Abstract. Accurately quantifying errors in soil moisture measurements from in situ sensors at fixed locations is essential for reliable state and parameter estimation in probabilistic soil hydrological modeling. This quantification becomes particularly challenging when the number of sensors per field or measurement zone (MZ) is limited. When direct calculation of errors from sensor data in a certain MZ is not feasible, we propose to pool systematic and random errors of soil moisture measurements for a specific measurement setup to derive a pooled error covariance matrix that applies across different fields and soil types. In this study, a pooled error covariance matrix was derived from soil moisture sensor measurements and soil moisture sampling campaigns conducted over three growing seasons, covering 93 cropping cycles in agricultural fields with diverse soil textures in Belgium. The MZ soil moisture estimated from soil samples, which showed a small standard error (0.0038 m³ m^‑3) and which was not correlated between different sampling campaigns since soil samples were taken at different locations, represented the ‘true’ MZ soil moisture. First, we established a pooled linear recalibration of the TEROS 10 (Meter Group, Inc., USA) manufacturer's sensor calibration function. Then, for each individual sensor as well as for each MZ, we identified systematic deviations and temporally varying residual deviations between the calibrated sensor data and sampling data. The autocovariance of the individual or the MZ-averaged sensor measurement errors was represented by the variance of the systematic deviations across all sensors or MZs whereas the random error variance was calculated from the variance of the pooled residual deviations. The total error variance was equal to the sum of the autocovariance and random error variance. Due to spatial sensor correlation, the variance and autocovariance of MZ-average sensor measurement errors could not be derived from the individual sensor error variances and covariances. The pooled error covariance matrix of the MZ-averaged soil moisture measurements indicated a significant sensor error autocorrelation of 0.518, as the systematic error standard deviation (σ_α- = 0.0327 m³ m^‑3) was similar to the random error standard deviation (σ_ε- = 0.0316 m³ m^‑3). These results demonstrate that the common assumption of uncorrelated random errors to determine parameter and model prediction uncertainty is not valid when measurements from sparse in situ soil moisture sensors are used to parameterize soil hydraulic models. Further research is required to assess to what extent the error covariances found in this study can be transferred to other areas, and how they impact parameter estimation in soil hydrological modeling.

中文翻译：

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佛兰德斯农田稀疏原位土壤湿度传感器测量的合并误差方差和协方差估计


摘要。在概率土壤水文建模中，通过固定位置的原位传感器准确量化土壤水分测量中的误差对于可靠的状态和参数估计至关重要。当每个区域或测量区 （MZ） 的传感器数量有限时，这种量化变得特别具有挑战性。当无法直接计算某个 MZ 中传感器数据的误差时，我们建议将特定测量设置的土壤水分测量的系统误差和随机误差汇集在一起，以得出适用于不同田地和土壤类型的合并误差协方差矩阵。在这项研究中，从土壤湿度传感器测量和三个生长季节进行的土壤湿度采样活动中得出一个汇总误差协方差矩阵，涵盖了比利时具有不同土壤质地的农田的 93 个种植周期。从土壤样本中估计的 MZ 土壤水分，显示一个小的标准误差（0.0038 m^{3 m-3}），并且由于土壤样本在不同的位置采集，因此在不同采样活动之间不相关，代表了“真实”MZ 土壤水分。首先，我们建立了 TEROS 10 （Meter Group， Inc.， USA） 制造商传感器校准函数的池化线性重新校准。然后，对于每个单独的传感器以及每个 MZ，我们确定了校准传感器数据和采样数据之间的系统偏差和时间变化的残差。单个或 MZ 平均传感器测量误差的自协方差由所有传感器或 MZ 的系统偏差方差表示，而随机误差方差则由合并残差的方差计算得出。 总误差方差等于自协方差和随机误差方差之和。由于空间传感器相关性，MZ 平均传感器测量误差的方差和自协方差不能从单个传感器误差方差和协方差中得出。MZ 平均土壤水分测量的合并误差协方差矩阵表明显著的传感器误差自相关为 0.518，因为系统误差标准差 （σ_α- = 0.0327 m^{3 m-3}） 与随机误差标准差 （_{σ ε}- = 0.0316 m^{3 m-3}） 相似。这些结果表明，当使用稀疏原位土壤湿度传感器的测量值来参数化土壤水力模型时，用于确定参数和模型预测不确定性的不相关随机误差的常见假设是无效的。需要进一步的研究来评估本研究中发现的误差协方差在多大程度上可以转移到其他领域，以及它们如何影响土壤水文建模中的参数估计。