To an observer standing at the side of a swimming pool, objects under the water appear to be nearer the surface than they really are. A similar effect can be seen when "looking through" glass or any other transparent substance.

A fish appears to be nearer the surface than it really is:

A straight stick appears to be bent when part in and part out of water:

Both these effects are caused by refraction of light at the surface of the water. Therefore the magnitude of the effect is related to the refractive index of the media involved.

The real depth of the fish is R and its apparent depth is A. It is clear that

If i and r are small angles then sini = tani and sinr = tanr.

and if n represents the refractive index of light going from air to water, we have

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