The Science Behind SAR

For an antenna of size L, receiving radio-waves of wavelength λ, it is impossible to get an angular resolution better than the diffraction limit:

In this formula, β is measured in radians. That means that at a distance S, the spatial resolution will be equal to:

There are multiple ways to make our resolution smaller:

1. Increase L
2. Decrease S
3. Decrease λ.

We cannot decrease λ too much because then we will get into infrared waves that do not pass through the atmosphere freely. We cannot decrease S greatly because the altitude of the plane cannot fall below 152 meters. In addition, a large S value is preferential for a larger field of view. Building too large antennas (to increase L) is impractical. For example, if we assume an altitude of H = 800km, a look angle θ = 30, a wavelength of λ = 0.03m, we find that an unreasonable antenna length of about L = 800m would be needed to achieve a ∆x = 60m resolution from space (See Fig. 1 and Fig. 2).

Instead of using a large antenna, SAR uses the motion of the radar to different locations(See Fig. 3). In other words, it synthesizes a large antenna with its motion. Hence, the name Synthetic Aperture radar.

In an electromagnetic wave, the electric field and the magnetic field oscillate. Polarization specifies in which direction the field oscillates. The SAR has three kinds of polarization(See Fig. 4, Fig. 5, Fig.6):

1. linear
2. circular
3. elliptical.

A linearly polarized wave is a wave in which the electric field oscillates along a line- vertically polarized, horizontally polarized, or wave polarized at angle θ. Linear polarization has a polarization angle. A circularly polarized wave rotates at a constant rate around the direction of propagation. The wave is left-handed(counterclockwise) if we look from the propagation direction, and right-handed (clockwise) if we look from the direction of propagation (from the viewpoint of the receiver). Elliptical polarization is similar to circular polarization, but in the shape of an ellipse. Elliptical polarization has these three properties: polarization angle, right/left-handness, and eccentricity.

Example: Grass is a conductor. It is also one of the most prominent anisotropic materials found in nature. Examples of anisotropy: Specific direction of the grass, other plants, artificial objects (light posts, wires, etc.). If we shine a polarized light at it, the grass' particles will react differently to vertically/horizontally polarized light. A vertically polarized wave will make changes oscillate along the grass. Hence, the grass reflects the vertically polarized wave very efficiently. On the other hand, a horizontally polarized wave makes the charges oscillate perpendicular to the grass. Hence, the grass reflects the horizontally polarized light non-efficiently. If we shine a circularly polarized microwave on the grass, the reflected microwaves will form an elliptically polarized light.

In Conclusion: The grass will reflect vertically polarized microwaves much more efficiently. The same consideration applies to any kind of anisotropic material. Since the wavelength range for SAR is around 3 cm-few meters, we are interested in anisotropy at these scales.

HH, HV, VH, and VV are types of radar polarizations used in remote sensing(See Fig. 7). HH means the radar signal is transmitted and received in horizontal polarization, which is good for detecting surface roughness, such as soil or ocean waves. VV means the signal is transmitted and received in vertical polarization, often used for studying smoother surfaces like calm water or flat terrain. HV means the signal is transmitted horizontally but received vertically, while VH is the opposite—transmitted vertically and received horizontally. These cross-polarizations (HV and VH) are especially sensitive to complex structures like vegetation, where the radar signal is scattered in many directions, and they help identify differences between natural surfaces and man-made objects.