# How are radar returns sorted in range space in Synthetic Aperture Radar?

I'm trying to understand how a SAR system is able to place returns it receives in range space.

Consider the scenario when an airplane is passing over terrain and is imaging it using SAR, as shown in this image from radartutorial.eu:

If it relies on the timing of the return, it seems that there would be ambiguous cases where a tall object further in the swath could have returns that are received earlier than a short object in the near swath. How would a SAR system ever sort out the elevation of the terrain being scanned in the swath?

From what I've found it doesn't seem like there is any sort of servo tracking the beam along the swath width (and it would have to be moving quite fast for this to work). This leads me to think that there must be some signal processing wizardy used to sort these returns that prevents timing ambiguities. What is going on?

As you've mentioned, all radar works by measuring the TOA (Time of Arrival) of a return pulse that has reflected back from an object. However, SAR uses a lot of special signal processing to compensate for the fact that the actual antenna is much smaller than the synthetic aperture.

The movement of the physical antenna from the time it emits an incident pulse to the time it receives a reflected pulse is what makes the synthetic aperture. So, among other things, the doppler effect of that movement on the frequency of the pulse and the fact that different frequency components of the incident pulse get reflected back to varying degrees (group delay) are all measured and compensated.

In addition, modern SAR is focused rather than unfocused. That is, similar to lenses in optics, phase adjustments of components of the radar return can be processed to effectively focus the pattern and increase the resolution.

I'm no expert by any means, but those are the basics. If you want to get a little deeper into it (with all kinds of lovely calculus equations), you might enjoy reading this book chapter specifically on that topic. That document is a couple of decades old, but the principles are the same.

If a picture is worth a thousand words, as the old saying goes, http://www.intro2radar.com/ is worth many, many thousands because it has not only pictures but animations. You may find the "ranging imaging" animation particularly helpful to answer your question. It's also worth pointing out that the animations are intended to complement the excellent book, Introduction to Microwave Remote Sensing by Dr. Iain H. Woodhouse.

• Thanks for the response and the link. But I still fear I'm a bit lost: what exactly is allowing range discrimination? Am I correct in my understanding that SAR is scanning perfectly perpandicular to the aircraft? If so, wouldn't we expect no doppler shift in the returned signal? Oct 21, 2014 at 19:49
• I've added another paragraph to my answer that may help answer your range question. Also remember that whenever there is relative motion between sender and receiver, there will be a doppler shift. Oct 21, 2014 at 20:31
• Start your SAR thinking assuming that the airplane is a helicopter with an omnidirectional antenna. Transmit a pulse, collect echoes, move to next position. Later, you can synthesise an aperture. Movement, directional antennas and phased arrays are all refinements on the basic idea. Jan 6, 2015 at 15:11

A long time ago i used to work on SAR imagery (without much understanding its principles).

As far as I understand there is no wizardy used to sort returns from higher object: In my case I was trying to estimate object's heights from dimensions of "shadows".

BTW a similar situation can take place in optical imagery too: imagine that you are illuminating obgect along one direction, but taking pictures from far away in perpendicular direction. You will have exactly the same situation.