## 3 Methodology
### Particle Filter Approach and Implementation  
A particle filter approach is implemented for the assimilation of SSM into the FAO-56 model in order to retrieve irrigation water amount and timing. The most popular data assimilation technique used in geophysical problems is the Kalman filter and its variants, an optimal solution when the problem is linear and the posterior density is Gaussian. Some variants of Kalman filters are developed to solve some nonlinear problems as well, such as extended Kalman filters, but with a Gaussian posterior density function. However, when these two conditions (linearity and Gaussianity) are not satisfied, the performance of the variants can be limited [56]. In this case, the particle filter technique was shown to be better suited than [PERSON]'s similar approaches [57; 58]. Particle filters are a sequential Monte Carlo data assimilation technique implementing a recursive Bayesian filter [59]. They are powerful and flexible techniques that can describe a wide range of problems [60], in particular they are not limited to Gaussian distribution, unimodal solution and linear problems [60; 61; 62]. The principle of particle filters can be summarized in two steps [63]: prediction and weights update. In the first step, a set of particles of the analyzed variable with equal weights is generated. The particles are propagated in the state space using the model (state equation) to generate predictions of the state variable until a new measurement (or observations) is available. The predictions are compared then to measurement and the weights are actualized based on the adequacy of predictions with the current observations thanks to the likelihood function [64]. In this study, the algorithm is run on assimilation windows of a duration equal to the data revisit time (6 days or 12 days for S1 SSM products depending on the site). A sensitivity analysis to the revisit time is conducted using in situ measurements by varying the windows length from 3 days (revisit time of SMOS product) to 12 days (revisit time of S1 when S1A only was available before 2016). An ensemble of N particles is stochastically generated at the beginning of each assimilation windows. Each particle is associated to an irrigation scenario generated stochastically. The amounts and the day of irrigation (timing) are generated stochastically using a uniform distribution as no information are available on both variables. For flood irrigation, the minimum time between two irrigations is defined to a value of n days (a sensitivity analysis to this constraint is carried out in the Section 4). Indeed, flooding is constraint by the channel network bringing water to the field and by available workforce leading to an irrigation event every 15 days at best. The amount ranges between 20 and 80 mm of water which are typical values applied for flooding irrigation. For drip irrigation, the irrigation is applied with values ranging from 0 to 40 mm and n (the minimum time between two irrigations) is defined to two days. The set of particles evolves using the FAO-56 from the beginning to the end of the window where an observation is available following:  
\[x_{i}^{k}=f\left(x_{i-1}^{k},I_{i}^{k},u\right)+\epsilon_{i}^{k} \tag{3}\]  
where \(x_{i}^{k}\) is the analyzed variable, (the water depletion only in our study; see Equation (2)), \(f\) is the FAO-56 model equation (Equation (2)), \(I_{i}^{k}\) is the corresponding irrigation scenario, \(u\) contains the model parameters (FAO-56 parameters, see Table A2 in the Appendix A) and \(\epsilon_{i}^{k}\) is the modeling error. After predicting a set of SSM states (N values) corresponding to the ensemble of irrigation scenarios, the particle that most closely matches the observation is sought.