and coordinate system of a multilayered half space
Schematic diagram of the generalized reflection-transmission coefficient method. ${{\rho }^{\left( j \right)}},{{\lambda }^{\left( j \right)}},{{\mu }^{\left( j \right)}}$ and ${{h}^{\left( j \right)}}$ are the density, the Lamé constants and the layer thickness of the $j$-th layer; ${{\mathbf{C}}^{\left( j \right)}} = {{\left[ \mathbf{C}_{d}^{\left( j \right)},\mathbf{C}_{u}^{\left( j \right)} \right]}^{T}}$ are the corresponding coefficients of these downgoing and upgoing waves of the $j$-th layer; ${{\mathbf{R}}^{\left( j \right)}} = {{[\mathbf{R}_{ud}^{\left( j \right)},\mathbf{R}_{du}^{\left( j \right)}]}^{T}}$ and ${{\mathbf{T}}^{\left( j \right)}} = {{\left[ \mathbf{T}_{u}^{\left( j \right)},\mathbf{T}_{d}^{\left( j \right)} \right]}^{T}}$ are the reflection coefficients and the transmission coefficients of the $j$-th layer
Inversion results using SIRW and LSM. (a) Data fitting for noiseless data: observed data (blue asterisks), dispersion curves corresponding to the initial model (brown dashed line), the models inverted from the noise-free data using SIRW (red solid line) and LSM (green solid line); (b) Inverted results with noiseless data: the true S wave velocity model (blue solid line), the initial S wave velocity model (blue dashed line), the S wave velocity models inverted by the SIRW (red solid line) and the LSM (green solid line) for the noise-free data; (c) Data fitting for noisy data: observed data (blue asterisks), dispersion curves corresponding to the initial model (brown dashed line), the models inverted from the noisy data using SIRW (red solid line) and LSM (green solid line); (d) Inverted results with noisy data: the true S wave velocity model (blue solid line), the initial S wave velocity model (blue dashed line), the S wave velocity models inverted by the SIRW (red solid line) and the LSM (green solid line) for the noisy data
])">Figure 4.  Inversion results for the field data (a) Seismic record; (b) Normalized power spectrum and the extracted fundamental dispersion curve (red solid line). (c) Initial S wave velocity model (green dashed line), S wave velocity structure inverted by the SIRW (red solid line) and S wave logging results (blue dots). (d) A geologic profile ([13])
Parameters of the five-layer synthetic model
 Layer S wave velocity P wave velocity Density Thickness numbers (m/s) (m/s) (g/cm3) (m) 1 300 560 1.7 2 2 340 640 1.8 5 3 400 750 1.8 3 4 360 680 1.9 4 5 500 900 2.0 $\infty$
 Layer S wave velocity P wave velocity Density Thickness numbers (m/s) (m/s) (g/cm3) (m) 1 300 560 1.7 2 2 340 640 1.8 5 3 400 750 1.8 3 4 360 680 1.9 4 5 500 900 2.0 $\infty$
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