72
2 Experiments in Pebble Flows
2.5.3.2 Analysis Methodology
Time Correlation
The correlation function is a measure of the order of a system. The mathematical
correlation function in statistical mechanics describes how the microscopic variables
at different positions are related. More specifically, the correlation function quantifies
how the microscopic variable correlates with another in space and time.
The most common definition of the correlation function is the average of the
scalar products between the two series of random variables at times t and t + τ . The
normalized autocorrelation of the vertical velocity is defined as
C hx (τ ) =
ΔV hx,t ΔV hx,t+τ
V hx,t
2
(2.3)
where ΔV hx,t = =V hx,t − −V hx , and the brackets ·· indicate the averaging operator.
Wavelets Analysis
In wavelet analysis, the combination of the Gaussian distribution wavelet function
can be used to present the velocity fluctuation signals. First, the wavelet transform
(WT) [21, 22] of a function f consists of decomposing it into elementary space-scale
contributions, associated with the wavelets constructed from the analyzing wavelet
φ by translations and dilations [23, 24]. The WT of the function f is defined as
W φ [ f ](b, a) =
1
a
+∞
−∞
¯
φ(
x − b
a
) f (x)dx,
(2.4)
where W φ [ f ](b, a) is the wavelet coefficient; φ is the basic wavelet function; a is a
scale parameter; and b is a space parameter [25].
Wavelet functions form a family of basic functions with high frequency and small
duration, which are normalized dilations and translations of a prototype “wavelet
base” function, i.e.,
φ a,b (t) =
1
√
|a|
φ(
x − b
a
).
(2.5)
An original signal is always decomposed into the approximations and details with
different frequency bands using the wavelet transform. The wavelet series of the
desired composition level J approximate a continuous signal x(t i ) expressed as
x(t i ) ≈ A j (t i ) + D j (t i ) + D j−1 (t i ) + · · · + D 1 (t i ),
(2.6)
where D 1 (t i ), D 1 (t i ), ..., and D j (t i ) stand for detail signals of the multi-resolution
decomposition at different resolution 2
j , and A j (t i ) the approximation signal of
decomposition at resolution 2
j .
2 Experiments in Pebble Flows
2.5.3.2 Analysis Methodology
Time Correlation
The correlation function is a measure of the order of a system. The mathematical
correlation function in statistical mechanics describes how the microscopic variables
at different positions are related. More specifically, the correlation function quantifies
how the microscopic variable correlates with another in space and time.
The most common definition of the correlation function is the average of the
scalar products between the two series of random variables at times t and t + τ . The
normalized autocorrelation of the vertical velocity is defined as
C hx (τ ) =
ΔV hx,t ΔV hx,t+τ
V hx,t
2
(2.3)
where ΔV hx,t = =V hx,t − −V hx , and the brackets ·· indicate the averaging operator.
Wavelets Analysis
In wavelet analysis, the combination of the Gaussian distribution wavelet function
can be used to present the velocity fluctuation signals. First, the wavelet transform
(WT) [21, 22] of a function f consists of decomposing it into elementary space-scale
contributions, associated with the wavelets constructed from the analyzing wavelet
φ by translations and dilations [23, 24]. The WT of the function f is defined as
W φ [ f ](b, a) =
1
a
+∞
−∞
¯
φ(
x − b
a
) f (x)dx,
(2.4)
where W φ [ f ](b, a) is the wavelet coefficient; φ is the basic wavelet function; a is a
scale parameter; and b is a space parameter [25].
Wavelet functions form a family of basic functions with high frequency and small
duration, which are normalized dilations and translations of a prototype “wavelet
base” function, i.e.,
φ a,b (t) =
1
√
|a|
φ(
x − b
a
).
(2.5)
An original signal is always decomposed into the approximations and details with
different frequency bands using the wavelet transform. The wavelet series of the
desired composition level J approximate a continuous signal x(t i ) expressed as
x(t i ) ≈ A j (t i ) + D j (t i ) + D j−1 (t i ) + · · · + D 1 (t i ),
(2.6)
where D 1 (t i ), D 1 (t i ), ..., and D j (t i ) stand for detail signals of the multi-resolution
decomposition at different resolution 2
j , and A j (t i ) the approximation signal of
decomposition at resolution 2
j .