Measuring snowpack from the ground and from space
2026-02-27
Every April, hydrologists across western North America wait for the same number: the 1 April SWE — the Snow Water Equivalent at a network of automated monitoring stations, taken on the date that historically provides the best predictor of spring and summer streamflow. Water utilities, irrigation districts, hydropower operators, and municipal planners use that number to plan reservoir releases, irrigation allocations, and flood risk preparations for the coming months. A high 1 April SWE means a full growing season and good late-summer flows for salmon. A low one means water restrictions, reduced hydropower, and stress for irrigators who planted crops in anticipation of normal allocation.
The challenge is that a mountain snowpack is not uniformly distributed. Elevation, aspect, wind exposure, and forest cover create enormous spatial variability — a north-facing bowl at 2,000 metres might hold three metres of snow while an adjacent south-facing ridge at the same elevation holds less than one. A handful of SNOTEL stations scattered across a basin captures this variability poorly. Satellite remote sensing offers spatial coverage but has its own limitations: passive microwave retrievals saturate at deep snowpacks; optical sensors cannot see through clouds and are blind beneath forest canopy; airborne LiDAR provides excellent snow depth maps but at high cost and limited frequency. SWE estimation across a large basin is therefore a data fusion problem — combining point observations, remote sensing, terrain analysis, and interpolation into a spatial estimate with quantified uncertainty. This model derives the measurement physics of each approach and the statistical methods used to combine them.
What’s the total water stored in snow across a 1000 km² mountain basin?
Three measurement approaches:
1. Ground observations (point data): - SNOTEL stations (automated) - Manual snow courses (monthly surveys) - Snow pillows (weight measurement)
2. Remote sensing (spatial coverage): - Passive microwave (AMSR-E, SSMIS) - Active microwave (SAR) - LiDAR (airborne snow depth) - Optical (MODIS snow cover extent)
3. Model-data fusion: - Combine observations with snow models - Assimilation techniques - Produce gridded SWE estimates
The mathematical question: How do we convert different measurements (microwave brightness temperature, snow depth, weight) into SWE, and how do we scale from points to basins?
Fundamental relationship:
\text{SWE} = \rho_{\text{snow}} \times d
Where: - \rho_{\text{snow}} = bulk snow density (kg/m³) - d = snow depth (m)
Challenge: Density varies (100-500 kg/m³) → must estimate.
Typical approach:
\rho_{\text{snow}} = \rho_0 + k_1 \text{DOY} + k_2 d
Where: - DOY = day of year (snow gets denser over time) - Coefficients calibrated regionally
Example (Colorado Rockies):
\rho = 250 + 0.5 \times \text{DOY} + 10 \times d
For March 15 (DOY 74), depth 1.5m:
\rho = 250 + 0.5(74) + 10(1.5) = 302 \text{ kg/m}^3
\text{SWE} = 0.302 \times 1.5 = 0.453 \text{ m} = 453 \text{ mm}
Snow emits microwave radiation based on: - Temperature - Depth - Grain size - Liquid water content
Brightness temperature:
T_B = T_{\text{snow}} \times (1 - e^{-\tau}) + T_{\text{ground}} \times e^{-\tau}
Where: - T_B = brightness temperature measured by satellite (K) - \tau = optical depth (function of SWE) - T_{\text{snow}} = physical temperature of snow (K) - T_{\text{ground}} = ground temperature (K)
Optical depth depends on SWE:
\tau = \beta \times \text{SWE}
Where \beta = scattering coefficient (~0.002-0.005 m²/kg at 37 GHz).
Key insight: Deep snow → more scattering → lower T_B
Use two frequencies: - 19 GHz (penetrates deep, insensitive to grain size) - 37 GHz (scattered by snow grains)
Spectral gradient:
\Delta T_B = T_B(19\text{ GHz}) - T_B(37\text{ GHz})
Empirical SWE retrieval:
\text{SWE} = a \times \Delta T_B
Where a \approx 3 mm/K (calibration coefficient).
Positive \Delta T_B: Snow present (37 GHz scattered more)
Near zero: Little/no snow
Measures weight of snowpack:
W = \rho_{\text{snow}} \times d \times A \times g
Where: - W = weight (N) - A = pillow area (m²) - g = 9.81 m/s²
SWE directly:
\text{SWE} = \frac{W}{A \times \rho_w \times g}
Advantage: Direct measurement (no density assumption)
Disadvantage: Point measurement, expensive infrastructure
Federal sampler: - Aluminum tube pushed through snowpack - Extract core - Weigh core
SWE calculation:
\text{SWE} = \frac{m_{\text{sample}}}{A_{\text{tube}} \times \rho_w}
Where: - m_{\text{sample}} = mass of snow in tube (kg) - A_{\text{tube}} = tube cross-section area (m²)
Typical: 10-12 sampling points along transect, average for site SWE.
Uncertainty: ±5-10% from sampling variability.
Chang et al. (1987) algorithm:
For dry snow:
\text{SWE} = c_1 \times (T_B^{19V} - T_B^{37V}) \times f_{\text{forest}}
Where: - c_1 = 1.59 mm/K - T_B^{19V} = 19 GHz vertical polarization (K) - T_B^{37V} = 37 GHz vertical polarization (K) - f_{\text{forest}} = forest correction factor
Forest correction:
f_{\text{forest}} = 1 - 0.8 \times f_c
Where f_c = forest cover fraction (0-1).
Limitations: - Saturates at ~150mm SWE (deep snow) - Fails with wet snow (liquid water absorbs microwaves) - Coarse resolution (25 km pixels)
Airborne laser scanning:
Snow-on flight: Measure surface elevation z_{\text{snow}}
Snow-off flight: Measure bare ground elevation z_{\text{ground}}
Snow depth:
d = z_{\text{snow}} - z_{\text{ground}}
Convert to SWE:
\text{SWE} = \rho(x,y) \times d(x,y)
Density from: - Regional relationships - Interpolated from SNOTEL - modelled
Resolution: 1-3 meters (excellent for terrain effects)
Cost: Expensive (aircraft), limited temporal coverage
Problem: Estimate basin-average SWE from multiple sources.
Data:
SNOTEL stations (3 sites): - Low elevation (2000m): SWE = 300 mm - Mid elevation (2500m): SWE = 450 mm - High elevation (3000m): SWE = 600 mm
Passive microwave (basin average): - \Delta T_B = 50 K - SWE = $1.59 = 79.5$ mm (underestimate due to saturation)
LiDAR (sample area, 2500m elevation): - Average depth: 1.8 m - Estimated density: 300 kg/m³ - SWE = $0.300 = 540$ mm
Basin characteristics: - Area: 500 km² - Elevation range: 1800-3200 m - Forest cover: 40%
Step 1: Elevation-based interpolation
Assume linear SWE-elevation relationship:
From SNOTEL: - 2000m → 300mm - 2500m → 450mm - 3000m → 600mm
Gradient: \frac{600 - 300}{3000 - 2000} = 0.30 mm/m
SWE(z):
\text{SWE}(z) = 300 + 0.30(z - 2000)
Step 2: Basin hypsometry
Assume elevation distribution: - 1800-2200m: 20% of area - 2200-2600m: 40% of area - 2600-3000m: 30% of area - 3000-3200m: 10% of area
Average SWE per band:
Band 1 (2000m): SWE = 300mm → 20% × 300 = 60 mm
Band 2 (2400m): SWE = 420mm → 40% × 420 = 168 mm
Band 3 (2800m): SWE = 540mm → 30% × 540 = 162 mm
Band 4 (3100m): SWE = 630mm → 10% × 630 = 63 mm
Basin average: 60 + 168 + 162 + 63 = 453 mm
Step 3: Validate with LiDAR
LiDAR at 2500m: 540mm
Predicted at 2500m: 450mm
Difference: 90mm (20% higher)
Possible reasons: - Local accumulation pattern - Density uncertainty - Measurement error
Step 4: Final estimate
Best estimate: 453mm (from elevation interpolation)
Uncertainty: ±15% (±68mm)
Total water volume:
V = \text{SWE} \times A = 0.453 \text{ m} \times 500 \times 10^6 \text{ m}^2 = 226.5 \times 10^6 \text{ m}^3
= 226,500 acre-feet or 279 million cubic meters
Below is an interactive SWE estimation tool combining multiple sensors.
<label>
SNOTEL low (mm):
<input type="range" id="snotel-low" min="100" max="500" step="50" value="300">
<span id="low-val">300</span>
</label>
<label>
SNOTEL mid (mm):
<input type="range" id="snotel-mid" min="200" max="700" step="50" value="450">
<span id="mid-val">450</span>
</label>
<label>
SNOTEL high (mm):
<input type="range" id="snotel-high" min="300" max="900" step="50" value="600">
<span id="high-val">600</span>
</label>
<label>
Microwave ΔTB (K):
<input type="range" id="delta-tb" min="0" max="100" step="5" value="50">
<span id="tb-val">50</span>
</label>
<div class="swe-est-info">
<p><strong>Basin avg SWE:</strong> <span id="basin-swe">--</span> mm</p>
<p><strong>Total volume:</strong> <span id="total-volume">--</span> million m³</p>
<p><strong>Uncertainty:</strong> <span id="uncertainty">--</span>%</p>
</div><canvas id="swe-est-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Try this: - Red dots: SNOTEL station measurements - Blue line: Interpolated SWE-elevation relationship - Green bars: Elevation band SWE values - Orange dashed: Microwave satellite estimate (underestimates deep snow) - Adjust SNOTEL values: See gradient change - Microwave ΔTB: Higher = more snow detected (saturates ~150mm) - Notice: Microwave underestimates in deep snow (saturation issue) - Basin average combines elevation bands weighted by area
Key insight: Multiple sensors needed—ground stations for accuracy, satellites for coverage, models to bridge gaps!
Automated stations: - Snow pillow (SWE) - Snow depth (ultrasonic) - Precipitation - Air temperature
US Western mountains: ~800 sites
Real-time data at https://www.nrcs.usda.gov/wps/portal/wcc/home/
Uses: - Water supply forecasting - Avalanche prediction - Climate monitoring - Model validation
Multi-sensor airborne campaign:
Sensors: - Passive microwave radiometer - Active microwave (SAR, scatterometer) - LiDAR snow depth - Ground-penetrating radar (snow stratigraphy)
Goals: - Improve satellite algorithms - Understand uncertainty sources - Develop next-generation products
Findings: - Forest significantly attenuates microwave signal - Snow grain size affects scattering - Wet snow problematic for all microwave sensors
Surface Water and Ocean Topography (2022 launch):
Ka-band radar interferometry: - Measures water surface elevation - Rivers, lakes, reservoirs
Snow applications: - Seasonal water storage changes in reservoirs - Flood wave propagation - Groundwater-surface water exchange
Complements SWE observations for total water budget.
Deep snow (>150mm SWE):
Microwave signal saturates → underestimate.
Example: - True SWE: 800mm - Microwave estimate: 120mm - Error: -85%!
Worse in: Maritime climates (deep, wet snow)
Solution: - Combine with model - Use passive + active microwave - Assimilation techniques
Liquid water absorbs microwaves.
Result: Brightness temperature increases (appears as less snow).
Example: - Dry snow: SWE = 300mm, ΔTB = 40K - After melt onset: Same SWE, ΔTB = 10K - Appears as: SWE decreased to 80mm (false!)
Solution: Freeze/thaw detection, temperature thresholding.
Dense canopy: - Intercepts snow (different properties) - Emits radiation (masks snow signal) - Attenuates satellite view
Correction factors crude, uncertain.
Better: Higher frequency (e.g., 89 GHz) penetrates better, or use model.
SNOTEL = point measurement (10m footprint)
Basin = 100s-1000s km²
Representativeness issues: - SNOTEL in clearings (higher SWE than forest) - Elevation bias (stations at accessible locations) - Aspect not captured
Solution: Distributed modelling with data assimilation.
Combine model with observations optimally.
Ensemble Kalman Filter (EnKF):
Steps: 1. Run snow model ensemble (100s of realizations with perturbed forcing) 2. Forecast SWE at observation time 3. Compare forecast to observations 4. Update ensemble using Kalman gain:
\mathbf{K} = \mathbf{P}^f \mathbf{H}^T (\mathbf{H}\mathbf{P}^f\mathbf{H}^T + \mathbf{R})^{-1}
Where: - \mathbf{K} = Kalman gain - \mathbf{P}^f = forecast error covariance - \mathbf{H} = observation operator - \mathbf{R} = observation error covariance
Analysis:
\mathbf{x}^a = \mathbf{x}^f + \mathbf{K}(\mathbf{y} - \mathbf{H}\mathbf{x}^f)
Result: Gridded SWE estimate that: - Respects physics (model) - Matches observations (data) - Quantifies uncertainty (ensemble spread)
Operational: NOAA National Water Model, NASA SWE reconstruction
Frequency ranges:
| Band | Frequency | Wavelength | Use |
|---|---|---|---|
| L | 1-2 GHz | 15-30 cm | Soil moisture |
| C | 4-8 GHz | 3.75-7.5 cm | Soil moisture, ice |
| X | 8-12 GHz | 2.5-3.75 cm | SAR imaging |
| Ku | 12-18 GHz | 1.67-2.5 cm | Precipitation |
| K | 18-27 GHz | 1.11-1.67 cm | Snow, water vapor |
| Ka | 27-40 GHz | 0.75-1.11 cm | Clouds, rain |
Snow remote sensing: 19 GHz and 37 GHz most common
Why microwave? - Penetrates clouds (all-weather) - Sensitive to snow (grain scattering) - Day/night capable (active radiation)
Blackbody emission:
B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{h\nu/kT} - 1}
Where: - B_\nu = spectral radiance (W/(m²·sr·Hz)) - h = Planck constant - \nu = frequency - k = Boltzmann constant - T = temperature (K)
Rayleigh-Jeans approximation (microwave regime):
B_\nu(T) \approx \frac{2\nu^2 kT}{c^2}
Brightness temperature: Temperature of blackbody that would produce observed radiance.