Journal of Geo-information Science >
Investigating the Impact of Violations in Orthogonality and Zero Cross-Correlation Assumption upon the Accuracy of Triple Collocation Methods
Received date: 2023-08-28
Revised date: 2023-11-10
Online published: 2024-03-31
Supported by
National Natural Science Foundation of China(41877158)
S&T Program of Hebei(19275408D)
Jiangsu Water Conservancy Science and Technology Project(2020040)
Triple Collocation (TC) is a technique for assessing the uncertainties of three samples individually without knowledge of the true values. This method is based on the assumptions of linearity, orthogonality, and zero cross-correlation. In practical use, these three assumptions are often difficult to achieve, particularly the orthogonality and zero cross-correlation assumptions, which often encounter significant violations. Moreover, we are uncertain about the impact of these assumption violations on the errors of the method's results. In this study, we simulated multiple sets of synthetic samples with varying degrees of two assumption violations to investigate the impact of assumption violations on the accuracy of the TC method. The results of synthetic samples experiment indicate that, in general, when there is an increase in the violation of orthogonality or zero cross-correlation assumptions, the error of the method's results increases linearly or quadratically. However, under certain specific conditions of assumption violation, there is a sudden and spike-like increase in the error of TC method results. This phenomenon is referred to as "outliers". To understand the origin of the outliers, we derived the complete mathematical relationship between the violation of assumptions and the errors of the results. This relationship exhibits a fractional structure rather than a linear one, contributing to the emergence of outliers. From the perspective of the difference notation, this fractional structure results from rescaling coefficients. Continuing to analyze this mathematical relationship, we can draw two conclusions. Firstly, merely ensuring the approximate independence of the three samples does not necessarily lead to improved method results. When the structural relationships among the three samples meet certain conditions, outliers emerge. Additionally, previous attempts at method improvement have aimed at overall reducing the sensitivity of this method to assumptions, neglecting the presence of outliers. Considering these factors, the key to suppressing outliers lies in better designing these rescaling coefficients. The paper presents two possible improvement methods:(1) Ignoring the additive bias, so that the rescaling coefficients are not affected by the orthogonality or zero cross-correlation assumptions. (2) Limiting the upper and lower bounds of the rescaling coefficients. We achieved favorable results in suppressing outliers by constraining the absolute values of the rescaling coefficients between 0.25 and 4. Both improvement methods can suppress the occurrence of outliers. However, when the additive bias is significant, the first improvement method generates substantial extreme errors due to its inherent structure, which is insufficient to eliminate outliers. The second method performs effectively even in complex scenarios. Lastly, we conducted a simple estimation of the probability of outliers occurring in practical usage, which was approximately 3.2%. In addition, we used SMOS, SMAP, and AMSR2 soil moisture data to validate the phenomenon of outliers and compared the two improved methods. According to real data, some outliers appear as negative values and are removed because the calculated results cannot be negative. Therefore, A portion of the outlier does not cause a significant deviation in the calculation result; instead, they simply prevent the calculation of meaningful results. Therefore, when employing the TC method with fewer repetitions for calculations (e.g., with fewer than 500 repetitions), the influence of outliers can be disregarded.
TAN Songlin , WANG Jie , JI Jingjing , LIU Meili , ZHAN Zhongyu , LIU Miao , WANG Lirong , HU Xiaodong . Investigating the Impact of Violations in Orthogonality and Zero Cross-Correlation Assumption upon the Accuracy of Triple Collocation Methods[J]. Journal of Geo-information Science, 2024 , 26(3) : 591 -603 . DOI: 10.12082/dqxxkx.2024.230502
表1 两组不相关性假设违背程度及下文简称Tab. 1 Quantification of deviation from orthogonality and zero cross-correlation assumption, and subsequent abbreviation |
所属假设 | 量化违背程度的指标 | 下文简称 |
---|---|---|
随机误差与真值不相关性假设 | ||
随机误差互不相关性假设 | ||
表2 虚拟样本实验方案Tab. 2 Synthetic samples experimental design |
实验组 | 实验名称 | 实验变量 | 值 | 值 | 其他参数 | ||
---|---|---|---|---|---|---|---|
A组 | 实验1 | 变化 | 每组实验中, 中的非实验变量设置为0.001,即假设满足不相关性假设。 为保证所设计的统计参数具有代表性,样本数量设计为200 000个 | ||||
实验2 | 变化 | ||||||
实验3 | 变化 | ||||||
B组 | 实验4 | 变化 | |||||
实验5 | 变化 | ||||||
实验6 | 变化 | ||||||
C组 | 实验7 | 变化 | |||||
实验8 | 变化 | ||||||
实验9 | 变化 | ||||||
X组 | 实验10 | 变化 | |||||
实验11 | 变化 | ||||||
实验12 | 变化 | ||||||
Y组 | 实验13 | 变化 | |||||
实验14 | 变化 | ||||||
实验15 | 变化 | ||||||
Z组 | 实验16 | 变化 | |||||
实验17 | 变化 | ||||||
实验18 | 变化 |
注:经大量实验验证,改变 值不会改变实验结果,故 值变化的情况在此不做讨论。 |
表3 影响异常点出现的各项参数的假定分布Tab. 3 Assumed distributions of various parameters affecting the occurrence of outliers |
参数 | 假定的分布或值 | ||
---|---|---|---|
情景1 | 情景2 | 情景3 | |
30 | |||
50 | |||
10 |
表4 各情景下误差概率分布Tab. 4 Probability distribution of errors for different scenarios (%) |
概率 | 情景1 | 情景2 | 情景3 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
TC | 改进1 | 改进2 | TC | 改进1 | 改进2 | TC | 改进1 | 改进2 | |||
29.56 | 28.70 | 29.13 | 29.20 | 30.30 | 29.00 | 29.24 | 36.49 | 28.78 | |||
6.76 | 5.00 | 5.53 | 6.50 | 7.60 | 5.40 | 6.88 | 10.38 | 5.41 | |||
3.27 | 1.35 | 1.77 | 3.20 | 3.90 | 1.80 | 3.06 | 5.21 | 1.65 | |||
1.25 | 0.18 | 0.12 | 1.70 | 1.70 | 0.50 | 1.19 | 2.21 | 0.20 | |||
0.76 | 0.05 | 0.01 | 1.20 | 0.70 | 0.10 | 0.70 | 1.43 | 0.01 | |||
0.56 | 0.03 | 0.01 | 1.00 | 0.70 | 0.00 | 0.49 | 1.22 | 0.00 |
感谢南京信息工程大学高性能计算中心为本研究提供的计算资源。
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