four equations, 



-^„Y„. + I a'-^Y^^ 



g-gi 



j=l 



g: 



, 



(i,g = 1,2) , 



(there are an infinite number of possible choices) . Define the numbers 

 C . by 



gi 



gi 



gi j^i gD 



-1/2 



and let 



= y C . (£./a.) 

 .^^ gi 1 1 



i=l 



Here, w^ and W2 are independent and Ew^ = EW2 = 

 2 2 



Then, 



y A. .{l./a.) (i./o.) = y wVx . 

 1,3 = 1 -^ g=l ^ ^ 



It is easily verified that the variance w //x~ is 



g g 



V = y C2./X , 



i=l 



gi g 



(g = 1,2) 



Let t equal w /A v 



g g g g 



Then, 



y A. .(a. /a.) (a./a.) = y V t2 , 



1,3=1 -" -^ g=l ^ ^ 



where t]^ , t2 are independent and standardized normal. Thus, 



11 



■2^2 



.2 



'2^1 



2 

 'l'^2 ^ 



approximately equals the probability that the submarine position is within 

 a distance R (on the ocean surface) from the estimated position (X,Y) . 



The procedure used to make these transformations is the method of 

 principal components (for example, see ref . 1) . 



Evaluation of Probability . 



Let V = min(vi , V2) and V = max(vi , V2)/v . The probability to be 

 approximately evaluated can be expressed in the form 



P(ti + Vt2 <. rVv) . 



One way of approximately evaluating probabilities of this form is by the 

 method of ref. 2. Specifically, the cumulative distribution function of 

 t2 + vt| can be expressed as 



y H c 



w=0 



w 2w+2 



454 



(X) , 



