Variational methods and core modes - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 476

a j

r ij =

f j / f j 2 .

(8.120)

4 Q f i 1 + −

For record segments of 12,000-hour length, product spectral estimates s i are spaced
at intervals of 1 hour along the frequency axis. In the subtidal band,

between the 2-hour and 8-hour period, there are 4,501 spectral estimates. For 25
spectral estimates centred on frequency f j with Q

/ 12,000 cycles /

= 100, the misfit of (8.120) to

spectral estimate s i is

ij = A j r ij − s i . (8.121)

The error energy of the fit is

i = j − 12

j + 12

2

I j =

ij .

(8.122)

Minimising the error energy of the fit gives

j + 12 j + 12

r ij s i r ij , A j = (8.123)

i = j − 12 i = j − 12

with minimum error energy

j + 12 j + 12

s i − A j r ij . I min = (8.124)

i = j − 12 i = j − 12

As a measure of the strength of a potential resonance of the form (8.120),we use the
parameter S j =

A j / I min . When a large well-fitted resonance is found, we expect
S j to be large, and if a small poorly fitted spectral feature is found, we expect S j
to be small. For each of the available 4477 frequencies, f j , in the subtidal band, we
set T 0 = 1/ T P from equation

(8.113). The values of S R , S A and S P of the resonance parameter S 2 at the dis-
crete frequencies nearest f R , f A and f P , respectively, are then multiplied together to
form the splitting product P j as an indicator of the presence of correctly split res-
onances. In Figure 8.18 we show the resulting probability density function (PDF)
for the splitting product computed at 4119 points along the frequency axis in the
subtidal band. The PDF shown in Figure 8.18 allows evaluation of the significance
of translational triplets along the frequency axis. In Figure 8.19 we show the split-
ting products found between the 2-hour and 10-hour periods. A very large value of
P is found at T 0

1/ f j and compute f R = 1/ T R , f C = 1/ T A and f P =−

3.7975 hours. From the PDF it is found that the probability of a
realisation of P j larger than the largest shown in Figure 8.19 is only 1 in 6.8

=

10 38 .
This appears to confirm the identification of the translational triplet in the product
spectrum.

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