Pooled Error Variance and Covariance Estimation of Sparse In Situ Soil Moisture Sensor Measurements in Agricultural Fields in Flanders

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.