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  • View in gallery

    Phase-space diagram (surface displacement vs acceleration) for three monochromatic waves (η1, η2, η3). The dark gray parallelogram, light gray hexagon, and dashed black line enclose all curves defined by the sum of η1 + η2, η1 + η2 + η3, and η1 + η3, respectively, for random phases.

  • View in gallery

    (left) Deployment site of the wave buoy (cross) off the coast of King George Island, near the Antarctic Peninsula. (right) A photograph of the deployed wave buoy mooring is shown, with the main marker buoy on the left side and the wave buoy floating on the right side of the image. The Po/V Almirante Maximiano can be seen in the background. Photo taken by Bruna Pavani.

  • View in gallery

    Sensitivity of spectral properties of vertical surface acceleration derived from surface elevation measurements: (a) generated JONSWAP spectrum; (b) acceleration spectrum; (c) cumulative sum of the acceleration spectrum. The gray and black lines correspond to time series with sampling rates of 2.5 and 10 Hz, respectively. In (b) and (c), the acceleration spectra of the dot–dashed and dotted lines are obtained through a three- and nine-point central difference scheme of the surface displacement time series, respectively; the solid line represents the exact spectrum S(f) = ω4E(f).

  • View in gallery

    The radius of the minor axis x2 of the ellipse against Eq. (11).

  • View in gallery

    Phase-space diagram for (a) a synthetic JONSWAP spectrum (Hs = 2 m, Tp = 6 s, Δt = 0.4 s) and (b) the same spectrum, but with a 0.6 m swell superimposed (T = 10 s). Ellipses are defined by Eqs. (5) and (11), with p = 2, 4, and 6, for the smallest to largest ellipses, respectively. The black line aligned with the major axis of the ellipse corresponds to a monochromatic wave with period Tm02 and amplitude 0.5Hs; the second black line corresponds to a monochromatic wave with period Tp and amplitude 0.5Hs. The probability density function of the normalized vertical displacement η/ση and vertical surface acceleration ∂2η/∂t2/σA are provided along horizontal and vertical axes (not to scale). Dotted lines correspond to twice the standard deviation of the dataset. Crosses identify observations where z > 4ση. Circles define points outside middle ellipse.

  • View in gallery

    Example observations of surface elevation measured by a moored Spotter wave buoy: (a) time series with one wave anomaly; (b) phase-space diagram before removal of wave anomalies; (c) wave energy spectrum before and after removal of wave anomaly (to calculate the spectrum after anomaly removal, the time series was shifted 5 min to avoid discontinuities in the time series). Wave 1 identifies the wave anomaly in (a) and (b). Ellipses in (c) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

  • View in gallery

    Example observations of surface elevation measured by a moored Spotter wave buoy: (a) time series with two significant wave events; (b) “wave 2” in the wave record where thick gray and black line marks sections outside middle and largest ellipses, respectively; (c) phase-space diagram before removal of wave anomalies where crosses and circles define “wave 1” and “wave 2,” respectively; (d) wave energy spectrum before and after removal of wave anomaly (to calculate the spectrum after anomaly removal, the time series was shifted five minutes to avoid discontinuities in the time series). Ellipses in (b) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

  • View in gallery

    Example observations of surface elevation measured from a fixed offshore platform using a laser altimeter, with Hs = 4.5 m and Tp = 11.4 s: (a) time series with two major and one minor spike; (b) phase-space diagram before removal of wave anomalies; (c) wave energy spectrum before and after removal of wave anomalies. Wave anomalies are marked with black circles. Ellipses in (b) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

  • View in gallery

    Comparison of (a) significant wave height Hs and (b) mean wave period Tm01 before and after removal of wave anomalies. Wave anomalies are defined based on a phase-space threshold of p = 6 (see section 5a).

  • View in gallery

    Geographical position of the wave buoy for the complete deployment (30 min intervals) relative to the central mooring. Gray scale defines the relative error of the significant wave height Hs, defined as the ratio of wave height before and after removal of erroneous observations in the time series.

  • View in gallery

    Phase-space diagram of synthetic wave data generated through a phase-resolved numerical model: (a) density contours of full time series without wave extremes; (b) extreme wave troughs and crests only where the black circles identify the minima/maxima and the gray dots the neighboring data points of these minima/maxima. In (a), the probability density functions of the normalized surface displacement and vertical acceleration are shown by the red curve, for comparison the black curve depicts a Gaussian distribution. In (b), the red line describes the linear relation between surface displacement and surface acceleration of a monochromatic wave with period T0 = (m0/m4)0.25 [see Eq. (8)]. The dashed line in (b) identifies the approximate condition of ∂2η/∂t2 = g.

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Wave Anomaly Detection in Wave Measurements

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  • 1 Department of Infrastructure Engineering, University of Melbourne, Melbourne, Victoria, Australia
  • | 2 Laboratory for Regional Oceanography and Numerical Modeling, Qingdao National Laboratory for Marine Science and Technology, Qingdao, China
  • | 3 Laboratory of Ocean and Atmosphere Studies, Earth Observation and Geoinformatics Division, National Institute for Space Research (OBT/INPE), São José dos Campos, Brazil
  • | 4 Institute of Energy and Environment, University of São Paulo, São Paulo, Brazil
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Abstract

Quality control measures for ocean waves observations are necessary to give confidence of their accuracy. It is common practice to detect anomalies or outliers in surface displacement observations by applying a standard deviation threshold. Besides being a purely statistical method, this quality control procedure is likely to flag extreme wave events erroneously, thereby impacting higher-order descriptions of the wave field. In this paper we extend the use of the statistical phase-space threshold, an established outlier detection method in the field of turbulence, to detect anomalies in a wave record. We show that a wave record in phase space (here defined as a diagram of displacement against acceleration) can be enclosed by a predictable ellipse where the major and minor axes are defined by the spectral properties of the wave field. By using the parameterized ellipse in phase space as a threshold to identify wave anomalies, this is a semiphysical filtering method. Wave buoy data obtained from a mooring deployed near King George Island, Antarctica [as part of the Antarctic Modeling Observation System (ATMOS)], and laser altimeter data obtained at the Northwest Shelf of Australia were used to demonstrate the functioning of the filtering methodology in identifying wave anomalies. Synthetic data obtained using a high-order spectral model are used to identify how extreme waves are positioned in phase space.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Joey Voermans, jvoermans@unimelb.edu.au

Abstract

Quality control measures for ocean waves observations are necessary to give confidence of their accuracy. It is common practice to detect anomalies or outliers in surface displacement observations by applying a standard deviation threshold. Besides being a purely statistical method, this quality control procedure is likely to flag extreme wave events erroneously, thereby impacting higher-order descriptions of the wave field. In this paper we extend the use of the statistical phase-space threshold, an established outlier detection method in the field of turbulence, to detect anomalies in a wave record. We show that a wave record in phase space (here defined as a diagram of displacement against acceleration) can be enclosed by a predictable ellipse where the major and minor axes are defined by the spectral properties of the wave field. By using the parameterized ellipse in phase space as a threshold to identify wave anomalies, this is a semiphysical filtering method. Wave buoy data obtained from a mooring deployed near King George Island, Antarctica [as part of the Antarctic Modeling Observation System (ATMOS)], and laser altimeter data obtained at the Northwest Shelf of Australia were used to demonstrate the functioning of the filtering methodology in identifying wave anomalies. Synthetic data obtained using a high-order spectral model are used to identify how extreme waves are positioned in phase space.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Joey Voermans, jvoermans@unimelb.edu.au

1. Introduction

Quality control (QC) of ocean wave measurements provides confidence of the accuracy of wave observations (e.g., IOOS 2019) for the purpose of model calibration and validation, and the assimilation of real-time data in operational forecasting models (e.g., Gilhousen 1994; Komen et al. 1996). QC procedures are applied at different stages of the data pipeline where the initial stages target the functioning of the sensors and the last stages commonly focus on the identification of erroneous data in a time series. Besides sensor failure, inaccuracies in a wave record often occur due to instrument design limitations, sensor deterioration, vandalism, transmission issues and/or environmental factors, and, ultimately, add unwanted noise to the signal. As this noise can be significantly different from the true signal, they form physical and/or statistical anomalies. To improve data quality, QC methods are designed to flag these anomalies.

By far the most commonly used QC method for wave analysis is the use of a cutoff threshold in terms of the standard deviation of a record. That is, all data in a time series X(t) exceeding p times its standard deviation σX with respect to the mean of the time series X¯ are flagged as outliers (i.e., |X(t)|>X¯+pσX). For wave measurements, a common value for p is 3 (e.g., NDBC 2009) or 4 (e.g., IOOS 2019). While perfectly able to flag spikes in a wave record, the standard deviation threshold method cannot distinguish between a rare but natural event and instrument problem. For example, rogue waves, typically defined as an individual wave crest exceeding about 5 times the standard deviation of the surface elevation (e.g., Dysthe et al. 2008; Candella 2016), are most likely to be flagged as an anomaly and, without interference, are at risk of being removed from any wave record. Though the removal of a single waves does not necessarily impact the first-order statistics of the wave field significantly, it does impact the higher-order statistics of the wave field, perhaps most notably, the probability distribution of extreme wave heights.

Alternative to, or, alongside the standard deviation threshold, equivalent threshold methods based on the measured or estimated surface velocity and surface acceleration can be used to identify anomalies, or the data quality can be defined based on the spectral properties of the wave record (Tucker 1993). More sophisticated anomaly detection methods include the use of wavelet analysis for wave measurements (e.g., Huang 2004), and the use of a phase-space threshold in turbulence data (e.g., Goring and Nikora 2002). Though originally designed for the despiking of time series of turbulent velocity components obtained through acoustic Doppler velocimeters, the phase-space threshold is able to detect (multipoint) spikes in a record based on the statistical properties of the time series. Unfortunately, all these methods are purely statistical such that the choice of the thresholds relies on sampling statistics or the experience of the user.

In this study we extend the use of the phase-space threshold method to the detection of wave anomalies in surface displacement observations. We show that the characteristics of the wave record in phase space are predictable and relate strongly to the physical properties of the wave field. Synthetic data are used to understand how a random wave field in the absence of erroneous data behaves in phase space. Wave buoy data, obtained as part of the Antarctic Modeling Observation System (ATMOS) near King George Island (Antarctica), and laser altimeter data obtained at the Northwest Shelf of Western Australia, are then used to define a threshold to identify anomalies in surface displacement time series. High-order spectral model simulations are used to identify how extreme waves fit into this framework.

2. Phase-space threshold

To understand the behavior of surface gravity waves in phase space, consider a monochromatic wave η(t) = a sin(ωt + ϕ) with amplitude a, phase ϕ, radian frequency ω = 2π/T, and wave period T. The corresponding acceleration of a water particle at the surface is then given by d2η(t)/dt2 = −ω2a sin(ωt + ϕ). As the surface displacement and acceleration are 180° out of phase, minima (maxima) in surface displacement correspond to maxima (minima) in acceleration. Thus, a monochromatic wave appears as a straight inclined line in an acceleration–displacement phase-space diagram (i.e., see η1, η2, and η3 in Fig. 1). The length and inclination angle of these lines are defined by the amplitude of the surface displacement (a) and acceleration (ω2a), that is, the wave period and wave amplitude of the harmonic. When two harmonic waves are added, the relation between surface displacement and acceleration becomes more complex. For any phase difference, the paths in the phase-space diagram resemble Lissajous curves. The set of curves, each with a random phase, is enclosed by a parallelogram (Fig. 1). The longest diagonal of the parallelogram is defined by the resultant of the maximum (minimum) acceleration and minimum (maximum) surface displacement of the individual harmonics. When three harmonics are added, the boundaries are defined by a hexagon, again with the longest diagonal defined by the resultant of the individual harmonics (e.g., Fig. 1). We also note that the width of the elongated polygons seems to be related to the range of periods of the individual harmonics. Hence, this suggests that when the properties of the harmonics are known, some properties of the polygon enclosing the wave record in phase space are known too.

Fig. 1.
Fig. 1.

Phase-space diagram (surface displacement vs acceleration) for three monochromatic waves (η1, η2, η3). The dark gray parallelogram, light gray hexagon, and dashed black line enclose all curves defined by the sum of η1 + η2, η1 + η2 + η3, and η1 + η3, respectively, for random phases.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

A more realistic wave field is, however, described by a superposition of a large number of harmonics (at least in a linear approximation), each with different period, amplitude, and phase (note that we have neglected directional information here):
η(t)=i=1naisin(ωit+ϕi),
d2η(t)dt2=A(t)=i=1nωi2aisin(ωit+ϕi).
When the number n of independent harmonics is large, the distributions of surface displacement and acceleration are Gaussian distributed (central limit theorem), implying that the wave record in phase space can be enclosed by an ellipse, the axes of which, are defined by the properties of the wave field. Here, we will define the major and minor axes of the enclosing ellipse and use the ellipse to identify anomalies in a wave record.

a. Major axis

Noting that the longest axis of the polygons described in Fig. 1 is defined by the resultant of the amplitudes of displacement and acceleration of the individual harmonics and that the variances of the surface displacement and surface acceleration are, respectively, defined by
ση2=1ni=1n12ai2,
σA2=1ni=1n12(ωi2ai)2,
the major axis of the ellipse is considered proportional to the standard deviations of the surface displacement η and acceleration A, where p defines the fraction of the wave record enclosed by the ellipse (e.g., for p = 1, about 68% of the wave record falls within the bounds of the ellipse). When the axes of the phase-space diagram are nondimensionalized by ση and σA, respectively, the radius of the major axis of the ellipse can be defined as
x1=2p
and is inclined under an angle −atan2(p, p) = −1/4π.
Although the description of x1 [Eq. (5)] can be interpreted as purely statistical, it is strongly related to the properties of the wave field. Let us consider a monochromatic wave η0(t), with corresponding acceleration A0(t), that characterizes the wave field described by Eqs. (1) and (2). In line with Eq. (5), we define its respective amplitudes, a0 and ω02a0, as p times the standard deviation of the surface displacement ση and acceleration σA of the wave record, respectively. Taking n → ∞, it follows that the variance of the surface displacement [Eq. (3)] can be written in terms of the wave energy spectrum E(f) of the wave record:
ση2=0E(f)df=m0,
and similarly for the variance of the vertical wave particle acceleration:
σA2=16π40f4E(f)df=16π4m4,
where m0 and m4 are also known as the zeroth- and fourth-order moment of the wave spectrum. After equating the amplitude of surface acceleration (ω02a0) to A, noting that we defined the wave amplitude as a0 = η, the wave period of the characteristic monochromatic wave T0 is defined as
T0=(m0m4)1/4.
After defining the significant wave height of the wave field as Hs=4m0, the major axis of the ellipse is equivalent to a monochromatic wave η0 = (p/4)Hs sin(2πt/T0). Hence, the major axis of the ellipse enclosing the wave record is critically defined by the properties of the wave field.

Though Eq. (8) is sound from a theoretical point of view, the dependence of T0 on m4, as we will show later in section 3c, is problematic from an experimental point of view.

b. Minor axis

Unlike the major axis of the ellipse, the dependence of the radius of the minor axis (x2) in terms of wave field properties is difficult to discern from Fig. 1. We therefore approximate this dependence, for now, by ignoring the inclination angle of the ellipse. When the ellipse is vertically oriented, the minor axis is defined by the distribution of the surface elevation under the condition that the acceleration is approximately zero. In the case of two harmonics with similar amplitude [i.e., η = η1 + η2 and η1/η2 = O(1)] it can be derived that the maximum surface displacement given ∂2η/∂t2 = 0 is ηmax=(1T22/T12)a1, where subscripts 1 and 2 in T and a refer to the period and amplitude of the two individual harmonics. Note that when T1 = Tm01 = m0/m1 and T2 = Tm02 = (m0/m2)1/2 are taken, this relation is equivalent to the product of a typical wave amplitude with the square of the spectral bandwidth υ, where υ as defined by Longuet-Higgins (1983):
υ=(m0m2m121)1/2.
Hence, it is expected that length of the minor axis x2 of the ellipse is a function of the typical wave height of the wave record, say Hs, and the spectral bandwidth υ. The exact relation will be determined empirically using our wave buoy data in section 4.

3. Methods

a. Experimental data

Data were obtained from a wave buoy mooring deployed near King George Island (Antarctica) in November 2019 (Fig. 2). This deployment belongs to the observational activity developed by ATMOS. ATMOS was a response to the Brazilian Antarctic Program (PROANTAR) and aims to improve our understanding of sea ice–ocean–atmosphere–waves interactions and turbulent fluxes exchanges in their interface, at micro- and mesoscales in the Atlantic sector of the Southern Ocean. At this time the ATMOS-1 research cruise was on board of the Brazilian Navy Polar Vessel (Po/V) Almirante Maximiano (H-41), and she facilitated the mooring of the wave buoy. The moored wave buoy (Spotter wave buoy, Sofar Ocean Technologies) was deployed for 4 months (November 2019–March 2020) off the coast of King George Island (62.2°S, 58.2°W, Fig. 2). The water depth at the site of deployment is approximately 80–90 m. The vertical component of the mooring system was 120 m in length and consisted of an anchor weight at the sea floor and a large marker buoy (170 kg buoyancy) at the sea surface. To limit the horizontal restriction of the wave buoy, the wave buoy was connected to the large marker buoy by two sections of 20 m rope split by a small marker buoy. The mooring system facilitated many other environmental sensors but only the wave data measured by the wave buoy is used in this study.

Fig. 2.
Fig. 2.

(left) Deployment site of the wave buoy (cross) off the coast of King George Island, near the Antarctic Peninsula. (right) A photograph of the deployed wave buoy mooring is shown, with the main marker buoy on the left side and the wave buoy floating on the right side of the image. The Po/V Almirante Maximiano can be seen in the background. Photo taken by Bruna Pavani.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

The Spotter is a small solar powered wave buoy (approximately 40 cm × 40 cm) with Iridium transmission capabilities, allowing for real-time monitoring of wave sea state. The wave buoy measures surface elevation η at a sampling rate of 2.5 Hz and has an accuracy of approximately 2 cm. For more details on the Spotter wave buoy and its performance against industry standard wave buoys, see Raghukumar et al. (2019) and Voermans et al. (2020).

Though wave energy spectra were transmitted every 2 h during the deployment, the wave buoy was retrieved in March 2020 providing access to the continuous time series of the wave measurements. As the deployment site is sheltered from the Drake Passage by the South Shetland Islands, waves measured were relatively mild, but nevertheless, significant wave height up to 4 m was recorded. Wave climate is considered to be a mixture of wind sea and swell. Though the restrictions of horizontal motion of the wave buoy were limited by the mooring design, infrequent occurrences of physically impossibly large waves were observed in the wave records. The resulting spectra associated with these large waves were characterized by added wave energy in the low-frequency range. We attribute these wave anomalies to mooring line restrictions and will show in section 4 that they are clearly distinguishable from the rest of the wave record in phase space.

To broaden the assessment of the anomaly detection method, the phase-space threshold method will also be applied to a time series containing a different type of error, namely, single and multipoint spikes. For this, we use wave observations obtained from an offshore platform on the Northwest Shelf of Western Australia. The Northwest Shelf is prone to tropical cyclones and is, as such, an ideal location to study air–sea interactions during extreme weather conditions. For observational and modeling purposes, the fixed platform contains a variety of atmospheric and oceanic instruments, including four laser altimeters in rectangular formation to measure ocean surface elevation and directional properties of the waves. The laser altimeters are located about 28 m above mean sea level and measure the distance between the laser and ocean surface at a frequency of 5 Hz through reflection of the laser light at the ocean surface. Contrary to a wave buoy mooring, noise from a laser altimeter is typically characterized by single- or multipoint spikes. The reader is referred to Voermans et al. (2019) for more details on these laser altimeter observations.

b. Synthetic data

Synthetic data are used to confirm whether a random wave field, without wave anomalies, can indeed be enclosed by an ellipse in phase space. To obtain time series of surface elevation in the absence of erroneous data, both linear and nonlinear synthetic wave data were generated. Following a linear approach, time series were generated using the Wave Analysis for Fatigue and Oceanography (WAFO) MATLAB toolbox (Brodtkorb et al. 2000) which produces a wave record from a predescribed spectrum. While it is in general sufficient to consider a linear wave field for the purposes of this study (i.e., we replicate a sea state through the superposition of many independent linear wave components), this is not necessarily the case for extreme waves.

To understand how extreme waves are positioned relative to the ellipses defined by the linear framework in section 2, we also generated synthetic data using HOS-Ocean, a high-order spectral code that solves the deterministic propagation of nonlinear wave fields in deep water (Ducrozet et al. 2016). This code has been used before to study extreme waves (e.g., Kirezci et al. 2020, manuscript submitted to Ocean Dyn.). A total of 210 simulations were performed to generate time series where the initial conditions follow a conventional JONSWAP spectrum with cosine squared directional distribution function G(θ):
G(θ)={1βcos2(2πθ4β),|θ|β0,|θ|>β.
The spectral shape is specified by JONSWAP shape parameters α ∈ [0.0059, 0.0098], γ ∈ [3, 6] and the directional spreading parameter β ∈ [0.8, 1.2]. In the HOS-Ocean model, the Taylor expansion of the solution is truncated at the fifth order, hence, the impact of high-order nonlinear interactions on surface waves were ensured. The random phase effect is also considered with multiple realizations of the simulations. The peak period Tp is kept constant at 5 s, the significant wave height varies slightly Hs = 1.29 ± 0.14. While the values of α, γ, and β define different sea states (though in the absence of swell), they do not impact the interpretation of the results nor the conclusions directly, thus, we do not further distinguish between these properties hereafter. From each simulation, only the time series with the largest minimum and maximum momentary surface elevation where selected and used in this study, providing 420 wave records, 210 of which include extreme wave crests in the range η/ση ∈ [4.2, 7.7] and the other 210 include extreme wave troughs in the range of η/ση ∈ [−5.7, −3.8]. For more information on the higher-order spectral model and its use in the study of extreme waves, the reader is referred to Ducrozet et al. (2016) and Kirezci et al. (2020, manuscript submitted to Ocean Dyn.), respectively.

c. Data processing

To place the wave record into phase space, both the surface displacement η(t) and surface acceleration A(t) time series need to be known. While most conventional wave buoys obtain η(t) through double integration of A(t), typically measured using accelerometers, for our datasets only η(t) is known. Hence, we focus here on the numerical approximation of A(t) from η(t) and the impact of sampling frequency and numerical method on the estimation of A(t) from η(t).

Two random time series were generated (one with a sampling frequency of Δt = 0.4 s, the other 0.1 s) with a spectral shape equal to that of a JONSWAP spectrum (Fig. 3a) using the WAFO MATLAB toolbox (Brodtkorb et al. 2000). The spectral properties of the time series are the same for both cases, with Hs = 2 m, Tp = 6. Beyond the peak frequency, the spectra are near identical and follow a f−5 relation. While we note that there are minor differences between the spectra of the two time series (for numerical reasons), these differences have negligible impact on the following interpretation. The value of A(t) was then calculated using a simple three-point and nine-point central difference scheme. In Fig. 3b the spectra of the different time series of A(t) are compared against the exact spectrum of the surface acceleration S(f), obtained from the wave energy spectrum S(f) = ω4E(f). As one would expect, the exact solution of S(f) for both the 2.5 and 10 Hz time series overlap, whereas the nine-point central difference scheme is more accurate than the three-point central difference scheme.

Fig. 3.
Fig. 3.

Sensitivity of spectral properties of vertical surface acceleration derived from surface elevation measurements: (a) generated JONSWAP spectrum; (b) acceleration spectrum; (c) cumulative sum of the acceleration spectrum. The gray and black lines correspond to time series with sampling rates of 2.5 and 10 Hz, respectively. In (b) and (c), the acceleration spectra of the dot–dashed and dotted lines are obtained through a three- and nine-point central difference scheme of the surface displacement time series, respectively; the solid line represents the exact spectrum S(f) = ω4E(f).

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

When considering the cumulative distribution of S(f) (Fig. 3c), it can be observed that, unlike the numerical approximations, the exact distribution does not reach an asymptotic value and implies that the true value of m4 or σA [i.e., the square root of the integral of S(f)] is strongly affected by the cutoff frequency. This is a direct result of the f−4 or f−5 dependency of the wave energy spectrum at high frequencies. Importantly, for both the 3- and 9-point central difference numerical approximations of A(t), an asymptotic value is reached, albeit generally with a (significantly) different value than the theoretical value.

While this error is unwanted, it does not necessarily negatively impact the description of the ellipse and data in the phase space. That is, as long as σA is based on one numerical approximation consistently, both A(t) and the derived ellipse scale with the resulting error systematically. However, it does mean that, in contrast to a theoretical phase-space diagram, the radius of the major axis x1 cannot be approximated by a monochromatic wave with period T0 = (m0/m4)1/4 from experiments as m4 is likely to be underestimated when derived from η (Fig. 3c). As the wave buoy measures surface elevation at a sampling frequency of 2.5 Hz, and therefore an error will always be made in approximating m4, hereafter, A(t) is simply calculated based on a 3-point central difference numerical scheme. While a more accurate value of σA can be determined from S(f) when derived from E(f) directly, this approach will lead to statistical inconsistencies when used to normalize the time series of A(t). Thus, we derive σA from A(t) instead.

Once A(t) and σA are known, the major axis of the ellipse can readily be determined through Eq. (5) (note that x1 applies to a normalized phase-space diagram). However, one problem with data anomalies is that they can dominate the statistical properties of the dataset considerably, and therefore the scales of the ellipse. We circumvent this impact by determining the standard deviation after removal of all observations beyond 4ση before a revised ση is calculated. This action is repeated until no points are outside the 4ση limit. Note that a typical wave record is half an hour long, which, at a sampling frequency of 2.5 Hz, has only one expected data point above this threshold if the surface elevation is assumed to be Gaussian distributed. To gain a proxy of the higher-order moments of the spectrum, the spectrum is determined based on the time series without points beyond 4ση. Note that this is required to determine the spectral bandwidth, necessary for the characterization of the minor axis (e.g., see section 2). With this information, the wave record can be placed in phase space and the ellipse can be drawn to identify the anomalies in the wave record.

4. Results

Before the wave records are presented in phase space, the minor axis of the enclosing ellipse needs to be defined. To do so, wave buoy measurements were placed in phase space and the minor axis of the ellipse was incrementally lengthened until 95% (i.e., corresponding to p = 2) of the wave record falls within the ellipse compared to an infinitely wide ellipse. If truly Gaussian, the minor axis can easily be scaled to obtain other values of p. As described in section 2b, it is expected that x2 is a function of the spectral bandwidth υ. Various nondimensional forms of dependency on spectral parameters were investigated and smallest root-mean-square error (RMSE) of x2 was observed when scaled with the υ and Tm02 (Fig. 4). Specifically, the radius of the minor axis is defined here as
x2=0.2υσAσηTm02.
Equation (11) is consistent with observations of the phase-space diagram in Fig. 1, where for a monochromatic wave (υ → 0) the ellipse becomes a line. Even though x2 is normalized by both ση and σA [and thus might be insensitive to the sampling frequency and different numerical schemes to determine A(t)], the dependence of x2 on the typical time scale Tm02 might differ (see section 3c). Moreover, Eq. (11) should be used with caution for more energetic sea states as we have only a limited number of observation where Hs > 3 m.
Fig. 4.
Fig. 4.

The radius of the minor axis x2 of the ellipse against Eq. (11).

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

Now that both axes of the ellipse are defined in terms of the wave field, we use a synthetic wave record (based on an idealized JONSWAP spectrum using the WAFO MATLAB toolbox; Brodtkorb et al. 2000) to confirm that a random wave field can indeed be described by an ellipse in phase space. The generated wave record has spectral properties Hs = 2 m, Tp = 6 s, υ = 0.38, and a sampling frequency Δt = 0.4 s (Fig. 5a). To verify the impact of swell on the distribution of the wave record in phase space, swell was superimposed onto the spectrum through a narrow Gaussian distribution in frequency space with Hs = 0.6 m and Tp = 10 s, giving υ = 0.45 (Fig. 5b). Note that in Fig. 5 three ellipses are drawn, corresponding to p = 2, 4, and 6. For both cases, the dense cloud of the wave record are well correlated to the predefined ellipse. Multiple points of the wave record are just outside the second ellipse. In Fig. 5a the cross identifies an extreme wave with z = 4.4ση (or z = 1.1Hs). It is worth mentioning that when one would apply the conventional 3p standard deviation threshold to flag anomalies, a total of 30 data points (0.3%) would have been removed from the synthetic wave record.

Fig. 5.
Fig. 5.

Phase-space diagram for (a) a synthetic JONSWAP spectrum (Hs = 2 m, Tp = 6 s, Δt = 0.4 s) and (b) the same spectrum, but with a 0.6 m swell superimposed (T = 10 s). Ellipses are defined by Eqs. (5) and (11), with p = 2, 4, and 6, for the smallest to largest ellipses, respectively. The black line aligned with the major axis of the ellipse corresponds to a monochromatic wave with period Tm02 and amplitude 0.5Hs; the second black line corresponds to a monochromatic wave with period Tp and amplitude 0.5Hs. The probability density function of the normalized vertical displacement η/ση and vertical surface acceleration ∂2η/∂t2/σA are provided along horizontal and vertical axes (not to scale). Dotted lines correspond to twice the standard deviation of the dataset. Crosses identify observations where z > 4ση. Circles define points outside middle ellipse.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

As discussed in section 2, T0 based on m4 is ill defined in a practical/experimental setting. Instead, we included two lines corresponding to a monochromatic wave with T = Tm02 and T = Tp, and for both a0 = 0.5Hs is used. A wave period of Tm02 tends to define the main axis of the ellipse well (Fig. 5).

In Fig. 6a a typical wave anomaly is shown as observed in the time series of our wave buoy records. Due to the significance of the anomaly in the time series, ηmax ≈ 20ση, almost any detection technique will identify this wave or part thereof as an anomaly. Nevertheless, we use this extremity as an example to show how such an anomaly looks like in phase space. When viewed in phase space (Fig. 6b), a large number of measurement points of the particular wave are located outside the largest ellipse (p = 6). The impact of this single wave on the wave energy spectrum is significant (Fig. 6c) and therefore strongly impacts the wave statistics. For example, the significant wave height before and after removal is 2.4 and 1.4 m, respectively, whereas the peak period Tp decreases from 26 to 5 s. Note, to obtain these spectra for comparison while avoiding discontinuities in the wave record after removal of the complete wave, we shifted the time series by 5 min.

Fig. 6.
Fig. 6.

Example observations of surface elevation measured by a moored Spotter wave buoy: (a) time series with one wave anomaly; (b) phase-space diagram before removal of wave anomalies; (c) wave energy spectrum before and after removal of wave anomaly (to calculate the spectrum after anomaly removal, the time series was shifted 5 min to avoid discontinuities in the time series). Wave 1 identifies the wave anomaly in (a) and (b). Ellipses in (c) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

When these anomalies are less extreme, it becomes less obvious whether the anomaly is simply a natural extreme or noise. In Fig. 7a a time series is shown containing two individual waves whose crest and/or trough exceeds 4ση relative to the mean surface elevation and would, therefore, normally be identified as anomalies. We name these two events “wave 1” and “wave 2,” respectively, and note that “wave 1” has a wave trough of ηmin ≈ −4.3ση and “wave 2” has a wave crest of ηmax ≈ 8ση. When this time series is considered in phase space, 48% of “wave 2” is outside the middle ellipse, whereas 33% is outside the largest ellipse (Figs. 7b,c). In contrast, the extreme wave trough of “wave 1” has no points outside the middle ellipse. Hence, even though a statistical anomaly, “wave 1” is expected to be a natural extreme as its characteristics are consistent with the properties of the wave field. This is in contrast to “wave 2,” which has properties distinctly different from the rest of the time series. We therefore consider only “wave 2” to be a erroneous measurement. Though the impact of this erroneous wave anomaly on the spectrum is considerably smaller than seen for the wave record in Fig. 6, the wave anomaly does increase the significant wave height and mean period by about 10% (Fig. 7c).

Fig. 7.
Fig. 7.

Example observations of surface elevation measured by a moored Spotter wave buoy: (a) time series with two significant wave events; (b) “wave 2” in the wave record where thick gray and black line marks sections outside middle and largest ellipses, respectively; (c) phase-space diagram before removal of wave anomalies where crosses and circles define “wave 1” and “wave 2,” respectively; (d) wave energy spectrum before and after removal of wave anomaly (to calculate the spectrum after anomaly removal, the time series was shifted five minutes to avoid discontinuities in the time series). Ellipses in (b) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

Contrary to our wave buoy measurements where anomalies occur continuously over the duration of a few wave periods, anomalies in the laser altimeter data generally take the form of spikes and are typically one or two points long. For consistency and accuracy reasons (see section 3b) the sampling resolution of the laser altimeter was reduced to 2.5 Hz. However, rather than resampling the 5 Hz time series of the laser altimeter, we split the wave record into two alternate time series with a sampling rate of 2.5 Hz instead. Figure 8a provides an example wave record measured by the laser altimeter with Hs = 4.5 m, showing one single-point minor spike (η ≈ −4.5ση) and two major multipoint spikes. The circles identify the observations outside the largest ellipse (Figs. 8a,b). In contrast to “wave 1” in Fig. 7, this spike with a maximum surface elevation 4.5ση from its mean would lie on the edge of an ellipse with p = 15, i.e., roughly 15 equivalent standard deviations in phase space. Contrary to the moored wave buoy, anomalies in the form of spikes in the laser altimeter wave record lead to high-frequency noise in the spectrum (Fig. 8c). This translates in a 5% error in Hs and a 36% error in mean period Tm01. Interestingly, also for this wave record, the major axis of the smallest ellipse corresponds roughly to a monochromatic wave with amplitude 0.5Hs and period Tm02. It is worth mentioning that the points directly adjacent to the spikes are outside the largest ellipse as well. However, this is an artifact of the numerical scheme used to calculate the surface acceleration, which uses information of neighboring points to estimate A(t).

Fig. 8.
Fig. 8.

Example observations of surface elevation measured from a fixed offshore platform using a laser altimeter, with Hs = 4.5 m and Tp = 11.4 s: (a) time series with two major and one minor spike; (b) phase-space diagram before removal of wave anomalies; (c) wave energy spectrum before and after removal of wave anomalies. Wave anomalies are marked with black circles. Ellipses in (b) are defined as per Fig. 5 and the thick solid represents a monochromatic wave with period Tm02.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

5. Discussion

a. Threshold to identify wave anomalies

So far, we have loosely identified wave anomalies in a wave record through the comparison of their relative position with respect to predefined ellipses in phase space. We believe this method is superior to the popular standard deviation threshold, as, for most false measurements, the correlation between surface displacement and acceleration is distinctly different from a random wave field. However, as with many other quality control methods, the phase-space method proposed here still relies on a user specified threshold to identify the anomalies.

To automate the process of anomaly detection using a phase-space threshold on the complete wave buoy dataset, we, without too much consideration on the exact value of the threshold p, use a threshold value of p = 6 here [i.e., equivalent to 6σ, Eq. (5)]. This is a very conservative value which, for a Gaussian distributed signal, corresponds to a 1 in 5 × 108 event, or once every 6 years for a sampling rate of 2.5 Hz. In contrast, noting that the surface elevation reasonably follows a Gaussian distribution, the commonly applied standard deviation threshold of p = 3 will flag about 0.3% of the data as anomaly, this corresponds to a total of 5 s for a half-hour record. We note that for p = 6, a single-point spike at η = 4.5ση is still flagged as an anomaly with the phase-space threshold (e.g., Fig. 8), while an extreme wave event of similar magnitude in the wave record is not (e.g., Fig. 7). However, by using a threshold of 6σ we naturally put a lower limit to the extreme waves that will be identified erroneously as anomalies.

As the anomalies in our wave buoy data are associated with the complete wave, rather than a single point, we first identify the wave (the surface profile between two successive upward or downward zero-crossings) associated with the data points outside the ellipse. If this particular wave has at least two points outside the ellipse defined by p = 6, the complete wave is flagged as an anomaly. To be able to compare the impact of the removal of these wave anomalies, we then recalculate the wave energy spectrum based on continuous wave records of 30 min without any flags to determine time series of Hs and Tm01. We then interpolate the integral wave properties to those of the original wave records, which were defined at the start and halfway every hour. Comparison of Hs and Tm01 before and after the removal of wave anomalies is shown in Fig. 9. Note that while the majority of the observations is unaffected by the filter (i.e., positioned around the 1:1 line), some records show significant differences, particularly for the mean wave period. Note that minor deviations from the 1:1 line exist due to the interpolation of the integral wave properties to match those of the integral properties which include the wave anomalies.

Fig. 9.
Fig. 9.

Comparison of (a) significant wave height Hs and (b) mean wave period Tm01 before and after removal of wave anomalies. Wave anomalies are defined based on a phase-space threshold of p = 6 (see section 5a).

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

To substantiate that restrictions in the horizontal movement of the wave buoy are a likely source of the detected wave anomalies, instances of a high relative error of Hs are compared to the geographic coordinates of the wave buoy with respect to the origin of the mooring (Fig. 10). Indeed, largest differences between Hs before and after wave anomaly removal are observed in east-northeast (ENE) direction at the maximum horizontal displacement of the wave buoy. Note that the ENE direction is parallel to the coast of King George Island (Fig. 2), and corresponds to the direction of the surface current at this site (e.g., Zhou et al. 2006). We hypothesize that when the mooring line is under strong tension, it applies a large force on the buoy when it attempts to ride a wave away from the central mooring (leading to a jerking motion) and/or causes significant over wash of the wave buoy, thereby hindering the GPS connectivity of the buoy and thus the surface motion measurements. However, without further study on the exact source of this behavior, this remains rather speculative.

Fig. 10.
Fig. 10.

Geographical position of the wave buoy for the complete deployment (30 min intervals) relative to the central mooring. Gray scale defines the relative error of the significant wave height Hs, defined as the ratio of wave height before and after removal of erroneous observations in the time series.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

b. Extreme waves

The linear framework that describes the ellipse that encloses the wave record might not necessarily hold for nonlinear waves, in particular, for extreme waves. We therefore use the synthetic wave data generated through a phase-resolved numerical model (see section 3b) to provide guidance on how nonlinear waves fit within the presented framework. In Fig. 11a, data of all synthetic time series without extreme waves are presented in a phase-space diagram. The majority of data is well aligned with the major axis of the ellipse. However, wave crests above approximately 2η/ση start to deviate from this axis. Based on the probability density function of surface displacement and acceleration, this deviation corresponds to a slightly skewed surface acceleration compared to that of the linear (Gaussian) counterpart and with fewer positive extremes but increased negative extremes. The distribution of surface elevation of the nonlinear waves compares well to that of a Gaussian distribution.

Fig. 11.
Fig. 11.

Phase-space diagram of synthetic wave data generated through a phase-resolved numerical model: (a) density contours of full time series without wave extremes; (b) extreme wave troughs and crests only where the black circles identify the minima/maxima and the gray dots the neighboring data points of these minima/maxima. In (a), the probability density functions of the normalized surface displacement and vertical acceleration are shown by the red curve, for comparison the black curve depicts a Gaussian distribution. In (b), the red line describes the linear relation between surface displacement and surface acceleration of a monochromatic wave with period T0 = (m0/m4)0.25 [see Eq. (8)]. The dashed line in (b) identifies the approximate condition of ∂2η/∂t2 = g.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0090.1

When the surface elevation and surface acceleration of the extreme wave crests are considered, these also deviate from the major axis of the predefined ellipse but are well aligned with an axis defined by the fourth-order moment of the spectrum [Eq. (8)]. Moreover, it is observed that the neighboring points of the extreme wave crests follow this axis before returning back to the smallest prescribed ellipse. In contrast to the extreme wave crests, the extreme wave troughs are all well aligned with the major axis of the ellipse and, as such, suggests that the extreme wave troughs behave as linear opposed to the nonlinear response seen for the extreme wave crests.

While these extreme waves are outside the ellipse defined by p = 6 (not unexpectedly, as a significant proportion of these extremes is larger than 6η/ση), this does not necessarily mean that the wave anomaly detection method erroneously identifies wave extremes. Specifically, if one would apply the same procedure as in section 5a to flag data as erroneous, only 6% of the extreme wave crests would be removed from the dataset as the vast majority of these extreme waves have only one data point outside the predefined ellipse. The strong correlation between the surface displacement and acceleration of the extreme wave crests, despite deviating from the major axis of the ellipses, seem to be very predictable in phase space and thus simple to exclude when being identified erroneously as wave anomaly.

c. Limitations

Although the presentation of wave time series in phase space suggest that the behavior of waves in phase space is predictable, even for extreme waves, there are some limitations to the presented framework that requires further study to improve universal application of the presented method to identify wave anomalies in surface displacement measurements. When the surface acceleration record is determined from the surface elevation time series, values of σA are sensitive to the measurement sampling frequency and numerical scheme used (here 2.5 Hz and three-point central difference scheme, respectively) (Fig. 3c). While this does not necessarily impact the properties of the major axis of the ellipse, which scales with the error systematically, it might impact the quantification of the minor axis (i.e., see Section c). Moreover, the radii of the minor axis was determined based on limited observations of energetic sea states and the method should thus be used with caution when Hs > 3 m. We also observe that the main axis of the ellipse corresponds closely to a monochromatic wave with period Tm02 when using a three-point central difference scheme to determine A(t) with Δt = 0.4. While this is an artifact of the numerical accuracy of A(t), we note that the correspondence does seem to occur consistently across the generated and measured data used here (Figs. 58). However, we observe that for the laser altimeter data at a sampling frequency of 5 Hz and the use of a higher-order numerical scheme to determine A(t), the main axis is represented by a smaller wave period (not shown here), and we expect this period to approach the theoretical period based on m4 [as in Eq. (8)] when higher accuracy is achieved for A(t).

6. Conclusions

We presented a method to identify wave anomalies in surface displacement measurements. In a phase-space diagram (surface displacement-acceleration), a wave record can be enclosed by an ellipse, the axis of which, are defined by the wave field properties. The major axis of the ellipse is defined by the standard deviations of the surface displacement and acceleration time series (physically representing a monochromatic wave with an amplitude of half the significant wave height and fourth-order moment wave period), whereas the minor axis is characterized by the spectral bandwidth and mean wave period of the wave field. The wave anomaly detection procedure can identify single and multipoint spikes in a wave record, as well as flagging extreme waves induced by wave mooring restrictions. While we observe that extreme wave crests depart from the linear behavior of waves their properties remain, nevertheless, predictable in phase space. Extreme wave troughs, on the other hand, are well aligned with the ellipse in phase space and, as such, behave as linear. Further study is, however, required to define the radius of the minor axis of the ellipse for other sampling approaches and numerical schemes to determine the acceleration record from surface displacement time series.

Acknowledgments

The authors acknowledge the funding support provided by the Brazilian agencies, for funding this study. Antarctic Modeling Observation System (ATMOS) project is funded by CNPq/CAPES/MCTIC (443013/2018-7). L. P. Pezzi is supported by a CNPq fellowship for scientific productivity (CNPq 304858/2019-6). We thank the captain and crew from Po/V Almirante Maximiano (H-41). The Brazilian Ministry of Science, Technology, Innovations and Communications (MCTIC), as well as the Brazilian Antarctic Program (PROANTAR) and the Brazilian Navy are acknowledged for making both the cruise under Antarctic Operation 38 (OP38) possible. Special thanks for Eliana Rosa, Ueslei Sutil, Fabiane Furlan, Regiane Moura, Gabriel Munchow, Cláudia Parise, and Denni de Morais for their assistance in the build and deployment of the wave buoy mooring. AVB was supported by the DISI Australia-China Centre through Grant AC SRF48199. JJV and AVB were supported by the Australian Antarctic Science Program under Project AAS4593. AVB acknowledges support from the U.S. Office of Naval Research Grant N00014-17-1-3021. JJV and AVB would also like to thank Paul Tinkler and Daniel Ierodiaconou for sharing their experience in building a mooring system for the Sofar Spotter wave buoy. The authors thank the three anonymous reviewers for their constructive comments and suggestions.

Data availability statement

The data used in creating the figures are available in a public repository (doi:10.5281/zenodo.3893967). Laser altimeter data were kindly provided by Woodside Ltd., and the authors can be approached regarding the laser altimeter data used in this study. We refer the reader to Ducrozet et al. (2016) for details on the HOC-Ocean model. Simulated data used in this study can be found at doi.org/10.5281/zenodo.4028014.

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