Also late to the post: but i think the only way to solve this, is by some kind of aproximation. I implemented a working aproximation for gerstner waves using newtonian style gradient decent with 4 iterations:

Your problem, generally formulated, is:

Given a position a,b - what are the coordinates x,y, such that

P(x,y,t)=a,b,c

-> this will result in c being the local hight for coordintes a and b.

I dont think, there is an analytic inversion of P derivalble, instead one can solve the minimization problem (for fixed t)

F(x,y)= P(x,y) - a,b = 0

Wich can iteratively aproximated, for example with:

x_1,y_1= F(x_0,y_0)/|det(J(F)(x_0,y_0))| - (x_0,y_0)

where J(F)(x,y) is the jacobian of F (wich is easily derivalble from the Gerstner Function definition)