SECT. 5] ANALYSIS AND STATISTICS 577 



types of measurement of the surface, we may deduce values for some of these 

 moments, or groups of moments, which may themselves give sufficient informa- 

 tion. In theory the complete set of moments of a function determines the 

 function uniquely, and Longuet-Higgins (1957), in fact, suggests a method 

 of obtaining an approximation to E"{u, v) by estimation of m^pq and rripq up to 

 finite order. However, this method does not seem very practicable, since high- 

 order moments are sensitive to instrumental noise, and the most practical 

 methods are those based on estimation of either the autocovariance function 

 or the angular harmonies of E{k, 6). The autocovariance can be defined by 



i/j{X, Y, r) = Ux, y, mx + X, y+ 7, t + r), (18) 



the bar denoting a mean over all x, y and t, although averaging over t alone 

 with fixed values of x and y is also satisfactory. From (6) it can be deduced 

 that ^(X, Y, t) is also expressible in the form 



Too /*« 



ijiiX, Y, t) = E"{u, v) cos {uX + vY- or) du dv, 



J-co J-co 



(19) 



as is well known. 



In some types of measurement (for example, Cote et al., 1960) only instan- 

 taneous values of i,{x, y, t) can be measured, from which one can evaluate 

 j/((Jl, Y, 0) by averaging over many values of a; and y. From this can be obtained 

 the even function F{u,v) = \{E"{u,v)-\-E"{ — u, —v)] by means of a Fourier 

 transform (Khintchine, 1934) : 



1 /'oo /^oo 



F{u, v) = -—\ MX, Y, 0) cos {uX + vY) dXdY. 



477'^ j-oo j-ao 



This may be sufficient if it can be assumed that E"{u, v) is zero over two 

 adjacent quadrants (that is, all the wave energy is being propagated within 

 + 90° of a single direction), but for a complete estimation of E"{u, v) we also 

 need the odd function G{u, v) = \{E"{u, v) — E"{ — u, — v)]. This may be obtained 

 by using appropriate values of t : 



G{u, v) = -i- P I i/r(X, Y, TTJ'Ia) sin {uX + vY) dX dY , (a - aiu, v)), 



477^ j-00 j 



or by measuring 8^1 dt and forming the covariance function 



0'(X, Y, 0) = U^, y, t){clctK{x + X, y+ Y, t) 



= g{u, v) E"{u, v) sin {uX-^vY) du dv, 



from which 



G{u,v) = -—4 : r U'iX, Y, 0) sin {uX + vY)dXdY, 



47T^a{u, V) j-00 j ^ 



since cr( — w, —v) = a{u, v). We then have, of course, 



E"{u, v) = F{u, v) + G{u, v). 



