

If our line of sight is crossed by a keel of width w at an angle of incidence of d, then the 

 apparent width of the keel is sec 6. The probability of crossing a randomly oriented keel at 

 angle d must He derived. 



This probability is proportional to cos d. By symmetry, 6 can be confined to the 

 interval from to 90 degrees. The probability distribution function is therefore cos 6, O<0<9O 

 degrees and zero elsewhere. The cumulative distribution function of d is therefore sin 6 over the 

 same interval. A random value of d from this distribution can now be picked by taking arcsin 

 (x) where x is uniformly random from to 1. The desired width of this randomly selected keel, 

 W, is then * sec [arcsinf.*)] or 



W - Ml -x*) 



2\- x A 



(1) 



This foi mula is repeated on Fig. 2. 



Figure 3 shows three sample keel sets that were used to estimate scattering. They will 

 be identified in the :• mainder of this report as sets A, B, and C Sigma indicated or. each one is 

 the standard deviation of ice depth in meters. As can be seen, there is considerable variation in 

 this parameter, although the three sets were generated by selecting random values from the 

 iame distributions. If the three sets are combined into one long set. it has a sigma of 2.1 m, or 

 nearly average lor observed Arctic ice roughness. 



10 



5 - 



: V ' U 



i 



If v 



« = 260m 



8 n 



i/v 



V 



y 



a - 1 49 m 





y 



y 



o = t60m 



SO 



500 



750 



1000 



1250 



R»ng«(ml 



Figjre 3 Thre« keel sets generated by random numbers The 

 standard deviation of ice depth is given on each set 



