Module 2

Physical Principles of Remote Sensing

Electromagnetic radiation, radiometric quantities, and the physics of SAR/InSAR

The Electromagnetic Spectrum for EO

Earth observation instruments exploit different parts of the electromagnetic spectrum, each sensitive to different surface and atmospheric properties. The choice of wavelength determines what physical quantity is measured and how the atmosphere affects the signal.

Band	Wavelength	Source	Applications	Sensors
VIS	0.4 - 0.7 um	Reflected solar	True color, chlorophyll, turbidity	S2, Landsat, MODIS
NIR	0.7 - 1.0 um	Reflected solar	Vegetation (NDVI), water bodies	S2, Landsat, Planet
Red Edge	0.7 - 0.78 um	Reflected solar	Vegetation stress, LAI, chlorophyll	S2 (B5-B7, B8a)
SWIR	1.0 - 2.5 um	Reflected solar	Soil moisture, minerals, fire scars	S2, Landsat, WV-3
MWIR	3 - 5 um	Mixed (emitted + reflected)	Active fire detection, hotspots	MODIS, VIIRS, SLSTR
TIR	8 - 14 um	Emitted thermal	LST, SST, urban heat islands	Landsat TIRS, MODIS, GOES
Microwave	1 mm - 1 m	Active (SAR) or passive	All-weather imaging, soil moisture, ice	S1, SMAP, AMSR-E

Atmospheric windows (regions of high transmittance) dictate which wavelengths reach the surface. Water vapor absorbs strongly at 1.4 um and 1.9 um; CO2 absorbs around 4.3 um and 15 um; ozone absorbs UV below 0.3 um.

Radiometric Quantities

Satellite sensors measure spectral radiance at the top of the atmosphere (TOA). To extract meaningful surface properties, we must convert raw digital numbers (DN) to physical quantities.

Planck's Blackbody Radiation Law

Every object at temperature $T$ emits radiation according to Planck's law:

$$B_\lambda(T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{hc/(\lambda k_B T)} - 1}$$

where $h = 6.626 \times 10^{-34}$ J s is Planck's constant,$c = 3 \times 10^8$ m/s is the speed of light, and$k_B = 1.381 \times 10^{-23}$ J/K is Boltzmann's constant. The peak wavelength follows Wien's displacement law:$\lambda_{max} = 2898 / T$ um.

TOA Radiance from DN

For Landsat 8/9, the conversion from quantized calibrated pixel values ($Q_{cal}$) to TOA spectral radiance uses band-specific gain and offset from the metadata:

$$L_\lambda = \frac{M_L \cdot Q_{cal} + A_L}{\sin(\theta_{SE})}$$

where $M_L$ is the radiance multiplicative scaling factor, $A_L$ is the additive offset, and $\theta_{SE}$ is the sun elevation angle. Units: W m^-2 sr^-1 um^-1.

TOA Reflectance

TOA reflectance normalizes for solar irradiance and sun-Earth distance, making it more comparable across dates and locations:

$$\rho_\lambda = \frac{\pi \cdot L_\lambda \cdot d^2}{ESUN_\lambda \cdot \cos\theta_z}$$

where $d$ is the Earth-Sun distance in astronomical units, $ESUN_\lambda$ is the mean exoatmospheric solar irradiance for that band, and $\theta_z$ is the solar zenith angle. Note: for Landsat 8/9, a direct reflectance rescaling formula is available that bypasses the radiance step.

Stefan-Boltzmann Law & Brightness Temperature

Total emitted power per unit area follows the Stefan-Boltzmann law:

$$M = \varepsilon \sigma T^4$$

In thermal remote sensing, we invert the Planck function to retrieve brightness temperature from measured radiance:

$$T_B = \frac{K_2}{\ln\left(\frac{K_1}{L_\lambda} + 1\right)}$$

where $K_1$ and $K_2$ are band-specific thermal constants provided in the metadata.

SAR & InSAR Physics

Synthetic Aperture Radar (SAR) is an active microwave sensor that transmits pulses and records the backscattered signal. Unlike optical sensors, SAR provides its own illumination and operates independently of weather and sunlight.

SAR Signal Model

Each SAR pixel stores a complex number encoding both amplitude and phase:

$$s = A \cdot e^{i\phi} \quad\text{where}\quad \phi = -\frac{4\pi}{\lambda} r + \phi_{scatt}$$

The amplitude $A$ depends on the backscattering coefficient $\sigma^0$, which is governed by surface roughness, dielectric constant, and incidence angle. The phase$\phi$ encodes the two-way travel distance$r$ plus a scattering phase term.

InSAR Phase Equation

Interferometric SAR (InSAR) exploits the phase difference between two SAR acquisitions from slightly different positions or times:

$$\phi_{InSAR} = \frac{4\pi}{\lambda}\Delta r + \phi_{topo} + \phi_{atm} + \phi_{noise}$$

The total interferometric phase contains contributions from: geometric baseline ($\Delta r$), topography ($\phi_{topo}$), atmospheric delay ($\phi_{atm}$), and decorrelation noise ($\phi_{noise}$).

DInSAR: Differential InSAR

By removing the topographic phase contribution (using an external DEM or a reference interferogram), DInSAR isolates the surface displacement signal:

$$\phi_{defo} = \phi_{InSAR} - \phi_{topo} = \frac{4\pi}{\lambda}\Delta r_{LOS}$$

Each fringe cycle (2pi) corresponds to $\lambda/2$ of line-of-sight displacement. For Sentinel-1 C-band ($\lambda = 5.55$ cm), one fringe = 2.775 cm of LOS motion.

Coherence

Coherence measures the quality of the interferometric phase estimate. It ranges from 0 (complete decorrelation) to 1 (perfect phase stability):

$$\gamma = \left|\frac{\langle s_1 \cdot s_2^* \rangle}{\sqrt{\langle|s_1|^2\rangle \langle|s_2|^2\rangle}}\right|$$

Low coherence occurs over vegetation (temporal decorrelation), water surfaces, or when the perpendicular baseline is too large (geometric decorrelation). Typical InSAR applications require $\gamma > 0.3$.

Planck Blackbody Curves & Atmospheric Windows

The following simulation computes Planck blackbody emission curves for the Sun (~5778 K) and the Earth (~288 K), and shows approximate atmospheric transmission windows to illustrate which spectral bands are usable for remote sensing.

Simulation

Python

script.py97 lines

import numpy as np
import matplotlib.pyplot as plt

# --- Planck function ---
h = 6.626e-34   # Planck constant (J s)
c = 3.0e8       # speed of light (m/s)
kB = 1.381e-23  # Boltzmann constant (J/K)

def planck(wavelength_m, T):
    """Spectral radiance B(lambda, T) in W m^-2 sr^-1 m^-1"""
    a = 2.0 * h * c**2 / wavelength_m**5
    b = h * c / (wavelength_m * kB * T)
    return a / (np.exp(b) - 1.0)

# Wavelength range: 0.1 um to 100 um
lam_um = np.linspace(0.1, 50, 2000)
lam_m = lam_um * 1e-6

# Compute curves for different temperatures
temperatures = [5778, 1500, 500, 288]
labels = ['Sun (5778 K)', 'Lava (1500 K)', 'Forest Fire (500 K)', 'Earth (288 K)']
colors = ['#FFD700', '#FF6347', '#FF8C00', '#00BFFF']

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(10, 10), gridspec_kw={'height_ratios': [3, 1]})

# --- Top panel: Planck curves ---
for T, label, color in zip(temperatures, labels, colors):
    B = planck(lam_m, T)
    # Normalize each curve to its peak for visibility
    B_norm = B / B.max()
    ax1.plot(lam_um, B_norm, color=color, linewidth=2, label=label)

# Shade EO spectral regions
bands = [
    (0.4, 0.7, 'VIS', '#7B68EE', 0.15),
    (0.7, 1.0, 'NIR', '#CD5C5C', 0.12),
    (1.0, 2.5, 'SWIR', '#DAA520', 0.10),
    (3.0, 5.0, 'MWIR', '#FF8C00', 0.10),
    (8.0, 14.0, 'TIR', '#DC143C', 0.10),
]
for bmin, bmax, bname, bcolor, balpha in bands:
    ax1.axvspan(bmin, bmax, alpha=balpha, color=bcolor)
    ax1.text((bmin + bmax)/2, 1.02, bname, ha='center', va='bottom',
             fontsize=8, color=bcolor, fontweight='bold', transform=ax1.get_xaxis_transform())

ax1.set_xlim(0.1, 30)
ax1.set_xlabel('Wavelength (um)', fontsize=12)
ax1.set_ylabel('Normalized Spectral Radiance', fontsize=12)
ax1.set_title('Planck Blackbody Curves at Different Temperatures', fontsize=14, fontweight='bold')
ax1.legend(loc='upper right', fontsize=10)
ax1.set_yscale('linear')
ax1.grid(True, alpha=0.2)

# --- Bottom panel: Approximate atmospheric transmission ---
# Simplified model of atmospheric transmission windows
lam_atm = np.linspace(0.1, 30, 3000)
transmission = np.ones_like(lam_atm) * 0.85

# H2O absorption bands
for center, width, depth in [(1.38, 0.15, 0.9), (1.87, 0.15, 0.85),
                              (2.7, 0.3, 0.95), (6.3, 1.5, 0.9)]:
    transmission *= 1 - depth * np.exp(-0.5 * ((lam_atm - center)/width)**2)

# CO2 absorption
for center, width, depth in [(4.3, 0.3, 0.8), (15.0, 2.0, 0.7)]:
    transmission *= 1 - depth * np.exp(-0.5 * ((lam_atm - center)/width)**2)

# O3 absorption (UV)
transmission *= 1 - 0.95 * np.exp(-0.5 * ((lam_atm - 0.25)/0.05)**2)

# General absorption increase beyond 25 um
transmission *= 1.0 / (1.0 + np.exp((lam_atm - 25) / 2))

transmission = np.clip(transmission, 0, 1)

ax2.fill_between(lam_atm, 0, transmission, alpha=0.4, color='#00BFFF')
ax2.plot(lam_atm, transmission, color='#00BFFF', linewidth=1.5)
ax2.set_xlim(0.1, 30)
ax2.set_ylim(0, 1.05)
ax2.set_xlabel('Wavelength (um)', fontsize=12)
ax2.set_ylabel('Transmission', fontsize=12)
ax2.set_title('Approximate Atmospheric Transmission', fontsize=14, fontweight='bold')
ax2.grid(True, alpha=0.2)

# Mark windows
for bmin, bmax in [(0.3, 1.3), (1.5, 1.8), (2.0, 2.5), (3.5, 4.2), (4.5, 5.0), (8.0, 14.0)]:
    ax2.axvspan(bmin, bmax, alpha=0.08, color='lime')

plt.tight_layout()
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0a0a1a', edgecolor='none')
plt.close()
print("Planck blackbody curves and atmospheric transmission windows plotted successfully.")
print("Key observations:")
print("- Sun peaks at ~0.5 um (visible), Earth peaks at ~10 um (TIR)")
print("- Atmospheric windows allow observation in VIS/NIR, SWIR, MWIR, and 8-14 um TIR")
print("- H2O, CO2, and O3 create absorption bands where the atmosphere is opaque")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Common Spectral Indices

Spectral indices combine bands to enhance specific surface features. They exploit known spectral response differences between materials:

NDVI (Vegetation)

$$NDVI = \frac{NIR - Red}{NIR + Red}$$

Range: -1 to +1. Healthy vegetation > 0.3

NDWI (Water)

$$NDWI = \frac{Green - NIR}{Green + NIR}$$

Water bodies > 0. Vegetation < 0.

MNDWI (Modified Water)

$$MNDWI = \frac{Green - SWIR}{Green + SWIR}$$

Better urban/water discrimination than NDWI

NBR (Burn Ratio)

$$NBR = \frac{NIR - SWIR_2}{NIR + SWIR_2}$$

Burn scars show strong negative NBR values

The Radar Equation

The power received by a SAR sensor from a distributed target is governed by the radar equation:

$$P_r = \frac{P_t G^2 \lambda^2 \sigma^0 A}{(4\pi)^3 R^4}$$

where $P_t$ is the transmitted power,$G$ is the antenna gain,$\sigma^0$ is the normalized radar cross-section (backscattering coefficient), $A$ is the resolution cell area, and $R$ is the slant range distance.

The backscattering coefficient $\sigma^0$ is the key geophysical parameter. It depends on surface roughness relative to the radar wavelength, the dielectric constant of the material, and the local incidence angle. Typical values: calm water $\sigma^0 \approx -20$ dB, forest $\sigma^0 \approx -8$ dB, urban $\sigma^0 \approx 0$ dB.

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