Modal wave front estimation is reviewed. A set of functions are derived which are orthogonal to the gradients of the modal functions, such as the Zernike polynomials. These functions can be used to project out any mode without cross coupling. Cross coupling is due to undermodeling and not due to a lack of orthogonality of the gradients. Cross coupling can be beneficial in that it provides a means of obtaining the best fit to a wave front with a given number of modes. Best fit estimation is distinguished from modal projection estimation.
W. H. Southwell,
"What's Wrong With Cross Coupling In Modal Wave-Front Estimation?", Proc. SPIE 0365, Adaptive Optics Systems and Technology, (23 August 1983); doi: 10.1117/12.934204; https://doi.org/10.1117/12.934204