# A Multi-Year Study on Rice Morphological Parameter Estimation with X-Band Pulsar Data
## 2 Methodology  
### Theoretical EM Model and Its Metamodel  
In this study, the EM model [11], \(\mathcal{M}(\bar{\xi})\), is employed to simulate the backscattering intensities, \(\bar{\sigma}^{o}\), from a set of rice plant morphology parameters, \(\bar{\xi}\), through Monte Carlo simulations for varying scatterer locations. The \(\bar{\xi}\) set includes stalk dimensions (height and diameter), leaf dimensions (length and width), panic dimensions (length and width), and their structural densities. The simulations are done for a unit area \(A\), having randomly placed non-overlapping cylindrical stalks with a specific height and diameter. Each stalk is modeled to have elliptical leaves with a fixed length and width. In this study, we assumed flooded ground and plant components with fixed complex dielectric constants for the complete growth cycle by relying on the sensitivity analysis of the EM model [12]. The \(\mathcal{M}(\bar{\xi})\), given in (1), provides the relation between an incident \(\bar{E}^{i}\) and a scattered wave \(\bar{E}^{s}\) through the coherent sum of four different scattering mechanisms (S\({}_{n}\)), as shown in Figure 2.  
\[\bar{\sigma}^{o}_{qq}=\mathcal{M}(\bar{\xi})=\frac{4\pi r^{2}}{A}\frac{\left\langle |E^{s}_{i}|^{2}\right\rangle}{|E^{i}_{i}|^{2}}=\left\langle\left|\frac{e^{ikr} }{r}\left(\sum_{n=1}^{4}S_{n}\right)\right|^{2}\right\rangle \tag{1}\]  
Figure 1: Block-diagram of the proposed stochastic inversion of metamodel-driven electromagnetic scattering (EM) model.  
Figure 2: The four different scattering mechanisms considered within the EM model [11]. \(S_{1}\): Direct scattering from the scatterers, \(S_{2}\): Scattering from the canopy followed by reflection from the ground, \(S_{3}\): Reflection from the ground followed by scattering from the canopy, \(S_{4}\): Reflection from the ground followed by scattering from the canopy and followed by reflection from the ground.  
In the EM model given in (1), \(q\) and \(p\) subscripts correspond to transmitted and received horizontal (H) and vertical (V) linear polarization channels. The parameters \(k\) and \(r\) represent the free-space wavenumber and the distance between the sensor and the target, respectively. The \(\bar{\sigma}_{qq}^{0}\) for the different polarimetric channels, \(qq\), are approximated from the ratio between \(E_{q}^{s}\) and \(E_{q}^{i}\).  
The \(\mathcal{M}(\xi)\) has high computation cost due to its multi-dimensional algorithm and Monte Carlo simulations. The computation costs of the algorithm were reduced using sparse Polynomial Chaos Expansion (PCE) metamodels. The PCE metamodels are spectral variance decompositions of the original model with low training cost and wide coverage in the parameter domain [13]. For the chosen \(\mathcal{M}(\xi)\), its PCE metamodel, \(\mathrm{PCE}_{\mathrm{EM}}(\xi)\), is developed from the (2), given below.  
\[\mathcal{M}(\xi)\cong\mathrm{PCE}_{\mathrm{EM}}(\xi)=\sum_{j=0}^{\infty}a_{j }\Psi_{j}(\xi)\cong\sum_{j=0}^{N}a_{j}\Psi_{j}(\xi). \tag{2}\]  
In (2), \(a_{j}\in\mathbb{R}\) is a set of scalar coefficients and the \(\Psi_{j}(\xi)\ \in\ \mathbb{R}\) form a polynomial orthonormal basis [14]. For practical reasons and to avoid over fitting conditions, the metamodels were limited to \(N\) (=20) expansions. In this study, the \(\mathrm{PCE}_{\mathrm{EM}}\) metamodel, \(Y(\xi)\), was implemented the UQLab toolbox [15] with Legendre polynomial family with the uniform [-1,+1] input distributions. Details of the metamodel implementation for the EM model can be found in [10].