So what's a "synthetic aperture radar?"

So what’s a synthetic aperture radar (SAR)? I should take a moment to emphasize that this is all based on the mathematics of combining different signals. It’s essentially “Digital Signal Processing” (DSP) but on reflected microwave signals not audio from a microphone or other source. 

Take a look at the antenna in the previous post again. Each one of those dark spots is a cut out slot in sheet metal, like the dual of a wire on a mounting plate that’s not conductive. A properly sized, non-conductive slot, in a conductive sheet becomes the antenna. I talk about this in a post on aircraft antennas (of all things). It's in the portion below the line about 3/4 the way down the post. Especially the links on slot antennas. 

Consider these slots being cut for 10 GHz.  A quarter wave (90 degrees) slot’s length can be found by (antenna length - in feet) = 234/(freq in MHz)

L = 234/10,000 MHz = 0.0234 feet or 0.2808 inch

so a quarter wave is 0.281 inch and each of those slots is positioned on the sheet so that the E- (electric) fields combine to produce the desired radiation pattern. You’ll note that in some parts of the (more or less) circular outline of the plate, particularly along the left and right sides (in the picture), the slots seem aligned so that any four slots look like a square, while closer to the center, they don’t look to be aligned that way. The squares look cattywampus, to use the southern word. 

Remember the radar is on a moving satellite, and since the satellite's pointing the radar down on the weather, it’s moving fast, but not toward or away from the weather – it’s moving at right angles to the ground on Earth. That’s an approximation, but good enough for now.

The motion of the radar source causes a Doppler shift, the change in returned frequencies higher as radar approaches the reflector and lower as the two bodies separate. It’s hard to catch this from all these mentions of speeds and distances involved, but the reflected wave reaches slots on the antenna different from the ones they were transmitted from because of the satellite's speed in orbit and the time for the radar to get to the ground and bounce back up to the antenna. "Radar reflections from discrete objects in a passing radar beam field each [have] a minute Doppler, or speed, shift relative to the antenna"  

Carl Wiley, working at Goodyear, Arizona, (which later became Goodyear Aerospace, and eventually Lockheed Martin Corporation) in 1951, suggested the principle that – because each object in the radar beam has a slightly different speed relative to the antenna – each object will have its own Doppler shift. A precise frequency analysis of the radar reflections will thus allow the construction of a detailed image

Colburn, Robert (10 March 2009). "Your Engineering Heritage: Synthetic Aperture Radar". IEEE USA InSight IEEE.

In order to realise this concept, electromagnetic waves are transmitted sequentially, the echoes are collected and the system electronics digitizes and stores the data for subsequent processing. As transmission and reception occur at different times, they map to different small positions. The well ordered combination of the received signals builds a virtual aperture that is much longer than the physical antenna width. That is the source of the term "synthetic aperture," giving it the property of an imaging radar.[5] The range direction is perpendicular to the flight track and perpendicular to the azimuth direction, which is also known as the along-track direction because it is in line with the position of the object within the antenna's field of view. 

An illustration of the aperture synthesized - each of the long rectangles is depicting one of the antenna slots in the antenna we show. The motion of the antennas is along the X direction, left to right.