1. Introduction

Internal waves (IWs) are caused by differences in the temperature of water in the ocean. The presence of such anomalies in the seawater creates time-dependent and spatial variations in the sound speed profiles (SSPs) and affects underwater sound propagation due to the downward refraction of sound waves (Katsnelson et al. 2021; Rouseff 2001). Usually, sparse representations of SSPs via empirical orthogonal functions (EOFs) are used to support inversion algorithms for sound speed (North 1984). However, effective sparse representation of SSPs can be compromised due to high perturbations in the water column. Recently, Bianco and Gerstoft (2017) have shown that dictionary learning (DL), an unsupervised machine learning method, is better suited to sparsely represent SSPs. In this paper, DL and EOFs are used to model and classify measured SSPs affected by IWs during the Shallow Water Experiment 2006 (SW06).

Internal waves can be thought of as four-dimensional phenomena because of their effects in both 3D spatial (x, y, z) and temporal (t) domains. Characterization of the behavior and statistical properties of IWs have been carried out using 3D mapping techniques (Badiey et al. 2013, 2016), resulting in the general understanding of the regimens in the propagation of IWs. Several studies have shown that internal waves create significant variability in the speed of sound, affecting how sound propagates through the ocean, due to the fluctuations in the acoustic modal behavior (Flatté and Tappert 1975; Rouseff 2001; Helfrich and Melville 2006). The variation due to the passing of IW packets affects the propagation and reception of acoustic signals underwater because of drastic changes in the acoustic channel (Huang et al. 2008).

EOF analysis has been commonly used to represent SSPs as a linear combination of few orthogonal basis functions that describe the statistics of the sound speed uncertainty (Xu and Schmidt 2006; Abiva et al. 2019). Those resulting sparse representations are employed to aid inversion procedures (Huang et al. 2008). However, strong variations in the SSPs occurring because the passing of IWs yield a dramatic decay in the reconstruction accuracy of SSPs using EOFs (Sun and Zhao 2020; Roundy 2015). As a result, different approaches such as 3D mapping (Badiey et al. 2013) or learning methods (Jain and Ali 2006; Bianco and Gerstoft 2017; Sun and Zhao 2020) have been studied for modeling and reconstruction of SSPs.

Dictionary learning, an unsupervised learning method, aims to find a set of nonorthogonal basis functions (referred to as atoms) that can sparsely reconstruct signals (Zhang et al. 2015). The DL framework has been extensively applied to dimensionality reduction (Tošić and Frossard 2011), pattern recognition (Wright et al. 2010) and sparse representation modeling (Rubinstein et al. 2010). Recent studies have been conducted to sparsely represent SSPs using learned dictionaries (LDs) (Kuo and Kiang 2020). Bianco and Gerstoft (2017) showed that LDs are well-suited to generate sparse representations of SSPs using few basis functions. Sun and Zhao (2020) tested the effectiveness of LDs using HYCOM data and supported the results found by Bianco and Gerstoft (2017), concluding that nonorthogonal atoms allow for more flexible dictionaries and produce better sparse representations of SSPs.

Because of the relaxation of the orthogonal requirements and the possibility of generating optimal sparse representations, LDs have also been applied to clustering (Sprechmann and Sapiro 2010) and classification tasks (Tang et al. 2019; Suo et al. 2014). In this approach, specific dictionaries are trained to retain most of the meaningful information of each class. Then, testing data are classified by selecting the dictionary yielding the sparse representation that generates the lowest error (Ramirez et al. 2010; Zhao et al. 2018). Here, classification via dictionary learning is extended to label data containing SSPs affected by low, medium, and high IW activity collected during the SW06 experiment.

In this paper, there are two main contributions. First, a comparison of the ability to sparsely model SSPs with few basis functions is made between EOFs, complete LDs, and overcomplete LDs. Second, a DL-based SSPs classification setting is proposed and assessed via LD. The proposed framework is compared with standard classification algorithms such as support vector machine (SVM) and k-nearest neighbors (KNN) classifier. The dictionary atoms are calculated with online dictionary learning, a stochastic gradient-descent approach introduced by Mairal et al. (2009), while sparse coding is performed using the orthogonal matching pursuit algorithm and least absolute shrinkage and selection operator (lasso) convex optimization. This paper is structured as follows: In section 2 preliminary concepts and notations are presented. Section 3 introduces the data collected during the SW06 experiment used in this paper. Both EOF analysis and dictionary learning frameworks are introduced in section 4, and results for modeling and classification of SSPs are shown in section 5, followed by the conclusions in section 6.

2. Preliminaries and notation

In this paper matrices are denoted in sans serif (except for greek letters) uppercase boldface type, vectors are in lowercase boldface type, and scalars are in lowercase and uppercase italic. The ℓ_p-norm of a vector or matrix is represented as ǁ⋅ǁ_p, with 1 ≤ p ≤ ∞. For any $\mathsf{X}$, if p = 2, ǁ⋅ǁ_p takes the name of Frobenius norm ǁ⋅ǁ_{$\mathcal{F}$}. Notice that these p-norms are entrywise rather than the induced norms.

Any square diagonalizable matrix $\mathsf{X}$ ∈ Ρ^n×n can be factorized into its canonical form via eigendecomposition satisfying the linear equation $\mathsf{X}$ = $\mathsf{Q}$Λ$\mathsf{Q}$^T, where $\mathsf{Q}$ = [q₁, …, q_n] ∈ Ρ^n×n is the matrix containing eigenvectors and Λ = diag(λ₁, …, λ_n) ∈ Ρ^n×n is a diagonal matrix with the set of eigenvalues. Similarly, any rectangular diagonizable matrix $\mathsf{X}$ ∈ Ρ^m×n accepts the factorization $\mathsf{X}$ = $\mathsf{U}$Σ$\mathsf{V}$^T via singular value decomposition (SVD) (Stewart 1993), where $\mathsf{U}$ ∈ Ρ^m×m and $\mathsf{V}$ ∈ Ρ^n×n are the matrices of left and right singular vectors, respectively, and Σ ∈ Ρ^m×n contains the singular values. For the cases $X\mathsf{X}$^T and $\mathsf{X}$^T$\mathsf{X}$, the diagonal matrix of the nonzero eigenvalues is related to the singular values of $\mathsf{X}$, such that Λ = Σ² = ΣΣ^T = Σ^TΣ.

3. Experimental data

The raw data introduced in this work were collected during the SW06 experiment performed off the coast of New Jersey from mid-July to mid-September in 2006 (Tang et al. 2007; Newhall et al. 2007). During the experiment, 62 acoustic and oceanographic moorings were deployed in a “T” geometry as sketched in Fig. 1. This T mooring conformation measured data on an almost constant bathymetry of 80 m in the along-shelf direction and a bathymetry across shelf starting at 600 m going shoreward to 60-m depth. The intersection of the two paths in the T was populated with a cluster of 16 moorings to measure the 3D environment. Most of the environmental moorings in the area consisted of temperature, conductivity, and pressure sensors that measured the physical oceanography in the water column.

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Some of the SW06 moorings along shelf and across shelf conforming to a T geometry. SSPs used in this work are derived from data collected at SW30 station, which is marked with a white star.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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This paper uses data from mooring SW30, deployed at 39°1.501′N, 73°4.007′W, to study the time-evolving SSPs via dictionary learning. The SW30 station was part of the 16-mooring cluster located at the intersection of the T geometry deployed in the SW06 experiment. The location of mooring SW30 inside the cluster is marked with a white star in Fig. 1. The SW30 station had 11 unevenly spaced sensors collecting conductivity and temperature profiles from 14 to 83.3 m in a water column with seafloor at 86-m depth. For this study, temperature, conductivity, and pressure data were extracted from 0000:00 UTC 1 August to 1600:00 UTC 5 September 2006 at the SW30 location.

During the SW06 experiment, IW activities were reported as initiating at the shelf break and propagating toward the shore after 17 August 2006 (Badiey et al. 2013). The transition of internal waves over the area caused highly anisotropic SSPs such as the ones depicted in Fig. 2a, where temperature profiles exhibited significant variations across the water column and caused notable changes in both the acoustic channel and sound transmission.

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IW event spotted from 2100:00 UTC 17 to 1000:00 UTC 18 Aug 2006. (a) Temporal evolution of temperature profiles at mooring SW30. (b) The square of the buoyancy frequency N², in terms of depth, divided into four sections corresponding to different regimens. (c) Standard deviation (shading) and mean of the SSPs presented in (a). (d) Individual SSP samples at each geotime t_gi, with colors matching the vertical dashed lines in (a). (e) Buoyancy frequency N² for each of the four regimens shown in (b).

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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The study of IWs is a difficult task because of the spatial resolution of the measurements in experiment areas (Badiey et al. 2016). In this paper, only temporal IW anomalies are used for the analyses and applications presented in subsequent sections. As an example, the temporal displacement of an IW event spotted from 2100:00 UTC 17 to 1000:00 UTC 18 August 2006 at the SW30 location is presented in Fig. 2. The temperature variability across the water column produced by the passing of IWs provokes changes in the acoustic duct and degrades the acoustic propagation underwater. In Fig. 2a, the beginning of each stage (approaching, onset, propagation, and tail) of the IW event is marked with a dashed line in Fig. 2a and labeled with a geotime t_gi (i.e., t_g0, t_g1, t_g2, and t_g3). These abrupt changes in temperature alter the isotropic properties of the SSPs and produce drastic variations in the SSPs as shown in Figs. 2c and 2d. Figure 2c shows the mean ${\mu }_{{\mathsf{y}}_i}$ and standard deviation ${\sigma }_{{\mathsf{y}}_i}$ values of the SSPs over the entire time window in Fig. 2a, whereas Fig. 2d depicts individual SSPs at geotimes t_gi, with colors matching the vertical lines in Fig. 2a.

An additional consequence of the IW passing is the instability of vertical displacements that affects acoustic propagation. The periodicity at which a vertically displaced small volume of water oscillates is measured by the buoyancy frequency N (s⁻¹). The oscillations in the water column are expressed as the square of the buoyancy frequency N² to obtain real values, as shown in Fig. 2b. The time interval has been divided into four sections, each categorized by the regimens 1–4 describing the IW behavior. These regimens are 1) the approaching, 2) the onset, 3) the propagation, and 4) the tail of the IW event. The squared buoyancy frequency N² is given by g²ρ(βdS_A − αdΘ)/dP, where g is the gravitational acceleration (m s⁻²), ρ is the density (kg m⁻³), S_A is the absolute salinity (g kg⁻¹), Θ is the conservative temperature (°C), P is the pressure (Pa), and β and α are respectively the saline contraction and thermal expansion coefficients evaluated at the average values of S_A, Θ, and P. The mean value of N² for each of the four regimens indicated in Fig. 2b is shown in Fig. 2e. The behavior of the buoyancy frequency clearly shows the magnitude of oscillations at different depths during the IW passing.

The computed SSPs were structured as a group of sample vectors y_i such that $\mathsf{Y}$ = [y₁, …, y_n] ∈ Ρ^m×n, with m discrete depth points (features) and n SSP samples in time. SSP samples were generated every 15 s during the entire period. To obtain a better representation of the sound speed in the water column, shape-preserving piecewise cubic interpolation was used along the depths from 14 to 40 m with a spacing of 0.5 m, yielding m = 53 depth-dependent features. The above resulted in n = 205 547 SSP samples for the entire dataset (from 0000:00 UTC 1 August to 1600:00 UTC 5 September 2006).

As shown in Figs. 2 and 3, IW events can be categorized as phenomena with low, medium, and high effects in the water column by considering the amplitude of the internal wave (variation of temperature vs depth). To distinguish between IW activities with low, medium, and high incidence, the extracted SSPs were labeled in four different classes (1–4), as detailed in Table 1. These resulting labels serve for the supervised framework presented in the following sections. The classes were inferred via k-medoids clustering with Euclidean distance as the measuring metric (Park and Jun 2009). This approach is more robust to noise and outliers as compared with k-means because it minimizes a sum of pairwise dissimilarities instead of a sum of squared Euclidean distances. The clustering strategy is composed of two steps, the build step and the swap step. In the build step each of k clusters is associated with a potential medoid, whereas in the swap step each point is tested as a potential medoid by checking the sum of within-cluster distances to define a new medoid. At each iteration, every point is then assigned to the cluster with the closest medoid until convergence.

Table 1

Four resulting classes after applying the k-medoids to SSP data extracted from SW30 (from 0000:00 UTC 1 Aug to 1600:00 UTC 5 Sep 2006). The columns in the table show the number of SSP samples per class, the mean (${\mu }_{\mathsf{Y}}$), and the standard deviation (${\sigma }_{\mathsf{Y}}$) for the SSPs in each class.

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The aforementioned process led to a full-labeled dataset with four distinguishable subsets Y_i, one per class: i ∈ [1, …, 4]. Of the total number of samples in the dataset (n = 205 547), class 1 has n = 32 992 SSPs, class 2 has n = 59 630 samples, class 3 has n = 67 536 SSPs, and class 4 has n = 45 389 samples. The distribution of classes can be observed in Fig. 4. Basic descriptive statistics were calculated for each subset, as shown in Table 1. The mean ${\mu }_{{\mathsf{Y}}_i}$ and the standard deviation ${\sigma }_{{\mathsf{Y}}_i}$ of all elements in class i were computed as

${\mu }_{{\mathsf{Y}}_i}=\frac{1}{mn}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{y}_{ij}}}\text{\hspace{1em}and}$

${\sigma }_{{\mathsf{Y}}_i}=\sqrt{\frac{1}{(n-1)(m-1)}{\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=1}^n{(y_{ij}-{\mu }_{{\mathsf{Y}}_i})}^2}}},$

respectively. The values of ${\mu }_{{\mathsf{Y}}_i}$ and ${\sigma }_{{\mathsf{Y}}_i}$ for each class subset provide a general view of the magnitude and variability of the SSPs within class i. As shown in Table 1 and Fig. 3, those classes with higher ${\mu }_{{\mathsf{Y}}_i}$ and ${\sigma }_{{\mathsf{Y}}_i}$ present SSPs with the larger fluctuations in depth (classes 1 and 4), whereas classes with lower values of ${\mu }_{{\mathsf{Y}}_i}$ and ${\sigma }_{{\mathsf{Y}}_i}$ exhibit less-variable SSPs (classes 2 and 3).

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(a)–(e) Sound speed profiles from SW30 station calculated using the nine-term equation (Mackenzie 1981). The SSPs are depth dependent as indicated with the y axis. The x axis shows the number of SSPs in each panel. These SSPs are extracted every 15 s from 0000:00 UTC 1 Aug to 1600:00 UTC 5 Sep 2006. In (a), a total of n = 205 547 SSPs are extracted during the period described. In this work, k-medoids clustering is used to label the data. The resulting classes from 1 to 4 that are shown in (b), (c), (d), and (e), respectively. Notice that the SSPs on each class are not ordered in time.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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The entire labeled dataset presented in Fig. 3a was split into training/testing sets to be used for supervised classification of SSPs. Uniform random sampling was used to split the data, where 80% of data were destined for training while the remaining 20% were for testing, and resulted in 164 438 and 41 109 samples, respectively. The training data are meant to train the classification algorithms, and the testing data are used to perform classification and measure the performance of each model. The distribution of training and testing sets for each class is shown in Fig. 4.

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Distribution of data into the four classes calculated using a k-medoids algorithm. The original data are split into training and testing sets, whose number of samples per class are also depicted in the plot.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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4. Sparse representation of SSPs

Sparse representation aims to describe a dataset as a linear combination of few elements (basis) (Rubinstein et al. 2010). These elements capture the relevant statistical information that best describes the data and are combined with a matrix of few nonzero coefficients calculated by imposing an ℓ_p constraint that controls the sparsity level or nonzero elements (Zhang et al. 2015; Wright et al. 2010). In ocean acoustics, SSPs inversions are often regularized by considering a sparse representation of SSPs using EOF analysis (Gerstoft and Gingras 1996; Huang et al. 2008) or DL (Bianco and Gerstoft 2017). This section introduces a method to implement EOF analysis and DL to represent SSPs as a linear combination of basis functions using measured data from the SW06 experiment (see section 3).

5. Results and discussion

To illustrate the practicability of the methods presented in section 4, both DL and EOF analysis are applied to data collected at SW30 station during the SW06 experiment (section 2). In this section, DL and EOFs are implemented to sparsely represent SSPs altered by the passing of IWs using OMP algorithm for sparse coding. In addition, dictionary learning is later employed for SSP classification following the scheme described in section 4d, using the LARS-lasso algorithm for sparse coding.

The matrix $\mathsf{Q}$ ∈ Ρ^m×k containing the EOFs is calculated via SVD [Eq. (2)], and the learned dictionary $\mathsf{D}$ ∈ Ρ^m×k is obtained using the convex optimization shown in Eq. (7), via online dictionary learning algorithm. Equation (7) is solved iteratively using the LARS algorithm with a tolerance parameter δ = 1 × 10⁻⁸, number of iterations T_max = 2000, and regularization parameter λ = 1 (Pedregosa et al. 2011). Furthermore, sparse coding via OMP algorithm with T_max = 2000 is used to compute the coefficient matrix $\mathsf{C}$ ∈ Ρ^k×n to compare performance of EOFs and DL (section 5a). The number of nonzero elements in c_i, computed via the OMP algorithm, are set to exactly n_nz = 3 nonzero coefficients. For classification, only DL is used and sparse coding is done via the LARS-lasso algorithm with λ = 1, η = 1, T_max = 2000, and δ = 1 × 10⁻⁸ (see section 5).

a. Dictionary learning and EOF analysis for SSPs sparse representation

Because of the reward for sparsity and the absence of an orthogonality requirement, LDs can provide an alternative for a sparse representation that might yield more accurate reconstructions than traditional methods, such as EOFs. Bianco and Gerstoft (2017) showed that LDs are suitable for representing sound speed profiles in underwater environments, even outperforming EOFs with few basis functions.

To test the versatility of dictionary atoms in highly fluctuating environments such as those with internal waves, both EOFs and DL are compared using a portion of data collected from 0000:00 UTC 17 to 2359:59 UTC 18 August 2006 at SW30 station during high internal wave activity. The data used to compare both methods are shown between the two black lines drawn in Fig. 4a, with n = 11 520 SSPs sampled every 15 s measured at m = 53 different depths. EOFs are calculated for k = 53 basis functions. Similarly, a complete dictionary with k = 53 and an overcomplete dictionary with k = 100 are also computed. Sparse coding based on both EOFs and LDs is performed using n_nz = 3 nonzero coefficients via OMP algorithm. The resulting EOFs and atoms from the LD are shown in Fig. 5. Each EOF/atom contains meaningful information about the sound speed variability in terms of depth. Notice only the leading-order EOFs in Fig. 5a capture the variability of SSPs, whereas the SSP variance is distributed on all atoms of both complete (Fig. 5b) and overcomplete dictionary (Fig. 5c).

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Basis functions computed via (a) EOF analysis, (b) complete DL, and (c) overcomplete DL. Only the first leading-order EOFs describe variability in the SSPs, whereas variance is distributed along all atoms in the LDs.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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The explained variance ratio per EOF/atom is shown in Fig. 6 and complements the findings presented in Fig. 5. Most of the variance of the SSPs is mainly concentrated in the first k = 5 leading-order EOFs (Fig. 6a), while in the case of LDs, the variance is shared among most of the basis functions (Figs. 6a,b). Given these findings, SSPs can be reconstructed effectively using only the first leading-order EOFs, while in the case of LDs, the SSPs can be sparsely represented by a sparse combination of almost any atom in the dictionary.

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SSP explained variance ratio for (a) EOFs and (b) complete LD entries with k = 53 and for (c) overcomplete LD with k = 100 atoms using n_nz = 3 nonzero coefficients.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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As stated in section 4, the EOFs in the matrix $\mathsf{Q}$ are computed via SVD and have the property of being normalized orthogonal basis functions. Conversely, atoms in a dictionary $\mathsf{D}$ are not required to be orthogonal and are restricted to have unit ℓ₂-norm as stated in Eq. (6). Coherence among column entries in $\mathsf{Q}$ and $\mathsf{D}$ can be calculated by computing the Gram matrix ${\mathsf{G}}_{\mathsf{Q}}$ = | $\mathsf{Q}$^T$\mathsf{Q}$ | for EOFs, and ${\mathsf{G}}_{\mathsf{D}}$ = | $\mathsf{D}$^T$\mathsf{D}$ | for LDs. For the EOF analysis, ${\mathsf{G}}_{\mathsf{Q}}$ = $\mathsf{I}$, as shown in Fig. 7a, because of the orthogonal properties of EOFs. In contrast, atoms in the complete and overcomplete LDs are not necessarily orthogonal, as displayed in Figs. 7b and 7c.

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Coherence matrices for (a) EOFs, (b) complete LD entries, and (c) overcomplete LD (see Fig. 5) using n_nz = 3 nonzero coefficients. Eigenvectors in the EOF dictionary are orthogonal, whereas those of the LDs are not.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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The relaxation of the orthogonality requirement for LD atoms leads to more flexible dictionaries and richer data representation. As a result of this, better compression of data is achieved using dictionary learning. For both LDs and EOFs, increasing the number of basis functions k and nonzero coefficients n_nz yields lower representation errors. To study sparse representation of SSPs, the OMP algorithm is used to compute sparse codes with 1 ≤ n_nz ≤ 12 for EOFs and LDs. The root-mean-square error (RMSE) between the sparse representation ${\widehat{\mathsf{Y}}}_i$ and the original data $\mathsf{Y}$ is measured as

$\text{RMSE}=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{(y_{ij}-{\widehat{y}}_{ij})}^2}}}{mn}},$(11)

where m are depths, and n is the number of SSPs in $\mathsf{Y}$. The RMSE values as a function of n_nz, shown in Fig. 8, provide a method for comparing the different sparse representations. It is clear that, for n_nz ≥ 8, the EOFs, complete LD, and overcomplete LD give the almost the same RMSE. For n_nz < 7, however, the RMSE using EOFs is larger than for LDs, with the overcomplete LD yielding the lowest RMSE for small values of n_nz.

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Root-mean-square error of EOFs and complete LD, both with k = 53, and overcomplete LD with k = 100 when different numbers of nonzero coefficients n_nz are used for sparse coding.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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To demonstrate the efficacy of both EOFs and LDs to sparsely represent SSPs, six individual random samples y_i are chosen from the dataset $\mathsf{Y}$ and are depicted with solid green lines in Fig. 9. The sparse representation of SSPs ${\widehat{y}}_i$ of the six samples using the EOFs and the complete LD, both with k = 53 and n_nz = 3 are shown in Fig. 9. The dashed orange line presents the time-mean value from the entire dataset, dotted blue line shows the SSP reconstructed using EOFs, whereas dash–dotted red line the reconstruction using LDs. In addition, the absolute error between each real sample y_i and the reconstruction ${\widehat{y}}_i$ in terms of depth was computed as $\left|{\mathbf{y}}_i-{\widehat{y}}_i\right|$, and is shown with color bars (to the right of each line plot) for both LD and EOFs. Dark colors in the bars correspond to small errors, whereas light colors represent high absolute errors. For each case, only with n_nz = 3 basis functions both EOFs complete LD yield a good representation of SSPs. Even with internal wave events passing by, complete LD provides a nearly perfect representation of the six SSPs and outperforms conventional EOF analysis in the sparse reconstruction of sound speed profiles.

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Sparse representation of six SSP samples using complete DL and EOF with n_nz = 3 nonzero coefficients and k = 53 atoms and EOFs. For each sample, color bars present the absolute error between the sample and the reconstruction in terms of depth for LD and EOFs, respectively.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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b. Classification of SSPs via dictionary learning

In previous sections is mentioned that learned dictionaries $\mathsf{D}$ can retain meaningful information from a dataset $\mathsf{Y}$ ∈ Ρ^m×n that can or not be labeled in a class ϕ_j, j = 1, …, l. However, limitations for data classification arise when few n_nz atoms are used because of the impossibility of representing the variability in the data. Overcomplete dictionaries, because of their redundant nature, are suitable for classification since they can perform data analysis with functions that are likely to match the characteristics in different classes of data.

In this section, the dictionary learning framework is used to label sound speed profiles following the classification setting introduced in section 4d. Complete and overcomplete LDs are employed to classify data with low, medium, and high occurrence of internal wave events in the ocean. For the best analysis, the data introduced in section 2 and Table 1 are used for to test the algorithms in this section.

As the extracted SSPs from the experiment are not initially labeled, each SSP sample is labeled into four classes (1–4) depending on the internal wave level using k-medoids algorithm, as shown in section 3. Once all data samples are labeled, the effectiveness of LD atoms to infer a class type for SSPs is studied using the training/testing sets described in section 3. Here, class-specific dictionaries are learned using training data, to then classify unlabeled SSPs samples in the testing set utilizing a dissimilarity metric $\widehat{\mathcal{R}}(\mathbf{y},\widehat{y})$ [Eq. (10)]. The scheme introduced in section 4d is implemented for complete (k = 53) and overcomplete (k = 100 and k = 300) dictionaries. In addition, dictionary learning is compared with SVM and KNN algorithms, both commonly used for supervised classification tasks. KNN is implemented using k = 100 neighbors and Euclidean distance as metric, whereas ℓ₂ regularized SVM with Gaussian kernel is employed with a one-vs-the-rest configuration for multiclass classification.

The LDs models, KNN, and SVM are trained using the training set with n = 164 438 samples and four different classes and then tested on the remaining n = 41 109 unseen samples, whose labels are inferred by the classification model. That is, the trained classifier h ∈ $\mathcal{H}$ aims to assign a label to unlabeled data ${\mathbf{y}}_i^{\text{test}}$, such that ${\widehat{j}}_i=h({\mathbf{y}}_i^{\text{test}}),\widehat{j}=1,\dots ,l$. Here, the accuracy is reported for each model and accounts for the number of correct classifications over the total number of samples in the testing set. This metric is calculated as

$\text{accuracy}=\frac{{\displaystyle \sum _{i=1}^{n}1({\widehat{j}}_i=j_i)}}{n}\times 100\%,$(12)

where $1$(⋅) is the indicator function, n is the total number of samples in the dataset, j_i is label or ground truth for the ith sample, and ${\widehat{j}}_i$ is the inferred class for the ith SSP, such ℓ that ${\widehat{j}}_i=h({\mathbf{y}}_i^{\text{test}})$.

The classification results for KNN, SVM, and LDs are shown in Table 2, where accuracy is reported for all the models using the entire testing data along with the performance for individual subsets corresponding to each independent class (j). As demonstrated in section 5a, the overcomplete LD yields lower reconstruction errors than complete LDs. This fact is also corroborated by results in Table 2, where the overcomplete LD with k = 300 reaches higher accuracy than LDs with k = 53 and k = 100 for all the four classes.

Table 2

Classification results for KNN, SVM, complete LD, and overcomplete LDs for testing data. Accuracy (%) is reported for each classification model. The first four rows in the table correspond to independent classes in the testing set, and the last row presents the overall accuracy for each model in the complete testing set. Boldface type indicates which model performed best for each class/set.

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It is notable that results of the overcomplete LD with k = 300 are comparable to SVM and KNN and are even better at differentiating between classes with high internal wave activity (classes 1 and 4). Notice that the misclassification of SSPs by LDs is due to the possible lack of sufficient information about the variability retrieved by the k atoms. It is clear to see that the more atoms the dictionary uses, the more the accuracy will be. Therefore, the classification results reported for LDs in Table 2 can be improved by increasing the number of atoms used. This study shows that classification of SSPs via dictionary learning is feasible and can be extended to large-context scenarios as long as exists labeled data to compute specific dictionaries for each class.

From results in Table 2, it is possible to conclude overcomplete dictionary learning offers a good alternative for classifying SSPs with high internal wave activity if using sufficient k atoms. The relaxation in the orthogonal constraint in the basis functions allows DLs to capture the most representative information from data. Each class-specific dictionary $\mathsf{D}$_j tends to have different patterns than others from different classes ($\mathsf{D}$_j′). The coherence ${\mathsf{G}}_{{\mathsf{D}}_j}$ for class-specific learned dictionaries with k = 53, k = 100, and k = 300 atoms is shown in Fig. 10. Notice that for different values of k and class, each class-specific dictionary presents different and distinctive patterns than are being learned from data.

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Coherence of individual class-specific learned dictionaries for classification of SSPs: (a)–(d) coherence for the complete LD (k = 53), (e)–(h) coherence of overcomplete LDs with k = 100 atoms for all the classes, and (i)–(l) coherence for LDs with k = 300 for classes (left) 1, (left center) 2, (right center) 3, and (right) 4.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-21-0090.1

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It is important to remark that if the number of k atoms increases, the dictionary will lead to higher accuracy with a cost of an increment in the complexity of the convex optimization. Therefore, it is important to consider the trade-off between accuracy and complexity when training LDs, as is done with any learning model, such as neural networks. In cases where there are not sufficient labeled training data, it is possible to apply data processing techniques such as data augmentation to increase the variability and the number of data samples within a class (Castro-Correa et al. 2021).

Similar to previous research (Bianco and Gerstoft 2017; Sun and Zhao 2020), overcomplete LDs perform well as a sparse representation algorithm for SSPs. Dictionary learning outperforms EOFs because it tends to distribute the SSPs energy among all the atoms and its ability to generate nonorthogonal functions. Furthermore, as few as n_nz nonzero coefficients are needed to consistently provide a complete enough representation to achieve very low training error even when dictionaries are trained on SSPs in the presence of internal waves. Results demonstrate competitive performance of LDs with respect to standard classification models. The amount of data, the sparsity level provided by n_nz, and the number of atoms k are crucial factors to obtain optimal sparse representations of SSPs.

6. Conclusions

Both dictionary learning and empirical orthogonal functions were implemented to sparsely represent SSPs disturbed by the passing of internal waves in the SW06 experiment. Because of their redundant nature and their ability to generate nonorthogonal basis functions, overcomplete learned dictionaries showed better performance for reconstructing SSPs than the EOFs and the complete LD when using the best n_nz = 3 combination of basis functions.

The presence of internal waves in the water column causes highly anisotropic SSPs. The variability in the SSPs induced by IWs was reflected in higher errors when both EOF and dictionary learning frameworks were applied. Under those circumstances, overcomplete dictionaries with k > m atoms were shown to achieve even better compression of SSPs than EOFs and the complete LD k = m. The improvement in the reconstruction of SSPs was produced due to the relaxation of the orthogonal requirements, and the number of atoms used in the dictionary.

In this paper, the classification of SSPs via dictionary learning was introduced. Here, specific dictionaries were built for each class of internal wave activity, and testing datasets were classified by finding the dictionary corresponding to the most accurate sparse representation. Results demonstrated that overcomplete learned dictionaries trained on labeled data are suitable to classify SSPs successfully. When the training data are representative and are labeled by a class, the resulting overcomplete dictionary can be effectively applied to other datasets to classify SSPs.

This work provides insights into the application of learned dictionaries for the representation and classification of SSPs. Further analyses are required to find the optimal number of nonzero coefficients n_nz and k atoms used to obtain the best possible sparse representation of SSPs. Future research needs to be conducted to find a better-suited dissimilarity cost $\widehat{\mathcal{R}}(\cdot )$ that can yield higher classification performance, while further studies are required to determine if undercomplete LDs (k < m) are suitable for sparse representation and dimensionality reduction of SSPs.

Sparse representation of sound speed profiles using dictionary learning promotes and expedites research into internal waves. A representative dictionary of basis functions is an efficient way to store and generate thousands of unique sound speed profiles, given that the dictionary succinctly and effectively represents the small-scale variability to model. Using learned dictionaries to generate realistic and variable training datasets may further the progress of machine learning in many undersea applications.

Data availability statement.

Data used in this work were collected during the Shallow Water Experiment 2006 (SW06). Because of privacy and ethical concerns, neither the data nor the source of the data can be made available.