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| Inversion of the Rayleigh wave dispersion curves based on the sine-cosine algorithm |
FU Yu1,2( ), AI Han-Bing1,2,3, YAO Zhen-An1,2(), MEI Zhu-Xu1,2, SU Ke-Jia1,2,4 |
1. Engineering Research Center for Seismic Disaster Prevention and Engineering Geological Disaster Detection of Jiangxi Province(East China University of Technology),Nanchang 330013,China
2. School of Geophysics and Measurement-control Technology,East China University of Technology,Nanchang 330013,China
3. School of Geophysics and Geomatics, China University of Geosciences(Wuhan),Wuhan 430074,China
4. Research Institute No.270,CNNC,Nanchang 330200,China |
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Abstract
Rayleigh wave is widely used in engineering investigation and surveys.The inversion of its dispersion curves allows for effectively obtaining stratigraphic information.However,conventional global optimization algorithms in the dispersion curve inversion have a slow convergence rate and low convergence precision and are prone to prematurity.Therefore,this study introduced a novel global optimization algorithm—the sine cosine algorithm (SCA)—to solve the problems mentioned above.Based on the mathematical properties of sine and cosine functions,the SCA can adjust the exploration and development capabilities during the optimization using multiple random parameters and adaptive variables.As a result,it can ensure a high convergence rate and great stability while obtaining high-accuracy solutions.First,the feasibility of the SCA for the dispersion curve inversion was verified using four noise-free models.Then,the strong anti-interference ability of the SCA was proved by adding 15% of random noise to the models.Afterward,it was verified that SCA can yield high-precision,high-stability solutions in the dispersion curve inversion by comparison with the particle swarm optimization (PSO) approach.Finally,the practicability of the SCA was confirmed using seismic data from Arnarbælidi in Iceland and Wyoming in the USA.As indicated by the calculation results of theoretical models and the analysis of measured data,the SCA has a high processing speed,precision,stability,and practicability and thus allows for effective quantitative interpretation of the Rayleigh wave dispersion curves.
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Received: 05 August 2022
Published: 23 January 2024
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Flow chart of SCA
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| 模型 | 层序号 | 模型参数 | | 搜索范围 | | vs/(m·s-1) | vp/(m·s-1) | ρ/(g·cm-3) | h/m | | vs/(m·s-1) | h/m | | 模型A | 1 | 200 | 780 | 1.9 | 5 | | 100~300 | 2.5~7.5 | | 均匀半空间 | 350 | 850 | 1.9 | ∞ | | 175~525 | ∞ | | 模型B | 1 | 200 | 663 | 1.9 | 2 | | 100~300 | 1~3 | | | 2 | 300 | 995 | 1.9 | 4 | | 150~450 | 2~6 | | 3 | 400 | 1327 | 1.9 | 6 | | 200~600 | 3~9 | | 均匀半空间 | 500 | 1658 | 1.9 | ∞ | | 250~750 | ∞ | | 模型C | 1 | 200 | 663 | 1.9 | 2 | | 100~300 | 1~3 | | | 2 | 160 | 673 | 1.9 | 4 | | 80~240 | 2~6 | | 3 | 300 | 1102 | 1.9 | 6 | | 150~450 | 3~9 | | 均匀半空间 | 400 | 1470 | 1.9 | ∞ | | 200~600 | ∞ | | 模型D | 1 | 150 | 498 | 1.9 | 2 | | 75~225 | 1~3 | | | 2 | 250 | 829 | 1.9 | 4 | | 125~375 | 2~6 | | 3 | 200 | 841 | 1.9 | 6 | | 100~300 | 3~9 | | 均匀半空间 | 400 | 1470 | 1.9 | ∞ | | 200~600 | ∞ | |
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Model parameter and search space
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Inversion results of noise-free data of model A
a—inverted response compared with the original one;b—inverted shear-wave velocity profile
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| 模型 | 参数 | 真实值 | | 不含噪声 | | | | 含噪声 | | | 反演均值 | 相对误差/% | 标准差 | | 反演均值 | 相对误差/% | 标准差 | | 模型A | vs1/(m·s-1) | 200 | 199.54 | 0.23% | 1.28 | | 199.69 | 0.15% | 2.71 | | vs2/(m·s-1) | 350 | 347.90 | 0.60% | 5.98 | | 350.79 | 0.23% | 9.07 | | H1/m | 5 | 4.76 | 4.76% | 0.18 | | 4.75 | 5.00% | 0.22 | | 模型B | vs1/(m·s-1) | 200 | 200.94 | 0.47% | 2.52 | | 200.95 | 0.47% | 3.82 | | vs2/(m·s-1) | 300 | 300.75 | 0.25% | 7.42 | | 304.07 | 1.36% | 15.60 | | vs3/(m·s-1) | 400 | 399.19 | 0.20% | 9.34 | | 391.88 | 2.03% | 25.47 | | vs4/(m·s-1) | 500 | 500.19 | 0.04% | 6.03 | | 499.62 | 0.08% | 6.80 | | H1/m | 2 | 2.05 | 2.52% | 0.11 | | 2.02 | 0.83% | 0.16 | | H2/m | 4 | 3.99 | 0.24% | 0.28 | | 3.96 | 1.12% | 0.18 | | H3/m | 6 | 5.59 | 6.76% | 0.53 | | 5.61 | 6.54% | 0.97 | | 模型C | vs1/(m·s-1) | 200 | 197.78 | 1.11% | 9.25 | | 199.90 | 0.05% | 14.71 | | vs2/(m·s-1) | 160 | 161.15 | 0.72% | 4.52 | | 160.55 | 0.34% | 5.35 | | vs3/(m·s-1) | 300 | 286.04 | 4.65% | 32.13 | | 285.78 | 4.74% | 22.89 | | vs4/(m·s-1) | 400 | 397.61 | 0.60% | 13.58 | | 397.19 | 0.70% | 18.96 | | H1/m | 2 | 1.97 | 1.57% | 0.17 | | 1.91 | 4.51% | 0.23 | | H2/m | 4 | 3.86 | 3.58% | 0.43 | | 3.85 | 3.71% | 0.30 | | H3/m | 6 | 5.62 | 6.28% | 1.09 | | 5.51 | 8.16% | 0.43 | | 模型D | vs1/(m·s-1) | 150 | 150.58 | 0.39% | 1.03 | | 149.80 | 0.13% | 2.92 | | vs2/(m·s-1) | 250 | 252.30 | 0.92% | 4.25 | | 251.41 | 0.56% | 7.38 | | vs3/(m·s-1) | 200 | 199.81 | 0.09% | 3.94 | | 202.01 | 1.01% | 4.41 | | vs4/(m·s-1) | 400 | 400.90 | 0.23% | 4.80 | | 400.46 | 0.12% | 6.36 | | H1/m | 2 | 2.03 | 1.41% | 0.07 | | 2.03 | 1.36% | 0.08 | | H2/m | 4 | 3.93 | 1.78% | 0.14 | | 3.98 | 0.44% | 0.24 | | H3/m | 6 | 6.09 | 1.45% | 0.23 | | 5.99 | 0.14% | 0.26 |
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Inversion results of noise-free and noise-contaminated data with model A,B,C and D
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Inversion results of noise-free data of model B,C,and D
a1,b1,c1—inverted dispersion curves obtained from model B,C and D respectively;a2,b2,c2—inverted shear-wave velocity profiles obtained from model B,C and D respectively
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Inversion results of noise-contaminated data of model A,B,C,and D
a1,b1,c1,d1—inverted dispersion curves obtained from model A,B,C and D respectively;a2,b2,c2,d2—inverted shear-wave velocity profiles obtained from model A,B,C and D respectively
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Inversion results of model C with multi-modal data
a—inverted responses compared with the original ones;b—inverted shear-wave velocity profile
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| 参数 | 真实值 | 基阶数据 | 多模数据 | | 反演均值 | 相对误差/% | 标准差 | 反演均值 | 相对误差/% | 标准差 | | vs1/(m·s-1) | 200 | 197.78 | 1.11% | 9.25 | 200.18 | 0.09% | 9.75 | | vs2/(m·s-1) | 160 | 161.15 | 0.72% | 4.52 | 158.98 | 0.64% | 4.60 | | vs3/(m·s-1) | 300 | 286.04 | 4.65% | 32.13 | 308.08 | 2.93% | 21.04 | | vs4/(m·s-1) | 400 | 397.61 | 0.60% | 13.58 | 393.26 | 1.68% | 15.64 | | H1/m | 2 | 1.97 | 1.57% | 0.17 | 1.99 | 0.70% | 0.23 | | H2/m | 4 | 3.86 | 3.58% | 0.43 | 3.99 | 0.28% | 0.30 | | H3/m | 6 | 5.62 | 6.28% | 1.09 | 6.08 | 1.38% | 0.43 |
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Inversion results of the fundamental-model and multi-model data with model C
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Comparison of the convergence rate between SCA and PSO on noise-free data from model D
a—comparison of convergence curves before zooming up;b—comparison of convergence curves after zooming up
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| 参数 | 真实值 | SCA | PSO | | 反演均值 | 相对误差/% | 标准差 | 反演均值 | 相对误差/% | 标准差 | | vs1/(m·s-1) | 150 | 150.58 | 0.39% | 1.03 | 148.06 | 1.29% | 9.91 | | vs2/(m·s-1) | 250 | 252.30 | 0.92% | 4.25 | 242.06 | 3.18% | 23.34 | | vs3/(m·s-1) | 200 | 199.81 | 0.09% | 3.94 | 176.28 | 11.86% | 37.69 | | vs4/(m·s-1) | 400 | 400.90 | 0.23% | 4.80 | 374.89 | 6.28% | 27.63 | | H1/m | 2 | 2.03 | 1.41% | 0.07 | 1.93 | 3.5% | 0.38 | | H2/m | 4 | 3.93 | 1.78% | 0.14 | 3.72 | 6.86% | 0.97 | | H3/m | 6 | 6.09 | 1.45% | 0.23 | 4.07 | 32.14% | 2.03 |
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Comparison of the inverted results generated by SCA and PSO
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Field data(a) and its dispersion image(b) in Arnarb?lidi [29]
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| 文献估计层厚 | 文献S波速度 | S波速度搜索范围 | 厚度搜索范围 | | /m | /(m·s-1) | /(m·s-1) | /m | | 0.8 | 78 | 60~90 | 0.1~1.5 | | 0.5 | 80 | 70~100 | 0.1~1.5 | | 0.7 | 92 | 80~110 | 0.1~1.5 | | 1.2 | 111 | 100~150 | 0.5~2.0 | | 1.9 | 141 | 120~200 | 0.5~2.0 | | 3.0 | 184 | 150~250 | 2.0~4.0 | | 4.7 | 230 | 200~300 | 3.0~5.5 | | 7.5 | 277 | 250~350 | 6.0~9.0 | | 5.2 | 350 | 300~400 | 4.0~6.0 | | ∞ | 350 | 320~450 | ∞ |
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Reference inversion model [29]parameters and search range[30] in Arnarb?lidi region
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Inversion results in Arnarb?lidi region
a—inverted response compared with the original one;b—the minimum value of the objective function changes with the number of iterations;c—inverted shear-wave velocity profile
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Field data(a) and its dispersion image(b) in Wyoming [32]
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| 层数 | vs/(m·s-1) | h/m | 泊松比 | ρ/(g·cm-3) | | 1 | 100~300 | 1~5 | 0.38 | 2.0 | | 2 | 100~400 | 1~5 | 0.38 | 2.0 | | 3 | 100~600 | 1~5 | 0.35 | 2.0 | | 4 | 200~600 | 1~5 | 0.35 | 2.0 | | 5 | 200~800 | 均匀半空间 | 0.30 | 2.0 |
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Inversion model parameters and search range in Wyoming region[32]
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Inversion results in Wyoming region
a—inverted response compared with the original one;b—the minimum value of the objective function changes with the number of iterations;c—comparison of the logging data and the inverted shear-wave velocity profile
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