大地测量地震断层同震滑动分布反演的两步解法

针对同震滑动分布反演中系数矩阵出现病态的问题,提出两步解法,并在两步解法反演过程中引入拉普拉斯二阶平滑矩阵进行平滑约束。该方法不仅改善了系数矩阵的病态问题,同时也很好地抑制了相邻断层面间出现大的梯度变化。在两步解法反演过程中,用L曲线法确定正则化参数。系统模拟实验表明,对于最大滑动量,该方法的反演结果较一步最小二乘法的反演结果精度提高了3.34%~19%;对于均方根误差,该方法的反演结果较一步最小二乘法减小了3.3%~13.3%。芦山地震反演结果表明,利用两步解法进行滑动分布反演是可行的。     

确定大地水准面的Tikhonov最小二乘配置法

LSC法（最小二乘配置法）因能融合不同种类重力观测数据确定大地水准面的特性而受到广泛关注,但由于协方差矩阵存在病态性,微小的观测误差将被协方差矩阵的小奇异值放大,导致计算的配置结果不稳定且精度偏低。本文提出Tikhonov_LSC法,即在LSC法中引入Tikhonov正则化算法,基于GCV法选择协方差矩阵的正则化参数,利用正则化参数修正协方差矩阵的小奇异值,以抑制其对观测误差的放大影响。基于Tikhonov_LSC法计算大地水准面,能有效提高其稳定性和精度。通过以EGM2008重力场模型分别计算山区、丘陵和海域重力异常作为基础数据确定相应区域大地水准面的实验,验证了该方法的有效性。     

采用联合平差法处理附有病态等式约束的反演问题

探讨了附有病态等式约束的反演问题,尝试用降秩处理方法解这类病态约束问题,通过算例验证了此种方法与截断奇异值方法是等价的。然后提出了一种联合平差方法,它不仅能解病态的约束问题,而且能解决主模型秩亏或病态同时约束模型病态的问题,增强了应用性。最后设计了多种方案进行计算和比较,验证了联合平差法的有效性和可行性。     

病态总体最小二乘问题的广义正则化

总体最小二乘(TLS)算法可以视为一个降正则化的过程,对比最小二乘算法,病态总体最小二乘方法的解受系数阵数据误差和观测值误差的影响将更为严重。本文探讨用广义正则化的方法降低病态性对总体最小二乘数值求解的影响,以提高求解结果的稳定性。通过多组算例结果表明,本文采用的广义正则化方法在处理病态总体最小二乘问题上具有明显的优势。     

顾及基线先验信息的GPS模糊度快速解算

采用GPS相位观测值进行快速定位时,其解算模型严重病态,最小二乘解得的浮点模糊度精度差且相关性大,导致整周模糊度搜索空间过大,难以正确固定。本文提出一种顾及基线先验信息和模糊度线性约束的整数条件的GPS模糊度快速解算方法,先用顾及基线先验信息的正则化算法解得精度较高且相关性较小的浮点模糊度,以减小整周模糊度的搜索空间;再综合利用整周模糊度间的线性约束的整数条件和基线先验信息,进一步有效地减小模糊度搜索空间,提高搜索效率。算例表明:顾及基线先验信息的正则化算法有效地改善了模糊度浮点解,模糊度线性约束的整数条件有效地提高搜索效率和成功率。     

Fast GNSS ambiguity resolution as an ill-posed problem

A linear observational equation system for real-time GNSS carrier phase ambiguity resolution (AR) is often severely ill-posed in the case of poor satellite geometry. An ill-posed system may result in unreliable or unsuccessful AR if no care is taken to mitigate this situation. In this paper, the GNSS AR model as an ill-posed problem is solved by regularizing its baseline and ambiguity parameters, respectively, with the threefold contributions: (i) The regularization parameter is reliably determined in context of minimizing mean square error of regularized solution where the covariance matrix of initial values of unknowns is used as an approximate smoothness term instead of the quadratic matrix of the true values of unknowns; (ii) The different models for computing initial values of unknowns are systematically discussed in order to address the potential schemes in real world applications; (iii) The superior performance of the regularized AR are demonstrated through the numerically random simulations as well as the real GPS experiments. The results show that the proposed regularization strategies can effectively mitigate the model’s ill-condition and improve the success AR probability of the observational system with a severely ill-posed problem.     

均方误差意义下的正则化参数二次优化方法

Tikhonov正则化法是大地测量中应用最为广泛的病态问题解算方法之一。影响正则化法解算效果的重要因素是正则化参数,然而,最优正则化参数的确定一直是正则化解算的难题,如L曲线法确定的正则化参数具有稳定性好、可靠性高的优点,但存在过度平滑问题,导致正则化法对模型参数估值精度改善较小。本文从均方误差角度分析了正则化参数对模型参数估计质量的影响。基于奇异值分解技术,提出了由模型参数投影值分块计算均方误差的方法,避免了均方误差迭代计算,并基于均方误差最小准则给出了正则化参数优化方法,实现了对L曲线正则化参数的优化。数值模拟试验与PolInSAR植被高反演试验结果表明,正则化参数优化方法有效改善了正则化法解算效果,提高了模型参数估计精度。     

GNSS垂直位移反演区域地表质量变化的模拟分析

连续密集的全球导航卫星系统(global navigation satellite system,GNSS)地表形变监测为反演精细的区域地表质量变化提供了有效技术手段.针对格林函数方法反演区域地表质量变化的病态问题,给出了一种改进的正则化拉普拉斯约束矩阵,讨论了广义交叉检验(generalized cross-vali...     

一种解算病态问题的方法--两步解法

提出了一种解算病态问题的方法———两步解法。在两步计算中,均采用L曲线法来确定正则化参数α。通过算例,比较了该方法和LS估计、岭估计及截断奇异值方法的效果。结果表明,该方法要明显优于LS估计、岭估计及截断奇异值法。     

Adaptive MAP sub-pixel mapping model based on regularization curve for multiple shifted hyperspectral imagery

Sub-pixel mapping is a promising technique for producing a spatial distribution map of different categories at the sub-pixel scale by using the fractional abundance image as the input. The traditional sub-pixel mapping algorithms based on single images often have uncertainty due to insufficient constraint of the sub-pixel land-cover patterns within the low-resolution pixels. To improve the sub-pixel mapping accuracy, sub-pixel mapping algorithms based on auxiliary datasets, e.g., multiple shifted images, have been designed, and the maximum a posteriori (MAP) model has been successfully applied to solve the ill-posed sub-pixel mapping problem. However, the regularization parameter is difficult to set properly. In this paper, to avoid a manually defined regularization parameter, and to utilize the complementary information, a novel adaptive MAP sub-pixel mapping model based on regularization curve, namely AMMSSM, is proposed for hyperspectral remote sensing imagery. In AMMSSM, a regularization curve which includes an L-curve or U-curve method is utilized to adaptively select the regularization parameter. In addition, to take the influence of the sub-pixel spatial information into account, three class determination strategies based on a spatial attraction model, a class determination strategy, and a winner-takes-all method are utilized to obtain the final sub-pixel mapping result. The proposed method was applied to three synthetic images and one real hyperspectral image. The experimental results confirm that the AMMSSM algorithm is an effective option for sub-pixel mapping, compared with the traditional sub-pixel mapping method based on a single image and the latest sub-pixel mapping methods based on multiple shifted images.     

NLS问题的不适定性及正则化解算方法

本文进一步完善定义了NLS问题的两种不适定性,对产生这两种不适定问题的现象进行了分析。借助于正则化理论,通过添加稳定泛函,结合高斯-牛顿法,构造了不适定NLS问题的正则化高斯-牛顿法求解公式;解决了普通高斯-牛顿法在迭代过程中其Jacobian矩阵是秩亏或者严重病态导致的不能收敛的问题;给出了非线性秩亏自由网平差的正则化高斯-牛顿法步骤;以几个经典NLS问题为例进行了数值实验,说明了本文所提方法的适用性。     

A semilabeled-sample-driven bagging technique for ill-posed classification problems

In this letter, a semilabeled-sample-driven bootstrap aggregating (bagging) technique based on a co-inference (inductive and transductive) framework is proposed for addressing ill-posed classification problems. The novelties of the proposed technique lie in: 1) the definition of a general classification strategy for ill-posed problems by the joint use of training and semilabeled samples (i.e., original unlabeled samples labeled by the classification process); and 2) the design of an effective bagging method (driven by semilabeled samples) for a proper exploitation of different classifiers based on bootstrapped hybrid training sets. Although the proposed technique is general and can be applied to any classification algorithm, in this letter multilayer perceptron neural networks (MLPs) are used to develop the basic classifier of the proposed architecture. In this context, a novel cost function for the training of MLPs is defined, which properly considers the contribution of semilabeled samples in the learning of each member of the ensemble. The experimental results, which are obtained on different ill-posed classification problems, confirm the effectiveness of the proposed technique.     

Reducing errors in the GRACE gravity solutions using regularization

The nature of the gravity field inverse problem amplifies the noise in the GRACE data, which creeps into the mid and high degree and order harmonic coefficients of the Earth’s monthly gravity fields provided by GRACE. Due to the use of imperfect background models and data noise, these errors are manifested as north-south striping in the monthly global maps of equivalent water heights. In order to reduce these errors, this study investigates the use of the L-curve method with Tikhonov regularization. L-curve is a popular aid for determining a suitable value of the regularization parameter when solving linear discrete ill-posed problems using Tikhonov regularization. However, the computational effort required to determine the L-curve is prohibitively high for a large-scale problem like GRACE. This study implements a parameter-choice method, using Lanczos bidiagonalization which is a computationally inexpensive approximation to L-curve. Lanczos bidiagonalization is implemented with orthogonal transformation in a parallel computing environment and projects a large estimation problem on a problem of the size of about 2 orders of magnitude smaller for computing the regularization parameter. Errors in the GRACE solution time series have certain characteristics that vary depending on the ground track coverage of the solutions. These errors increase with increasing degree and order. In addition, certain resonant and near-resonant harmonic coefficients have higher errors as compared with the other coefficients. Using the knowledge of these characteristics, this study designs a regularization matrix that provides a constraint on the geopotential coefficients as a function of its degree and order. This regularization matrix is then used to compute the appropriate regularization parameter for each monthly solution. A 7-year time-series of the candidate regularized solutions (Mar 2003–Feb 2010) show markedly reduced error stripes compared with the unconstrained GRACE release 4 solutions (RL04) from the Center for Space Research (CSR). Post-fit residual analysis shows that the regularized solutions fit the data to within the noise level of GRACE. A time series of filtered hydrological model is used to confirm that signal attenuation for basins in the Total Runoff Integrating Pathways (TRIP) database over 320 km radii is less than 1 cm equivalent water height RMS, which is within the noise level of GRACE.     

选权拟合法解的特性探讨

选权拟合法是解决大地测量中的不适定问题的一种方法,是对吉洪诺夫正则化方法的改造。在推导一般正则化解的偏差计算公式并回顾了选权拟合法的基本原理和公式的基础上,推导了选权拟合法解的一个重要性质:只要被约束的部分参数估值无偏,其余的也无偏;该性质说明利用选权拟合进行参数估计的结果是有条件无偏的。这个不同于一般正则化解的重要特性可以用于设计加快GPS短基线快速定位双差模糊度解算策略。恰当利用选权拟合法,用实测数据算例分析了GPS基线分量的先验信息的偏差大小对模糊度解算的影响。     

非线性参数估计的自适应松弛正则化算法

非线性方程参数估计存在的弊端在于非线性观测方程存在不适定问题时,以线性化平差估计和高斯牛顿为代表的经典数值算法会产生较强的不稳定特征。因此,针对传统非线性最小二乘求解不稳定且可靠性低的特点,基于稳定泛函极小准则最优化思想,提出了一种自适应松弛正则化数值算法。该算法采用正则化参数几何递增计算方法和残差最小步长准则,实现了正则参数和迭代步长计算的完全自适应,提高了非线性迭代收敛效率。以病态仿真数据和水下实测数据为例,验证了该方法的数值收敛解优于线性平差估计解,收敛效率优于迭代Tikhonov正则化方法。     

误差限的病态总体最小二乘解算

大地测量和地球物理数据解算中时常会涉及病态问题的处理。基于客观的观测精度,利用设计矩阵与观测向量的误差限制,一方面降低了病态性对求解造成的波动;另一方面避免引入正常数,从而提高整个解算过程的客观性与可靠性。计算表明,本文提出的方法可以有效地处理病态总体最小二乘问题,并且具有较高的稳定性。     

单频GPS静态定向中整周模糊度的快速解算

针对GPS快速静态定向中法矩阵严重病态的特点,采用了载波和伪距联合解算及Tikhonov正则化方法,改善了法矩阵的病态性,实际算例表明该方法是有效的、可行的.     

融合海域多源重力数据的正则化点质量方法

针对传统点质量方法在融合处理多源重力数据过程中可能出现的病态性问题,特别引入Tikhonov正则化方法,对点质量法计算模型进行正则化改造,建立了相应的正则化点质量解算模型。使用EGM2008位模型模拟产生航空重力和海面船测重力数据进行了融合处理仿真试验。实际验证结果表明,正则化处理方法能够有效抑制病态系数矩阵小奇异值放大噪声对点质量解的污染,提高解算结果的精度和稳定性。     

地震同震滑动分布反演的总体最小二乘方法

同震滑动分布参数与地表形变间的线性关系依赖于格林函数矩阵的构造,格林函数矩阵元素与破裂面位置、几何参数、破裂方式及位错模型假设等因素有关。本文尝试考虑格林函数矩阵元素的误差来补偿上述原因在一定程度上对反演参数的影响,采用同时顾及系数矩阵(格林函数矩阵)和观测向量两者误差的总体最小二乘方法反演同震滑动分布。首先确定了系数矩阵元素和观测向量的协因数矩阵,考虑到格林函数矩阵的病态性(秩亏),借助拉普拉斯二阶平滑得到正则化矩阵,采用总体最小二乘正则化法反演同震滑动分布。并对2009年意大利中部拉奎拉(L’Aquila)Mw6.3级地震实例进行同震滑动分布反演研究。结果表明,拉奎拉地震的走向为144.37°,倾角为59.06°,滑动分布的最大滑动量为0.95m,平均滑动角为-96.4°,主要滑动深度为4~15km的范围,地震矩为3.63×10~(18)N·m,对应的矩震级为Mw6.34。总体最小二乘与最小二乘法的滑动分布解存在一定差别,但差别的量级在10-4以内。     

Simultaneous estimation of cross-validation errors in least squares collocation applied for statistical testing and evaluation of the noise variance components

The cross-validation technique is a popular method to assess and improve the quality of prediction by least squares collocation (LSC). We present a formula for direct estimation of the vector of cross-validation errors (CVEs) in LSC which is much faster than element-wise CVE computation. We show that a quadratic form of CVEs follows Chi-squared distribution. Furthermore, a posteriori noise variance factor is derived by the quadratic form of CVEs. In order to detect blunders in the observations, estimated standardized CVE is proposed as the test statistic which can be applied when noise variances are known or unknown. We use LSC together with the methods proposed in this research for interpolation of crustal subsidence in the northern coast of the Gulf of Mexico. The results show that after detection and removing outliers, the root mean square (RMS) of CVEs and estimated noise standard deviation are reduced about 51 and 59%, respectively. In addition, RMS of LSC prediction error at data points and RMS of estimated noise of observations are decreased by 39 and 67%, respectively. However, RMS of LSC prediction error on a regular grid of interpolation points covering the area is only reduced about 4% which is a consequence of sparse distribution of data points for this case study. The influence of gross errors on LSC prediction results is also investigated by lower cutoff CVEs. It is indicated that after elimination of outliers, RMS of this type of errors is also reduced by 19.5% for a 5 km radius of vicinity. We propose a method using standardized CVEs for classification of dataset into three groups with presumed different noise variances. The noise variance components for each of the groups are estimated using restricted maximum-likelihood method via Fisher scoring technique. Finally, LSC assessment measures were computed for the estimated heterogeneous noise variance model and compared with those of the homogeneous model. The advantage of the proposed method is the reduction in estimated noise levels for those groups with the fewer number of noisy data points.