基于模型空间压缩的大地电磁三维反演研究

本文提出了一种基于模型空间压缩技术的大地电磁三维反演方法.该方法在传统大地电磁三维反演理论的基础上,通过小波变换将待反演的空间域模型参数映射到小波域进行反演,获得小波域更新模型后再通过小波逆变换得到空间域反演模型.由于小波变换具有压缩特性和多尺度分辨能力,本文反演方法可在一定程度上提高反演分辨率.为了提高反演效率,我们针对基于L_1范数的模型约束求解不易收敛的反演问题,提出了一种基于模型粗糙度的简单有效的预条件处理技术.为验证本文算法的有效性,本文首先对经典的"棋盘"模型进行三维反演测试.反演结果表明本文算法的反演效率与传统方法相当,但对于深部异常体具有更好的分辨能力.最后,我们通过对实测数据反演进一步验证本文算法的有效性.     

基于小波变换的重力压缩正演和多尺度反演研究

解决多解性和计算效率等问题是重力反演中的主要难点。针对计算效率问题,本文基于小波压缩理论,提出一种改进的重力矩阵压缩正演方法。该方法通过引入一种新的灵敏度矩阵排序规则,降低灵敏度峰值和波动、减少高频信息,在保证精度的同时改善小波变换后灵敏度矩阵的稀疏性、提高压缩比,从而降低内存需求,提高正演速度。正演模拟分析结果表明,本文提出的正演矩阵压缩方法在保证正演精度的前提下压缩比可达300以上。其次,本文将基于小波变换的多尺度反演应用到三维重力反演中,通过将重力反演问题分解成不同尺度的子问题,充分利用不同尺度的数据信息来改善重力反演的多解性和稳定性问题。最后,我们分别对理论数据和实测数据进行常规聚焦反演和多尺度反演以证明多尺度反演在重力反演中的有效性。     

基于曲波变换的大地电磁二维稀疏正则化反演

为了提高二维大地电磁反演对异常体边界的刻画能力,我们引入曲波变换建立一种新的稀疏正则化反演方法.与传统的在空间域中对模型电阻率参数求解的方式不同,我们借助曲波变换将二维电阻率模型转换为曲波系数,并采用L1范数约束以保证系数的稀疏性.曲波变换是一种多尺度分析方法,其系数分为粗尺度系数和精细尺度系数,粗尺度的系数代表电阻率模型的整体概貌,而精细尺度中较大系数代表目标体的边缘细节.此外,曲波变换的窗函数满足各向异性尺度关系,并具有多方向性,因此曲波变换可以近似最佳地提取目标体的边缘特征信息,这为我们在反演中恢复边界提供有利条件.通过对大地电磁的理论模型合成数据和实测数据反演,验证了基于曲波变换稀疏正则化反演对异常体边界的刻画能力优于常规的L2范数和L1范数反演方法.     

基于K-SVD字典学习的稀疏约束编码多震源方向:全波形反演（英文）

全波形反演(FWI)是一种较为重要的速度建模方法,但计算量巨大是阻碍其实用化。业已证明通过多震源策略减少模拟单炮次数,可以大大提高全波形反演计算效率,但引入了交叉串扰噪音。为解决上述问题,本文提出一种基于K-SVD字典学习的稀疏约束编码多震源全波形反演方法。首先,增加不同单炮的差异性引入极性编码策略减少串扰噪音;其次基于FWI不同迭代次数反演结果特征引入K-SVD字典学习方法计算变换基函数,推导了基于稀疏约束的目标泛函;进一步我们引入基于维纳滤波的时间域多尺度反演方法,提高反演方法的稳定性。最后,通过洼陷模型和Marmousi模型测试验证表明:1)本文的基于K-SVD字典学习的多震源编码反演方法,在减少全波形反演计算量的同时,能有效克服反演串扰噪音,提高反演精度;2)新方法能灵活的与时间域多尺度反演方法结合,降低反演过程陷入局部极小值,增强反演稳定性,对复杂模型也具有较好的适应性。     

频率多尺度全波形速度反演

以二维声波方程为模型,在时间域深入研究了全波形速度反演.全波形反演要解一个非线性的最小二乘问题,是一个极小化模拟数据与已知数据之间残量的过程.针对全波形反演易陷入局部极值的困难,本文提出了基于不同尺度的频率数据的"逐级反演"策略,即先基于低频尺度的波场信息进行反演,得出一个合理的初始模型,然后再利用其他不同尺度频率的波场进行反演,并且用前一尺度的迭代反演结果作为下一尺度反演的初始模型,这样逐级进行反演.文中详细阐述和推导了理论方法及公式,包括有限差分正演模拟、速度模型修正、梯度计算和算法描述,并以Marmousi复杂构造模型为例,进行了MPI并行全波形反演数值计算,得到了较好的反演结果,验证了方法的有效性和稳健性.     

小波变换多尺度地震波形反演

完全非线性地震波形反演问题是石油地球物理勘探领域中一个非常重要而又难度很大的问题。本文提出了多尺度地震波形反演的小波变换方法,对于一维非线性地震波形反演问题,将该方法和已有的简单迭代法及多重网格法作了比较,数值实验结果表明,本方法效果较好。     

大地电磁的多尺度反演

对于迭代方式的参数化反演方法,如何使反演结果稳定地收敛到整体极小仍是目前大地电磁（MT）反演中急需解决的问题．本文利用小波变换理论中的多尺度分析方法将大地电磁反问题分解为依赖于尺度变量的反问题序列,然后按尺度从大到小的次序依次求解,求解过程中前一个尺度反问题的解作为下一个尺度反问题的初始模型,直到来出对应于尺度为0的原反问题的解为止．该方法称为多尺度反演方法．数值试验和实际资料的反演结果表明,该方法可有效改善传统广义逆反演方法易陷入局部极小的弊端．     

频率域全波形反演方法研究进展

全波形反演方法利用叠前地震波场的运动学和动力学信息重建地下速度结构,具有揭示复杂地质背景下构造与岩性细节信息的潜力.根据研究需要,全波形反演既可在时间域也可在频率域实现.频率域相对于时间域反演具有计算高效、数据选择灵活等优势.近十几年来频率域全波形反演理论在波场模拟方法、反演频率选择策略、目标函数设置方式、震源子波处理方式、梯度预处理方法等方面取得了进展.目标函数存在大量局部极值的特性是影响反射地震全波形反演效果的重要内在因素之一.如果将Laplace域波形反演、频率域阻尼波场反演、频率域波形反演三种方法有机结合,可以降低反演的非线性程度.     

基于双级并行的弹性波频率域全波形多尺度反演方法（英文）

弹性波场的复杂性使得反演问题非线性增强,容易陷入局部极值,需要采用合理的多尺度反演策略降低非线性。在逐频组多尺度反演的基础上引入第二级别的反演策略,即基于阻尼波场的层剥离方法,能够改善反演过程的稳定性。针对频域全波形反演计算效率低、内存占用大的问题,采用双级并行算法:(i)利用多波前大规模并行直接解法(MUMPS)软件包,多节点并行实现波场正演;(ii)基于MPI实现频组内各频率并行计算梯度和步长等,使得多尺度反演算法在提高精度的前提下,保证了计算效率,提高了算法的实用性。Overthrust模型的数值实验表明,本文反演算法能够在有效改善反演稳定性的前提下,高效地获得精度较高的反演结果。     

时间域平面波全波形反演

全波形反演同时利用地震波的振幅和相位信息,基于最小二乘思想反演介质参数,具有高分辨率、高精度等优点.但是,在地震数据的处理过程中运用全波形反演的运算成本非常高.为了解决这个问题,本文通过线性时移函数将多个炮记录压缩为若干个平面波记录,实现了一种基于平面波编码的时间域多尺度全波形反演方法.通过对Marmousi模型的反演测试以及与常规全波形反演对比分析,验证了本文方法的较高的运算效率和较强的抗噪声能力.     

Gravity compression forward modeling and multiscale inversion based on wavelet transform

The main problems in three-dimensional gravity inversion are the non-uniqueness of the solutions and the high computational cost of large data sets. To minimize the high computational cost, we propose a new sorting method to reduce fluctuations and the high frequency of the sensitivity matrix prior to applying the wavelet transform. Consequently, the sparsity and compression ratio of the sensitivity matrix are improved as well as the accuracy of the forward modeling. Furthermore, memory storage requirements are reduced and the forward modeling is accelerated compared with uncompressed forward modeling. The forward modeling results suggest that the compression ratio of the sensitivity matrix can be more than 300. Furthermore, multiscale inversion based on the wavelet transform is applied to gravity inversion. By decomposing the gravity inversion into subproblems of different scales, the non-uniqueness and stability of the gravity inversion are improved as multiscale data are considered. Finally, we applied conventional focusing inversion and multiscale inversion on simulated and measured data to demonstrate the effectiveness of the proposed gravity inversion method.     

L1范数约束被动源数据稀疏反演一次波估计

对于被动源地震数据,运用常规的互相关算法得到的虚拟炮记录中,不仅含有一次波反射信息,还包括了表面相关多次波.然而,通过传统的被动源数据稀疏反演一次波估计(EPSI)方法,可以求得只含有一次波,不含表面相关多次波的虚拟炮记录.本文改进了传统的被动源数据稀疏反演一次波估计问题的求解方法,将被动源稀疏反演一次波估计求解问题转化为双凸L1范数约束的最优化求解问题,避免了在传统的稀疏反演一次波估计过程中用时窗防止反演陷入局部最优化的情况.在L1范数约束最优化的求解过程中,又结合了2DCurvelet变换和小波变换,在2DCurvelet-wavelet域中,数据变得更加稀疏,从而使求得的结果更加准确,成像质量得到了改善.通过简单模型和复杂模型,验证了本文提出方法的有效性.     

基于归一化能量谱目标函数的全波形反演方法

基于L2范数的常规全波形反演目标函数是一个强非线性泛函,在反演过程中容易陷入局部极小值.本文提出归一化能量谱目标函数来缓解全波形反演过程中的强非线性问题,同时能够有效地缓解噪声和震源子波不准等因素的影响.能量谱目标函数是通过匹配观测数据与模拟数据随频率分布的能量信息来实现最小二乘反演的,其忽略了地震数据波形与相位变化的细节特征,这在反演的过程中能够有效缓解波形匹配错位等问题.数值测试结果表明,基于归一化能量谱目标函数在构建初始速度模型、抗噪性和缓解震源子波依赖等方面都优于归一化全波形反演目标函数.金属矿模型测试结果表明,即使地震数据缺失低频分量,基于归一化能量谱目标函数的全波形反演方法在像金属矿这样的强散射介质反演问题上同样具有一定的优势.     

Linear inversion via the discrete wavelet transform pseudoinverse

Classical least‐squares techniques (Moore–Penrose pseudoinverse) are covariance based and are therefore unsuitable for the solution of very large‐scale linear systems in geophysical inversion due to the need of diagonalisation. In this paper, we present a methodology to perform the geophysical inversion of large‐scale linear systems via the discrete wavelet transform. The methodology consists of compressing the linear system matrix using the interesting properties of covariance‐free orthogonal transformations, to design an approximation of the Moore–Penrose pseudoinverse. We show the application of the discrete wavelet transform pseudoinverse to well‐conditioned and ill‐conditioned linear systems. We applied the methodology to a general‐purpose linear problem where the system matrix has been generated using geostatistical simulation techniques and also to a synthetic 2D gravimetric problem with two different geological set‐ups, in the noise‐free and noisy cases. In both cases, the discrete wavelet transform pseudoinverse can be applied to the original linear system and also to the linear systems of normal equations and minimum norm. The results are compared with those obtained via the Moore–Penrose and the discrete cosine transform pseudoinverses. The discrete wavelet transform and the discrete cosine transform pseudoinverses provide similar results and outperform the Moore–Penrose pseudoinverse, mainly in the presence of noise. In the case of well‐conditioned linear systems, this methodology is more efficient when applied to the least‐squares system and minimum norm system due to their higher condition number that allows for a more efficient compression of the system matrix. Also, in the case of ill‐conditioned systems with very high underdetermined character, the application of the discrete cosine transform to the minimum norm solution provides very good results. Both solutions might differ on their regularity, depending on the wavelet family that is adopted. These methods have a general character and can be applied to solve any linear inverse problem arising in technology, particularly in geophysics, and also to non‐linear inversion by linearisation of the forward operator.     

实数编码遗传算法在大地电磁测深二维反演中的应用

作为全局非线性优化的新方法之一的遗传算法,近年来已从生物工程流行到大地电磁测深资料解释中.然而,大地电磁反演问题具有不适定性,解的非唯一性.通过结合求解不适定问题的Tikhonov正则化方法,本文采用实数编码遗传算法求解大地电磁二维反演问题.此算法在构建目标函数时引入正则化的思想,利用遗传算法求解最优化问题.常规的基于局部线性化的最优化反演方法易使解陷入局部极小值,而且严重的依赖初始模型的选择.与传统线性化的迭代反演方法相比,实数编码遗传算法能够克服传统方法的不足且能获得更好的反演结果.通过对大地电磁测深理论模型进行计算,结果表明:该算法具有收敛速度快、解的精度高和避免出现早熟等优点,可用于大地电磁资料解释.     

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.     

A two-step wavelet-based regularization for linear inversion of geophysical data

Regularization methods are used to recover a unique and stable solution in ill-posed geophysical inverse problems. Due to the connection of homogeneous operators that arise in many geophysical inverse problems to the Fourier basis, for these operators classical regularization methods possess some limitations that one may try to circumvent by wavelet techniques.
In this paper, we introduce a two-step wavelet-based regularization method that combines classical regularization methods with wavelet transform to solve ill-posed linear inverse problems in geophysics. The power of the two-step wavelet-based regularization for linear inversion is twofold. First, regularization parameter choice is straightforward; it is obtained from a priori estimate of data variance. Second, in two-step wavelet-based regularization the basis can simultaneously diagonalize both the operator and the prior information about the model to be recovered. The latter is performed by wavelet-vaguelette decomposition using orthogonal symmetric fractional B-spline wavelets.
In the two-step wavelet-based regularization method, at the first step where fully classical tools are used, data is inverted for the Moore-Penrose solution of the problem, which is subsequently used as a preliminary input model for the second step. Also in this step, a model-independent estimate of data variance is made using nonparametric estimation and L-curve analysis. At the second step, wavelet-based regularization is used to partially recover the smoothness properties of the exact model from the oscillatory preliminary model.
We illustrated the efficiency of the method by applying on a synthetic vertical seismic profiling data. The results indicate that a simple non-linear operation of weighting and thresholding of wavelet coefficients can consistently outperform classical linear inverse methods.