 Solution

For near-normal viewing through the flat top surface, the refracted ray appears to originate at an apparent depth $h_{\mathrm{app}}$. A ray at horizontal offset $x$ has $x=h\tan {\theta }_w=h_{\mathrm{app}}\tan {\theta }_a$, and Snell's law gives $n\sin {\theta }_w=\sin {\theta }_a$. In the paraxial limit,At an oblique viewing angle the exact ray-based apparent depth is $h\tan {\theta }_w/\tan {\theta }_a$; the simple ratio presumes the usual near-normal top view.