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.