Earth’s rotation: observations and theory - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 280

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m 1 in 0.01 Greenwich

Figure 4.3 The pole path for the year 2004 determined by the VLBI technique.
Standard error bars, inflated by a factor of 100 for clarity, are overplotted.

N

1
M

g j = G k e i 2π( k / M ) t j , (4.12)

k =− N

where M is the length of the record segment and G k is the DFT sequence. The
DFT is found by least squares adjustment of g j to g j weighting by the inverse of
the square of the standard errorσ j . The conditional equations for the DFT sequence
(2.187) have a Toeplitz coe cient matrix with elements

L

1
σ

e − i 2π( m / M ) t j C m = , (4.13)

2
j j = 1

while the components of the right-hand side vector are

L

g j
σ

e − i 2π( l / M ) t j d l = M (4.14)

2
j j = 1

with the sums over the L sample points.
The conditional equations for the DFT for unequally spaced time sequences can
be solved by the methods of Section 2.3.6. The coe cient matrix is first subjec-

ted to singular value decomposition (SVD). Singular values are then eliminated,
starting with the smallest and working upward, until Parseval's relation is satisfied
as closely as possible. The method is quite general, although here we consider its
application to VLBI observations of polar motion and nutation.