⟨S_{grain}S_{leaf}^{∗}⟩ = ∑_{m=1}^{4} ∑_{n=1}^{4}⟨S_{grain}^{m}S_{leaf}^{n∗}⟩

S_{grain}^{m} = S_{grain}^{m′}(s̄) Φ^{m}(p̄)

s̄ = State variables inclding: moisture

size shape radar extinction

p̄ = Propagation variables inclding: soil moisture

position

stalk state radar extinction

⟨S_{grain}S_{leaf}^{∗}⟩ = ∑_{m=1}^{4} ∑_{n=1}^{4} ∫ S_{grain}^{m′}(s̄_{g}) p(s̄_{g}) ds̄_{g} ∫ S_{leaf}^{n′∗}(s̄_{l}) p(s̄_{l}) ds̄_{l}

∫ Φ_{grain}^{m}(p̄_{g})Φ_{leaf}^{n}(p̄_{l})^{∗}p(p̄_{g}, p̄_{l})dp̄

⟨S_{stalk}S_{leaf}^{∗}⟩ = ∑_{m=1}^{4} ∑_{n=1}^{4}⟨S_{stalk}^{m′}Φ_{leaf}^{n∗}⟩⟨S_{leaf}^{n′∗}⟩