Remote determination of the profiles of the atmospheric structure constant and wind velocity along a line-of-sight path by a statistical inversion procedure

A statistical inversion procedure is developed for remotely determining the average transverse wind velocity and the atmospheric structure constant at
N
arbitrary points along a line-of-sight path. Linear integral equations are given relating the amplitude correlation function and the amplitude and phase structure functions with the unknown structure constant and wind velocity. It is shown that the standard matrix inversion leads to large variations, as high as
10^{11}
percent errors, in the unknown for small data error of one percent; thus the problem is ill posed. The errors are reduced to a suitable level by a statistical procedure that is dependent upon a priori knowledge of the statistics of the unknowns. For the structure constant, a procedure developed by Franklin and others is used, while the wind velocity is determined by modifying the method to include the errors in the structure constant. Computer simulation is used to estimate the error. It is shown that, with an input error of one percent, the rms error in the unknown is only on the order of ten percent. To show the effectiveness of the method, experimental data obtained by a Stanford group are used to determine the wind velocity and the structure constant along the path, yielding reasonable results.