The effect of the random errors on Q can then be determined under 

 the assumption that the different types of errors are small and independent. 



<ll-3) Qpq = Qpq(true) + ^oq + %o + S^o 



where S^ = E(6j''')^ if p = for any q and is zero if p ;^ ; 



S = E(6j^') if q = for any p and is zero if q ^ ; 



S = E(6.t )^ if p = and q = and is zero if p ^ and q ?( . 

 The effect of a random error along a column of the data is thus to cause a 

 constant error to be added to every value on the vertical axis of the coordinate 

 system of the covariance surface; an error along a row adds a constant error 

 to each value on the horizontal axis; and a random error over the whole plane 

 is concentrated as a spike at the origin. 



The values of L,(r, s) can then be found from the values of Q(pq) 

 (11.4) L(r, s) = L(r. s)^^,,^^^) + W^^ + W^o + ^rs 



where W^g = -jq E(e.*)^ if r = for any s and zero if r ?< ; 



Wj.Q = -yFT' E(€j') if s = for any r and zero if s ^ ; «• 



1 =!< 2 



and W = — ^^- E(€., ) , for every value of r and s . 

 rs 800 ^jk' ' ^ 



Thus, random errors along columns in the original data show up as a 

 constant error along the horizontal coordinate axis in the Ij(r, s) plane; er- 

 rors along rows show up as a constant error along the vertical axis, and ran- 

 dom errors show up as a constant error at each point in the spectral plane. 

 Of course, since the data are really a finite sample, there will be fluctu- 

 ations from point to point in the Li(r, s) plane. 



167 



