## 1. Introduction

*T*

_{peak}is typically considered to be a measure for the period of the dominant, or most energetic, waves. This follows from its definition as the period

*T*, corresponding to the peak of the wave variance (∝ energy) spectrum

*E*(

*f*):

*f*denoting the frequency and

*T*= 1/

*f*denoting the wave period. This interpretation, combined with the ease of definition and calculation, is why it often serves to characterize the wave field. For example,

*T*

_{peak}is used to predict wave run-up based on offshore wave conditions (Stockdon et al. 2006), characterize peak orbital velocities for sediment transport estimates (van der A. et al. 2013), estimate stability of rubble mound structures (van der Meer and Daemen 1994), estimate Stokes drift (Breivik et al. 2014), and characterize the role of the wave field in ice break up (Voermans et al. 2020). Given the broad utilization of

*T*

_{peak}, real-time estimates derived from wave buoy observations [such as the SOFAR Spotter Raghukumar et al. (2019)] are of particular value.

The peak period measure suffers from two deficiencies. Foremost, spectral measurements are noisy, both due to observational error and due to inherent fluctuations in irregular waves. Consequently, for multimodal wave spectra, a dominant wave system cannot always be unambiguously and robustly defined, and peak period estimates may oscillate between systems of comparable strength for successive observations. This well-known issue is fundamental to the definition and, while undesirable, can only be ameliorated by the use of different period measures [e.g., mean period and zero crossing period; see, for example, Holthuijsen (2010)]. This is arguably a fundamental short-coming of the definition.

Second, the finite duration of observations and the use of the discrete Fourier transform in spectral estimation imply that spectra are discretely sampled on a regular frequency grid defined as *f _{n}* =

*n*Δ

*f*, where in the present context (for reasons discussed below), Δ

*f*≈ 0.01 Hz. Consequently, at swell frequencies (

*f*< 0.1 Hz), the period resolution at

*O*(1) s is relatively coarse—and catastrophically so in the lowest resolvable bins. This loss of resolution is not fundamental to the definition but results from practical limitations imposed by spectral analysis.

To ensure quasi-stationary wave conditions (a basic assumption of spectral analysis), spectral estimates are typically constrained to records of 30 min–1 h duration *D*. To reliably estimate the spectrum, one typically uses Welch’s method (Welch 1967) and subdivides the duration into *N* overlapping windows (50% overlap) with duration *D _{w}* ∝

*D*/

*N*(implying Δ

*f*∝

*N*/

*D*) to generate an ensemble of spectral realizations, from which the final estimate is defined as the mean. For larger

*N*, the certainty of spectral estimates is increased at the cost of increased Δ

*f*. Specifics may vary (windowing, overlap, etc.), but given typical values of

*D*and the dual requirements of sufficiently independent of ensemble members and sufficiently large

*N*, the effective frequency bin size Δ

*f*is restricted to

_{e}*O*(0.01) Hz.

Alternatives (zero padding and frequency-domain smoothing) to conventional spectral analysis do exist, but these invariably are more computationally involved (a barrier to on-device implementation) and require access to time-domain data to regenerate spectral estimates. The latter is not always possible for historical records, or for active but remote instruments such as the 550+ Spotter drifters currently deployed in SOFAR’s open ocean network. Here, we therefore consider a different approach to improve peak period estimates. Specifically, justified based on the relation between total energy contained within a finite-frequency band and the discrete estimate thereof, we propose to define the spectral density as the derivative of a spline-based interpolation of the spectral distribution function. The resulting spectrum is continuous and consequently permits a continuous definition of the peak period. In what follows, we first discuss spectral analysis and subsequently motivate our spectral definition. Next, we validate the method with artificial data and with reprocessed spectral data observed at Ocean Beach, San Francisco, California. Finally, we end with a summary of findings and a discussion of implications and limitations.

## 2. Spectral density estimation

*ζ*(

*t*). Following Hasselmann et al. (1963), this zero-mean signal can be considered as a random, stationary process and related via the Fourier–Stieltjes integral to the Fourier–Stieltjes amplitude

*dZ*(

*f*) as

*ω*= 2

*πf*. To note, the finite approximation of

*dZ*(

*f*) is the discrete Fourier transform of the vertical displacement, to be discussed in the following section. Proceeding with continuous variables for now, the power spectral density of the vertical displacement time series

*dZ*(

*f*) with the Wiener–Khinchin theorem as

*f*→ ∞. Due to the symmetry of

*f*= 0, it is convenient to introduce the one-sided

*V*(

*f*) and

*E*(

*f*) as

*E*(

*f*) as our formal definition of the energy spectrum estimated from observations. The connection from the buoy observation to the spectral distribution and finally the estimated

*E*(

*f*) becomes more explicit when considering finite-length discrete signals in the following section.

### Finite-length discrete signals

Observed signals are always inherently discretely sampled and of a finite length. Let the sampling frequency *f _{s}* equal 1/(Δ

*t*) with Δ

*t*being the sampling interval. For a signal of finite length, the longest resolvable frequency is limited to Δ

*f*= 1/

*T*, with

*T*being the observed period.

*E*(

*f*) in terms of discrete, finite-length observations. To that end, we replace the differential on the left-hand side of Eq. (4) with its discrete analog:

*d*Z

*as*

_{n}*ζ*. The discrete variance (energy) within a given frequency band can then be related to the Fourier series of the free-surface elevation using the discrete version of Eq. (4) as follows:

_{n}*f*,

_{n}*dV*/

*df*. This approximation effectively represents

*E*(

*f*) as the mean gradient of

_{n}*f*

_{n}_{−1/2}≤

*f*<

*f*

_{n}_{+1/2}.

Within the ocean wave community, the energy density spectrum *E _{n}*(

*f*) is thus typically defined following Eq. (8). While formally the density thus defined is piecewise constant (to ensure conservation of total energy), the continuous

*E*(

*f*) is often obtained from linear interpolation of the discretely sampled estimates. Further, in practice, the formal ensemble is replaced with a finite-length temporal ensemble through splitting an observation record into multiple records and assuming statistics are stationary and ergodic. This is invariably combined with sample windowing (to reduce sidelobes) and partial overlap, which improve spectral estimates, but which we will not discuss in further detail to avoid unnecessary complexity. The necessity to subdivide the observation record into multiple records introduces the tension between frequency resolution and statistical confidence.

*V*that is directly sampled and not

_{n}*E*(

*f*). Further, we may estimate the discretely sampled

*V*(

*f*) from the cumulative sum, so that the discretely sampled spectral distribution is approximated by

## 3. Interpolation of spectra and determination of peak period

*V*, we express

_{n}*V*and

*E*as sum over cubic splines:

*f*to the start

*C*

^{2}continuity for

*V*(smooth spectrum, no “kinks”), respects monotone behavior of

*V*(

*E*≥ 0), and is consistent with the observations, i.e.,

*V*(

*f*

_{n}_{+1/2}) =

*V*

_{n}_{+1/2}. For a given set of nodes, a spline satisfying all these conditions may not exist. Here, we therefore consider different interpolations that (approximately) satisfy these conditions.

### a. Piecewise constant and linear interpolation

*E*as direct samples of

_{n}*E*(

*f*), and interpolating curves are anchored on nodes defined by

*f*,

_{n}*E*. Using the nearest neighbor interpolation, the spectrum

_{n}*E*becomes piecewise constant, or the splines linear, i.e.,

*f*

_{n}_{+1/2}so that the distance

*E*linearly, resulting in a quadratic spline

_{n}*f*. Both methods interpolate

_{n}*E*monotonically, ensuring that no new extrema are created so that the peak period is effectively restricted to 1/

*f*. Hence, in both the piecewise and linear cases, the peak period is simply determined based on the maximum of the discrete

_{n}*E*, the standard approach that is described by Eq. (1). The resulting estimates for

_{n}*V*(

*f*) are strictly monotone functions [

*E*(

*f*) ≥ 0]. The linear spline leads to noncontinuous (piecewise constant) spectra that over the interval

*f*

_{n}_{−1/2}≤

*f*≤

*f*

_{n}_{+1/2}exactly integrate to Δ

*V*(as observed). Conversely, the quadratic spline is continuous but does not preserve this integration constraint.

_{n}### b. Natural cubic spline interpolation

*V*

_{f}_{+1/2}instead of

*E*automatically ensures that we adhere to the integration constraint Eq. (10). Further, if we use cubic splines, we can guarantee

_{n}*C*

^{2}continuity by ensuring the function and its first two derivatives are continuous along spline borders, i.e.,

*C*

^{3}continuity at the end points, i.e.,

*a*

_{1}=

*a*

_{2}and

*a*

_{M}_{/2}=

*a*

_{M}_{/2 − 1}(referred to as “not-a-knot” boundary conditions). Because

*E*=

*dV*/

*df*is a quadratic polynomial, peak spectral values are no longer constrained to nodes so that peak frequency becomes a continuous variable. However, natural splines are not necessarily monotone, and consequently, spectral estimates may take on nonphysical negative values. Because

*V*tracks out a relatively smooth curve and splines do not in general suffer unduly from over- or undershoot, negative values are expected to be rare and restricted to regions of sharp curvature (regions next to peaks).

_{i}*E*(

*f*) are found by solving for

*d*

^{2}

*S*/

_{n}*df*

^{2}= 0 for each spline, or equivalently,

*n*if |

*a*| > 0 and if the maximum lies within the spline interval, i.e., if

_{n}### c. Monotone spline interpolation

*S*is monotone by ensuring that gradients at the endpoints are within the monotone region, i.e.,

_{n}*C*

^{2}natural spline may not exist. Here, we therefore follow Wolberg and Alfy (2002) and find the

*C*

^{1}spline that is “closest” to

*C*

^{2}out of all possible

*C*

^{1}splines by reformulating the problem into a constrained minimization problem. Specifically, we minimize for the squared difference of

*d*

^{2}

*S*/

_{n}*df*

^{2}between splines

*C*

^{1}continuous,

*E*> 0) and consistency with the integral constraint. Once a solution is found, maxima are determined in the same way as for the natural spline.

## 4. Test data

### a. Synthetic spectra

To test approaches, we will consider synthetic spectra with different spectral shapes. Synthetic spectra allow us to focus purely on errors related to the reconstruction, since there is no statistical noise. Specifically, we consider Gaussian-shaped spectra characterized by peak frequency *f _{p}* and standard deviation and further consider a JONSWAP (Hasselmann et al. 1973) and Pierson–Moskowitz (PM; Pierson and Moskowitz 1964) proposed spectra with default shape parameters [expressions may be found in Holthuijsen (2010)]. Narrow Gaussian spectra serve as a proxy model for swell, whereas JONSWAP and PM shapes are often used parametric models for developing young and mature sea states. Note that for fitting purposes, only the shape, and not the total area under the curve, matters. Consequently, total variance is arbitrarily set to 1 m

^{2}here (approximately 4-m significant wave height). We will refer to these continuous spectra as the target shapes or

*E*

^{target}.

*f*= 0.01 Hz, integrals are evaluated discretely with the midpoint rule using a mesh size of Δ

*f*/100, and spectral estimates are reconstructed using “observed” Δ

*V*.

_{n}### b. Observed spectra

To compare reconstruction to observed data, including sampling and instrument noise, we use field data obtained from a SOFAR Spotter wave buoy [see Raghukumar et al. (2019) for a description of the Spotter’s performance characteristics and data quality]. The buoy was deployed during September 2022 at Ocean Beach, San Francisco, in approximately 20 m of water. During September, Ocean Beach is typically exposed to persistent low-energy [wave height *O*(1) m] long-period swells (>10 s) from the Southwest, originating from storms in the Southern Ocean.

Hourly spectra are obtained from vertical displacement data sampled at 2.5 Hz. Standard spectral processing onboard of Spotter is based on Welch’s method using a 256 sample long Hann (102.4 s) with 50% overlap, resulting in 69 realizations, or approximately 131° of freedom (DOF), and a spectral resolution of Δ*f* ≈ 0.01. These spectra serve as input to the interpolation methods.

To obtain target spectra for comparison of various interpolation methods, the same dataset was reprocessed to obtain higher-resolution spectral estimates. Specifically, we used the same spectral analysis but with 2048 sample long windows (factor eight longer) to generate spectra with Δ*f* ≈ 0.0012 Hz resolution. In this case, only seven realizations are available within an hour, or 14 DOF. To increase confidence, we averaged spectral levels through convolution with a nine-point centered rectangular window, which (if frequency estimates are independent) raises to DOF with a factor nine (in practice, it will be less). These spectra will serve as target estimates, or ground truth.

## 5. Results

### a. Synthetic spectra

The characteristic spectral width of the peak compared to Δ*f* determines how well the peak is resolved and consequently is expected to play a role in performance of any reconstruction. This is confirmed if we consider Gaussian shapes with a peak frequency of 5.25Δ*f* (we will comment on this choice below) of different widths (Fig. 1). If the width is on the order of Δ*f*, a piecewise (or linear) reconstruction fairly describes bulk energy distribution (sufficiently well for calculation of bulk moments), though the peak is underestimated and details on the peak location are lost. In this case, the underlying data are still sufficiently smooth that the natural spline is monotone, and consequently, the natural and monotone splines are identical.

For decreasing width, the piecewise/linear interpolation methods poorly resolve the shape and negative sidelobes appear with the natural spline. The monotone interpolation fares somewhat better as the monotone constraint essentially add additional information to the system. Once the underlying distribution essentially becomes a subgrid feature of which only total energy is known (width of 0.1Δ*f*), all approximations are essentially equally bad—though negative values and extended footprint disqualify in particular the natural spline. The peak frequency (due to symmetry) falls in the center of the bin, so that only if peak frequency happens to be a multiple of 0.5Δ*f*, the guess will be correct (motivating the choice of *f*_{peak} = 5.25Δ*f* here).

Similar results are found for more realistic spectral shapes (Fig. 2). For both JONSWAP and PM shapes, the peak width is effectively a function of frequency. Consequently, errors are larger when target peak frequencies are extremely low (*f*_{peak} = 0.55 Hz), for example, resulting in negative energies for natural spline reconstruction (Figs. 2a–c). For sea with lower (and arguably more realistic) peak periods of about 10 s, the *C*^{2} spline is monotone, and consequently both spline methods are identical, perform equally well, and in the case of the JONSWAP spectrum (Fig. 2b), appear to improve spectral estimate.

While shape improvements are typically minor, peak period estimates do markedly improve (Fig. 3). For a generally well-resolved peak (e.g., Gaussian with standard deviation Δ*f*), the spline methods have errors <0.25 s for target peak frequencies in 0.05 ≤ *f*_{peak} ≤ 0.15 Hz. Errors are bigger if the peak is not well resolved (e.g., JONSWAP spectrum at 0.55 Hz; see also Fig. 2a), though errors are still at least half of the conventional approximation. The asymmetry of the JONSWAP peak is likely responsible for the bias toward overestimation of the peak period. Last, it is noteworthy that peak periods derived from natural spline reconstruction are comparable with those derived from the monotone spline reconstruction. It appears that peak location is thus generally not negatively impacted by the occurrence of negative (and nonphysical) spectral values.

### b. Observed spectra

For observed spectra, severe loss of *T*_{peak} resolution is particularly noteworthy on 24 September, when 18-s forerunners of a swell system arrive at Ocean Beach (Fig. 4, top). From 24th to 29th, the discrete *T*_{peak} estimates (gray lines) for this swell system are 20.5, 17.1, and 14.6 s, with oscillation between the latter values around the 27th as the swell peak frequency is approximately halfway in between bins. In contrast, natural and monotone spline-based estimates generally follow results from target spectral estimates well.

While the change in the period between 24 and 25 September is large (about 19–16 s), the peak frequency change is *O*(Δ*f*) (cf. peaks between 24 and 25 September, Fig. 4, lower two panels). That the spline-based methods are able to track peak frequency on subgrid scales for observed (and noisy) data is a testament to the robustness of interpolating the distribution function. Further, for practical purposes, the natural splines performs equivalent to the monotone spline reconstruction for peak period estimation. Negative values do occur (e.g., Fig. 4, lower left panel), though very infrequently.

Bimodal seas of comparable magnitude do introduce divergences. For instance, during 17 and 23 September, swell and local sea peaks are of comparable magnitude and peak period generally alternates between the systems. The spline-based *T*_{peak} estimates may diverge from the target, though as argued before, this is mostly an issue of the volatility of the measure in this case. Note that peak location is generally well approximated for spline-based methods though they may disagree on which peak is “dominant” with target spectra. If continuous series of the swell peak period are of interest, partitioning spectra (e.g., Portilla et al. 2009) may alleviate such volatility. This is nontrivial though, as the number of peaks is not constant, and the notion of an individual wave system is not well defined.

## 6. Discussion and conclusions

We considered interpolation methods to improve spectral estimates from observations and in particular improve resolution of peak period estimates for long-period waves. Specifically, we argue that interpolation of the distribution function, as opposed to the density function, can lead to improved estimates of peak location. The method is targeted to cases when spectra of sufficient resolution are not available, and reprocessing of data is not viable (e.g., data from the SOFAR Spotter network).

Our findings show that peak periods are generally improved. Especially, errors for long-period waves are significantly reduced. Further, the interpolation allows for continuous rather than discrete estimates. This allows us to retrieve the well-known decay of swell period after first arrivals which is related to the distance of the observer to the generating storm (Munk et al. 1963). There is however little difference between monotone and natural spline interpolation. While monotone interpolation will avoid negative-spectral densities, peak location is generally unaffected. Since natural spline interpolation methods are readily available (e.g., in “SciPy” in Python, Van Rossum and Drake 1995; Virtanen et al. 2020), this may be preferable if only peak periods are of interest.

Arguably slight improvement in the spectral levels may also be found through interpolation. But changes are typically marginal, especially so when applied to observed data. The total energy, by design, remains conserved in the interpolation. Changes are localized around the spectral peaks, and the tail of the spectrum remains unchanged. Moments of the spectrum are therefore (virtually) unchanged, as was confirmed through comparison to other spectral parameters (mean period, zero crossing period, etc., not shown). Consequently, unless spectral levels near the peak are critical, the spectral interpolation does not add benefits beyond visually smoother results.

The proposed method for peak period estimation is not without limitations. In general, spectral peak period estimates may be corrupted by uncertainty in the energy spectra from which they are derived. However, assuming the standard *χ*^{2} form for spectral confidence intervals, which are not frequency dependent (Bendat and Piersol 2011), the peak period derived from the higher-order spline interpolation is no more vulnerable to uncertainty than the traditional linear or piecewise estimates. The circumstances where the new peak period estimate diverged most noticeably from the target were the case of the bimodal energy spectrum shown in Fig. 4. This emphasizes that a coarser frequency resolution has inherent drawbacks that interpolation cannot always compensate for. Nevertheless, the natural and monotone spline estimates are a significant improvement over traditional techniques in those cases. Moreover, the computational and data transmission costs required to produce the target estimates may be too high for remote systems that rely on onboard processing and telemetry.

Finally, we point out that the method is also applicable to wave spectra obtained from operational third-generation spectral wave models (Booij et al. 1999; Tolman et al. 2019; Komen et al. 1996). These models invariably operate on discrete frequency grids in the spectral domain, and (depending on numerics) spectral values may be interpreted in the finite-volume sense to represent cell averages or equivalently the mean slope of the distribution function [analogous to Eq. (8)]. In practice, operational models typically distribute frequencies as a geometric series, and consequently Δ*f* is typically reduced at low frequencies. For example, Δ*f* = 0.005 Hz at *f* = 0.05 Hz for a typically used growth factor of 1.1 (e.g., Smit et al. 2021), leading to a period delta of about 1.8 s, which may be compared to a delta of 3.3 s for Δ*f* = 0.01. While reduced, this nevertheless represents a potential 10% error so that postprocessing may be of interest in select cases.

## Acknowledgments.

The authors acknowledge support from the Office of Naval Research through Grants N00014-20-1-2439 and N00014-22-1-2405.

## Data availability statement.

Data and Python scripts necessary to reproduce results in this work can be found at https://github.com/sofarocean/peak-period-estimation.

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