Post on 22-Jan-2021
transcript
Bicoherence Validation Test for
HOME Data Analysis
Martin D. Guiles
SOEST, University of Hawaii at Manoa
1
1. Bicoherence Test for the Low Frequency Internal Wave
Band
One powerful tool for determining non-linearity in a record is the test
for Bispectra/Bicoherence. This technique has been utilized successfully
in oceanographic applications to determine possible wave interactions at
many scales and frequencies. As it pertains to internal waves, the usage
was perhaps maligned for its inability to enlighten the underlying mech-
anisms which create the internal wave spectra and its representations,
specifically the GM spectra. A quote from McComas [6]: ”Computations
using a Garrett and Munk spectral model demonstrate the futility of bis-
pectral analysis for indicating ocean internal wave interactions.” Indeed,
a lack of care in interpreting higher order statistics will inevitably lead
to assumptions that have little evidentiary substance. One must consider
that the tool is useful when applied to waranted situations. It is clear that
a statistical measure that relies upon multiple realizations (such as bis-
pectral analysis) will be inadequate to elicit useful interaction information
from a empirical spectral description such as GM by definition.
A more pressing issue arises in the paper mentioned above, and that is
the subject termed ”kinematic contamination”, and referred to henceforth
as the advection mechanism. With bispectral analysis, true nonlinear
interactions can have the same signature as simple advection when viewed
in a Eulerian frame. It has been speculated that a result of this phenomena
explains some of the common empirical spectral representations. We will
look closely at the difference between the advective mechanism and true
nonlinearity as it applies to tidal and near inertial internal waves.
Carter and Gregg [1] recently identified near inertial peaks in bicoher-
ence, specifically at possible subharmonic interaction frequencies.
(REF REF REF)
One difficulty with techniques that rely on expectation values of Fourier
decompositions is the necessary trade off between temporal resolution and
frequency resolution. This can be alleviated somewhat by various taper
applications, but the underlying problem becomes quite difficult when
the data set contains event oriented forcing at frequencies of interest.
This is the case when investigating inertial interactions, as the inertial
band is dominated by storm generated wind events of a random nature.
For investigating non-stationary data like near inertial oceanic velocity
records, we can utilize wavelet transforms.
Lien et. al. ([5]) utilized wavelet decomposition to identify a turbulent
event in the high frequency internal wave band.
Further related use of wavelets has matured in the area of tidal analysis
as described Flinchem [2] and more recently Jay [4]
Atmospheric gravity wave event dynamics have been analyzed with
wavelet decomposition successfully by Zhang et. al. [7].
Wavelet bicoherence has been used in the aeromechanical field to iden-
tify nonlinear interactions in Gurley [3].
(TRANSITION)
However, for the purposes of determining interactions between low fre-
quency high energy motions in the internal wave field there is little prior
work. We intend to specify the signature of different types of interactions
and as part of that differentiation, bicoherence can yield valuable insight.
A example of bicoherence for ADCP velocity data at mooring C2 of the
HOME mooring deployment is shown in figure 1. The depth of the record
is 536m. The peaks indicated do not exceed sixty percent. There are
questions that arise regarding this plot and the interactions that interest
us, primarily the possible interaction between the inertial and semidiurnal
frequencies.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 10−4
−2
−1
0
1
2
x 10−4
IWISCf M2
M2+f
IWISC
f
M2
M2+f
IWISC
f
M2
M2+f
Figure 1. Bicoherence of meridional velocity at the C2 mooring with
depth 536m showing peaks in interaction between the indicated frequen-
cies.
To determine precisely what we should be observing in the bicoherence
plots, a synthetic time series was constructed that closely emulates the
observed inertial and semi-diurnal tide motions. For the inertial, a varying
amplitude is applied throughout the record, as depicted in figure 2. The
semi-diurnal is represented by the first four constituents, whose relative
amplitude ratios are taken for this location from the TPXO6.2 tidal model
(REF). In the figure the length of data used in the bicoherence estimates
is indicated by vertical dotted lines. A snapshot of the resultant velocities
within the two bands is shown in the lower panel.
50 100 150 200 250 300 350
0.5
1
1.5
2
2.5
3
3.5
Days
U E
nvel
ope
(cm
/s)
InertialSemi−Diurnal
210 212 214 216 218 220 222−5
0
5
Days
U (
cm/s
)
Figure 2
The key feature in this model is the vertical advection of horizontal in-
ertial motion. We allow the semi-diurnal tide to heave the inertial motion
and replicate one of the possible mechanisms for the spectral peaks ob-
served in many oceanic velocity records. The spectra of the synthetic time
series is shown in figure 3. A curious feature for this record is the spectral
peak at the frequency 2M2 − f . The peaks associated with the advection
mechanism in this regard are the M2−f (IWISC) and M2 +f frequencies.
The semi-diurnal peak signature is reflected in these advection peaks. To
10−4
100
101
102
103
104
105
106
107
ω (rad/s)
Rot
ary
PS
D
M2−f M2+f
M2f
2M2+f2M2−f
95%
CCWCW
10−4
100
101
102
103
104
105
106
107
ω (rad/s)
Rot
ary
PS
D
M2−f M2+f
M2f
2M2+f2M2−f
95%
CCWCW
Figure 3. Spectra for two time periods of the Synthetic Model. One is
the whole two year record, the other is for the analysis period.
validate the bicoherence test we look at the case where the bicoherence is
a purely Lagrangian record with no interaction between frequencies. This
test is shown in figure 4. There is only a peak involving the zero frequency
currents.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
x 10−4
−2
−1
0
1
2
x 10−4
IWISCf M2
M2+f
IWISC
f
M2
M2+f
IWISC
f
M2
M2+f
Figure 4. Bicoherence of a record from the Synthetic Model without advetion.
The final case is for a advected record as shown in figure 5. This record is
during a elevated inertial period indicated earlier. And, for comparison,
a example of the auto bispectra is shown for the record with elevated
inertial as used previously (figure 6).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
x 10−4
−2
−1
0
1
2
x 10−4
IWISCf M2
M2+f
IWISC
f
M2
M2+f
IWISC
f
M2
M2+f
Figure 5. Bicoherence of the zonal velocity for the advected record indi-
cated with high inertial event.
The primary constraint on the bicoherence estimates usefullness in this
situation with non-stationary sources in the record is the overwhelming
0.5
1
1.5
2
2.5
x 10−4
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
x 10−4
−2
−1
0
1
2
x 10−4
IWISCf M2
M2+f
IWISC
f
M2
M2+f
IWISC
f
M2
M2+f
Figure 6. Bispectra of zonal velocity for the Synthetic Model
contribution to the expectation value from a very few records. This creates
a deceptive situation, and can be illustrated using wavelet analysis. Using
the same sub-record indicated above, a wavelet decomposition was done.
In figure 7, the wavelet decomposition is shown in ’frequency’ space, where
the scale of the wavelet has been related to its equivalent Fourier frequency.
The wavelet itself is a second order complex frequency b-spline. This
is especially suited to frequency representation and use in higher order
techniques like the wavelet bispectra (figure 8.
205 210 215 220 225 230 235 240
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
x 10−4
Figure 7. Frequency B-Spline wavelet decomposition of zonal velocity
for the Synthetic Model
References
[1] G. S. Carter and M. C. Gregg, Persistent near-diurnal internal waves observed above a site of m-2
barotropic-to-baroclinic conversion, Journal Of Physical Oceanography 36 (2006), no. 6, 1136–1147.
[2] E. P. Flinchem and D. A. Jay, An introduction to wavelet transform tidal analysis methods, Estuarine,
Coastal and Shelf Science 51 (2000), 177–200.
Figure 8. Wavelet bispectra of zonal velocity for the Synthetic Model
[3] K. Gurley, T. Kijewski, and A. Kareem, First- and higher-order correlation detection using wavelet
transforms, ASCE Journal of Engineering Mechanics 129 (2003), no. 2, 188–201.
[4] D. A. Jay and Tobias Kukulka, Revising the paradigm of tidal analysis – the uses of non-stationary
data, Ocean Dynamics 53 (2003), no. 3, 110–125.
[5] R. C. Lien, E. A. D’Asaro, and M. J. McPhaden, Internal waves and turbulence in the upper central
equatorial pacific: Lagrangian and eulerian observations, J. Phys. Oceanogr. 32 (2002), 2619–2639.
[6] C. H. McComas and M. G. Briscoe, Bispectra of internal waves, JOURNAL OF FLUID MECHAN-
ICS 97 (1980), no. 1, 205–213.
[7] F. Zhang, S. E. Koch, C. A. Davis, and M. L. Kaplan, Wavelet analysis and the governing dynamics
of a large-amplitude mesoscale gravity-wave event along the east coast of the united states, Quarterly
Journal of the Royal Meteorological Society 127 (2001), no. 577, 2209–2245.