## 3 Methodology
### Experimental Design  
#### 3.3.1 Synthetic Experiments  
The objective of the synthetic experiments is to investigate the sensitivity of the algorithm to some important parameters of the approach including (i) errors on the SSM measurements, (ii) window length or observation frequency and, (iii) irrigation timing constraints. For this purpose, a set of synthetic observations of SSM were generated using the FAO-56 model by perturbing the model's parameters. Two experiments were carried out, one for flood and a second for drip irrigation. For the flood experiment, 6 irrigation events of 60 mm were applied throughout the season with an interval of 15 days between two successive events, which corresponds to the typical frequency of the classical flooding technique. For the drip experiments, irrigation data recorded on D1 during the 2016-2017 growing season were used.  
_Errors on SSM_: The final objective of our study is to rely on remote sensing SSM products. Products derived from microwave remote sensing are overall retrieved with an error [17; 39; 81; 82]. This error is assumed to decrease the approach performance and thus results in errors in the state estimation. To assess the effect of error on the proposed approach, five levels of error are investigated in this study: 0 (no error), 0.02, 0.04, 0.06 and 0.1 m\({}^{3}\)/m\({}^{3}\). This frames the error value of most satellite products as well as the error threshold that defines good quality of SSM products (0.04 m\({}^{3}\)/m\({}^{3}\); [83]). Errors are first randomly generated by a Gaussian PDF with a mean equal to zero and a standard deviation equal to the level of error, and then added to the synthetic observation data set.  
_Window length_: It is the frequency of observation availability. In this study, the window length corresponds to the revisit time of the satellite, which implies that only one observation is available per window. The effect of this parameter is important given that the weights are kept constant along the window, which means that a large window (large difference between two successive observations) is less accurate in tracking the state variable than a small window. In the context of this study: irrigation estimation, a long window will miss the effect of irrigation because the soil dries out after a long period of time. Therefore, if the following SSM observation is low, the algorithm assumes that no irrigation has taken place. In this study, we considered the investigation of three windows, being 3-, 6- and 12-day long.  
_Irrigation timing constraints_: The aim is to evaluate the effect of adding some constraints on the timing of irrigation application in the case of flood irrigation. It consists of adding a condition to the approach when it detects an irrigation event. A 5-day constraint for example implies that when an event is detected, no irrigation will take place within the next 5 days. For this study, two constraints of 5 days and 10 days were tested, in addition to no constraint, i.e., the approach can retrieve irrigation events every day.  
For each set of algorithmic parameters (SSM error, window length and timing constraint), the data assimilation is run 10 times to take into account the effect of random errors and particle distribution.