τ-ω Radiative Transfer model

Model Inputs

Brightness Temperatures

Tb vertical 210.0 K

Tb horizontal 170.0 K

Incidence angle θ 40 °

Physical Temperatures

Soil temperature Ts 27.0 °C

Canopy temperature Tc 22.0 °C

Sky downwelling Tsky 5.0 K

Calibratable Parameters fit these in the Calibration tab

RMS height h_rms 0.010 m

Roughness exponent n 1.00

Q cross-pol mixing 0.100

Single-scatter albedo ω 0.050

Retrieval Mode

Fixed τ (known) held constant

Fixed SMC (m³/m³) held constant

Initial Guesses

τ initial guess 0.50

SMC initial guess 0.25 m³/m³

Soil moisture content

—

m³ m⁻³ volumetric

—

Vegetation optical depth τ

—

nadir, dimensionless

—

Emissivity eV

—

vertical pol.

Emissivity eH

—

horizontal pol.

Transmissivity γ

—

at θ, dimensionless

Dielectric ε'

—

soil permittivity

Simulated TbV

—

Simulated TbH

—

ΔTbV

—

K residual

RMSE

—

K total misfit

●Initializing…

Forward model equations

Tb_p = Tc·(1−ω)·(1−γ_p)·(1+r_p·γ_p) + Ts·(1−r_p)·γ_p + Tsky·r_p·γ_p²
γ_p = exp(−τ/cosθ) | e_p = 1 − r_p | p ∈ {V, H}
r_p_rough = [(1−Q)·r_p_smooth + Q·r_q_smooth]·exp(−h_r·cosⁿθ)
h_r = (2k·h_rms)² k = 2πf/c f = 1.4 GHz
ε(SMC, Ts) via Wang & Schmugge (1980) → Fresnel → r_V, r_H
Inversion: Nelder-Mead simplex minimising (TbV_obs−TbV_sim)² + (TbH_obs−TbH_sim)²

Calibratable Parameters

RMS height h_rms

bounds 0 – 0.05 m

Roughness exponent n

bounds 0 – 2

Q cross-pol mixing

bounds 0 – 0.2

Single-scatter albedo ω

bounds 0 – 0.2

Shared Conditions

Sky downwelling Tsky

Observation Data (measured)

#	Tb V (K)	Tb H (K)	Ts (°C)	Tc (°C)	θ (°)	SMC meas (m³/m³)

h_rms (m)

—

RMS surface height

—

roughness exponent

—

cross-pol mixing

—

single-scatter albedo

●—

Per-Observation Fit

#	SMC meas	τ fitted	TbV obs	TbV sim	ΔTbV	TbH obs	TbH sim	ΔTbH