On the top picture why is the second rainbows colors reversed? I have observed a rainbows that was huge from behind the horizon that began vertically and the small ones made with a sprayer hose. I cannot manage to make a double rainbow with a sprayer hose. The farther ones relative to the objects on the ground and in the air in front of it appeared bigger. Why do rainbows appear smaller the closer it is? The size of the raindrop makes a difference, but how? The last picture is a fog bow and notice the absence in color.
Rainbows are produced by sunlight entering a raindrop or other drop of water in the atmosphere. When it enters, the light is first refracted (bent) due to entering a denser medium. Some of the light exits the back of the drop but some is also reflected off the back surface of the drop. Upon re-encountering the front face of the drop, much of the light exits the drop where it is again refracted. The angle that it makes from the sun to the drop to your eye is a consequence of the angle of refraction. This, in turn, is related to the difference in velocity of the light in water as compared to air. Different colors undergo difference velocity changes in the water and so the angle of refraction is different for these different colors. In this manner, the raindrop acts much like a prism.
For a particular color, the sun-drop-eye angle is always the same. It doesn't matter whether the drops are 1 meter away, as in a lawn sprinkler, or 1 kilometer away, as in a rainbow, this angle is fixed for each color. For example, you will see red when the sun-drop-eye angle is 42°. Since this angle is different for different colors, you have to look in a different portion of the sky to see different colors. Other drops are also reflecting/refracting red but your eye is not in a position (correct angle) to see it.
Your upper images shows a double rainbow. The upper portion is produced from a second reflection inside the drop. Here's a diagram: http://www.nilvalls.com/supernumerary-rainbow/ Search "double rainbow diagram" for many other diagrams.
Image Source : Double Rainbows
The size of the drops makes no difference, (they are all equal in size) hopefully the diagram showing the path from the light refracted by the drop to your eye might help.
The raindrops act as a prism, reflecting the light at a different angle for each frequency. Part of that light also "bounces" a second time twice on the surface of the droplet, and this is what produces second (and much fainter) second rainbow
Since it bounces twice, the colours are reversed on the fainter secondary.
From: Wikipedia Rainbows
Secondary rainbows are caused by a double reflection of sunlight inside the raindrops, and are centred on the sun itself. They are about 127° (violet) to 130° (red) wide. Since this is more than 90°, they are seen on the same side of the sky as the primary rainbow, about 10° above it at apparent angles of 50–53°. As a result of the "inside" of the secondary bow being "up" to the observer, the colours appear reversed compared to the primary bow. The secondary rainbow is fainter than the primary because more light escapes from two reflections compared to one and because the rainbow itself is spread over a greater area of the sky. Each rainbow reflects white light inside its coloured bands, but that is "down" for the primary and "up" for the secondary. The dark area of unlit sky lying between the primary and secondary bows is called Alexander's band, after Alexander of Aphrodisias who first described it.