2 

 where y is the arithmetic mean of n and o is the variance. Intui- 

 tively n is as likely to be positive as negative, so let us assume that 

 Prob (n(xQ, Yq, t^) < 0) = 1/2. Since n is Gaussian distributed, and 

 is thus synmaetric about its mean, we have Prob(ri(xo, yo> to) < p) = 1/2 

 or that M ■ 0. For a^ , we have (using expected value notation) 



whoio we ;u-Q thinking of n as a random variable. 



A Gaussian process is completely defined statistically if we know 

 the toni> of the mean 



K(n(x,y,t)) and the covariances 



where X, Y, and T are space and time separations, respectively. Refer 

 to Parzen (1962 - pages 88-89). We have assumed E(n(x,y,t)) = y = and 

 that the process is stationary (only weakly stationary is necessary) . 

 Hence, by definition of weak stationarity, we have for each (x,y,t), 

 and yet independent of the particular x,y,t values, the covariance form 



R(xXT)sE[n(^,v»,t)Y^(^-.X,ii+Y>t+T)] 



(3.3) 



All of the properties of the stationary Gaussian process ri(x,y,t) are 

 implicit in R(X,Y,T), just as a knowledge of y and a^ for a single 

 Gaussian random variable completely defines such a random variable. 

 Here it is important to understand that we are discussing expected 

 values across all possible realizations at a point (x,y,t); i.e., 

 across the ensemble of all possible sea wave shapes at (x,y,t) for a 

 given S(i!,,m,f ) . 



There is a simple and unique relationship between R(x,y,t) and 

 S(J!,,m,f). Consider a single real wave element (from Equation (2.16) and 

 (3.2)) as a random process and write n(x,y,t) = [EXP(i2T7(£xH-my+ft+a)) 



