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Article(id=1322916751503131623, tenantId=1146029695717560320, journalId=1281212831689347082, issueId=1322916705915249052, articleNumber=null, orderNo=null, doi=10.14075/j.jgg.2025.11.389, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=research-article, receivedDate=1762963200000, receivedDateStr=2025-11-13, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1793337532162, onlineDateStr=2026-10-30, pubDate=1784044800000, pubDateStr=2026-07-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1793337532162, onlineIssueDateStr=2026-10-30, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1793337532162, creator=13701087609, updateTime=1793337532162, updator=13701087609, issue=Issue{id=1322916705915249052, tenantId=1146029695717560320, journalId=1281212831689347082, year='2026', volume='46', issue='7', pageStart='790', pageEnd='912', issueExtLink='null', onlineDate='null', pubDate='1784044800000', pubDateStr='2026-07-15', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1793337521294, creator='13701087609', updateTime=1793337521294, updator='13701087609', preIssue=null, nextIssue=null, articleTotal=null, ext=null, issueFiles=null, downloadFileDto=null}, startPage=842, endPage=850, ext={EN=ArticleExt(id=1322916752723674088, articleId=1322916751503131623, tenantId=1146029695717560320, journalId=1281212831689347082, language=EN, title=Inversion of Rayleigh Wave Dispersion Curve Based on Improved Whale Migration Algorithm, columnId=null, journalTitle=Journal of Geodesy and Geodynamics, columnName=null, runingTitle=null, highlight=null, articleAbstract=

To address the challenges of strong dependence on initial models, low convergence efficiency, and susceptibility to local optima in traditional Rayleigh wave dispersion curve inversion for subsurface shear-wave velocity structure, this study proposes two enhanced algorithms based on the whale migration algorithm (WMA). By introducing Lévy flight and spiral flight strategies, we develop the Lévy flight whale migration algorithm (LF-WMA) and the spiral flight whale migration algorithm (SF-WMA) to strengthen global search capability and local exploitation ability. The performance of five algorithms-damped least squares (Levenberg-Marquardt method, LM), whale optimization algorithm (WOA), WMA, LF-WMA, and SF-WMA-was systematically evaluated using Rastrigin function tests, theoretical geological models, random noise tests, and field data applications. The results demonstrate that SF-WMA outperforms the other methods in inversion accuracy, stability, convergence speed, and noise robustness. It achieves the optimal solution of 4.74×10-10 in the Rastrigin test, exhibits the smallest error and fastest convergence in noisy models, and, when applied to real seismic data, clearly reveals stratigraphic structures within the 0 to 30 m depth range. The inversion results are in good agreement with

, authors=Zhifu YANG1, Renfei TIAN1, 2, *, Wende MA3, Xianqiong CHENG1, authorsList=Zhifu YANG, Renfei TIAN, Wende MA, Xianqiong CHENG, authorCompany=null, correspAuthors=Renfei TIAN, authorNote=null, correspAuthorsNote=null, copyrightStatement=All rights reserved. Unauthorized reproduction is prohibited., copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, fund=null), CN=ArticleExt(id=1322916762332824568, articleId=1322916751503131623, tenantId=1146029695717560320, journalId=1281212831689347082, language=CN, title=基于改进鲸鱼迁徙算法的瑞雷波频散曲线反演, columnId=1281909276545356011, journalTitle=大地测量与地球动力学, columnName=大地测量学, runingTitle=null, highlight=null, articleAbstract=

针对传统瑞雷波频散曲线反演地下横波速度结构存在对初始模型依赖强、收敛效率低和易陷入局部最优解等问题, 以鲸鱼迁徙算法(WMA)为基础, 通过引入Lévy飞行与螺旋飞行2种策略, 构建Lévy飞行鲸鱼迁徙算法(LF-WMA)和螺旋飞行鲸鱼迁徙算法(SF-WMA), 以增强全局搜索能力和局部开发能力。通过Rastrigin函数、理论地质模型、随机噪声测试和实际资料应用, 系统对比分析阻尼最小二乘法(LM)、鲸鱼优化算法(WOA)、WMA、LF-WMA和SF-WMA五种算法的性能。结果表明, SF-WMA在反演精度、稳定性、收敛速度和抗噪性方面均表现最优, 其在Rastrigin测试中获得最优解4.74×10-10, 在含噪模型中误差最小, 收敛速度最快。将SF-WMA应用到实际地震资料反演中, 反演结果清晰地揭示了0~30 m深度范围内的地层结构, 反演结果与地质资料吻合良好。

, authors=杨植富1, 田仁飞1, 2, *, 马文德3, 程先琼1, authorsList=杨植富, 田仁飞, 马文德, 程先琼, authorCompany=null, correspAuthors=田仁飞, authorNote=

杨植富, 硕士生, 主要从事面波反演研究, E-mail:

, correspAuthorsNote=
田仁飞, 副教授, 主要从事综合地球物理研究, E-mail:
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figureFileSmall=gDC/+A6SCJ8yIOW8Isxhrg==, figureFileBig=KOEjBi5xX7F/YQDRt6QDJQ==, tableContent=null), ArticleFig(id=1322916778006937649, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=EN, label=Tab. 1, caption=

Test results of Rastrigin function

, figureFileSmall=null, figureFileBig=null, tableContent=
函数名称
WOA WMA LF-WMA SF-WMA
Rastrigin
最优解		 2.81×10-9 3.25×10-9 2.86×10-9 4.74×10-10
), ArticleFig(id=1322916778128572466, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=CN, label=表1, caption=

Rastrigin函数测试结果

, figureFileSmall=null, figureFileBig=null, tableContent=
函数名称
WOA WMA LF-WMA SF-WMA
Rastrigin
最优解		 2.81×10-9 3.25×10-9 2.86×10-9 4.74×10-10
), ArticleFig(id=1322916778229235763, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=EN, label=Tab. 2, caption=

Model 1 and its inversion search rang

, figureFileSmall=null, figureFileBig=null, tableContent=
层序号 模型参数 搜索空间
VS/m·s-1 VP/m·s-1 ρ/g·cm-3 h/m VS/m·s-1 Δh/m
1 100 269 2 3 40~160 1.2~4.8
2 150 404 2 3 60~240 1.2~4.8
3 200 539 2 4 80~320 1.6~6.4
4 300 808 2 ∞ 120~480 ∞
), ArticleFig(id=1322916778325704756, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=CN, label=表2, caption=

模型1及反演搜索范围

, figureFileSmall=null, figureFileBig=null, tableContent=
层序号 模型参数 搜索空间
VS/m·s-1 VP/m·s-1 ρ/g·cm-3 h/m VS/m·s-1 Δh/m
1 100 269 2 3 40~160 1.2~4.8
2 150 404 2 3 60~240 1.2~4.8
3 200 539 2 4 80~320 1.6~6.4
4 300 808 2 ∞ 120~480 ∞
), ArticleFig(id=1322916778434756661, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=EN, label=Tab. 3, caption=

Inversion statistical results of model 1

, figureFileSmall=null, figureFileBig=null, tableContent=
参数 Vs1 Vs2 Vs3 Vs4 h1 h2 h3
WOA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.26 m/s 161.86 m/s 195.36 m/s 297.59 m/s 3.20 m 2.73 m 3.47 m
相对误差 0.26% 7.90% 2.32% 0.80% 6.10% 8.97% 13.33%
LM 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.10 m/s 153.42 m/s 205.07 m/s 299.61 m/s 2.93 m 2.71 m 3.70 m
相对误差 0.10% 2.28% 2.54% 0.13% 2.33% 9.67% 7.60%
WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.03 m/s 152.96 m/s 206.65 m/s 300.36 m/s 3.04 m 3.34 m 3.73 m
相对误差 0.03% 1.97% 3.32% 0.12% 1.40% 11.40% 6.80%
LF-WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.01 m/s 151.90 m/s 206.76 m/s 300.39m/s 3.03 m 3.26 m 3.91 m
相对误差 0.01% 1.27% 3.38% 0.13% 0.83% 8.53% 2.35%
SF-WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.00 m/s 150.07 m/s 204.53 m/s 300.33 m/s 3.00 m 3.07 m 4.17 m
相对误差 0.00% 0.04% 2.27% 0.11% 0.00% 2.43% 4.23%
), ArticleFig(id=1322916778522837046, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=CN, label=表3, caption=

模型1反演结果统计

, figureFileSmall=null, figureFileBig=null, tableContent=
参数 Vs1 Vs2 Vs3 Vs4 h1 h2 h3
WOA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.26 m/s 161.86 m/s 195.36 m/s 297.59 m/s 3.20 m 2.73 m 3.47 m
相对误差 0.26% 7.90% 2.32% 0.80% 6.10% 8.97% 13.33%
LM 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.10 m/s 153.42 m/s 205.07 m/s 299.61 m/s 2.93 m 2.71 m 3.70 m
相对误差 0.10% 2.28% 2.54% 0.13% 2.33% 9.67% 7.60%
WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.03 m/s 152.96 m/s 206.65 m/s 300.36 m/s 3.04 m 3.34 m 3.73 m
相对误差 0.03% 1.97% 3.32% 0.12% 1.40% 11.40% 6.80%
LF-WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.01 m/s 151.90 m/s 206.76 m/s 300.39m/s 3.03 m 3.26 m 3.91 m
相对误差 0.01% 1.27% 3.38% 0.13% 0.83% 8.53% 2.35%
SF-WMA 真实值 100 m/s 150 m/s 200 m/s 300 m/s 3.0 m 3.0 m 4.0 m
反演均值 100.00 m/s 150.07 m/s 204.53 m/s 300.33 m/s 3.00 m 3.07 m 4.17 m
相对误差 0.00% 0.04% 2.27% 0.11% 0.00% 2.43% 4.23%
), ArticleFig(id=1322916778644471863, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=EN, label=Tab. 4, caption=

Initial model

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分层深度 VP/m·s-1 VS/m·s-1
0~3 m 150~600 114~456
3~7 m 272~1 088 144~576
7~11 m 408~1 632 184~736
11~17 m 552~2 208 314~1 256
17~30 m 690~2 760 364~1 456
), ArticleFig(id=1322916778749329464, tenantId=1146029695717560320, journalId=1281212831689347082, articleId=1322916751503131623, language=CN, label=表4, caption=

初始模型

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分层深度 VP/m·s-1 VS/m·s-1
0~3 m 150~600 114~456
3~7 m 272~1 088 144~576
7~11 m 408~1 632 184~736
11~17 m 552~2 208 314~1 256
17~30 m 690~2 760 364~1 456
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基于改进鲸鱼迁徙算法的瑞雷波频散曲线反演
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杨植富 1 , 田仁飞 1, 2, * , 马文德 3 , 程先琼 1
大地测量与地球动力学 | 大地测量学 2026,46(7): 842-850
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大地测量与地球动力学 |大地测量学 2026 , 46 (7) : 842 -850
基于改进鲸鱼迁徙算法的瑞雷波频散曲线反演
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杨植富1 , 田仁飞1, 2, * , 马文德3, 程先琼1
作者信息
  • 1 成都理工大学地球物理学院, 成都, 610059
  • 2 地球勘探与信息技术教育部重点实验室(成都理工大学), 成都, 610059
  • 3 中铁二院工程集团有限责任公司地勘岩土工程设计研究院, 成都, 610032
通讯作者:
田仁飞, 副教授, 主要从事综合地球物理研究, E-mail:
作者简介:

杨植富, 硕士生, 主要从事面波反演研究, E-mail:

Inversion of Rayleigh Wave Dispersion Curve Based on Improved Whale Migration Algorithm
Zhifu YANG1 , Renfei TIAN1, 2, * , Wende MA3, Xianqiong CHENG1
Affiliations
  • 1 College of Geophysics, Chengdu University of Technology, Chengdu 610059, China
  • 2 Key Lab of Earth Exploration and Information Techniques (Chengdu University of Technology), Ministry of Education, Chengdu 610059, China
  • 3 Institute of Design and Research of Geotechnical Engineering, China Railway Eryuan Engineering Group Co Ltd, Chengdu 610032, China
出版时间: 2026-07-15 doi: 10.14075/j.jgg.2025.11.389
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摘要
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针对传统瑞雷波频散曲线反演地下横波速度结构存在对初始模型依赖强、收敛效率低和易陷入局部最优解等问题, 以鲸鱼迁徙算法(WMA)为基础, 通过引入Lévy飞行与螺旋飞行2种策略, 构建Lévy飞行鲸鱼迁徙算法(LF-WMA)和螺旋飞行鲸鱼迁徙算法(SF-WMA), 以增强全局搜索能力和局部开发能力。通过Rastrigin函数、理论地质模型、随机噪声测试和实际资料应用, 系统对比分析阻尼最小二乘法(LM)、鲸鱼优化算法(WOA)、WMA、LF-WMA和SF-WMA五种算法的性能。结果表明, SF-WMA在反演精度、稳定性、收敛速度和抗噪性方面均表现最优, 其在Rastrigin测试中获得最优解4.74×10-10, 在含噪模型中误差最小, 收敛速度最快。将SF-WMA应用到实际地震资料反演中, 反演结果清晰地揭示了0~30 m深度范围内的地层结构, 反演结果与地质资料吻合良好。

瑞雷波  /  频散曲线  /  鲸鱼迁徙算法  /  Lévy飞行  /  螺旋飞行

To address the challenges of strong dependence on initial models, low convergence efficiency, and susceptibility to local optima in traditional Rayleigh wave dispersion curve inversion for subsurface shear-wave velocity structure, this study proposes two enhanced algorithms based on the whale migration algorithm (WMA). By introducing Lévy flight and spiral flight strategies, we develop the Lévy flight whale migration algorithm (LF-WMA) and the spiral flight whale migration algorithm (SF-WMA) to strengthen global search capability and local exploitation ability. The performance of five algorithms-damped least squares (Levenberg-Marquardt method, LM), whale optimization algorithm (WOA), WMA, LF-WMA, and SF-WMA-was systematically evaluated using Rastrigin function tests, theoretical geological models, random noise tests, and field data applications. The results demonstrate that SF-WMA outperforms the other methods in inversion accuracy, stability, convergence speed, and noise robustness. It achieves the optimal solution of 4.74×10-10 in the Rastrigin test, exhibits the smallest error and fastest convergence in noisy models, and, when applied to real seismic data, clearly reveals stratigraphic structures within the 0 to 30 m depth range. The inversion results are in good agreement with

Rayleigh wave  /  dispersion curve  /  whale migration algorithm  /  Lévy flight  /  spiral flight
杨植富, 田仁飞, 马文德, 程先琼. 基于改进鲸鱼迁徙算法的瑞雷波频散曲线反演. 大地测量与地球动力学, 2026 , 46 (7) : 842 -850 . DOI: 10.14075/j.jgg.2025.11.389
Zhifu YANG, Renfei TIAN, Wende MA, Xianqiong CHENG. Inversion of Rayleigh Wave Dispersion Curve Based on Improved Whale Migration Algorithm[J]. Journal of Geodesy and Geodynamics, 2026 , 46 (7) : 842 -850 . DOI: 10.14075/j.jgg.2025.11.389
正文
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瑞雷波频散反演能够获得地下横波速度结构，在地球物理勘探、工程地质勘察、岩土工程检测和环境地质调查等领域得到广泛的应用[1]。瑞雷波勘探的主要步骤是野外数据采集、频散曲线提取和频散曲线反演。其中，频散曲线反演是瑞雷波勘探中最关键的环节[2]。然而，频散曲线反演在本质上具有高度的非线性、多参数及非唯一性等特征，传统的阻尼最小二乘(LM)法严重依赖于初始模型的选择[3]，容易陷入局部极值，从而影响反演结果的可靠性。为克服以上不足，智能优化算法被逐步引入到频散曲线反演中，如遗传算法[4]、模拟退火法[5]和粒子群算法[6]等。这类方法对初始模型的依赖较小，求解精度也有所提高，但在处理复杂地层数据时，仍然容易出现收敛速度慢、计算成本高等问题。
近年来，受自然界生物群体智能行为启发的新型元启发式算法[7-8]，为解决面波频散反演问题提供了新的思路，如萤火虫算法[9]、鲸鱼优化算法[10]、野马优化算法[11]和秃鹰搜索算法[12]等。鲸鱼迁徙算法(WMA)[13]是一种元启发式优化方法，属于全局非线性反演方法，其灵感来源于座头鲸的协作迁徙行为。该算法通过模拟座头鲸的迁徙和捕食行为，实现了在优化过程中的高效搜索。WMA通过整合领导者-追随者动态和自适应迁徙策略，平衡探索和开发之间的关系，从而提高了算法避免局部最优和有效收敛的能力。但WMA在处理高维、复杂的地球物理反演问题时，仍存在种群多样性易丢失、优化能力不平衡等问题。
因此，本文在传统的WMA算法中加入Lévy飞行策略, 形成Lévy飞行鲸鱼迁徙算法(LF-WMA)，另加入螺旋飞行策略形成螺旋飞行鲸鱼迁徙算法(SF-WMA)。为验证改进算法的稳定性和准确性，采用模拟实验和实测数据进行测试，并与LM、WOA、WMA等算法进行对比分析。结果表明，2种改进算法在面波频散曲线反演中均表现出更优性能，其中SF-WMA的综合效果更为突出。
1 方法原理
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1.1 频散曲线反演目标函数
在瑞雷波面波勘探中，基阶波能量较强、易观测、分布广，易于从频散图中提取[14]，故本文采用基阶频散曲线进行反演。反演过程中以实测与理论计算的瑞雷波相速度之间的RMS作为目标函数：
$\mathit{Φ}=\sqrt{\sum\limits_{r=1}^m\left(\boldsymbol{V}_{\mathrm{R}}^{\text {obs }}-\boldsymbol{V}_{\mathrm{R}}^{\text {theo }}\right)^2 / m}$
式中，m为频点数；VRobs为m×1维实测瑞雷波相速度向量；VRtheo为m×1维理论计算的瑞雷波相速度向量。
瑞雷波频散曲线对横波速度VS和层厚h这2个参数最为敏感[14]，故将两者作为反演的核心参数。初始纵波速度VP根据参考文献[1]中的经验公式获得，密度ρ设置为常数。
1.2 鲸鱼迁徙算法
鲸鱼迁徙算法的核心思想在于模拟鲸群在迁徙中的集体智慧，通过引入领导者-追随者动态机制与自适应迁移策略，有效平衡全局探索与局部开发，从而提升算法在复杂优化问题中的寻优能力与收敛效率[12-13]。该方法基本原理如下：
1) 种群初始化。在搜索空间内随机生成初始鲸鱼群：
$\boldsymbol{W}_i=\boldsymbol{L}+\operatorname{rand}(1, D) \odot(\boldsymbol{U}-\boldsymbol{L})$
式中，Wi为第i个鲸鱼个体的位置；L和U分别为搜索空间的下界和上界；rand(1, D)表示生成维度为D的随机向量；⊙为哈达玛积[13]。
2) 计算领导者的均值位置。在每一群迁徙鲸鱼中，经验更丰富、位置知识更优越、目标函数值更高的个体(领导者)会引导和指导群体，带领其他鲸鱼前往目的地，这有助于加快收敛速度并逃离局部最优，其公式为：
$ \boldsymbol{W}_{\text {Mean }}=\frac{1}{N_L} \sum\limits_{j=1}^{N_L} \boldsymbol{W}_j $
式中，WMean为领导者的平均位置，即鲸鱼群的迁徙方向，前NL个最优个体即为领导者。
3) 跟随者的位置更新。经验不足的鲸鱼(跟随者)由经验丰富的鲸鱼(领导者)指导。若代表领导者平均位置的WMean与全局最优解WBest之间的距离开始缩短，则表明整个经验丰富的鲸群正在向最优区域靠拢。此时，经验不足的鲸鱼(幼鲸)也需沿相同方向移动。其位置更新公式如下：
$\begin{gathered}\boldsymbol{W}_i^{\text {new }}=\boldsymbol{W}_{\text {Mean }}+\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{i-1}-\boldsymbol{W}_i\right)+ \\\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_{\text {Mean }}\right) \\i=N_L+1, \cdots, N_{\text {pop }}\end{gathered}$
式中，WMean设置为当前NL个领导者位置的平均值，参数NL表示更有经验的鲸鱼(领导者)的数量元素Wi的定性前体；rand(1, D)是维度为问题D的区间[0, 1]内的随机数向量。
4) 领导者位置更新。由更有经验的鲸鱼(领导者)发现和搜索新领域，负责识别和选择到达目的地的最佳路线，其更新公式如下：
$ \begin{gathered}\boldsymbol{W}_i^{\text {new }}=\boldsymbol{W}_i+\boldsymbol{r}_1 \odot \boldsymbol{L}+\boldsymbol{r}_1 \odot \boldsymbol{r}_2 \odot(\boldsymbol{U}-\boldsymbol{L}) \\i=1, \cdots N_L\end{gathered} $
式中，r1和r2是维度为D的区间[0, 1]内的随机数向量；L、U表示位置向量; U-L表示相对方向向量。
1.3 Lévy飞行鲸鱼迁徙算法
Lévy飞行最初在果蝇飞行行为研究中被发现，是指果蝇在直线飞行的过程中会突然出现90°转向，并在多次短距离移动间偶尔出现一次长距离跳跃，这种飞行方式称为Lévy飞行。Lévy飞行可以增强全局探索能力[13]，将这种飞行方式引入WMA中，称为Lévy飞行鲸鱼迁徙算法(LF-WMA)。Lévy飞行随机步长生成公式为：
$\text { Lévy }(\beta) \sim 0.1 \times \frac{\mu}{|\nu|^{1 / \beta}}$
式中，β为指数参数，通常取值范围为1.0~2.0[9]，测试发现，当β取值1.5时效果最优，因此将其设置为1.5；μ~N(0, σ2), ν~N(0, 1)，其中σ定义为：
$\sigma=\left(\frac{\tau(1+\beta) \sin (\pi \beta / 2)}{\tau((1+\beta) / 2) \cdot \beta \cdot 2^{(\beta-1) / 2}}\right)^{1 / \beta}$
式中，τ为标准Gamma函数。
在LF-WMA中，追随者和领导者的位置更新时引入了Lévy飞行策略。即在原有WMA的基础上，该算法80%使用原始WMA更新公式，20%使用Lévy飞行的更新公式(该比例经本文实验对比确定，能够在维持算法原有开发能力的同时有效增强全局探索能力，取得相对更优的整体性能)。改进后的跟随者更新公式为：
$\boldsymbol{W}_i^{\text {new }}=\left\{\begin{array}{l}\boldsymbol{W}_{\text {Mean }}+\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{i-1}-\boldsymbol{W}_i\right)+ \\\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_{\text {Mean }}\right), \operatorname{rand}>0.2 \\\boldsymbol{W}_i+\operatorname{Lévy} \odot\left(\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_i\right), \operatorname{rand}＜0.2\end{array}\right.$
式中，WMean设置为当前NL个领导者位置的平均值；rand(1, D)是维度为D的区间[0, 1]内的随机数向量。改进后的领导者更新公式为：
$\boldsymbol{W}_i^{\text {new }}=\left\{\begin{array}{l}\boldsymbol{W}_i+\boldsymbol{r}_1 \odot \boldsymbol{L}+\boldsymbol{r}_1 \odot \boldsymbol{r}_2 \odot(\boldsymbol{U}-\boldsymbol{L}), \text { rand }>0.2 \\\boldsymbol{W}_i+\text { Lévy } \odot\left(\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_i\right), \text { rand }＜0.2\end{array}\right.$
式中，i=1、…、NL；r1和r2是维度为D的区间[0, 1]内的随机数向量；L为位置向量；U-L为相对方向向量。
1.4 螺旋飞行鲸鱼迁徙算法
基于鲸鱼优化算法(WOA)中螺旋路径对局部开发能力的提升作用[15]及自适应对数螺旋路径萤火虫算法(ALSL-FA)中螺旋路径在平衡全局探索与局部开发方面的优势，提出一种螺旋飞行改进策略，构建螺旋飞行鲸鱼迁徙算法(SF-WMA)。该算法在WMA的迭代过程中引入螺旋飞行机制，旨在进一步提升其全局寻优性能。在原有WMA的基础上，90%使用原始WMA更新公式，10%使用加入螺旋飞行的更新公式(该比例是本文通过实验对比得出的最佳比例)。改进后跟随者的位置更新公式为：
$\boldsymbol{W}_i^{\text {new }}=\left\{\begin{array}{l}\boldsymbol{W}_{\text {Mean }}+\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{i-1}-\boldsymbol{W}_i\right)+\operatorname{rand}(1, D) \odot\left(\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_{\text {Mean }}\right), \text { rand }>0.1 \\\boldsymbol{W}_{\text {Best }}+\mathrm{e}^{z \cdot \boldsymbol{L}} \cdot \cos (2 \pi \boldsymbol{L}) \cdot\left|\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_i\right|, \operatorname{rand}＜0.1\end{array}\right.$
改进后领导者的位置更新公式为：
$\boldsymbol{W}_i^{\text {new }}=\left\{\begin{array}{l}\boldsymbol{W}_i+\boldsymbol{r}_1 \odot \boldsymbol{L}+\boldsymbol{r}_1 \odot \boldsymbol{r}_2 \odot(\boldsymbol{U}-\boldsymbol{L}), \text { rand }>0.1 \\\boldsymbol{W}_{\text {Best }}+\mathrm{e}^{z \boldsymbol{L}} \cdot \cos (2 \pi \boldsymbol{L}) \cdot\left|\boldsymbol{W}_{\text {Best }}-\boldsymbol{W}_i\right|, \text { rand }＜0.1\end{array}\right.$
式(10)和式(11)中的参数含义与式(8)和式(9)中相同。
本文改进的鲸鱼迁徙算法的实现流程，可以归纳为如下4个步骤(图 1)：
1) 初始化种群。随机生成鲸鱼种群位置和适应度，并记录当前最优解。
2) 计算出领导者的均值位置。对种群适应度进行排序，计算出前NL个领导者的平均位置，并判断其是否为领导者。
3) 跟随者与领导者的位置更新。跟随者位置更新，判断是否使用螺旋飞行的位置更新公式；否则，使用标准WMA跟随者位置更新公式。领导者位置更新公式，判断是否使用螺旋飞行的位置更新公式；否则，同样使用标准WMA领导者位置更新公式，计算新位置适应度并更新全局最优解。运用螺旋飞行策略，提升算法全局寻优能力和局部开发能力。
4) 终止判断。判断迭代次数是否满足终止条件，输出最终结果。
1.5 算法测试
Rastrigin函数[9](图 2)是优化算法领域经典的非线性多峰测试函数，其定义域内分布着大量局部极小值点，但全局最小值唯一且位于坐标原点，函数值为0。其作用是检验算法处理复杂搜索空间、跳出局部最优解的能力，是评估算法性能的重要工具，可以表示为:
$\begin{gathered}f\left(x_i\right)=\sum\limits_{i=1}^2\left[x_i^2-10 \cos \left(2 \pi x_i\right)+20\right], \\x_i \in[-5.12, 5.12]\end{gathered}$
使用WOA、WMA、LF-WMA、SF-WMA四种算法对Rastrigin函数进行测试，结果见图 3和表 1。
从表 1可以看出，WOA、WMA、LF-WMA、SF-WMA的最优解分别是2.81×10-9、3.25×10-9、2.86×10-9、4.74×10-10，其中SF-WMA的最优解最小，最接近0值。从图 3可知，SF-WMA的收敛速度显然快于其他3种算法，最先趋近于0值附近，证明SF-WMA有更强的寻优能力和更快的收敛速度。
2 理论模型测试
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为了检验WMA、LF-WMA、SF-WMA这3种算法反演瑞雷波频散曲线的准确性和稳定性，设计一个理论4层速度递增地质模型(模型1，参数见表 2)进行测试分析，并与传统阻尼最小二乘法(LM)进行对比。
试算中，采用基阶模式频散曲线进行反演，所有模型的算法种群数及最大迭代次数设置相同，将鲸鱼种群数量设置为85，最大迭代次数为100次。为了防止单次反演算法结果出现的随机性及反演结果偏差较大和运算失败等情况，取10次独立反演结果的平均值作为算法最终的反演结果。
本文频散曲线正演计算基于传播矩阵法完成，具体采用了整合于MuLTI框架中的Geopsy算法[16]。该算法通过求解层状介质模型的全局传播矩阵特征值来高效计算理论频散曲线，以满足反演过程对大量正演计算的速度需求。
此外，由于WOA和WMA也是基于鲸鱼行为的全局非线性优化算法，在机理上具有较强的对比性，因此将本文所改进的算法与WOA、WMA的反演结果进行对比。
2.1 模型1
图 4为模型1的横波速度反演结果。从5种算法获得的频散曲线拟合情况(图 4(a))可以看出，5种算法获得的频散曲线均与理论频散曲线高度吻合。
在横波速度剖面(图 4(b))方面，第1层的厚度和横波速度参数与理论值基本一致，对于其余3层，WOA、LM、WMA、LF-WMA均存在不同程度的偏差，仅SF-WMA的反演结果与理论值最为接近(表 3)。尽管SF-WMA在第3层厚度的反演精度略低于LF-WMA，但优于WOA、LM和WMA方法，表明SF-WMA具有更高的反演精度和稳定性。
从反演迭代收敛曲线(图 4(c))来看，在前10次迭代中，WOA和LM收敛速度缓慢，WMA和LF-WMA的收敛速度基本一致，SF-WMA的收敛速度更快；在10~20次迭代中，SF-WMA最先收敛趋近0值，其次是LF-WMA、WMA，而WOA和LM直至20次迭代之后才收敛。由此可见，SF-WMA收敛速度快于其他4种算法。
从厚度差值(图 4(d))和横波速度来看，SF-WMA在第1、2层厚度的差值最小，分别为0.00 m和0.07 m；在第3层，SF-WMA的差值略大于LF-WMA，两者仅相差0.08 m，二者差值小于其他3种算法。对于横波速度差值(图 4(e))，SF-WMA在各层均最小，分别为0.00 m/s、0.07 m/s、4.53 m/s、0.33 m/s。总体而言，SF-WMA的稳定性最好。
从反演统计结果(表 3)来看，WOA、LM、WMA、LF-WMA和SF-WMA反演获得横波速度的平均相对误差分别是2.82%、1.26%、1.36%、1.20%、0.61%，反演获得层厚平均相对误差分别为9.42%、6.53%、6.47%、3.97%、2.19%。将5种算法获得的所有地层参数平均相对误差进行统计，WOA的平均相对误差为5.65%，LM的平均相对误差为3.52%，WMA的平均相对误差为3.58%，LF-WMA的平均相对误差为2.39%，而SF-WMA的平均相对误差仅为1.29%。上述统计结果表明，SF-WMA的反演精度高于其他4种算法。
2.2 含随机噪声地质理论模型试算
野外采集瑞雷波时会存在一定程度的随机噪声，影响反演结果的准确性和稳定性。因此，有必要进行随机噪声测试。为了检验算法的抗噪能力，本文在模型1的频散曲线中加入信噪比(SNR)5~50 dB的随机噪声，以5 dB为间隔进行系统测试，并评估反演的横波速度和厚度的平均相对误差。
从横波速度的平均相对误差对比结果(图 5(a))可以看出，随着信噪比的增大，WOA的结果并没有出现明显下降趋势，且数值浮动较大，说明WOA稳定性较差。WMA、LF-WMA、SF-WMA趋势基本一致，均随信噪比的增大相对误差逐渐减小。其中，SF-WMA比LF-WMA、WMA的平均相对误差更小。从厚度的平均相对误差对比(图 5(b))来看，WOA在5~25 dB的信噪比范围内误差最低；在30~50 dB区间，WOA数值浮动较大，表明其抗噪性能较差。相比之下，WMA、LF-WMA、SF-WMA仍保持随着信噪比增大而平均相对误差减小，且SF-WMA平均相对误差最小。
综合横波速度和厚度这2个反演参数的平均相对误差可知，虽然WOA的抗噪效果在低噪音(5~25 dB)时较好，但其整体稳定性较差。对比WMA、LF-WMA、SF-WMA这3种算法，SF-WMA的抗噪能力优于其他2种算法。
与无噪声模型1相比，有噪声时，结果的厚度和横波速度均受到不同程度的影响，但统计结果表明，SF-WMA受噪声影响最小，具有更强的稳定性和鲁棒性。
3 实测资料应用
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为了进一步验证改进的WMA算法的实用性，选取四川省德阳市罗江区白马镇采集的地震数据进行分析[17]。该套数据采用中间激发的观测系统，道间距为2 m，激发点距为6 m，时间采样间隔为0.5 ms，最小炮检距为1 m，最大炮检距为239 m，单炮240道接收，共设计356个炮点。激发震源为GeoPenLVIB-10A液压伺服可控震源，采集仪器为Sercel 428XL地震数据采集仪器车和XS-5GX单点检波器。
根据实验工区当地的地层条件及参考前人在工区的一些研究结果，设置5层初始模型(表 4)。本文采用刘俊杰[17]提供的356条高质量频散曲线数据，这些频散曲线是采用高分辨率Radon变换对原始炮数据逐炮处理，通过提取频散能量图获得的。基于此频散曲线数据，运用本文改进的WMA方法进行反演计算，并与WOA、WMA的反演结果进行比较。在参数设置方面，鲸鱼种群数量设为95，最大迭代次数设为80次。此外，针对第341炮数据，经参数优化后的具体设置为：初始Lévy步长因子为0.1，Lévy飞行参数β为1.5，Lévy飞行随机选择概率设定为20%，螺旋飞行步长为1，螺旋飞行随机选择概率为10%。第341炮反演结果如图 6所示。
从频散曲线拟合情况(图 6(a))来看，4种算法获得的频散曲线均与实测频散曲线基本一致，但在低频部分，WMA、LF-WMA、SF-WMA的频散曲线存在一定差异。从横波速度剖面对比(图 6(b))来看，WMA、LF-WMA、SF-WMA的反演结果在0~8 m深度较为接近；对于8 m以下深度，WMA、LF-WMA反演结果较为接近，SF-WMA与这2种算法存在一定偏差。从理论模型试算和含随机噪声地质理论模型试算结果来看，SF-WMA的反演结果均比其他2种算法更接近理论值，说明SF-WMA反演结果更可靠。WOA反演结果与其他3种算法的反演结果存在一定偏差。从4种算法迭代收敛对比(图 6(c))看出，在整个迭代过程中，SF-WMA的收敛速度最快，最先趋近于0值，表明其在实际应用中具有较好的适用性。
采用SF-WMA依次对整条测线逐炮进行反演处理，获得二维横波速度剖面(图 7)。该剖面清晰地揭示了测区浅层30 m的地层结构特征：0~5 m深度内，横波速度范围在150~370 m/s，对应淤泥质黏土；5~10 m深度内，横波速度范围在470~670 m/s，反映含泥岩与粉砂岩风化破碎层；10~30 m深度内，横波速度范围在670~990 m/s，表征为白垩系(K)完整的岩体。反演结果与区域地质特征[18]高度一致，充分证明了SF-WMA算法在实际应用中的有效性与可靠性。
4 结语
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本文在传统鲸鱼迁徙算法(WMA)的基础上，分别引入Lévy飞行和螺旋飞行策略，提出了LF-WMA和SF-WMA两种改进算法。通过函数测试和瑞雷波频散曲线反演实验，结合理论模型、抗噪测试和实际资料应用，得出以下结论：
1) Lévy飞行和螺旋飞行策略的引入显著增强了算法的全局探索能力和跳出局部极值的能力。测试结果表明，2种改进算法在收敛速度、反演精度和稳定性方面均优于传统WMA和WOA。其中，SF-WMA表现最优，在Rastrigin函数测试中获得4.74×10-10的最优解，且收敛速度最快。
2) 在含噪理论模型测试中，SF-WMA在5~50 dB信噪比范围内始终保持最小的速度和厚度反演误差，表现出良好的鲁棒性。实际资料处理中，SF-WMA反演得到的横波速度剖面与区域地质结构吻合良好，清晰地揭示了0~30 m深度范围内的地层结构，验证了其在噪声环境下的可靠性和工程适用性。
基金
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  • 国家自然科学基金(41774095)
文献
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2026年第46卷第7期
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doi: 10.14075/j.jgg.2025.11.389
  • 接收时间：2025-11-13
  • 首发时间：2026-10-30
  • 出版时间：2026-07-15
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  • 收稿日期：2025-11-13
基金
国家自然科学基金(41774095)
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    1 成都理工大学地球物理学院, 成都, 610059
    2 地球勘探与信息技术教育部重点实验室(成都理工大学), 成都, 610059
    3 中铁二院工程集团有限责任公司地勘岩土工程设计研究院, 成都, 610032

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田仁飞, 副教授, 主要从事综合地球物理研究, E-mail:
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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