直流激电反演中的线性与非线性方法研究
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摘要
地球物理反演理论和方法发展至今,凝聚了历代科学家和实践者的智慧,已成为今天人们用于揭示地球奥秘的有效工具。地球物理反演作为地球物理学科的重要组成部分,已成为地球物理勘探数据处理中的一项核心技术,其反演结果为地质推断解释提供了有利的依据。直流激电法作为地球物理勘探方法的一个重要分支,在能源、矿产及工程勘查中发挥着重要作用。近年来,随着高密度、高精度电法仪器发展,其应用领域也在逐步拓宽,观测位置也从地面向水中、坑道及井中等复杂情况发展。与此同时,人们对其勘探和解释精度的要求也在不断提高,那么作为直流激电数据处理的一项核心技术——反演,也将面临着新的挑战。本文主要针对直流激电数据反演中的线性与非线性方法进行研究,着重探讨如何提高反演结果的分辨率,减少反演的多解性,以及如何增强反演过程的稳定性和健全性等问题。
在第二章中,首先从最小二乘广义线性反演方程出发,引出正则化的概念及正则化技术的基本思想;然后从正则化参数的选择、稳定化泛函的构造、引入某种先验约束及修正迭代步长等几个方面探讨了广义线性反演的正则化技术,以此改善反演方程的病态特征,抑制反演过程的不稳定性,最终达到提高反演分辨率的目的。
在第三章中,考虑到在地球物理勘探过程中,由于受非高斯噪音的影响,使采集的数据含有突变噪音,如果仍然采用基于最小二乘准则的广义线性反演方法进行反演,将会造成反演假象,进而可能导致错误解释。据此,笔者提出了混合范数的最优化方法,即根据观测数据品质的优劣,对数据空间和模型空间分别采用不同的范数进行测度,达到压制干扰突出有用异常的目的。由于混合范数的引入,增加了线性反演方程的复杂性,对其求解更加困难。因此,通过将加权矩阵规范化,并采用混合范数下迭代再加权阻尼共轭梯度算法进行求解,使得问题得到解决。最后,通过对含有和不含有猝发噪音的模拟电阻率数据进行反演,验证了混合范数下最优化反演方法的可行性和有效性。
在第四章中,从最小二乘广义线性反演的角度,探讨了直流激电数据的广义线性反演方法;在偏导数矩阵的计算过程中,通过将互换原理与Broyden拟牛顿技术相结合的计算方式,可大大提高反演的运算速度。然后对三种观测方式的直流激电数据反演中所涉及到一些技术问题加以解决和完善。对于垂直激电测深(VES)二维反演,主要从正反演所采用的双网格系统、正演模拟的误差校正以及反演中初始模型的给定三个方面探讨如何提高正演的精度和反演的分辨率;对于电阻率最小二乘间歇反演,提出了一种同时反演所有不同时刻观测数据的新方法,即通过一次反演过程来获得所有不同时刻的地下模型。在反演过程中,引入时间和空间稳定化泛函,将不同时刻的观测数据联系起来,可极大的提高反演的分辨率和减少反演的多解性;对于复杂条件下的三维直流激电数据反演,可以说它是一个复杂的系统工程,要涉及到多个重要环节,如起伏地表大量网格节点的高程插值、地下介质的网格剖分、正演中刚度矩阵元素的存储及其方程的求解以及正演的计算精度和效率问题等。笔者对这些问题逐一进行了研究,使三维直流激电反演中所涉及的问题得到进一步完善。最后,对起伏地形条件下的三维直流激电数据进行反演,验证方法的有效性。
在第五章中,由于地球物理反问题大多都具有多参数、多极值和非线性的特点,线性或拟线性反演方法往往得到的只是局部最优解,并且反演结果常常与初始模型有关,如果初始模型选择的合适,它可以得到较好的结果,反之,还可能造成反演的假象。而以随机过程为基础的全局最优化反演方法不受确定规则的限制,自由的在模型空间内随机搜索,这种较强的搜索能力能够保证反演结果基本上是全局最优或次最优解。本章主要对非线性全局最优化方法中的模拟退火算法和遗传算法进行研究。在简单介绍方法原理的基础上,着重对全局与局部优化方法以及全局与全局优化方法相结合的混合方法进行深入研究,以提高算法的全局寻优性能。最后通过对直流激电测深曲线进行反演,验证了非线性混合反演方法的有效性。
Theory and methods of geophysical inversion developed up to now, agglomerating the intelligence of many scientists and practicers, and it had became a valid tool that people revealed the profound mystery of the Earth. Geophysical inversion was an important section of geophysical branch of knowledge, and it had become a core technique of the geophysical data processing, the inverted result provided a beneficial basis for the geological interpretation. Direct current induced polarization method was an important branch of geophysical exploration methods, which played an important part in the energy, mineral resources and engineering exploration. In recent years, with the development of the high density and high precision instrument of electrical method, the applied field of electrical surveying is in continuous evolution, the survey space is gradually expanding from ground to some complex quomodo of underwater, sap and borehole etc. At the same time, people's request for the exploration and interpretation precision is also raising continuously, so, the inversion acting as a core technique of data processing of direct current induced polarization, which will be also facing with new challenge. In this dissertation, we will carry on an investigation in the linear and nonlinear inversion methods of direct current induced polarization measurements, emphasizing to study how to improve the resolution and reduce the multisolution of inverted result, and how to enhance the stabilization and robustness etc.
In the chapter 2, firstly, the concept of regularization and idea of regulation technique were introduced beginning with the generalized linear inversion equation of least squares method. Then, we studied the regularization techniques of generalized linear inversion from several aspects, for instance, the choice of regularization parameter, the design of stabilizing functionals, the introduce of some prior restriction and the modification of iteration step etc, whicd can improve the ill-posedness of inversion equation and restrain the non-stability of inversion process and heighten the resolution of inversion result finally.
In the chapter 3, we considered that geophysical exploration work was often influenced by the non-Gaussian noise and made the data set of acquisition contain some false data, it could lead to the artifact of inversion and the mistake of interpretation if the data set containing non-Gaussian noise was inverted using generalized linear inversion method based on the least squares principle. Thereby we put forward the optimized inversion method using mixed norm, that is to say, according to the good or poor quality of observed data, the data and model space is measured by different norm, which can gain the ends of inhibiting the disturbance and giving prominence to the useful abnormity. Because of the introduction of mixed norm, which increases the complexity of linear inversion equation, it is more difficult to solve the linear equation than ever. Therefore the problem was settled by normalizing the weighted matrix and solving the linear inversion equation of mixed norm using the iteratively reweighted damping conjugate gradient algorithm. At last, we verified that the optimized inversion method based on mixed norm is workable and effective through inverting the simulated resistivity data containing and not containing abrupt noise.
In the chapter 4, we discussed the generalized linear inversion method of direct current IP data from the angle of the least squares method. The computed speed of inversion can be greatly raised by calculating the partial derivatives matrix combining the reciprocity principle with Broyden's quasi-Newton method. Then we solved and improved some technique problems concerning dicrect current IP data inversion for three observed methods. For the 2D inversion of vertical IP sounding data, we studied how to raise the precision of forward and resolution of inversion from three aspects of using dual-grid system, modifing the forward error and giving the initial inversion model. For the least squares time-lapse inversion of resistivity data, we put forward a new method to invert all data sets collected at different time simultaneously, namely get all earth models of different time through only one inversion process. In the inversion, we combined all data sets acquired at different time by introducing the stability functional in time and space to objective function, which could greatly improve the stability and reduced the multi-solution. For the 3D inversion of direct current IP data in complex earth condition, we can say that it is a complex systemic engeering being involved to many important taches, for example, the elevation interpolation of large numbers of gridding nodes on the rolling ground, gridding divition of underground, the storage of rigid matrix elements and the solution of its equation in the forward and the precision and efficiency of forward simulation etc.We studied these problems one by one, which made some problems concerning 3D inversion of direct current IP data be improved further. At last, we verified the effectiveness of method by means of inverting the dicrect current IP data of 3D rolling terrain.
In the chapter 5, linear and quasi-linear inversion method can only get the local optimum solution because the geophysical inverse problems are of the characters of multi-parameters, multi-extrema and non-linearity, and the inversion result is related to the initial modeling, we can get good solution if the initial model choose is suitable, otherwise can lead to the artifact of inversion. Whereas the global optimum inversion method based on random process is not limited by the fixed rule, it can freely search the optimized solution in the model space, this powerfully searching ability ensures that the inversion solution basically is the global optimum or hypo-optimum solution. This chapter mainly studied the simulated annealing algorithm and genetic algorithm belonging to non-linear global optimum method. We deeply studied the mixed methods of combining the global with the global and the global with the global optimum method on the foundation of simply introducing the methods amd principles, the mixed method can raise the global searching ability of algorithm. At last, we verified the effectiveness of mixed inversion method by inverting the dicrect current IP sounding data.
在第二章中,首先从最小二乘广义线性反演方程出发,引出正则化的概念及正则化技术的基本思想;然后从正则化参数的选择、稳定化泛函的构造、引入某种先验约束及修正迭代步长等几个方面探讨了广义线性反演的正则化技术,以此改善反演方程的病态特征,抑制反演过程的不稳定性,最终达到提高反演分辨率的目的。
在第三章中,考虑到在地球物理勘探过程中,由于受非高斯噪音的影响,使采集的数据含有突变噪音,如果仍然采用基于最小二乘准则的广义线性反演方法进行反演,将会造成反演假象,进而可能导致错误解释。据此,笔者提出了混合范数的最优化方法,即根据观测数据品质的优劣,对数据空间和模型空间分别采用不同的范数进行测度,达到压制干扰突出有用异常的目的。由于混合范数的引入,增加了线性反演方程的复杂性,对其求解更加困难。因此,通过将加权矩阵规范化,并采用混合范数下迭代再加权阻尼共轭梯度算法进行求解,使得问题得到解决。最后,通过对含有和不含有猝发噪音的模拟电阻率数据进行反演,验证了混合范数下最优化反演方法的可行性和有效性。
在第四章中,从最小二乘广义线性反演的角度,探讨了直流激电数据的广义线性反演方法;在偏导数矩阵的计算过程中,通过将互换原理与Broyden拟牛顿技术相结合的计算方式,可大大提高反演的运算速度。然后对三种观测方式的直流激电数据反演中所涉及到一些技术问题加以解决和完善。对于垂直激电测深(VES)二维反演,主要从正反演所采用的双网格系统、正演模拟的误差校正以及反演中初始模型的给定三个方面探讨如何提高正演的精度和反演的分辨率;对于电阻率最小二乘间歇反演,提出了一种同时反演所有不同时刻观测数据的新方法,即通过一次反演过程来获得所有不同时刻的地下模型。在反演过程中,引入时间和空间稳定化泛函,将不同时刻的观测数据联系起来,可极大的提高反演的分辨率和减少反演的多解性;对于复杂条件下的三维直流激电数据反演,可以说它是一个复杂的系统工程,要涉及到多个重要环节,如起伏地表大量网格节点的高程插值、地下介质的网格剖分、正演中刚度矩阵元素的存储及其方程的求解以及正演的计算精度和效率问题等。笔者对这些问题逐一进行了研究,使三维直流激电反演中所涉及的问题得到进一步完善。最后,对起伏地形条件下的三维直流激电数据进行反演,验证方法的有效性。
在第五章中,由于地球物理反问题大多都具有多参数、多极值和非线性的特点,线性或拟线性反演方法往往得到的只是局部最优解,并且反演结果常常与初始模型有关,如果初始模型选择的合适,它可以得到较好的结果,反之,还可能造成反演的假象。而以随机过程为基础的全局最优化反演方法不受确定规则的限制,自由的在模型空间内随机搜索,这种较强的搜索能力能够保证反演结果基本上是全局最优或次最优解。本章主要对非线性全局最优化方法中的模拟退火算法和遗传算法进行研究。在简单介绍方法原理的基础上,着重对全局与局部优化方法以及全局与全局优化方法相结合的混合方法进行深入研究,以提高算法的全局寻优性能。最后通过对直流激电测深曲线进行反演,验证了非线性混合反演方法的有效性。
Theory and methods of geophysical inversion developed up to now, agglomerating the intelligence of many scientists and practicers, and it had became a valid tool that people revealed the profound mystery of the Earth. Geophysical inversion was an important section of geophysical branch of knowledge, and it had become a core technique of the geophysical data processing, the inverted result provided a beneficial basis for the geological interpretation. Direct current induced polarization method was an important branch of geophysical exploration methods, which played an important part in the energy, mineral resources and engineering exploration. In recent years, with the development of the high density and high precision instrument of electrical method, the applied field of electrical surveying is in continuous evolution, the survey space is gradually expanding from ground to some complex quomodo of underwater, sap and borehole etc. At the same time, people's request for the exploration and interpretation precision is also raising continuously, so, the inversion acting as a core technique of data processing of direct current induced polarization, which will be also facing with new challenge. In this dissertation, we will carry on an investigation in the linear and nonlinear inversion methods of direct current induced polarization measurements, emphasizing to study how to improve the resolution and reduce the multisolution of inverted result, and how to enhance the stabilization and robustness etc.
In the chapter 2, firstly, the concept of regularization and idea of regulation technique were introduced beginning with the generalized linear inversion equation of least squares method. Then, we studied the regularization techniques of generalized linear inversion from several aspects, for instance, the choice of regularization parameter, the design of stabilizing functionals, the introduce of some prior restriction and the modification of iteration step etc, whicd can improve the ill-posedness of inversion equation and restrain the non-stability of inversion process and heighten the resolution of inversion result finally.
In the chapter 3, we considered that geophysical exploration work was often influenced by the non-Gaussian noise and made the data set of acquisition contain some false data, it could lead to the artifact of inversion and the mistake of interpretation if the data set containing non-Gaussian noise was inverted using generalized linear inversion method based on the least squares principle. Thereby we put forward the optimized inversion method using mixed norm, that is to say, according to the good or poor quality of observed data, the data and model space is measured by different norm, which can gain the ends of inhibiting the disturbance and giving prominence to the useful abnormity. Because of the introduction of mixed norm, which increases the complexity of linear inversion equation, it is more difficult to solve the linear equation than ever. Therefore the problem was settled by normalizing the weighted matrix and solving the linear inversion equation of mixed norm using the iteratively reweighted damping conjugate gradient algorithm. At last, we verified that the optimized inversion method based on mixed norm is workable and effective through inverting the simulated resistivity data containing and not containing abrupt noise.
In the chapter 4, we discussed the generalized linear inversion method of direct current IP data from the angle of the least squares method. The computed speed of inversion can be greatly raised by calculating the partial derivatives matrix combining the reciprocity principle with Broyden's quasi-Newton method. Then we solved and improved some technique problems concerning dicrect current IP data inversion for three observed methods. For the 2D inversion of vertical IP sounding data, we studied how to raise the precision of forward and resolution of inversion from three aspects of using dual-grid system, modifing the forward error and giving the initial inversion model. For the least squares time-lapse inversion of resistivity data, we put forward a new method to invert all data sets collected at different time simultaneously, namely get all earth models of different time through only one inversion process. In the inversion, we combined all data sets acquired at different time by introducing the stability functional in time and space to objective function, which could greatly improve the stability and reduced the multi-solution. For the 3D inversion of direct current IP data in complex earth condition, we can say that it is a complex systemic engeering being involved to many important taches, for example, the elevation interpolation of large numbers of gridding nodes on the rolling ground, gridding divition of underground, the storage of rigid matrix elements and the solution of its equation in the forward and the precision and efficiency of forward simulation etc.We studied these problems one by one, which made some problems concerning 3D inversion of direct current IP data be improved further. At last, we verified the effectiveness of method by means of inverting the dicrect current IP data of 3D rolling terrain.
In the chapter 5, linear and quasi-linear inversion method can only get the local optimum solution because the geophysical inverse problems are of the characters of multi-parameters, multi-extrema and non-linearity, and the inversion result is related to the initial modeling, we can get good solution if the initial model choose is suitable, otherwise can lead to the artifact of inversion. Whereas the global optimum inversion method based on random process is not limited by the fixed rule, it can freely search the optimized solution in the model space, this powerfully searching ability ensures that the inversion solution basically is the global optimum or hypo-optimum solution. This chapter mainly studied the simulated annealing algorithm and genetic algorithm belonging to non-linear global optimum method. We deeply studied the mixed methods of combining the global with the global and the global with the global optimum method on the foundation of simply introducing the methods amd principles, the mixed method can raise the global searching ability of algorithm. At last, we verified the effectiveness of mixed inversion method by inverting the dicrect current IP sounding data.
引文
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