filter used in the one-dimensional case as shown by the following scheme 

 where the product of the two one-dimensional smoothing filters give the filter 

 values over a square grid of nine points. 



Table 8.1. The Smoothing Filter 

 0. 23 0, 54 0. 23 



0.23 0.053 0.124 0.053 



0. 54 0„ 124 0. 292 0. 124 



0. 23 0.053 0. 124 0.053 



The smoothed spectral estimates are finally obtained frona equation (8.23), 

 (8.22) U(r, s) - 0.053[L,(r+l, s+1) + L(r+1, s-1) + L(r-1, s+1) + L(r-1, s-1)] 

 + 0.124[L{r, s+1) + L(r, s-1) + L(r+1> s) + L(r-1, s)] 

 + 0.292[L(r, s)] 

 where again U(r, s) = U(-r, -s) and r = 1, 2, , . . . , 20 



s = -20, ..... +20 . 

 The U's are esl.'mates of that contribution to the total variance of tl).e sea sur- 

 face made by wa^■es with frequency componerats between 2T7(r -— )/<!OAx and 



2ir(r + — )/40 Ax in the r direction, and between 2tt(s - -i) /40 Ax and 

 ^ 2 



1 + 



2it(s +— )/40Ax in the s direction. 



One difficulty with estimating power spectral by these techniques is ^hat 



the operations on the original data described by the above equations do not 



guarantee that the spectral estimates will be positive, and yet in the theory 



they should he. This is because a term of the form 



sin cX sin (3Y 

 X ' Y 



operates on the spectrum in the complete derivation when X and Y are kept 



^ 



Except at the borders— see Part 11. 70 



