Thermal Infrared Remote Sensing
modelling Level 3

Thermal Infrared Remote Sensing

What is the surface temperature of that field? How much water is being lost through evaporation? Thermal infrared sensors measure emitted radiation in 8-14 μm window, enabling temperature mapping and energy flux estimation. This model derives brightness temperature conversion, implements split-window atmospheric correction, calculates evapotranspiration from thermal data, and maps urban heat islands.

Prerequisites: planck law, stefan boltzmann, emissivity, radiative transfer, energy balance

Updated 13 min read

1. The Question

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)

2. The Conceptual Model

Thermal Radiation Fundamentals

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)

Emissivity

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 vs Kinetic 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.

3. Building the Mathematical Model

At-Sensor Radiance

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

Split-Window Algorithm

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.

Surface Energy Balance

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

Evapotranspiration Estimation

SEBAL algorithm:

Surface Energy Balance Algorithm for Land

Steps:

  1. Calculate net radiation from reflectance and TIR data
  2. Estimate ground heat flux: $G \approx 0.3 R_n$ (bare soil)
  3. Identify “hot” (dry) and “cold” (wet) pixels
  4. Derive sensible heat as linear function of $T_s$
  5. Solve for $LE$ as residual: $LE = R_n - H - G$

Evapotranspiration:

\[ET = \frac{LE}{\lambda}\]

Where $\lambda$ = latent heat of vaporization (2.45 MJ/kg)

Output: ET in mm/day

4. Worked Example by Hand

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.

Solution

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.

5. Computational Implementation

Below is an interactive thermal remote sensing simulator.

Surface temp: -- °C

Brightness temp: -- K

Emissivity: --

Net radiation: -- W/m²

Sensible heat: -- W/m²

Latent heat: -- W/m²

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

6. Interpretation

Urban Heat Islands

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

Agricultural Water Stress

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.

Volcanic Monitoring

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.

Sea Surface Temperature

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.

7. What Could Go Wrong?

Emissivity Uncertainty

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

Atmospheric Effects

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

Mixed Pixels

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

Diurnal Sampling

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

8. Extension: Temperature-Emissivity Separation

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:

  1. Assume initial emissivity (from NDVI)
  2. Retrieve temperature
  3. Update emissivity using spectral shape
  4. Iterate until convergence

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

9. Math Refresher: Blackbody Radiation

Stefan-Boltzmann Law

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.

Wien’s Law

Peak wavelength:

\[\lambda_{max} T = 2898 \text{ μm·K}\]

Explains why:

  • Hot objects glow red (shorter wavelengths)
  • Room temperature objects emit in infrared (invisible)

Kirchhoff’s Law

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.

Summary

  • Thermal infrared remote sensing measures emitted radiation in 8-14 μm atmospheric window
  • Land surface temperature derived from brightness temperature requires emissivity correction
  • Split-window algorithms use differential atmospheric absorption between two bands for correction
  • Surface energy balance partitions net radiation into sensible, latent, and ground heat fluxes
  • Urban heat islands mapped with temperature differences of 10-30°C between materials
  • Evapotranspiration estimated from thermal data enables precision irrigation and drought monitoring
  • Applications span urban climate, agriculture, volcanology, oceanography, and wildfire detection
  • Challenges include emissivity uncertainty, atmospheric effects, mixed pixels, and diurnal sampling
  • MODIS, Landsat TIRS, ASTER, and ECOSTRESS provide operational thermal data at various resolutions
  • Critical tool for energy balance studies and temperature-dependent processes

References