First‐arrival traveltime tomography with modified total‐variation regularization

First‐arrival traveltime tomography is a robust tool for near‐surface velocity estimation. A common approach to stabilizing the ill‐posed inverse problem is to apply Tikhonov regularization to the inversion. However, the Tikhonov regularization method recovers smooth local structures while blurring the sharp features in the model solution. We present a first‐arrival traveltime tomography method with modified total‐variation regularization to preserve sharp velocity contrasts and improve the accuracy of velocity inversion. To solve the minimization problem of the new traveltime tomography method, we decouple the original optimization problem into the two following subproblems: a standard traveltime tomography problem with the traditional Tikhonov regularization and a L₂ total‐variation problem. We apply the conjugate gradient method and split‐Bregman iterative method to solve these two subproblems, respectively. Our synthetic examples show that the new method produces higher resolution models than the conventional traveltime tomography with Tikhonov regularization, and creates less artefacts than the total variation regularization method for the models with sharp interfaces. For the field data, pre‐stack time migration sections show that the modified total‐variation traveltime tomography produces a near‐surface velocity model, which makes statics corrections more accurate.     

Two dimensional combined inversion of short- and long-normal dc resistivity well log data

A method is presented for the two-dimensional combined inversion of short- and long-normal tool direct current resistivity data with symmetry. The forward problem is solved using the finite element method in the cylindrical coordinates system. The inverse problem is solved using a conjugate gradient technique with the partial derivatives obtained using reciprocity. The parameters were obtained by means of both conjugate gradient relaxation and conventional conjugate gradient method. The solution of this highly underdetermined inverse problem is stabilized using Tikhonov regularization and the scheme yields a blurred image of the subsurface. The scheme is tested using synthetic data and field data. Tests using synthetic data suggest that traces of the horizontal boundaries are delineated in the range of the exploration distance while the resolution of vertical boundaries depends upon the solution regularization. Application to field data shows that additional information is necessary for resolving the resistivity structure when there are low resistivity contrasts between formation units.     

Comparison of methods for a 3-D density inversion from airborne gravity gradiometry

In this study, we apply Tikhonov’s regularization algorithm for a 3-D density inversion from the gravity-gradiometry data. To reduce the non-uniqueness of the inverse solution (carried out without additional information from geological evidence), we implement the depth-weighting empirical function. However, the application of an empirical function in the inversion equation brings the bias problem of the regularization factor when a traditional Tikhonov’s algorithm is applied. To solve the bias problem of regularization factor selection, we present a standardized solution that comprises two parts for solving a 3-D constrained inversion equation, specifically the linear matrix transformation and Tikhonov’s regularization algorithm. Since traditional regularization techniques become numerically inefficient when dealing with large number of data, we further apply methods which include the Simultaneous Iterative Reconstruction Technique (SIRT) and the wavelet compression combined with Least Squares QR-decomposition (LSQR). In our simulation study, we demonstrate that SIRT as well as the wavelet compression plus LSQR algorithm improve the computation efficiency, while provide results which closely agree with that obtained from applying Tikhonov’s regularization. In particular, the algorithm of wavelet compression plus LSQR shows the best computing efficiency, because it combines the advantages of coefficients compression of big matrix and fast solution of sparse matrix. Similar findings are confirmed from the vertical gravity gradient data inversion for detecting potential deposits at the Kauring (near Perth, Western Australia) testing site.     

1D SNMR吉洪诺夫正则化反演方法研究

地面核磁共振(SNMR)技术是目前世界上直接用来寻找地下水的技术。本文在讨论反演基本问题的基础上,用共轭梯度法实现了核磁共振的一维正则化反演。通过反演理论数据、噪声数据和实测数据,说明该方法的可靠性。     

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.     

Imaging of elastic seismic data by least-squares reverse time migration with weighted L2-norm multiplicative and modified total-variation regularizations

Least-squares reverse time migration has the potential to yield high-quality images of the Earth. Compared with acoustic methods, elastic least-squares reverse time migration can effectively address mode conversion and provide velocity/impendence and density perturbation models. However, elastic least-squares reverse time migration is an ill-posed problem and suffers from a lack of uniqueness; further, its solution is not stable. We develop two new elastic least-squares reverse time migration methods based on weighted L2-norm multiplicative and modified total-variation regularizations. In the proposed methods, the original minimization problem is divided into two subproblems, and the images and auxiliary variables are updated alternatively. The method with modified total-variation regularization solves the two subproblems, a Tikhonov regularization problem and an L2-total-variation regularization problem, via an efficient inversion workflow and the split-Bregman iterative method, respectively. The method with multiplicative regularization updates the images and auxiliary variables by the efficient inversion workflow and nonlinear conjugate gradient methods in a nested fashion. We validate the proposed methods using synthetic and field seismic data. Numerical results demonstrate that the proposed methods with regularization improve the resolution and fidelity of the migration profiles and exhibit superior anti-noise ability compared with the conventional method. Moreover, the modified-total-variation-based method has marginally higher accuracy than the multiplicative-regularization-based method for noisy data. The computational cost of the proposed two methods is approximately the same as that of the conventional least-squares reverse time migration method because no additional forward computation is required in the inversion of auxiliary variables.     

位场向下延拓的改进迭代维纳滤波法

根据维纳滤波理论导出的位场向下延拓滤波器为最佳下延滤波器,但因其实现需要已知待求位场和噪声的功率谱而在实际应用中受到限制.针对该问题,本文首先提出一种基于位场径向平均功率谱的位场噪声水平估计方法,进而利用偏差准则求取正则化参数,实现位场正则化向下延拓;然后将位场正则化下延结果的功率谱作为待求位场功率谱的估计初值,采用带修正项的迭代维纳滤波方法来更新对待求位场功率谱的估计,最后提出本文的位场向下延拓改进迭代维纳滤波方法.基于理论重力模型数据及航磁实测数据进行了向下延拓对比试验,结果表明,改进迭代法具有较好的收敛性,且下延精度优于Tikhonov正则化法和递增型维纳滤波法.     

水平层状介质中双侧向测井资料的迭代Tikhonov正则化反演

根据非线性反演理论与Morozov偏差原理研究建立从双侧向测井(DLL)资料中同时重构地层原状电阻率、侵入带电阻率、侵入半径、层界面位置以及井眼泥浆电阻率的迭代正则化算法.首先利用Tikhonov正则化反演理论将双侧向测井资料的反演问题转化为含有稳定泛函的非线性目标函数的极小化问题,并利用Gauss-Newton算法确定极小化解.为得到稳定的反演结果并有效实现测井资料的最佳拟合,在迭代过程中将Morozov偏差原理和Cholesky分解技术相结合,建立了一套后验选择正则化因子的方法.最后通过理论模型和大庆油田实际测井资料的处理结果,验证了该算法能够取得更为满意的反演效果.     

反演地表质量变化的附有空间约束的三维加速度点质量模型法

重力卫星可以在相同误差尺度下对全球质量变化进行连续重复观测,并在近十余年来取得了巨大成功,探索重力卫星数据精化处理方法和相关应用研究具有重要意义.本文基于三维加速度点质量模型法的基本原理,进一步发展建立了时变重力场模型球谐位系数的变化和地面点质量变化的关系,可有效考虑地表质量变化导致的负荷形变的影响;引入等权形式、线性形式、指数形式和高斯形式的空间约束方法处理南北条带噪声和向下延拓导致的病态问题,并与零阶Tikhonov正则化方法进行对比分析.采用模拟数据和一个月的实测GRACE时变重力场模型计算全球质量变化,对三维加速度点质量模型法和几种空间约束方法进行对比分析验证.计算结果表明,对于3°等面积的全球格网质量点,高斯和指数形式空间约束方法的最优相关距离约为500 km,等权和线性形式空间约束方法的最优相关距离约为600 km,各方法均可有效处理条带噪声的影响,四种空间约束方法的计算效果优于零阶Tikhonov正则化方法,本文的相关方法为进一步利用三维加速度点质量模型法监测全球质量变化提供了借鉴.

     

航空重力测量数据向下延拓方法比较

首先介绍了空中重力异常向下延拓的直接代表法、正则化法、点质量法和基于球内Dirichlet问题的调和解法,分析了它们各自的特点.然后利用这些方法对中国大同航空重力测量数据进行向下延拓计算.通过对延拓结果与地面实测数据的比较分析,比较客观地评价了不同延拓方法的计算精度、可靠性及适应性.实际计算结果表明,直接代表法和正则化法的延拓结果具有较高的精度和可靠性,而球内Dirichlet问题调和解法的精度较差.     

基于时空域混合法利用Kaula正则化精确和快速解算GOCE地球重力场

为了研究卫星重力梯度技术对中高频地球重力场反演精度的影响,本文基于时空域混合法,利用Kaula正则化反演了250阶GOCE地球重力场.模拟结果表明:第一,时空域混合法是精确和快速求解高阶地球重力场的有效方法;第二,Kaula正则化是降低正规阵病态性的重要方法;第三,基于改进的预处理共轭梯度迭代法可快速求解大型线性方程组...     

基于Extrapolation Tikhonov正则化算法的重力数据三维约束反演

通过研究重力数据三维反演解的病态性,利用基于拉格朗日插值方法的Extrapolation Tikhonov正则化方法来解决反演中解的不唯一性和不稳定性问题,该方法最大限度的减少了因正则化参数的引入而在反演结果中介入的误差,同时详细讨论了基于三种选择原则的正则化双参数的具体选择方法,模型试算结果表明,与原Tikhonov方法相比,该方法提高了反演的拟合精度.其次,为了消除核函数随深度增加而快速衰减对反演结果的影响,本文改进了前人的重力数据三维反演深度加权函数,改进后的加权函数与原函数相比能更好的识别异常体底部密度分布特征,对于埋深较深的异常体具有较好的识别效果,更好的解决了由近地面趋肤效应作用引起的密度分布不均的问题.同时,利用上下限约束函数限制每一个立方体的密度差范围,并应用于多组人工合成模型.结果表明:该反演方法能准确地获得正演模型的预设参数范围和位置.     

GOCE Data Processing: The Spherical Cap Regularization Approach

Due to the sun-synchronous orbit of the satellite gravity gradiometry mission GOCE, the measurements will not be globally available. As a consequence, using a set of base functions with global support such as spherical harmonics, the matrix of normal equations tends to be ill-conditioned, leading to weakly determined low-order spherical harmonic coefficients. The corresponding geopotential strongly oscillates at the poles. Considering the special configuration of the GOCE mission, in order to stabilize the normal equations matrix, the Spherical Cap Regularization Approach (SCRA) has been developed. In this approach the geopotential function at the poles is predescribed by an analytical continuous function, which is defined solely in the spatially restricted polar regions. This function could either be based on an existing gravity field model or, alternatively, a low-degree gravity field solution which is adjusted from GOCE observations. Consequently the inversion process is stabilized. The feasibility of the SCRA is evaluated based on a numerical closed-loop simulation, using a realistic GOCE mission scenario. Compared with standard methods such as Kaula and Tikhonov regularization, the SCRA shows a considerably improved performance.     

Impact of GOCE Level 1b data reprocessing on GOCE-only and combined gravity field models

The reprocessing of Gravity field and steady-state Ocean Circulation Explorer (GOCE) Level 1b gradiometer and star tracker data applying upgraded processing methods leads to improved gravity gradient and attitude products. The impact of these enhanced products on GOCE-only and combined GOCE+GRACE (Gravity Recovery and Climate Experiment) gravity field models is analyzed in detail, based on a two-months data period of Nov. and Dec. 2009, and applying a rigorous gravity field solution of full normal equations. Gravity field models that are based only on GOCE gradiometer data benefit most, especially in the low to medium degree range of the harmonic spectrum, but also for specific groups of harmonic coefficients around order 16 and its integer multiples, related to the satellite’s revolution frequency. However, due to the fact that also (near-)sectorial coefficients are significantly improved up to high degrees (which is caused mainly by an enhanced second derivative in Y direction of the gravitational potential — V_YY), also combined gravity field models, including either GOCE orbit information or GRACE data, show improvements of more than 10% compared to the use of original gravity gradient data. Finally, the resulting gradiometry-only, GOCE-only and GOCE+GRACE global gravity field models have been externally validated by independent GPS/levelling observations in selected regions. In conclusion, it can be expected that several applications will benefit from the better quality of data and resulting GOCE and combined gravity field models.     

基于SVD分解的二维空间数据融合方法

建立地球全空间数据库需要研究空间数据融合的方法和智能技术.本文以布格重力异常数据的融合为例,介绍一种基于奇异值分解的二维空间数据融合方法.此方法在原理上可以实现融合后的数据集误差模不大于融合前的子数据集的最小误差模,相较于传统的融合处理方法原理更加先进.在算法中应用了权重矩阵,可使融合之后数据区域与原始数据的边界上梯度变化较小;运用有限差分法计算重力水平梯度,检验数据融合边界的效果.理论模型和实际数据试验表明,此方法可以保证数据隐含的重要属性完整保存,具有计算过程简单、适用范围广、融合效果好等特点,可以应用于位场等地球物理数据融合,也可以用于遥感和其他地理信息的数据融合.     

使用小震震源机制解研究山东地区背景应力场

基于山东地区1970—2012年5月387次中小地震震源机制解数据,使用Gephart和Forsyth的 FMSI方法计算了该区的平均构造应力方向; 使用区域应力张量阻尼反演方法,结合共轭梯度法求解了应力张量的阻尼最小二乘最优化问题,并根据Lund和Townend推导给出的由构造应力张量计算最大水平主应力方向的公式,反演计算了山东地区构造应力场的空间分布特征,并讨论了该区构造应力场的变化及其与郯庐断裂带的关系．     

Sparse calibration of subsurface flow models using nonlinear orthogonal matching pursuit and an iterative stochastic ensemble method

We introduce a nonlinear orthogonal matching pursuit (NOMP) for sparse calibration of subsurface flow models. Sparse calibration is a challenging problem as the unknowns are both the non-zero components of the solution and their associated weights. NOMP is a greedy algorithm that discovers at each iteration the most correlated basis function with the residual from a large pool of basis functions. The discovered basis (aka support) is augmented across the nonlinear iterations. Once a set of basis functions are selected, the solution is obtained by applying Tikhonov regularization. The proposed algorithm relies on stochastically approximated gradient using an iterative stochastic ensemble method (ISEM). In the current study, the search space is parameterized using an overcomplete dictionary of basis functions built using the K-SVD algorithm. The proposed algorithm is the first ensemble based algorithm that tackels the sparse nonlinear parameter estimation problem.     

Backus — Gilbert’s method in gravity and magnetic inversion

In this paper, the application of Backus—Gilbert’s inversion method to the potential field anomalies for evaluating gravity and magnetic inversion solutions is discussed. Errors in data and singularity of kernels in the equations result in difficulties in solving equations. The application of regularization method similar to spectral expansion method makes calculation fast and easily. To make solution stable, constraints are used, which make the spread of solutions become narrow, standard deviation become small and iterative computations of inversion become fast. Finally, the author analyses specifically two profiles of Yunchen basin and calculated the Moho interface and the Curie isotherm of these two profiles.     

预条件共轭梯度法在地震数据重建方法中的应用

基于最小平方的Fourier地震数据重建方法最终转化为求解一个线性方程组, 其系数矩阵是Toeplitz矩阵,可以用共轭梯度法求解该线性方程组.共轭梯度法的迭代次数受系数矩阵病态程度的影响,地震数据的非规则采样程度越高,所形成的系数矩阵病态程度越高,就越难收敛和得到合理的计算结果.本文研究了基于Toeplitz矩阵的不同预条件的构造方法,以及对共轭梯度法收敛性的影响.通过预条件的使用,加快了共轭梯度法的迭代速度, 改进了共轭梯度算法的收敛性,提高了计算的效率.数值算例和实际地震数据重建试验证明了预条件共轭梯度法对计算效率有很大的提高.     

Parallel iterative solution for h and p approximations of the shallow water equations

A p finite element scheme and parallel iterative solver are introduced for a modified form of the shallow water equations. The governing equations are the three-dimensional shallow water equations. After a harmonic decomposition in time and rearrangement, the resulting equations are a complex Helmholz problem for surface elevation, and a complex momentum equation for the horizontal velocity. Both equations are nonlinear and the resulting system is solved using the Picard iteration combined with a preconditioned biconjugate gradient (PBCG) method for the linearized subproblems. A subdomain-based parallel preconditioner is developed which uses incomplete LU factorization with thresholding (ILUT) methods within subdomains, overlapping ILUT factorizations for subdomain boundaries and under-relaxed iteration for the resulting block system. The method builds on techniques successfully applied to linear elements by introducing ordering and condensation techniques to handle uniform p refinement. The combined methods show good performance for a range of p (element order), h (element size), and N (number of processors). Performance and scalability results are presented for a field scale problem where up to 512 processors are used.