# A Multi-Year Study on Rice Morphological Parameter Estimation with X-Band Pulsar Data
## 2 Methodology
### Probabilistic Particle Swarm Optimization  
The inversions of multi-dimensional EM models are ill-posed problems with the higher number of unknowns compared to the number of equations. For such problems, optimization algorithms are used to reach the optimum solution in the parameter space using different constraints. However, the optimization of multi-dimensional EM model inversions may result in multiple optimum solutions since different inputs can lead to similar outputs. The presence of multiple optimum solutions prevents the use of deterministic optimization algorithms that focus on a single solution. For the existence of multiple solutions, stochastic optimization algorithms can be considered as an alternative. In stochastic optimization, the procedure is initiated several times to obtain all local solutions in a given parameter space based on the defined set of rules.  
In this study, the Particle Swarm Optimization (PSO) algorithm [16] is utilized, which is based on updating the position of the particles, i.e., possible solutions, until they converge to an optimum solution within a parameter space. In each iteration, the locations of the particles are updated according to the position of the particle with the best position. The iterations continue until the particles converge to a solution that agrees with the defined constraints. For the estimation of rice morphology parameters from copolar X-band SAR backscattering intensities, the \(\mathrm{PCE}_{\mathrm{EM}}\) metamodel is inverted with the stochastic PSO algorithm.  
The fitness function for the PSO is given in (3) for HH and VV polarimetric channels and an arbitrary \(\mathrm{i}^{th}\) iteration. The fitness function is defined to minimize the difference between measured \(\sigma_{\mathrm{HH,VV}}^{o}\) and estimated \(\bar{\sigma}_{\mathrm{HH,VV}}^{o}\) values. The consistency of the solutions for different polarimetric channels is provided by considering the same input vector for both HH and VV polarimetric channels.  
\[\min\mathbb{C}_{i}=(\sigma_{\mathrm{HH}}^{0}-\bar{\sigma}_{\mathrm{HH}_{i}}^{0 })^{2}+(\sigma_{\mathrm{VV}}^{0}-\bar{\sigma}_{\mathrm{VV}_{i}}^{0})^{2}\ \ \ \ \ i=1\,...\,K \tag{3}\]  
Optimization problems need constraints to simplify the problem by reducing the dimensional complexity. In \(\mathrm{PCE}_{\mathrm{EM}}\) metamodel inversion, three plant morphology dependent constraints were established that are based on the natural limitations excerpted from the available rice morphology data.  
**Positivity constraint** ensures positive and real morphological estimations for all iterations.  
**Min-Max constraint** limits the morphological estimations based on the phenological phase boundaries. Phenological phases are defined by the International Rice Research Institute (IRRI) [17].  
The scale divides the growth cycle to five major phases. Figure 3 presents the boundaries obtained from ground data, for the chosen morphological parameters.  
**Natural limitations** provide non-linear relationships among rice morphology parameters during their development. These relations eliminates the solutions which might be impossible for a healthy plant, such as a 10-cm-tall stalk having two 50-cm-length leaves. This condition restricts the parameter space with a convex hull. Convex hull defines non-linear boundaries in a parameter space according to the experimental ground measurements that involve the agronomically possible solutions.  
The stochastic PSO optimization provides distributions for each morphology parameter. For the accuracy analysis, the mean value of the resulting stochastic distributions is assigned as the estimated dimensions of the rice morphological parameters. For a single stochastic PSO optimization, a total number of 200 iterations was found to be sufficient for the optimization convergence which changes less than 0.1% regarding the mean of the estimated values.