538 



We can combine these measurements to get 



M^{d) + M^(d) = 21^(6) +1^(6) +1^(6). (12) 



If we could find the doubly symmetric component, I^, we could find the value of 



M^(0) + M^(e) - /,,(9) = /,,(0) +/,„(0) + I,,(e). (13) 



In fact, the value of the doubly symmetric component, I^^, is implicit in either of the sets of measure- 

 ments, M^ or M^, thus: 



4/55(6) = 2M^(9) + 2Af^(7r -6) = 2^2(9) + 2M2(-e) 



= M^(0) + Mj(7r - 0) + M^{9) + M^i- 6). (14) 



Equation (13) is the closest we can ever come to a reconstruction of the original intensity field, 

 1(d), since, as shown in Eqs. (10) and (11), neither measurement set M^(d) or M^iO) contains any in- 

 formation whatsoever about the doubly anti-symmetric component, I^^. This missing term represents an 

 irreduceable bias in any estimate of field I{0 ) reconstructed from these measurements. Any claim to do 

 better without additional information is an exaggeration, for the characteristics of the component I^ are 

 totally unrelated to the data, either deterministically or statistically. 



Six Equally Spaced Headings 



The same technique can be carried out with any number of equally spaced array headings, but 

 symmetry and anti-symmetry arguments are not very helpful. It is easier to revert to the formulation in 

 Fourier components of Eq. (4). Let us suppose that the array headings with respect to an arbitrary 

 reference direction are 0, it/6, 27r/6, ..., rrni/G, where m takes on the values 0-5, inclusive. Then we may 

 set 



e = <p+ — , (15) 



where 6 is the angle with respect to the original reference system and is the angle with respect to the 

 array axis. With this substitution, we can rewrite 1(6) as 



,,., "o v^ r ^ '"""■ ^ ■ '"""■ 



1(0) = Y "^ 2_ rn ^°^ "'^ '^°^ ~6~ ' "" ^^ ^'" ~6~ 

 n=l 



, . mmr , ^ mniri ,,„. 



-•■ o„ cos n0 sm —z~ + o„ sm n0 cos . (lb) 



The array oriented in the direction mir/6 measures 



°0 v^ , / mmr , mmr\ .,_, 



.(^)=Y"^^ cos n0 (a„ cos -^ + 6^ sjn — j . (17) 



Mm' 



When « is a multiple of 6, sin mnir/G is identically zero, and the term containing 6„ vanishes identically, 

 regardless of the value of m which characterizes the steering angle. 



Using the methods described in Appendix A of Ref. 1, we can reconstruct from the measurements 

 the following function (the hat or circumflex * signifying an estimating function): 



894 



