Currently, there is no airborne in situ method to reconstruct with high fidelity the instantaneous elevation of a dynamically rough surface of a turbulent flow. This work proposes a holographic method that reconstructs the elevation of a one-dimensional rough water surface from airborne acoustic pressure data. This method can be implemented practically using an array of microphones deployed over a dynamically rough surface or using a single microphone which is traversed above the surface at a speed that is much higher than the phase velocity of the roughness pattern. In this work, the theory is validated using synthetic data calculated with the Kirchhoff approximation and a finite difference time domain method over a number of measured surface roughness patterns. The proposed method is able to reconstruct the surface elevation with a sub-millimeter accuracy and over a representatively large area of the surface. Since it has been previously shown that the surface roughness pattern reflects accurately the underlying hydraulic processes in open channel flow [e.g., Horoshenkov, Nichols, Tait, and Maximov, J. Geophys. Res. 118(3), 1864-1876 (2013)], the proposed method paves the way for the development of non-invasive instrumentation for flow mapping and characterization that are based on the acoustic holography principle.