带先验知识的波阻抗反演正则化方法研究

针对波阻抗反演中存在的不适定性问题,本文提出了一种带先验知识的正则化重开始共轭梯度法.该方法的内层循环采用修改的共轭梯度法,并使用重开始技巧;外层循环使用Morozov偏差准则作为停机准则.正则参数的选取采用连续几何选取法.克服了传统共轭梯度法迭代不足或迭代过度的缺点,将迭代步数控制在了合适的范围,使算法能够更快速更准确的收敛.同时考虑了用最速下降法计算先验解和对解施加非均一的规范约束.通过理论模型试算和实际资料处理,并与共轭梯度法进行对比,表明该算法具有精度高、抗病态能力强,运算速度快的优点,具有实用性.     

倾子资料三维共轭梯度反演研究

在对倾子响应和共轭梯度算法深入分析的基础上,我们实现了倾子资料三维共轭梯度反演算法.基于倾子资料的三维共轭梯度反演研究,探讨了利用倾子资料进行三维反演定量解释的方法.通过对理论模型合成数据 进行反演试算,验证了所实现的倾子资料三维共轭梯度反演算法的有效性和稳定性.该反演算法可用于对大地电磁测深和地磁测深(地震地磁台站进...     

从瞬变电磁扩散场到拟地震波场的全时域反变换算法

将瞬变电磁满足的扩散方程转变为波动方程,然后利用地震类成像方法实现瞬变电磁虚拟波场成像,是实现瞬变电磁三维反演的有效手段之一.为了实现由扩散场到虚拟波场的转换,文中采用预条件正则化共轭梯度法求解波场反变换问题.首先,对几种离散方式进行比较,采用条件数最小的离散方式进行离散;然后选择最优的正则化参数,并利用超松弛预条件技术对系数矩阵进行预条件处理;最后,利用共轭梯度法进行迭代求解.超松弛预条件有效降低了系数矩阵的条件数,正则化方法使得反变换得到的波场稳定、可靠,共轭梯度法能够保证计算快速收敛.将反变换结果与已知虚拟波场函数对比,证明算法稳定、可信.将文中算法结果与前人研究结果进行对比,说明方法效果.通过实测数据的波场变换处理给出了文中方法的实际应用效果.结合反变换算法,对不同参数模型进行分析,总结了虚拟波场在色散介质中的传播规律.     

AN ALTERNATIVE STRATEGY FOR NON-LINEAR INVERSION OF SEISMIC WAVEFORMS1

A common example of a large-scale non-linear inverse problem is the inversion of seismic waveforms. Techniques used to solve this type of problem usually involve finding the minimum of some misfit function between observations and theoretical predictions. As the size of the problem increases, techniques requiring the inversion of large matrices become very cumbersome. Considerable storage and computational effort are required to perform the inversion and to avoid stability problems. Consequently methods which do not require any large-scale matrix inversion have proved to be very popular. Currently, descent type algorithms are in widespread use. Usually at each iteration a descent direction is derived from the gradient of the misfit function and an improvement is made to an existing model based on this, and perhaps previous descent directions. A common feature in nearly all geophysically relevant problems is the existence of separate parameter types in the inversion, i.e. unknowns of different dimension and character. However, this fundamental difference in parameter types is not reflected in the inversion algorithms used. Usually gradient methods either mix parameter types together and take little notice of the individual character or assume some knowledge of their relative importance within the inversion process. We propose a new strategy for the non-linear inversion of multi-offset reflection data. The paper is entirely theoretical and its aim is to show how a technique which has been applied in reflection tomography and to the inversion of arrival times for 3D structure, may be used in the waveform case. Specifically we show how to extend the algorithm presented by Tarantola to incorporate the subspace scheme. The proposed strategy involves no large-scale matrix inversion but pays particular attention to different parameter types in the inversion. We use the formulae of Tarantola to state the problem as one of optimization and derive the same descent vectors. The new technique splits the descent vector so that each part depends on a different parameter type, and proceeds to minimize the misfit function within the sub-space defined by these individual descent vectors. In this way, optimal use is made of the descent vector components, i.e. one finds the combination which produces the greatest reduction in the misfit function based on a local linearization of the problem within the subspace. This is not the case with other gradient methods. By solving a linearized problem in the chosen subspace, at each iteration one need only invert a small well-conditioned matrix (the projection of the full Hessian on to the subspace). The method is a hybrid between gradient and matrix inversion methods. The proposed algorithm requires the same gradient vectors to be determined as in the algorithm of Tarantola, although its primary aim is to make better use of those calculations in minimizing the objective function.     

Optimized staggered-grid finite-difference operators using window functions

The staggered-grid finite-difference (SGFD) method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.     

基于初至波走时层析成像的Tikhonov正则化与梯度优化算法

初至波走时层析成像是利用地震初至波走时和其传播的射线路径来反演地下介质速度的技术.该问题本质上是一个不适定问题,需要使用正则化方法并辅之以适当的最优化技巧.本文从数值优化的角度介绍了初至波走时层析成像的反演原理,建立了Tikhonov正则化层析成像反演模型并提出求解极小化问题的加权修正步长的梯度下降算法.该方法可以从速度模型的可行域中迭代找到一个最优解.数值试验表明,该方法是可行和有应用前景的.     

可控源音频大地电磁三维共轭梯度反演研究

可控源音频大地电磁法在资源勘探等领域中发挥着重要的作用.我们把有限差分数值模拟方法用于可控源音频大地电磁三维正演,结合正则化反演方案和共轭梯度反演的思路,将反演中的雅可比矩阵计算问题转为求解两次"拟正演"问题,得到模型参数的更新步长,形成反演迭代,实现了可控源音频大地电磁三维共轭梯度反演算法.该反演算法可用于对有限长度电偶源激发下采集到的可控源音频大地电磁全区(近区、过渡区和远区)视电阻率和相位资料进行三维反演定量解释,获得地下三维模型的电阻率结构.理论模型合成数据的反演算例验证了所实现的可控源音频大地电磁三维共轭梯度反演算法的有效性和稳定性.     

Error control of iterative linear solvers for integrated groundwater models

An open problem that arises when using modern iterative linear solvers, such as the preconditioned conjugate gradient method or Generalized Minimum RESidual (GMRES) method, is how to choose the residual tolerance in the linear solver to be consistent with the tolerance on the solution error. This problem is especially acute for integrated groundwater models, which are implicitly coupled to another model, such as surface water models, and resolve both multiple scales of flow and temporal interaction terms, giving rise to linear systems with variable scaling. This article uses the theory of "forward error bound estimation" to explain the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed by the US Geological Survey and the California State Department of Water Resources, we observe that this error bound guides the choice of a practical measure for controlling the error in linear systems. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive Over-Relaxation (SOR) method, the most widely known iterative solver for nonsymmetric coefficient matrices. With forward error control, GMRES can easily replace the SOR method in legacy groundwater modeling packages, resulting in the overall simulation speedups as large as 7.74×. This research is expected to broadly impact groundwater modelers through the demonstration of a practical and general approach for setting the residual tolerance in line with the solution error tolerance and presentation of GMRES performance benchmarking results.     

Comparison of several methods for the solution of the Helmholtz equation

Summary A series of Helmholtz equations has to be solved in short-range weather forecast models which use a splitting scheme of integration. For these purposes the successive overrelaxation, the Gauss-Seidel relaxation, the conjugate gradient method, the steepest descent method, the full-multigrid method and the direct method based on the minimum degree algorithm were used and their efficiencies were compared. It was found that the full-multigrid method was the most efficient among the iterative methods in terms of computational time, and that the effect rapidly increased with the grid size. The direct method may be an appropriate approach if the solution is repeated for various right-hand sides, but it requires large auxiliary computer memory. The selection of the optimum method depends on the concrete problem being solved and on the computer memory available.     

Two-dimensional inversion of magnetotelluric data with consecutive use of conjugate gradient and least-squares solution with singular value decomposition algorithms

I investigated the two‐dimensional magnetotelluric data inversion algorithms in studying two significant aspects within a linearized inversion approach. The first one is the method of minimization and second one is the type of stabilizing functional used in parametric functionals. The results of two well‐known inversion algorithms, namely conjugate gradient and the least‐squares solution with singular value decomposition, were compared in terms of accuracy and CPU time. In addition, magnetotelluric data inversion with various stabilizers, such as L2‐norm, smoothing, minimum support, minimum gradient support and first‐order minimum entropy, were examined. A new inversion algorithm named least‐squares solution with singular value decomposition and conjugate gradient is suggested in seeing the outcomes of the comparisons carried out on least‐squares solutions with singular value decomposition and conjugate gradient algorithms subject to a variety of stabilizers. Inversion results of synthetic data showed that the newly suggested algorithm yields better results than those of the individual implementations of conjugate gradient and least‐squares solution with singular value decomposition algorithms. The suggested algorithm and the above‐mentioned algorithms inversion results for the field data collected along a line crossing the North Anatolian Fault zone were also compared each other and results are discussed.     

Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravity-gradiometry data

With the continuous development of full tensor gradiometer (FTG) measurement techniques, three-dimensional (3D) inversion of FTG data is becoming increasingly used in oil and gas exploration. In the fast processing and interpretation of large-scale high-precision data, the use of the graphics processing unit process unit (GPU) and preconditioning methods are very important in the data inversion. In this paper, an improved preconditioned conjugate gradient algorithm is proposed by combining the symmetric successive over-relaxation (SSOR) technique and the incomplete Choleksy decomposition conjugate gradient algorithm (ICCG). Since preparing the preconditioner requires extra time, a parallel implement based on GPU is proposed. The improved method is then applied in the inversion of noisecontaminated synthetic data to prove its adaptability in the inversion of 3D FTG data. Results show that the parallel SSOR-ICCG algorithm based on NVIDIA Tesla C2050 GPU achieves a speedup of approximately 25 times that of a serial program using a 2.0 GHz Central Processing Unit (CPU). Real airborne gravity-gradiometry data from Vinton salt dome (southwest Louisiana, USA) are also considered. Good results are obtained, which verifies the efficiency and feasibility of the proposed parallel method in fast inversion of 3D FTG data.     

Long period multiple suppression by predictive deconvolution in the x–t domainl1

There are two forms of systematic error in conventional deconvolution as applied to the problem of suppressing multiples with periodicities longer than a hundred milliseconds. One of these is the windowing effect due to the assumption that a true autocorrelation function can be computed from a finite portion of data. The second form of error concerns the assumption of periodicity, which is strictly true only at zero offset for a 1D medium. The seriousness of these errors increased with the lengthening of the multiple period. This paper describes and illustrates a rigorous 2D solution to the predictive deconvolution equations that overcomes both of the systematic errors of conventional 1D approaches. This method is applicable to both the simple or trapped system and to the complex or peg-leg system of multiples. It does not require that the design window be six to ten times larger compared to the operator dimensions and it is accurate over a wide range of propagation angles. The formulation is kept strictly in the sense of the classical theory of prediction. The solution of normal equations are obtained by a modified conjugate gradient method of solution developed by Koehler. In this algorithm, the normal equations are not modified by the autocorrelation approximation. As with all linear methods, approximate stationary attitude in the multiple generating process is assumed. This method has not been tested in areas where large changes in the characteristic of the multiple-generating mechanism occur within a seismic spread length.     

复杂岩性解释模型稳定性和可靠性评价

对解释模型进行评价以及如何选择合适的解释模型是复杂岩性解释中非常重要的问题。本文应用统计学的理论以及非线性优化技术,系统研究了在线性约束条件下对储层参数的置信区域和误差大小进行估计的问题,通过将梯度投影算法与奇异值分解技术相结合的方法给出了计算储层参数置信区域与误差大小的公式,以及对复杂岩性解释模型的稳定性和可靠性进行合理评价的具体方法。最后通过数值计算实例研究了一种能够有效提高解释模型稳定性和可靠性的措施,数值计算结果证明解释模型响应函数的最小非零奇异值大小是影响稳定性和可靠性的一个重要参数,当最小非零奇异值较大时,模型的稳定性往往较好,否则较差,因此通过对解释模型进行适当组合并选取最小非零奇异值较大的综合解释模型,可以大大提高其稳定性和可靠性。     

基于数据空间和稀疏约束的三维重力和重力梯度数据联合反演

随着重力和重力梯度测量技术的日趋成熟,基于重力和重力梯度数据的反演技术得到了广泛关注.针对反演多解性严重、计算效率低和内存消耗大等难点问题,本文开展了三维重力和重力梯度数据的联合反演研究,该方法结合重力和重力梯度两种数据,将L0范数正则化项加入到目标函数中,并在数据空间下采用改进的共轭梯度算法求解反演最优化问题.同时,...     

井地磁异常模量联合反演

三维反演是磁测数据定量解释的重要方法,在金属矿勘探中扮演着重要的角色.但是在实际矿区的应用中,传统的磁总场异常反演方法依然存在两个问题：一是地面磁异常反演的深度分辨率较低,深部场源体的成像效果差;二是金属矿中可能包含强剩磁,反演结果可能是完全错误的.尽管前人对上述两个问题分别进行了广泛的研究,但尚未尝试同时解决这两个问题.本文在前人研究的基础上,提出了一种井地磁异常模量联合反演方法,该方法需要的控制参数少,无需加入额外的地质信息,且可用于多场源复杂磁异常的反演,具有较强的适用性.本文方法首先将地面和井中磁异常转化为模量数据,然后利用基于核函数或距离的加权函数将井地模量数据结合起来,使得该方法适用于联合反演.我们利用井地多种异常参量进行反演的模型试验表明,在强剩磁存在时,本文方法的效果优于其他方法,在减少剩磁影响的同时,也改善了深部成像效果,具有良好的应用前景.     

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.     

大地电磁全信息资料三维共轭梯度反演研究(英文)

在对张量阻抗数据、倾子数据和共轭梯度算法深入分析的基础上,我们实现了大地电磁全信息资料三维共轭梯度反演算法。基于全信息资料的三维共轭梯度反演研究,探讨了同时利用五个电磁场分量整理得到的大地电磁资料进行三维反演定量解释的方法以及全信息数据在三维反演中的作用。理论模型合成数据的反演结果表明,在三维反演中使用张量阻抗和倾子数据结合的全信息数据的反演结果优于只使用张量阻抗数据(或只使用倾子数据)的反演结果,提高了反演结果的分辨率和可信度。合成数据的反演算例也验证了所实现的大地电磁全信息资料三维共轭梯度反演算法的正确性和稳定性。     

Preconditioned non‐linear conjugate gradient method for frequency domain full‐waveform seismic inversion

We present preconditioned non‐linear conjugate gradient algorithms as alternatives to the Gauss‐Newton method for frequency domain full‐waveform seismic inversion. We designed two preconditioning operators. For the first preconditioner, we introduce the inverse of an approximate sparse Hessian matrix. The approximate Hessian matrix, which is highly sparse, is constructed by judiciously truncating the Gauss‐Newton Hessian matrix based on examining the auto‐correlation and cross‐correlation of the Jacobian matrix. As the second preconditioner, we employ the approximation of the inverse of the Gauss‐Newton Hessian matrix. This preconditioner is constructed by terminating the iteration process of the conjugate gradient least‐squares method, which is used for inverting the Hessian matrix before it converges. In our preconditioned non‐linear conjugate gradient algorithms, the step‐length along the search direction, which is a crucial factor for the convergence, is carefully chosen to maximize the reduction of the cost function after each iteration. The numerical simulation results show that by including a very limited number of non‐zero elements in the approximate Hessian, the first preconditioned non‐linear conjugate gradient algorithm is able to yield comparable inversion results to the Gauss‐Newton method while maintaining the efficiency of the un‐preconditioned non‐linear conjugate gradient method. The only extra cost is the computation of the inverse of the approximate sparse Hessian matrix, which is less expensive than the computation of a forward simulation of one source at one frequency of operation. The second preconditioned non‐linear conjugate gradient algorithm also significantly saves the computational expense in comparison with the Gauss‐Newton method while maintaining the Gauss‐Newton reconstruction quality. However, this second preconditioned non‐linear conjugate gradient algorithm is more expensive than the first one.     

层状TI介质中微地震定位和各向异性速度结构同时反演

微地震监测被广泛应用于非常规油气资源的水力压裂作业、油藏描绘和水驱前缘监测工程中.微地震定位采用的初始速度模型一般是基于地震测井记录和射孔数据建立,该速度模型的不准确性易引起定位误差.为降低这种定位误差,本文发展了一种微地震定位和各向异性速度结构同时反演的方法.研究对象为1-D的层状TI介质,其中对称轴方向任意.利用改进的分区多步最短路径算法计算qP、qSV和qSH波的到达时间和射线路径,结合共轭梯度法求解带约束的阻尼最小二乘问题.数值模拟结果表明,该算法能同时进行各向异性速度结构模型(每层的Thomsen参数和界面深度)和微震震源参数(空间坐标和发震时刻)的反演,并且对随机噪声不敏感,有利于实际工程应用.     

非线性波动方程地震反演的方法原理及问题

在解反射地震的非线性反问题时,目前都采用各种迭代算法(如梯度法、最速下降法及共轭梯度下降法等),并以拟合差取极小为准则.本文对这方面具有代表性的波动方程反演理论作分析评述,指出这种经典性方法的缺点和局限,以及发展非线性波动方程地震反演的方向.