Geoscience Reference
In-Depth Information
3.0
×10
-7
×10
-7
2.5
2.5
2.0
2.0
1. 5
1. 5
1. 0
1. 0
0.5
0.5
13.54
0
05 0 5
20
5
30
13.56
13.58
13.62
13.60
Time (s)
Time (s)
Figure 15.10.
(a) Typical amplitude of the recorded down-mixed signal bandpass filtered around the Doppler peak (between
frequencies
f
c
and
F
c
as shown in Figure 15.9). Peaks of high amplitude correspond to events where a particle travels into the mea-
surement volume and scatters the incident acoustic wave toward the receiver. (b) Zoom on one such isolated event. From
Qureshi
[2009].
signal recorded in the wind tunnel experiment by
Qureshi
et al.
[2007]. The peak at zero frequency corresponds
to the emitting frequency
ν
0
(which was 80 kHz in this
experiment) and the secondary peak (around 7 kHz) cor-
responds to the Doppler shift from the wave scattered by
moving tracers (note that the central peak at the emit-
ting frequency is enlarged by aerodynamic effects). The
interest of heterodyne downmixing is that recording the
original signal would require very high sampling rates, as
the Doppler frequency would be here around
ν
s
=
ν
0
+
δν
particle event.) Second, the portion of signal correspond-
ing to each individual event is analyzed with dedicated
time-frequency tools to extract the instantaneous Doppler
shift. For the same reasons previously discussed in the
context of optical ELDV, the instantaneous Doppler shift
can be efficiently extracted using an AML algorithm
where after downmixing the signal scattered by one parti-
cle and recorded by the receiver is modeled in the form
i
2
π
t
0
δν
i
(t
)
d
t
]
z
i
(t)
=
A
i
(t)
exp
[
+
n(t)
,
(15.5)
87 kHZ), while it is only at 7 kHz after downmixing.
Figure 15.10a shows the amplitude
A(t)
of the
down-mixed signal versus time; each peak of amplitude
corresponds to the passage of a particle in the measure-
ment zone. The spectrum previously discussed was cal-
culated from the entire times series and hence all time
information has been lost: The observed Doppler peak
corresponds to the superposition of spectral contribu-
tions from thousands of successive scattered traveling
through the measurement volume. It is only interesting to
extract global informations, such as the average velocity
of the particles (given by central frequency of the Doppler
peak) or the typical level of velocity fluctuation (given by
the width of the Doppler peak). However, accessing the
Lagrangian dynamics of the particles requires to extract
the instantaneous Doppler shift
δν(t)
for each individual
particle.
This is achieved in two steps: First, each event cor-
responding to the passage of a particle is detected
from the amplitude signal as shown in Figure 15.10a.
(Figure 15.10b shows a zoom on such an individual
where
A
i
(t)
is the amplitude of the scattered acoustic
wave,
δν
i
(t)
is the instantaneous frequency of the signal
(subscript
i
indicates that we consider particle number
i
),
and
n(t)
represents an additive experimental noise. The
AML algorithm determines for each particle
i
the best
functions
A
i
(t)
and
δν
i
(t)
so that the model given by equa-
tion (15.5) matches as close as possible the actual recorded
signal (
n(t)
is assumed to be a random Gaussian noise
whose amplitude is fixed according to the actual noise
level of the experimental signal). Such a parametric algo-
rithm is capable of overcoming the Heisenberg limitation
(which would severely constrain the resolution of the mea-
surement) due to the extra information given by the a pri-
ori imposed shape of the signal in equation (15.5) added
in the signal processing. The gain in resolution offered by
the AML method is illustrated by Figure 15.11. Also of
interest is the fact that the AML algorithm yields a quan-
titative indicator of the relevance of expression (15.5) for
the actual modeling of the scattering signal. This allows to
discard spurious events from the statistical ensemble, for
