I'm new in dealing with compression noise in post, I have been able to reduce it using internal or 3rd party plug-ins. However, I was not able to get rid of those wavy looking artifacts, they become more visible after colour correction and grading.
Here is an example, look at the sky how sharp the gradient layers are:
Please help me in identifying this kind of noise and how I can remove it.
Think of a gradient or feathered vignette like a ramp -- a smooth surface rising from low to high over a certain run. However, computers, being digital instead of analog, don't do smooth. Think of a computer's representation of that gradient as a staircase, with a series of discrete steps from low to high over that same distance.
If you use only a few steps, they must be very deep to get you all the way across the run, and they must be very steep you get you all the way up the rise. As a result, you get big bumps going from one deep and steep step to another.
If you use many, many steps, they must each be shallower to fit within the run and shorter to fit within the height. A great many very shallow, short steps approximates a smooth ramp.
"Bit depth" or "color depth" is the number of steps. Because of the way computers count, using 8 bits gets you only 256 separate shades of gray between black and white, but using 16 bits gets you 32,768 separate shades of gray.
Practically speaking, just bumping your project's color depth from 8bpc to 16bpc should improve the banding, as AE will then attempt to dither the gradient (adding noise to smooth it out) when outputting to an 8-bit format.
You may also manually add some noise. GenArts Sapphire has S_Deband, which tries to intelligently dither banded areas.