Directional Recording of Swell from Distant Storms - Department of ...

DIRECTIONAL RECORDING OF SWELL FROM DISTANT STORMS 523At any prescribed frequencyfo, C-i Q can therefore be thought of as a space correlogramwith complex values, whose transform is the 'directional spectrum' E(l,m,fo) for thatfrequency. The functions C and Q are precisely those computed from the recordings(?4(d)).7. INTERPRETATION OF THE CROSS-SPECTRAL MATRIXThe next step in the analysis is to deduce the directional spectrum E(l, mf) from theobserved values, C(X, Y,f) and Q (X, Y,f). The inverse of (6.1 1) isE(l, m, f) = f _00 _00j'0 [C(X, Y,f) -iQ(X, Y,f)] exp [-i2nff(lX+mY)] dXdY, (7.1)and it would be a straightforward matter to calculate E(l, m,f) if one knew the C's and Q'sas continuous functions ofXand Y. Unfortunately one knows the C's and Q's only at discretevalues of X and Y, namely those intervals between the various instruments taken in pairs.The present section discusses ways of evading this practical limitation.The interval distances will now be written more briefly as(X., Yn) and (- Xn, -Y) (n = 1 ... N)and the corresponding C's and Q's as Cn, Qn. If there are M instruments the number Nis ofcourse 'M(M- 1).(a) Conventional treatmentOne conventional but unconvincing assumption would treat all the unknown values ofC -i Q as zero and weight the known values by delta functions of arbitrary weights bn.With this treatment, the true power distribution E(l, m,f) in 7-1 becomes an approximatedistribution E'(l, m,f)n=NE'(l, m, f)-bo Co + 2 bn Cn cos 2ir(lXn + m Y) 2 E bn Qn sin 2ir(lXn + mYn).1 1Thus, the spectrum is pictured as the sum of N sinusoids plus a constant. In the presentexperiments there are only three instruments (M 3) and consequently only three instrumentpairs (N 1 3) so the spectrum is made up of only three sinusoids. One is at libertyto choose the weights bn in any way that seems best. There is some advantage in choosingthem each to be -1. If, for example, the C's and Q's are in fact due to only a single welldirectedswell of power Ao and wave numbers 1, mo, then from (6 9) and (6.10):n-NCo. - AO)A.~~~~~~~72Cn AO cos 2ir(10 Xn + mO Yn), (7(2)QnAO sin 2ir(10 Xn + m0 Yn).JThe calculated distribution E' is then (bn being .)1r 7n-3 n=3E'(1, m f) = 7IC0?2 E Cn cos 2r(1Xn + m YJ )-2 I Qn sin 2nr(IX7?mY3)I . (7.3)For the stated values of the C's and 9's this reduces toE' (l, m,fE) = A4 ( 1 +2 Ecos 2X77[3(l-10) + Y73(m-mo)]}) (7.4)64-2