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APPENDIX: 
MULTIPLE-WINDOW HARMONIC ANALYSIS 
We shortly summarise the basic assumptions and formulas of 
this multiple-window harmonic analysis technique which was 
introduced by Thomson (1982) and extended by Park et al. 
(1987), Lindberg and Park (1987). 
The purpose of an harmonic analysis is the detection of 
spectral harmonic line components and the measurement of 
their frequencies and amplitudes. It is assumed that the 
spectrum of a time-series x(t) consist of harmonic line 
components and a continuous background spectrum. In the 
frequency interval (f+W.f-W) with sufficiently small values of 
W the record x(t) can be written 
i2nft 
x(t) =e + e(t) (1) 
where |t is a complex amplitude and e(t) is an error term. The 
error term e(t) consist of other sinusoids and noise. In practice 
we have N measurements of x(t). The time between successive 
samples is assumed to be 1 so that the frequency f is defined 
on its principal domain (-1/2,1/2]. 
K different complex eigenspectra yx(f) are produced in the 
frequency domain by the discrete Fourier transformations 
N-1 : 
yp(f)= X ef yy, (p 
tz 
k=0,1,..,K-1 (2) 
where the vi(t) are the discrete prolate spheroidal sequences. 
The discrete prolate spheroidal sequences v(t) can be 
calculated by taking a singular value decomposition of a sinc 
matrix and are optimal windows for concentrating the energy 
of sinusoids in the frequency interval (f+W,f-W). They 
maximise the functional 
f+W 
J lye? av 
f-— 
z 
(3) 
yi (V) dv 
aX 
  
DS 
With W chosen to be 4/N the functional (3) has a value close to 
one for the first eight discrete prolate spheroidal sequences 
vo(t), vi(t), … v7(t), and rapidly drops off thereafter. Since only 
vw(t) with good spectral leakage resistance should be 
considered in the computations, we set K equal to 8. 
749 
The absolute squares of yx(f) 
Si (f) = yy (f k=0,1,..K-1 (4) 
can be regarded as individually direct spectrum estimates. 
However, the novelty of the multiple taper method is that is 
utilises more data than conventional methods by using all 
complex eigenspectra y«(f) with good spectral leakage 
resistance. Applying regression techniques to the y(f) an 
estimate of the complex amplitude p is obtained 
K-1 
2 Vox Yi (F) 
Bf) » ES —— (5) 
X Vi 
k=0 
where 
N-1 
Vok = 2 V(t). (6) 
t=0 
The multiple-window method also provides a statistical F-test 
to test the fit of the sinusoid model. The random variable 
  
a Le 2 
(K-D|à([ X vj. 
F(f) = — pel 5 (7) 
X y. (D - if) Vo | 
kzl 
  
follows an F-distribution with 2 and 2K-1 degrees of freedom. 
For significance level y the hypothesis p=0 is rejected if 
F(f)ZF25»x.1:.. In our case (K-8) the 95% confidence level of 
the f-test statistic is at 3.74 and the 9996 confidence level is at 
6.51. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B7. Vienna 1996