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You can refer to the available literature on the topic. The papers I am aware of are:

Ray Tracing in Non-Constant Media

Interactive Rendering of Non-Constant, Refractive Media Using the Ray Equations of Gradient-Index Optics

and

Refractive radiative transfer equation

The general algorithm for handling a varying index of refraction is ray-marching, where the size of the steps is related to the accuracy of your solution.