半参数模型中影响正则化参数的因素分析

通过模拟算例,比较了L曲线法、GCV法（广义交叉核实法）和虚拟观测法确定出的正则化参数,并对影响正则化参数的因素进行了分析,得出结论：正则化参数的确定与信噪比密切相关,当信噪比增大时,各种方法确定的正则化参数变化趋势不同,不同的情况确定正则化参数的适用方法也会有所差异。     

航空重力向下延拓的多参数正则化方法

为了克服航空重力向下延拓解算的病态性影响,介绍了一种多参数正则化方法,以均方误差最小为目标函数,设计了选取正则化参数的迭代算法,并比较了基于L曲线法、广义交叉核实(generalized cross-validation,GCV)方法选取正则化参数的Tikhonov正则化方法,同时给出了均方误差意义下多参数正则化解优于最小二乘估计的条件。基于EGM2008地球重力场模型进行了仿真试验,计算结果表明,多参数正则化方法能够保证向下延拓结果的可靠性和稳定性,并优于现有的Tikhonov正则化方法,验证了多参数方法在航空重力向下延拓中的可行性。     

联合GOCE卫星轨道和重力梯度数据严密求解重力场的模拟研究

论述了联合卫星轨道和重力梯度数据严密求解重力场的方法及数据处理方案,研究了GOCE重力场反演中有色噪声的AR去相关滤波、病态法方程的Kaula正则化和观测值最优加权的方差分量估计等关键问题。模拟结果表明:①极空白问题会降低法方程求解的稳定性,导致低次位系数的求解精度较低,而Kaula正则化可有效用于GOCE病态法方程的求解,并得到合理稳定的解;②重力梯度有色噪声会降低GOCE重力场求解的整体精度,特别是对低阶位系数的影响最为明显,而AR去相关滤波法可有效处理有色噪声,但解算结果仍含有低频误差;③方差分量估计可有效确定SST和SGG两类观测值的最优权比,并且有色噪声造成的低频误差经过联合求解后得到了抑制;④利用30d、5s采样的GOCE模拟数据恢复200阶次的重力场模型,其大地水准面和重力异常精度在纬度±83°范围内分别为±3.81cm和±1.056mGal。     

航空重力数据向下延拓的波数域迭代Tikhonov正则化方法

航空重力数据向下延拓是重力场数据联合处理的重要步骤。研究了Tikhonov正则化方法以及迭代Tikhonov正则化方法在航空重力数据向下延拓中的应用,指出Tikhonov正则化方法在利用GCV法或L曲线法选取正则化参数时存在的不足以及空间域迭代法迭代不收敛的问题。波数域Tikhonov迭代正则化法的引入,有效解决了上述问题。算例结果表明,波数域迭代法迭代过程收敛,且计算精度高、速度快,值得广泛应用于航空重力数据的向下延拓。     

利用GOCE卫星轨道数据恢复地球重力场模型方法的分析

欧空局早期公布的时域法和空域法解算的GOCE模型均采用能量守恒法处理轨道数据, 但恢复的长波重力场信号精度较低, 而且GOCE卫星在两极存在数据空白, 利用其观测数据恢复重力场模型是一个不适定问题, 导致解算的模型带谐项精度较低, 需进行正则化处理。本文分析了基于轨道数据恢复重力场模型的方法用于处理GOCE数据的精度, 对最优正则化方法和参数的选择进行研究。利用GOCE卫星2009-11-01—2010-01-31共92 d的精密轨道数据, 采用不依赖先验信息的能量守恒法、短弧积分法和平均加速度法恢复GOCE重力场模型, 利用Tikhonov正则化技术处理病态问题。结果表明, 平均加速度法恢复模型的精度最高, 能量守恒法的精度最低, 短弧积分法的精度稍差于平均加速度法。未来联合处理轨道和梯度数据时, 建议采用平均加速度法或短弧积分法处理轨道数据, 并且轨道数据可有效恢复120阶次左右的模型。Kaula正则化和SOT处理GOCE病态问题的效果最好, 并且两者对应的最优正则化参数基本一致, 但利用正则化技术不能完全抑制极空白问题的影响, 需要联合GRACE等其他数据才能获得理想的结果。     

利用GOCE卫星轨道数据恢复地球重力场模型的方法

欧空局早期公布的时域法和空域法解算的GOCE模型均采用能量守恒法处理轨道数据,但恢复的长波重力场信号精度较低,而且GOCE卫星在两极存在数据空白,利用其观测数据恢复重力场模型是一个不适定问题,导致解算的模型带谐项精度较低,需进行正则化处理。本文分析了基于轨道数据恢复重力场模型的方法用于处理GOCE数据的精度,对最优正则化方法和参数的选择进行了研究。利用GOCE卫星2009-11-01—2010-01-31共92d的精密轨道数据,采用不依赖先验信息的能量守恒法、短弧积分法和平均加速度法恢复GOCE重力场模型,利用Tikhonov正则化技术处理病态问题。结果表明,平均加速度法恢复模型的精度最高,能量守恒法的精度最低,短弧积分法的精度稍差于平均加速度法。未来联合处理轨道和梯度数据时,建议采用平均加速度法或短弧积分法处理轨道数据,并且轨道数据可有效恢复120阶次左右的模型。Kaula正则化和SOT处理GOCE病态问题的效果最好,并且两者对应的最优正则化参数基本一致,但利用正则化技术不能完全抑制极空白问题的影响,需要联合GRACE等其他数据才能获得理想的结果。     

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。     

基于L曲线法和GCV法的有理多项式参数求解

在求解有理函数模型的多项式参数时,通常采用的是最小二乘估计方法进行求解,但若控制点分布不均匀或模型过度参数化,则法方程矩阵很容易产生病态,不能得到正确的解。使用Tikhonov正则化方法可以较好地改善法方程的状态,使方程解趋于稳定。通过影像数据,分别采用了L曲线法和GCV法进行试验,试验证明该方法使RPC参数的解算精度有了显著提高,验证了该种方法在求解病态矩阵误差方程中的正确性。     

向下延拓航空重力数据的Tikhonov双参数正则化法

为了避免正则化参数对向下延拓过程可靠成分的修正影响,提出了Tikhonov双参数正则化法。引进截断参数,将法矩阵的奇异值分为相对较大的奇异值（可靠部分）和相对较小的奇异值（不可靠部分）;引进正则化参数,只对法矩阵的小奇异值进行修正,以抑制高频误差对向下延拓解的影响。采用改进的广义交互确认法（GCV）确定截断参数和正则化参数。基于EGM2008重力场模型仿真了一组航空重力数据,验证了该方法对航空重力数据向下延拓过程的有效性。     

利用最优正则化方法确定Tikhonov正则化参数

基于均方误差最小意义下运用最优正则化方法确定正则参数,推导了计算最优正则参数的公式,并结合算例分析比较了求解病态方程的L-曲线法、GCV法等常用的方法,算例表明,基于最小均方误差的Tikhonov正则化参数优化选取方法是一种可行有效的方法。     

Computation of spherical harmonic coefficients from gravity gradiometry data to be acquired by the GOCE satellite: regularization issues

The issue of optimal regularization is investigated in the context of the processing of satellite gravity gradiometry (SGG) data that will be acquired by the GOCE (Gravity Field and Steady-State Ocean Circulation Explorer) satellite. These data are considered as the input for determination of the Earths gravity field in the form of a series of spherical harmonics. Exploitation of a recently developed fast processing algorithm allowed a very realistic setup of the numerical experiments to be specified, in particular: a non-repeat orbit; 1-s sampling rate; half-year duration of data series; and maximum degree and order set to 300. The first goal of the study is to compare different regularization techniques (regularization matrices). The conclusion is that the first-order Tikhonov regularization matrix (the elements are practically proportional to the degree squared) and the Kaula regularization matrix (the elements are proportional to the fourth power of the degree) are somewhat superior to other regularization techniques. The second goal is to assess the generalized cross-validation method for the selection of the regularization parameter. The inference is that the regularization parameter found this way is very reasonable. The time expenditure required by the generalized cross-validation method remains modest even when a half-year set of SGG data is considered. The numerical study also allows conclusions to be drawn regarding the quality of the Earths gravity field model that can be obtained from the GOCE SGG data. In particular, it is shown that the cumulative geoid height error between degrees 31 and 200 will not exceed 1 cm. AcknowledgmentsThe authors thank Dr. E. Schrama for valuable discussions and for computing the orbit used to generate the long data set. They are also grateful to Prof. Tscherning and two anonymous reviewers for numerous valuable remarks and suggestions. The orbit to generate the short data set was kindly provided by J. van den IJssel. Computing resources were provided by Stichting Nationale Computerfaciliteiten (NCF), grant SG-027.     

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.     

Regularization of gravity field estimation from satellite gravity gradients

 The performance of the L-curve criterion and of the generalized cross-validation (GCV) method for the Tikhonov regularization of the ill-conditioned normal equations associated with the determination of the gravity field from satellite gravity gradiometry is investigated. Special attention is devoted to the computation of the corner point of the L-curve, to the numerically efficient computation of the trace term in the GCV target function, and to the choice of the norm of the residuals, which is important for the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) in the presence of colored observation noise. The trace term in the GCV target function is estimated using an unbiased minimum-variance stochastic estimator. The performance analysis is based on a simulation of gravity gradients along a 60-day repeat circular orbit and a gravity field recovery complete up to degree and order 300. Randomized GCV yields the optimal regularization parameter in all the simulations if the colored noise is properly taken into account. Moreover, it seems to be quite robust against the choice of the norm of the residuals. It performs much better than the L-curve criterion, which always yields over-smooth solutions. The numerical costs for randomized GCV are limited provided that a reasonable first guess of the regularization parameter can be found. Received: 17 May 2001 / Accepted: 17 January 2002     

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。     

重力与磁力测量数据向下延拓中最优正则化参数确定方法研究

向下延拓是重、磁测量数据处理的关键步骤之一,然而,向下延拓是一个典型的不适定问题,需要采用正则化方法实现有效延拓,因此,正则化参数的确定是重、磁测量数据向下延拓正则化方法研究中最重要内容。本文根据观测面和延拓面测量数据的Poisson积分平面近似关系,结合快速傅里叶变换算法,将其转换到频率域进行计算,提高了计算速度,为了克服计算的不稳定性并进一步提高计算结果的精度,引入Landweber正则化迭代法,在此基础上采用L曲线法研究了最优正则化参数的确定,最后采用模型磁测数据验证了所确定的正则化参数的有效性,并取得了较好的延拓结果。     

基于 EIGEN6C2模型的 Kaula 规则精化

高精度重力场模型精化Kaula规则其要点是将Kaula规则乘上一个与位系数阶数项相关的二阶有理函数,并基于EIGEN6C2重力场模型解算有理函数模型的系数。精化后的Kaula规则与EIGEN6C2模型和EGM2008模型的逼近误差都只是原来Kaula规则的0.26%。因此,精化后的Kaula规则更能正确表示各阶引力位的实际能量,对于重力场模型的解算提供更加合理的约束。     

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.     

短基线集形变模型反演的正则化解算方法

针对短基线集形变模型反演中法方程系数矩阵呈病态的问题,提出一种正则化稳健解算方法。该方法基于Tikhonov正则化理论,将形变速率求解问题转化为极小化问题,根据L-曲线法选取正则化参数,考虑最小二乘残差各个分量间的关系选取正则化矩阵,实现短基线集形变模型反演的稳健解算。分别采用LS法、岭估计法和Tikhonov正则化法对覆盖北京地区的29景ENVISAT ASAR数据进行处理,反演出研究区沉降速率图。通过对代表不同沉降情况的21个点的均方误差值和时间相干值、整个研究区的均方误差图等的对比分析,表明本文提出的短基线集形变模型反演的正则化稳健解算方法可获取更可靠的形变监测结果。