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9 Experimental Methods in Fluid Mechanics
9.5.2 Standardization of Data, Trend Removal and Filtering
Prior to calculating the frequency spectrum of variable (, various preliminary
operations are usually applied to the data, i. e. data standardization, trend
removal and filtering. To standardize the ( values we present them in a nondimensional form:
(9.42)
where:
(9.43)
For a stationary ergodic process (for definition see Massel, 1996a), the quantity
( is an unbiased estimate of the true mean value. The unbiased estimate of the
standard deviation of data (n is given by:
(9.44)
Especially for the case of wave orbital velocity measurements in the coastal
zone, in regions with large tidal motion, removal of the spurious trend or low
frequency components with wavelengths longer than the record length, is usually required. The most common technique for trend removal is to fit a loworder polynomial to the data using the 'least squares method'. Thus, we assume
that the original data {(n} can be approximated by a polynomial of order K:
K
-
k
(" = L bk(n.6.t) n = 1,2, ... , N.
(9.45)
k=O
A 'least squares' fit provides a system of equations for unknown coefficients bk
as (Bendat and Piersol, 1986):
K
N
N
L bk L (n.6.t)k+m = L (n(n.6.t)m, m = 0, 1,2, ... , = K.
(9.46)
k=O n=l
n=l
Assuming that K = 1, we obtain:
b _ 2(2N + 1) 2:;;'=1 (n - 62:;;'=1 n(n
0N(N - 1)
,
(9.47)
b _ 122:;;'=1 n(n - 6(N + 1) ~;;'=l (n
1 -
.6.tN(N - 1)(N + 1)
.
(9.48)