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In this work, we obtain the hydrostatic approximation by taking the smallaspect ratio limit to the Navier-Stokes equations. The aspect ratio (the ratioof the depth to horizontal width) is a geometrical constraint in general largescale geophysical motions that the vertical scale is significantly smaller thanhorizontal. We use the versatile relative entropy inequality to proverigorously the limit from the compressible Navier-Stokes equations to thecompressible Primitive Equations. This is the first work to use relativeentropy inequality for proving hydrostatic approximation and derive thecompressible Primitive Equations.

On the hydrostatic approximation of compressible anisotropic Navier-Stokes equations -- rigorous justification