DOI 10.1007/s10652-010-9172-1
O R I G I NA L A RT I C L E
Using discriminant analysis to determine the breaking
criterion for an ISW propagating over a ridge
Chen-Yuan Chen
Received: 10 September 2009 / Accepted: 21 April 2010 / Published online: 7 May 2010 © Springer Science+Business Media B.V. 2010
Abstract This study aims to develop methods that are capable of deciding the breaking criterion for an internal solitary wave (ISW) propagating over a submarine ridge. Laboratory experiments were conducted in a wave tank to measure ISWs propagating over a submarine ridge. The results suggest that the ISW-ridge interaction can be grouped according to three degrees of magnitude based on the blockage parameterζ and the degree of blocking B. For classification reasons, we first present an alternative decision model for evaluating the inter-action of ISWs with an underwater ridge in a two-layer system. This approach is based on a multivariate statistical method and discriminant analysis. Information obtained from the eigenvalues is used to combine different ratio measures which are defined according to every single input and output. The discriminant model effectively classifies units into distinct predefined groups. An experimental simulation is conducted to demonstrate the practical implementation of the ISW-ridge interaction. The results of the method applied in this exam-ple are statistically significant, demonstrating the effectiveness of the ISW-ridge interaction classification method.
Keywords Internal solitary wave· Wave dispersion · Energy loss · Profile change · Discriminant analysis
1 Introduction
Internal solitary waves (ISWs) have been detected at the interface of a stratified fluid system comprised of layers of different densities such as can be found in many areas in the lakes and oceans around the world. Large ISWs can affect oil drilling operations [1], produce tur-bulent mixing [2,25,28], and cause nutrient pumping that will induce fish foraging [17,30]. Wang et al. [32] observed the regulation of ecological processes by ISWs propagating over the Dongsha Atoll, an isolated reef in the South China Sea (SCS). Others have also observed
C.-Y. Chen (
B
)Department of Computer Science, National Pingtung University of Education, No. 4-18, Ming Shen Rd., Pingtung 90003, Taiwan
how pilot whales follow ISWs in the SCS [27]. The mixing and dissipation generated by internal waves also have important effects on cross slope exchange processes, enhancement of bottom stress, and generation of nepheloid layers.
It has recently been proposed that internal waves may also make a significant contribution to internal oceanic mixing and hence have an important influence on climate change. Given the afore-mentioned reasons, it is clear why it is important to scrutinize the interaction of nonlinear internal solitary waves (ISWs) with the seabed topography [5–7,4,15,16,21,23,24,
26,33]. However, it should also be noted that since energy dissipation plays such an important and varied role in the movement of water and sediment in coastal seas [3], we need to include this when developing a better fitting and more appropriate model for predicting ISW prop-agation and the breaking criterion. A statistics-based methodology based on internal wave data is earnestly needed. Zheng et al. [34] utilized data collected from the years 1995 to 2001 for their pioneering statistical analysis of IW occurrence. They made field measurements of sea surface wind, sea state and vertical temperature profiles which were used for analyzing IW generation and for generating synthetic aperture radar (SAR) imaging conditions.
Soon after this, the effects of weighted parameters on the amplitude and reflection of energy-based ISWs from uniform slopes in a two layered fluid system were investigated [8]. The results of that study were quite consistent with other experimental results, and are applicable to the naturally occurring reflection of ISWs from sloping bottoms. More recently, Chen et al. [12] concluded that the goodness-of-fit and predictive ability of the cumulative logistic regression models make them better than binary logistic regression models for the modeling of the ISW-ridge interaction. After removing outliers and influential observations from the data, the remaining observations are refitted to find the goodness-of-fit of the revised model [13]. However, the data in these cases are so small that some observations have pro-portions close to zero or one; furthermore, inferences based on the asymptotic distribution of the change in deviance must also be taken into account [14].
Researchers have classified the level of wave-ridge interaction by means of visual descrip-tion based on previous experience [9]. The traditional method calls for further improvement by statistical manipulation. This study is primarily aimed at making improvements using data collected from the Ref. [9]. The existing strategy for classification of wave-ridge inter-actions into three degrees of magnitude is reviewed and a statistical classification scheme is outlined which might provide a frame-of-reference for future endeavors. The rest of this paper is organized the paper as follows. In Sect.2, we describe the experimental set-up and theoretical background needed to understand the hydrodynamic interaction. We also dis-cuss the physical parameters utilized in the study. Section3is devoted to the derivation of a mathematical model and the attempt to adapt a system model for data analysis using this statistical method. In Sect.4, some results are stated and illustrated. The methodology is obviously efficient and significant. Finally, some conclusions are offered and extensions for future research highlighted.
2 Experiments and physical system
The data are obtained from experimental measurement. Some physical parameters are acquired via sequential analyses of digital signal processing and mathematical manipulation.
2.1 Setup
0 η 1 ρ 2 ρ H a
Fig. 1 Representation of the wave tank setup and physical parameters
to generate internal solitary waves which would move over a triangular ridge mounted on the bottom which represented an oceanic terrain (see Fig.1). A two-layer fluid system was placed in the tank. It contained an upper layer of fresh water with a densityρ1and a depth
H1, and a lower layer of briny water with a densityρ2 and a depth H2. The movement of
the depression-type ISW could be altered by adjusting the interfacial level on either side of the sluice gate. Lifting the sluice gate generated a leading solitary wave which propagated through the main section of the wave flume [10]. Before encountering the isolated triangu-lar submarine ridge, the initial wave form had stable soliton features dependent upon the thickness ratio(H1/H2) used [11].
2.2 Parameters
Internal wave breaking occurs when the particle velocity at the top of the crest exceeds the wave celerity. Three key parameters critical to the breaking events were observed: speed of wave propagation, degree of blocking, and the blockage parameter. The wave celerity was calculated using a linear relation for the interfacial wave [33]
C = gH, (1)
where H= H1H2/(H1+ H2), and g= g(ρ2 − ρ1)/ρ1. The water depth varied in relation
to the seabed topography. Therefore, Eq.1can be rewritten as
˜C =
g˜h1˜h2
˜h1 + ˜h2
, (2)
where ˜hi is the modified water depth in the i-th layer given by ˜h1 = H1 + a, and ˜h2 =
H2− hs− a, respectively. As shown in Fig.2, the relation between wave amplitude and ridge
height is included. Thus, Eq.2can be used to calculate the celerity of an ISW propagating over a variable seabed.
A dimensionless blockage parameter
ζ = (a + H1) / (H1+ H2− hs) (3)
is used to quantify the relationship between the water depths of the two layers in a strat-ified system [31]. Whenζ = 1, the wave crest touches the apex of the ridge. If ζ > 1, the height of the ridge intrudes on the ISW. Therefore, the incident wave speed ˜C given by Eq.2
becomes a function of parameterζ . The level of interaction between a ridge and an ISW can be determined from the wave speed ˜C and the blockage parameterζ .
The degree of blocking B is defined as the height of the obstruction divided by the depth of the lower layer, i.e., hs/H2. It basically indicates how much the layers are obstructed by the
2 ρ 1 ρ 1 ~ h 2 ~ h s h 2 H 1 H
Fig. 2 Definition of the physical variables
B> 1, the tip reaches into the interface. It was found that for very high blocking, say B ≥ 1.2, virtually no signal was transmitted and the entire wave train was reflected [33]. On the other hand, for small B, the entire wave was transmitted and there was virtually no reflection.
3 Discriminant analysis
Discriminant analysis (DA) is suitable for the classification and grouping of large data sets for various experiments. Prior studies indicate that DA is suitable when a high classification rate is needed; see references [19,18,20]. DA is a multivariate statistical method and group-ing technique used to find the linear combination of ratios which best discriminate between the groups which are being classified. The difference between two or more groups is the discriminant function; only one discriminant function is required for two groups. When there are more than two groups, one uses the minimum number of discriminant function(s) needed to best represent the difference between them [29]. DA is the appropriate statistical technique for estimating both categorical and metric variables attributed to dependent and independent variables. The DA function can be written as [22]
Zj k = b + aiXi k, (4)
where∀Zj k is the Z score of discriminant function j for object k; b is the intercept; ai is
the weight of independent variable I; Xi kis the independent variable i for object k. Like in a
regression model, aiis the discriminant coefficient for maximizing the discriminant criterion
λ, in which λ = aAa
aW a;∀A is among the group consisting of the sums of the square and
cross products matrix (SSCP); W is pooled within the group SSCP matrix. To establish the DA function one also has to focus on the maxλ (see AppendixI).
The discriminant rule is also used to predict the classification of the variables as in the following function (for a two group example):
c = N2Z1+ N1Z2 N1 + N2 ,
(5)
where N is the variable amount; Zi is the average value of the DA score.
If Zi < c, then it is classified as part of Group 1.
We have to know the correct rate of the DA for the function. This is called the hit rate (h) and is used for estimating the discriminant validation rate:
h = I
i= 1Jj= 1Ni j
N . (6)
A higher h value means better predictive capacity of the discriminant function.
4 Analytical results
In a natural environment, internal waves retain their features, but appear to become deformed when encountering an oceanic bottom ridge. This interaction can be defined [9] as follows: weak interaction, moderate interaction, or wave breaking between the ISW and bottom ridge. In the present study, we determine these three classifications based on the effects of DA.
4.1 Three classifications based on the dimensionless blockage parameter
All observations for wave speed and the blockage parameter obtained from the SPSS dis-criminant analysis procedure are clustered into 3 groups with centers at 0.4566, 0.6598 and 0.8773, as shown in Table1. An F-value of 10.726 is statistically significant(p < .0000) and indicates group equality. The canonical discriminant function is Z = 1.035ζ + 0.042C and its eigenvalue is 6.816. A summary of the classification results is shown in Table2giving an overall classification rate of 100%.
The relationship between dimensionless wave speed and the blockage parameter is indi-cated in Fig.3. The three groups are derived using the canonical discriminant function. When ζ < 0.55, the ridge has little influence on the motion of the ISW. When ζ ranges between 0.55 and 0.78, the ridge has an apparent influence on the incident waveforms. Whenζ > 0.78, the ridge blocks the incident wave motion and induces large vortex induced wave breaking. Thanks to the effects of DA, there is major improvement in the data index as indicated by a, b, c- h in Fig.3, leading to a larger statistical significance.
Table 1 Group statistics for
dimensionless wave speed and blockage parameter
Table 2 Classification results for dimensionless wave speed and blockage parameter
Cluster number of cases Predicted group membership Total Weak interaction Moderate interaction Wave breaking 1 Original
Count Weak interaction 14 0 0 14
Moderate interaction 0 29 0 29 Wave breaking 0 0 25 25 % Weak interaction 100.0 0 0 100.0 Moderate interaction 0 100.0 0 100.0 Wave breaking 0 0 100.0 100.0 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 weak interaction moderate interaction wave breaking ζ
Fig. 3 Terrain map of dimensionless wave speed and blockage parameterζ
4.2 Three classifications of the degree of blocking
We next investigate the degree of blocking versus dimensionless wave amplitude. There are three obvious group centers located at 0.1205, 0.1238 and 0.1470, as shown in Table3. An F-value of 5.791 is significant(p < .0000), indicating that equality of groups means statis-tical significance. The canonical discriminant function is Z= 0.756 B + 0.102ai/H2and its
eigenvalue is 3.924. Table4presents a summary of the classification results giving an overall classification rate of 98.7%.
Table 3 Group statistics for the
degree of blocking versus dimensionless wave amplitude
Cluster number of cases Mean Std. deviation Weak interaction B 0.1205 0.07391 ai/H2 0.4453 0.15117 Moderate interaction B 0.1238 0.05271 ai/H2 0.6537 0.06376 Wave breaking B 0.1470 0.04894 ai/H2 0.7854 0.06517
Table 4 Classification results for the degree of blocking versus dimensionless wave amplitude
Cluster number of cases Predicted group membership Total
Weak interaction Moderate interaction Wave breaking 1 Revised
Count Weak interaction 16 0 0 16
Moderate interaction 1 21 0 22 Wave breaking 0 0 37 37 % Weak interaction 0 100.0 0 100.0 Moderateinteraction 100.0 0 0 100.0 Wave breaking 4.5 95.5 0 100.0 ai/ 2 s / 2 0.25 0.2 0.15 0.1 0.05 0.2 0.3 0.4 0.5 0.7 0.8 0.9 1.0 0.6
*
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*
weak interaction wave breaking*
moderate interaction h H H C B A b aamplitude, there is no effect on the internal solitary wave. The encounter between wave and ridge can be regarded as a weak interaction. On the other hand, for a large blockage, e.g., B> 0.6, the ridge can block the propagation of the incident wave to cause vortex induced wave breaking. The experimental results show that although the amplitude of the ISW in the wave tank is small (e.g., ai/H2of about 0.06), the waveform is still affected by the topography
of the seabed and results in breaking.
5 Conclusions
We demonstrate the successful application of a multivariate statistical method for classify-ing the evolution of an ISW over a submarine ridge in a stratified two-layer fluid system. Although classification by visual examination may be more convenient, this method lacks a mathematical basis and scientific reproducibility. Statistical data analysis techniques can improve method validation and ease of interpretation. Multivariate functions are more suit-able for discriminating among groups in large sample sets than is visual assessment. The results of this study are a reminder to researchers that DA is a better choice to check or classify exact experimental data experimental.
The ideal rate for classifying the relationship between a dimensionless wave speed and the blockage parameter is 100%; the rate is 98.7% (one exception) for the relationship between the degree of blocking versus dimensionless wave amplitude. Accuracy of classification is rather high for ISW evolution over a submarine ridge in a stratified two-layer fluid system. The recommended cut-off points for the classification criterion for discriminating between “weak interaction”, “moderate interaction” and “wave breaking” are 0.5 and 0.78 and are obtained using a single experimental measurement of the blockage parameter. Another method for stationary estimation is to look at the relationship between the degree of blocking and dimen-sionless wave amplitude. We utilize two empirical functions to classify data into three groups. The functions are self-parallel and linear. The results of the study should also be of benefit in the fields of ocean engineering and oceanography for the examination of a stratified two-layer fluid system.
The main objective of the present study is to classify the flow regimes associated with the submarine ridge encounter. Statistical methods are used to investigate and classify the degree of ISW-ridge interaction. The results clearly indicate that DA plays a role in estimating degrees of ISW-ridge interaction events. Because of the complex nature of the physical obser-vation process, neither numerical analyses nor laboratory experiments in nature seem enough clearly describe the wave features. Rather than assuming classification efficiency based on the visible mechanism, specific effort is needed to directly estimate the classification efficiency of such ISW-ridge interaction events.
Further challenges suggested by this current work include extending the data obtained in the field to cover the full range of natural phenomenon. Also, the effects of breaking ISWs on the energy spectrum need to be addressed; this could potentially help clarify and identify which slope areas are likely to yield significant mixing due to breaking waves.
Appendix I: Proof of the maxλ maxλ = a Aa aW a (A1) s.t. a× a = 1,
We adapt the Lagrangian method, letting the differential result equal zero. Then,
∂λ ∂a = ∂aAa aW a ∂a = 2Aa aW a −aAa W a aW a 2 = 0 letλ = a Aa aW a (A2)
We now have2[Aaa− λWaW a ] = 0.
Multiply both sides of the function byaW a2 . We now have Aa− λWa = 0. Next multiply the W−1by both sides, that is the function ofW−1A− λI a = 0 ⇒ a = 0 orW−1A− λI = 0. Since we have W−1A − λI = 0, the function of detW−1A− λI = 0. We
now have an equation with k unknown quantities, which means that we have k solutions of λ value in the Lagrange multiplier with eigenvalues of λ1 ≥ λ2 ≥ · · · ≥ λk. Thus we can
put the maxλ1into
W−1A− λI a = 0, so that we have a of the discriminant coefficient to be put into s.t. a × a = 1 to build the discriminant function.
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