modelling Level 5

SAR Fundamentals and Applications

How do we image Earth through clouds and darkness? Synthetic Aperture Radar (SAR) transmits microwave pulses and processes returns to achieve high-resolution imaging independent of weather and illumination. This model derives range and azimuth resolution equations, implements backscatter analysis, demonstrates polarimetric decomposition, and shows applications from flood mapping to ship detection.

Prerequisites: doppler processing, range resolution, azimuth resolution, backscatter modelling

27 Feb 2026 Updated 12 Mar 2026 12 min read

1. The Question

Can we monitor flooding in real-time even when clouds obscure optical satellites?

SAR (Synthetic Aperture Radar):

Active microwave sensor that creates its own illumination.

Key advantages:

All-weather: Microwaves penetrate clouds, rain, smoke

Day/night: Active sensor, no sunlight needed

Penetration: Sees through vegetation canopy (long wavelengths)

Coherent: Phase information enables interferometry

Wavelength bands:

X-band: 3 cm (9.6 GHz) - High resolution, urban
C-band: 6 cm (5.4 GHz) - General purpose, Sentinel-1
L-band: 24 cm (1.3 GHz) - Vegetation penetration, ALOS-2
P-band: 70 cm (430 MHz) - Deep penetration, biomass

Applications:

Flood mapping (water appears dark)
Oil spill detection (dampens ocean waves)
Ship detection (bright against dark ocean)
Deformation monitoring (InSAR, Model 57)
Sea ice mapping
Agriculture (crop type, soil moisture)
Disaster response (rapid mapping)

2. The Conceptual Model

Radar Fundamentals

Transmitted pulse:

\[P_t = P_{\text{peak}} \times \tau\]

Where:

$P_t$ = transmitted power (W)
$\tau$ = pulse duration (seconds)

Range resolution:

\[\delta_r = \frac{c \tau}{2}\]

Where $c$ = speed of light.

Example: $\tau = 10$ μs

\[\delta_r = \frac{3 \times 10^8 \times 10 \times 10^{-6}}{2} = 1500 \text{ m}\]

Too coarse!

Pulse compression:

Use chirped pulse (frequency modulation).

Achieves:

\[\delta_r = \frac{c}{2B}\]

Where $B$ = bandwidth.

Example: $B = 100$ MHz

\[\delta_r = \frac{3 \times 10^8}{2 \times 100 \times 10^6} = 1.5 \text{ m}\]

Much better resolution!

Synthetic Aperture

Real aperture: Physical antenna size limits resolution.

Azimuth resolution (real aperture):

\[\delta_a = \frac{\lambda R}{L}\]

Where:

$\lambda$ = wavelength
$R$ = range (distance to target)
$L$ = antenna length

Problem: For $\lambda = 0.06$ m (C-band), $R = 800$ km, $L = 10$ m:

\[\delta_a = \frac{0.06 \times 800000}{10} = 4800 \text{ m}\]

4.8 km resolution - poor!

Synthetic aperture:

Platform motion creates long virtual antenna.

Processing multiple pulses coherently.

Achieves:

\[\delta_a = \frac{L}{2}\]

Independent of range!

For $L = 10$ m: $\delta_a = 5$ m

Orders of magnitude improvement.

Backscatter Coefficient

Radar cross-section normalized by area:

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

Where:

$\sigma^0$ = backscatter coefficient (unitless, often dB)
$G$ = antenna gain
$A$ = illuminated area

Typical values (dB):

Water (calm): -25 to -15 dB (dark)
Vegetation: -15 to -5 dB
Urban: -5 to +5 dB (bright)
Metal (corner reflector): +20 to +40 dB (very bright)

Image formation:

Pixel brightness ∝ $\sigma^0$

Interpretation:

Bright: Strong backscatter (rough, metallic)
Dark: Weak backscatter (smooth, water)

3. Building the Mathematical Model

Radar Range Equation

Received power:

\[P_r = \frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 R^4}\]

Key: $R^{-4}$ dependence (two-way path loss)

Solving for range:

\[R = \left(\frac{P_t G^2 \lambda^2 \sigma}{(4\pi)^3 P_r}\right)^{1/4}\]

Maximum range:

Limited by signal-to-noise ratio:

\[SNR = \frac{P_r}{P_n} > SNR_{\text{threshold}}\]

Typically $SNR_{\text{threshold}} = 10$ dB

Doppler Processing

Doppler shift from platform motion:

\[f_d = \frac{2v_r}{\lambda}\]

Where:

$v_r$ = radial velocity component
$\lambda$ = wavelength

For satellite moving at 7 km/s, C-band:

\[f_d = \frac{2 \times 7000}{0.06} = 233 \text{ kHz}\]

Doppler history of point target creates chirp.

Azimuth compression:

Match filter to Doppler chirp → focus image.

Resolution:

\[\delta_a = \frac{L}{2}\]

Half the real antenna length!

Polarimetry

Transmit/receive polarization:

HH: Horizontal transmit, horizontal receive
VV: Vertical transmit, vertical receive
HV: Horizontal transmit, vertical receive
VH: Vertical transmit, horizontal receive

Scattering matrix:

\[\mathbf{S} = \begin{bmatrix} S_{HH} & S_{HV} \\ S_{VH} & S_{VV} \end{bmatrix}\]

Different targets, different polarizations:

Surface scattering (water): HH ≈ VV, HV ≈ 0
Double-bounce (urban): HH ≈ VV, strong
Volume scattering (forest): HV strong
Single-bounce (bare soil): HH > VV

Decomposition:

Pauli basis:

\[\mathbf{k} = \frac{1}{\sqrt{2}} \begin{bmatrix} S_{HH} + S_{VV} \\ S_{HH} - S_{VV} \\ 2S_{HV} \end{bmatrix}\]

Components:

Surface
Double-bounce
Volume

4. Worked Example by Hand

Problem: Calculate SAR resolution and classify target.

SAR parameters:

Wavelength: 5.6 cm (C-band)
Antenna length: 12 m
Bandwidth: 150 MHz
Range to target: 850 km

Target backscatter:

HH: -8 dB
VV: -9 dB
HV: -18 dB

Calculate resolution and identify scattering mechanism.

Solution

Step 1: Range resolution

\[\delta_r = \frac{c}{2B} = \frac{3 \times 10^8}{2 \times 150 \times 10^6} = 1.0 \text{ m}\]

Step 2: Azimuth resolution

\[\delta_a = \frac{L}{2} = \frac{12}{2} = 6 \text{ m}\]

Step 3: Compare to real aperture

\[\delta_{a,\text{real}} = \frac{\lambda R}{L} = \frac{0.056 \times 850000}{12} = 3967 \text{ m}\]

Synthetic aperture improvement: 3967 / 6 = 661×

Step 4: Analyze polarimetry

HH ≈ VV (similar, both -8 to -9 dB)
HV weak (-18 dB, 10 dB below co-pol)

Ratio:

\[\frac{\sigma_{HV}}{\sigma_{HH}} = 10^{(-18 - (-8))/10} = 10^{-1} = 0.1\]

Cross-pol 10% of co-pol

Step 5: Classification

Similar HH/VV → surface or double-bounce
Weak HV → not volume scattering

If bright (positive dB): Urban/double-bounce
If moderate (near 0 dB): Bare soil/surface
If dark (negative dB): Smooth surface

Given -8 dB: Moderately rough surface (agricultural field, rough terrain)

Not: Forest (would have strong HV)
Not: Water (would be much darker, -20 dB)
Not: Urban (would be brighter, +5 dB)

5. Computational Implementation

Below is an interactive SAR simulator.

Wavelength band: Scene type: Incidence angle (°): 35

Range resolution: -- m

Azimuth resolution: -- m

Backscatter (HH): -- dB

Backscatter (VV): -- dB

Scattering type: --

Observations:

Urban areas show strong backscatter (bright in SAR images)
Water appears dark (specular reflection away from sensor)
Forest shows moderate backscatter with volume scattering
Incidence angle affects backscatter magnitude
Resolution independent of range (synthetic aperture advantage)
Different wavelengths penetrate differently

Key insights:

SAR brightness reveals surface properties
Polarimetry distinguishes scattering mechanisms
All-weather capability critical for operational monitoring
Synthetic aperture achieves fine resolution from space

6. Interpretation

Flood Mapping

Water detection:

SAR backscatter from water: -25 to -15 dB (dark)

Flooded areas: Very dark compared to surrounding land

Automated detection:

Threshold: $\sigma^0 < -15$ dB → Water

Challenges:

Wind roughens water (brighter)
Vegetation emergence (wet but not dark)
Urban flooding (double-bounce from buildings)

Sentinel-1 operational:

6-day repeat (2 satellites)
Free and open data
Copernicus Emergency Management Service

Example - 2017 Texas flooding:

SAR mapped extent when clouds obscured optical
Enabled emergency response routing

Ship Detection

Ships = bright targets against dark ocean.

CFAR (Constant False Alarm Rate):

\[T = \mu + k\sigma\]

Where:

$T$ = threshold
$\mu$ = background mean
$\sigma$ = background standard deviation
$k$ = constant (typically 3-5)

Pixel > $T$ → Potential ship

False alarms:

Waves (in high seas)
Oil platforms
Icebergs

Discrimination:

Size filter (ships 10-400 m)
Shape analysis (elongated)
AIS correlation (automatic identification system)

Applications:

Maritime surveillance
Illegal fishing detection
Search and rescue
Traffic monitoring

Agriculture

Crop monitoring:

Backscatter varies with:

Crop type (structure)
Growth stage (biomass)
Soil moisture (dielectric)
Roughness

VV polarization: Sensitive to vertical structure (stems)

HH polarization: Sensitive to horizontal elements (leaves)

Temporal analysis:

Track backscatter through season:

Planting (low, bare soil)
Growth (increasing)
Maturity (peak)
Harvest (decrease)

Crop type classification:

Combine temporal signature with optical data.

Accuracy: 85-95% for major crops

7. What Could Go Wrong?

Speckle Noise

Coherent imaging produces granular noise.

Speckle: Random constructive/destructive interference

Standard deviation = mean for single-look

Reduction:

Multi-looking: Average independent looks

\[\sigma_{\text{speckle}} = \frac{\sigma}{\sqrt{N}}\]

Where $N$ = number of looks.

Trade-off: Noise reduction vs resolution degradation

Filtering:

Lee filter
Frost filter
Gamma-MAP

Geometric Distortions

Layover:

Targets closer to sensor than base → appear reversed

Common: Mountains toward sensor

Shadow:

Terrain blocks radar → no signal return

Common: Mountains away from sensor

Foreshortening:

Slopes compressed in range direction

Correction:

DEM required for geometric terrain correction (GTC)

Temporal Decorrelation

Phase coherence required for InSAR.

Decorrelation sources:

Vegetation motion (wind)
Soil moisture changes
Snow melt/accumulation

Coherence:

\[\gamma = \frac{|\langle s_1 s_2^* \rangle|}{\sqrt{\langle |s_1|^2 \rangle \langle |s_2|^2 \rangle}}\]

Where $s_1$, $s_2$ = complex signals from two acquisitions.

$\gamma = 1$: Perfect coherence
$\gamma = 0$: Complete decorrelation

L-band better: Less decorrelation than C-band

Ambiguities

Range ambiguities:

Signal from wrong swath enters image

Azimuth ambiguities:

Doppler aliasing creates ghost targets

Mitigation:

Careful PRF (pulse repetition frequency) selection
Antenna pattern shaping
Azimuth filtering

8. Extension: Polarimetric SAR

Full polarimetry:

Measure complete scattering matrix $\mathbf{S}$

Freeman-Durden decomposition:

\[\langle T \rangle = f_s T_s + f_d T_d + f_v T_v\]

Where:

$f_s$ = surface scattering fraction
$f_d$ = double-bounce fraction
$f_v$ = volume scattering fraction

RGB composite:

Red: Double-bounce (urban)
Green: Volume (vegetation)
Blue: Surface (bare soil, water)

Applications:

Land cover classification
Crop type mapping
Wetland characterization
Snow/ice discrimination

Sensors:

ALOS-2 PALSAR-2 (L-band)
RADARSAT-2 (C-band)
Upcoming: NISAR (L+S band)

9. Math Refresher: Doppler Effect

Frequency Shift

Approaching source:

\[f' = f \frac{c + v_r}{c}\]

Receding source:

\[f' = f \frac{c - v_r}{c}\]

For $v_r \ll c$:

\[\Delta f = f \frac{v_r}{c}\]

Two-Way Doppler (Radar)

Signal travels to and from target:

\[f_d = \frac{2v_r}{\lambda}\]

Independent of transmitted frequency!

Example:

Satellite velocity: 7 km/s
Wavelength: 6 cm (C-band)

\[f_d = \frac{2 \times 7000}{0.06} = 233 \text{ kHz}\]

Phase history creates azimuth signal.

Summary

Synthetic Aperture Radar achieves fine resolution through coherent processing of multiple pulses
Range resolution determined by bandwidth while azimuth resolution equals half antenna length
All-weather capability enables imaging through clouds, rain, and darkness
Backscatter coefficient reveals surface roughness and material properties
Polarimetry distinguishes surface, double-bounce, and volume scattering mechanisms
Applications span flood mapping, ship detection, agriculture monitoring, disaster response
Speckle noise inherent to coherent imaging requires multi-looking or filtering
Geometric distortions (layover, shadow) require DEM-based correction
SAR interferometry enables millimeter-scale deformation measurement
Critical technology for operational Earth observation and emergency response