Dual Frequency Bayesian SWE Retrievals

• The paper introduces a Bayesian framework that fuses dual-frequency SAR observations with multilayer snow hydrology and explicit canopy parameterizations to estimate SWE.
• It employs a Metropolis–Hastings MCMC inversion with physically consistent priors and coupled radiative-transfer models to constrain snowpack parameters.
• Validation against ground and airborne data demonstrates improved spatial accuracy and variance capture over traditional hydrology-only simulations.

Dual Frequency Bayesian Snow Water Equivalent (SWE) Retrievals represent a physically informed statistical framework for estimating SWE and snow depth in forested landscapes by combining dual-frequency (X- and Ku-band) Synthetic Aperture Radar (SAR) observations with multilayer snow hydrology modeling, microwave radiative-transfer, and explicit canopy parameterizations. The approach leverages physically consistent priors derived from ensemble snowpack simulations, and incorporates semi-empirical canopy and soil models to enable robust retrievals in heterogeneous, partially forested terrain. Validation with ground-based and airborne datasets demonstrates improved spatial accuracy and variance capture over hydrology-only simulations, with operational extensibility to future satellite SAR missions (Singh et al., 23 Nov 2025).

1. Bayesian Inference Framework

The retrieval algorithm formulates the SWE estimation problem using Bayesian statistics, where the unknown snowpack and land-surface state is represented by a vector

$$\mathbf{x} = \begin{bmatrix} \text{SWE},\, D_\text{snow},\, \rho_\text{snow},\, \ell_\text{corr},\, T_\text{snow},\, T_\text{soil},\, M_v,\, A,\, B,\, \theta_\text{soil},\, \omega_\text{soil} \end{bmatrix}^T$$

encompassing SWE, snow depth $D_\text{snow}$, density $\rho_\text{snow}$, grain correlation length $\ell_\text{corr}$, snow/soil temperature $T_\text{snow}, T_\text{soil}$, vegetation water content $M_v$, Water Cloud Model parameters $A, B$, and soil roughness/moisture $\theta_\text{soil}, \omega_\text{soil}$. The observed vector

$$\mathbf{y} = \begin{bmatrix} \sigma^0_{X\text{-VV}(\theta)},\, \sigma^0_{Ku\text{-VV}(\theta)} \end{bmatrix}^T$$

consists of co-polarized backscatter at X- and Ku-bands for incidence angle $\theta$. The posterior is given by

$$p(\mathbf{x}|\mathbf{y}) \propto p(\mathbf{y}|\mathbf{x}) p(\mathbf{x})$$

with $p(\mathbf{x})$ supplied by the physically-based prior (from MSHM) and the likelihood $p(\mathbf{y}|\mathbf{x})$ computed via a coupled radiative-transfer/backscatter model (Singh et al., 23 Nov 2025).

2. Prior Construction via Multilayer Snow Hydrology Model

Priors for snowpack properties are generated from MSHM, forced with high-resolution numerical weather prediction (HRRR, 3 km grid), incorporating precipitation, radiation, and wind data. MSHM outputs ensembles of multilayer snowpack profiles $\{\rho_k, z_k, \ell_k, T_k \}_{k=1}^K$, which are then collapsed to two-layer equivalent priors per pixel: $\{D_i, \rho_i, \ell_i, T_i \}_{i=1,2}$, providing mean and variance estimates for each parameter. The use of MSHM-driven prior ensembles regularizes retrievals and constrains solutions within physically plausible bounds dictated by large-scale weather forcing and snow physics.

3. Forward Microwave Model and Likelihood Formulation

Total observed backscatter at each pixel is partitioned into microwave contributions from (a) vegetation canopy, (b) snow volume, and (c) ground-snow interface according to

$$\sigma^0(f,\theta) = \sigma^0_{\rm veg}(f,\theta) + T^2_{\rm veg}(f,\theta) [\,\sigma^0_{\rm vol}(f,\theta) + T^2_{\rm vol}(f,\theta)\, \sigma^0_{\rm bkg}(f,\theta)\,]$$

where:

• $\sigma^0_{\rm veg}(f,\theta) = A(f) M_v \cos\theta$
• $$T_{\rm veg}(f,\theta) = \exp[-2 B(f) M_v \sec\theta]$$
• $\sigma^0_{\rm vol}$: snow volume backscatter (from MEMLS, Proksch et al. 2015)
• $T_{\rm vol} = \exp[-2\,\kappa_{\rm snow} D]$
• $\sigma^0_{\rm bkg}$: interface backscatter (IEM, function of $\omega_\text{soil}$, $\theta_\text{soil}$)

The likelihood is computed as

$$p(\mathbf{y}|\mathbf{x}) \propto \exp\left\{ -\frac{1}{2} \left\| \sigma^0_{\rm obs} - \sigma^0(\mathbf{x}) \right\|^2_{\mathbf{R}^{-1}} \right\}$$

where $\mathbf{R}$ encodes total SAR noise and model residual covariance (Singh et al., 23 Nov 2025).

4. Canopy and Soil Parameterization

Canopy closure $C_c$ is derived from MODIS Leaf Area Index (LAI) and GLAD tree height datasets via

$C_c = 0.29\, \log(\mathrm{LAI}) + 0.55$

with LAI resampled to 30 m using forest fraction and tree height. For frozen ground, $A, B, M_v$ initialization proceeds by holding snow parameters fixed ($D_{\rm snow} \sim 1$ mm, negligibly small), inverting Ku-HH backscatter to estimate canopy parameters, and bounding their values using physical constraints (e.g., $T_{\rm veg} \ge 1 - C_c$, $\sigma^0_{\rm veg} \le \sigma^0_{\rm obs}$). Soil roughness and dielectric parameters are retrieved from open areas (X-HH with minimal snow cover), with priors interpolated into forested pixels by ordinary kriging (95% confidence) (Singh et al., 23 Nov 2025).

5. Inversion Algorithm and Regularization

A Metropolis–Hastings Markov Chain Monte Carlo (MCMC) is executed per pixel with Gaussian proposals. The algorithm iteratively proposes state vector updates, evaluates the corresponding simulated backscatter using the physically coupled RTM, and accepts or rejects each candidate according to the posterior probability ratio. Posterior means and standard deviations are derived from the last half of the converged chains. Regularization is achieved by enforcing Gaussian priors (with MSHM-derived variance) and explicit parameter bounds (e.g., positivity, constraints from canopy closure). Pixels with relative residual backscatter (RRB) $<$ 30% and local incidence angle $30^\circ$–$50^\circ$ are retained (Singh et al., 23 Nov 2025).

6. Empirical Validation and Performance Metrics

Validation against SnowEx’17 snow pit SWE and Airborne Snow Observatory (ASO) LiDAR snow depth data demonstrates:

• SWE RMSE $= 0.033$ m ($< 8\%$ of max pit SWE) over 90 m forested pixels at $N \sim 12$ pit sites
• LiDAR snow depth RMSED $\sim 0.11\, \text{--}\,0.20$ m per flight
• Bhattacharyya coefficient BC $= 0.78\,\text{--}\,0.94$ for retrieval, $0.68\,\text{--}\,0.78$ for MSHM alone
• $62\%$ of pixels have $| \Delta \text{depth} | < 10\%$, $82\%$ have $< 20\%$ error (90 m)
• At 30 m, retrieval yields RMSE $\sim 0.20$ m, BC $= 0.87$, and $78\%$ increase in fraction below 10% error threshold

Error patterns are spatially correlated with forest–grassland/mixed-pixel boundaries, wetlands, small water bodies, and steep terrain, reflecting residual uncertainties in canopy closure, subpixel land cover heterogeneity, and NWP precipitation input. LiDAR underestimation beneath dense canopy is noted as an additional uncertainty source (Singh et al., 23 Nov 2025).

7. Operational Implications and Prospects

Dual-frequency Bayesian SWE retrievals robustly estimate mean and variance in snowpack state across partially forested regions, surpassing hydrology-only snow simulations. The methodology's physically informed priors and explicit canopy/soil parameterization (modified WCM) enable scalable operations in complex terrain and direct extension to orbital SAR missions, obviating the need for low-altitude specific incidence angle filters. Prospective improvements include higher-resolution LAI and tree-height maps for subpixel land cover, time-series inversion of WCM parameters using multi-temporal SAR, and synergistic assimilation of active/passive microwave and LiDAR data. A plausible implication is improved accuracy and reliability of SWE retrievals in operational hydrologic and climate monitoring of snow-dominated ecosystems (Singh et al., 23 Nov 2025).