# Delay-Doppler maps in space science

Maybe this question might be more physics related, but I've read about mostly in context of space science:

What is a delay-doppler map (DDM)?

I can find it

I know about Doppler and also about delay in the sense of Time-of-Flight, so I only can imagine that a DDM has something to do with measuring both of them for GNSS reflectometry, although I have problems working it out. In the end it comes down, that I'm not sure how such a figure has been achieved and how to read: Delay Doppler map (DDM) from the CYGNSS mission

The DDM term is mostly used in the GNSS Reflectometry (GNSS-R) scope, but is conceptually equivalent to the radar ambiguity function. In fact, a GNSS-R system can be understood as a bistatic radar, that is to say, a configuration in which the transmitter (the GPS/Galileo/GLONASS/Beidou satellite) and the receiver (which receives the reflected signal) are different vehicles.

The bistatic scattering has its own complexities, but a GNSS-R instrument can operate as a radar altimeter, scatterometer, or even as a SAR radar [1]. People get often confused with the GNSS-R concept, but essentially the idea behind GNSS-R is to use the GNSS reflected signals as "a signals of opportunity". It is unrelated with the original navigation/location purposes of GNSS. In fact, in GNSS-R, the navigation messages are not demodulated, except maybe, in order to remove phase jumps due to data bit transitions.

The term DDM comes from the fact that reflections are mapped into a delay-Doppler plane. Reflections over different areas but a the same distance to the receiver and with the same Doppler shift, are mapped into the same "pixel" in the DDM. This is why in radar this function is called ambiguity function, because in principle, it is not possible to locate the reflection in the spatial domain, except for the specular reflection. Although there are methods to resolve or to reduce this ambiguity.

[1] V. U. Zavorotny, S. Gleason, E. Cardellach, and A. Camps, “Tutorial on Remote Sensing Using GNSS Bistatic Radar of Opportunity,” Geoscience and Remote Sensing Magazine, IEEE, vol. 2, no. 4, pp. 8–45, dec 2014 https://ieeexplore.ieee.org/document/6985926

• +1 There are plenty more questions tagged delay-doppler, several still have no accepted answers and there's always room for additional answers. – uhoh Sep 23 at 16:11
• @uhoh I procrastinated too much already today... But I will have have a look to other questions soon. – danipascual Sep 23 at 16:16

The DDM measures both the delay that a reflected signal takes to get back to you and the doppler (frequency) shift of that signal.

In your left most figure, what you are looking at is a picture of how a GPS signal reflected off of the ocean. The bright red spot is the point of "maximum specular reflection", that is, the point directly underneath the spacecraft that also happens to be the most reflective.

As the ocean waves make the ocean more...wavy...the reflection becomes less and less specular. Wind speeds also have something to do with it, among other effects that change both the frequency and the amount of signal reflected, and in which directions.

Since we know how the signal propagates (through an atmosphere or through space), we know the orbital parameters and transmitting powers/receiving antenna properties of the satellites generating/receiving we can create a theoretical model for what the surface should reflect like, given the unknowns like wind speed, wave height etc. (in asteroids, it would be things like albedo and surface roughness). One can then, through a LOT of signal processing, find those unknown parameters that best fit the reflected actual data. This can be used then to infer surface roughness (and really whatever else, like wind speed, that you have a model for).