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。     

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。     

Smooth inversion for ground surface temperature histories: estimating the optimum regularization parameter by generalized cross-validation

The inversion of recent borehole temperatures has proved to be a successful tool to determine ancient ground surface temperature histories. To take into account heterogeneity of thermal properties and their non-linear dependence on temperature itself, a versatile 1-D inversion technique based on a finite-difference approach has been developed. Regularization of the generally ill-posed problem is obtained by an appropriate version of Tikhonov regularization of variable order. In this approach, a regularization parameter has to be determined, representing a trade-off between data fit and model smoothness. We propose to select this parameter by generalized cross-validation. The resulting technique is employed in case studies from the Kola ultradeep drilling site, and another borehole from northeastern Poland. Comparing the results from both sites corroborates the hypothesis that subglacial ground surface temperatures as met in Kola often are much higher than the ones in areas exposed to atmospheric conditions (Poland).     

一种改进的不适定问题奇异值分解法

综合截断奇异值法和奇异值修正法的特点,提出了一种改进的不适定问题奇异值分解法,先用截断参数将奇异值分为相对较大的奇异值和相对较小的奇异值;再用修正参数对相对较小的奇异值进行修正,以抑制其对观测噪声的放大作用。通过位场延拓中航空重力测量数据向下延拓的仿真实验,验证了改进的奇异值分解法要优于截断奇异值法、奇异值修正法和Tikhonov正则化法。     

混凝土结构缺陷检测的探地雷达资料波场反演方法

本文针对工程建筑中的混凝土无损探测问题,提出了一种新的反演成像方法.它避免了传统方法观察偏移探测数据的情况.这种方法以探地雷达为探测手段,以电磁场原理为理论依据,其结果更具定量化和可视性.具体算法是将阻尼高斯牛顿法和GCV方法(Generalized Cross Validation)结合起来,其中阻尼牛顿法可以避免传统牛顿法计算Hessian矩阵的困难,GCV方法可以自适应地选择正则化参数,既减小了计算量又克服了噪声及不适定性的影响.数值算例验证了算法的有效性.     

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。     

Tikhonov正则化方法在GOCE重力场求解中的模拟研究

本文在阐述Tikhonov正则化方法基本原理的基础上,给出了四类可用于重力场解算的正则化矩阵(零次、一次、二次和Kaula),以及用于确定正则化参数的L曲线法和GCV方法的数学模型。基于SA方法利用模拟数据分析讨论了零次、一次以及Kaula正则化矩阵应用于GOCE全球重力场模型确定的有效性,并由Kaula正则化矩阵分析了L曲线法和GCV方法确定正则化参数的可行性。数值结果表明三类正则化矩阵获得的最优解(以大地水准面MSE最小为准则确定)的精度水平相近,关键在于相应正则化参数的确定,数值结果同时说明了GCV方法和L曲线法可用于确定正则化参数,且前者较后者具有更好的稳定性。     

GPS速度场球面小波模型中的吉洪诺夫正则化方法

针对用小波框架表示GPS速度场可能会产生病态问题,该文提出了采用吉洪诺夫正则化方法及相应的3种正则化参数选择方法(广义交叉检验法、L曲线法和留一交叉验证法)进行模型求解。该方法通过引入合适的正则化参数及正则化矩阵的方式,来克服GPS速度场球面小波模型难于得到惟一解的问题。基于中国地壳运动观测网络两个局部区域的GPS速度场数据和亚洲太平洋地区地球动力学计划局部区域的GPS速度场数据的实验结果表明:使用该方法可以得到模型的稳定解,且以外部检核均方误差最小为准则时,3种正则化参数选择方法获得的解的精度水平相当。     

Bilinear surface smoothing for spatial interpolation with optional incorporation of an explanatory variable. Part 2: Application to synthesized and rainfall data

ABSTRACT

The non-parametric mathematical framework of bilinear surface smoothing (BSS) methodology provides flexible means for spatial (two dimensional) interpolation of variables. As presented in a companion paper, interpolation is accomplished by means of fitting consecutive bilinear surface into a regression model with known break points and adjustable smoothing terms defined by means of angles formed by those bilinear surface. Additionally, the second version of the methodology (BSSE) incorporates, in an objective manner, the influence of an explanatory variable available at a considerably denser dataset. In the present study, both versions are explored and illustrated using both synthesized and real world (hydrological) data, and practical aspects of their application are discussed. Also, comparison and validation against the results of commonly used spatial interpolation methods (inverse distance weighted, spline, ordinary kriging and ordinary cokriging) are performed in the context of the real world application. In every case, the method’s efficiency to perform interpolation between data points that are interrelated in a complicated manner was confirmed. Especially during the validation procedure presented in the real world case study, BSSE yielded very good results, outperforming those of the other interpolation methods. Given the simplicity of the approach, the proposed mathematical framework’s overall performance is quite satisfactory, indicating its applicability for diverse tasks of scientific and engineering hydrology and beyond.

Editor Z. W. Kundzewicz; Associate editor A. Carsteanu     

基于B spline和正则化算法的低轨卫星轨道平滑

本文提出了一个利用纯几何轨道和力模型的新算法来计算精确且相对平滑的卫星轨道. 该法将一个纯几何轨道表达为一个B spline的线性组合，线性组合的系数可以由最小二乘法估计获得. 力模型通过计算加速度来附加约束. 为了平衡几何轨道的点位误差和加速度的不精确，一个基于“广义交互确认（GCV，generalized cross validation）”的正则化算法运用其中. 由于B spline的本地控制性，该方法的计算效率相当高. 本文的数值分析表明了该法的有效性. 模拟计算的结论是：带加速度约束较不带加速度约束的平滑效果好. 力模型越精确，平滑的轨道就越精确. 三个月的CHAMP实测轨道数据处理结果表明，平滑后的轨道改进了重力场模型.