Measuring temperature and heat flux from emitted radiation
2026-02-27
In the early hours of 18 July 2022, an Aqua satellite passed over Sicily during the most severe heat wave in the island’s recorded history. The MODIS thermal infrared sensor recorded land surface temperatures of 47–49°C over the bare limestone hillsides and agricultural fields — temperatures hot enough to kill vegetation, accelerate wildfires, and push human physiology to its limits. On the same day, a fire that had started near Palermo burned through 37,000 hectares in 48 hours. The thermal imagery captured both the extreme surface heating and the active fire fronts — distinguishable from the background because their brightness temperatures exceeded 500°C, far beyond the normal surface range, saturating the sensor.
Thermal infrared remote sensing measures the electromagnetic radiation emitted by surfaces in the 8–14 µm wavelength range. Unlike visible and near-infrared sensors that detect reflected sunlight, thermal sensors detect radiation that the surface itself emits because it has temperature above absolute zero — in accordance with Planck’s law and the Stefan-Boltzmann relation. The brightness temperature measured by the sensor is not exactly the surface kinetic temperature: it is modified by the surface emissivity (how efficiently the surface radiates compared to a perfect blackbody) and by atmospheric absorption and emission along the path. Converting from brightness temperature to actual land surface temperature requires knowing the emissivity — which varies by surface type from about 0.90 for dry sand to 0.995 for water — and correcting for atmospheric effects using a split-window algorithm that exploits the differential absorption in two adjacent thermal channels. This model derives the complete chain from Planck’s law through brightness temperature to land surface temperature retrieval.
How hot is that parking lot compared to the adjacent park?
Thermal infrared (TIR) remote sensing measures emitted radiation from Earth’s surface.
Wavelength range: - Thermal infrared: 8-14 μm (micrometers) - Also called longwave infrared or far-infrared - Atmospheric window: Minimal atmospheric absorption
Key difference from visible/NIR: - Visible/NIR: Reflected solar radiation (daytime only) - Thermal IR: Emitted thermal radiation (day or night)
Applications: - Land surface temperature mapping - Urban heat island quantification - Evapotranspiration estimation - Volcanic hot spot detection - Building heat loss assessment - Wildfire monitoring - Sea surface temperature
Sensors: - Landsat 8/9: TIRS (100m resolution) - MODIS: TIR bands (1km resolution) - ASTER: TIR (90m resolution) - ECOSTRESS: (70m resolution, diurnal coverage) - GOES-R: ABI (2km resolution, geostationary)
Planck’s Law:
Every object above absolute zero emits electromagnetic radiation.
Spectral radiance:
L_\lambda(T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc/\lambda k T} - 1}
Where: - L_\lambda = spectral radiance (W/m²/sr/μm) - h = Planck constant (6.626 × 10⁻³⁴ J·s) - c = speed of light (3 × 10⁸ m/s) - k = Boltzmann constant (1.381 × 10⁻²³ J/K) - \lambda = wavelength (m) - T = temperature (K)
Wien’s Displacement Law:
Peak emission wavelength:
\lambda_{max} = \frac{2898}{T}
Where T in Kelvin, \lambda_{max} in μm.
Example: - Sun (5800 K): \lambda_{max} = 0.5 μm (visible, green) - Earth (288 K): \lambda_{max} = 10 μm (thermal infrared)
Not all objects are perfect blackbodies.
Spectral emissivity:
\varepsilon_\lambda = \frac{L_\lambda^{actual}}{L_\lambda^{blackbody}}
Typical values (8-14 μm): - Water: 0.98-0.99 - Vegetation: 0.96-0.98 - Soil (moist): 0.95-0.97 - Soil (dry): 0.92-0.95 - Asphalt: 0.93-0.96 - Concrete: 0.88-0.92 - Metal (polished): 0.05-0.30
Lower emissivity → reflects more, emits less → appears cooler than true temperature.
Brightness temperature T_B:
Temperature a blackbody would have to produce observed radiance.
Kinetic temperature T_s:
Actual surface temperature.
Relationship:
T_s = \frac{T_B}{\varepsilon^{1/4}}
(Approximate, valid for \varepsilon near 1)
Example:
Concrete with \varepsilon = 0.90, T_B = 300 K:
T_s = \frac{300}{0.90^{0.25}} = \frac{300}{0.974} = 308 \text{ K}
Actual temperature 8 K warmer than brightness temperature.
Radiative transfer equation:
L_{sensor} = \varepsilon L_{surface} \tau + L_{atm}^{\uparrow} + (1-\varepsilon) L_{atm}^{\downarrow} \tau
Where: - L_{sensor} = radiance at sensor - L_{surface} = surface emission (Planck function at T_s) - \tau = atmospheric transmittance - L_{atm}^{\uparrow} = upwelling atmospheric emission - L_{atm}^{\downarrow} = downwelling atmospheric emission - (1-\varepsilon) = reflectance (Kirchhoff’s law)
Three terms: 1. Surface emission attenuated by atmosphere 2. Atmosphere emits upward 3. Downwelling atmospheric radiation reflected off surface
Uses two TIR bands to correct for atmospheric water vapor.
Landsat 8 TIRS: - Band 10: 10.6-11.2 μm - Band 11: 11.5-12.5 μm
Algorithm:
T_s = T_{10} + c_1(T_{10} - T_{11}) + c_2(T_{10} - T_{11})^2 + c_0
Where: - T_{10}, T_{11} = brightness temperatures in bands 10, 11 - c_0, c_1, c_2 = coefficients (depend on atmospheric conditions)
Typical coefficients:
T_s \approx 1.38 T_{10} - 0.38 T_{11} + 0.5
Principle:
Differential absorption by water vapor between two bands enables atmospheric correction.
Net radiation partitioning:
R_n = H + LE + G
Where: - R_n = net radiation (W/m²) - H = sensible heat flux - LE = latent heat flux (evapotranspiration) - G = ground heat flux
Sensible heat parameterization:
H = \rho c_p \frac{T_s - T_a}{r_a}
Where: - \rho = air density - c_p = specific heat of air - T_s = surface temperature (from TIR) - T_a = air temperature - r_a = aerodynamic resistance
Large T_s - T_a → large H → low LE → water stressed
Applications: - Irrigation scheduling - Drought monitoring - Crop water stress detection
SEBAL algorithm:
Surface Energy Balance Algorithm for Land
Steps:
Evapotranspiration:
ET = \frac{LE}{\lambda}
Where \lambda = latent heat of vaporization (2.45 MJ/kg)
Output: ET in mm/day
Problem: Calculate land surface temperature from Landsat thermal data.
At-sensor brightness temperatures: - Band 10: T_{10} = 300 K - Band 11: T_{11} = 298 K
Surface properties: - Emissivity: \varepsilon = 0.96 (vegetation)
Atmospheric correction coefficients: - c_0 = 0.5, c_1 = 1.38, c_2 = 0 (simplified)
Calculate land surface temperature.
Step 1: Split-window correction
T_s = c_1 T_{10} - (c_1 - 1) T_{11} + c_0
= 1.38(300) - 0.38(298) + 0.5
= 414 - 113.24 + 0.5 = 301.26 \text{ K}
Step 2: Convert to Celsius
T_s = 301.26 - 273.15 = 28.1°\text{C}
Step 3: Emissivity correction (if needed)
For vegetation with \varepsilon = 0.96, already accounted for in split-window coefficients.
If not using split-window:
T_s = \frac{T_B}{\varepsilon^{0.25}} = \frac{300}{0.96^{0.25}} = \frac{300}{0.990} = 303.0 \text{ K} = 29.8°\text{C}
Summary: - Split-window LST: 28.1°C - Single-channel with emissivity correction: 29.8°C - Difference reflects atmospheric correction importance
Interpretation:
Surface temperature of ~28°C typical for vegetated surface on warm day.
Below is an interactive thermal remote sensing simulator.
<label>
Surface type:
<select id="surface-type">
<option value="water">Water</option>
<option value="vegetation">Vegetation</option>
<option value="soil-wet">Soil (wet)</option>
<option value="soil-dry">Soil (dry)</option>
<option value="asphalt">Asphalt</option>
<option value="concrete">Concrete</option>
<option value="metal">Metal roof</option>
</select>
</label>
<label>
Air temperature (°C):
<input type="range" id="air-temp" min="15" max="35" step="1" value="25">
<span id="air-val">25</span>
</label>
<label>
Solar radiation (W/m²):
<input type="range" id="solar-rad" min="0" max="1000" step="50" value="600">
<span id="solar-val">600</span>
</label>
<label>
Time of day:
<input type="range" id="time-hour" min="0" max="23" step="1" value="12">
<span id="time-val">12</span>:00
</label>
<div class="thermal-info">
<p><strong>Surface temp:</strong> <span id="surf-temp">--</span> °C</p>
<p><strong>Brightness temp:</strong> <span id="bright-temp">--</span> K</p>
<p><strong>Emissivity:</strong> <span id="emissivity">--</span></p>
<p><strong>Net radiation:</strong> <span id="net-rad">--</span> W/m²</p>
<p><strong>Sensible heat:</strong> <span id="sens-heat">--</span> W/m²</p>
<p><strong>Latent heat:</strong> <span id="lat-heat">--</span> W/m²</p>
</div><canvas id="thermal-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Observations: - Red line shows surface temperature varying through day - Blue dashed line shows constant air temperature for reference - Surface temperature peaks in afternoon, not at noon (thermal inertia) - Asphalt reaches much higher temperatures than vegetation - Metal roofs show extreme temperatures due to low emissivity - Water shows minimal temperature variation (high thermal capacity) - Temperature difference (Ts - Ta) drives sensible heat flux - Energy partitioning varies by surface type affecting local climate
Key findings: - Different surfaces exhibit different thermal behaviors - Urban materials (asphalt, concrete) create heat islands - Vegetation moderates temperature through evapotranspiration - Surface emissivity affects both measured brightness temperature and actual cooling rate - Diurnal cycle reveals thermal properties and energy partitioning
Phenomenon:
Cities warmer than surrounding rural areas.
Magnitude: - Daytime: 1-3°C warmer - Nighttime: 3-5°C warmer (sometimes 10°C)
Thermal remote sensing reveals: - Hot spots: Parking lots, roofs, roads - Cool spots: Parks, water bodies, tree canopy - Spatial patterns: Correlate with land use
Example - Phoenix, Arizona:
Landsat TIR shows: - Asphalt parking lots: 65-70°C - Vegetated parks: 35-40°C - Difference: 30°C!
Health impacts: - Heat-related mortality - Air quality degradation - Energy demand for cooling
Mitigation strategies: - Cool roofs (high albedo, high emissivity) - Urban tree canopy - Green roofs - Reflective pavements
Crop Water Stress Index (CWSI):
CWSI = \frac{(T_s - T_a) - (T_s - T_a)_{LL}}{(T_s - T_a)_{UL} - (T_s - T_a)_{LL}}
Where: - LL = lower limit (well-watered) - UL = upper limit (water-stressed)
Range: 0 (no stress) to 1 (maximum stress)
Application:
ECOSTRESS provides 70m resolution thermal data with diurnal coverage enabling: - Within-field stress mapping - Irrigation scheduling - Yield prediction
Economic value:
Precision irrigation based on thermal data reduces water use 20-30% while maintaining yield.
Thermal anomalies indicate: - Active lava flows - Lava lake temperature - Fumarole activity - Dome growth
MODIS thermal bands: - Nightly global coverage - Detect temperature increases weeks before eruption - Track eruption intensity
Example - Kilauea, Hawaii 2018:
MODIS detected thermal anomaly increase before major eruption.
Enabled evacuation planning and hazard assessment.
MODIS SST product: - Daily global coverage - 1 km resolution - Accuracy: ±0.5°C
Applications: - Ocean circulation mapping - El Niño monitoring - Coral bleaching prediction (thermal stress) - Fisheries (temperature gradients concentrate fish)
Coral bleaching threshold:
Sustained SST > 1°C above climatological maximum triggers bleaching.
Thermal remote sensing provides early warning.
Problem:
Don’t know exact emissivity of surface.
Error propagation:
\Delta T_s \approx \frac{T_s}{\varepsilon} \Delta\varepsilon
Example:
T_s = 300 K, \varepsilon = 0.95 \pm 0.03:
\Delta T_s \approx \frac{300}{0.95} \times 0.03 = 9.5 \text{ K}
Large uncertainty!
Solution: - Use emissivity databases (ASTER spectral library) - Estimate from NDVI (empirical relationships) - Temperature-emissivity separation algorithms
Water vapor absorption (especially 8-9 μm):
Can reduce apparent temperature by 5-10 K.
Aerosols:
Scatter and absorb, further complicating retrieval.
Solution: - Split-window algorithms - Atmospheric correction with radiosonde data - In-situ calibration/validation
Landsat thermal: 100m resolution
Pixel may contain: - 50% vegetation (25°C) - 50% bare soil (45°C)
Measured: ~35°C (area-weighted average)
But: Not representative of either component.
Problem for applications:
Crop stress detection needs pure vegetation pixels.
Solution: - Unmix thermal signal (if know components) - Use higher resolution (ECOSTRESS 70m) - Aggregate to coarser resolution
Most satellites: Fixed overpass time (e.g., Landsat ~10:30 AM)
Misses: - Peak afternoon temperature - Nighttime cooling - Full diurnal cycle
Problem:
Can’t distinguish thermal inertia differences.
Solution: - Geostationary satellites (GOES, MSG) - hourly - ECOSTRESS - variable overpass times - Model diurnal cycle from limited observations
Challenge: Retrieve both T_s and \varepsilon from single thermal measurement.
Underdetermined problem: One equation, two unknowns.
Solution: Use multiple TIR bands with different emissivity contrast.
ASTER TIR: 5 bands in 8-12 μm
Algorithm:
Emissivity spectrum reveals: - Quartz: Peak at 8.6 μm - Carbonates: Trough at 11.2 μm - Vegetation: Flat, high emissivity
Applications: - Mineral mapping from emissivity - Surface composition without field work
Total emitted power:
M = \sigma T^4
Where: - M = exitance (W/m²) - \sigma = 5.67 \times 10^{-8} W/(m²·K⁴) - T = temperature (K)
Integration of Planck function over all wavelengths.
Peak wavelength:
\lambda_{max} T = 2898 \text{ μm·K}
Explains why: - Hot objects glow red (shorter wavelengths) - Room temperature objects emit in infrared (invisible)
At thermal equilibrium:
\alpha_\lambda = \varepsilon_\lambda
Where: - \alpha = absorptivity - \varepsilon = emissivity
Good absorbers are good emitters.
Corollary:
\rho_\lambda = 1 - \varepsilon_\lambda
Where \rho = reflectivity (for opaque surfaces).
Low emissivity → high reflectivity → shiny surfaces appear cooler.