## 1. Introduction

In recent years, remote sensing approaches based on lidar technology (Brodie et al. 2015; Martins et al. 2017b, 2020b) or stereo-video imagery (de Vries et al. 2011; Guimarães et al. 2020) have seen tremendous developments and now allow the collection of accurate and direct measurements of the sea surface elevation in nearshore areas. Yet, subsurface pressure sensors and current velocity meters remain the most robust, easy-to-deploy, and versatile solutions for measuring the transformation of wind-generated waves as they propagate shoreward. The relationship between subsurface wave-induced hydrodynamics and the free surface elevation is, however, not straightforward and has been a subject of research for several decades.

*p*is known in both time and space, linear wave theory allows an estimation of the free surface elevation

*ζ*as

**k**is the wavenumber,

*t*is the time variable,

*K*

_{p}refers to the pressure transfer function, and

*h*is the mean water depth. The “

*s*” subscripts used in Eq. (1) for the hydrostatic (

*ζ*

_{hyd,s}) and linear (

*ζ*

_{L,s}) reconstructions of

*ζ*are intended to highlight the

*spatial*character of these reconstructions. The hydrostatic reconstruction

*ζ*

_{hyd,s}is simply related to the spatially varying pressure field

*p*as

*X*denotes the two-dimensional spatial coordinate,

*p*

_{a}is the atmospheric pressure,

*g*is the acceleration of gravity, and

*ρ*is the water density. In this context, the linear wave theory also provides a direct relation between the time- and spatially varying orbital wave velocities

*u*at the bottom and

*ζ*

_{L,s}:

*K*

_{u}refers to the transfer function for orbital wave velocities. In a similar way that

*p*needs to be spatially known in Eq. (1), Eq. (4) requires knowledge of the spatial evolution of

*u*.

_{t}is the partial time derivative operator and the “NL,

*s*” subscript is used to refer to the nonlinear spatial reconstruction of

*ζ*. Accounting for nonlinearities in the wave field leads to a better reconstruction of

*ζ*, with an accurate description of energy levels at high harmonics (Bonneton and Lannes 2017; Mouragues et al. 2019). Since the linear spatial reconstruction

*ζ*

_{L,s}requires knowledge of the temporal and spatial evolution of

*p*(or

*u*), the application of Eq. (6) in the field is somewhat limited in its current form. However, it is important to note that the formula does not involve any spatial derivative. Thus, it can be used locally (at a fixed position

*X*), provided that an approximation of

*ζ*

_{L,s}is available.

*ω*= 2

*πf*is the angular frequency and

*κ*

_{L}denotes the (single-valued) wavenumber magnitude solution to the linear dispersion equation (

*κ*will, hereafter, denote the wavenumber magnitude). The terms

*ζ*

_{hyd}refer to the linear and hydrostatic

*temporal*reconstructions of

*ζ*at a single location

*X*in space, respectively.

Since the TFM relies on the linear dispersion relation, it suffers from its limitations in intermediate and shallow water depths, where nonlinear interactions between triads of frequencies intensify (Phillips 1960; Elgar and Guza 1985a). The nonlinear energy transfers associated with these interactions cause large deviations of dominant wavenumbers *κ* at high frequencies from theoretical values by the linear dispersion relation (Thornton and Guza 1982; Elgar and Guza 1985b; Herbers et al. 2002; Martins et al. 2021). At a given frequency, these deviations were recently shown to increase with the relative amount of forced energy and the efficiency of nonlinear energy transfers, which is strongly controlled by the spectral bandwidth (Martins et al. 2021). The large overestimation of dominant wavenumbers at high frequencies combined with the exponential growth of *K*_{p,L} [see Eq. (8)] lead to the blow-up of reconstructed sea surface energy levels in nearshore areas. Here, it is important to remind that this blow-up only occurs for temporal reconstructions employing the linear dispersion relation [Eq. (9)], and not for linear reconstructions in space (Bonneton and Lannes 2017; Mouragues et al. 2019). A pragmatic solution to this blow-up consists in applying a cutoff frequency. The energy missed in the sea surface spectrum tail with the TFM explains why this approach fails at correctly describing the nonlinear shape and peaky character of nearshore waves. Recent work have reported errors on the significant wave height typically between 5% and 10% near or within the surf zone, while these can reach up to 30% and 60% for individual wave heights and third-order parameters, respectively (Martins et al. 2017a; Bonneton et al. 2018; Mouragues et al. 2019; Martins et al. 2020b). Martins et al. (2021) observed that nonlinear effects on the dispersion relation of irregular waves could be important in water depths characterized by *μ* = (*κ*_{p}*h*)^{2} up to 0.5, where *κ*_{p} is the peak wavenumber. Thus, nonlinear effects in the transfer functions are also expected much farther seaward of the breaking point. At these depths, nonlinear effects can be better accounted with Eq. (6), however, applying the nonlinear formula to an approximation of *ζ*_{L,s} based on the linear dispersion relation (i.e.,

*κ*(

*ω*). Furthermore, the reconstruction has the benefit of not requiring a cutoff frequency, except for preventing noise overamplification at high frequencies. The linear (SL) and nonlinear (SNL) weakly dispersive formula derived by Bonneton et al. (2018) read

*μ*up to 0.2 in Mouragues et al. (2019),

*μ*~ 0.24 in this study].

Even when the most energetic components of a typical incident wave field enter shallow waters (*μ* typically between 0.15 and 0.3, in the presence of relatively strong wind seas or for wave conditions characterized by broad or bimodal spectra. In such situations, the conditions become too dispersive at high frequencies for the weakly dispersive assumption to remain valid. A more general approach is then required to account for the presence of both free components at high frequencies and intense nonlinear effects in the shoaling region.

The objective of this paper is to propose and assess a nonlinear fully dispersive reconstruction of the free surface elevation from subsurface hydrodynamic measurements (pressure or wave orbital velocities). This nonlinear fully dispersive *temporal* reconstruction is obtained via a local application of Eq. (6) with an approximation for the linear *spatial* reconstruction *ζ*_{L,s}. This approximation is based on the determination of the dominant wavenumber spectra *κ*(*ω*) characterizing the wave field. We show that such information are a key determinant for accurately estimating sea surface energy levels at high frequencies, wherever nonlinear effects are significant. Using the dominant wavenumber spectra *κ*(*ω*) allows to accurately quantify the relative contribution—in an energy-averaged sense—of both free and forced components. It also appears critical to correctly quantify both linear and nonlinear effects in the pressure and wave orbital velocities transfer functions for nonlinear nearshore waves.

After describing the theoretical and practical aspects of the different temporal reconstruction methods (section 2), the experimental datasets used herein are briefly presented (section 3). The nonlinear fully dispersive reconstruction is first assessed (section 4) with a laboratory dataset collected at high spatial and temporal resolution [Gently Sloping Beach Experiment (GLOBEX); Ruessink et al. 2013]. This dataset represents an ideal situation, where unidirectional waves are considered and *κ* spectra can be accurately determined from the highly resolved surface elevation measurements. The present approach is then tested in the field (section 5) for the more challenging case of directionally spread waves (Anglet experiment; Mouragues et al. 2020a). In this case, the reconstructions use an approximation for *κ* derived from the Boussinesq theory of Herbers et al. (2002), applied either to the hydrostatic or the directly measured surface elevation signal. The results are discussed in section 6, with a particular focus on the role of the spectral bandwidth on nonlinear effects in the transfer functions. Section 7 summarizes the results of this study and provides some perspectives.

## 2. Theoretical and practical aspects of the reconstruction methods

*ω*,

*κ*(

*ω*) is uniquely defined. For the TFM, such a relation is provided by the linear relation dispersion [Eq. (9)], which considers the wave field to be composed by free components only. In the nearshore region, however, both forced and free components can coexist at a given frequency meaning that energy is spread across different wavenumbers. A dominant wavenumber, representative of this mixed sea state in an energy-averaged sense, thus needs to be used for temporal reconstruction methods. This is a fundamental difference with spatial reconstructions [cf. Eqs (1), (4), and (6)], which directly address the energy repartition across several wavenumbers. If

*ζ*(i.e., irregular waves)—is known, such a representative wavenumber can be defined through the root-mean-square average wavenumber

*κ*

_{rms}(Herbers et al. 2002):

*κ*(

*ω*) spectra are determined with two different methods. For unidirectional irregular waves propagating in the laboratory (GLOBEX), dominant wavenumber spectra

*κ*

_{meas}are estimated directly from cross-spectral analysis between adjacent wave gauges. In practice, we observe that such spectral estimates represent the energy spread across several wavenumbers (see, e.g., appendix A of Martins et al. 2021) and are hence equivalent to

*κ*

_{rms}[Eq. (12)]. In the field,

*κ*

_{rms}is estimated from the pressure and directly measured surface elevation data collected from single sensors using the Boussinesq theory of Herbers et al. (2002).

### a. Linear reconstruction methods

*κ*(

*ω*) can be determined from the linear wave dispersion relation [Eq. (9),

*κ*=

*κ*

_{L}]. In this case, the transfer function for

*u*measured at

*δ*

_{m}above the seabed reduces to

*ζ*with the TFM will be denoted with the “

*L*,

*κ*

_{L}” subscript.

*κ*

_{meas}) or predicted (

*κ*

_{rms}) dominant wavenumbers to approximate the spatial reconstruction

*ζ*

_{L,s}in Eq. (6). This is done by directly providing these dominant wavenumbers to either

*K*

_{p}[Eq. (2)] or

*K*

_{u}[Eq. (5)], depending on which hydrodynamic variable is measured, and replace the spatial Fourier transform with a temporal one. This ensures the correct representation of energy spreading across several wavenumbers (e.g., presence of both free and forced components at a particular frequency), and is the key element behind the present fully dispersive reconstruction. The second linear reconstruction, denoted with the “

*L*,

*κ*

_{meas}” subscript, only concerns the laboratory dataset (GLOBEX) for which direct estimates of dominant wavenumbers are available (

*κ*

_{meas}). The corresponding reconstruction problem reads:

*L*,

*κ*

_{rms}” subscript, provides the transfer function

*K*

_{p}[Eq. (2)] and

*K*

_{u}[Eq. (5)] with the approximation for

*κ*

_{rms}given by Herbers et al. (2002):

*E*(

*ω*) and

*B*(

*ω*

_{1},

*ω*

_{2}) are the spectral and bispectral densities of

*ζ*, respectively, and Re{⋅} denotes the real part. The bispectrum is here defined following Kim and Powers (1979):

*A*(

*ω*) are the complex Fourier coefficients, * denotes the complex conjugate, and

*ε*=

*H*

_{s}/2

*h*, where

*H*

_{s}is the significant wave height, this condition is met for Ursell numbers Ur =

*ε*/

*μ*~

*O*(1). Herbers et al. (2002) further assume the water depth to be slowly varying in the direction of propagation, which with our current notation writes

*β*is a characteristic bottom slope. For the cases investigated next, this is always fulfilled (

The leading-order term of Eq. (17) (*κ*_{sw}) represents the wavenumber given by the dispersion relation for nondispersive shallow-water waves [Eq. (18)] while *β*_{fr} and *β*_{am} are second-order frequency and amplitude dispersion terms, respectively. If the amplitude dispersion effects are neglected [*β*_{am} = 0 in Eq. (17)], we can also define a linear Boussinesq wavenumber *κ*_{B}. Only real values of the bispectrum contribute to amplitude dispersion effects at the order considered in this theory [*O*(*μ*, *ε*), see Herbers et al. 2002]. Although the present approach is fully dispersive with *κ*_{meas} (i.e., no hypothesis required on the wave field dispersive regime), the approximation of *κ*_{rms} is only weakly dispersive. In the following, the linear reconstruction based on *κ*_{rms} will hence be described as moderately dispersive, in order to avoid confusions with the weakly dispersive approaches of Bonneton et al. (2018).

### b. Weakly nonlinear reconstruction methods

The nonlinear reconstruction formula provided in Eq. (6) is based on a linear *spatial* reconstruction of *ζ* computed either from the pressure [Eq. (1)] or orbital wave velocities [Eq. (4)] measured at the sea bottom. Bonneton and Lannes (2017) demonstrated that for linear waves, the spatial Fourier transform in these reconstructions can be replaced with the temporal one so that the spatial [Eqs. (1) and (4)] and temporal [Eqs. (7) and (15)] reconstructions are equivalent. This replacement can also be performed in the case of nonlinear waves of permanent forms (all modes propagating at a speed *C*), as long as *κ*(*ω*/*C*) is used instead of the linear wave dispersion relation (Bonneton and Lannes 2017).

*κ*(

*ω*) is not unique. For replacing the spatial Fourier transform with a temporal one in such cases, we here assume that using the dominant wavenumbers

*κ*

_{rms}is sufficient to correctly represent the energy spreading across different wavenumbers and pass from a

*spatial*to

*temporal*reconstruction problem. With the hypothesis that

*ζ*

_{L,s}at a given location in space, a local and temporal application of Eq. (6) then becomes possible. The nonlinear reconstruction based on directly measured dominant wavenumbers (subscript “NL,

*κ*

_{meas}”) or those predicted with Boussinesq theory (subscript “NL,

*κ*

_{rms}”) are obtained by replacing

*ζ*

_{L,s}in Eq. (6) with the corresponding linear reconstruction, described in the previous section. For the reconstruction that uses directly measured dominant wavenumbers

*κ*

_{meas}, this reads:

*ζ*. Note that for the case of

*p*or

*u*measured at an arbitrary height above the seabed, an extra term arises in Eqs. (6) and (22); the reader is referred to Eq. (C4) of Bonneton and Lannes (2017) for the complete formula.

Equation (6) is valid for *κ* approximates well the magnitude of the wavenumber projected in the main direction of propagation. For large directional spreading angles or crossing seas, nonlinear energy transfers toward high harmonics become slightly less efficient (Herbers and Burton 1997; de Wit et al. 2020). Thus, in such situations, Eqs. (6) and (22) could slightly overestimate the intensity of nonlinear interactions.

In the following, we show that both *ω*_{c} in the linear reconstructions

Synthesis of the temporal reconstruction methods investigated in this study and their range of application.

## 3. Experimental datasets

### a. GLOBEX

The laboratory dataset was collected during GLOBEX, which was performed in a 110-m-long, 1-m-wide, and 1.2-m-high wave flume located in the Scheldegoot in Delft, the Netherlands (Ruessink et al. 2013). During these experiments, free surface elevation and current velocity data were collected at high spatial and temporal resolution in order to study the transformation of short and infragravity waves propagating over a mildly sloping beach (e.g., see de Bakker et al. 2015; Tissier et al. 2015; Rocha et al. 2017). The high spatial resolution was obtained by repeating each wave test 10 times and moving the wave gauges (sampling at 128 Hz) across the 1:80 concrete beach (see Fig. 1). In addition to wave gauges, five electromagnetic current meters (ECM) were fixed to the movable trolleys and allowed the collection of current velocities at numerous cross-shore locations. An acoustic Doppler velocimeter (ADV) was also deployed at four different locations, as shown in Fig. 1.

This study concentrates on the 70-min-long irregular wave tests of the A series (A1, A2, A3; see Ruessink et al. 2013) during which JONSWAP spectra were imposed at the wave paddle. These wave tests reproduced moderate to energetic sea conditions characterized by broadband to narrowband sea surface spectra; see Table 2. For these experiments, dominant wavenumber *κ* were computed via cross-spectral analyses of the highly resolved surface elevation dataset (Martins et al. 2021). Besides their relevance for verifying the proposed fully dispersive approach, these direct measurements are particularly useful to assess the accuracy of the Boussinesq predictions of *κ*_{rms} [Eq. (17)] (Herbers et al. 2002) and the corresponding reconstructions. Indeed, in most typical field situations, measurements of dominant wavenumbers *κ* are not available and Eq. (17) remains the only option to get an estimation of this quantity. For the GLOBEX dataset, bispectra in Eq. (17) were computed on the free surface elevation signals downsampled to 16 Hz by averaging estimates from 126 records of 128 s, which were overlapping by 75%. Statistical stability was increased by merging estimates over three frequencies (e.g., see Elgar and Guza 1985a) yielding approximately 150 equivalent degrees of freedom, and a spectral resolution of 0.023 Hz. The measured and predicted dominant wavenumbers are used in the following to reconstruct the free surface elevation from cross-shore current velocities measured with the ECMs. As the ADV data contained more noise at high frequencies, the data from this sensor was not used for the reconstruction but only to calibrate current velocities as measured by the ECMs (see online supplemental material). Energy spectra of *u* and *ζ* were computed by averaging estimates from 63 Hann-windowed records of 128 s, which were overlapping by 50%, yielding approximately 70 equivalent degrees of freedom, and a spectral resolution of 0.008 Hz.

Significant wave height *H*_{s} and discrete peak wave frequency *f*_{p} imposed during the three irregular wave tests of GLOBEX. The peak enhancement *γ* of the JONSWAP spectra characterizes its spectral bandwidth: a value of 3.3 corresponds to broad spectra while a narrowbanded spectra is imposed with *γ* = 20.

### b. Anglet field experiment

In addition to the GLOBEX dataset, we also use data collected in Anglet, France, during field experiments conducted at La Petite Chambre d’Amour beach from 3 to 26 October 2018 (Mouragues et al. 2020a,b). These experiments aimed at studying the wave-induced circulation occurring along this geologically constrained beach under varying incident wave conditions. The data from two particular current profilers are used: SIG1, a Nortek Signature 500 kHz continuously sampling the near-bottom pressure and Eulerian fluid velocities across the vertical at 4 Hz, and SIG2, a Nortek Signature 1000 kHz that sampled similar quantities at 8 Hz. Additionally, both instruments continuously tracked the free surface elevation by acoustic means [acoustic surface tracking (AST)]. The SIG1 and SIG2 pressure sensors were deployed at 0.6 and 0.69 m above the seabed in mean water depths of 13.9 and 8.1 m, respectively. The depth contours around both instruments are nearly parallel to the coastline [see Mouragues et al. (2020a) for further details].

Similar to the GLOBEX experiments, two contrasting events in terms of incident energy and frequency bandwidth were selected among the nearly two weeks during which both instruments collected data simultaneously. Both events consider shoaling waves outside the surf zone so that the measurements from the AST remain fully reliable. The offshore significant wave height *H*_{s} and peak wave period *T*_{p} measured at the nearest wave buoy [Centre d'Archivage National des Données de Houle In Situ (CANDHIS) buoy 06402, located 3.5 km off the coast in 50-m water depth] were quite similar during both events with *H*_{s} ~ 2–2.5 m and *T*_{p} ~ 13 s. Case A (low tide at 0100 local time 15 October 2018) corresponds to a moderate energy event characterized by (frequency) broadband sea surface spectra, while case B (high tide at 1800 local time 13 October 2018) is a relatively more energetic event characterized by narrowband sea surface spectra. The wave directional spreading angle measured offshore was weak during both events, with mean values of 27° and 20° during case A and B, respectively (20° at the energy peak for both events). The offshore mean wave direction was estimated between 290° and 300° W, which corresponds to a mean incident angle smaller than 10° relative to the beach orientation. The temporal mean depth-averaged currents *U* observed at the two sensors location were <0.1 m s^{−1} during both events which, at the highest frequency of interest here (5.5*f*_{p} = 0.43 Hz, with *f*_{p} the peak wave frequency), corresponds to mean current to (linear) wave phase speed ratio *U*/*c* <0.03 at all times. The background mean current is therefore neglected in the present study. Other relevant parameters for these events are given in Table 3.

Relevant parameters for the two events from the Anglet experiments (Mouragues et al. 2020a): significant wave height *H*_{s}, peak wave period *T*_{p}, mean water depth *h*, dispersion parameter *μ*, nonlinearity parameter *ε* = *H*_{s}/2*h*, and Ursell number Ur = *ε*/*μ*.

Dominant *κ*_{rms} were computed from two different sources: *ζ*_{hyd} and *ζ*_{AST}, the free surface directly measured with the AST. Bispectra were computed from 2.2-h-long detrended time series using 61 records of 512 s, which were overlapping by 75%. Stationarity was ensured by selecting ~1.1 h before and after low (case A) or high tide (case B), the maximal water depth variation being only 0.36 m. Energy spectra of *ζ* were computed by averaging estimates from 61 Hann-windowed records of 512 s, which were overlapping by 75%. Statistical stability was increased by averaging spectral and bispectral estimates over three frequencies yielding approximately 103 and 73 equivalent degrees of freedom, respectively, and a frequency resolution of 0.0059 Hz for both estimates.

## 4. Assessment of the nonlinear fully dispersive reconstruction methods

In this section, the different reconstruction methods presented in section 3 are assessed using the dataset collected during GLOBEX, for which measurements of *κ*(*ω*) are available [denoted *κ*_{meas}(*ω*)]. We focus on two of the four regimes of propagation (I–IV) described in Martins et al. (2021): a shoaling situation (stage II) and near-breaking situation (stage III). Besides being of little interest in the present context, the linear regime (stage I) occurs in the deepest section of the flume (up to ~*x* = 15 m) and hence cannot be addressed here since the first current meter was located at *x* = 24.86 m. Similarly, the stage corresponding to surf zone conditions (stage IV) is not covered here due to limitations and uncertainties in the measurements collected ~1 cm from the flume bottom, in the vicinity of the boundary layer [see also Aubrey and Trowbridge (1985) for a discussion of ECMs in such conditions]. Nonetheless, the weakly dispersive reconstruction of Bonneton et al. (2018) was already shown to accurately describe both the sea surface energy content at all frequencies and the wave-by-wave characteristics in surf zones (Martins et al. 2020a,b).

The current velocity density spectra *E*_{u}(*ω*) corresponding to stages II and III are displayed in Fig. 2. The two wave tests compared here (A2 and A3) differ in the energy and spectral bandwidth imposed at the paddle (see also Table 2): A2 corresponds to energetic conditions with broadband spectra while for A3, the conditions are typical of a swell (moderate energy and narrowband spectrum). For both tests, the cutoff frequency for not overamplifying noise in the measurements was determined at 3.35*ω*_{p}, with *ω*_{p} the peak wave angular frequency, based on the slope changes occurring on *E*_{u}(*ω*) around this frequency.

Cross-shore current velocity energy density spectra *E*_{u}(*ω*) computed at the location in the wave flume corresponding to the stages II (Ur ~ 0.3) and III (Ur ~ 0.7) described in Martins et al. (2021). The frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test (left) A2 and (right) A3. The value of *k*_{p}*h* for each test and propagation stage is indicated in the legend. The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. The cutoff frequency *ω*_{c} applied to the fully and moderately dispersive linear reconstructions is indicated by the vertical red dashed line.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Cross-shore current velocity energy density spectra *E*_{u}(*ω*) computed at the location in the wave flume corresponding to the stages II (Ur ~ 0.3) and III (Ur ~ 0.7) described in Martins et al. (2021). The frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test (left) A2 and (right) A3. The value of *k*_{p}*h* for each test and propagation stage is indicated in the legend. The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. The cutoff frequency *ω*_{c} applied to the fully and moderately dispersive linear reconstructions is indicated by the vertical red dashed line.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Cross-shore current velocity energy density spectra *E*_{u}(*ω*) computed at the location in the wave flume corresponding to the stages II (Ur ~ 0.3) and III (Ur ~ 0.7) described in Martins et al. (2021). The frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test (left) A2 and (right) A3. The value of *k*_{p}*h* for each test and propagation stage is indicated in the legend. The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. The cutoff frequency *ω*_{c} applied to the fully and moderately dispersive linear reconstructions is indicated by the vertical red dashed line.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

### a. Shoaling of weakly nonlinear waves (II)

During stage II, nonlinear energy transfers between principal components promote the dominance of bound high harmonics in wavenumber spectra (see Figs. 3a,b). The differences in dispersive properties between wave tests A2 and A3 are mainly explained by differences in forced-to-free energy ratio at high harmonics, as estimated in Martins et al. (2021). Although less energetic than for A2, the narrowbanded condition imposed during A3 led to more effective nonlinear energy transfers by sum interactions [(*ω*_{p}, *ω*_{p}) → 2*ω*_{p}; (*ω*_{p}, 2*ω*_{p}) → 3*ω*_{p}], so that 70% and 90% of the energy was estimated to be forced around 2*ω*_{p} and 3*ω*_{p}, respectively. These intense energy transfers explain why measured dominant *κ*(*ω*) at high harmonics (e.g., 2*ω*_{p}, 3*ω*_{p}, and 4*ω*_{p}) follow the simple relation *ω*/*c*(*ω*_{p}) (Fig. 3b), suggesting that these components mostly propagate at the same speed as the peak component *ω*_{p}. In contrast, during A2, dominant *κ*(*ω*) at high frequencies lie between *ω*/*c*(*ω*_{p}) and values predicted by the linear wave dispersion relation (Fig. 3a), which is explained by the weaker amounts of forced energy found during this test (around 20% and 30% at 2*ω*_{p} and 3*ω*_{p}, respectively). The weakly nonlinear approximation of Herbers et al. (2002) [Eq. (17)] provides very accurate estimates of the dominant wavenumbers and describes well its variation across infragravity and gravity frequency bands up to 2.5*ω*_{p} for both tests. Components at higher frequencies are too dispersive for Eq. (17) to be accurate and a higher-order theory is probably required. These results are consistent with those from Herbers et al. (2002) who focused on weakly dispersive waves for their comparisons at *ω*_{p}, 2*ω*_{p}, and 3*ω*_{p} (data for which 3*f*_{p} > 0.25 Hz were removed for the model assessment).

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage II (*h* = 0.56 m for A2, *h* = 0.44 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructions. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage II (*h* = 0.56 m for A2, *h* = 0.44 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructions. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage II (*h* = 0.56 m for A2, *h* = 0.44 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructions. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

In such dispersive conditions (*μ* = 0.5 and 0.4 for A2 and A3, respectively), the large discrepancies of wavenumbers predicted by the linear wave dispersion relation have dramatic effects on the surface elevation signal reconstructed with the TFM at high frequencies (see *ζ*_{wg}), the TFM induces significant overestimation of energy levels starting from 2.5*ω*_{p}, where in typical applications, a cutoff frequency should be applied. Up to 2.5*ω*_{p}, this overestimation leads to errors on the significant wave height *H*_{s} and mean wave period *T*_{m02} lower than 2% and 4%, respectively (see appendix for the definition of these bulk parameters), which is considerably lower than those found in the field, for relatively more energetic and directionally broad wave fields (section 5). In contrast with *ζ*_{wg} tend to increase with frequencies but remain small. Around 2*ω*_{p}, the *ω*_{p} and 3.5*ω*_{p}, where the Boussinesq approximation of *κ*_{rms} starts diverging from measurements (Figs. 3a,b). Due to noise in the ECM data, a cutoff frequency at *ω*_{c} = 3.35*ω*_{p} was applied for both wave tests. The extension of *ω*_{c} shows that noise in the ECM data only results in a plateau starting around 3.5*ω*_{p} (see Figs. 3c,d). This eventually leads to a slight overestimation of energy levels after 5*ω*_{p}. In contrast, the reconstruction *κ*_{rms} from observed dominant wavenumbers at frequencies higher than 3*ω*_{p}.

For both wave tests, the nonlinear reconstructions *ω*_{p} for A2). Since the cutoff frequency for the linear reconstructions *ω*_{p} (*ω*_{c} in Figs. 2–4), energy levels beyond this frequency principally originate from quadratic interactions between components around the peak and the second harmonic (Bonneton and Lannes 2017). For the present case, the inaccuracies of the Boussinesq approximation of Herbers et al. (2002) after ~3*ω*_{p} hence have little impact on the nonlinear reconstruction. The accuracy of *κ*_{rms} predictions up to and slightly beyond the second harmonic allow good performances of the reconstruction

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage III (*h* = 0.41 m for A2, *h* = 0.29 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructionsy. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage III (*h* = 0.41 m for A2, *h* = 0.29 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructionsy. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Measured and predicted dimensionless wavenumber spectra during stage III (*h* = 0.41 m for A2, *h* = 0.29 m for A3). Measured wavenumber *κ*_{meas} is computed from cross-spectral analysis between adjacent wave gauges while *κ*_{rms} is predicted with Eq. (17); *κ*_{L} is the solution to the linear wave dispersion [Eq. (9)]. (c)–(f) The surface elevation energy density spectra *E*(*ω*) of the different linear and nonlinear reconstructionsy. For both *ω*_{c} (vertical gray dashed line at 3.35*ω*_{p}). The separation between the infragravity and gravity band of frequencies (0.6*ω*_{p}) is indicated by the vertical black dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 2.79 Hz for both wave test A2 (left panels) and A3 (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

### b. Shoaling of nonlinear waves in the vicinity of the breaking point (III)

The next regime of propagation (III) considers nonlinear waves approaching the mean breaking point location. Stage III differs from II in two principal aspects: 1) amplitude effects become increasingly important and induce deviations of *O*(10%–15%) between *κ*(*ω*_{p}) and predictions by the linear dispersion relation; 2) the wavenumber and phase velocity spectra become less frequency dependent. As a result, wavenumbers are relatively well described by the simple relation *ω*/*c*(*ω*_{p}) for both tests (Figs. 4a,b). At these depths (*μ* ~ 0.36 and 0.25 for A2 and A3, respectively), it is worth noting that this relation almost coincides with the shallow-water dispersion relation [Eq. (18)]. The Boussinesq approximation of Herbers et al. (2002) [Eq. (17)] provides very accurate estimates of dominant wavenumbers up to typically 2.5*ω*_{p} for A2 and 3*ω*_{p} for A3, which is explained by the less dispersive conditions found for A3 at stage III.

Despite the conditions being less dispersive at this stage, the inability of the linear wave dispersion relation to predict wavenumbers at high frequencies still strongly reflects from the energy density spectra of *ω*_{p} and 3.5*ω*_{p} and then increase at higher frequencies. As in stage II, the extension of

The nonlinear reconstructions *ω*_{p} and 5.5*ω*_{p}, differences between reconstructed and directly measured surface elevation energy spectra are typically within the 95% confidence interval for A3. Compared to the linear reconstructions, the nonlinear formula [Eq. (6)] substantially improves the description of energy levels at 2*ω*_{p} and 3*ω*_{p} (see inserts in Fig. 4f in comparison with Fig. 4d), suggesting strong nonlinear effects. Similar to stage II, differences between *κ*_{rms} values diverge from measurements only after the cutoff frequency *ω*_{c}. This is quite a promising result considering that *κ*_{rms} can be estimated in the field from a single sensor.

### c. Wave shape and wave-by-wave statistics

The comparison of

Figures 5a and 5b show the reconstructed surface elevations *κ*_{p}*h* = 0.7). This example exhibits a moderately energetic group (between 1000 and 1015 s) along with an extreme wave (1040–1045 s, crest elevation ~1.25*H*_{s}), and a much calmer period in between. The contribution of nonlinear interactions to the reconstructed *ζ*, which can be directly estimated from Eq. (22) as *ζ* (Fig. 5c). When the local amplitude increases, such as during the first wave group or for the extreme wave,

Comparison of directly measured (*ζ*_{wg}) and reconstructed (*a*_{0} = 0.1 m at stage II during the wave test A2 (*κ*_{p}*h* = 0.72, *h* = 0.56 m). (a) A 50-s-long sample and (b) a zoom-in around the extreme wave event occurring around 1042 s (crest elevation ~1.25*H*_{s}). (c),(d) The contribution of the nonlinear terms to *ζ*, directly estimated from Eq. (22) as *a*_{0} are shown below the corresponding panel. A low-pass filter at 2.65*ω*_{p} was applied to the linear reconstruction

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Comparison of directly measured (*ζ*_{wg}) and reconstructed (*a*_{0} = 0.1 m at stage II during the wave test A2 (*κ*_{p}*h* = 0.72, *h* = 0.56 m). (a) A 50-s-long sample and (b) a zoom-in around the extreme wave event occurring around 1042 s (crest elevation ~1.25*H*_{s}). (c),(d) The contribution of the nonlinear terms to *ζ*, directly estimated from Eq. (22) as *a*_{0} are shown below the corresponding panel. A low-pass filter at 2.65*ω*_{p} was applied to the linear reconstruction

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Comparison of directly measured (*ζ*_{wg}) and reconstructed (*a*_{0} = 0.1 m at stage II during the wave test A2 (*κ*_{p}*h* = 0.72, *h* = 0.56 m). (a) A 50-s-long sample and (b) a zoom-in around the extreme wave event occurring around 1042 s (crest elevation ~1.25*H*_{s}). (c),(d) The contribution of the nonlinear terms to *ζ*, directly estimated from Eq. (22) as *a*_{0} are shown below the corresponding panel. A low-pass filter at 2.65*ω*_{p} was applied to the linear reconstruction

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

These qualitative observations are complemented here with the comparison of two bulk parameters computed from both reconstructed surface elevation time series: the wave skewness *S*_{k} (see appendix) and the mean elevation of the 1/10th largest wave crests *ζ*_{wg}. This is consistent with the observations made on Fig. 5. Closer to the breaking point (see Table 5), or for narrowband conditions, this underestimation becomes less significant. The improved predictions of *S*_{k}. The errors made on this third-order parameter with

Surface elevation skewness *S*_{k} and mean elevation of the 1/10th largest wave crests *κ*_{p}*h* = 0.72 m for A2, *κ*_{p}*h* = 0.63 m for A3). The relative errors of these parameters are provided between parentheses.

Surface elevation skewness *S*_{k} and mean elevation of the 1/10the largest wave crests *κ*_{p}*h* = 0.60 m for A2, *κ*_{p}*h* = 0.50 m for A3). The relative errors of these parameters are provided between parentheses.

The relatively better performances obtained during A3 can be explained by the narrowbanded conditions, which promoted the dominance of forced harmonics at high frequencies. Although both free and forced components can be present, the phase information retained in a Fourier analysis of the subsurface signal per frequency is unique. In the nearshore region, this phase should often be biased toward forced components due to the weaker attenuation expected for forced components across depth compared to free ones. For instance, this could explain the slight overestimation of *S*_{k} and

## 5. Application of the nonlinear fully dispersive reconstruction to field conditions

The previous section demonstrates that when dominant *κ* spectra of the wave field are known, the sea surface spectrum associated with nonlinear shoaling waves can be accurately estimated at all frequencies with a local application of the nonlinear formula of Eq. (6) in both shallow (*μ* > 0.3). This is explained by the fact that *temporal* reconstructions that use measured *κ* spectra (here *spatial* reconstruction. In most typical field situations, however, measurements of *κ*(*ω*) are not available. The Boussinesq approximation of Herbers et al. (2002) for *κ*_{rms} [Eq. (17)] remains for now the only possibility to account for nonlinear effects in the dispersive properties of a wave field. In this section, the potential for field applications of the nonlinear moderately^{1} dispersive reconstruction based on *κ*_{rms} is investigated with the data collected during the Anglet experiments.

The accuracy of *κ*_{rms} spectral estimates not only depends on the range of validity of the Boussinesq approach employed in Herbers et al. (2002), but also on the nature of the signal used to compute this quantity. In the field, *κ*_{rms} is typically computed from the pressure signal, since the sea surface elevation is the unknown. Figures 6a and 6b compare *κ*_{rms} spectral estimates computed at the deepest location (SIG1) from two different sources: the hydrostatic reconstruction *ζ*_{hyd} and the directly measured surface elevation *ζ*_{AST}. The deviations of *κ*_{rms} from *κ*_{B} suggest significant nonlinear effects in the dispersive properties during both events. These effects are strongest for the narrowband spectra (Fig. 6b), during which forced high harmonics dominate [e.g., see around 2*fp* where *κ*_{rms} ~ *ω*/*c*(*ω*_{p})]. The two estimates of *κ*_{rms} differ by less than 2% at most frequencies for case A, with the largest errors (~5%) observed around 4*ω*_{p}, where predictions of *κ*_{rms} are less reliable in such water depths. For case B, *κ*_{rms} values computed from *ζ*_{hyd} are 8%–15% lower than those computed from *ζ*_{AST} between 1.5*ω*_{p} and 3*ω*_{p}, which suggests a negative bias due to the dominance of forced components in *ζ*_{hyd} at these frequencies. As they concentrate at high frequencies, where sea surface energy levels are low, differences in the computations of *κ*_{rms} have a relatively small impact on the reconstructed signals (Figs. 6c,d).

(a),(b) Dimensionless wavenumber spectra predicted at SIG1 (*h* = 13.1 m for case A, *h* = 15.3 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Dimensionless wavenumber spectra predicted at SIG1 (*h* = 13.1 m for case A, *h* = 15.3 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Dimensionless wavenumber spectra predicted at SIG1 (*h* = 13.1 m for case A, *h* = 15.3 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

The performances of the nonlinear reconstructions at SIG1 are quite mixed, with an accurate description of energy levels typically up to 2.5*ω*_{p}, where the predictions of *κ*_{rms} are reliable. At higher frequencies, sea surface energy levels are underestimated in both broadband and narrowband cases. In such water depths and wave conditions [*U*_{r} ~ *O*(0.2), see Table 3], the application of Boussinesq theory starts to be questionable. This most likely explains the limitations of the Boussinesq approximations of Herbers et al. (2002) for describing the dispersive properties of the wave field at high frequencies and in particular the dominance of free components observed at these depths. This is consistent with the smaller underestimations observed for the narrowbanded conditions, which promote the dominance of forced harmonics at high frequencies. For the two energetic cases considered here, the TFM exhibits a very similar behavior as that observed in laboratory conditions: energy levels are overestimated in a region starting around 2*ω*_{p} (see insert in Fig. 6d) and up to 3–3.5*ω*_{p}, before the reconstructions blow up.

The larger deviations of *κ*_{rms} from *κ*_{B} observed at SIG2 (Figs. 7a,b) compared to those at SIG1 suggest that nonlinear amplitude effects in the dispersive properties of the wave field [*β*_{am} in Eq. (17)] intensify closer to shore during both events. For the narrowband case B, dominant wavenumbers follow the simple dispersion relation *ω*/*c*(*ω*_{p}), suggesting that the energy at high frequencies is predominantly forced. Similar to the situation at SIG1, *κ*_{rms} estimates computed from the two sources (*ζ*_{hyd} and *ζ*_{AST}) during case A differ by less than 2% except for frequencies larger than 3.5*ω*_{p} (maximum difference of ~6%). However, the *κ*_{rms} spectra computed from *ζ*_{hyd} provide quite unreliable values at frequencies between 2.5*ω*_{p} and 4.5*ω*_{p} for the narrowbanded wave conditions, likely due to more intense forced components. These estimates can be easily improved with an iterative process: starting from the wavenumber spectra *ζ*_{hyd}, the wavenumbers *j* can be recomputed from the nonlinear fully dispersive reconstruction of *ζ* that is based on *κ*_{rms} (*ζ*_{hyd} and *ζ*_{AST}, Fig. 7b) to around 3%.

(a),(b) Dimensionless wavenumber spectra predicted at SIG2 (*h* = 7.2 m for case A, *h* = 9.5 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Dimensionless wavenumber spectra predicted at SIG2 (*h* = 7.2 m for case A, *h* = 9.5 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

(a),(b) Dimensionless wavenumber spectra predicted at SIG2 (*h* = 7.2 m for case A, *h* = 9.5 m for case B). *κ*_{rms} spectra predicted with Eq. (17) are computed from two different sources: *ζ*_{AST} and *ζ*_{hyd}. For the sake of readability, only one in two points is shown. (c),(d) Comparison of the energy density spectra of the different reconstructions with those estimated from the directly measured surface elevation *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstructions *ω*_{p}) is indicated by the black vertical dashed line. In all panels, the frequency axis is normalized by the peak frequency *ω*_{p}, which equals 0.48 Hz for both case A (left panels) and B (right panels).

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

In the Boussinesq regime (Ursell number Ur ~ 0.8), the estimations of *κ*_{rms} from the AST are highly reliable and explain the excellent performances reached by the nonlinear reconstruction *ω*_{p} or 5*ω*_{p} depending on the case). Most importantly, these overestimations occur at frequencies below the cutoff frequency that would typically be applied for estimating bulk parameters from this reconstruction: 4.5*ω*_{p} (0.35 Hz) and 4*ω*_{p} (0.3 Hz) for cases A and B, respectively. The excess of energy in *H*_{s} by 7% and 5% for cases A and B, respectively, while the mean wave period *T*_{m02} is underestimated by 11% in both cases. It is worth noting that for optimal choices of cutoff frequency for *H*_{s} was observed in the surf zone (Martins et al. 2020b), with the surface elevation variance being consistently underestimated with the TFM due to relatively higher levels of energy missed at high frequencies. Since large fractions of the sea surface energy levels at high frequencies originate from interactions between primary components and with their first harmonics, the choice of *ω*_{c} has only a weak influence on the computation of bulk parameters from the

At the wave-by-wave scale, the overestimation of sea surface energy levels by the TFM at relatively high frequencies leads to the appearance of unrealistic oscillations in *ζ* (Fig. 8b) shows a slightly different behavior. At the trough levels, the contribution from nonlinear terms is weak, which suggests that the spurious oscillations in

Example of wave group extracted during case B at SIG2 (*κ*_{p}*h* = 0.49, *h* = 9.5 m; 1845 local time 13 Oct 2018): (a) the directly measured (*ζ*_{AST}) and reconstructed (*ζ*, directly estimated from Eq. (22) as *ω*_{p} was applied to the linear reconstruction *ω*_{p} or 4.25*ω*_{p}) results in even larger unrealistic oscillations at the trough levels. For this comparison, *κ*_{rms} values computed from *ζ*_{AST}.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Example of wave group extracted during case B at SIG2 (*κ*_{p}*h* = 0.49, *h* = 9.5 m; 1845 local time 13 Oct 2018): (a) the directly measured (*ζ*_{AST}) and reconstructed (*ζ*, directly estimated from Eq. (22) as *ω*_{p} was applied to the linear reconstruction *ω*_{p} or 4.25*ω*_{p}) results in even larger unrealistic oscillations at the trough levels. For this comparison, *κ*_{rms} values computed from *ζ*_{AST}.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Example of wave group extracted during case B at SIG2 (*κ*_{p}*h* = 0.49, *h* = 9.5 m; 1845 local time 13 Oct 2018): (a) the directly measured (*ζ*_{AST}) and reconstructed (*ζ*, directly estimated from Eq. (22) as *ω*_{p} was applied to the linear reconstruction *ω*_{p} or 4.25*ω*_{p}) results in even larger unrealistic oscillations at the trough levels. For this comparison, *κ*_{rms} values computed from *ζ*_{AST}.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

## 6. On the influence of the spectral bandwidth and directional spreading

By controlling the efficiency of nonlinear energy transfers between triads of frequencies, both the spectral bandwidth and directional spreading are expected to influence nonlinear effects on the dispersion relation of irregular waves (Elgar and Guza 1985b; Herbers et al. 2002, 2003; Martins et al. 2021). Boussinesq model-based predictions by Herbers and Burton (1997) suggest only a weak influence from the directional spreading angle on the efficiency of sum interactions, which are those of interest here. For spreading angles as large as 60°, these authors predict a weaker growth of high-harmonic bound waves by only 10%–20%. More recently, de Wit et al. (2020) noted variations of approximately 10% for smaller directional spreading angles, which are more commonly found in the nearshore region. As during GLOBEX, where unidirectional waves are considered, the differences in dispersive properties observed for cases A and B in Anglet are hence principally explained by the contrasting spectral bandwidth characterizing these events (Herbers et al. 2002; Martins et al. 2021). The influence of the spectral bandwidth on the transfer functions *K*_{p} and *K*_{u} is here further analyzed using the field data collected at SIG2, where *κ*_{rms} estimates are most reliable. In particular, the contrasting wave conditions in terms of spectral bandwidth provide good examples for discussing the application range of the weakly dispersive method of Bonneton et al. (2018) and relate it to the moderately and fully dispersive reconstructions proposed in this study.

Figure 9 examines, for the two contrasting cases A and B, the transfer functions *K*_{p} corresponding to the different moderately and weakly dispersive reconstructions investigated here. These are compared with *ζ*_{hyd} to *ζ*_{AST}. The overestimation of dominant wavenumbers by the dispersion relation observed in section 5 results in the consistent overestimation of *K*_{p} at all frequencies (6%–8% at 2*ω*_{p} and up to a factor 2 around 4*ω*_{p}) for both broadbanded (Fig. 9a) and narrowbanded (Fig. 9b) incident wave conditions. This is consistent with the deviations obtained with the TFM from observed attenuation of *u* or *p* across the vertical previously reported in shallow water depths (e.g., see Constantian 1999; Elgar et al. 2001).

Transfer functions corresponding to the different linear (*ζ* for the broadband case A (*κ*_{p}*h* = 0.42, *h* = 7.2 m) and narrowband case B (*κ*_{p}*h* = 0.49, *h* = 9.5 m) at SIG2. These are compared with *ζ*_{hyd} to *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstruction *ω*_{p}, which equals 0.48 Hz for both case (a) A and (b) B.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Transfer functions corresponding to the different linear (*ζ* for the broadband case A (*κ*_{p}*h* = 0.42, *h* = 7.2 m) and narrowband case B (*κ*_{p}*h* = 0.49, *h* = 9.5 m) at SIG2. These are compared with *ζ*_{hyd} to *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstruction *ω*_{p}, which equals 0.48 Hz for both case (a) A and (b) B.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Transfer functions corresponding to the different linear (*ζ* for the broadband case A (*κ*_{p}*h* = 0.42, *h* = 7.2 m) and narrowband case B (*κ*_{p}*h* = 0.49, *h* = 9.5 m) at SIG2. These are compared with *ζ*_{hyd} to *ζ*_{AST}. The gray vertical dashed lines indicate the cutoff frequency *ω*_{c} used for the moderately dispersive linear reconstruction *ω*_{p}, which equals 0.48 Hz for both case (a) A and (b) B.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

For the broadband case A, nonlinear amplitude effects in the dispersion relation [quantified through *β*_{am}, Eq. (20)] are weak as evidenced by the slight divergence of *κ*_{rms} from *κ*_{B} (Fig. 7a). The transfer function for the linear reconstruction *K*_{p} as long as accurate *κ* are used (here up to ~3.5*ω*_{p}, see Fig. 9a). Conversely, nonlinear amplitude effects are much stronger for the narrowband case B and balance second-order frequency ones around high harmonics [i.e., *β*_{fr} ~ *β*_{am} in Eq. (17)]. As a consequence, most high-frequency components are forced and follow the dispersion relation for nondispersive shallow-water waves [Eq. (18)]. Around 2*ω*_{p} and 3*ω*_{p}, *K*_{p} at these frequencies is much more important than for case A (see contrasting differences between *K*_{p}.

As mentioned in the introduction, the weakly dispersive formula [Eqs. (10) and (11)] are not appropriate when high-frequency components are principally free (i.e., conditions are too dispersive). This is the case here for case A, during which *ζ*_{SNL} underestimates high-frequency energy levels (Fig. 9a). This occurs even though wave components around the peak frequency approach the shallow water limit (*μ* ~ 0.18, see Table 3). In contrast, when forced components dominate as during case B (*μ* ~ 0.24), the linear weakly and fully dispersive reconstructions closely match around 2*ω*_{p} and 3*ω*_{p} (i.e., *ω*_{p}. The weakly dispersive approach is thus recommendable only when dominant wavenumbers closely match those of nondispersive shallow-water waves *κ*_{sw}, i.e., when most wave components travel at the speed of nondispersive shallow-water waves. For broadbanded incident wave spectra, this occurs approximately for

## 7. Conclusions and perspectives

This study proposes and assesses a nonlinear fully dispersive method for reconstructing the free surface elevation from pressure or wave orbital velocities collected under nonlinear nearshore waves. It relies on the knowledge of dominant wavenumber spectra *κ*(*ω*) of the wave field considered in order to account for varying degrees of nonlinearities (i.e., varying amounts of forced energy at a particular frequency). In laboratory conditions, where measurements of *κ*(*ω*) are available (*κ*_{meas}), the nonlinear fully dispersive reconstruction demonstrates excellent skills in diverse nonlinear and dispersive conditions [*μ* ranging from 0.25 to 0.5, Ur ~ *O*(0.1–1)], and for both broadbanded and narrowbanded incident wave conditions. In the field, where measurements of *κ*(*ω*) are rarely available, the reconstruction can use a Boussinesq approximation of *κ* (*κ*_{rms}; see Herbers et al. 2002). The accuracy of this moderately dispersive reconstruction is then directly dependent on that of the Boussinesq approximation for *κ*_{rms}. Overall, the present results suggest that the relation between *p*, *u* and *ζ* in nearshore nonlinear waves strongly depends on the relative importance of forced components at high frequencies (forced-to-free energy ratio), and our capacity to predict it through, for instance, predictions of *κ*_{rms}.

Figure 10 synthesizes the range of validity of the different temporal reconstructions investigated in this study. The analysis of field data with contrasting wave conditions in terms of spectral bandwidth clearly illustrates the application range for the weakly dispersive formula of Bonneton et al. (2018). For broad spectra, typically with low fractions of forced energy at high frequencies in intermediate water depths, the application of Eqs. (10) and (11) should be limited to around the breaking point and in the surf zone (*κ*_{meas} through cross-spectral analysis is not straightforward (e.g., see Herbers et al. 2002) and sometimes impossible simply due to limited experimental resources. As an alternative, the Boussinesq approximation of *κ*_{rms} (Herbers et al. 2002) can be used locally for estimating the dominant dispersive properties of the wave field considered: here, similar performances as achieved in the laboratory were reached for Ur typically greater than 0.7–0.8. As seen in section 6, both the weakly and moderately dispersive approaches converge in the shallow water limit and provide the same results. In such cases, however, the weakly dispersive formula has the advantage to work in the temporal domain and does not require the computation of bispectra products, which can be challenging to obtain in shallow water depths (e.g., in macrotidal environments).

Synthesis of the temporal reconstruction methods investigated in this study and their range of validity; taken from Mouragues et al. (2019) and updated with the knowledge developed in Martins et al. (2020a,b) and in the present study. The *x* and *z* represent the cross-shore and the vertical axes, respectively.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Synthesis of the temporal reconstruction methods investigated in this study and their range of validity; taken from Mouragues et al. (2019) and updated with the knowledge developed in Martins et al. (2020a,b) and in the present study. The *x* and *z* represent the cross-shore and the vertical axes, respectively.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Synthesis of the temporal reconstruction methods investigated in this study and their range of validity; taken from Mouragues et al. (2019) and updated with the knowledge developed in Martins et al. (2020a,b) and in the present study. The *x* and *z* represent the cross-shore and the vertical axes, respectively.

Citation: Journal of Physical Oceanography 51, 11; 10.1175/JPO-D-21-0061.1

Although currently limited to *κ* will be required to improve the performances in conditions characterized by Ur numbers smaller than typically 0.5.

## Acknowledgments

Kévin Martins greatly acknowledges the financial support from the University of Bordeaux, through an International Postdoctoral Grant (Idex, 1024R-5030), and from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement 887867 (*lidBathy*). The GLOBEX project was supported by the European Community’s Seventh Framework Programme through the Hydralab IV project, EC Contract 261520. The Anglet field study has received financial support from Region Nouvelle-Aquitaine (Contract 2017-1R20107) and was carried out as part of the project MEPELS (Contract 18CP05), performed under the auspices of the DGA, and led by SHOM. We thank F. Feddersen and one anonymous reviewer for making very helpful comments and suggestions.

## Data availability statement

The GLOBEX data used in this research can be accessed on Zenodo at https://zenodo.org/record/4009405 and can be used under the Creative Commons Attribution 4.0 International license. The bispectral analysis tools developed and used in this study are accessible from the first author GitHub repository at https://github.com/ke-martins/bispectral-analysis.

## APPENDIX

### Computation of First-, Second-, and Third-Order Parameters

*j*th moment of the surface elevation energy density spectra is defined as

*f*

_{p}and 5.5

*f*

_{p}, with

*f*

_{p}the discrete peak frequency, so that the reconstruction is assessed in the shortwave frequency band. For

*T*

_{m02}is defined as

*H*

_{s}and

*T*

_{m02}, a low-pass filter at 5.5

*f*

_{p}is first applied to

*ζ*.

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