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A method for updating GNSS satellite ultra-rapid clock offsets and orbits with the aid of a covariance intersection algorithm 基于协方差交点算法的GNSS卫星超快时钟偏移和轨道更新方法
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-02-16 DOI: 10.1007/s40328-022-00374-8
Chao Hu, Zhongyuan Wang, Weicai Lv, Tong Cheng

Global navigation satellite system (GNSS) ultra-rapid clock offsets and orbit products are essential for near-real-time and real-time uses. To meet the requirements of accuracy and timeliness in high-accuracy applications, the issuing rates of ultra-rapid products are increased to six or three hours. However, there is an appreciable fluctuation of positioning residuals during the period of updating of products. To improve the performance of GNSS rapid services including the GPS, GLONASS, GALILEO and BeiDou, this paper proposes a method for updating satellite ultra-rapid clock offsets and orbits based on the covariance intersection algorithm, where kernel tricks are used to model the series in the position domain. Moreover, the parameter characteristics of the clock and orbit and the unknown inter-series correlation are considered in the model of satellite ultra-rapid products. Meanwhile, a sparse strategy is used in solving the model coefficients; i.e., the least absolute shrinkage and selection operator (LASSO) strategy. Several experimental schemes show that 1) jumps and gross errors in the GNSS ultra-rapid clock offset affect the modeling and services and should be detected and repaired before high-accuracy applications; 2) the updating of clock offsets and orbits reduces the steadiness of product services, while position residual fluctuations are introduced using GNSS precise point positioning solutions; 3) improved clock offset and orbit series can be obtained using the covariance intersection algorithm and kernel tricks; 4) the LASSO strategy can automatically and effectively choose and estimate the model coefficients of clock offset and orbit series; and 5) the proposed method can smooth the short-term arcs of GNSS products and positioning services by 29.8–99.5% for Multi-GNSS Experiment stations compared with original series. It is thus concluded that the proposed updating strategy is meaningful for improving GNSS satellite ultra-rapid products.

全球导航卫星系统(GNSS)的超快速时钟偏移和轨道产品对于近实时和实时应用至关重要。为满足高精度应用中对准确性和及时性的要求,超高速产品的发卡速度提高到六到三个小时。但是,在产品更新期间,定位残差有明显的波动。为了提高GPS、GLONASS、GALILEO和北斗等GNSS快速业务的性能,提出了一种基于协方差相交算法的卫星超快速时钟偏移和轨道更新方法,该方法利用核技巧在位置域对序列进行建模。此外,在卫星超快产品模型中考虑了时钟和轨道的参数特性以及未知的序列间相关性。同时,采用稀疏策略求解模型系数;即,最小的绝对收缩和选择算子(LASSO)策略。几种实验方案表明:GNSS超快时钟偏移中的跳变和粗误差影响建模和服务,在高精度应用前应进行检测和修复;2)时钟偏移和轨道的更新降低了产品服务的稳定性,而GNSS精密点定位方案引入了位置残留波动;3)利用协方差交点算法和核技巧可以得到改进的时钟偏移量和轨道序列;4) LASSO策略可以自动有效地选择和估计时钟偏移和轨道序列的模型系数;5)多GNSS实验站的GNSS产品和定位服务的短期弧线平滑度较原系列提高29.8 ~ 99.5%。因此,提出的更新策略对提高GNSS卫星超快产品质量具有重要意义。
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引用次数: 0
Effects of high temperature on the linear thermal expansion coefficient of Nanan granite 高温对南安花岗岩线性热膨胀系数的影响
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-02-07 DOI: 10.1007/s40328-022-00375-7
Zhennan Zhu, Shengqi Yang, Ren Wang, Hong Tian, Guosheng Jiang, Bin Dou

The evaluation of physico-mechanical characteristics of rocks after thermal treatment is a key issue in underground rock engineering projects such as exploitation of geothermal resources and geological disposal of nuclear waste. In this research, the lengths of cylindrical Nanan granite specimens were obtained before, during and after thermal treatment (up to 1000 °C) to investigate their linear thermal expansion coefficients, and the variation mechanisms were revealed by optical microscopic observations. According to the experimental results collected from the extensive corresponding literature, the relationships between the linear thermal expansion coefficients of various granites were also elaborated. The experimental results demonstrated that the linear thermal expansion coefficients of the granite in this study both under and after thermal treatment increase with temperature. Meanwhile, the linear thermal expansion coefficients increase rapidly above 500 °C, which is because of the quartz phase transition from α–phase to β–phase. The increase of linear thermal expansion coefficients of granite under and after thermal treatment closely relates to the thermal expansion of mineral crystals and the development and coalescence of intergranular and transgranular microcracks. The experimental results are expected to provide a reference to analytical calculations of thermophysical processes in granite.

岩石热处理后的物理力学特性评价是地热资源开发、核废料地质处置等地下岩石工程的关键问题。本研究对南安花岗岩柱状试样在热处理前、热处理中、热处理后(温度高达1000℃)的长度进行了研究,并通过光学显微镜观察揭示了其线性热膨胀系数的变化机理。根据大量相关文献收集的实验结果,阐述了各种花岗岩的线性热膨胀系数之间的关系。实验结果表明,本文研究的花岗岩在热处理前后的线性热膨胀系数都随着温度的升高而增大。同时,在500℃以上,线性热膨胀系数迅速增大,这是由于石英相由α -相转变为β -相。花岗岩热处理前后线性热膨胀系数的增大与矿物晶体的热膨胀以及晶间和穿晶微裂纹的发育和贯通密切相关。实验结果可为花岗岩热物理过程的解析计算提供参考。
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引用次数: 2
Magnetotelluric resistivity imaging of the Baribis fault zone’s Majalengka segment in West Java, Indonesia 印度尼西亚西爪哇巴里比斯断裂带Majalengka段大地电磁电阻率成像
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-25 DOI: 10.1007/s40328-022-00372-w
Djedi Setyo Widarto, Febty Febriani, Boko Nurdiyanto, Syuhada Syuhada, Titi Anggono, Cinantya Nirmala Dewi, Aditya Dwi Prasetio

This work discusses the application of the magnetotelluric (MT) method to observing and delineating a local fault in the Majalengka Regency, West Java, Indonesia. This fault is part of the well-known Baribis fault segment. Phase tensor and induction vector analysis were applied to all MT data to reveal the dimensionality, geoelectric strike, and geological conditions of the study area, with 12 MT sites composing the studied profile. The estimated skew angle (β) value is − 3º < β < 3º enabling the subsurface structure modeling using the 2-D inversion. The calculated geoelectric strike of the study area of N15oE was used to rotate the impedance tensor of all MT observation points before modeling. The induction vector analysis revealed that the vectors did not lie in a particular direction. It can be possibly related to the volcanic products which dominate the surroundings of the study area. The 2-D subsurface electrical resistivity model suggested the presence of a very conductive zone (C1 ≤ 10 Ωm), which may be related to the existence of the targeted fault. The subsurface model also showed the resistivity contrasts between C1 (≤ 10 Ωm) and R1 (ρ ≥ 500 Ωm) as well as C1 (≤ 10 Ωm) and R2 (ρ = 50–100 Ωm). These notable contrasts are represented by the models’ block boundaries, and it is suggested that these may become a future earthquake epicenter.

本文讨论了大地电磁(MT)方法在印度尼西亚西爪哇岛Majalengka Regency的局部断层观测和圈定中的应用。这条断层是著名的巴里比斯断层段的一部分。利用相位张量和感应矢量分析对所有大地电磁学数据进行分析,揭示了研究区大地电磁学的维数、地电走向和地质条件。估计的倾斜角度(β)值为- 3º< β < 3º,可以使用二维反演进行地下结构建模。利用计算得到的N15oE研究区的地电走向,旋转各MT观测点的阻抗张量进行建模。感应矢量分析表明,矢量不位于一个特定的方向。这可能与研究区周围主要的火山产物有关。二维地下电阻率模型显示存在非常导电带(C1≤10 Ωm),可能与目标断层的存在有关。地下模型也显示了C1(≤10 Ωm)和R1 (ρ≥500 Ωm)以及C1(≤10 Ωm)和R2 (ρ = 50-100 Ωm)之间的电阻率差异。这些显著的对比由模型的块边界表示,这些块边界可能成为未来的地震震中。
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引用次数: 1
Ionospheric irregularities during disturbed geomagnetic conditions over Argentinian EIA region 阿根廷EIA区域扰动地磁条件下的电离层不规则性
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-20 DOI: 10.1007/s40328-021-00370-4
Gilda de Lourdes González

Ionospheric irregularities can severely degrade radio communication and navigation systems. Geomagnetic storms may affect the generation of these irregularities in a way that is not yet fully understood. To improve the forecasting of this phenomenon, we need to study the ionosphere in different regions of the world, and in particular in the equatorial ionization anomaly (EIA) where irregularities are usually more intense. This study analyses the effect of geomagnetic storms on ionospheric irregularities. We examined the occurrence of irregularities at the southern crest of the EIA in Argentina (Tucumán, 26.9°S, 294.6°E, dip latitude 15.5°S) during three intense and one moderate geomagnetic storm of different solar sources, between 2015 and 2018. We used data from an ionosonde, a Global Positioning System receiver and magnetometers. Ionogram spread-F, the F-layer bottom side (h'F), the critical frequency of the F2-layer (foF2), the rate of TEC index and the S4 scintillation index were analysed. The data show irregularities were present as range spread-F and moderate TEC fluctuations in one storm: 27 May 2017 (a coronal mass ejection CME-driven storm occurred on local winter), and were absent in the other events. We suggest that eastward disturbance dynamo electric field and over-shielding prompt penetration electric fields may create favourable conditions for developing these irregularities, whereas westward storm time electric fields might inhibit the growth of irregularities during the other storms considered. During co-rotating interaction region CIR-driven storms, the westward disturbance dynamo electric field may be associated with the non-occurrence of irregularities.

电离层不规则会严重降低无线电通信和导航系统的性能。地磁风暴可能以一种尚未完全了解的方式影响这些不规则现象的产生。为了提高对这一现象的预测,我们需要对世界不同地区的电离层进行研究,特别是对赤道电离异常(EIA)进行研究,因为赤道电离异常通常是较为强烈的。本文分析了地磁风暴对电离层不规则性的影响。我们研究了2015 - 2018年不同太阳源的三次强地磁风暴和一次中地磁风暴在阿根廷EIA南波峰(Tucumán, 26.9°S, 294.6°E,倾角15.5°S)发生的不规则性。我们使用了来自电离层探空仪、全球定位系统接收器和磁力计的数据。分析了离子图扩展f、f层底边(h'F)、f - 2层临界频率(foF2)、TEC指数速率和S4闪烁指数。数据显示,在2017年5月27日的一次风暴(日冕物质抛射cme驱动的风暴发生在当地冬季)中,不规则性表现为范围扩展f和中度TEC波动,而在其他事件中则不存在。东向扰动发电机电场和过屏蔽提示穿透电场可能为这些不规则性的发展创造有利条件,而西向风暴时电场可能抑制其他风暴期间不规则性的发展。在同向旋转相互作用区中,向西扰动的发电机电场可能与不规则性的不发生有关。
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引用次数: 2
Magnetotelluric responses of a vertical inhomogeneous and anisotropic resistivity structure with a transitional layer 具有过渡层的垂直非均匀各向异性电阻率结构的大地电磁响应
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-18 DOI: 10.1007/s40328-022-00373-9
Linjiang Qin, Changfu Yang, Weifeng Ding

The theoretical magnetotelluric responses of a vertical inhomogeneous and anisotropic resistivity structure with a transitional layer in which the resistivity is a linear function of depth are investigated. The expressions of the tangential electric and magnetic fields at the surface of the Earth model and the corresponding impedance have been evaluated. The influence of some model parameters such as the anisotropic dipping angles, the anisotropic coefficients and the resistivity contrast as well as the thickness of the transitional layer on the apparent resistivity and impedance phase are treated in detail. The results are graphically illustrated in the form of apparent resistivity and impedance phase curves, and they may be used in the interpretation of magnetotelluric sounding data in some specified geologic situations.

研究了具有电阻率随深度线性变化的过渡层的非均质各向异性垂直电阻率结构的理论大地电磁响应。计算了地球模型表面切向电场和磁场的表达式以及相应的阻抗。详细讨论了各向异性倾角、各向异性系数和电阻率对比等模型参数以及过渡层厚度对视电阻率和阻抗相位的影响。结果以视电阻率和阻抗相位曲线的形式进行了图解说明,可用于特定地质条件下大地电磁测深资料的解释。
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引用次数: 1
Gauss process regression for real-time ionospheric delay estimation from GNSS observations 基于GNSS观测的实时电离层延迟估计高斯过程回归
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-12 DOI: 10.1007/s40328-021-00368-y
Balazs Lupsic, Bence Takacs

The number of devices equipped with global satellite positioning has exceeded seven billion recently. There are a wide variety of receivers regarding their accuracy and reliability. Low cost, multi-frequency units have been released on the market latterly; however, the number of single-frequency receivers is still significant. Since their measurements are influenced by ionospheric delay, accurate ionosphere models are of utmost importance to reduce the effect. This paper summarizes how Gauss process regression (GPR) can be applied to derive near real-time regional ionosphere models using raw Global Navigation Satellite System (GNSS) observations of permanent stations. While Gauss process is widely used in machine learning, GPR is a nonparametric, Bayesian approach to regression. GPR has several benefits for ionosphere monitoring since it is quite robust and efficient to derive a grid model from data available in irregular set of ionospheric pierce points. The corresponding instrumental delays are estimated by a parallel Kalman filter. The presented algorithm can be applied near real-time, however the results are offline calculated and are compared to two high quality TEC map products. Based on the analysis, the accuracy of the GPR modell is in 2 TECu range. The developed methods could be efficiently applied in the field of autonomous vehicle navigation with meeting both accuracy and integrity requirements.

最近,全球卫星定位设备的数量超过了70亿部。关于接收机的准确性和可靠性,有各种各样的接收机。低成本、多频单元最近已投放市场;然而,单频接收器的数量仍然很大。由于它们的测量受到电离层延迟的影响,因此精确的电离层模型对于减少这种影响至关重要。本文总结了高斯过程回归(GPR)如何利用全球导航卫星系统(GNSS)永久站原始观测数据推导出近实时区域电离层模型。虽然高斯过程广泛应用于机器学习,但GPR是一种非参数贝叶斯回归方法。探地雷达对电离层监测有几个好处,因为它是相当鲁棒和有效地从一组不规则的电离层穿透点的数据推导网格模型。用并行卡尔曼滤波器估计相应的仪器时延。该算法可以接近实时地应用,但结果是离线计算的,并与两个高质量的TEC地图产品进行了比较。经分析,GPR模型的精度在2tecu范围内。该方法在满足精度和完整性要求的前提下,可以有效地应用于车辆自动导航领域。
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引用次数: 0
Comparisons of autoregressive integrated moving average (ARIMA) and long short term memory (LSTM) network models for ionospheric anomalies detection: a study on Haiti (Mw = 7.0) earthquake 自回归综合移动平均(ARIMA)和长短期记忆(LSTM)网络模式在电离层异常探测中的比较——以海地7.0级地震为例
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-11 DOI: 10.1007/s40328-021-00371-3
Mohd Saqib, Erman Şentürk, Sanjeev Anand Sahu, Muhammad Arqim Adil

Since ionospheric variability changes dramatically before the major earthquakes (EQ), the detection of ionospheric anomalies for EQ forecasting has been a hot topic for modern-day researchers for the last couple of decades. Therefore, there is a need to identify highly accurate, advance, and intelligent models to identify these anomalies. In the present study, we have discussed artificial intelligence techniques e.g. autoregressive integrated moving average (ARIMA), and long short-term memory (LSTM) network, to detect ionospheric anomalies using the total electron content (TEC) time series over the epicenter of Mw 7.0 Haiti EQ on January 12, 2010. We have considered 20 days of TEC data with a daily 2-h interval and trained the models with an accuracy of 1.28 and 0.07 TECU for ARIMA and LSTM, respectively. Both ARIMA and LSTM results showed that the negative anomalies are recorded 5 days before the EQ (January 7), while strong positive anomalies are recorded 1–2 days before the EQ (January 11–12) that are consistent with the findings of previous studies. Moreover, the quiet space weather conditions during the analyzed period indicate that the observed variations could be considered precursors to the impending Haiti EQ. Our analysis suggests that the performance of the LSTM model is more robust as compared to the ARIMA model in terms of detection of seismoionospheric anomalies.

由于电离层变率在大地震之前发生了巨大的变化,因此电离层异常的探测用于大地震预报已经成为近几十年来现代研究人员的一个热门话题。因此,需要识别高度准确、先进和智能的模型来识别这些异常。在本研究中,我们讨论了人工智能技术,如自回归综合移动平均(ARIMA)和长短期记忆(LSTM)网络,利用2010年1月12日7.0级海地EQ震中的总电子含量(TEC)时间序列检测电离层异常。我们考虑了20天的TEC数据,每天间隔2小时,并对ARIMA和LSTM分别以1.28和0.07 TECU的精度训练模型。ARIMA和LSTM结果均显示,负异常出现在EQ前5天(1月7日),而强正异常出现在EQ前1-2天(1月11-12日),与前人研究结果一致。此外,在分析期间的安静空间天气条件表明,观测到的变化可以被认为是即将到来的海地EQ的前兆。我们的分析表明,在检测地震电离层异常方面,LSTM模式的性能比ARIMA模式更稳健。
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引用次数: 9
Assessment of point-mass solutions for recovering water mass variations from satellite gravimetry 从卫星重力测量中恢复水质量变化的点质量解的评估
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2022-01-11 DOI: 10.1007/s40328-021-00369-x
Yanan Yang, Vagner Ferreira, Kurt Seitz, Thomas Grombein, Bin Yong, Bernhard Heck

Previous studies have shown the feasibility of point-mass modellings for deriving terrestrial water storage (TWS) from the harmonic solutions of the Gravity Recovery And Climate Experiment (GRACE) mission at regional scales (e.g., Greenland and Antarctica). However, a thorough assessment of point-mass modelling approaches at the global and river basin levels is still necessary. Therefore, this study’s objective is to assess the implementation and performance of the point-mass modelling approaches based on simulations using as inputs the TWS from Global Land Data Assimilation System (GLDAS). First, the approximate solutions of Newton’s integral using the Taylor series expansion, such that the zeroth-order approximation is equivalent to the “original point-mass” (OPM) and the third-order approximation to the “improved point-mass” (IPM) modellings are presented. Second, numerical comparisons of the gravitational potential forwarded by the IPM and OPM are carried out at which both approaches show errors smaller than the GRACE uncertainties for the potential differences ((sim 7.6times 10^{-4}) (hbox {m}^2)/(hbox {s}^2)). Nevertheless, the spatial patterns of the OPM’s errors still assemble the TWS’s spatial variations. Finally, simulations showed that considering OPM’s deviations from IPM improves the root-mean-square-difference (RMSD) of the inverted TWS up to 50% at the global and basin scales if the edge effects are neglected. After accounting for the edge effects, the IPM solution presented an RMSD of 6.44 mm with an enhancement of about only 20% regarding the OPM. Although the present study confirms the suitability of point-mass approaches for recovering TWS, further investigations regarding its advantages compared to GRACE spherical harmonic synthesis are still necessary.

以往的研究表明,在区域尺度(如格陵兰岛和南极洲),用点质量模型从重力恢复和气候实验(GRACE)任务的调和解中推导陆地储水量(TWS)是可行的。然而,在全球和流域水平上对点质量模拟方法进行彻底评估仍然是必要的。因此,本研究的目的是评估基于模拟的点质量建模方法的实施和性能,该方法使用全球土地数据同化系统(GLDAS)的TWS作为输入。首先,利用泰勒级数展开式给出了牛顿积分的近似解,使得零阶近似等价于“原始点质量”模型,三阶近似等价于“改进点质量”模型。其次,对IPM和OPM转发的重力势进行了数值比较,两种方法对势能差的误差都小于GRACE的不确定性((sim 7.6times 10^{-4})(hbox {m}^2) / (hbox {s}^2))。尽管如此,OPM误差的空间模式仍然集合了TWS的空间变化。最后,仿真结果表明,考虑OPM与IPM的偏差后,反演TWS的均方根差(RMSD)可提高到50% at the global and basin scales if the edge effects are neglected. After accounting for the edge effects, the IPM solution presented an RMSD of 6.44 mm with an enhancement of about only 20% regarding the OPM. Although the present study confirms the suitability of point-mass approaches for recovering TWS, further investigations regarding its advantages compared to GRACE spherical harmonic synthesis are still necessary.
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引用次数: 2
A new ridge estimation method on rank-deficient adjustment model 秩缺平差模型的脊估计新方法
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2021-11-29 DOI: 10.1007/s40328-021-00366-0
Yingchun Song, Wenna Li, Caihua Deng, Xianqiang Cui

In this paper, we present a new ridge estimation method for solving rank-deficient least squares problems, in which a rank-deficient matrix is regarded as an almost rank-deficient. First, we give an algebraic derivation that the optimal solution can in fact be obtained by solving a related regularized problem on the optimal worst-case residual. Second, we give a new iterative algorithm to solve ridge parameter and prove its convergence. Finally, examples are given to demonstrate the efficiency of our new method. It is shown that the proposed algorithm can not only assess the stability of solution but also use additional prior information to guarantee the uniqueness of solutions to the problem of rank-deficient free-network adjustment.

本文提出了一种求解秩缺乏型最小二乘问题的脊估计方法,将秩缺乏型矩阵看作几乎秩缺乏型矩阵。首先,我们给出了一个代数推导,证明了最优解实际上可以通过求解最优最差残差的相关正则化问题得到。其次,给出了一种新的求解脊参数的迭代算法,并证明了其收敛性。最后,通过算例验证了该方法的有效性。结果表明,该算法不仅可以评估解的稳定性,而且可以利用附加的先验信息来保证缺秩自由网络平差问题解的唯一性。
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引用次数: 0
Tikhonov-regularized weighted total least squares formulation with applications to geodetic problems tikhonov正则化加权总最小二乘公式及其在大地测量问题中的应用
IF 1.4 4区 地球科学 Q2 Earth and Planetary Sciences Pub Date : 2021-10-05 DOI: 10.1007/s40328-021-00365-1
M. M. Kariminejad, M. A. Sharifi, A. R. Amiri-Simkooei

This contribution presents the Tikhonov regularized weighted total least squares (TRWTLS) solution in an errors-in-variables (EIV) model. The previous attempts had solved this problem based on the hybrid approximation solution (HAPS) within a nonlinear Gauss-Helmert model. The present formulation is a generalized form of the classical nonlinear Gauss-Helmert model, having formulated in an EIV general mixed observation model. It is a follow-up to the previous work throughout the WTLS problems formulated rely on a standard least squares (SLS) theory. Two cases, namely the EIV parametric model and the classical nonlinear mixed model, could be considered special cases of the general mixed observation model. These formulations are conceptually simple; because they are formulated based on the SLS theory, and subsequently, the existing SLS knowledge can directly be applied to the ill-posed mixed EIV model. Two geodetic applications have then adopted to illustrate the developed theory. As a first case, 2D affine transformation parameters (six-parameter affine transformation) for ill-scattered data points are adeptly solved by the TRWTLS method. Second, the circle fitting problem as a nonlinear case is not only tackled for well-scattered data points but also tackled for ill-scattered data points in a nonlinear mixed model. Finally, all results indicate that the Tikhonov regularization provides a stable and reliable solution in an ill-posed WTLS problem, and hence an efficient method applicable to many engineering problems.

本文提出了误差变量(EIV)模型中的Tikhonov正则化加权总最小二乘(TRWTLS)解。以前的尝试是基于非线性高斯-赫尔默特模型中的混合近似解(HAPS)来解决这个问题。本公式是经典非线性高斯-赫尔默特模型的推广形式,在EIV一般混合观测模型中进行了表述。这是之前的工作的后续,整个WTLS问题的制定依赖于标准最小二乘(SLS)理论。EIV参数模型和经典非线性混合模型两种情况可视为一般混合观测模型的特例。这些公式在概念上很简单;因为它们是基于SLS理论制定的,因此,现有的SLS知识可以直接应用于病态混合EIV模型。然后采用两个大地测量应用来说明发展的理论。作为第一种情况,利用TRWTLS方法熟练地求解了病态散射数据点的二维仿射变换参数(六参数仿射变换)。其次,作为非线性情况的圆拟合问题不仅解决了非线性混合模型中良散数据点的拟合问题,而且解决了良散数据点的拟合问题。最后,所有结果都表明,Tikhonov正则化为病态WTLS问题提供了稳定可靠的解,因此是一种适用于许多工程问题的有效方法。
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引用次数: 1
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