This section summarises the results of an experimental field validation of the proposed algorithms at an on-production farm in Dos Hermanas (Spain). The farm features a drip irrigation system and the soil at the site is classified as silty loam, a texture that combines moderate levels of sand, silt, and clay, making it ideal for agricultural activities due to its good water retention and drainage properties. This specific soil type was considered to refine the initial estimations of soil parameters within the model.
3.2.6. Results of the Field Application Case
The outcomes of the data assimilation process are illustrated in a series of figures.
Figure 3 and
Figure 4 illustrate the data used in the assimilation process for both methods for
values of 0.01, 0.02 and 0.03, respectively, and
r = 100, as an example. In each figure, three time series are plotted: the sensor readings (green), the corrected readings (
) using the calibration coefficients from the data assimilation process (orange) and the VWC time series simulated by the Hydrus model using the hydraulic parameters obtained in the data assimilation process (blue).
In all cases, even with different values of r, notable peaks in the sensor data occur during irrigation events, reflecting the immediate and rapid increase in soil moisture from the drip irrigation system. While the Hydrus 1D simulation does not fully capture the sharp peaks following these events, it shows good alignment with the corrected sensor data during the intervals between irrigations, particularly after the water redistributes and moisture levels stabilise. This indicates that the model effectively simulates the longer-term moisture trends after irrigation events, even if it misses some of the immediate dynamics.
One of the possible reasons for the mismatch during irrigation events stems from the limitation of the Hydrus 1D model, which only simulates vertical water movement and does not account for lateral water spread, which is significant in drip irrigation systems. In drip irrigation, water disperses both vertically and laterally, creating a wetting bulb around the emitter. This localised water distribution leads to rapid increases in soil moisture near the emitter, which are quickly detected by the nearby sensors. These sensors are highly sensitive to localised changes in soil moisture, particularly those near the surface, and are able to capture the sharp peaks right after irrigation. However, because Hydrus 1D cannot simulate lateral water movement, it smooths out these localised increases, resulting in a more gradual moisture curve that misses the sharp spikes.
Despite these limitations, the Hydrus 1D model shows good performance in simulating the longer-term moisture behaviour, particularly after the effects of the irrigation peaks dissipate. The corrected sensor readings align well with the Hydrus 1D simulation once the water has redistributed deeper into the soil and the short-term effects of irrigation have subsided. This alignment shows that Hydrus 1D is effective in capturing the general water infiltration and redistribution trends.
As for the validation analysis,
Table 7 shows the validation results for the IES and PF methods across different values of
r (25, 50 and 100) and
(0.01, 0.02 and 0.03). For the IES method, the results show minimal variation in corrected RMSE as the
r value increases. For instance, at an
of 0.01, the RMSE slightly changes from 0.0483 (ensemble size 25) to 0.0533 (ensemble size 100). However, the corrected RMSE shows a slight decrease as the
value increases, with the best result being 0.0384 for an
r value of 50 and an observational error of 0.03. In contrast, the PF method exhibits a more pronounced improvement in performance, particularly as the
increases. At an
r of 100, the PF achieves a significantly lower RMSE of 0.0309 (
= 0.01) compared to the IES method for the same value, which further reduces to 0.0164 for an
of 0.03, this being the best result achieved.
Both the IES and the PF methods lead to a significant improvement in accuracy compared to the uncorrected sensor, which has an RMSE of 0.108. For the IES method, the lowest corrected RMSE achieved is 0.0384, representing a 64.4% improvement in accuracy. Similarly, the PF, which demonstrates the best performance with an RMSE of 0.0164, achieves an even more substantial improvement of 84.8%. These results highlight that, while both methods considerably enhance the sensor’s precision, the PF outperforms the IES method, particularly at higher observational errors and smaller values of r.
The performance of the IES and PF methods varies depending on the characteristics of the data. With synthetic data, the IES demonstrated superior performance compared to the PF. This advantage is likely due to the synthetic data’s linear bias and absence of random noise, which align well with the IES’s assumptions of linearity and Gaussian error. These conditions enable the IES to efficiently converge towards the true model and bias parameters.
However, the situation changes with real-world data, which often include non-Gaussian noise, non-linearity, and un-modelled discrepancies such as soil heterogeneity, temporal variability and spatial variability due to drip irrigation patterns. Under these conditions, the PF method outperforms the IES. Unlike the IES, the PF method does not rely on linear approximations or Gaussian assumptions. Instead, it employs weighted particles to represent the posterior distribution, allowing it to capture complex, non-linear relationships and adapt to non-Gaussian noise. This flexibility makes the PF method more robust against model discrepancies and uncertainties, resulting in better agreement with validation data.
The superior performance of the PF with real-world data can be attributed to its ability to handle the inherent complexities of field observations, such as non-linearity and variability in irrigation patterns. Conversely, the IES’s reliance on its assumptions limits its effectiveness in these scenarios. These differences highlight how the characteristics of the data influence the comparative performance of the two methods. While the IES excels in synthetic scenarios where its assumptions hold, the PF adaptability gives it an advantage in the challenging context of real-world soil moisture measurements.
Figure 5 and
Figure 6 show the time series of moisture from the sensor readings, the ThetaProbe sensor readings and the corrected readings for both methods, with different values of
r and
. As can be observed, the sensor readings show a significant bias compared to those of the ThetaProbe. In all cases, the corrected readings are closer to the ThetaProbe readings.
Table 8 and
Table 9 present the predicted soil parameter values using the IES and the PF methods, respectively, for an ensemble size of 100 and
values of 0.01, 0.02 and 0.03, with initial parameter values provided for comparison. Although the true values of the soil parameters are not known, the predicted results across both methods do not deviate significantly from the initial values, which are consistent with the expected ranges for a silty loam soil.
For the IES method (
Table 8), the predicted values show slight adjustments from the initial parameters. The parameter
remains close to the initial value (0.067), fluctuating between 0.0647 and 0.0678 across the different noise levels. The other soil parameters, such as
,
n and
, also exhibit relatively small changes, with
ranging from 0.0012 to 0.0063 and
n showing a small upward trend as the noise level increases.
The Particle Filter method (
Table 9) shows similar trends, with slight variations in the predicted values. For example, the parameter
varies from 0.0659 to 0.0739, while
and
n show minimal changes from the initial values. Interestingly,
and
are slightly more variable in the PF results compared to IES, but still remain within a reasonable range for silty loam soils.
Table 10 shows that there is no significant difference in execution time between the IES and PF methods, especially for smaller ensemble sizes. For an ensemble size of 25, both methods have similar runtimes (22 min for IES and 19 min for PF). Although the execution time increases with ensemble size, the difference remains modest (84 min for IES and 104 min for PF at size 100). The primary delay in both methods is attributed to the Hydrus 1D software employed for prediction calculations, rather than the convergence time of the algorithms themselves.
Although the data assimilation algorithms require significant execution times, especially for larger ensembles, real-time correction is unnecessary in this context. Soil moisture sensor drift occurs gradually over long periods of time [
30], meaning that, once corrected, sensors maintain improved accuracy for an extended period. Additionally, the methods provide updated calibration coefficients that can be used to adjust future sensor readings without reapplying the computational algorithms. Thus, these techniques function as an efficient offline calibration tool rather than a real-time correction mechanism.