What are the physical quantities in a SAR image?

When and Why to Calibrate in a Training Database?

8 min readAug 24, 2024

The Two Key Quantities

To fully exploit radar images, it is essential to understand the physical quantities they measure and how these measurements are calibrated. Two key concepts in this context are the Radar Cross Section (RCS) and the backscatter coefficient.

Radar Cross Section (RCS)

The Radar Cross Section, often referred to by its acronym RCS ( SER for ‘Surface Équivalente Radar’ in French), is a measure of the amount of radar energy reflected by a target back to the radar sensor. In simple terms, the RCS quantifies an object’s ability to reflect incident radar waves. It is expressed in square meters (m²) and can vary significantly depending on the target’s size, shape, composition, and orientation.

Definition: The RCS is defined as the area of a perfect sphere that would reflect the same amount of energy back to the radar as the observed target. More formally, if Pr is the power of the echo received by the radar and Pi is the power of the incident wave, the RCS (σ) is given by:

where R is the distance between the radar and the target. A high RCS indicates that the target reflects radar waves well, while a low RCS means that the target scatters little radar energy

Backscatter coefficient

The backscatter coefficient, denoted by σ⁰ (sigma-nought), is a normalized measure of the radar echo power returned by a unit area of the target’s surface. Unlike the RCS, which is a global measure for an entire target, the backscatter coefficient is a localized measure that describes how a small portion of the target surface reflects radar waves.

Definition: The backscatter coefficient is defined as the ratio of the radar echo power returned by a unit area of the target to the incident wave power per unit area, normalized by the ground area of that unit. It is often expressed in decibels (dB) for convenience, and the mathematical relationship is

σ⁰=σ/A

where σ is the RCS of the small portion of the surface and A is the area of that portion. In terms of the signal received by the radar, this can be expressed as:

σ⁰(dB) = 10 log⁡10 (σ/A)

The backscatter coefficient allows for the comparison of different surfaces independently of their size, which is particularly useful for detailed analyses of surface characteristics.

Other calibration coefficients: β⁰, γ⁰

The backscatter coefficient β⁰ (or beta-nought) is the ratio of the radar echo power returned by a unit area perpendicular to the radar beam to the incident wave power. Unlike σ⁰, β⁰ does not take into account the ground geometry but focuses on a surface perpendicular to the radar beam, Aβ.

The backscatter coefficient γ⁰ (or ‘gamma-nought’) is the ratio of the radar echo power returned by a unit area on the ground to the incident wave power, corrected for the radar’s incidence angle. In other words, γ⁰ normalizes the echo power by the area projected perpendicularly to the radar beam, Aγ, taking into account the incidence angle.

The coefficient β⁰ is useful for analyses focused on the reflected power independently of the ground area, particularly in applications where the incidence angle may vary significantly. The coefficient γ⁰ is especially useful for analyses where the geometry of the scene plays an important role. Accounting for the incidence angle allows for the comparison of the backscattering properties of surfaces with a correction for the acquisition geometry.

From: D. K. Atwood, D. Small and R. Gens, “Improving PolSAR Land Cover Classification With Radiometric Correction of the Coherency Matrix,” in IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 5, no. 3, pp. 848–856, June 2012, doi: 10.1109/JSTARS.2012.2186791.

Calibration of Radar Images

Why calibrating ?

Calibrating radar images is a crucial process to enable the extraction of physical measurements from the image. Calibration ensures that radar data can be used for quantitative comparisons, making it essential if one wishes to compare multiple images:

Within a time series
Between two sensors.

For example, imagine you want to mix images from TerraSAR-X with UMBRA data in your training dataset. It might be useful to calibrate them beforehand to ensure the correspondence between the two images.

This involves correcting the radar data for instrumental and environmental effects so that the obtained values accurately reflect the physical properties of the observed targets.

Calibration procedures include:

Radiometric Calibration using Reference Targets: To adjust the received power values based on the transmitted power and losses along the path, often using reference targets (typically trihedrals), which have known RCS, to validate and adjust the measurements.
Antenna Pattern Correction: To compensate for variations in radar antenna sensitivity, ensuring uniformity of values across the entire image.

In practice ?

The use of dB

The use of decibels does not correspond to a change of unit, but simply a change of scale: to handle large dynamic ranges, we switch to a logarithmic scale. If I consider the RCS, which is homogeneous to an area, I can convert an area to dB. In this case, the unit is noted as dB m². If I consider a backscatter coefficient, which is dimensionless, I can also convert it to dB. The ‘dB’ indicates a scale (logarithmic), not a physical dimension!

When should I use 10 log10 or 20 log10 to convert the value to dB?

Typically, we use 10 log10 when we want to convert a physical quantity that is homogeneous to a power (or intensity, or the square of amplitude, or σ⁰, RCS…), and 20 log10 when converting a physical quantity that is homogeneous to a complex field amplitude! In practice, in a complex SLC image, if I consider a complex quantity Z representing the electric field, then |Z| = A is homogeneous to an amplitude, |Z|² = A² = I is homogeneous to an intensity.

💡 10 log10(I) = 10 log10(A²) = 20 log10(A), it’s like magic!

I sincerely apologize to those who find this obvious. But every time I explain this concept to a group of students, there are always a few who suddenly have a spark of understanding — and gratitude — in their eyes… ‘aaaaaaaah’… So, just for them, I’m writing it out again here.

Usage and order of magnitudes

In practice, if I am interested in targets such as planes or ships, I focus on bright points and RCS values. If I am interested in natural environments, I focus on backscatter coefficients.

The Radar Cross Section (RCS) is highly dependent on factors such as frequency, incidence angle, and target characteristics. At X-band (10 GHz), typical RCS values are:

Car: between 1 m² and 10 m².
Small Boat (like a yacht): between 10 m² and 100 m².
Large Boat (like a cargo ship): between 100 m² and 1000 m².
Airplane: for a fighter jet, between 1 m² and 10 m²; for a commercial airplane, between 10 m² and 100 m².

For backscatter coefficients (σ⁰), which are also frequency-dependent:

Forests at P-band (300 MHz — 1 GHz): between −10 dB and −20 dB.
Forests at L-band (1 GHz — 2 GHz): between −5 dB and −12 dB.
Forests at X-band (8 GHz — 12 GHz): between −3 dB and −8 dB.
Lake or Calm Water Surface at C-band (5 GHz): Very low, typically between −20 dB and −30 dB.
City or Urban Area at C-band (5 GHz): Typical values between 0 dB and +10 dB.

How can I extract physical measurements from a SAR image?

For Sentinel-1, the sin(θ) factor is explicitly used to normalize the intensity based on the incidence angle. For TerraSAR-X, this correction is often implicitly included in the provided calibration coefficients, thus simplifying the formula for end users.

I retrieve the calibration key K from the metadata. If my data is intensity data I, I calculate K⋅I. If the data is from Sentinel-1, UMBRA, etc., you need to further multiply the result K⋅I by sin⁡(θ), where θ is the incidence angle. This is not the case for TSX, which has included this component in the calibration key!
If the data is complex, I can be derived as I= |A|². To calculate the RCS (σ) from the same data, if you know the area (A) of the reflective surface corresponding to each pixel, you can then obtain the RCS using the relation σ=σ⁰⋅A.

Will the values be preserved if I resample my image?

Where the comparison of images between sensors becomes challenging is that, in addition to using different calibration keys, the images are often sampled at different resolutions. Therefore, we need to resample some of them. But will this process preserve my calibration?

⚠️It is important to know that there are two possible, yet incompatible, conventions:

Either you want to keep a constant instantaneous value in a pixel when changing the sampling rate. This is relevant if you are more interested in RCS (Radar Cross Section), meaning the amplitude or intensity values of bright points.
Or you want to maintain a constant average value over a speckle area when changing the sampling rate. This approach is used when focusing on sigma° (sigma-naught) coefficients, i.e., when the natural areas are of interest.

In all cases, you need to think in terms of the spectrum’s energy content. The energy of the spectrum is carried by its area.
Let Z be your SLC (complex) signal Image. Let F be the spectrum of Z, that means the bidimensional Fourier transform of Z.

Let f be the factor between the initial area and the final area of the spectrum. For example, if my spectrum F has 100x100 meaningful pixels, (by meaningful, I mean the spectrum is full), and I select a 20x20 sub-section of the spectrum, the factor f is f=(20x20)/(100x100).

If I want to keep the RCS constant, then after selecting a sub-section of my spectrum, I’ll divide the amplitude obtained by f. (and therefore the intensity by f²). In the example below, the initial bright spot has a maximum value of 1. After setting part of the spectrum to zero and applying the inverse Fourier transform, the resulting bright spot becomes more spread out. However, the maximum value of the main lobe remains 1 if we divide the amplitude by a factor of f.

If I want to preserve the average value of my backscatter coefficient, I must divide the final intensity by f (and therefore the amplitude by √f). In this new example, the initial image represents a fully developed speckle pattern, as does its spectrum. After setting part of the spectrum to zero and applying the inverse Fourier transform, the speckle grains become larger. However, the mean intensity remains unchanged if the intensity is divided by f.

It is crucial to be mindful of the convention you are using to ensure the accurate interpretation of physical values.

Summary:
There are two primary physical quantities in SAR images: the RCS, used to describe a bright point, and sigma nought, used to characterize a natural surface or volume.
To compare images in terms of physical values, it is necessary to apply appropriate calibration keys. However, calibration conventions can vary: some take the incidence angle into account, while others do not.
If, after calibration, spatial resampling of the images is considered, it is crucial to account for potential changes in the size of the spectral support. A choice of convention must then be made: should local bright point values or speckle backscatter averages be preserved?