层状介质中瑞利面波多模式性质及其在正反演中的应用
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摘要
Rayleigh面波法是一种新兴的地球物理勘探方法,具有能量大,信噪比高等特点,近些年来越来越多的被应用于浅层地球物理勘探,深层地震学研究,超声波无损检测等多个领域。它涉及到瑞利面波的频散曲线的提取、正演理论及反演这三个方面的研究。目前人们对频散曲线对应的振动特征和一些形状特征的研究仍然不够深入,而对此进行研究将对频散曲线的正演和反演有一定的指导意义。本文对层状介质中Rayleigh波多模式频散曲线形状特征及其对应波的振动特征进行了研究,并且将其应用到Rayleigh波的正演计算和反演中,主要内容和成果如下所述:
将久期函数族的概念应用到本征函数的计算中,以此研究了Rayleigh波三种基本模式(R模、R型周期模和S型周期模)的振动特征。首先将久期函数族的概念应用到广义反射-透射系数方法计算本征函数中,避免了广义反射-透射系数方法在计算含低速层模型对应的Rayleigh波本征函数时可能会遇到的数值不稳定问题。利用这一方法计算得到频散曲线上点对应的竖直本征位移曲线,通过对三个模型对应的不同模式Rayleigh波的本征振动的分析,研究了频散曲线对应的三种基本模式的振动特征。发现R模的振动主要集中在地表,随深度迅速衰减,穿透深度约为波长的一半;R型周期模的振动主要集中在前几层,穿透深度与波长几乎无关;S型周期模的振动主要集中在低速层,穿透深度与波长几乎无关。通常R模只在地表附近有一个振幅极值点,而R型和S型周期模的振幅极值点个数与其对应的阶数近似相等,对这一现象进行了理论证明。从公式角度较深入地分析了广义反射-透射系数法计算含低速层模型对应频散曲线时丢根的原因,发现这是Rayleigh波在相邻两层能量差别过大导致广义反射-透射系数的求解数值不稳定造成的。基于频散曲线对应基本模式的振动特征,明确地验证了只要用地表和低速层对应的久期函数组成久期函数族即可避免漏根问题。
研究了含低速层模型对应频散曲线的“交叉点”附近的Rayleigh波模式以及交叉点频率与介质参数之间的关系。利用竖直本征位移曲线研究了频散曲线“交叉点”附近Rayleigh波模式的变化情况,发现频散曲线“交叉”现象通常是由于同一相速度分区内存在两种不同模式的Rayleigh波造成的。当频散曲线实际上不相交时,“交叉点”附近的频散曲线对应模式经历了由一种模式经由耦合模式逐渐变为另一种模式的过程;当频散曲线相交时,交叉点附近的频散曲线模式没有明显的变化,只在交点附近很小的范围内存在较少的耦合模式。利用高频近似和δ矩阵方法推导了频散曲线交叉点位置的计算公式并分析了公式的误差。利用这一公式研究了频散曲线交叉点对应频率与介质参数之间的关系,对实例进行计算发现当低速层越明显时,频散曲线的各交叉点对应频率就越低。
利用本征位移曲线从力学角度解释了频散曲线对于介质各层横波速度的敏感性,由此提出一种考虑频散曲线敏感性的反演方法。通过实例研究了频散曲线敏感性与本征位移的关系,发现大多数情况下频散曲线的敏感性与其在各层的本征位移,特别是水平方向的本征位移有着较好的对应关系。通常情况下,Rayleigh波在一层水平方向的本征位移越大,则它对该层横波速度的敏感性就越强,反之亦然。这一规律在高频时尤其明显。这一结论使得Rayleigh波频散曲线对各层横波速度的敏感性与Rayleigh波的基本模式建立了联系。基于这一结论,提出了一种考虑Rayleigh波频散曲线敏感性的反演方法。这种方法将频散点按敏感性分类进行反演。通过反演模拟并与近几年受到较多关注的多阶模方法结果进行对比发现该方法在一定程度上可以避免普通局部优化方法陷入局部极小值的问题,但是有时需要比其他方法更多的频散数据才能得到较好的结果,而且反演稳定性仍有待提高。
研究了当层状介质含低速层时频散曲线的“之”字形回折现象。在三种不同的震源下改变给定模型的各个参数,计算其对应频散曲线起跳点受各个参数的影响情况,以此研究了“之”字形频散曲线起跳点的位置与介质参数及偏移距的关系。发现“之”字形频散曲线起跳点对应频率与检波器和震源的距离几乎无关,但与震源和介质各层参数有关。在相同震源下,对频散曲线的起跳点频率影响最大的是低速层和低速层以上层的横波速度,其次是这些层的层厚。总的来说,低速层越明显,频散曲线的起跳点频率就越低。该结论可以作为地下低速层存在情况的定性分析应用于路基压实度检测等。利用两层递增型介质中Rayleigh波频散曲线关于横波速度的近似线性性质提出了一种快速反演两层介质参数的方法。针对前人提出的各向同性半空间Rayleigh波相速度公式中普遍包含虚数计算的问题,提出一种不包含虚数计算的公式,为理论研究和工程中的应用提供了方便。
Rayleigh-wave-method is a new method in geophysical prospecting. Rayleigh wave usually has more energy and higher signal to noise ratio than body waves, so there are more and more applications of Rayleigh wave in geophysical prospecting, seismology and ultrasonic nondestructive testing and so on recent years. It involves three aspects: data collection, forward-deduction and inversion. Till now, the study on the formation and the vibration characteristics corresponding to the dispersion curves is not enough. And the study will be meaningful for the forward-deduction and inversion of Rayleigh waves. In this dissertation, the formation and the vibration characteristics corresponding to the dispersion curves are studied, and the results are applied in Rayleigh wave forward-deduction and inversion. The main contribution are as follows:
The secular function family is applied into the computation of eigenfunctions. Then the eigenfunctions are used to study the vibration characteristics of the three basic modes of Rayleigh wave (R-mode、R-period-mode、S-period-mode). Firstly, the secular function family is applied into the computation of eigenfunctions with generalized R/T coefficients method. Through this, the numerical instability in computing eigenfunctions with generalized R/T coefficients method can be avoided. Then the vertical eigendisplacement curves corresponding to the dispersion points can be obtained. By analyzing the eigendisplacements corresponding to different modes of Rayleigh wave in three models, the vibration characteristics of the three basic modes of Rayleigh wave are studied. It is found that the vibration of R mode is mainly near the surface and attenuates with depth rapidly, and the penetrate depth is about half of the wave length, the vibration of R-period-mode is in the first several layers, and the penetrate depth is independent of the wave length, the vibration of S-period-mode is in the low velocity layers, and the penetrate depth is independent of the wave length. Normally, the eigendisplacement curve corresponding to R mode has only one extreme, while the amounts of extremes of eigendisplacement curves corresponding to R-period-mode or S-period-mode approximately equals the mode order. The reason of root lost during computing dispersion curves with generalized R/T coefficients method is studied more deeply from the formula. It is found that when the energy gap is too large, the computation of generalized R/T coefficients will be instable, which causes the root lost. Based on the vibration characteristics of the basic modes, it is demonstrated clearly that the secular function family can be reduced into the secular functions corresponding to surface and low velocity layers.
The modes of Rayleigh waves near the‘cross point’of dispersion curves and the relationship between the frequencies of the cross points and the parameters are studied. The variation of Rayleigh waves near the‘cross’points is analyzed with vertical eigendisplacement curves. It is found that the‘cross’phenomenon is usually related to the existence of two different modes of Rayleigh wave in the same velocity zone. If the two dispersion curves are not cross, the transformation between two different modes via coupled mode happens to the mode on each curve. And if the two dispersion curves are cross, the mode corresponding to the same dispersion curve remains unchanged, though there may be some coupled mode very close to the cross point. With approximation in high frequencies andδmatrix method, the formula to the position of cross points is deduced and the error is analyzed. With this formula, the relationship between the frequencies of the cross points and the parameters is studied. It is found that the more obvious the low velocity layer is, the lower the frequency of each the cross point is.
The sensitivity of dispersion curves to the S-wave velocities is explained with horizontal eigendisplacement curves, and an inversion method concerning the sensitivity is presented based on it. Two examples are used to study the relationship between the sensitivity and the eigendisplacements. It is found that usually the sensitivity is related to the eigendisplacements of the Rayleigh wave in each layer, especially the horizontal eigendisplacements. Normally, if the horizontal eigendisplacement corresponding to one dispersion point is larger in one layer than another layer, the Rayleigh wave velocity will be more sensitive to the S-wave velocity in this layer, and vice versa. This characteristic is more obvious in high frequencies. The result shows the relationship between sensitivity with the basic modes of Rayleigh waves. Based on this, a kind of inversion method concerning about sensitivity is presented. Inversion simulation is conducted and the results are compared with that obtained from multimode inversion method. It is found that the method sometimes avoids the problem of local minimum in normal local optimizations. However, the method often needs more dispersion points, and the stability also needs to be improved.
The‘zigzag’phenomenon of dispersion curves when there is low velocity layer is studied. The parameters of a given medium are changed under three different sources to study the influence of the parameters on the‘jump point’frequencies. The frequencies showed little relationship with the offset. It is found that although the‘jump point’frequencies depend on the source style, there are the same rules under each source: the S-wave velocities of the low velocity layer and the layer above it are the dominant parameters that influence the‘jump point’frequencies, and followed by their thickness, the other parameters have less effect on the‘jump point’frequencies, the more obvious the low velocity layer is, the lower the‘jump point’frequency is. The result can be used as a qualitative analysis of low velocity layer in the cases like compaction test of subgrade. Rayleigh wave dispersion curves in two layer media can be approximately seen to be linear corresponding with S-wave velocities. Based on this, a quick inversion method of Rayleigh wave fundamental mode dispersion curves for two layer media is presented. All the formulas of the Rayleigh wave velocity in isotropic half space before contain complex computation. In this paper, a new formula without complex computation is presented. It makes the computation of Rayleigh wave velocities more convenient.
将久期函数族的概念应用到本征函数的计算中,以此研究了Rayleigh波三种基本模式(R模、R型周期模和S型周期模)的振动特征。首先将久期函数族的概念应用到广义反射-透射系数方法计算本征函数中,避免了广义反射-透射系数方法在计算含低速层模型对应的Rayleigh波本征函数时可能会遇到的数值不稳定问题。利用这一方法计算得到频散曲线上点对应的竖直本征位移曲线,通过对三个模型对应的不同模式Rayleigh波的本征振动的分析,研究了频散曲线对应的三种基本模式的振动特征。发现R模的振动主要集中在地表,随深度迅速衰减,穿透深度约为波长的一半;R型周期模的振动主要集中在前几层,穿透深度与波长几乎无关;S型周期模的振动主要集中在低速层,穿透深度与波长几乎无关。通常R模只在地表附近有一个振幅极值点,而R型和S型周期模的振幅极值点个数与其对应的阶数近似相等,对这一现象进行了理论证明。从公式角度较深入地分析了广义反射-透射系数法计算含低速层模型对应频散曲线时丢根的原因,发现这是Rayleigh波在相邻两层能量差别过大导致广义反射-透射系数的求解数值不稳定造成的。基于频散曲线对应基本模式的振动特征,明确地验证了只要用地表和低速层对应的久期函数组成久期函数族即可避免漏根问题。
研究了含低速层模型对应频散曲线的“交叉点”附近的Rayleigh波模式以及交叉点频率与介质参数之间的关系。利用竖直本征位移曲线研究了频散曲线“交叉点”附近Rayleigh波模式的变化情况,发现频散曲线“交叉”现象通常是由于同一相速度分区内存在两种不同模式的Rayleigh波造成的。当频散曲线实际上不相交时,“交叉点”附近的频散曲线对应模式经历了由一种模式经由耦合模式逐渐变为另一种模式的过程;当频散曲线相交时,交叉点附近的频散曲线模式没有明显的变化,只在交点附近很小的范围内存在较少的耦合模式。利用高频近似和δ矩阵方法推导了频散曲线交叉点位置的计算公式并分析了公式的误差。利用这一公式研究了频散曲线交叉点对应频率与介质参数之间的关系,对实例进行计算发现当低速层越明显时,频散曲线的各交叉点对应频率就越低。
利用本征位移曲线从力学角度解释了频散曲线对于介质各层横波速度的敏感性,由此提出一种考虑频散曲线敏感性的反演方法。通过实例研究了频散曲线敏感性与本征位移的关系,发现大多数情况下频散曲线的敏感性与其在各层的本征位移,特别是水平方向的本征位移有着较好的对应关系。通常情况下,Rayleigh波在一层水平方向的本征位移越大,则它对该层横波速度的敏感性就越强,反之亦然。这一规律在高频时尤其明显。这一结论使得Rayleigh波频散曲线对各层横波速度的敏感性与Rayleigh波的基本模式建立了联系。基于这一结论,提出了一种考虑Rayleigh波频散曲线敏感性的反演方法。这种方法将频散点按敏感性分类进行反演。通过反演模拟并与近几年受到较多关注的多阶模方法结果进行对比发现该方法在一定程度上可以避免普通局部优化方法陷入局部极小值的问题,但是有时需要比其他方法更多的频散数据才能得到较好的结果,而且反演稳定性仍有待提高。
研究了当层状介质含低速层时频散曲线的“之”字形回折现象。在三种不同的震源下改变给定模型的各个参数,计算其对应频散曲线起跳点受各个参数的影响情况,以此研究了“之”字形频散曲线起跳点的位置与介质参数及偏移距的关系。发现“之”字形频散曲线起跳点对应频率与检波器和震源的距离几乎无关,但与震源和介质各层参数有关。在相同震源下,对频散曲线的起跳点频率影响最大的是低速层和低速层以上层的横波速度,其次是这些层的层厚。总的来说,低速层越明显,频散曲线的起跳点频率就越低。该结论可以作为地下低速层存在情况的定性分析应用于路基压实度检测等。利用两层递增型介质中Rayleigh波频散曲线关于横波速度的近似线性性质提出了一种快速反演两层介质参数的方法。针对前人提出的各向同性半空间Rayleigh波相速度公式中普遍包含虚数计算的问题,提出一种不包含虚数计算的公式,为理论研究和工程中的应用提供了方便。
Rayleigh-wave-method is a new method in geophysical prospecting. Rayleigh wave usually has more energy and higher signal to noise ratio than body waves, so there are more and more applications of Rayleigh wave in geophysical prospecting, seismology and ultrasonic nondestructive testing and so on recent years. It involves three aspects: data collection, forward-deduction and inversion. Till now, the study on the formation and the vibration characteristics corresponding to the dispersion curves is not enough. And the study will be meaningful for the forward-deduction and inversion of Rayleigh waves. In this dissertation, the formation and the vibration characteristics corresponding to the dispersion curves are studied, and the results are applied in Rayleigh wave forward-deduction and inversion. The main contribution are as follows:
The secular function family is applied into the computation of eigenfunctions. Then the eigenfunctions are used to study the vibration characteristics of the three basic modes of Rayleigh wave (R-mode、R-period-mode、S-period-mode). Firstly, the secular function family is applied into the computation of eigenfunctions with generalized R/T coefficients method. Through this, the numerical instability in computing eigenfunctions with generalized R/T coefficients method can be avoided. Then the vertical eigendisplacement curves corresponding to the dispersion points can be obtained. By analyzing the eigendisplacements corresponding to different modes of Rayleigh wave in three models, the vibration characteristics of the three basic modes of Rayleigh wave are studied. It is found that the vibration of R mode is mainly near the surface and attenuates with depth rapidly, and the penetrate depth is about half of the wave length, the vibration of R-period-mode is in the first several layers, and the penetrate depth is independent of the wave length, the vibration of S-period-mode is in the low velocity layers, and the penetrate depth is independent of the wave length. Normally, the eigendisplacement curve corresponding to R mode has only one extreme, while the amounts of extremes of eigendisplacement curves corresponding to R-period-mode or S-period-mode approximately equals the mode order. The reason of root lost during computing dispersion curves with generalized R/T coefficients method is studied more deeply from the formula. It is found that when the energy gap is too large, the computation of generalized R/T coefficients will be instable, which causes the root lost. Based on the vibration characteristics of the basic modes, it is demonstrated clearly that the secular function family can be reduced into the secular functions corresponding to surface and low velocity layers.
The modes of Rayleigh waves near the‘cross point’of dispersion curves and the relationship between the frequencies of the cross points and the parameters are studied. The variation of Rayleigh waves near the‘cross’points is analyzed with vertical eigendisplacement curves. It is found that the‘cross’phenomenon is usually related to the existence of two different modes of Rayleigh wave in the same velocity zone. If the two dispersion curves are not cross, the transformation between two different modes via coupled mode happens to the mode on each curve. And if the two dispersion curves are cross, the mode corresponding to the same dispersion curve remains unchanged, though there may be some coupled mode very close to the cross point. With approximation in high frequencies andδmatrix method, the formula to the position of cross points is deduced and the error is analyzed. With this formula, the relationship between the frequencies of the cross points and the parameters is studied. It is found that the more obvious the low velocity layer is, the lower the frequency of each the cross point is.
The sensitivity of dispersion curves to the S-wave velocities is explained with horizontal eigendisplacement curves, and an inversion method concerning the sensitivity is presented based on it. Two examples are used to study the relationship between the sensitivity and the eigendisplacements. It is found that usually the sensitivity is related to the eigendisplacements of the Rayleigh wave in each layer, especially the horizontal eigendisplacements. Normally, if the horizontal eigendisplacement corresponding to one dispersion point is larger in one layer than another layer, the Rayleigh wave velocity will be more sensitive to the S-wave velocity in this layer, and vice versa. This characteristic is more obvious in high frequencies. The result shows the relationship between sensitivity with the basic modes of Rayleigh waves. Based on this, a kind of inversion method concerning about sensitivity is presented. Inversion simulation is conducted and the results are compared with that obtained from multimode inversion method. It is found that the method sometimes avoids the problem of local minimum in normal local optimizations. However, the method often needs more dispersion points, and the stability also needs to be improved.
The‘zigzag’phenomenon of dispersion curves when there is low velocity layer is studied. The parameters of a given medium are changed under three different sources to study the influence of the parameters on the‘jump point’frequencies. The frequencies showed little relationship with the offset. It is found that although the‘jump point’frequencies depend on the source style, there are the same rules under each source: the S-wave velocities of the low velocity layer and the layer above it are the dominant parameters that influence the‘jump point’frequencies, and followed by their thickness, the other parameters have less effect on the‘jump point’frequencies, the more obvious the low velocity layer is, the lower the‘jump point’frequency is. The result can be used as a qualitative analysis of low velocity layer in the cases like compaction test of subgrade. Rayleigh wave dispersion curves in two layer media can be approximately seen to be linear corresponding with S-wave velocities. Based on this, a quick inversion method of Rayleigh wave fundamental mode dispersion curves for two layer media is presented. All the formulas of the Rayleigh wave velocity in isotropic half space before contain complex computation. In this paper, a new formula without complex computation is presented. It makes the computation of Rayleigh wave velocities more convenient.
引文
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