Observations of Surface Gravity Wave Spectra from Moving Platforms
1. Introduction
Ocean surface gravity waves play a crucial role in Earth’s climate system by mediating the exchange of energy, momentum, heat, and gases between the ocean and atmosphere (Cavaleri et al. 2012). This is due in part to the ubiquity and dominance of surface waves at the air–sea interface. Quantifying the influence surface waves have on these exchanges is needed to improve our understanding of the physics driving the ocean–atmosphere system, such as the active two-way coupling between winds, waves, and currents. These improvements in physical understanding will lead to advancements in forecasting Earth’s climate in numerical simulations through physics-driven parameterizations of air–sea processes.
To achieve this, acquiring high-quality measurements of the directional surface wave field, especially down to small scales O(1) m, where most of the stress between the ocean and atmosphere is supported (Mitsuyasu 1985; Melville 1996), is needed. Autonomous surface vehicles (ASVs) measure the wave spectrum as they move across the ocean surface from the vehicle motion by assuming the platform motion closely follows the sea surface displacement of the surface waves (see, e.g., Lenain and Melville 2014; Thomson et al. 2018; Grare et al. 2021). Despite the limitations of this assumption, for example, the platform’s heave response damping of high-frequency waves [see Thomson et al. (2015) for a related discussion on high-frequency attenuation from biofouling on a wave buoy and (Alvarez 2015) for a discussion on Slocum gliders], ASVs have been shown to collect robust estimates of directional wave measurements and bulk wave statistics (Lenain and Melville 2014; Grare et al. 2021). With the recent advancements in the development of ASVs, surface waves may be observed in remote areas over long-duration deployments. Moreover, measurements taken on board these vehicles span broad spatiotemporal scales. This allows these platforms to capture the space and time evolution of the wave field.
Several theoretical, laboratory, and field studies have demonstrated the variety of ways a moving platform can impact measurements of surface waves (e.g., Kats and Spevak 1980; Longuet-Higgins 1986; McAllister and Van Den Bremer 2019). Here, we focus on the impact of platform motion on the observed period of the waves. Differences in observed wave period exist depending on the reference frame of the observer. Longuet-Higgins (1986) showed for example that for steep waves in deep water, the observed wave period measured by a Lagrangian observer (fluid-particle-following reference frame), and an Eulerian observer (fixed-location reference frame), may differ as much as 38% due to a Doppler shift. Wave-following ASVs measure the wave field from a pseudo-Lagrangian perspective because they do not strictly follow fluid particles due to the dynamic response of the platform to the wave forcing and its mechanical propulsion system. The autonomous platform motion relative to the incoming waves causes the observed frequencies to be Doppler shifted. These Doppler effects impact spectral quantities derived from the time series of platform motion by shifting the power spectral density to higher or lower frequencies depending on the platform’s velocity relative to the waves.
To obtain wave spectra independent of the platform motion, a frequency mapping to a reference frame free from Doppler effects is required. Previous studies have explored different approaches to the mapping (Drennan et al. 1994; Hanson et al. 1997; Cifuentes-Lorenzen et al. 2013; Collins et al. 2017; Amador et al. 2022) based on the theoretical work of Kats and Spevak (1980). Frequency spectra, hereafter referred to as the one-dimensional (1D) spectrum, have been mapped assuming a unidirectional wave field (usually using a head-sea wave direction or wave direction derived from the platform’s drift direction, wind direction, wave model, buoy measurement, or marine radar measurements; see Hanson et al. 1997; Cifuentes-Lorenzen et al. 2013; Collins et al. 2017) or using a frequency-dependent mean wave direction (computed from the first- and second-order Fourier moments of the directional distribution of wave energy; see Lenain and Melville 2014; Amador et al. 2022). A weakness in these methods is that they ignore the full directionality of the wave field, which may cause the mapping to erroneously project spectral energies into wave directions that are not captured by a mean direction characterization (Amador et al. 2022).
In this study, we present a general mapping method, building upon the work of Drennan et al. (1994), that can be applied to any autonomous surface vehicle to account for Doppler effects. Our approach is validated and tested using unique datasets collected by a fleet of Liquid Robotics instrumented Wave Gliders (Grare et al. 2021). We then compare our method with the approaches used in Hanson et al. (1997), Cifuentes-Lorenzen et al. (2013), and Collins et al. (2017) to demonstrate which best removes Doppler effects due to platform motion which is a key assessment missing in previous studies. In addition, we provide detailed descriptions and open-source software tools for implementing the general mapping method. Note that frequency ambiguities when the platform moves in the direction of wave propagation limit our ability to observe the shortest measurable waves from the platform. Here, we provide physical insight into why and where these ambiguities in frequency occur and guidance on how to select the correct branch of the different solution scenarios.
This paper is structured as follows: The overview of the experiments, including the instrumentation used and the range of environmental conditions encountered, is given in section 2. In section 3, the theoretical background of the frequency mapping between reference frames, the mapping method of the observed two-dimensional spectrum, and the spectral analysis techniques are presented. Section 4 illustrates and discusses the Doppler effects in the observed wave spectra using data collected by instrumented Wave Gliders. A general method for accounting for these Doppler effects is presented and applied to our particular dataset. Finally, section 5 presents a summary of the conclusions and ends with a reference to the open-source code for implementing this method.
2. Experiments, instrumentation, and environmental conditions
a. Experiments
In this study, observations collected from two experiments are investigated. The first was conducted as part of the ONR Task Force Ocean (TFO) initiative. Observations from two Wave Gliders, referenced as “Planck” and “Stokes,” were collected off the coast of Del Mar, California (see Fig. 1), during a 3-day deployment from 9 to 11 September 2020 (referred hereon as DELMAR2020). The two Wave Gliders moved along two tight square tracks: a large square with a 1000-m edge length and a small square with a 500-m edge length. Here, we focus on data from the small square where Doppler effects on the frequency spectrum are more apparent due to rapid changes in platform heading.
(a) Trajectories of Wave Gliders Planck (blue) and Stokes (red) during the DELMAR2020 experiment for the time period of 0230 UTC 9 Sep–1610 UTC 11 Sep 2020. (b) The geographic region with the location of the experiment site (white box). (c) Mean platform heading, (d) wind speed at 10 m above the ocean surface (solid line) and wind direction at 1 m above the ocean surface (triangular markers), and (e) significant wave height measured by Planck (blue) and Stokes (red). Here, the directional convention is clockwise, reference north, and going toward.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
The second experiment was conducted as part of the NASA Submesoscale Ocean Dynamics Experiment (S-MODE) program, a project that aims to determine whether submesoscale ocean dynamics make important contributions to the vertical exchange of physical and biological variables in the upper ocean using a combination of aircraft-based remote sensing, research vessels, and autonomous oceanographic platforms measurements along with numerical modeling (Farrar et al. 2020). The pilot experiment considered here was conducted off the coast of San Francisco, California (see Fig. 2), from 29 October to 4 November 2021 (referred hereon as SMODE2021). In this work, we focus on the observations from one of the Wave Gliders deployed in the experiment, referenced throughout as WHOI43.
(a) Trajectory of Wave Glider WHOI43 (blue) during the SMODE2021 experiment for the time period of 0000 UTC 29 Oct–2300 UTC 3 Nov 2021. (b) The geographic region with the location of the experiment site (white box). (c) Mean platform heading, (d) wind speed at 10 m above the ocean surface (solid line) and wind direction at 1 m above the ocean surface (triangular markers), and (e) significant wave height measured by WHOI43 (blue). Here, the directional convention is clockwise, reference north, and going toward.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
b. Instrumentation
Instrumented Boeing Liquid Robotics SV3 Wave Gliders are used to collect ocean and atmospheric observations at the air–sea interface (Grare et al. 2021, see their Fig. 1). Included in the present study are atmospheric measurements from a Vaisala WXT (model 530) mounted ∼1 m above the ocean surface that records wind speed and direction among other atmospheric parameters. The wind speed 10 m above the ocean surface is estimated using the modified version of the Charnock relation (Charnock 1955) used in the TOGA COARE model (Fairall et al. 2003). Surface wave observations are derived from the motion of the platform (Lenain and Melville 2014; Thomson et al. 2018; Grare et al. 2021) measured by a coupled GPS–inertial motion unit (IMU) system (Novatel SPAN OEM7720–Epson EG320N for the DELMAR2020 experiment and a Vectornav VN-300 GPS/IMU for the SMODE2021 experiment) (Hodges et al. 2023). Uncertainties in these observations measured by Wave Gliders are discussed in detail by Grare et al. (2021), Thomson et al. (2018), and Lenain and Melville (2014). In these studies, wind and wave observations have been shown to agree well with independent ground truth data over a range of environmental conditions thus demonstrating the high accuracy and precision of these measurements.
c. Environmental conditions
Environmental conditions for the DELMAR2020 and SMODE2021 experiments are shown in Figs. 1d, 1e, 2d, and 2e. During the DELMAR2020 experiment, the sea state was moderate, with 0.8 to 1.4 m significant wave height and variable winds from 1 to 9 m s−1 generally coming from the north-northwest. During the SMODE2021 experiment, significant wave height ranged from 1.5 to 4.5 m and wind speed varied from 1 to 11 m s−1, coming from the northwest at the beginning of the experiment and shifting toward the south for the rest of experiment. Data from both experiments are investigated to increase the range of environmental conditions.
3. Methods
a. Mapping from observed to intrinsic frequency
Consider two reference frames, R′ and R, from which a platform may observe the surface wave field. In R′, the reference frame moves with the platform at a constant velocity u. In R, the reference frame is stationary. Waves measured by the moving observer in reference frame R′ are subject to a Doppler shift of their frequencies. That is when the observer is moving against the waves (i.e., in the opposite direction of wave propagation), the observer passes more wave crests in a unit of time compared to a stationary observer, leading to a higher observed frequency or a lower observed wave period. On the other hand, when the observer is moving with the waves (i.e., in the direction of wave propagation), the observer passes fewer wave crests leading to a lower observed frequency or a higher observed wave period. Waves measured by the stationary observer in reference frame R are free of platform-induced Doppler shifts because the observer velocity relative to the direction of wave propagation vanishes. In contrast to the reference frame dependence of the frequency, the wavenumber vector is invariant between reference frames. Note that this Doppler effect is opposite from the Doppler shift that arises when waves ride upon currents (Peregrine 1976). In the reference frame of a stationary observer, waves advected by a current with uniform velocity are subject to a Doppler shift. For a moving platform, it is the reference frame moving with the platform that experiences a Doppler shift in frequencies.
The k solutions for (5) using variable names from Hanson et al. (1997) in column 1. Columns 2 and 3 specify the form of the solution and the sign in front of the square root, respectively, in (6) and (8). Columns 4 and 5 specify the range of values for the relative direction of platform propagation with respect to the waves and the platform speed projected onto the wave direction; c and cg are the phase and group speeds, respectively. Column 6 provides a description of the platform–wave system, specifically their speed and direction relative to each other. The phase speed is considered when discussing waves in column 6.
Using the five solutions from Table 1, Fig. 3 displays the solutions of Eq. (7) for three cases of platform speed: U = 0.5, 1, and 2 m s−1, each with θr ranging from 0° (platform moving with waves) to 180° (platform moving against waves).
Mapping from observed to intrinsic frequency for a platform moving at (a) 0.5, (b) 1, and (c) 2 m s−1 with line color corresponding to the platform’s direction relative to the waves. The platform moves with the waves when θr ranges from 0° ≤ θr < 90° (red curves) and moves against the waves when θr ranges from 90° < θr ≤ 180° (blue curves). The platform moves perpendicular to the waves when θr = 90°. The gray dashed line is the cutoff frequency fc with intersecting gray circles being fc for a given direction. The black dashed line is the one-to-one line occurring when the projected speed vanishes (θr = 90°).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
When the platform moves against the waves for 90° < θr ≤ 180° (blue-colored curves), fob is mapped to a lower fin. These observed waves correspond to the solution ka. A defining characteristic of these curves is their curvature which increases as the platform speed in the direction of wave propagation increases. This curvature reflects the dispersive nature of deep water surface waves. That is, the nonlinearity of (1) causes the slope of (7) to vary with fob. This can be visualized in Fig. A1 (see appendix for details). Note that in the case of nondispersive waves, the relationship between fob and fin is linear. This reflects the fact that the slope of the dispersion relation is constant. Due to the downward shift in frequency, the highest resolved frequency in observed frequency space, the observed Nyquist frequency, is reduced to a lower intrinsic Nyquist frequency (Cifuentes-Lorenzen et al. 2013). This places a high-frequency limit on the wave frequencies resolved by a platform moving against the waves.
Turning to θr = 90° (green curve), the platform moves perpendicularly to the waves and the observed frequency is equal to the intrinsic frequency. These waves are represented by the solution kp. Here, the platform speed in the direction of wave propagation vanishes, indicating the transition between following and head seas as denoted by the dashed black line. Thus, the platform acts as a stationary observer.
Last, when the platform moves with the waves for 0° ≤ θr < 90° (red-colored curves), fob is mapped to a higher fin. These waves correspond to three solutions, namely, klt, kst, and kr. Solution klt corresponds to long-wavelength waves whose phase and group speed in R are faster than the platform speed in the direction of wave propagation. In Fig. 3c for θr = 0, this solution corresponds to fin values before the inflection point where fin bends back toward lower frequencies. This inflection point is also known as the bifurcation point between klt and kst. Solution kst corresponds to short-wavelength waves whose phase speed is faster than the platform speed in the direction of wave propagation, but whose group speed is slower. This solution corresponds to fin values after the inflection point but before the zero crossing. Solution kr corresponds to even shorter-wavelength waves whose phase and group speeds are both slower than the platform speed in the direction of wave propagation. This solution corresponds to fin values after the zero crossing. Note that kr is obtained using negative observed frequencies in (6).
The existence of the cutoff frequency states that the motion of the platform relative to the waves limits the high-frequency waves that a moving platform can observe. That is, when moving in the direction of wave propagation, the platform cannot resolve frequencies above the cutoff frequency (11) which is illustrated in the frequency ambiguity in the mapping (7). Figure 4 shows fc as a function of U and θr (Fig. 4a) along with a slice of the polar plot for θr = 0 (Fig. 4b). It shows that as the measurement platform increases speed and θr approaches 0°, the cutoff frequency decreases. For example, at U = 3 m s−1, the cutoff frequency is as low as 0.26 Hz. Thus, the spectral range reduces considerably and limits the highest-frequency waves that may be measured. The typical speed range for Saildrones (Gentemann et al. 2020), a type of ASV equipped to measure surface waves, extends up to 3 m s−1 with an average speed of 1.25 m s−1. Autonauts (Dallolio et al. 2019), another type of ASV, can achieve similar speeds ranging up to 2 m s−1. Careful consideration of the speed and direction of these and other ASVs is recommended during experimental planning to maximize the highest frequency resolved.
(a) Cutoff frequency fc as a function of platform speed and the relative angle between platform heading and wave propagation direction. Relative angles range from 0° to 90° corresponding to the platform moving with the waves. The typical range of speeds for Wave Glider and Saildrone platforms are shown along the radial axis. (b) Cutoff frequency as a function of platform speed for a platform moving strictly in the direction of wave propagation (θr = 90°; green curve).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
The frequency ambiguity in the case of the following seas presents a dilemma for mapping the 1D and 2D wave spectra. Here, the two-dimensional (2D) spectrum refers to the directional-frequency spectrum. For sufficiently large values of U cos(θr), we are faced with an ambiguity in determining how to allocate energy from one observed frequency to two intrinsic frequencies. To resolve this, we must choose from one of the two intrinsic frequencies from the mapping, namely, klt or kst. To do so, we solve (5) using the Newton–Raphson root-finding algorithm (Verbeke and Cools 1995) with an initial guess for k when the platform’s velocity vanishes (i.e., k = kp). Using this numerical approach, we find that the klt solution should be used. See the appendix for more information.
b. A practical 2D method to account for Doppler effects in wave spectra
Explicitly, the 2D method procedure (Fig. 5) consists of the following steps:
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Compute the observed 2D spectrum Sob(fob, θ) and Doppler shift velocity
. -
Project the Doppler shift speed onto the wave direction θi. Here, we define this projection as
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Map from the observed frequency to the intrinsic frequency using the projected speed
. -
Compute the Jacobian of the mapping
using finite differencing. -
Map the observed 2D spectrum in the θi direction Sob(fob, θi) into intrinsic frequency space. Here, we define the intrinsic 2D spectrum in the wave direction θi as
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Interpolate the intrinsic 2D spectrum in the θi direction onto a regular grid.
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Repeat steps 2–6 for each wave direction θi in the observed 2D spectrum.
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Compute the intrinsic 1D spectrum Sin(fin) by azimuthally integrating the intrinsic 2D spectrum Sin(fin, θ):
Flowchart of the 2D method procedure.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
The highest frequency resolved in the intrinsic 1D spectrum, defined as fb, is set by the Nyquist frequency, the bifurcation frequency, the Jacobian, and the interpolation process. This high-frequency cutoff leads to a reduced spectral range. See online supplementary material for more information. Note, for step 2, the mean speed of the Doppler shift velocity
c. Spectral analysis
Figure 6 shows a typical observed directional wave spectrum computed from Wave Glider Stokes during a single leg of the large 1000-m box trajectory in the DELMAR2020 experiment. The directions reported are the directions from which waves are propagating (“coming from” convention). The observed directional spectrum has two large peaks, indicating a local wind sea from the west and a lower-frequency swell from the south. One-dimensional and saturation spectra computed from the directional wave spectra shown in Fig. 6 are plotted in Figs. 7a and 7b, respectively, which further illustrate the presence of mixed-seas conditions with a swell peak of fob = 0.07 Hz and a wind sea peak around fob = 0.2 Hz. Note that the transition frequency between the saturation and equilibrium regimes fn, shown with the solid vertical line calculated following Lenain and Pizzo (2020), is resolved with
An observed directional wave spectrum computed from measurements collected on board Wave Glider Stokes during a single fixed heading leg of the large 1000-m box trajectory in the DELMAR2020 experiment (at 2357:06 UTC 9 Sep–0025:18 UTC 10 Sep 2020).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
Observed (a) 1D and (b) saturation spectra computed from the directional wave spectrum in Fig. 6. The dashed and dot–dashed lines a have
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
Observed (a) 1D and (b) saturation spectrograms computed from measurements collected on board the Wave Glider Stokes during the DELMAR2020 experiment. The dot–dashed vertical lines outline the time period from which spectral quantities in Figs. 6 and 7 are derived. The dashed rectangle outlines the time period when the Wave Glider Stokes traversed in its small 500-m box formation (Fig. 1).
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
The analysis of spectral quantities discussed above is split into two steps. In step 1, we aim to demonstrate the significant impact Doppler effects have on the saturation spectra, especially at high frequencies. To accomplish this, we focus on the small 500-m box pattern from the DELMAR2020 experiment (Fig. 1a) and compute spectra over time records when the Wave Glider is moving in a constant heading along the legs of the box. By doing so, we isolate the effect that a constant heading has on the spectra. The small box is orientated such that the platform moves almost directly against the waves on the north-to-south legs while it moves with the waves on the south-to-north legs. Note that a single leg of the small box took about 21 min to complete on average, in turn minimizing the natural spatial and temporal variability of the wave field. In the second step, where we aim to characterize the efficiency of the 2D mapping method, the saturation spectrogram is computed over constant time intervals (10-min records) independent of the heading of the platform for the entire experiment.
4. Results
(a) Wave Glider Stokes’s heading (using the coming from directional convention) for the small 500-m box legs. Red triangles denote the legs where the platform is moving in the direction of wave propagation for high-frequency waves (fob > 10−1 Hz). Blue triangles denote the legs where the platform is moving against the direction of wave propagation for high-frequency waves. These high-frequency waves have a mean wave direction coming from the northwest (approximately 300°). (b) Saturation spectrogram for Wave Glider Stokes’s repeated small 500-m box formation with red and blue colored ovals along the top axis specifying the relative angle between platform heading and mean wave direction for high-frequency wave using the same color code as in Fig. 9a. Saturation spectra within the two time periods outlined by dot–dashed vertical lines correspond to the averaged spectra in Fig. 9c. (c) Two averaged saturation spectra showing against (red curve) and with (blue curve) wave cases.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
Figure 10a shows the same observed saturation spectrogram presented in Fig. 9b along with the Doppler-corrected saturation spectrograms computed using the (Fig. 10b) 1D and (Fig. 10c) 2D methods. Note that the 1D method results assume a unidirectional wave field, taking the mean wind direction as the wave direction. The saturation spectrograms in Figs. 10a and 10b are frequency limited such that the intrinsic frequencies that are greater than fb are not resolved. We also find that the directional fluctuations in the saturation spectrogram are still present when using the 1D method. Here, this projection leads to the overcompensation of the power spectral density such that the saturation spectral level is mapped to either too high (with-wave legs) or too low (against-wave legs) of levels. In contrast, the 2D method used in Fig. 10c removes nearly all fluctuations. This is particularly evident in the shape of the contour line of the saturation spectrogram taken at B = 6 × 10−10 m2 Hz4 (shown in all panels of Fig. 10). This saturation spectral level was chosen because it is close to the frequency regime where fluctuations are most pronounced.
Saturation spectrogram for Wave Glider Stokes’s repeated small 500-m box trajectory in (a) observed frequency space and intrinsic frequency space using the (b) 1D and (c) 2D methods. The black curve is the 6 × 10−10 m2 Hz4 saturation spectral level contour to emphasize fluctuation.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
To quantify this improvement, we compute the correlation coefficient between the time series of the Wave Glider’s projected speed in the mean wave direction at high frequency and the mean saturation 〈B〉 defined here as the average saturation spectral level over the frequency band from 0.1 to 1 Hz in both observed and intrinsic frequency space. The low limit of this frequency band is chosen such that the fluctuations around 0.2 Hz are included in the mean saturation estimate. We find the correlation to be r = −0.546, 0.407, and 0.196 for observed, 1D, and 2D methods, respectively. These findings support the argument that the 2D method presented here is the most effective approach to accounting for Doppler effects in 1D spectra. The method correctly projects the spectral energy onto frequencies such that fluctuations in power spectral density over time are removed across all frequencies.
Last, we applied the 2D method to the saturation spectrogram computed over 10-min intervals for the entire DELMAR2020 and SMODE2021 experiments independent of platform heading (Fig. 11). Here, only spectrograms from Wave Gliders Stokes and WHOI43 are shown (similar results were found with Wave Glider Planck). We find that the 2D method performs well, removing any spurious fluctuations. Note that the fb for the SMODE2021 experiment is particularly low, especially from 1 to 2 November (minimum fb = 0.5 Hz). This is due to the high velocity of the platform during this time period (mean platform speed
(a),(c) Observed and (b),(d) intrinsic saturation spectrograms during the (a),(b) DELMAR2020 and (c),(d) SMODE2021 experiments for Wave Gliders (a),(b) Stokes and (c),(d) WHOI43. Spectrograms were computed over 10-min intervals independent of the Wave Glider heading.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
5. Discussion and summary
Autonomous surface vehicles play a crucial role in investigating air–sea interactions at the space and time scales of surface waves and in learning how surface waves influence Earth’s climate system. This better understanding could lead to improved forecasting capabilities of current weather and climate models. However, observations from moving vehicles present challenges in characterizing the properties of high-frequency surface waves. The platform speed and direction relative to the incoming waves cause the observed wave period to be Doppler shifted, particularly at high frequencies.
In this work, building upon past work on this topic (Kats and Spevak 1980; Drennan et al. 1994; Hanson et al. 1997; Cifuentes-Lorenzen et al. 2013; Collins et al. 2017; Amador et al. 2022) we present a general approach to correct Doppler effects in the directional wave spectrum so that computed spectral wave properties are independent of the platform’s motion. This method accounts for the full directionality in the wave field to properly map observed frequencies into the intrinsic frequency space, a reference frame free of Doppler effects that can be applied to any ASV. The significance of the spectral cutoff frequency is discussed by illustrating its dependence on the platform speed and relative angle. In particular, we show that it is crucial to consider the speed and direction of the platform relative to the waves during experimental planning if, for example, accurate characterization of the high-frequency portion of the wave spectrum is needed.
Last, we demonstrate the usefulness of this method with a unique dataset collected from instrumented Wave Gliders off the coast of Southern California which exhibits fluctuations in the spectral level of a wave spectrogram that are dependent upon the wave frequency, the platform speed, and the relative angle between wave and platform propagation directions. Using the 2D method, we show that Doppler effects are removed whereas fluctuations are still presented when using the 1D method. This illustrates that the intrinsic frequency reference frame, mapped using the 2D method, provides a coherent way to compare wave measurements from moving platforms and provide accurate measurements of directional surface waves down to short spatial scales [O(1) m].
Central to this work is the published open-source code that automates the 2D method for Doppler-correcting directional wave spectra. The code is a collection of documented functions for MATLAB 2022 software which are publicly available on GitHub and Zenodo (see data availability statement and supplementary material for more details). This code provides easy-to-use functions to implement the methods discussed here to data collected from any ASV, thus enabling the scientific community to accurately observe and investigate short-wavelength surface waves.
Acknowledgments.
The authors are grateful to Tom Farrar and the Upper Ocean Physics Laboratory at Woods Hole Oceanographic Institution for sharing their data with us. The authors are grateful to the crews of the R/V Oceanus and R/V Beyster for their support during the DELMAR2020 and SMODE2021 field deployments, respectively. This research was supported by ONR (Grant N00014-19-1-2635), and NASA (Grant 80NSSC19K1688).
Data availability statement.
The data used for this study are available through the University of California, San Diego, library’s Digital Collections at https://doi.org/10.6075/J0C829GC and it has been published as Colosi et al. (2023). The source code used to produce the figures can be accessed in the following GitHub repository https://github.com/lcolosi/WaveSpectrum as well as at https://doi.org/10.5281/zenodo.8102880.
APPENDIX
On the Solutions of k
For the mapping from observed to intrinsic frequency, we need to solve the deep-water surface wave dispersion relation (5) for k. The analytic solutions of (5) are given by (6) and (8) along with a breakdown of them in Table 1. Here, two natural questions arise: Why do multiple solutions exist and what does each solution correspond to physically? More specifically, why in a following seas scenario are two waves with distinct frequencies in the stationary reference frame R observed at one frequency in the moving reference frame R′?
(a) An illustration of the surface of revolution A2 [blue (+σ) and red (−σ) surfaces] and the plane A1 (green surface) with ω = 3.1412 rad s−1, U = 0.55 m s−1, and ϕ = 0°. (b),(c) Values of A2 and A1 on the surfaces of constant wavenumber for l =0 and k = 0, respectively. The k solutions of the dispersion relation, using variables defined in Table 1, are at intersection points of A2 and A1 (shown as the vertical black lines). The platform’s velocity vector U is shown in the k plane.
Citation: Journal of Atmospheric and Oceanic Technology 40, 10; 10.1175/JTECH-D-23-0022.1
Four solution points found at the intersections between traces in the l = 0 plane are labeled in Fig. A1b using the variables defined in Table 1. Note that the wavenumber components k and l are equal to plus or minus the wavenumber magnitude in the l = 0 and k = 0 planes, respectively, where the sign in front of k is set by the wave direction. The l = 0 plane corresponds to θr = 0° and 180°, and the k = 0 plane corresponds to θr = 90° and 270°.
The solution ka corresponds to a wave with a wavenumber vector k in the direction opposite of platform propagation [i.e., an approaching wave with U cos(θr) < 0], as seen by its negative k component, so that the observed frequency ω is greater than the intrinsic frequency σ. Similarly, the solution klt represents a wave with a wavenumber vector k in the direction of platform propagation [i.e., U cos(θr) > 0] so that ω is less than σ. These demonstrate the well-known Doppler effect; the next two solutions for k do not exist in the absence of platform velocity or for nondispersive waves.
Much like klt, the solution kst corresponds to a wave propagating in the same direction as the platform. However, its crests move faster than and overtake the platform while its energy moves slower and is outran. This can be seen by looking at the slope of (A2) at the intersection point. By inspection, the slope of (A2) (i.e., the group velocity), evaluated with the intrinsic frequency associated with the intersection, is less than the slope of (A1) (i.e., the projected platform speed). However, the slope of the plane is not so great that it is twice the group velocity (i.e., the phase speed). Therefore, we arrive at cg < U cos(θr) < c, which is consistent with the projected speed domain for kst in Table 1. A similar argument can be made for the klt solution showing that the slope of the dispersion relation at the intersection point is greater than the plane’s slope. That is, 0 < U cos(θr) < cg thus corroborating the projected speed domain for klt in Table 1. klt corresponds to longer-wavelength waves with higher group and phase speeds compared to the short-wavelength waves represented by kst. However, both long- and short-wavelength solutions correspond to waves with phase speed overtaking the platform. The intrinsic frequency σ associated with kst is greater than the observed frequency ω more so than in the scenario for klt because the magnitude of the Doppler shift in frequency increases with wavenumber magnitude.
In Fig. A1b, the trace of plane (A1) forms a secant line to the surface (A2) for positive k. As we consider higher observed frequencies in the R′ reference frame for the same speed U, we translate the trace of plane (A1) up the m axis and the resulting secant lines approach a tangent line where cg = U cos(θr). At a sufficiently large ω, given by (10), the solutions klt and kst coalesce. This transition from secant to tangent line corresponds to the bifurcation point in our solutions for k. At the bifurcation point, the wave energy is stationary relative to the platform’s velocity while the wave crests will be progressing forward ahead of the platform.
The solution kr corresponds to a wave propagating in the direction of the platform velocity but whose crests are overrun by the platform [i.e., c < U cos(θr)]. This causes the waves to have negative observed frequencies (ω < 0) and appear to move in the opposite direction of the platform, hence the intersection with the −σ surface and the negative wavenumber component k. Note that in reference frame R, the intrinsic frequency is in fact positive as shown in Fig. 3.
Two solution points found at the intersections between traces in the k = 0 plane are labeled in Fig. A1c. The superscript in kp denotes the sign of the wavenumber magnitude where − and + signs correspond to waves propagating in directions θr = 270° and 90°, respectively. Solutions
When the platform is approaching or moving perpendicular to the waves, k is determined uniquely. However, in the case of following seas (i.e., 0° ≤ θr < 90° and 270° ≤ θr < 360°), multiple solutions for k exist leading to the frequency ambiguity. To resolve this, we need to pick one of the two possible solutions: klt or kst. We make this choice using the Newton–Raphson root-finding (NRRF) algorithm, an iterative process that solves for the roots of a given real-valued function using the function’s first derivative and a carefully selected initial guess (Verbeke and Cools 1995). We use the solution for k in the absence of platform forward motion (kp) as the initial guess. Here, we recognize that at some point in time, the platform is at a state of rest on the ocean surface with U = 0 and observed frequencies fall along the dispersion relation given by (1) such that ω = σ(k). From this rest state, the platform speed projected onto the wave direction U cos(θr) is perturbed continuously as the magnitude and direction of the platform velocity fluctuate. This, in turn, leads to continuous deformation from the dispersion relation in reference frame R, i.e., σ(k), to the dispersion relation in reference frame R′, i.e., ω(k). Therefore, beginning with kp will lead us to the proper solution for k. We find with this numerical approach that the solution of (5) is the lowest wavenumber magnitude branch klt thus providing an answer of which solution of k to pick.
REFERENCES
Alvarez, A., 2015: Assessment of sea wave spectra using a surfaced glider. Deep-Sea Res. I, 102, 135–143, https://doi.org/10.1016/j.dsr.2015.04.015.
Amador, A., S. T. Merrifield, and E. J. Terrill, 2022: Assessment of atmospheric and oceanographic measurements from an autonomous surface vehicle. J. Atmos. Oceanic Technol., 40, 305–326, https://doi.org/10.1175/JTECH-D-22-0060.1.
Bendat, J. S., and A. G. Piersol, 2011: Random Data: Analysis and Measurement Procedures. John Wiley and Sons, 640 pp.
Bretherton, F. P., and C. J. R. Garrett, 1968: Wavetrains in inhomogeneous moving media. Proc. Roy. Soc. London, 302A, 529–554, https://doi.org/10.1098/rspa.1968.0034.
Brodtkorb, P. A., P. Johannesson, G. Lindgren, I. Rychlik, J. Rydén, and E. Sjö, 2000: WAFO—A MATLAB toolbox for analysis of random waves and loads. 10th Int. Offshore and Polar Engineering Conf., Seattle, WA, ISOPE, 343–350, https://onepetro.org/ISOPEIOPEC/proceedings-abstract/ISOPE00/All-ISOPE00/ISOPE-I-00-264/7046.
Cavaleri, L., B. Fox-Kemper, and M. Hemer, 2012: Wind waves in the coupled climate system. Bull. Amer. Meteor. Soc., 93, 1651–1661, https://doi.org/10.1175/BAMS-D-11-00170.1.
Charnock, H., 1955: Wind stress on a water surface. Quart. J. Roy. Meteor. Soc., 81, 639–640, https://doi.org/10.1002/qj.49708135027.
Cifuentes-Lorenzen, A., J. B. Edson, C. J. Zappa, and L. Bariteau, 2013: A multisensor comparison of ocean wave frequency spectra from a research vessel during the Southern Ocean Gas Exchange experiment. J. Atmos. Oceanic Technol., 30, 2907–2925, https://doi.org/10.1175/JTECH-D-12-00181.1.
Collins, C. O., III, and Coauthors, 2017: Doppler correction of wave frequency spectra measured by underway vessels. J. Atmos. Oceanic Technol., 34, 429–436, https://doi.org/10.1175/JTECH-D-16-0138.1.
Colosi, L., N. Pizzo, L. Grare, N. Statom, and L. Lenain, 2023: Data from: Observations of surface gravity wave spectra from moving platforms. UC San Diego Library Digital Collections, https://doi.org/10.6075/J0C829GC.
Dallolio, A., B. Agdal, A. Zolich, J. A. Alfredsen, and T. A. Johansen, 2019: Long-endurance green energy autonomous surface vehicle control architecture. Oceans 2019 MTS/IEEE Seattle, Seattle, WA, IEEE, https://doi.org/10.23919/OCEANS40490.2019.8962768.
Drennan, W. M., M. A. Donelan, N. Madsen, K. B. Katsaros, E. A. Terray, and C. N. Flagg, 1994: Directional wave spectra from a swath ship at sea. J. Atmos. Oceanic Technol., 11, 1109–1116, https://doi.org/10.1175/1520-0426(1994)011<1109:DWSFAS>2.0.CO;2.
Fairall, C. W., E. F. Bradley, J. E. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571–591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.
Farrar, J. T., and Coauthors, 2020: S-MODE: The Sub-Mesoscale Ocean Dynamics Experiment. IEEE Int. Geoscience and Remote Sensing Symp., Waikoloa, HI, IEEE, 3533–3536, https://doi.org/10.1109/IGARSS39084.2020.9323112.
Gentemann, C. L., and Coauthors, 2020: Saildrone: Adaptively sampling the marine environment. Bull. Amer. Meteor. Soc., 101, E744–E762, https://doi.org/10.1175/BAMS-D-19-0015.1.
Grare, L., N. M. Statom, N. Pizzo, and L. Lenain, 2021: Instrumented wave gliders for air-sea interaction and upper ocean research. Front. Mar. Sci., 8, 664728, https://doi.org/10.3389/fmars.2021.664728.
Hanson, K. A., T. Hara, E. J. Bock, and A. B. Karachintsev, 1997: Estimation of directional surface wave spectra from a towed research catamaran. J. Atmos. Oceanic Technol., 14, 1467–1482, https://doi.org/10.1175/1520-0426(1997)014<1467:EODSWS>2.0.CO;2.
Hodges, B., L. Lenain, T. Farrar, L. Grare, K. Matsuyoshi, N. Pizzo, and N. Statom, 2023: Evaluation of ocean currents observed from autonomous surface vehicles. J. Atmos. Oceanic Technol., https://doi.org/10.1175/JTECH-D-23-0066.1.
Kats, A., and I. Spevak, 1980: Reconstruction of the sea-wave spectra from measurements of moving sensors. Izv. Acad. Sci. USSR, Atmos. Oceanic Phys., 16, 194–200.
Lenain, L., and W. K. Melville, 2014: Autonomous surface vehicle measurements of the ocean’s response to Tropical Cyclone Freda. J. Atmos. Oceanic Technol., 31, 2169–2190, https://doi.org/10.1175/JTECH-D-14-00012.1.
Lenain, L., and N. Pizzo, 2020: The contribution of high-frequency wind-generated surface waves to the Stokes drift. J. Phys. Oceanogr., 50, 3455–3465, https://doi.org/10.1175/JPO-D-20-0116.1.
Longuet-Higgins, M. S., 1986: Eulerian and Lagrangian aspects of surface waves. J. Fluid Mech., 173, 683–707, https://doi.org/10.1017/S0022112086001325.
Lygre, A., and H. E. Krogstad, 1986: Maximum entropy estimation of the directional distribution in ocean wave spectra. J. Phys. Oceanogr., 16, 2052–2060, https://doi.org/10.1175/1520-0485(1986)016<2052:MEEOTD>2.0.CO;2.
McAllister, M., and T. van den Bremer, 2019: Lagrangian measurement of steep directionally spread ocean waves: Second-order motion of a wave-following measurement buoy. J. Phys. Oceanogr., 49, 3087–3108, https://doi.org/10.1175/JPO-D-19-0170.1.
Melville, W. K., 1996: The role of surface-wave breaking in air-sea interaction. Annu. Rev. Fluid Mech., 28, 279–321, https://doi.org/10.1146/annurev.fl.28.010196.001431.
Mitsuyasu, H., 1985: A note on the momentum transfer from wind to waves. J. Geophys. Res., 90, 3343–3345, https://doi.org/10.1029/JC090iC02p03343.
Peregrine, D., 1976: Interaction of water waves and currents. Adv. Appl. Mech., 16, 9–117, https://doi.org/10.1016/S0065-2156(08)70087-5.
Phillips, O. M., 1985: Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech., 156, 505–531, https://doi.org/10.1017/S0022112085002221.
Thomson, J., and Coauthors, 2015: Biofouling effects on the response of a wave measurement buoy in deep water. J. Atmos. Oceanic Technol., 32, 1281–1286, https://doi.org/10.1175/JTECH-D-15-0029.1.
Thomson, J., J. B. Girton, R. Jha, and A. Trapani, 2018: Measurements of directional wave spectra and wind stress from a Wave Glider autonomous surface vehicle. J. Atmos. Oceanic Technol., 35, 347–363, https://doi.org/10.1175/JTECH-D-17-0091.1.
Verbeke, J., and R. Cools, 1995: The Newton-Raphson method. Int. J. Math. Educ. Sci. Technol., 26, 177–193, https://doi.org/10.1080/0020739950260202.
