Stilwell 



f = J J cp2 dk^ dk^ = q"^ J J cp2 dxdy (18) 



Since the finite aperture utilized in the scene plane limits the 

 spatial frequency resolution^ an elemental area in the transform 

 plane (reciprocally related to the gaussian beam cross section in 

 the scene plane) has constant c£^ . Denoting that area by 



A = H^ ^ (^)2 (19) 



The integral is trivial and one obtains the photographic spectrum 



f = ^ $3 (20) 



The constant H expresses that factor of the gaussian half width 

 cr which is effective in the transform process. Further analysis 

 is reqiiired to obtain the actiial value of H but it is of the order 

 of /"S which is the eqirLvalent square pulsewidth of a gaussian 

 function. 



The expression thus far derived has not involved the 

 size of the sea photographed, only the photographic density 

 variation. It is possible to envision a situation in which a 

 sinusoidal wave of some fixed angular excursion propagating on the 

 surface could be photographed and result in a photo exactly 

 identical to a wave of different wavelength with the same angular 

 amplitude but photographed from a different height. Using 



0(k,t) = (2n)"^ |_ Tl(xo,t) Tl(xo+x,t) e'^^-'^dx (2l) 

 •^x 



from Kinsman^ one can determine that the spectrum is proportional 

 to the area analyzed. Thus the ratio of two spectra is propor- 

 tional to the ratio of the respective areas. Extending this 

 analogy to the photographic spectrum one can write that the sea 

 spectrum is proportional to the ratio of the sea area to photo- 

 graph area multiplied by the photographic spectrum. Or then 



= c^ P^ f ' (22) 



where p is linear reduction factor from the sea to photographic 



177 



