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 and derive a pooled error covariance matrix that applies to this setup across different fields and soil types. In this study, a pooled error covariance matrix was derived using soil moisture sensor measurements from three TEROS 10 (Meter Group, Inc., USA) sensors per MZ 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 a composite of nine soil samples with a small standard error (0.0038 m³ m⁻³) was considered the “true” MZ soil moisture. Based on these measurement data, we established a pooled linear recalibration of the TEROS 10 manufacturer's sensor calibration function. Then, for each individual sensor as well as for each MZ, we identified systematic offsets and temporally varying residual deviations between the calibrated sensor data and sampling data. Sensor deviations from the “true” MZ soil moisture were defined as observational errors and lump both measurement errors and representational errors. Since a systematic offset persists over time, it contributes to the temporal covariance of sensor observational errors. Therefore, we estimated the temporal covariance of observational errors of the individual and the MZ-averaged sensor measurements from the variance of the systematic offsets across all sensors and MZ averages, while the random error variance was derived from the variance of the pooled residual deviations. The total error variance was then obtained as the sum of these two components. Due to spatial soil moisture correlation, the variance and temporal covariance of MZ-averaged sensor observational errors could not be derived accurately from the individual sensor error variances and temporal covariances, assuming that the individual observational errors of the three sensors in a MZ were not correlated with each other. The pooled error covariance matrix of the MZ-averaged soil moisture measurements indicated a significant autocorrelation of sensor observational errors of 0.518, as the systematic error standard deviation (${\mathit{\sigma }}_{\stackrel{\mathrm{‾}}{\mathit{\alpha }}}=$ 0.033 m³ m⁻³) was similar to the random error standard deviation (${\mathit{\sigma }}_{\stackrel{\mathrm{‾}}{\mathit{ϵ}}}=$ 0.032 m³ m⁻³). To illustrate the impact of error covariance in probabilistic soil hydrological modeling, a case study was presented incorporating the pooled error covariance matrix in a Bayesian inverse modeling framework. 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 hydrological 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.