Numerical modelling of the equilibrium profile in Valencia (Spain)

Ocean Engineering 123 (2016) 164–173

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Numerical modelling of the equilibrium profile in Valencia (Spain) L. Aragonés a,n, Y. Villacampa b, F.J. Navarro-González b, I. López a a b

Department of Civil Engineering, University of Alicante, San Vicent del Raspeig s/n, 03690 Alicante, Spain Department Applied Mathematics, University of Alicante, San Vicent del Raspeig s/n, 03690 Alicante, Spain

art ic l e i nf o

a b s t r a c t

Article history: Received 28 October 2015 Received in revised form 11 April 2016 Accepted 15 July 2016 Available online 21 July 2016

In this paper, mathematical models for obtaining the parameter A of the potential function, which describes the equilibrium beach profile, were generated. The aim of the research was to analyse the most important variables related to the equilibrium beach profile, so that the models are used only those variables that are easy to obtain. On the other hand, the sensitivity of the models must be taken into account, since small changes in variables can cause significant deviations in the models results. For this reason, a detailed research of the sensitivity of the studied models in relation to the different variables has been performed. The parameters considered for the evaluation and selection of the optimal model are: the volume error, the Pearson coefficient (R2), the relative mean square error (MSE/Var), the mean error (ϵ) and the relative percentage error (δ) and the model stability under changes in the input variables of up to 20% by a Monte Carlo simulation. The best models have been validated using data corresponding to beaches from outside the area where the dataset has been collected. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Numerical model Potential function Sediment Maritime climate

1. Introduction Geomorphology and local climatic conditions are determinant factors when considering the characteristics of coastal zones. The morphological changes on beaches under the action of hydrodynamic agents are closely related to the concept of the equilibrium beach profile (EBP). Equilibrium beach profiles are mathematical models which try to reproduce the phenomena that model the offshore seabed. Sometimes, these models consist of one or a set of equations which can predict the real behaviour of the profile with different degrees of accuracy. A good approximation of real values is the basis of successful model for the equilibrium beach profile. The models must be validated using datasets different from those used for their calculation. The models obtained from lab test data do not usually fit well to local conditions due to their general characteristics (Kaiser and Frihy, 2009; Sierra et al., 1994). There are different methodologies to build models from experimental data. The two key characteristics that determine their success and usefulness are easy interpretation and simplicity of use. For the beaches studied in this paper, the model that best fits the shape of the cross-shore profiles is that defined by the potential function of Eq. (1) (Aragonés et al., 2016). n

Corresponding author. E-mail addresses: [email protected] (L. Aragonés), [email protected] (Y. Villacampa), [email protected] (F.J. Navarro-González), [email protected] (I. López). http://dx.doi.org/10.1016/j.oceaneng.2016.07.036 0029-8018/& 2016 Elsevier Ltd. All rights reserved.

y=A⋅x2/3

(1)

Where y is the depth, x is the offshore distance and A is a scale parameter. For some authors such as Bodge (1992), Dean (1977), Moore (1982) and Pilkey et al. (1993) the parameter A only depends on the median sediment size (D50). Others such as Larson et al. (1999) consider that a variable size must be considered along the profile length. However, assuming that the equilibrium profile depends only on sediment grain size is also considered an error, as is remarked by Pilkey et al. (1993). Also, Stockberger and Wood (1990) suggest that this dependence on the sediment size does not exist. Another key parameter is the wave height; Kaiser and Frihy (2009), Karunarathna et al. (2009) and Romanczyk et al. (2005) consider that this wave height must be representative of the period during which the profiles was generated. Boon and Green (1988) and Turker and Kabdasli (2006) state that the main parameter of their formulations is influenced by both the incident swell and the sediment properties. Several approaches to the problem of determining parameter A exist. The easiest and most used is that proposed by Dean (1987), which relates the median size (D50) to the sediment fall velocity.

A = 0. 067⋅ω0.44

(2)

Where ω is the sediment fall velocity in cm/s. Eq. (2) can also be obtained using Hallermeier (1981), expressing A as directly dependent on the median sediment size D50, as can be seen in Eq. (3).

L. Aragonés et al. / Ocean Engineering 123 (2016) 164–173

A = 0. 21⋅D500.48

(3)

Kriebel et al. (1991) proposed the relationship of Eq. (4) to determine the parameter A, which is better suited when the sediment fall velocity is between 1 and 10 cm/s.

⎛ ω2 ⎞1/3 A = 2. 25⋅⎜ ⎟ ⎝ g ⎠

(4)

Turker and Kabdasli (2006) introduced the effect of water density and the dissipation of swell energy, proposing the Eq. (5).

A=

a1

⎡ 3 2 −1/2 ⎤2/3 2 3/2 H h h Γ ⋅ ⋅ + ⋅ ⎢ ⎥⎦ b b b 2/3 ⎣ 5

( κ ⋅X ) 2

L

(5)

Where a1 is a proportionality factor (3.285), к the bottom breaking coefficient, Ґ the wave decay constant, Hb the breaking wave height, hb the wave breaking depth and XL the average displacement of sedimenting particles until dynamic stability is achieved. As seen, there are models described by a single equation (Dean, 1987; Hallermeier, 1981), or using a group of them (Turker and Kabdasli, 2006), predicting reality with different degrees of accuracy. However, the expressions used to model parameter A have evolved to more complex equations, as Eq. (5). Moreover, in more complex formulations, the uncertainty of the inputs must be considered. Frequently, coastal engineers use the easier expressions such as Eq. (3) due to the difficulty of obtaining some of the parameters involved (breaking wave height) or the lack of a clear definition of others (median sediment size D50 or wave height). Regarding the mathematical formulation and the determination of models, several methodologies exist in scientific literature. The statistical software S-Plus2000 (1999) and SPSS12.0. (2003) can determine linear and multi-linear relations (Camacho Rosales, 2002). Another approach obtains non-linear relations by considering expressions depending on unknown coefficients which are calculated by minimizing an error function, which is usually the sum of the squared errors. The software tool Models can be used to perform an automatic search between non-linear models (Cortés et al., 2000; Villacampa et al., 1999), while the methodology implemented in Poli model (Verdú and Villacampa, 2008) generates families of models which are non-linear in the parameters. Also noteworthy the methodologies presented in NavarroGonzález and Villacampa (2012), Pérez-Carrió et al. (2009) and Villacampa et al. (2009), which are related to numerical modelling, enabling the generation of bi-dimensional and multidimensional numerical models as well as analysing their stability. The aim of this study was to generate a model for the parameter A of the potential function (Eq. (1)) that describes the equilibrium beach profile in the study area. To do this, first, the variables that may influence the formation of the profile were studied. Some of the considered variables are related to the maritime climate (wave height Hs,12,, period and probability of occurrence (frequency) of the most energetic, most frequent and perpendicular waves) and to the sediment properties (median sediment size (D50) and sample porosity). Finally, some mathematical models are generated, considering their sensibility to small changes in the experimental data and selecting those most suitable for the studied area.

2. Study area The studied geographical zone is the province of Valencia, Spain. Valencia is on the east of the Iberian Peninsula (39°28′30.7′′ 00°22′33′′W), and is enclosed in the Gulf of Valencia (also called the Valencian oval), the greatest morphodynamical unit on the

165

natural Spanish coast. The Valencian coast has a length of 96,690 m and is divided into 45 beaches. Among them are sandy and soft bed beaches, usually of great length and shallow waters, where the predominant processes are the accumulation of sediments composed of sand, gravel and silt (ECOLEVANTE, 2006). The study has been performed on 28 beaches not influenced by breakwaters or capes. Beaches are not isolated entities, they are connected to the rest of the coast and any change to one point affects other places in different ways. Therefore, beaches are conditioned by the entire coastal system. However, the coastal system can be divided into sectors with similar dynamic characteristics, that is, each section reacts to changes in a similar way. The coastal system also has offshore limits, related to the maximum depth where sediment can be moved by climatic processes. This is the depth of closure (DoC). However, the coastal system is not restricted to the zone under the influence of marine agents; it extends to zones that affect its behaviour, such as river basins and marine drains where sediments settle. Four rivers empty into the Gulf of Valencia: Serpis, Júcar, Turia and Palancia. These rivers have been used to distinguish five zones in the area of study (Fig. 1). These rivers are an important source of sediments, and allow the longitudinal and cross-shore classification of the sand beaches depending on the distance to the river mouth. However, the Valencia port, Sagunto port and Gandía port act as barriers to the longshore sediment transport. On the other hand, Mediterranean tides are not constant over time, they can be classified, by considering the sea type into diurnal, semidiurnal and mixed (ECOLEVANTE, 2006). They are also affected by meteorological factors, creating tidal surges greater (up to 75 cm) than those caused by purely astronomic tildes (between 20 and 30 cm).

3. Materials and methods In the developed methodology, the first section analyses the variables which are related to equilibrium profile generation (maritime climate and sedimentology), while in the second part, some mathematical models are generated to represent it. 3.1. Maritime climate A detailed analysis of the data which can be of interest when characterizing incoming swell in the studied zone was made. The data corresponding to the deep water swell has been obtained from the information provided by “Departamento de Clima Marítimo de Puertos del Estado” corresponding to the Valencia “REDCOS” 1619 buoy at a depth of 50 m with coordinates 0.20°W, 39.51°N. The data comprises information such as: wave height Hs,12 and wave period Ts,12 (wave height exceeded 12 h a year and its period), wave direction and probability of occurrence (frequency). Another factor to consider is the length of the period studied. For Kaiser and Frihy (2009), Karunarathna et al. (2009) and Romanczyk et al. (2005), the wave height must be representative of the period in which the profile is studied. The authors have analysed the wave heights for the period studied and the relationship between the increase in period and wave height. Data on which the study has been based includes values from 08-06-2005 to 0111-2013. Later, the methodology introduced by Aragonés et al. (2016) has been followed to obtain the data corresponding to deep water swells for two reasons: a) the data source, the buoy, is located in intermediate water and b) the deep water data shows greater directional uniformity to be used as variables in the beaches studied.

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Fig. 1. Valencia location. a) Palancia river mouth. b) Port of Valencia and Turia river mouth. c) Jucar river mouth. d) Gandia port and Serpis river mouth.

All the waves represented in the data set corresponding to intermediate water were translated to deep water using the methodology proposed by the Recommendation for Marine Works (ROM), allowing data consistency. From these data, the waves of higher energy, higher occurrence and perpendicular to the coast were obtained. The last one avoids diffraction of refractive effects caused by the presence of ports or capes. 3.2. Experimental data A matter of concern when developing this research was the analysis of the sedimentological parameters in the studied zone. According to Bodge (1992), Dean (1977) and Moore (1982) the shape of the profile depends only on the median sediment size (D50) or on the shape and weight of the grain. The objective was to analyse the main factors affecting the dynamics coast of the province of Valencia. The beaches studied include 63.52% of the total length of the coast of the province (61.4 km). For the 28 studied beaches, 393 samples of the dry beach (14 or 15 sample per beach) and 540 of the wet beach (19 or 20 samples per beach) up to the bathymetric – 12 m were analysed. The data come from a report published by Dirección General de Costas (ECOLEVANTE, 2006). The median sediment size (D50), real sample density (ρm), the material density (ρs) and the porosity (p) were calculated using the dry beach samples. Later, the value of D50 is obtained at the DoC (Birkemeier, 1985). The experimental data were analysed in order to choose the

variables most related to parameter A for the model defined by the potential function in the area (Aragonés et al., 2016). The sample extraction was done using the following procedure: using a Van Veen's grab the sample is extracted and saved in a bucket to be labelled. After being packed into bags the samples are transported in an icebox to the laboratory, where the granulometric tests are carried out (ECOLEVANTE, 2006). The size of 50% of the particles D50,weight has been considered the most useful parameter (Aragonés et al., 2016). 3.3. Modelling The correlations between the main independent variables (D50, D50,weight) on the dry beach and at DoC, porosity (p), wave steepness Ho/L0 of the most energetic, most frequent and perpendicular waves, the wave frequency (f) and the dependent variable (parameter A) were analysed using the statistical software SPSS. From the results several subsets of variables were selected to generate models. To avoid heterogeneity in the values, they have been typified using Eq. (6).

arithmetic,

t = ( b − μ) / σ where t is the typified value, b the variable value, average and s the standard deviation.

(6)

μ the sample

3.3.1. Numerical modelling In the study and modelling of some systems it is necessary to analyse and determine the relationship between the variables defined by Eq. (7) and of which only the experimental data are

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relative percentage error (δ), being similar to those calculated in the empirical fit done in Aragonés et al. (2016) for the studied area. The models were validated using data from beaches with similar characteristics to the experimental data, that is, they are sandy beaches without breakwaters or Posidonia oceanica and with orientations similar to the studied area. The validation beaches are: Deveses, Rebollo, Guardamar, Moncayo and Torre la Mata, situated in the province of Alicante, south of the study area, and at a distance less than 200 km from the area of study.

Table 1 List of beaches. No. Beach

No. Beach

No. Beach

1 2 3 4

11 12 13 14

El Rey Mareny San Lorenzo El Dosel

21 22 23 24

Bellreguard Miramar Piles Oliva-Terranova

5

Almardá Canet Berrenguer Alboraya El CabañalMalvarrosa Pinedo

15

25

Oliva-Pau Pi

6

El Saler

16

Tabernes de Valladigna Jaraco

26

7 8 9 10

La Dehesa El Recatí El Perelló Mil Palmeras

17 18 19 20

L'Ahuir Grao de Gandía Venecia Daimuz

Oliva L'Aigua Blanca Oliva Rabdels Oliva-Les Deveses

27 28

4. Results

known (Eq. (8)).

z=u( x1, x2,…. ,x n)

{z , x , x i

1 i

2 i ,

xi3…. ,xin

(7)

}

i= 1,2, … ., p

(8)

In the literature there are different methodologies to obtain the relationship between the models of the Eq. (7) from the experimental data of the Eq. (8). Thus, models can be analytically (mathematical equations) or numerically defined. The models numerically defined, are defined by their value in a finite number of points, from which the value at any point can be obtained. In this paper several methodologies were used to determine the relationship between parameter A and the sets of selected independent variables. From all the obtained models, the authors selected those with a lower volume error. First linear models were obtained using S-Plus2000 (1999) and SPSS12.0. (2003). Also mathematical equations were obtained using the methodologies developed by Cortés et al. (2000), Verdú and Villacampa (2008) and Villacampa et al. (1999). However, the use of these models produced substantial errors in the estimated values and/or the derived volume error. For this reason, the numerical models presented in Navarro-González and Villacampa (2012) and Villacampa et al. (2009) were used with results which improve volume error. The numerical methodology developed by Navarro-González and Villacampa (2012) and Villacampa et al. (2009) generates n-dimensional representation models for the Eq. (7). The methodology is based on the definition and generation of a geometric model of finite elements (Villacampa et al., 2009). The experimental data are normalized to the n-dimensional n hyper-cube given by Ω=⎡⎣ 0,1⎤⎦ . Each interval [0,1] is divided in c subintervals (c is called the complexity of the model (Comp)). This n generates a set of c n elements and ( c + 1) nodes where an estimation of the relationship (7) can be calculated. To each model, a sum of squared errors (SSE) corresponds, that can be calculated and used as an index of quality of fit for the model, with a dependence SSE(c). Comparing the values of SSE(c) an optimum value for c* can be selected under the condition: SEE c* = minc SEE( c ). Two kind of models were obtained: one considering D50 on dry beach with and without porosity (p), and another using D50 at the DoC (Birkemeier, 1985). The other independent variables were: wave height “H0”, wave length “L0” and frequency “f” for deep water swell perpendicular to the coast. For the selection of the most appropriated models, several parameters were studied: the minimization of the volume error and relative average squared error (MSE/Var), mean error (ϵ) and

( )

167

Following the methodology explained, the results of the research will be detailed. First, the variables which affect the value of parameter A, such as the wave height Hs,12 and the median sediment size (D50), were studied. Subsequently, the numerical models which provide best results for the studied area are presented. Finally, the results for the validation of the selected models are explained. The beaches are represented in the following figures from North to South using the order shown in Table 1. 4.1. Maritime climate To obtain a wave height representative of the studied period, the values of Hs,12. were taken from 1 to 7 years for each direction and time interval in the studied beaches (Fig. 2). The curve tends toward an asymptote as the number of years increases. Also, the study of maritime climate includes a comparison of the most representative swells for each area. The most frequent, the most energetic and perpendicular swells were considered. The wave steepness and frequency of deep water waves are represented in (Fig. 3). In order to compare units of different dimensions, all results obtained have been typified, comparing them with the experimental results for parameter A. Fig. 3 shows that in 82.05% of cases the wave steepness Hs,12/L0 coincides with the most frequent and energetic swells. 87.17% of the most energetic and frequent swells have the same value. This contrasts with the observed variability of parameter A in the study area. However, for the results of swell perpendicular to the beach, the most repeated value is only 30.76% higher than parameter A. Some correlation between parameter A and the frequency and Ho/ Lo perpendicular to the beach is also seen, since when the frequency decreases and steepness increases, parameter A increases (21–28), and when frequency increases and steepness decreases, parameter A decreases (5–14). 4.2. Characteristics of the beach sediments Using the samples obtained in ECOLEVANTE (2006), the median sizes (D50) (average, minimum and maximum) on the dry beach and the median sizes (D50) at DoC (Birkemeier, 1985) and the porosity of the dry beach samples were calculated. The results are shown in Table 2. The properties of the sediment are represented by the porosity and the D50. As in the last point, the data is typified and compared with the results obtained for the dependent variable (parameter A). In Fig. 4, some relations between parameter A and the porosity and D50 at the dry beach can be observed. When D50 on the dry beach increases, its porosity likewise increases and parameter A decreases. However, no clear relationship can be seen between D50 at DoC and parameter A.

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Fig. 2. Variation of wave height Hs,12 in Valencia over the studied period for different directions.

Fig. 3. Typified values for wave heights for the most energetic, frequent and perpendicular swells and parameter A. Table 2 Average, maximum and minimum values of D50 on dry beach, D50 at DoC and porosity at the beaches of study. D50

D50, Dry Beach (mm)

D50, DoC (mm)

Porosity of Dry Beach

Average Maximun Minimun

0.239 0.325 0.167

0.154 0.204 0.116

0.6025 0.7180 0.5500

4.3. Numerical modelling Before obtaining the models, the correlations between the variables were studied using the statistical software SSPS. The results showed that parameter A had a high correlation with the variables which depend on swell energy represented by steepness (H0/L0) and frequency, being the correlation between the D50,weight and the sample porosity (Fig. 5) much lower. After analysing all the variables and the study of correlations, numerical models were generated. The independent variables used in the models are: a) Related to the swell energy, the wave steepness (H0/L0) and its

associated frequency, for the wave height Hs,12 perpendicular to the coast, were considered. b) Related to the sediment, two kinds of models, one including the porosity (p) and the other without it, were obtained due to their minimal variability in all samples considered. c) The median sediment size (D50) used for the modelling was D50,weight taken as representative of the sample (Aragonés et al., 2016). Although the correlation between D50 and the porosity in parameter A is low, it is included, leaving the task of determining its influence in profile generation to the models. Different models were generated with D50 at the dry beach and with D50 at DoC. Fig. 6 shows a good correlation between the experimental values and those calculated with the numerical models for parameter A. Aunque existe una diferencia aparente entre los valores experimentales y estimados del parámetro A (Fig. 6a), esto resulta en una variación insignificante en EBP (Fig. 6b). Fig. 7 shows the Pearson coefficients of each of the models and complexity (Comp) that were studied. Moreover, an analysis of the stability of the models was performed considering a Monte Carlo simulation. Changes in the

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Fig. 4. Variability of D50 and the porosity in dry beach and D50 at DoC related with the parameter A (typified values).

Fig. 7. Pearson's coefficient for the models with D50 on dry beach with and without porosity and at DoC for complexities 50, 70 and 100.

Fig. 5. Correlation between the independent (D50, Porosity (p), wave steepness (H0/ L0) and frequency (f)) and the dependent variable (parameter A).

experimental data of up to 20% produce average variations in the models of less than 0.3% (Fig. 8). As complexity increases stability decreases, being all models very stable. Fig. 9 shows the volume error for each model of D50 on the dry beach with and without porosity and D50 at DoC. As can be seen, the higher the complexity, lower the volume error, being the models with D50 on the dry beach and porosity that best results achieved.

Following the methodology proposed by Aragonés et al. (2016), for each model the corresponding relative average squared error (MSE/Var), mean error (ϵ), and the relative percentage error (δ) were determined (Table 3). In all the considered models these errors were similar to those obtained in the experimental fit of parameter A in the study of Aragonés et al. (2016). 4.4. Results validation The generated models were checked using data from other beaches different to those in the set corresponding to the

Fig. 6. Numerical model for complexity 70 with D50 (dry beach and DoC) and porosity (p).

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Fig. 8. Stability for the models for D50 at dry beach (a), D50 at dry beach and porosity (b) and D50 at DoC (c).

experimental dataset. The volume error for the four models can be seen in Fig. 10. As can be seen, best models are those with D50 on dry beach and porosity, being the volume error committed by the best model (D50 on dry beach and porosity for complexity 100) 272 times higher (8.12 m3) than the error of real data. While the Dean function makes an error of 30.66 m3 higher. The relative average squared error, mean error, and relative percentage error were also obtained as can be seen in Table 4. The Dean function does not fit well, with MSE/Var being multiplied by 5, and the model with D50 at DoC is which best fits.

5. Discussion The parameter A of the potential function proposed by Bruun (1954) and developed by Dean (1991) is the variable most frequently used (for easy use and few necessary variables) by coastal engineers as a tool to calculate the shape of the equilibrium beach profile, focused on, for example, estimating the volumes needed for a regeneration. However, being a general function that takes no account of the local factor, many volume errors are committed in nourishments (as was demonstrated by Aragonés et al. (2016)). There are numerous research papers related to the calculation of parameter A, however, no method for accurately developing a practical way of obtaining the variables related to the calculation exists. The equilibrium beach profile is generated by the incident swell, giving it a concave shape. The model that achieves the best fit for the analysed beaches was obtained using a potential function (Aragonés et al., 2016). The next step is to obtain a model for parameter A that appears in the function described in Eq. (1). Most of the models use simple functions, however, the most recent

models (Turker and Kabdasli, 2006) have a complex expression that makes difficult the measurement of the independent variables due to the uncertainty in their definition. In this study, the first attempts at modelling parameter A were made using linear functions and models defined by mathematical equations. However, the obtained errors when considering their use to estimate the experimental data were significant, therefore numerical models were tested. The generated numerical models offer a great advantage, its ease of use compared to other models that can be found in the literature. Generated models are intended to be as simple as possible, using variables obtained easily and inexpensively, and which take into account local factors. It is intended that these models will be easy tool for coastal engineer. In addition, it is intended that the error made, when using these models to determine the equilibrium beach profile in future nourishments, is as small as possible to reduce the costs. From the results of the analysis of the independent variables relative to the maritime climate it can be deduced that: a) The values of Hs,12 must be taken during a minimum period of seven years for the studied area (Fig. 2). After this period the curve trends to an asymptote and the error made in characterizing the wave climate is minimal. b) The most suitable swell direction for characterizing parameter A is the perpendicular to the coast, as can be seen in Fig. 3, since the waves for the most energetic and most frequent swell coincide in 82.05% of cases, which contrasts with the variability of parameter A. Fig. 1 shows bathymetric lines almost parallel to the coast, therefore the perpendicular direction avoids the effects of refraction and diffraction, being only affected by shoaling.

Fig. 9. Differences in the volume error between the function and the numerical models.

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Table 3 Relative mean square error, mean error and relative percentage error for each model. MSE/Var (%)

Real data

Mean error (m)

Relative percentage error (%)

Mean (%)

Maximun (%)

Minimun (%)

Mean

Maximun

Minimun

Mean (%)

Maximun (%)

Minimun (%)

1.22

6.21

0.09

0.343

0.802

0.101

6.5

28.3

1.8

Comp50

D50.Dry beach and Porosity D50.Dry beach D50.DoC

1.25 1.29 1.27

6.22 6.23 6.22

0.11 0.10 0.10

0.350 0.359 0.354

0.818 0.956 0.819

0.104 0.105 0.105

6.7 6.8 6.8

28.4 28.8 28.8

1.9 1.9 1.9

Comp70

D50.Dry beach and Porosity D50.Dry beach D50.DoC

1.24 1.26 1.25

6.22 6.22 6.22

0.11 0.11 0.10

0.348 0.353 0.350

0.804 0.858 0.807

0.104 0.105 0.105

6.7 6.7 6.7

28.4 28.8 28.8

1.9 1.9 1.9

Comp100

D50.Dry beach and Porosity D50.Dry beach D50.DoC

1.24 1.24 1.24

6.22 6.22 6.22

0.11 0.11 0.10

0.347 0.349 0.348

0.803 0.822 0.806

0.104 0.105 0.105

6.7 6.7 6.7

28.4 28.8 28.8

1.9 1.9 1.9

Fig. 10. Volume error for real data, D50 at dry beach, D50 at DoC and Dean (1991).

Previous to the study of the models, the variables which can affect the value of parameter A were analysed and the correlations between them were obtained. The wave steepness (0.627) and the swell frequency (0.446) are the independent variables which best correlate with variable A (Fig. 5). When considering the formulas used by Boon and Green (1988), Dean (1991), Kriebel et al. (1991), Pilkey et al. (1993), Rodriguez-Peces et al. (2011b) and Vellinga (1984) there is a tendency to relate the equilibrium beach profile to the median sediment size (D50). If the EBP is the result of the sediment positioning at the different depths caused by the swell, it would be necessary to determine the exact point at which obtain the sample. However, none of the authors provides a clear description of a method to measure this field parameter, which presents a high sensibility to the errors in all the formulations, being, in this way, the origin of errors in the models (Aragonés et al., 2016). The study of the characteristics of the sediments that form the profile gives the following results: a) The porosity, as can be seen in Fig. 4, shows little variability compared with parameter A, with a low correlation (0.192) as Fig. 5 shows. b) The D50 is the most frequently parameter use by researchers to obtain the equilibrium profile. Fig. 4 shows that there is not a direct relation between this variable measured on the dry beach and/or at the DoC, and the value of A. Fig. 5 shows how the correlation between them is the lowest among the

dependent variables (0.113). From the correlation results it can be deduced that the wave steepness and the wave frequency must be included in the models, and the porosity and the D50 do not show any direct correlation. However, the existence of a low correlation, only implies that the linear relationship between the variable and the parameter A does not exist, does not mean that this variable does not influence indirectly. As we can see when we analyse the models, they improve when we add the porosity although the correlation was low ( 0.192), and moreover, the model with D50 on dry beach fits better than the model with D50 at DoC, even though the correlation between D50 at DoC and parameter A was higher (0.411 vs 0.113). Once variables were determined, mathematical models were obtained. The use of linear regression shows that the model must be nonlinear, due to the low value for the determination coefficient (0.3). To improve the fit between the experimental and the estimated values, obtaining better values for the volume error, models were calculated based on the methodologies of NavarroGonzález and Villacampa (2012) and Villacampa et al. (2009). Families of numerical models including the D50,weight, porosity (p), wave steepness (H0/L0) and frequencies from swell perpendicular to the coast were studied: 1. Steepness and frequency of the swell perpendicular to the coast, with porosity and median size (D50) on dry beach.

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Table 4 Errors in the validation beaches.

MSE/Var (%)

Mean error (m)

Relative percentage error (%)

Rebollo

Guardamar

Moncayo

Torre la Mata

Les Deveses

Real data Function of Dean

2.60% 10.59%

8,93% 9.16%

4.66% 21.77%

8.06% 8.17%

1.41% 89.59%

Comp70

D50.Dry beach and Porosity D50.Dry beach D50.DoC

4.49% 6.68% 2.66%

9.09% 10.82% 12.32%

14.63% 6.50% 9.14%

20.59% 16.51% 13.06%

1.76% 6.77% 5.04%

Comp100

D50.Dry beach and Porosity D50.Dry beach D50.DoC

4.71% 6.69% 3.91%

9.08% 10.35% 9.15%

14.66% 6.25% 16.02%

20.63% 17.17% 8.07%

1.86% 6.75% 6.09%

Real data Function of Dean

0.065 0.798

0.160 0.978

0.138 1.579

0.154 0.593

0.110 3.570

Comp70

D50.Dry beach and Porosity D50.Dry beach D50.DoC

0.520 0.634 0.395

0.974 1.063 0.977

1.294 0.863 0.731

0.947 0.848 0.597

0.500 0.982 0.447

Comp100

D50.Dry beach and Porosity D50.Dry beach D50.DoC

0.532 0.635 0.485

0.974 1.040 0.966

1.296 0.846 1.355

0.948 0.865 0.593

0.514 0.980 0.931

6.4% 13.0%

13.6% 13.8%

9.9% 21.3%

12.2% 12.3%

5.8% 46.4%

Real data Function of Dean

Comp70

D50.Dry beach and Porosity D50.Dry beach D50.DoC

8.5% 10.3% 16.0%

13.7% 15.0% 128.5%

17.5% 11.7% 104.6%

19.3% 17.3% 56.8%

6.5% 12.8% 71.5%

Comp100

D50.Dry beach and Porosity D50.Dry beach D50.DoC

8.5% 10.3% 7.9%

13.7% 14.6% 13.8%

17.5% 11.4% 18.3%

19.3% 17.6% 12.2%

6.5% 12.7% 12.1%

2. Steepness and frequency of the swell perpendicular to the coast and median size (D50) on dry beach. 3. Steepness and frequency of the swell perpendicular to the coast and median size (D50) at the DoC. The results can be seen in Fig. 6, where the dependent variable A is compared with the values estimated by the numerical models with complexity 70. The error is small, verifying Pearson's coefficient (Fig. 7). The variable is very dependent on small changes (Aragonés et al., 2016). For this reason, parameters such as the stability of the models, the volume errors, the mean error, the relative percentage error and the relative average squared error were studied following the procedure presented in Aragonés et al. (2016). The results for models 2 and 3 demonstrate their stability, as changes up to 20% in data produce average changes of 0.25% in the estimated values (Fig. 8). The volume error (Fig. 9) for the three models shows models 1 and 2 performed best (complexities 70 and 100). By observing the mean error, the relative percentage error and the relative average squared error, it can be seen that their behaviour is similar to the real data error (Table 3). Therefore, models 1 and 2, with complexities 70 and 100, are the most suitable. Finally, beaches outside the studied area were validated. The results of this validation demonstrate that models with complexity 70 are the best option. Fig. 10 shows that the volume error is high, however, it continues to be better than in the results obtained by expressions 1 and 2 (Dean, 1987).

6. Conclusions To obtain the parameter A of the potential function that describes the equilibrium beach profile, three numerical models were considered: two for D50 at the dry beach with and without porosity (since this parameter shows low variability in the area studied) and another for D50 at the depth of closure. The linear models were discarded due to their bad fitting properties. The criteria used for model selection were: – The values of the volume error, the Pearson coefficient (R2), the mean error (ϵ), the relative percentage error (δ) and the relative average squared error (MSE/Var). – The stability of the model when parameters are changed by up to 20% in a random Monte Carlo simulation. – The validation of the models at beaches with similar characteristics outside the study area. Considering these criteria, the selected model was that of the complexity 70, and with the following variables: steepness and frequency of swell perpendicular to the coast, the porosity and the median size (D50,weight) on the dry beach. This model has a Pearson coefficient of 0.95, its stability is less than 1% when data is modified using a Monte Carlo simulation with changes of up to 20%, and the used variables are easy to obtain. However, the model gives unsatisfactory results when it is used to estimate values outside the study area, despite being better than those obtained from other known models. For this reason, to estimate the behaviour in other places, the best option is the use of

L. Aragonés et al. / Ocean Engineering 123 (2016) 164–173

sample data obtained from that area. The results presented are useful as a basis for a general method to be used in beaches similar to those from which the experimental data has been obtained. The general approach, which does not account for local variability, provides inaccurate results. The most frequently used functions have a high sensitivity to changes in input parameters. In addition, emphasizes that a relatively simple model was obtained employing local variables which improved the model results. This is an advantage for the working coastal engineer, compared to more complicated models that require more parameters, some of which may be poorly defined and/or expensive to obtain. Therefore, the proposed model is a user friendly tool that can be used to determine the equilibrium beach profile on sandy beaches with similar characteristics to those studied beaches, without committing big volume errors in future regenerations.

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