Explain Mirage formation?

Mirage

In hot deserts the sands become quickly hot and thus the layer of air very near to the sand becomes less dense. The other layers are successively denser upwards. As we know refractive index of a material varies with its density, thus rarer layer of air has the lowest refractive index and refractive index will increase successively as we go up. Thus when a ray of light from any object (like a tree in the fig) passes through these layers, because of change in refractive index rays suffer refraction. Since in this case rays have to pass from denser to rarer medium, they deviate away from the normal. After passing through a lot of layers successively, when the ray reaches an incident angle beyond the critical angle for the medium. It suffers total internal reflection and traces back its path from rarer to denser layers, thus moving towards the normal. When these rays fall in the eyes of a traveler in desert he sees an inverted image on the ground. Because of the shimmering of the image thus formed, he may think it were a lake. This is mirage.