Subhorizon Halos ~ Imaged near Stratford, Texas by Dean A Ketelsen (blog). At right is a colourless subsun and at left a prismatic subparhelion. A hint of subparhelic circle trailis away from the subparhelion. A thin layer if ice crystas beneath the aircraft formed the halos. ©Dean A Ketelsen, shown with permission

Odd or even?

Whether a sun ray forms a sundog or subparhelion depends on a count of internal reflections.  

Sundogs or 22 degree parhelia are said to arise from rays entering a side face of a horizontal plate ice crystal in cirrus cloud and leaving through a second side face inclined 60 degrees to the first.    That is, like the curate's egg, true in parts. Unless the sun is low on the horizon or the plate is unusually thick events are not quite that straightforward.  

Instead, once inside the ice, most rays intercept the lower crystal face and internally reflect.   They can reflect up and down between the upper and lower large faces several times before eventually leaving through a side face.

An even number of internal reflection gives a sundog above the horizon.   An odd number produces a subparhelion.

Subhorizon halos are, like their above horizon counterparts, produced by each sun ray interacting only once with a crystal. They are not halos produced from the light of other halos.

A HaloSim computer ray tracing looking straight downwards. It shows subhorizon halos that might result from a layer of plate crystals beneath you. All have been observed:

subparhelic circle

sub 120° parhelion

sub-Liljequist parhelia

sub CZA - circumnadiral arc

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They can be blindingly bright. Almost as though the sun is reflected by a mirror-like lake. There is not one mirror but millions.

Plate crystals are again responsible and sunlight reflects either externally from their top hexagonal faces or internally from the lower one.

The subsun is usually vertically elongated because the plates are wobbling as they drift in the cloud air currents. As the wobble increases the subsun transforms into a lower sun pillar. The ray paths are the same.