Comparison of deep-water-parameter-based wave overtopping with wirewall field measurements and social media reports at Crosby (UK)

Wave overtopping formulae, which often underlie coastal hazard early warning systems, are typically para-meterised using wave conditions at the toe of the structure. For very shallow conditions where significant wave breaking occurs over the foreshore, this usually requires computationally-demanding numerical models — and practitioners skilled in their application — to accurately transform offshore waves to the structure toe. An additional concern is that overtopping formulae are scarcely validated in the field due to the very limited availability of in-situ overtopping data obtained at actual structures. Here, we validate a set of deep-water-parameter-based formulae for mean overtopping discharge ( q ) at smooth slopes, which remove the need for nearshore measurements or additional numerical modelling but require that a single representative foreshore slope angle ( m ) be defined. The validation is carried out against field data gathered at Crosby (UK) using two novel approaches: i) a new overtopping measurement system called “ WireWall ” ; and ii) crowd-sourced data in the form of overtopping images obtained from a community Facebook page (social media). A method is introduced to define m for irregular bathymetries, based on the location where the local water depth is equal to the offshore significant wave height. The overtopping formulae proved accurate — with estimates of q being within a factor of 4 of observations — when compared to both 1-h averaged and 15-min averaged overtopping data, suggesting that the approach can be used for both design and assessment and now-casting hazard information. Finally, hindcasts made using the newly validated formulae for the events reported by the community indicate that q can exceed 10 l/s/m under yearly winter conditions, posing a serious hazard to pedestrians. This highlights the pressing need to update the current hazard warning system at Crosby, which estimates q to be a factor of 3 lower than the deep-water-parameter-based approach, on average.


Background
In coastal hazard forecasting, practitioners must accurately estimate the volume of water that passes over sea defences due to wave action-referred to as wave overtopping-that may cause flooding, damage structures or injure people.This is often achieved using empirical formulae that relate wave conditions at the toe of the struc-ture, and the structure's geometry, to a mean overtopping discharge, q (EurOtop, 2018).The main drawback to this approach lies in the fact that accurate wave measurements immediately in front of the structure are rarely available, compared to deep-water wave parameters which are often available via offshore buoys or reliable ocean-scale wave models (e.g., WaveWatch III (Tolman, 1999)).As a result, numerical models are typically used to transform wave conditions in deep water (offshore) to the structure toe.
The bathymetry seaward the structure (i.e. the foreshore) may be classified by the relative water depth at the toe of the structure-defined as the ratio of the local water depth to the deep-water significant wave height (h toe /H m0,deep ) (Hofland et al., 2017).For simple open coasts, with h toe /H m0,deep > 1, phase-averaged models (e.g.SWAN (Booij et al., 1999)) can quickly provide accurate estimates of wave conditions at the structure toe.However, for more shallow foreshores (h toe / H m0,deep ≤ 1), phase-resolving wave models (e.g.XBeach (Roelvink et al., 2009) or FUNWAVE (Shi et al., 2012)) are needed to accurately capture the nonlinear effects of wave shoaling and breaking.This presents two challenges: first, because of their high computational demand, these models are generally impractical for large-scale applications; and second, such numerical models require practitioners with the specialized knowledge and skill to optimize their various parameters.
To overcome these challenges, Mase et al. (2013) proposed a set of deep-water-parameter-based formulae for seawalls with shallow foreshores.The formulae, which were later refined by Yuhi et al. (2021), directly relate wave run-up to overtopping and make use of an imaginary seawall slope that considers both the foreshore and structure slopes.While physically justified, the formulae first require the user to accurately estimate a characteristic wave run-up value before estimating wave overtopping.Furthermore, the method used to determine the imaginary slope is highly sensitive to the estimated water depth at the onset of wave breaking-which can vary depending on the wave transformation method or model applied.
More recently (Lashley et al., 2021b), derived a set of empirical formulae for very shallow foreshores that use deep-water wave conditions to directly predict the mean overtopping discharge at smooth, uniform slopes and vertical structures.Similar to the early wave overtopping design diagrams of Goda et al. (1975), these formulae account for the nonlinear effects of wave breaking using two terms that describe the foreshore: h toe /H m0,deep and the foreshore slope angle (m).These two parameters have been successfully shown to determine the magnitude of several nearshore parameters for structures with shallow foreshores, including: the increase in mean water level referred to as wave setup (Goda, 2000;Gourlay, 1996;Lashley et al., 2021b); the significant wave height: H m0 (Lashley et al., 2021b) and H 1/3 (Goda, 2000); the spectral wave period (T m− 1,0 ) (Hofland et al., 2017); and the height of infragravity waves, also referred to as "surfbeat" or "dynamic setup" (Goda, 2000;Lashley et al., 2020a).
The overtopping formulae developed by Lashley et al. (2021b) proved accurate when applied to the physical (scale) model tests used to derived them-with simple uniformly sloping foreshore and structure slopes.However, like many other empirical overtopping models, the formulae have yet to be validated in the field.Field validation of empirical methods is necessary because field conditions (e.g., structure geometry and foreshore bathymetry) often vary from the controlled laboratory conditions used to derive them.This process not only quantifies the potential differences but also presents an opportunity to adjust or extend the methods to better bridge the gap between the laboratory and reality.Despite their importance, field measurements of wave overtopping are very rare due to the following reasons: i) storm conditions that result in significant wave overtopping are very infrequent (with very low annual probabilities of occurrence); and ii) deployment of the collection tanks typically used to measure wave overtopping in the field is very costly due to their size and weight (Briganti et al., 2005;Pullen et al., 2012;Troch et al., 2004).
Acknowledging these challenges, Van der Meer et al. (2019) proposed a unique approach at Eemshaven (the Netherlands) where overtopping tanks were cut and placed into the dike, essentially lowering the effective crest level and allowing for overtopping measurements under yearly storm conditions over the course of 12 years.However, such a method that requires the direct cutting and modification of the structure may not be easily implemented at other locations.On the other hand, Oosterlo et al. (2021) proposed a more flexible approach, which makes use of two laser scanners (LIDARs), attached to a relocatable pole, to measure the front velocity and layer thickness of wave run-up.These measurements may then be used to calculate a "virtual" overtopping discharge by assuming a hypothetical crest level and a relation between the maximum run-up volume and the actual overtopping volume.While this approach is readily adaptable, the reliability of the approach is subject to weather conditions (affecting the reflected signal intensity), blockage (e.g., a person standing in front of the beam), and the assumptions made.
More recently, two novel methods have been applied at Crosby (Northwest England), where the 900-m long coastal defence experiences overtopping every spring tide with an onshore wind.The first makes use of crowd-sourcing (or citizen science) methods, where residents of Crosby indicate-with photos via social media-when and where wave overtopping occurs (Brown et al., 2021).This approach is similar to the use of "YouTube" videos by Sandoval and Bruce (2018) to identify and analyse wave overtopping events.It recognises the utility of social media for the rapid collection (and dissemination) of information during extreme and more frequent nuisance overtopping events-providing a valuable data source for the qualitative validation of flood models (Khan et al., 2022;Smith et al., 2017).The second and more innovative approach at Crosby uses low-cost materials and an easily relocatable frame that supports a system of capacitance wires, termed "WireWall" (Section 2.2.1).The system, which has been validated using an extensive set of tests in a 2D wave basin, is capable of accurately measuring both individual wave overtopping volumes and mean overtopping discharge (Yelland et al., 2022).
The coastal defence at Crosby includes a stepped revetment and recurve wall, fronted by a shallow sandy beach with a ridge system on the upper beach approximately 50 m before the structure.At that site, wave overtopping alerts and warnings are issued by the Environment Agency (EA) when the estimated mean overtopping discharge-calculated using 15-min averaged now-cast offshore wave data-exceeds 2 l/s/m and 25 l/s/m, respectively.However, alerts were rarely issued for notable overtopping events that were reported via social media during the 2013 to 2018 period (Brown et al., 2020b).This suggests that the approach used by the EA to predict wave overtopping, based on SWAN and an older version of the EurOtop (2007) manual using a past beach level (that is higher than that seen during the relevant period), may not be suitable for the shallower present-day conditions at Crosby.
Recently, SWAN-Bayonet GPE (Pullen et al., 2018)-a meta-modelling approach comparable to a neural network-has been applied to estimate wave overtopping at Crosby (Brown et al., 2020a).This approach is based on the latest EurOtop (2018) manual, which is considered the industry standard for wave overtopping globally.However, as the name suggests, this approach still requires that the practitioner first use SWAN to transform waves from offshore to the structure toe.While SWAN may perform well for conditions with h toe /H m0,deep ≥ 1, it is expected to underestimate the wave height and period at the structure toe for shallower cases, as demonstrated in several modelling studies (Buckley et al., 2014;Lashley et al., 2020b;Salmon and Holthuijsen, 2015;van der Westhuysen, 2010).
In light of this, it is our primary aim here to use the data gathered using the above-mentioned novel techniques to assess the accuracy and viability of the deep-water-parameter-based approach proposed by Lashley et al. (2021b) (Equation (1) of this paper), as an alternative to SWAN-Bayonet GPE for very shallow conditions.Our secondary objective is to compare the overtopping discharge predicted using the newly validated formulae to those made using the older SWAN and EurOtop (2007) approach (which represents the current EA method) for the range of offshore wave and water level combinations when overtopping was reported by the community via social media.In this way, we aim to identify any potential limitations of the current EA method to help improve the coastal hazard early warning system at that location.

Outline
This paper is organized as follows: Section 2 describes the study site and the field data collected using the WireWall system and social media reports.Section 2 also details the deep-water-parameter-based approach (Lashley et al., 2021b), along with the SWAN Bayonet GPE (Pullen et al., 2018) and SWAN-EurOtop (2007) approaches, which are used here for benchmarking.In Section 3, the overtopping estimates are compared quantitatively to the WireWall field measurements and qualitatively to the photographic reports made via social media.The validity of the deep-water-parameter-based approach is discussed along with the likelihood of wave overtopping at Crosby for non-extreme offshore wave and water level combinations.Section 4 concludes the paper by addressing the overall research objectives, stating limitations and identifying areas for future work.

Study site
The study site is located at Crosby in the Northwest of England (Brown et al., 2020b) (Fig. 1), which is exposed to a tidal range of 10 m and deep-water significant wave heights (H m0,deep ) exceeding 3.4 m, during annual storms (as frequent as once every three months).The coastal defence includes a 1:2.5 sloping stepped revetment and a recurve wall with a crest height of approximately 6.4 m AOD (above ordnance datum) (Fig. 2).In front of the structure, a wide, gently sloping beach with a bar and trough system-also referred to as a ridge and runnel system-functions as a shallow foreshore by initiating wave breaking during storms (Fig. 3).Considering the difference between the maximum and minimum bed levels (Δz) recorded from September 1996 to January 2019 (a total of 32 profiles available from: https://coastalmo nitoring.org),the lower beach (x < − 200 m) experiences less variability (Δz = 0.35 m, on average) than the ridge system of the upper beach (x > − 200 m with Δz = 0.93 m, on average).In addition to long-term variability, it is important to note that the beach profile can experience both erosion and deposition over the course of single storm.However, this short-term variability is not considered here, as no incremental profile measurements were taken during the events.

WireWall
The WireWall system (Brown et al., 2020b;Yelland et al., 2022) was developed to be a relatively low-cost tool in field measurements of wave overtopping to help optimize the design of coastal defences and early warning processes.The system incorporates a three-dimensional grid of vertical capacitance wires that record the length of the wires in contact with water, and the speed of the water passing through the grid (Fig. 4).In this way, the system can measure the volume and speed of individual waves as they overtop the structure.
WireWall was deployed during a few spring high tides at Hall Road Crosby in winter 2018-2019, where it recorded wave overtopping on January 25th, 2019 from 13:21 to 14:21.During this period, the average water depth at the structure toe (h toe ), H m0,deep , offshore spectral wave period (T m− 10,deep ), and directional spreading (σ) were 1.6 m, 1.7 m, 5.7 s   and 19.7 • , respectively (Fig. 5).The standard deviations associated with these values were: 0.12 m, 0.04 m, 0.06 s and 2.89 • , respectively.
The water level data was obtained from the Gladstone Liverpool Dock tide gauge (Fig. 1 and available from: Tide Data), while the offshore wave data (in 24 m water depth) was obtained from the CEFAS database for Liverpool Bay WaveNet Site (Fig. 1 and available from: Wave Data).The above parameter values indicate a wind-sea wave climate and a very shallow condition at the structure toe: h toe / H m0,deep varied from 0.91 to 1.04 (average of 0.94) during the 1-h period that overtopping occurred.A beach profile collected during low water on January 25th, 2019, was used to estimate the local water depth.
The wave-by-wave overtopping volume measurements obtained by the WireWall system were averaged over 15-min periods and converted to a 15-min discharge rate (q), which ranged from 3 to 10 l/s/m (available from: WireWall Data (Brown et al., 2020a)).This 15-min period is consistent with the 15-min intervals used by the EA to now-cast hazard information.As the variation in forcing was minor during the hour when overtopping occurred-consistent with a single sea-state-the measurements were further averaged over that entire period to produce a 1-h averaged q of 4 l/s/m.

Social media reports
In December 2013, an open-access community Facebook page was established for Crosby with the title, "I'm at Crosby beach and the weather is …" (Facebook Page) and community members were able to report when wave overtopping of any magnitude occurred (Brown et al., 2021).Using the photographs from the page (Fig. 6) along with others supplied by local coastal managers/contractors (for the period January 2013 to December 2017) to identify the dates of overtopping events, the corresponding foreshore profiles (available from: https://coastalmonito ring.org),water levels and wave conditions were extracted from the national monitoring networks.The data was extracted at 15-min intervals for the 4-h periods centred over each high tide on the date reported.The resulting dataset included a total of 1 080 wave and water level combinations (over 41 days) that satisfied the shallow to emergent toe conditions (− 0.2 < h toe /H m0,deep ≤ 1.5), where Equation ( 1) is valid.

Deep-water-parameter-based wave overtopping 2.3.1. Main formulae
Here, we apply the deep-water-parameter-based formulae developed by Lashley et al. (2021b) to estimate q at smooth, sloping structures with shallow foreshores, including the effects of infragravity waves.The formulae, presented below, were derived using data from 389 physical (scale) model tests with varying wave, water level, foreshore slope angle and structure slope angle conditions (Table 1): ) . (1) For very shallow cases (regime 1), with 0.5 ≤ h toe /H m0,deep ≤ 1: (3) and, where R c is the crest freeboard, s om− 1,0 is the wave steepness based on H m0,deep and the spectral wave period in deep water (T m− 1,0,deep ), α is the structure slope angle and m is the foreshore slope angle (Fig. 7).For cases with highly irregular foreshores, m is taken as a single hypothetical slope (defined in Section 2.3.2).

Hypothetical foreshore slope
Given the irregular nature of the foreshore at Crosby, it is important that the foreshore slope angle parameter (m) be accurately prescribed in Equations 1 to 7, 13 and 14-which were developed using physical and numerical tests with uniformly sloping foreshores.Here, m is defined simply as the (hypothetical) constant slope angle, extending from a depth, h = H m0,deep to the structure toe (Fig. 7).
While the use of hypothetical slopes with empirical methods is common, the approach used here differs largely from other methods that aim to define a single slope to represent both the foreshore and structure together (Altomare et al., 2016;Mase et al., 2013;Saville Jr, 1957).The characteristic depth (h = H m0,deep ) is considered the point where the foreshore begins to significantly affect the magnitude of nearshore/shoreline parameters, such as: H m0,toe , T m− 1,0,toe , wave setup and q (Hofland et al., 2017;Lashley et al., 2021b).This is also made evident by the Goda (2000) design diagrams for estimating the wave height (H 1/3 ) in the surf zone-where the relationship between the change in wave height (H 1/3 /H ′ 0 ) and relative water depth becomes linear when h/ H ′ 0 ≤ 1, with H ′ 0 considered to be the same as H m0,deep .In the above approach, m varies depending on the offshore wave height and local water depth conditions.As often done in practice, a maximum foreshore slope of 1:10 is considered (EurOtop, 2018).The appropriateness of this approach was assessed and verified using the XBeach Non-hydrostatic and SWAN numerical models by comparing the characteristics of wave propagation for: i) the irregular (real) foreshore and ii) the hypothetical foreshore slope, as defined here.The results of that analysis are presented and discussed in Appendix A.

Influence factors
As the above formulae were developed for simple structures with smooth slopes, the estimated overtopping discharge must be modified to account for the influence of the stepped revetment and parapet wall.These influence factors (γ) are typically obtained by comparing physical model tests for a structure with a particular feature (e.g.stepped revetment or recurve wall) to a reference case (usually a straight slope) (EurOtop, 2018).They are usually expressed as: ) , (8)

Influence of stepped revetment
The influence of the stepped revetment can be interpreted as an increased roughness or friction compared to a slope without any roughness elements or features.EurOtop (2018) recommends minimum and maximum friction factors (γ f ) of 0.75 and 0.9, respectively for roughness elements.The manual also notes that the influence varies depending on the water level, as elements below the still water line tend to have less influence.To account for this, a variable γ f was used here.A value of 0.9 was applied for cases where the water level reached the top step and 0.75 was applied when the water levels were at the toe, with linear interpolation between these values for intermediate water levels.It should be noted that the Kerpen et al. (2019) approach, where γ f is a function of α, step height and wave height at the toe, was also considered and yielded similar γ f values (not shown) to the variable approach taken here.

Influence of recurve wall
Similarly, we adopted the EurOtop (2018) approach to account for the influence of the recurve wall (referred to as a "wall + bullnose" in the manual).The reduction factor (γ * ) is given by: where, where h wall (= 0.97 m) is the height of the wall; h n (= 0.47 m) is the thickness of the bullnose; ε is the bullnose angle (taken here as 45 • ) (Fig. 8); and s tm− 1,0 is the wave steepness calculated using the significant wave height at the toe (H m0,toe ) and the wave period at the structure toe (T m− 1,0,toe ).The parameter validity range for Equations ( 9)-( 12) are provided in Table 2.
Here, H m0,toe was estimated using the following empirical expression (Equation ( 13)), which accounts for the influence of wave setup and infragravity waves generated by waves breaking over shallow foreshores (Lashley et al., 2021b): where HIG is the estimated ratio of the infragravity to sea-swell wave heights at the structure toe (Appendix B).Equation ( 14) is considered valid for 1:1 000 ≤ tan(m) ≤ 1:10.Note that an alternative method to estimate T m− 1,0,toe by Hofland et al. (2017) could also be applied here and is expected to produce similar results.
Alternative approaches to account for the recurve wall, such as of those Owen and Steele (1993) and Coeveld et al. (2007), were also considered.However, the structure and forcing characteristics here were found to be outside their respective validity ranges.

Alternative overtopping methods for benchmarking
To benchmark to the performance of the deep-water-parameterbased approach (Section 2.3), two additional overtopping methods were considered for comparison: i. SWAN-Bayonet GPE, a calculation tool based on the latest wave overtopping guidance (EurOtop, 2018); and ii.SWAN-EurOtop (2007), which represents the method used by the EA to issue alerts and warnings for the period of January 2013 to December 2017.This approach is based on an older version of EurOtop (2007) manual.

SWAN-bayonet GPE
Bayonet GPE (Pullen et al., 2018) is a free empirical wave overtopping tool (available from: www.overtopping.co.uk).It uses the Gaussian process emulation (GPE) technique to essentially interpolate between the 13,500 known data points that underlie the EurOtop (2018) manual.The approach uses the phase-averaged wave model, SWAN (Booij et al., 1999) in 1D mode to transform offshore wave conditions to the structure toe.These nearshore parameters, along with those that describe the geometry of the structure, are then fed into Bayonet GPE to predict q.
SWAN-Bayonet GPE wave overtopping predictions (15-min averaged) were hindcasted for the January 25th , 2019 event at Crosby and published online (Brown et al., 2020a).Like with the deep-water-parameter-based formulae, the stepped revetment and recurve wall were considered using the influence factors outlined in Section 2.4.

SWAN-EurOtop (2007)
The EA's flood forecasting service uses a similar approach based on an older version of EurOtop (2007) manual, which is still considered the best practice for flood forecasting in England and Wales (Brown et al., 2020b).To represent the EA's hazard predictions for the period of January 2013 to December 2017, estimates of q were made using SWAN and the EurOtop (2007) empirical formulae, which follow: for ξ m− 1,0 < 5, with a maximum of, The reliability of Equations ( 15) and ( 17) is described by taking the coefficients 4.75 and 2.6 as normally distributed stochastic parameters with means of 4.75 and 2.6 and standard deviations of 0.5 and 0.35, respectively (EurOtop, 2007).
For waves breaking over shallow foreshores, with ξ m− 1,0 > 7, EurOtop (2007) recommends: where the coefficient c has a mean value of − 0.92 and a standard deviation of 0.24.For 5 < ξ m− 1,0 < 7, the user should use linear interpolation between Equation ( 15) or 17 and Equation ( 18).Note that for this SWAN-EurOtop (2007) approach, the H m0,toe and T m− 1,0,toe parameters used in Equations ( 15)-( 18) were obtained using SWAN.The stepped revetment and recurve wall were also considered here using the influence factors outlined in Section 2.4

WireWall data averaged over the full overtopping duration (1 h)
We first assess the ability of the deep-water-parameter-based approach (Equations (1)-( 7)) to predict q, averaged over the 1 h that wave overtopping occurred during the WireWall deployment on January 25th, 2019 (Yelland et al., 2022).Without including the overtopping reduction effects of the stepped revetment and recurve wall, q is initially overestimated by a factor of 10 (Fig. 9).However, when the influence factors are incorporated, the estimated q agrees well with the 1:3 to 1:2 Fig. 9. Ratio to measured (WireWall) mean overtopping discharge (q pred /q meas ) averaged over the 1 h that wave overtopping occurred on January 25th, 2019.Error bars represent the uncertainty (95% confidence intervals) for the overtopping predictions.

C.H. Lashley et al.
WireWall measurements-with the ratio of predicted to measured q (q pred /q meas ) within a factor of 2. The performance of the deep-water-parameter-based approach was also on-par with the SWAN-Bayonet GPE approach (Fig. 9), which represents the industry standard (EurOtop, 2018).
As both approaches use the same influence factors to account for the stepped revetment and recurve wall, their comparison here is considered a fair assessment of how well q was estimated-i.e., even without influence factors.The main difference between the two approaches is how the effects of the foreshore are considered.In the deep-water-parameterbased approach, the foreshore is directly considered by the hypothetical slope angle (m) and relative water depth (h toe /H m0,deep ).On the other hand, the SWAN-Bayonet GPE approach does not include m as an input parameter but assumes that the foreshore effects are accurately modelled by SWAN.For the conditions here, where only minor wave breaking takes place on the foreshore directly in front of the structure, the SWAN model performed well.However, for shallower conditions with heavy wave breaking over the foreshore, infragravity waves (also known as "surfbeat") are expected to play a significant role-resulting somewhat higher H m0,toe and significantly higher T m− 1,0,toe values than those estimated by SWAN because the model does not directly resolve these long-wave motions.Therefore, the performance of the SWAN-Bayonet GPE approach is expected to decrease significantly with relative water depth unless the recently-developed SWAN SurfBeat version (Reniers and Zijlema, 2022) or empirical correction is applied (Lashley et al., 2021a).

Sensitivity of the deep-water-parameter-based approach to foreshore slope
As the deep-water-parameter-based approach uses the hypothetical foreshore slope angle (m) as one of its main input parameters, it is important to assess the sensitivity of the predicted q to potential errors or uncertainty in the determination of m.Fig. 10 shows that q increases with foreshore slopes.Furthermore, for the same wave and water level forcing conditions, a change in tan(m) from 1:10 to 1:25 reduces the estimated q by one order of magnitude.This highlights the need to accurately define the representative foreshore slope for irregular bathymetries, particularly when assessing wave overtopping using a deep-water-parameter-based approach.Fig. 10 also demonstrates that the hypothetical foreshore slope (tan(m) = 1:10) obtained using the approach outlined in Section 2.3.2 and Appendix A, was a reasonable approximation of the actual foreshore; yielding a reasonably accurate estimate of q for the January 2019 event (Fig. 9).

WireWall data averaged over 15-min intervals
Formulae for q, such as those applied here, are typically developed by dividing the total volume of overtopping water collected during physical model tests by the duration of the experiment.Given the random nature of an irregular wave climate, the selected duration needs to be sufficiently long (≥500 waves) such that the mean overtopping discharge obtained is statistically valid and comparable between tests (see Romano et al. (2015) for a detailed analysis on how the number of waves influences the overtopping discharge).Considering an average T m− 1,0,deep of 5.2 s, the 1-h period of wave overtopping-where wave and water level conditions showed little variation at Crosby-corresponds to roughly 700 waves.This suggests that the deep-water-parameter-based approach can be reliably compared to the measured 1-h averaged q.
To assess the influence of storm duration and verify whether the approach is also suitable for very short-range (e.g., 15 min) forecasting, referred to as "nowcasting", we also compare the results from the deepwater-parameter-based approach to the WireWall measurements averaged over a 15-min period (Fig. 11a).As can be seen in Fig. 11a, the deep-water-parameter-based approach estimates q with reasonable accuracy (geometric mean and standard deviation of 0.46 and 1.9, respectively) despite the shorter (15-min) duration.It also performs marginally better than the SWAN-Bayonet GPE approach (geometric mean and standard deviation of 0.30 and 3.24, respectively).This is likely due to the SWAN model slightly underestimating the wave height and period at the toe of the structure at lower relative water depths (see Appendix A, for example) due to its exclusion of infragravity wave dynamics.This results in a minor underestimation of q (Fig. 11a), when the water level is lower (Fig. 11b).
It should be noted that neither the deep-water-parameter-based formulae nor the SWAN-Bayonet GPE approach were able to capture  C.H. Lashley et al. the drop in measured q at time 13:51 (Fig. 11a).However, as this decrease does not coincide with a change in forcing-i.e.h toe (Fig. 11b), H m0,deep or T m− 1,0,deep (Fig. 11c)-we attribute this to the random nature of irregular waves.As wind is known to notably affect low overtopping volumes, a further check was made to determine whether a change in wind speed or direction explained the observed decrease in q using wind data from the local meteorological station (Brown et al., 2020b).However, wind speed only decreased slightly over the 1-h duration (Fig. 11b) while wind direction remained constant.
While the predictions compared well to the 15-min averaged observations, both the deep-water-parameter-based approach and SWAN-Bayonet GPE both estimated some wave overtopping for conditions at times 13:06 and 14:36, but no overtopping was recorded at the WireWall (Fig. 11a).This discrepancy is attributed to the shorter (15-min) duration where no waves overtopped the structure, while both prediction methods assume a duration ≥500 waves-where the opportunity for some waves to overtop the structure is greater.The takeaway here is that while the approach proved accurate when compared to observations here, empirical overtopping methods that assume a duration ≥500 waves may not match observations for shorter durations.

Social media reports
Next, we assess the ability of the deep-water-parameter-based approach to hindcast q for the 1 080 wave height and water level conditions on the days when wave overtopping was reported by the Crosby community, between January 2013 and December 2017 (Fig. 12a).Estimates were also made using the SWAN-EurOtop (2007) (Fig. 12b), which represents the method used by the EA to nowcast hazard information during the same period.
Given the logarithmic nature of the overtopping formulae, it is necessary to define a threshold below which the calculated overtopping discharge is considered zero.Using a threshold of 0.03 l/s/m for a nonzero overtopping discharge prediction (EurOtop, 2018), the deep-water-parameter-based approach estimated q > 0.03 l/s/m for 35 of the 41 days (85%) when wave overtopping was recorded via photographs.The exceptions occurred on days when the water levels (<3.6 m AOD) and offshore wave heights (H m0,deep ≤ 0.9 m) were particularly low (Fig. 12a).On the other hand, the SWAN-EurOtop ( 2007) approach estimated non-zero wave overtopping (q > 0.03 l/s/m) on only 30 of the 41 days (73%) reported.This is because the method only estimates non-zero overtopping discharges when the water level >4.0 m AOD and H m0,deep > 3.5 m (Fig. 12b).
The validity of the deep-water-parameter-based approach at Crobsy, compared to SWAN-EurOtop (2007), is verified here by two notable features of Fig. 12. First, the variation in the deep-water-parameterbased q with water level and H m0,deep matches well (in terms of magnitude) with the WireWall measurements made during the January 2019 event (Fig. 12a).This is however not the case for the SWAN-EurOtop (2007) estimates, which suggest one order of magnitude lower discharges for the same water level and H m0,deep conditions (Fig. 12b).Second, for the event reported via Facebook on February 8th, 2016, showing significant wave overtopping (Fig. 6), the deep-waterparameter-based approach estimated 10 l/s/m ≤ q ≤ 33 l/s/m (Fig. 12a), indicating that a hazard warning should have been issued (since q > 25 l/s/m).However, a warning would not have been issued for the event using the SWAN-EurOtop (2007) approach, where 2 l/s/m ≤ q ≤ 16 l/s/m (Fig. 12b)-despite the danger to pedestrians, evidenced by Fig. 6.
The lower estimates of q obtained using the SWAN-EurOtop ( 2007) approach are mainly due to the underestimation of T m− 1,0,toe by SWAN (Appendix A), which is known to significantly affect the predicted overtopping discharge.This was demonstrated by Lashley et al. (2020b) who found that correcting the T m− 1,0,toe calculated by SWAN using the Hofland et al. (2017) empirical formula, as recommended by EurOtop (2018), increased the estimated q by a factor of 10.
What is also noteworthy about Fig. 12a, is that q in excess of 10 l/s/m regularly occurs at Crosby when the water depth and offshore wave heights exceed 4.5 m AOD and 1.5 m, respectively.These conditions correspond to non-extreme (yearly) events and highlight immediate need to improve hazard warning services at Crosby.

Critical foreshore width for wave overtopping at crosby
For the 1 080 wave and water level conditions reported via social media and their corresponding foreshore profiles, the offshore distance (critical foreshore width) used to determine the hypothetical foreshore slope angle (m) varied from immediately in front of the structure to over 550 m (Fig. 13).However, the distribution of this critical foreshore width suggests that the 200 m in front of the structure mostly influences wave overtopping at Crosby, with the first 30 m being the most critical-with tan(m) = 1:10 (or cot(m) = 10).This finding supports the conclusion of Brown et al. (2021) who found that the height of the beach within about 10 m of the structure toe had a strong influence on wave overtopping for any given wave and water level conditions.As the first 200 m in front of the structure is also the most dynamic (Fig. 3), coastal managers at Crosby can focus their surveying efforts on this limited beach area and still provide valuable information for flood risk assessments and forecasting.

Conclusions
Given high cost and high return periods for major storms, field measurements of wave overtopping are very limited.As a result, many empirical overtopping formulae are developed and validated using physical model tests and numerical simulations-without actual verification in the field.Moreover, many of the widely used overtopping formulae require wave parameters at the toe of the structure as input.For shallow foreshores where the nonlinear effects of wave breaking can dominate, this often requires that computationally demanding numerical models, and persons skilled in their application, be used to transform offshore waves to the structure toe.
In the present study, we validated a set of formulae for wave overtopping discharge (q) based on easily obtainable deep-water wave parameters.The validation was carried out using data gathered for the coastal defence (stepped revetment and recurve wall) and shallow foreshore at Crosby, UK.First, the formulae were compared quantitatively with wave overtopping measurements obtained using the Wire-Wall system of capacitance wires in January 2019.Next, the deep-waterparameter-based approach was compared qualitatively to photographic records of wave overtopping obtained via community photos (from January 2013 to December 2017).For the WireWall data, the predicted q was within a factor of 2 of the measurements averaged over the full 1-h period when wave overtopping was measured and conditions were relatively constant, and within a factor of 4 of those averaged over 15 min.This suggests that the approach is appropriate for design and assessments-using the total duration of the event-and forecasting the time-varying hazard (at 15-min intervals) for operational hazard response management at Crosby.
Additionally, the hindcasts made for the events reported via Facebook corroborated the community reports, which indicated that overtopping discharges-under typical winter conditions-can exceed 10 l/ s/m and pose a serious hazard to pedestrians.This finding, also reported by Brown et al. (2021), highlights the pressing need to improve the hazard forecasting and nowcast services at Crosby.
While the deep-water-parameter-based approach proved to be accurate here, it is still subject to certain limitations.For instance, the results highlighted the need to correctly define the representative foreshore slope angle (m), which is used as input to the formulae.The hypothetical foreshore slope method proposed here-where m is defined as a constant slope angle extending from a depth, h = H m0,deep to the structure toe-produced good results and may prove useful to practitioners who seek to parameterize surf zone processes (e.g. using the methods of Stockdon et al. (2006)) or make use of the well-known surf-similarity parameter (or "Iribarren number") (Battjes, 1974;Iribarren, 1949) in the field.However, it is recommended that the hypothetical foreshore slope method be further verified with physical model tests of different foreshore geometries (e.g., concave versus convex bottom profiles).Likewise, as our study was limited to the conditions at Crosby (UK), it is also recommended that the accuracy of the overall deep-water-parameter-based approach to estimate q be verified at field sites with different hydrodynamic and geomorphological features, such as those with wider, milder foreshore slopes or those with larger offshore waves.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
• For XBeach Non-hydrostatic (version XBeach X), the maximum breaking wave steepness parameter, λ = 0.8 following Lashley et al. (2020b) who found that λ varied from 0.7 to 0.9 for shallow foreshores under very mild and very steep wave conditions, respectively.A Manning's roughness coefficient (n = 0.012 s/m 1/3 ) is used to represent a smooth bottom.A constant cross-shore grid spacing (Δx = 0.5 m), which corresponds to > 100 grid cells per deep-water wavelength, was applied.The model was forced with JONSWAP wave spectra (spectra with a peak enhancement factor of 3.3) at its boundary.• For SWAN (version 41.31), friction is specified using a Nikuradse geometrical roughness of 0.3 × 10 − 3 m (smooth bottom), in line with the Lashley et al. (2020b) benchmarking study for shallow foreshores.Consistent with the XBeach setup, a constant x = 0.5 m was applied, and the model was forced with JONSWAP wave spectra (spectra with a peak enhancement factor of 3.3) at its boundary.It should be noted that a coarser grid could have been used without loss of accuracy.  . 3 show that H m0 begins to decrease on the foreshore slope due to wave breaking, where larger waves (with higher H m0,deep values) break further offshore.In the surf zone (h ≤ H m0,deep ), wave breaking is dominant, and the waves are now considered depth-limited-where H m0 is directly dependent on the water depth and, hence, the foreshore profile.On the other hand, T m− 1,0 increases from offshore to the structure toe (Figure A. 1 to Figure A. 3) due to the reduction in wave energy at high frequencies and the growth of energy at low (infragravity) frequencies during wave breaking.
Considering the XBeach simulations for the conditions observed on January 25th , 2019, with H m0,deep = 1.7 m, h toe = 1.6 m (Figure A. 1), the modelled change in H m0 and T m− 1,0 for the irregular bathymetry is similar to that of a constant hypothetical foreshore slope angle (m) of 1:10.Simulations for both the irregular profile and tan(m) = 1:10 show little difference between H m0,deep and H m0,toe , as only minor wave breaking occurs directly in front of the structure toe (Figure A. 1a).The modelled change in T m− 1,0 show similar agreement between the two XBeach simulations; though the simulation with tan(m) = 1:10 tends to underestimate the growth of T m− 1,0 as waves shoal over the irregular profile offshore (x < − 20 m), resulting in a minor underestimation of T m− 1,0,toe (Figure A. 1b).
When H m0,deep is increased to 2.7 m (while maintaining the same wave steepness), wave breaking initiates further offshore, with h = H m0,deep occurring ~60 m from the structure toe (Figure A. 2).Under these conditions, the characteristics of wave breaking are more comparable to those of a constant 1:60 foreshore slope.In like manner, for H m0,deep = 4.0 m, the tan(m) that produces a similar reduction in H m0 (Figure A. 3a) and growth in T m− 1,0 (Figure A. 3b) to the irregular bathymetry becomes 1:75-with h = H m0,deep occurring ~170 m offshore (Figure A. 3).
It should be noted that the differences observed between the irregular bathymetry and constant slope simulations at x > − 40 m for higher H m0,deep values (Figure A. 2a and Figure A. 3a) are due to the presence of the offshore trough (in the irregular profile), which greatly reduces wave breaking as the water depth becomes deeper.This process is not captured by the constant uniform slope.However, despite this minor disagreement, the H m0,toe and T m− 1,0,toe values obtained using the constant slope were still within 20% of those obtained using the irregular bathymetry and are therefore considered acceptable.
Compared to XBeach, SWAN predicts a much earlier onset of wave breaking, resulting in much lower estimates of H m0 at the location where h = H m0,deep (Figure A. 1a,Figure A. 2a and Figure A. 3a).This discrepancy increases as H m0,deep increases, with SWAN predicting ~1 m lower wave heights than XBeach for the simulations with H m0,deep = 4.0 m (Figure A. 3a).Despite these major differences offshore, both XBeach and SWAN tend to converge at the structure toe for lower h toe /H m0,deep values (Figure A. 2a and Figure A. 3a).This is attributed to the depth-limited conditions immediately in front of the structure.
Additionally, SWAN does not capture the growth in T m− 1,0 from offshore to the structure toe (Figure A. 1b,Figure A. 2b and Figure A. 3b).This increase, which grows rapidly as h toe /H m0,deep decreases (Figure A. 2b and Figure A. 3b), is due to the growth of long-period (infragravity) waves as the higher frequency (sea and swell waves) shoal and break over the foreshore (Longuet-Higgins and Stewart, 2006;Symonds et al., 1982).As this process is not resolved in SWAN, it consistently predicts almost no change in T m− 1,0 .Equation ( 14) also shows no increase in T m− 1,0 for h toe / H m0,deep = 0.9 (Figure A. 1b) but does capture the change for h toe /H m0,deep = 0.4-where the ratio of the spectral wave period at the toe to offshore, T m− 1,0,toe / T m− 1,0,deep ≈ 2.5 (Figure A. 3b).
The overall trends observed in Figure A. 1 to Figure A. 3 highlight the dependence of the representative foreshore slope on the relative water depth (h/H m0,deep ) for highly irregular bathymetries.It also demonstrates that the approach taken here to define m may be considered reasonable.This is further supported by Equation ( 13): where the estimates of H m0,toe obtained using m compare well (within 15%) to those modelled by XBeach using the irregular bathymetry ( As no nearshore measurements were available for comparison, the above model results are considered conceptual and used here to: i) rationalise the approach taken to define m; and ii) compare the performance of Equations ( 13) and ( 14) in estimating wave conditions at the toe with the much more computationally demanding numerical modelling approaches.No further discussion on the accuracy of the predictions is dealt with here.It should also be acknowledged that it is more accurate to simulate the effects of an irregular bathymetry using XBeach Non-hydrostatic (or similar phase-resolving numerical model) compared to empirical estimates.However, the hypothetical foreshore slope-developed for use with the deepwater-parameter-based formulae-may be considered a practical and reasonably accurate alternative.13)) and T m− 1,0,toe (Equation ( 14)) are also provided with error bars ( ± 2 standard deviations). .13)) and T m− 1,0,toe (Equation ( 14)) are also provided with error bars ( ± 2 standard deviations).

Appendix B. Relative Magnitude of Infragravity Waves
The relative magnitude or significance of infragravity waves at the toe of a structure ( HIG ) may be expressed as (Lashley et al., 2020a): C.H.Lashley et al.

Fig. 1 .
Fig. 1.Map of the study site showing the locations of the tide gauge and offshore wave buoy used in this study.Inset shows the location of the study site relative to the wider UK.

Fig. 2 .
Fig. 2. Photo of the coastal defence at Crosby with foreshore exposed at low tide.

Fig. 3 .
Fig. 3.The average foreshore profile recorded from September 1996 to January 2019, showing the bar-trough system (− 200 m < x < − 30 m), with bed level given in meters above ordnance datum (m AOD).The shaded area represents the maximum and minimum bed levels recorded during that period.Mean high water spring (MHWS) and mean low water spring (MLWS) levels are shown for reference.

Fig. 7 .
Fig. 7. Schematic representation of the foreshore and structure at Crosby (UK) showing the hypothetical foreshore slope angle (m) used in this study.

Fig. 8 .
Fig. 8. Schematic representation of the structure at Crosby (UK) comprising a stepped revetment and recurve wall, highlighting key variables: structure slope angle (α), bullnose thickness (h n ) and height of the recurve wall (h wall ).

Fig. 11 .
Fig. 11.a) Measured and predicted q at 15-min intervals on January 25th, 2019, showing b) the water depth at the structure toe (h toe ) and wind speed; and c) offshore significant wave height (H m0,deep ) and spectral wave period (T m− 1,0,deep ).Note that all observations and SWAN-Bayonet predictions are resampled to the 15-min intervals that match the WireWall data.Dotted lines indicate the 1-h period when wave overtopping occurred.

Fig. 12 .
Fig. 12. Predicted mean overtopping discharge for the wave heights (H m0,deep ) and water levels corresponding to photographic records at Crosby from 2013 to 2018, using: a) the deep-water-parameter-based approach; and b) SWAN-EurOtop (2007); where the solid line corresponds to h toe /H m0,deep = 1.5.Joint probability curves (dashed lines) are provided for reference (Halcrow Group Limited, 2011).The measured 15-min averaged discharges for the January 2019 event are also provided for reference.

Fig. 13 .
Fig.13.Scatter plot with histogram of the critical foreshore width (x-axis) used to determine the hypothetical foreshore slope, cot(m) (y-axis) and predict q using the deep-water-parameter-based approach, for the 1 080 wave and water level conditions reported.
C.H.Lashley et al.

Figure
Figure A. 1 to Figure A. 3  show that H m0 begins to decrease on the foreshore slope due to wave breaking, where larger waves (with higher H m0,deep values) break further offshore.In the surf zone (h ≤ H m0,deep ), wave breaking is dominant, and the waves are now considered depth-limited-where H m0 is directly dependent on the water depth and, hence, the foreshore profile.On the other hand, T m− 1,0 increases from offshore to the structure toe(Figure A. 1 to Figure A. 3) due to the reduction in wave energy at high frequencies and the growth of energy at low (infragravity) frequencies during wave breaking.Considering the XBeach simulations for the conditions observed on January 25th , 2019, with H m0,deep = 1.7 m, h toe = 1.6 m (Figure A. 1), the modelled change in H m0 and T m− 1,0 for the irregular bathymetry is similar to that of a constant hypothetical foreshore slope angle (m) of 1:10.Simulations for both the irregular profile and tan(m) = 1:10 show little difference between H m0,deep and H m0,toe , as only minor wave breaking occurs directly in front of the structure toe(Figure A. 1a).The modelled change in T m− 1,0 show similar agreement between the two XBeach simulations; though the simulation with tan(m) = 1:10 tends to underestimate the growth of T m− 1,0 as waves shoal over the irregular profile offshore (x < − 20 m), resulting in a minor underestimation of T m− 1,0,toe(Figure A. 1b).When H m0,deep is increased to 2.7 m (while maintaining the same wave steepness), wave breaking initiates further offshore, with h = H m0,deep occurring ~60 m from the structure toe(Figure A. 2).Under these conditions, the characteristics of wave breaking are more comparable to those of a constant 1:60 foreshore slope.In like manner, for H m0,deep = 4.0 m, the tan(m) that produces a similar reduction in H m0(Figure A. 3a) and growth in T m− 1,0 (Figure A. 3b) to the irregular bathymetry becomes 1:75-with h = H m0,deep occurring ~170 m offshore (Figure A. 3).It should be noted that the differences observed between the irregular bathymetry and constant slope simulations at x > − 40 m for higher H m0,deep values(Figure A. 2a and Figure A. 3a) are due to the presence of the offshore trough (in the irregular profile), which greatly reduces wave breaking as the water depth becomes deeper.This process is not captured by the constant uniform slope.However, despite this minor disagreement, the H m0,toe and T m− 1,0,toe values obtained using the constant slope were still within 20% of those obtained using the irregular bathymetry and are therefore considered acceptable.Compared to XBeach, SWAN predicts a much earlier onset of wave breaking, resulting in much lower estimates of H m0 at the location where h = H m0,deep(Figure A. 1a, Figure A. 2a and Figure A. 3a).This discrepancy increases as H m0,deep increases, with SWAN predicting ~1 m lower wave heights than XBeach for the simulations with H m0,deep = 4.0 m (Figure A. 3a).Despite these major differences offshore, both XBeach and SWAN tend to converge at the structure toe for lower h toe /H m0,deep values(Figure A. 2a and Figure A. 3a).This is attributed to the depth-limited conditions immediately in front of the structure.Additionally, SWAN does not capture the growth in T m− 1,0 from offshore to the structure toe(Figure A. 1b, Figure A. 2b and Figure A. 3b).This increase, which grows rapidly as h toe /H m0,deep decreases(Figure A. 2b and Figure A. 3b), is due to the growth of long-period (infragravity) waves as the higher frequency (sea and swell waves) shoal and break over the foreshore(Longuet-Higgins and Stewart, 2006;Symonds et al., 1982).As this process is not resolved in SWAN, it consistently predicts almost no change in T m− 1,0 .Equation (14) also shows no increase in T m− 1,0 for h toe / H m0,deep = 0.9 (Figure A. 1b) but does capture the change for h toe /H m0,deep = 0.4-where the ratio of the spectral wave period at the toe to offshore, T m− 1,0,toe / T m− 1,0,deep ≈ 2.5(Figure A. 3b).The overall trends observed in Figure A. 1 to Figure A. 3 highlight the dependence of the representative foreshore slope on the relative water depth (h/H m0,deep ) for highly irregular bathymetries.It also demonstrates that the approach taken here to define m may be considered reasonable.This is further supported by Equation (13): where the estimates of H m0,toe obtained using m compare well (within 15%) to those modelled by XBeach using the irregular bathymetry (Figure A. 1 to Figure A. 3); and Equation (14): where the estimates of T m− 1,0,toe are within 20% of those modelled by XBeach (Figure A. 1 to Figure A. 3).As no nearshore measurements were available for comparison, the above model results are considered conceptual and used here to: i) rationalise the approach taken to define m; and ii) compare the performance of Equations (13) and (14) in estimating wave conditions at the toe with the much more computationally demanding numerical modelling approaches.No further discussion on the accuracy of the predictions is dealt with here.It should also be acknowledged that it is more accurate to simulate the effects of an irregular bathymetry using XBeach Non-hydrostatic (or similar phase-resolving numerical model) compared to empirical estimates.However, the hypothetical foreshore slope-developed for use with the deepwater-parameter-based formulae-may be considered a practical and reasonably accurate alternative.
Figure A. 1 to Figure A. 3  show that H m0 begins to decrease on the foreshore slope due to wave breaking, where larger waves (with higher H m0,deep values) break further offshore.In the surf zone (h ≤ H m0,deep ), wave breaking is dominant, and the waves are now considered depth-limited-where H m0 is directly dependent on the water depth and, hence, the foreshore profile.On the other hand, T m− 1,0 increases from offshore to the structure toe(Figure A. 1 to Figure A. 3) due to the reduction in wave energy at high frequencies and the growth of energy at low (infragravity) frequencies during wave breaking.Considering the XBeach simulations for the conditions observed on January 25th , 2019, with H m0,deep = 1.7 m, h toe = 1.6 m (Figure A. 1), the modelled change in H m0 and T m− 1,0 for the irregular bathymetry is similar to that of a constant hypothetical foreshore slope angle (m) of 1:10.Simulations for both the irregular profile and tan(m) = 1:10 show little difference between H m0,deep and H m0,toe , as only minor wave breaking occurs directly in front of the structure toe(Figure A. 1a).The modelled change in T m− 1,0 show similar agreement between the two XBeach simulations; though the simulation with tan(m) = 1:10 tends to underestimate the growth of T m− 1,0 as waves shoal over the irregular profile offshore (x < − 20 m), resulting in a minor underestimation of T m− 1,0,toe(Figure A. 1b).When H m0,deep is increased to 2.7 m (while maintaining the same wave steepness), wave breaking initiates further offshore, with h = H m0,deep occurring ~60 m from the structure toe(Figure A. 2).Under these conditions, the characteristics of wave breaking are more comparable to those of a constant 1:60 foreshore slope.In like manner, for H m0,deep = 4.0 m, the tan(m) that produces a similar reduction in H m0(Figure A. 3a) and growth in T m− 1,0 (Figure A. 3b) to the irregular bathymetry becomes 1:75-with h = H m0,deep occurring ~170 m offshore (Figure A. 3).It should be noted that the differences observed between the irregular bathymetry and constant slope simulations at x > − 40 m for higher H m0,deep values(Figure A. 2a and Figure A. 3a) are due to the presence of the offshore trough (in the irregular profile), which greatly reduces wave breaking as the water depth becomes deeper.This process is not captured by the constant uniform slope.However, despite this minor disagreement, the H m0,toe and T m− 1,0,toe values obtained using the constant slope were still within 20% of those obtained using the irregular bathymetry and are therefore considered acceptable.Compared to XBeach, SWAN predicts a much earlier onset of wave breaking, resulting in much lower estimates of H m0 at the location where h = H m0,deep(Figure A. 1a, Figure A. 2a and Figure A. 3a).This discrepancy increases as H m0,deep increases, with SWAN predicting ~1 m lower wave heights than XBeach for the simulations with H m0,deep = 4.0 m (Figure A. 3a).Despite these major differences offshore, both XBeach and SWAN tend to converge at the structure toe for lower h toe /H m0,deep values(Figure A. 2a and Figure A. 3a).This is attributed to the depth-limited conditions immediately in front of the structure.Additionally, SWAN does not capture the growth in T m− 1,0 from offshore to the structure toe(Figure A. 1b, Figure A. 2b and Figure A. 3b).This increase, which grows rapidly as h toe /H m0,deep decreases(Figure A. 2b and Figure A. 3b), is due to the growth of long-period (infragravity) waves as the higher frequency (sea and swell waves) shoal and break over the foreshore(Longuet-Higgins and Stewart, 2006;Symonds et al., 1982).As this process is not resolved in SWAN, it consistently predicts almost no change in T m− 1,0 .Equation (14) also shows no increase in T m− 1,0 for h toe / H m0,deep = 0.9 (Figure A. 1b) but does capture the change for h toe /H m0,deep = 0.4-where the ratio of the spectral wave period at the toe to offshore, T m− 1,0,toe / T m− 1,0,deep ≈ 2.5(Figure A. 3b).The overall trends observed in Figure A. 1 to Figure A. 3 highlight the dependence of the representative foreshore slope on the relative water depth (h/H m0,deep ) for highly irregular bathymetries.It also demonstrates that the approach taken here to define m may be considered reasonable.This is further supported by Equation (13): where the estimates of H m0,toe obtained using m compare well (within 15%) to those modelled by XBeach using the irregular bathymetry (Figure A. 1 to Figure A. 3); and Equation (14): where the estimates of T m− 1,0,toe are within 20% of those modelled by XBeach (Figure A. 1 to Figure A. 3).As no nearshore measurements were available for comparison, the above model results are considered conceptual and used here to: i) rationalise the approach taken to define m; and ii) compare the performance of Equations (13) and (14) in estimating wave conditions at the toe with the much more computationally demanding numerical modelling approaches.No further discussion on the accuracy of the predictions is dealt with here.It should also be acknowledged that it is more accurate to simulate the effects of an irregular bathymetry using XBeach Non-hydrostatic (or similar phase-resolving numerical model) compared to empirical estimates.However, the hypothetical foreshore slope-developed for use with the deepwater-parameter-based formulae-may be considered a practical and reasonably accurate alternative.

Figure A1 .
Figure A1.Modelled change in a) H m0 and b) T m− 1,0 , from offshore to the structure toe using XBeach Nonhydrostatic and SWAN for h toe = 1.6 m and H m0,deep = 1.7 m (h toe /H m0,deep = 0.9); with c) the bed levels for the irregular bathymetry at Crosby and constant hypothetical foreshore slopes.Dotted vertical lines indicate h = H m0,deep .Empirical estimates of H m0,toe (Equation (13)) and T m− 1,0,toe (Equation (14)) are also provided with error bars ( ± 2 standard deviations). .

Figure A2 .
Figure A2.Modelled change in a) H m0 and b) T m− 1,0 , from offshore to the structure toe using XBeach Nonhydrostatic and SWAN for h toe = 1.6 m and H m0,deep = 2.7 m (h toe /H m0,deep = 0.6); with c) the bed levels for the irregular bathymetry at Crosby and constant hypothetical foreshore slopes.Dotted vertical lines indicate h = H m0,deep .Empirical estimates of H m0,toe (Equation (13)) and T m− 1,0,toe (Equation (14)) are also provided with error bars ( ± 2 standard deviations).

Figure A3 .
Figure A3.Modelled change in a) H m0 and b) T m− 1,0 , from offshore to the structure toe using XBeach Nonhydrostatic and SWAN for h toe = 1.6 m and H m0,deep = 4.0 m (h toe /H m0,deep = 0.4); with c) the bed levels for the irregular bathymetry at Crosby and constant hypothetical foreshore slopes.Dotted vertical lines indicate h = H m0,deep .Empirical estimates of H m0,toe (Equation (13)) and T m− 1,0,toe (Equation (14)) are also provided with error bars ( ± 2 standard deviations).