Wave run-up is the final expression of waves travelling from deep to shallow water and is directly associated with coastal hazards like flooding or erosion. Wave run-up height can be defined from water level elevation time series at the shoreline η(t) as the sum of two distinguished components: the wave set-up (the temporal mean of the time series 〈η〉 relative to the still water level) and the swash η′(t) (the vertical fluctuation of the water level around the wave set-up). Understanding and predicting swash characteristics is critical for researchers seeking to understand the dynamics of fluid motions and sediment transport in the swash zone (e.g. Elfrink and Baldock, 2002; Masselink and Puleo, 2006), and for managers and practitioners addressing hazard setbacks, risk and coastal vulnerability (e.g. Bosom and Jiménez, 2011; Vousdoukas et al., 2012). Wave run-up (and therefore swash excursion) is a key component used to evaluate inundation hazards and vulnerability to storm impacts (e.g. Bosom and Jiménez, 2011; Stockdon et al, 2007; Serafin et al., 2017). Stockdon et al. (2007) found that the wave action counted for about 48 % of the maximum total water level during two hurricanes along the USA coast. The problem of accurate predictions of wave run-up and swash on sandy beaches has been a research topic for over 50 years but today we still struggle to provide reliable quantitative predictions.

The first predictors of these phenomena were developed in the context of coastal structures (Miche, 1951; Hunt, 1959) and the formulas proposed, usually developed for steep slopes and under the assumption that run-up motions reflect the standing component of the incident wave field. Overall these formulas suggested a dependence of the uprush elevation on wave steepness and structure slope. A variety of predictors have since then been developed for vertical run-up (R) and swash (S) on sandy beaches (e.g. Guza and Thornton, 1982; Holman and Sallenger, 1985; Holman, 1986; Ruessink et al., 1998; Stockdon et al., 2006), with details of the parameterizations depending on different combinations of deepwater significant wave height (Hs0), deepwater wavelength (L0) and beach slope (β). Guza and Thorton (1982) proposed a linear relationship between the significant run-up (Rs) and Hs0: Rs=cHs0, where c=0.7. Guza and Thornton (1982) also first distinguished between infragravity and incident swash components, indicating that the swash component related to low frequencies (infragravity, RIg) depends only on significant wave height (therefore excluding the beach slope), while the incident component can saturate as a result of the dissipative processes occurring in the surf zone. Their findings were later confirmed by several other studies, although different dependencies on environmental parameters were suggested (e.g. Holman and Sallenger, 1985; Ruessink et al., 1998).

Holman and Sallenger (1985) studying an intermediate to reflective beach (Duck, North Carolina, USA) described Rs as Rs=cξ0Hs0, where c is a constant, Hs0/L0 is the wave steepness and ξ0=tanβHs0/L0, where β is the foreshore beach slope and ξ0 is the surf similarity index which is also often used for beach classification – beaches are classified as dissipative for values of ξ0 < 0.23, reflective for ξ0 > 1 and intermediate (Short, 1999).

Stockdon et al. (2006) used 10 experiments from different locations to generate new parameterizations of wave run-up on natural beaches. The 2 % exceedance value of wave run up R2 was defined as R2=1.1〈η〉+STot2, where 〈η〉 is the maximum set-up elevation and STot is the total swash defined as STot=Sin2+SIg2, where Sin and SIg are the incident and infragravity components of swash. Stockdon et al. (2006) used regression techniques to obtain relationships for Sin and SIg: Sin=0.75βH0L0, and SIg=0.06H0L0. Stockdon et al. (2006) is the most commonly used empirical parameterization of run-up but, as can be noted by comparing Eqs. (6) and (7), the beach slope is missing from the predictor of the infragravity component of swash. The dependency (or not) of SIg on beach slope is a topic that has been debated but not solved and some authors (e.g. Ruessink et al., 1998) have indicated that infragravity swash is independent from the beach slope, while a weak dependence on beach slope has instead been reported by others (e.g. Ruggiero et al., 2004). Cohn and Ruggiero (2016) suggested a bathymetric control of the infragravity swash component through 1-D and 2-D numerical simulations performed using Xbeach (where incident swash contribution is excluded) and compared them with previous formulas (Ruggiero et al., 2001; Stockdon et al., 2006) and field data on dissipative beaches. They suggested that beach morphology (> -2 m m.s.l.) influences the infragravity component of run-up more than the nearshore morphology (< -2 m m.s.l.) and indicated that including the foreshore beach slope in the formulation of SIg improves predictability. Overall, it remains unclear if and when SIg depends on beach slope. In addition the similarity in the temporal scales of wave set-up and infragravity motions could also be a confounding factor in measurements. Finally, a number of other studies have also proposed other predictors that introduce other parameters to account for the cross-shore wind component and the tidal range (Vousdoukas et al., 2012), the presence of nearshore sandbars (Cox et al., 2013) or the sediment mean grain size for the case of gravel beaches (Poate et al., 2016). The above-mentioned empirical run-up formulas have been developed primarily with classic regression approaches (e.g.; Ruessink et al., 1998; Ruggiero et al., 2001; Stockdon et al., 2006; Vousdoukas et al., 2012).

Because of the importance of accurate predictions of swash excursion, the predictors provided by Stockdon et al. (2006) have been tested by various authors on beaches ranging from reflective to dissipative (e.g. Vousdoukas et al., 2012; Cohn and Ruggiero, 2016; Atkinson et al., 2017). Predictions using Stockdon et al. (2006) are certainly sound (especially considering the task of generating a universal formula for vertical swash excursion), even though differences between measurements and predictions, possibly associated with local conditions, are inevitably found. More importantly, the regression approach of multiple data sets first proposed by Stockdon et al. (2006) paves the way for our working hypothesis: can powerful data-driven techniques be used to provide robust, reliable and realistic predictions of swash excursion?

When enough data exist, machine learning (ML) is a viable approach to regression problems. ML is a subdiscipline of computer science focused on techniques that allow computers to find insightful relationships between variables involved in swash processes, learning at each iteration (algorithm training and validation) from the provided data set. A key goal of ML is to develop predictors that are generalizable (able to describe the physical process beyond the training data set itself). Many different data-driven techniques fall under the purview of machine learning (e.g. decision trees, artificial neural networks, Bayesian networks and evolutionary computation), all of which have shown applicability in coastal settings (e.g. Pape et al., 2007; Knaapen and Hulscher, 2002; Dickson and Perry, 2016; Yates and Le Cozannet, 2012). Previous machine learning work has focused on predicting run-up and swash, but only for engineered structures, impermeable slopes and/or for laboratory experiments (e.g. Kazeminezhad and Etemad-Shahidi, 2015; Bonakdar and Etemad-Shahidi, 2011; Bakhtyar et al., 2008; Abolfathi et al., 2016) and not on natural beaches apart from Vousdoukas et al. (2011), which used artificial neural networks (ANNs) for shoreline contour elevation (which includes the wave run-up), on a natural beach in Portugal. In this study we focus on the use an evolutionary technique, genetic programming (GP), to solve the symbolic regression problem of developing new, optimized swash predictors.

In this contribution we first develop a swash excursion predictor using the original data set of Stockdon et al. (2006), one of the most comprehensive studies in this area of research. In addition, we use data from Guedes et al. (2011, 2012, 2013) and Senechal et al. (2011) to broaden the parameter space and to test the new swash equations. The data used in this work cover a broad range of swash excursion including extreme wave conditions (maximum H0= 6.4 m in Senechal et al., 2011). High swash excursions, generated by extreme storms, are of particular interest when studying coastal hazards because they relate to flooding, beach and dune erosion (Bosom and Jimenez, 2011; Stockdon et al., 2007). The new ML-derived results are also compared to the most widely used predictors from Stockdon et al. (2006). Finally, we discuss the physical interpretation of the GP predictors and how we can use ML to gain knowledge of physical processes related to the infragravity swash component.

This work is based on two published video image-derived run-up data sets – 13 field experiments in total. The first data set (referred to here as the “original data set”) is composed of 491 swash measurements from 10 experiments aggregated by Stockdon et al. (2006). The second data set (referred to here as the “new data set”) consists of 145 swash measurements compiled for this work from three experiments performed by Guedes et al. (2011, 2013) and Senechal et al. (2011). For details about the access to data, the reader is referred to the section data availability.

The compiled data set of total swash is plotted in Fig. 1. The compilation of a large data set deriving from 13 different experiments requires merging data collected using different techniques and equipment. Details of each experiment can be found in the original references. Looking at the environmental forcing conditions, Fig. 1 shows that the original and new data set cover similar ranges of beach slope, while they differ in significant wave height (the new data set includes wave heights over 6 m) and peak period (the original data set includes more short period waves).

Both data sets include recordings of infragravity swash (SIg; m), total swash (STot; m), beach slope (β) and associated offshore wave characteristics: significant wave height (H0; m) and peak period (Tp; s). From these measurements the offshore significant wavelength L0;m wave steepness (H0/L0) and Iribarren number (ξ0) are calculated. Experiments are located in North America, Europe and Oceania and cover a large range of the environmental condition (see Table 1 and Fig. 1).

Both data sets include all beach types, from dissipative to reflective. The two data sets also have a similar range of STot (although the original data set records a larger swash, 0.2–3.3 m vs. 0.24–2.5 m of the new data set), SIg (about 0.2–2.4 m for both) and β (about 0.01–0.1 for both). The two data sets differ in the range of offshore wave conditions – in the original data set H0 and Tp range over 0.4–4.1 (m) and 3.7–17 (s), while in the new data set the ranges are 0.6–6.4 (m) and 8.1–16.4 (s).

The dissipative beaches of the original data set (Fig. 2d, h) are Terschelling (Netherlands) and Agate (USA), and for the new data set on Ngarunui in New Zealand (although, during the experiment, the beach also experienced intermediate conditions). The purely intermediate beaches for the original and new data set are Scripps (USA) and Truc Vert (France). Some beaches from the original data set (USA) represent both intermediate and reflective conditions: Duck 94, Gleneden, Sandy Duck, Delilah and Duck 82. San Onofre for the original and Tairua (New Zealand) for new data set are reflective.

The large amount of data available (636 field swash records), including multidimensional variables, supports the feasibility of a ML approach. The data cover a wide range of environmental conditions (including extreme storms) and beach types, ensuring the applicability of our results to sandy beaches spreading from dissipative to reflective. We now outline the methods of the study. In Sect. 3.1, we present the supervised ML approach. In Sect. 3.2 we present the data preprocessing technique used to decide what data are shown to the ML algorithm. In Sect. 3.3 we discuss the techniques used to test the results from the ML algorithm against the testing data.

In this work we use data compiled by Stockdon et al. (2006) to build new predictors by the use of GP for both total and infragravity swash elevations. We then test the generalizability of these new predictors using new data (including some extreme conditions). This is different from many previous applications of ML in coastal settings in two ways: first, we are testing the ML-derived predictor on data that are collected from a different setting (compared to the training data) – three beaches not included in the training data. Second, the testing data includes events that are outside the data range of the training data. We are extrapolating the ML-derived predictor as a test of its generalizability. We do not assume a single criteria for the selection of the best predictors, but we find a compromise between error reduction (on the testing data set) and the physical interpretability of the results.

Results demonstrate that the GP predictors proposed in this work perform better than existing formulas and that ML can identify non-linear relationships between the variables of this problem. Specifically, Eq. (14) introduces the dependence of SIgon the beach slope, but also its non-linear relationship with the wavelength. Furthermore, solutions for SIg found by the GP algorithm with the smaller size (not shown) show a simple linear dependence on H0 with a constant (identical to early formulation of wave run-up, e.g. of Guza and Thornton, 1982). More complex predictors add a dependence on L0, H0L0 (similar to Eq. (7) of this paper from Stockdon et al., 2006) and H0L0.

The GP algorithm found solutions for SIg that include the beach slope (β), a variable that is never excluded from predictors of further increasing size. Because the candidate solutions resulted from GP experiments follow a Pareto front distribution in which the increase in fitting (smaller MSE) grows as the size of the formula rises, the continuous inclusion of β for more complex predictors implies that including β in SIg formulation reduces prediction error. The improvement of classic empirical techniques, by innovation in data-driven methodologies, has already been discussed (e.g. the case of depth-averaged velocities over model vegetation by Tinoco et al., 2015). Experiments based on GP also highlighted a way to focus on and add dependencies in predictors describing coastal processes (e.g. grain size in the case of prediction of ripple wavelength by Goldstein et al., 2013). The predictors proposed in this work perform well on a wide range on environmental conditions, including, as defined by Nicolae et al. (2016), the highest stormy condition data set (Truc Vert beach) recorded in the field and available in the literature. Furthermore the work here demonstrates that ML-derived results, when physically plausible, may be generalizable beyond the limits of the training data, extrapolating to a novel, out-of-sample data set.

Looking at the limitation of the proposed models, the variables taken into account (H0, Tp, L0, β) are easily accessible but also oversimplify the processes that affect swash. For instance, we do not include the influence of the wave-directional spread (Guza and Feddersen, 2012), the cross-shore wind component and the tidal range (Vousdoukas et al., 2012). However, in order to include these and other aspects (e.g. role of underwater vegetation, nearshore bathymetry) it is necessary to perform more field experiments that record swash, run-up and other relevant variables. An additional limitation is that the swash formulas obtained in this study approach a non-zero value as wave height approaches zero. While this is physically incorrect, the data used in the analysis do not include the limited condition of “no waves – no swash”. Consequently, even if the GP formulas obtained do not correctly predict the limit condition corresponding to a no wave scenario, the prediction for both data sets has smaller errors compared to commonly used formulas. Generally the results from the machine learning technique are strictly related to the range of the training and validation data sets (original data set in Fig. 1). This work demonstrated that the applicability of the predictors can sometimes be used beyond the range of the testing data set (new data set in Fig. 1). However it is unknown how predictors will perform in settings beyond those in the present work – future tests on new field data are therefore recommended. Furthermore, parameterizations always work better when free parameters are optimized to a given site by using existing data and should be considered when proposing universal parameterizations.

Our results contribute to the discussion on the role of beach slope on the prediction of the infragravity component of swash. The GP algorithm found an SIg dependence on beach slope and increasingly more complicated formulas (i.e. more precise predictions) found by the GP all include beach slope as one of the predictive variables. This result is in line with studies such as Ruggiero et al. (2001, 2004) and in contrast with Stockdon et al. (2006), Senechal et al. (2011) and Ruessink et al. (1998). Although difficult to quantify and extremely simplified (this parameter together with sediment diameter should integrate the effect of the entire cross-shore profile), our results suggest that some parameter involving the beach profile should be considered when predicting run-up characteristics.

Our results are relevant for a variety of applications where the errors related to empirical formulation obtained by classic regression techniques could be reduced. For instance in the case of coastal hazards, Stockdon et al. (2006) formulation for wave run-up is used by Serafin and Ruggiero (2014) for their extreme total water level estimation and by Bosom and Jimenez (2011) in their framework for coastal hazards assessment. Accuracy in a run-up formulation has consequences for risk and vulnerability assessment in coastal management maps (De Muro et al., 2017; Perini et al., 2016) and other several studies regarding sediment transport (Puleo et al., 2000), swash zone hydrodynamics and morphodynamics (Puleo and Torres-Freyermut, 2016).

Starting from a large data set covering a wide range of swash, beach and wave field characteristics, we developed two new predictors for total and infragravity swash elevations, using the machine learning technique of genetic programming. We tested and compared our new formulas with previously developed and largely accepted parameterizations of swash (e.g. Stockdon at al., 2006) using independent published data sets. Results of the two GP predictors selected (one for total and one for infragravity swash) show better performance compared with the formulation of Stockdon et al. (2006), evaluated using an independent (unknown to the algorithm) data set (which included extreme highly energetic wave storm, particularly relevant for coastal hazards). This work contributes to reducing the uncertainty in predicting the swash excursion and consequently in assessing the coastal vulnerability and hazards (e.g. inundation) which depend in part upon wave swash (Bosom and Jimenez, 2011). A better prediction of swash excursion could also influence retreat or accommodation strategies and integrated planning for the mitigation of coastal hazards. Furthermore, GP results indicate that the beach slope influences the infragravity component of the swash – GP predictors improve in performance when the beach slope was included. We therefore conclude that beach slope is a relevant parameter when predicting the infragravity component of the swash elevation, even though this is contrary to several previous studies (e.g. Stockdon et al., 2006; Ruessink, 1998; Senechal et al., 2011). ML, and specifically GP, can be a useful tool for data-rich problems, providing robust predictors and possibly also physical insight. The role and importance of the scientist is not reduced or substituted by the machine but instead improved thanks to a powerful data analysis tool.