- Email: [email protected]
Wave runup prediction using M5′ model tree algorithm
Ocean Engineering 112 (2016) 76–81
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Wave runup prediction using M50 model tree algorithm S. Abolfathi a,n, A. Yeganeh-Bakhtiary b,c, S.M. Hamze-Ziabari b, S. Borzooei d a
Flow Measurement & Fluid Mechanics Research Centre, Coventry University, CV1 5FB Coventry, UK School of Civil Engineering, Iran University of Science & Technology (IUST) , Tehran 16884, Iran Civil Engineering Program, Faculty of Engineering, Institut Teknologi Brunei (ITB), Brunei Darussalam d Politecnico di Torino, DIATI, C.so Degli Abruzzi 24, 10129 Torino, Italy b c
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2015 Accepted 8 December 2015
In recent years, soft computing schemes have received increasing attention for solving coastal engineering problems and knowledge extraction from the existing data. In this paper, capabilities of M50 Decision Tree algorithm are implemented for predicting the wave runup using existing laboratory data. The decision models were established using the surf similarity parameter (ξ), slope angle (cot α), beach permeability factor (Sp), relative wave height (H/h), wave spectrum (Ss) and wave momentum flux (m). 451 laboratory data of the wave runup were utilized for developing wave runup prediction models. The performance of developed models is evaluated with statistical measures. The results demonstrate the strength of M50 model tree algorithm in predicting the wave runup with high precision. Good agreement exists between the proposed runup formulae and existing empirical relations. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Wave runup Model tree M50 algorithm Nearshore hydrodynamics
1. Introduction Runup of waves on beaches and coastal structures has been studied extensively by coastal engineers, due to the importance of runup height in design of coastal structures (U.S. Army Corps of Engineers, 2002). Nearshore zone is the most dynamic part of the coastal area where wave activities result in complicated hydrodynamic behaviours with varying time and length-scale. One of the most important parameters in dealing with the nearshore hydrodynamics is the wave runup height. During the uprush of water on the beach/structure face or so-called wave runup, the remaining kinetic energy of incident broken waves dissipates and transforms into potential energy as it runs up. Wave runup height is a major concern for design of coastal infrastructure to minimize the occurrence of overtopping. Wave runup height depends on number of parameters such as the incident wave climate and specifically wave height (H), material and the slope. Fig. 1 is the definition sketch of wave runup. The wave runup height is a complex time-varying function of intersection location between the ocean and the beach/structure which depends on nonlinear transformation of waves in breaking zone, refraction of waves, porosity and roughness of slope materials, topography of the nearshore zone, three-dimensional n
Corresponding author. E-mail address: [email protected] (S. Abolfathi).
http://dx.doi.org/10.1016/j.oceaneng.2015.12.016 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
velocity field in the shallow water column and turbulent kinetic energy budget of the incident waves. Therefore, due to such complications in nearshore hydrodynamics and morphodynamics and also due to lack of precise field measurements, our understanding of underlying physical mechanisms of nearshore processes is limited and there is no rigorous theoretical approach to predict the runup height. Hence, the existing formulae for runup prediction are only based on empirical approaches of experimental data.
2. Previous studies Previous studies of the wave runup can be generally categorized to monochromatic and irregular wave conditions based on laboratory measurements. This section aims to review the existing widely accepted approaches to determine the wave runup for regular and irregular wave conditions. 2.1. Regular wave Hunt (1959) studied wave runup on smooth and rough bed conditions and proposed empirical relations for runup prediction based on the shape of breaking wave in surf zone (Eqs. (1) and (2)). Ru 3 H
ð1Þ
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Ru2% ¼ 4:5 0:2ξop for 2:5 r ξop r 9:0 H mo
Incident wave
z R x
y α
Fig. 1. Schematic sketch of wave runup.
Ru tan α R ¼ rffiffiffiffiffiffiffiffiffi ; H ¼ ξ H H
ð2Þ
L0
where Ru is the maximum vertical runup from still water level, L0 is the deep water wave length [L0 ¼ 2gπ T 2 ], α is the beach slope, H is the wave height and ξ is the surf similarity or Iribarren number. Eq. (1) predicts the runup height for surging or standing waves on steep slope and Eq. (2) is valid for spilling or plunging breaker type on mild slope. Both Eqs. (1) and (2) are valid for impermeable beach type. 2.2. Irregular waves One of the first widely used formula to predict irregular wave runup is proposed by Wassing (1957) which relates the runup to the effective wave height of incident waves (H s ) and the bed slope ( tan α) (Eq. (3)). Ru2% ¼ 8 H s tan α; tan α o
1 3
ð3Þ
Ru2% represents the wave runup height exceeded by highest 2% of runup and H s is the average of highest 1/3 waves in an irregular wave terrain. Eq. (3) is only valid for mild uniform slopes. Battjes (1974) modified Hunt (1959) formula for irregular wave conditions (Eq. (4)): Ru2% ¼ C m ξom Hs
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 1=3 =Lom ; Lom is wave length in deep where ξom ¼ tan α= water using mean irregular wave period (Tm). Constant Cm varies between 1.33–2.86. Ahrens (1981) performed laboratory measurements for irregular wave runup on impermeable bed with the slope range of tanα ¼ ¼ to 1 and proposed Eq. (5) for runup height of irregular waves. ! !2 R2% H mo H mo ¼ C1 þ C2 ð5Þ þ C 3 H mo gT 2p gT 2p C1, C2 and C3 are the coefficients derived from the best fitting method of the data, Hmo is zero-moment wave height and Tp is the wave period associated with spectrum peak frequency. For spilling and plunging breaker waves, on impermeable bed and slope range of 1/1–1/4, Ahrens (1981) derived Eq. (6) using deep water Iribarren number ξop , based on spectrum peak frequency wave period (Tp). 2:26ξop R2% for ξop r 2:5 ¼ H mo ð1 þ 0:234ξop Þ
ð6Þ
Coastal Engineering Manual (U.S. Army Corps of Engineers, 2002) employed Ahrens (1981) data and proposed two equations for irregular wave runup (Eqs. (7) and (8)). Ru2% ¼ 1:6ξop for ξop r 2:5 H mo
ð8Þ
Mase (1989) conducted series of laboratory tests for irregular wave runup measurements with the wave steepness range of 0.007–0.07 and slope range of 1:5–1:30. Eqs. (9)–(11) summarize the formulae proposed by Mase (1989) for the maximum runup (Rmax ), the 2% average runup (R2% ) and the mean runup (̅R ).
Runup level
Rundown level
77
ð7Þ
Rmax 0:77 ¼ 2:32ξ H0
ð9Þ
R2% 0:71 ¼ 1:86ξ H0
ð10Þ
R 0:69 ¼ 0:88ξ H0
ð11Þ
Van der Meer and Stam (1992) employed regression method and proposed runup based on an extensive series of irregular wave runup measurements. They examined the effects of slope angle, permeability of slope, incident wave climate and spectral shape (SS) on the wave runup. Van der Meer and Stam (1992) performed most of their measurements with a Pierson-Moskowitz spectrum, however, some data recorded with wide spectrum and narrow spectrum. Van der Meer and Stam (1992) stated that use of mean period (Tm) for evaluating the Iribarren number (ξom) gives similar maximum runup, unless the spectrum is very narrow. Van der Meer and Stam (1992) empirical model combined the effect of wave period, wave height and slope angle into Iribarren number and did not look at these parameters independently. Also, the effect of spectral shape was not considered on the runup prediction in their model. In the current study, the spectral shape was employed as a quantitative input parameter for wave runup prediction (Section 4). Hughes (2004) used a new concept of wave momentum flux (Mf) to improve empirical correlations for waves and nearshore coastal processes. He employed Fourier approximation wave theory and proposed an empirical relation for the wave momentum flux (Eq. (12)). ! ! A1 Mf h ¼ A0 ð12Þ gT 2 ρgh2 max where ρ is the water density, h is the water depth, g is the gravitational acceleration, A0 and A1 are the empirical coefficients as a function of relative wave height parameter (H/h). The wave momentum flux parameter was obtained for the experimental data of Granthem (1953), Saville (1955), Ahrens (1981) and Mase M (1989) and used as input parameter ( f 2 ¼ m) in development of ρgh
M50 MTs (Section 4). Table 1 summarizes the empirical relations proposed for all the data used in this paper.
3. Soft computing approach In the last decade, there has been a growing trend in the use of Decision Tree (DT) algorithms such as Classification And Regression Trees (CART) and Model Trees (MTs). Decision trees are a Machine Learning (ML) technique with the capability of discovering and extracting the knowledge from a data set and derive a set of rules amongst data set parameters (Solomatine and Dulal, 2003). Model trees (Quinlan, 1992) fall into ML category and are promising numerical prediction method. The efficiency and robustness of MT algorithms in prediction has been well proved (Solomatine and Dulal, 2003; Bhattacharya and Solomatine, 2005). A MT is an inverted tree with a root node located at the top of the tree and the leaves at the bottom. The data instants are first
78
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Table 1 Summary of empirical runup relations proposed based on this study data. Wave action
Study
Proposed runup relation
Irregular
Ahrens (1981)
Ru2% H mo
¼ 1:6ξop
ξop r 2:5
Ru2% H mo Ru2% H mo Ru2% Hs Ru2% Hs Ru2% Hs Ru2% Hs Ru2% Hs
¼ 4:5 0:2ξop
2:5 r ξop r 9
¼ 1:86ðξop Þ0:71
ξop r 3
tan α ¼ 1:5; 1:10; 1:20 & 1:30
¼ 0:96ξm
1 o ξm r 1:5
Impermeable
¼ 1:17ðξm Þ0:46
ξm 4 1:5
tan α ¼ 1:2 1:6
¼ 0:96ξm
1 o ξm r 1:5
Permeable
¼ 1:17ðξm Þ0:46
1:5 o ξm r 3:1
tan α ¼ 1:1:5 1:3
¼ 1:97
3:1 o ξop o 7:5
Mase (1989) Van der Meer and Stam (1992)
Hughes (2004)
Ru2% h
h i1=2 M ¼ 1:75 1 e j1:3cot αj tan α f 2
Ru2% h
¼ 4:4ð tan αÞ0:7
h
1 4r
Plunging/spilling waves
tan α r 11 ðHmo =LP o 0:0225Þ 1 2 5 r tan α r 3ðH mo =LP 4 0:0225Þ
Plunging/ spilling waves
tan α ¼ 1:5 1:30
0:35 r H=h r 1
Impermeable tan α ¼ 1:1 1:10
Non-breaking, surging & collapsing waves
i1=2
Mf 2
ρgh
h Mf i1=2 0:7
¼ 4:4 tan α 2 pgh h i1=2 R m ¼ 3:84 tan α 2 h
Hughes (2004)
tan α ¼ 1:1 1:4 Permeable & Impermeable Impermeable
pgh
Ru2% h
Regular
Accepted range
ρgh
y
Training Dataset New instance
y >z No
y
y
Yes No
Model 1
Model 2
Model 4
y >z
3
y
Model 2
Model 3
4 No
Yes
Yes
2
Yes
Model 1
2
No
Model 6
Model 4
No
Model 5
1
Model 6
Model 3
Model 5 1
Output
2
3
4
5
6
y
1
Y (output) Fig. 2. Prediction procedures for a new instance in M5 model tree. Fig. 3. Splitting space in M5 model tree algorithm.
entering to the root node as an input. In the MT root a test is carried out on the input data and the result of this test causes the tree to be split into branches, each of these branches are representing a possible answer. The MT will continue to split into branches until all the data in different classes have been classified. Model trees (MT) are employed to solve regression and classification problems. Model trees (MTs) and regression trees are a form of DT which has been developed for regression problems (Quinlan, 1992). The key difference between MTs and regression trees is that the leaves of regression trees produce a constant value, while MTs yield into linear models in their leaves (Samadi et al., 2014). These linear models enable MTs to predict numeric values for a given data sample. MTs are efficient for large data set and have higher prediction accuracy in comparison with regression trees (Solomatine and Siek, 2006). 3.1. M50 model tree A complex modelling problem can be divided into a number of simple sub-tasks and the answer would be combining the solution of all these tasks. M5 MTmethod divides the data space into smaller subspaces by divide-and-conquer method (Bhattacharya and Solomatine, 2005). This method use the hard (i.e. yes–no) splits of input space into regions and narrowing the input parameter space into subspaces and builds a linear regression model in each of these subspaces. As a result of this splitting procedure, M5 MT produces a hierarchy (a tree) with splitting rules in nonterminal nodes and the expert models in leaves (Solomatine and Dulal, 2003). Fig. 2 illustrates M5 MT splitting procedures to make a prediction for a new instance.
Fig. 3 shows how M5 MT algorithm split the input space (ψ 1 ψ 2 ), each model tree's leaf represents a linear regression model (Y ¼ ζ 0 þ ζ 1 ψ 1 þ ζ 2 ψ 2 ). M5 algorithm was originally proposed by Quinlan (1992) and further developed by Wang and Witten (1997) to so-called M50 algorithm. The M50 algorithm operates based on partitioning the input space into many subspaces and fitting a linear regression model (LRM) in each of the sub-spaces by using if-then rules. M50 builds a regression tree by recursive splitting based on treating the standard deviation (σ ) of the class values that reach a node as a measure of the error at the node. The attribute that maximizes the expected error reduction is selected for splitting at the node. Quinlan (1992) proposed the standard deviation reduction (σ R ) as a measure of the error at each node (Eq. (13)).
σ R ¼ σ ðT Þ
X jT i j i
jT j
σ ðT i Þ
ð13Þ
where T is the set of instances that reach the node, Ti are the sets resulted from splitting the node according to a given attribute and split value (Wang and Witten, 1997). If the output values of all instances that reach the node have slight variation or when few instances remain, the splitting procedure will automatically terminate. The splitting process may frequently impose too much accuracy on the training data set and hence may over-fit the training data which could result in poor performance of MT when running for unseen data (Bhattacharya and Solomatine, 2005). In this study a pruning technique is employed to overcome the problem of over-training the dataset. Pruning technique basically merges some of the lower sub-trees into one node. Pruning the MT can cause sharp discontinuities between the adjacent linear
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
models in the leaves of the pruned tree. Smoothing procedures has been considered to compensate for the discontinuities of pruned tree and improve the accuracy of MT predictions. In the present paper, capability of M50 model tree algorithm for prediction of the wave runup height is explored. 3.2. Statistical error parameters To evaluate the performance of M50 MT, statistical measures are utilized. In this paper, Correlation Coefficient (R2 ), Scattering Index (SI), bias and RMSE are calculated for wave runup predictions proposed by M50 MT. Eqs (14)–(17) present the statistical error parameters, which have been considered in this study. Bias ¼
N X
1 p oi Ni¼1 i
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X 2 p oi RMSE ¼ t Ni¼1 i SI ¼
RMSE Om
R2 ¼
100
ð14Þ
ð15Þ
ð16Þ
PN
i ¼ 1 ðOi Om Þðpi pm Þ 0:5 P 0:5 N 2 2 i ¼ 1 ðOi Om Þ i ¼ 1 ðpi pm Þ
PN
ð17Þ
Oi represents the observed value, P i is the predicted value, N is the number of observation data, O m is the mean observation value and P m represents the mean prediction value.
4. Data set description The wave runup is predicted by application of M50 MT algorithm. MTs were trained and tested using six effective parameters in the wave runup including surf similarity parameter (ξ), slope (cot α), bed permeability (SP), relative wave height (H/h), wave momentum flux (m) and wave energy spectrum (SS). To achieve the highest possible accuracy and best configuration of input parameters atrial & error procedure utilized. The data set implemented in this study consists of 451 runup data including monochromatic and irregular wave conditions. The irregular wave runup data were collected from Ahrens (1981), Mase (1989) and Van der Meer and Stam (1992) laboratory measurements. The runup data for monochromatic waves are obtained from Hughes (2004). Irregular wave runups on smooth permeable and impermeable slopes have been studied using Ahrens (1981). For the irregular wave runup on gentle impermeable slopes, Mase (1989) laboratory records with the slope range of 1/5–1/30 and Van der Meer and Stam (1992) runup data with slopes range of 1/ 1.5–1/4 have been utilized in the modelling procedures. For the regular wave runup on smooth, impermeable slopes with the slope range 2/3 r tanα r1/30, Hughes (2004) dataset have been utilized which include existing monochromatic wave runup measurements by Granthem (1953) and Saville (1955). The input variables and data range used for the development of M50 MTs are shown in Table 2. Permeability of slope material is a geomorphological parameter which could play a major role in determining the wave runup height. Beaches and structures made from porous materials (e.g., gravel beach) could results in higher kinetic energy dissipation and therefore decreasing runup height. This paper employed Van der Meer and Stam (1992) methodology to present permeability as a quantitative parameter (SP) based on the beach materials. Fig. 4
79
Table 2 Modelling parameters & range. Parameter Surf similarity parameter Permeability Bed slope Relative wave height Wave momentum flux Wave energy spectrum Relative wave runup Relative wave runup
ξop SP cot α H/h m SS Ru2%/H Ru2%/h
Variable type
Data range
Input Input Input Input Input Input Output Output
0.13–18.21 0.1–0.6 1–30 0.06–1.48 0.01–3.43 0.00–1.00 0.46–4.75 0.02–1.2
shows the permeability conditions that were considered for the data of this study. The effect of wave spectral shape (SS) on the wave runup is evaluated with M50 MTs as a quantitative input. Narrow band spectrum is assumed to be 1; Pierson-Moskowitzis 0.5 and wideband spectrum is assigned to 0. Although Iribarren number comprises the bed slope parameter (cot α), the wave runup height is a function of both Iribarren number and slope angle due to the interactions between individual runup bores (Bakhtyar et al., 2008). When a wave hits coastal structures or reaches the shoreline and transforms its kinetic energy into potential by moving up the beach/structure face, before it complete the runup cycle, the next wave will hit the shoreline. For the second wave two possible scenarios exist, either the wave overtakes the first wave during its uprush or collides with the first wave during the backwash. Such interactions between the incident and reflected waves always exist. Therefore, maximum wave runup is not corresponding to the uprush of the highest wave in the train. This behaviour is directly influenced by the foreshore slope. For the mild slopes the water film have longer time to uprush and backwash, however, in the case of steeper slope the runup time for each wave is more limited (Bakhtyar et al., 2008). Hence, for the irregular wave climate with complex wave interactions the effect of slope in wave runup prediction is significant. Ahrens (1981) and Van der Meer and Stam (1992) have studied relatively sharp slopes and Mase (1989) recorded the runup data for mild slopes (Table 3). Therefore, the data used in this paper cover wide range of slopes and both ξ and cotα parameters were considered separately in the modelling procedures to account for the effect of wave interactions on the wave runup.
5. Evaluation of M50 model trees predictions M50 MT algorithm is used for predicting the wave runup height. 451 laboratory runup data have been gathered and nondimensionalized for the purpose of developing model trees. Wave runup predicted based on dimensionless 2% wave runup. Two sets of MTs were developed to predict the relative wave runup based on significant wave height (Ru2%/H) and water depth (Ru2%/h). Selection of appropriate combinations of inputs which produce the best prediction, carried out by a trial and error procedure, as preferred in most MT applications (Bhattacharya and Solomatine, 2005; Kumar and Singh, 2010 and Samadi et al., 2014). Many M50 MTs were constructed with the 6 effective variables (Section 4) to achieve the best wave runup predictions. The input parameters are arranged according to their degree of influence in the wave runup phenomena. The relation between the input parameters and relative runup is non-linear; however, M50 MT algorithm is only capable of producing linear relationship between the input and output parameters. In order to overcome this limitation, all models were developed using log (inputs) and log
80
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
i- Impermeable Slope
ii- Permeable Slope
Sp = 0.1
Sp = 0.4
iii- Permeable Slope
iv- Homogeneous Slope
Sp = 0.5
S p = 0.6
Fig. 4. Permeability conditions based on bed materials.
4
M50 MTs
MT 1 R=H MT 2 R=H MT 3 R=H MT 4 R=H MT 5 R=H MT 6 R=H MT 7 R=H
Input parameters
ξ ξ, ξ, ξ, ξ, ξ, ξ,
SP cot α, H/h SP,cot α,Ss SP, H/h, Ss SP,cot α,Ss,m SP, cot α, H/h,Ss, m
Statistical Measures R2
RMSE
SI (%)
0.82 0.85 0.84 0.88 0.86 0.89 0.92
0.18 0.17 0.16 0.12 0.14 0.11 0.09
14 12.6 11.1 10.2 12.4 7.6 6.3
(outputs) and therefore a nonlinear prediction in the form of outp ut ¼ γ input 1 ψ 1 input 2 ψ 2 … input N ψ N was achieved. All model trees are constructed and trained with the same randomly chosen 70% training data. An algorithm has been developed to choose the training and test data set randomly. Efforts have been made to prevent from over-training the dataset by adopting pruning and smoothing procedures (Section 3). The trained M50 models are tested by the 30% testing dataset. None of the test dataset employed in training phase. Performance of M50 model trees are evaluated and compared with the measured runup by use of statistical error measures (Section 3.2). 5.1. Relative wave runup (Ru2% /H)
Predicted relative wave runup Ru2%/Hmo
Table 3 Input parameters and statistical measures for Ru2%/H model trees.
3
2
1 Input parameters: ξ, SP, cotα, H/h, Ss, m
0
3 1 2 Measured relative wave runup Ru2%/Hmo
Fig. 5. Measured relative wave runup versus predicted values for MT 5 R=H .
Table 4 Input parameters and statistical measures for Ru2%/h model trees. M50 MTs
Input parameters
0
451 experimental data was utilized for developing M5 MTs, 70% of the data were used for training and 30% applied for the testing set. Several models were developed and examined to achieve the best possible accuracy and extract meaningful relations between the input parameters and relative runup (Ru2%/H). 7 different combinations of input variables were tested with M50 MT algorithm. Accuracy of proposed models is evaluated with statistical error measures. The laboratory runup measurements are used as the benchmark for evaluating the performance of the proposed relations by M50 MTs. Table 3 presents the structure of proposed model trees and their performance. The results show that M50 MTs employs hydrodynamic parameters (ξ, H/h) as the primary contributor in wave runup prediction relations. The results demonstrated that including geomorphological parameters such as slope of the beach (cot α) and permeability of bed materials (SP) could improve the predictions of M50 MTs. The model with the input of both hydrodynamics and geomorphological parameters (MT 7 R=H ) demonstrates the best prediction performance with R2, RMSE & SI of 0.92, 0.09 and 6.3, respectively. Fig. 5 indicates the scatter diagram of observed and predicted Ru2%/H values by M50 algorithm for MT 7 R=H . 5.2. Modelling based on Ru2%/h The relationship between relative wave runup (Ru2%/h), hydrodynamic and geomorphological parameters is investigated
4
MT 1 R=h MT 2 R=h MT 3 R=h MT 4 R=h MT 5 R=h
ξ ξ, ξ, ξ, ξ,
SP cot α SP, cot α, Ss SP, cot α, Ss, m
Statistical Measures R2
RMSE
SI(%)
0.88 0.90 0.91 0.94 0.97
0.16 0.14 0.10 0.08 0.05
11.9 10.4 8.7 6.9 5.3
using M50 MT algorithm and laboratory data described in Section 4. MTs were developed, using randomly data split for training and testing set. 316 data points were included in the training dataset (70% data) and the rest of the data was employed for testing procedures (135 data points). M50 MTs input and performance are presented in Table 4. The results show the accuracy of M50 MTs improves by increasing number of input parameters in the model. MT 5 R=h achieved the best prediction accuracy amongst the tested models with R2 of 0.97 and scattering index 5.3. Fig. 6 presents the scattered diagram of measured and predicted relative runup (Ru2%/h) for the case of MT 5 R=h . The results confirm good agreements between the runup measurements and predicted values from M50 MTs relations. The advantage of model tree algorithms on other soft computing methods such as ANN, is its ability to produce meaningful relation between the input and output parameters. MT 5 R=h proposed number of rules to predict the relative wave runup (Ru2%/h) based on the data described in Section 4 (Table 5). The proposed rules by
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Predicted relative wave runup Ru2%/h
4
3
2
1 Input parameters: ξop, Sp, cotα , Ss , m
0
1 2 3 Measured relative wave runup Ru2%/h
4
Fig. 6. Measured relative wave runup versus predicted values for MT 5 R=h . Table 5 M50 MTs prediction rules for relative wave runup (Ru2%/h). M50 MT proposed formulae for relative runup (Ru2%/h) prediction
Valid range
MT 5 R=h
1:55ξ 1:06 cotα 0:23 Sp 0:05 m 0:37
ξ r 0:4
0:4ξ 1:16 cotα0:09 Sp 0:05 m 0:24
0.4 o ξ r 0.64
0:44ξ 1:38 cotα 0:16 Sp 0:05 m 0:43
0.64o ξ r 1.2 & cotα r 4.47 0.64o ξ r 1.2 & cotα 44.47 1.2 o ξ r 3.4, cotα o 4.47 & SP o 0.22 ξ 4 3.4 & SP o 0.22
4:42ξ 1:22 cotα 0:87 Sp 0:03 m0:13
2:38ξ 1:33 cotα 0:41 Sp 0:03 m0:09 0:12ξ 0:32 cotα0:54 Sp 1:09 m0:15 0:71ξ 0:15 cotα 0:08 Sp 0:23 m0:16
develop model trees. To achieve accurate relative wave runup (Ru2%/H & Ru2%/h) prediction, 12 model trees were developed in total. Different combinations of 6 input parameters was tested and analysed to find the best prediction. Model trees were construction and trained with the same randomly chosen training data. 316 data were used in training set and 135 data employed for the testing set. To evaluate the performance of developed M50 MTs statistical error parameters is used. The M50 MTs with 5 input parameters (ξ, SP, cotα, Ss, m) and the output of relative runup as a function of water depth (Ru2%/h) yields the best result out of all developed models. The results demonstrate the promising ability of M50 MT algorithm in dealing with complex multi parameters input–output systems with non-linear relationship between the input and output. It is also evident that M50 MT algorithm has great ability to learn from data and predict the runup with better accuracy than existing regression models. Unlike soft computing approaches such as ANN, M50 MT algorithm does not require many trial and error procedures to build the model and therefore it is a time-efficient approach. Also, in contrast with ANN approach, M50 MT algorithm is capable of producing meaningful and simple relations between the input and output parameters. Therefore, M50 MT rules could provide some information regarding the importance of each parameter in prediction of wave runup height.
References
Model
0:38ξ 1:49 cotα 0:13 Sp 0:05 m0:25
81
ξ 4 1.21, SP 4 0.22 & m r 0.12 ξ 4 1.21, SP 4 0.22 & m40.12
M50 MTs algorithm are meaningful from engineering sense and in good agreement with the existing empirical relations. The proposed rules for MT 5 R=h (Table 5) and statistical measures (Table 4) show that M50 MT algorithm is capable of predicting wave runup with high precision without any understanding of the complex physical processes associated with runup phenomena.
6. Concluding remarks Wave runup height is the consequence of numerous complex multifaceted time and space-dependent hydrodynamic and geomorphological parameters. The available formulae for prediction of the wave runup are solely based on statistical regression of the experimental data. Empirical approaches based on regression models have drawbacks mainly due to linearity, normality and variance constancy. In this study, M50 model tree algorithm was used to predict the maximum wave runup height based on hydrodynamics and geomorphological parameters obtained from laboratory measurements. 451 experimental data was utilized to
Ahrens, J.P., 1981. Irregular Wave Run up on Smooth Slopes. 1981. Coastal Engineering Research Center, Ft. Belvoir (VA), p. 57, CETA No. 81-17. U.S. Army Corps of Engineers. Bakhtyar, R., Yeganeh-Bakhtiary, A., Ghaheri, A., 2008. Application of neuro-fuzzy approach in the prediction of run-up in swash zone. Appl. Ocean. Res. 30, 17–27. Battjes, J.A., 1974. Surf similarity. In: Proceedings of the 14th International Coastal Engineering Conference, Vol 1, American Society of Civil Engineers, pp. 466– 479. Bhattacharya, B., Solomatine, D.P., 2005. Neural networks and M5 model trees in modelling water level–discharge relationship. Neuro Comput. 63, 381–396. Granthem, K.N., 1953. A Model Study of Wave Run-up on Sloping Structures. 3. Institute of Engineering Research, University of California, Berkeley, California, Technical Report, Series. Hughes, S.A., 2004. Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter. Coast. Eng. 51, 1085–1094. Hunt, I.A., 1959. Design of Seawalls and Breakwaters. Journal of the Waterway, Port, Coastal, and Ocean Division, 85. ASCE, pp. 123–152. Kumar, K.S., Singh, M.V.P., 2010. Estimation of mean annual flood in Indian catchments using Backpropagation neural network & M5 model tree. J. water Resour. Manag. (24), 2007–2019. Mase, H., 1989. Random wave runup height on gentle slope. J. Waterw., Port., Coast., Ocean. Eng. 115 (5), 649–661. Quinlan, J.R., 1992. Learning with Continuous Classes. World Scientific, pp. 343–348 Proceedings of AI'92 (Adams and Sterling. Samadi, M., Jabbari, E., Aza,athulla, H.M., 2014. J. Neural Comput. Appl. 24, 357–366. Saville Jr., T., 1955. Laboratory Data on Wave Run up and Overtopping on Shore Structure, ,. Beach Erosion Board, U.S. Army Corps of Engineers, Washington DC, Technical Memorandum No. 64. Solomatine, D.P., Dulal, K.N., 2003. Model trees as an alternative to neural networks in rainfall-runoff modeling. J. Hydro Sci. 48 (3), 399–411. Solomatine, D.P., Siek., M.B., 2006. Modular learning models in forecasting natural phenomena. Neural Netw. J. 19 (2), 215–224. U.S. Army Corps of Engineers, 2002. Coastal Engineering Manual. Engineer Manual 1110-2-1100. U.S. Army Corps of Engineers, Washington, D.C., in 6 volumes. Van der Meer, J.W., Stam, C.-J.M., 1992. Wave runup on smooth and rock slopes of coastal structures. J. Waterw., Port., Coast., Ocean. Eng. 118 (5), 534–550. Wang, Y., Witten, I.H., 1997. Induction of model trees for predicting continuous lasses, Proceedings of the Poster Papers of the European Conference on Machine Learning. University of Economics, Faculty of Informatics and Statistics, Prague. Wassing, F., 1957. Model investigation on wave run-up carried out in the Netherlands during the past twenty years. In: Proceedings of the 6th International Coastal Engineering Conference, American Society Civil Engineers, pp. 700–714.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Wave runup prediction using M50 model tree algorithm S. Abolfathi a,n, A. Yeganeh-Bakhtiary b,c, S.M. Hamze-Ziabari b, S. Borzooei d a
Flow Measurement & Fluid Mechanics Research Centre, Coventry University, CV1 5FB Coventry, UK School of Civil Engineering, Iran University of Science & Technology (IUST) , Tehran 16884, Iran Civil Engineering Program, Faculty of Engineering, Institut Teknologi Brunei (ITB), Brunei Darussalam d Politecnico di Torino, DIATI, C.so Degli Abruzzi 24, 10129 Torino, Italy b c
art ic l e i nf o
a b s t r a c t
Article history: Received 8 May 2015 Accepted 8 December 2015
In recent years, soft computing schemes have received increasing attention for solving coastal engineering problems and knowledge extraction from the existing data. In this paper, capabilities of M50 Decision Tree algorithm are implemented for predicting the wave runup using existing laboratory data. The decision models were established using the surf similarity parameter (ξ), slope angle (cot α), beach permeability factor (Sp), relative wave height (H/h), wave spectrum (Ss) and wave momentum flux (m). 451 laboratory data of the wave runup were utilized for developing wave runup prediction models. The performance of developed models is evaluated with statistical measures. The results demonstrate the strength of M50 model tree algorithm in predicting the wave runup with high precision. Good agreement exists between the proposed runup formulae and existing empirical relations. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Wave runup Model tree M50 algorithm Nearshore hydrodynamics
1. Introduction Runup of waves on beaches and coastal structures has been studied extensively by coastal engineers, due to the importance of runup height in design of coastal structures (U.S. Army Corps of Engineers, 2002). Nearshore zone is the most dynamic part of the coastal area where wave activities result in complicated hydrodynamic behaviours with varying time and length-scale. One of the most important parameters in dealing with the nearshore hydrodynamics is the wave runup height. During the uprush of water on the beach/structure face or so-called wave runup, the remaining kinetic energy of incident broken waves dissipates and transforms into potential energy as it runs up. Wave runup height is a major concern for design of coastal infrastructure to minimize the occurrence of overtopping. Wave runup height depends on number of parameters such as the incident wave climate and specifically wave height (H), material and the slope. Fig. 1 is the definition sketch of wave runup. The wave runup height is a complex time-varying function of intersection location between the ocean and the beach/structure which depends on nonlinear transformation of waves in breaking zone, refraction of waves, porosity and roughness of slope materials, topography of the nearshore zone, three-dimensional n
Corresponding author. E-mail address: [email protected] (S. Abolfathi).
http://dx.doi.org/10.1016/j.oceaneng.2015.12.016 0029-8018/& 2015 Elsevier Ltd. All rights reserved.
velocity field in the shallow water column and turbulent kinetic energy budget of the incident waves. Therefore, due to such complications in nearshore hydrodynamics and morphodynamics and also due to lack of precise field measurements, our understanding of underlying physical mechanisms of nearshore processes is limited and there is no rigorous theoretical approach to predict the runup height. Hence, the existing formulae for runup prediction are only based on empirical approaches of experimental data.
2. Previous studies Previous studies of the wave runup can be generally categorized to monochromatic and irregular wave conditions based on laboratory measurements. This section aims to review the existing widely accepted approaches to determine the wave runup for regular and irregular wave conditions. 2.1. Regular wave Hunt (1959) studied wave runup on smooth and rough bed conditions and proposed empirical relations for runup prediction based on the shape of breaking wave in surf zone (Eqs. (1) and (2)). Ru 3 H
ð1Þ
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Ru2% ¼ 4:5 0:2ξop for 2:5 r ξop r 9:0 H mo
Incident wave
z R x
y α
Fig. 1. Schematic sketch of wave runup.
Ru tan α R ¼ rffiffiffiffiffiffiffiffiffi ; H ¼ ξ H H
ð2Þ
L0
where Ru is the maximum vertical runup from still water level, L0 is the deep water wave length [L0 ¼ 2gπ T 2 ], α is the beach slope, H is the wave height and ξ is the surf similarity or Iribarren number. Eq. (1) predicts the runup height for surging or standing waves on steep slope and Eq. (2) is valid for spilling or plunging breaker type on mild slope. Both Eqs. (1) and (2) are valid for impermeable beach type. 2.2. Irregular waves One of the first widely used formula to predict irregular wave runup is proposed by Wassing (1957) which relates the runup to the effective wave height of incident waves (H s ) and the bed slope ( tan α) (Eq. (3)). Ru2% ¼ 8 H s tan α; tan α o
1 3
ð3Þ
Ru2% represents the wave runup height exceeded by highest 2% of runup and H s is the average of highest 1/3 waves in an irregular wave terrain. Eq. (3) is only valid for mild uniform slopes. Battjes (1974) modified Hunt (1959) formula for irregular wave conditions (Eq. (4)): Ru2% ¼ C m ξom Hs
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H 1=3 =Lom ; Lom is wave length in deep where ξom ¼ tan α= water using mean irregular wave period (Tm). Constant Cm varies between 1.33–2.86. Ahrens (1981) performed laboratory measurements for irregular wave runup on impermeable bed with the slope range of tanα ¼ ¼ to 1 and proposed Eq. (5) for runup height of irregular waves. ! !2 R2% H mo H mo ¼ C1 þ C2 ð5Þ þ C 3 H mo gT 2p gT 2p C1, C2 and C3 are the coefficients derived from the best fitting method of the data, Hmo is zero-moment wave height and Tp is the wave period associated with spectrum peak frequency. For spilling and plunging breaker waves, on impermeable bed and slope range of 1/1–1/4, Ahrens (1981) derived Eq. (6) using deep water Iribarren number ξop , based on spectrum peak frequency wave period (Tp). 2:26ξop R2% for ξop r 2:5 ¼ H mo ð1 þ 0:234ξop Þ
ð6Þ
Coastal Engineering Manual (U.S. Army Corps of Engineers, 2002) employed Ahrens (1981) data and proposed two equations for irregular wave runup (Eqs. (7) and (8)). Ru2% ¼ 1:6ξop for ξop r 2:5 H mo
ð8Þ
Mase (1989) conducted series of laboratory tests for irregular wave runup measurements with the wave steepness range of 0.007–0.07 and slope range of 1:5–1:30. Eqs. (9)–(11) summarize the formulae proposed by Mase (1989) for the maximum runup (Rmax ), the 2% average runup (R2% ) and the mean runup (̅R ).
Runup level
Rundown level
77
ð7Þ
Rmax 0:77 ¼ 2:32ξ H0
ð9Þ
R2% 0:71 ¼ 1:86ξ H0
ð10Þ
R 0:69 ¼ 0:88ξ H0
ð11Þ
Van der Meer and Stam (1992) employed regression method and proposed runup based on an extensive series of irregular wave runup measurements. They examined the effects of slope angle, permeability of slope, incident wave climate and spectral shape (SS) on the wave runup. Van der Meer and Stam (1992) performed most of their measurements with a Pierson-Moskowitz spectrum, however, some data recorded with wide spectrum and narrow spectrum. Van der Meer and Stam (1992) stated that use of mean period (Tm) for evaluating the Iribarren number (ξom) gives similar maximum runup, unless the spectrum is very narrow. Van der Meer and Stam (1992) empirical model combined the effect of wave period, wave height and slope angle into Iribarren number and did not look at these parameters independently. Also, the effect of spectral shape was not considered on the runup prediction in their model. In the current study, the spectral shape was employed as a quantitative input parameter for wave runup prediction (Section 4). Hughes (2004) used a new concept of wave momentum flux (Mf) to improve empirical correlations for waves and nearshore coastal processes. He employed Fourier approximation wave theory and proposed an empirical relation for the wave momentum flux (Eq. (12)). ! ! A1 Mf h ¼ A0 ð12Þ gT 2 ρgh2 max where ρ is the water density, h is the water depth, g is the gravitational acceleration, A0 and A1 are the empirical coefficients as a function of relative wave height parameter (H/h). The wave momentum flux parameter was obtained for the experimental data of Granthem (1953), Saville (1955), Ahrens (1981) and Mase M (1989) and used as input parameter ( f 2 ¼ m) in development of ρgh
M50 MTs (Section 4). Table 1 summarizes the empirical relations proposed for all the data used in this paper.
3. Soft computing approach In the last decade, there has been a growing trend in the use of Decision Tree (DT) algorithms such as Classification And Regression Trees (CART) and Model Trees (MTs). Decision trees are a Machine Learning (ML) technique with the capability of discovering and extracting the knowledge from a data set and derive a set of rules amongst data set parameters (Solomatine and Dulal, 2003). Model trees (Quinlan, 1992) fall into ML category and are promising numerical prediction method. The efficiency and robustness of MT algorithms in prediction has been well proved (Solomatine and Dulal, 2003; Bhattacharya and Solomatine, 2005). A MT is an inverted tree with a root node located at the top of the tree and the leaves at the bottom. The data instants are first
78
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Table 1 Summary of empirical runup relations proposed based on this study data. Wave action
Study
Proposed runup relation
Irregular
Ahrens (1981)
Ru2% H mo
¼ 1:6ξop
ξop r 2:5
Ru2% H mo Ru2% H mo Ru2% Hs Ru2% Hs Ru2% Hs Ru2% Hs Ru2% Hs
¼ 4:5 0:2ξop
2:5 r ξop r 9
¼ 1:86ðξop Þ0:71
ξop r 3
tan α ¼ 1:5; 1:10; 1:20 & 1:30
¼ 0:96ξm
1 o ξm r 1:5
Impermeable
¼ 1:17ðξm Þ0:46
ξm 4 1:5
tan α ¼ 1:2 1:6
¼ 0:96ξm
1 o ξm r 1:5
Permeable
¼ 1:17ðξm Þ0:46
1:5 o ξm r 3:1
tan α ¼ 1:1:5 1:3
¼ 1:97
3:1 o ξop o 7:5
Mase (1989) Van der Meer and Stam (1992)
Hughes (2004)
Ru2% h
h i1=2 M ¼ 1:75 1 e j1:3cot αj tan α f 2
Ru2% h
¼ 4:4ð tan αÞ0:7
h
1 4r
Plunging/spilling waves
tan α r 11 ðHmo =LP o 0:0225Þ 1 2 5 r tan α r 3ðH mo =LP 4 0:0225Þ
Plunging/ spilling waves
tan α ¼ 1:5 1:30
0:35 r H=h r 1
Impermeable tan α ¼ 1:1 1:10
Non-breaking, surging & collapsing waves
i1=2
Mf 2
ρgh
h Mf i1=2 0:7
¼ 4:4 tan α 2 pgh h i1=2 R m ¼ 3:84 tan α 2 h
Hughes (2004)
tan α ¼ 1:1 1:4 Permeable & Impermeable Impermeable
pgh
Ru2% h
Regular
Accepted range
ρgh
y
Training Dataset New instance
y >z No
y
y
Yes No
Model 1
Model 2
Model 4
y >z
3
y
Model 2
Model 3
4 No
Yes
Yes
2
Yes
Model 1
2
No
Model 6
Model 4
No
Model 5
1
Model 6
Model 3
Model 5 1
Output
2
3
4
5
6
y
1
Y (output) Fig. 2. Prediction procedures for a new instance in M5 model tree. Fig. 3. Splitting space in M5 model tree algorithm.
entering to the root node as an input. In the MT root a test is carried out on the input data and the result of this test causes the tree to be split into branches, each of these branches are representing a possible answer. The MT will continue to split into branches until all the data in different classes have been classified. Model trees (MT) are employed to solve regression and classification problems. Model trees (MTs) and regression trees are a form of DT which has been developed for regression problems (Quinlan, 1992). The key difference between MTs and regression trees is that the leaves of regression trees produce a constant value, while MTs yield into linear models in their leaves (Samadi et al., 2014). These linear models enable MTs to predict numeric values for a given data sample. MTs are efficient for large data set and have higher prediction accuracy in comparison with regression trees (Solomatine and Siek, 2006). 3.1. M50 model tree A complex modelling problem can be divided into a number of simple sub-tasks and the answer would be combining the solution of all these tasks. M5 MTmethod divides the data space into smaller subspaces by divide-and-conquer method (Bhattacharya and Solomatine, 2005). This method use the hard (i.e. yes–no) splits of input space into regions and narrowing the input parameter space into subspaces and builds a linear regression model in each of these subspaces. As a result of this splitting procedure, M5 MT produces a hierarchy (a tree) with splitting rules in nonterminal nodes and the expert models in leaves (Solomatine and Dulal, 2003). Fig. 2 illustrates M5 MT splitting procedures to make a prediction for a new instance.
Fig. 3 shows how M5 MT algorithm split the input space (ψ 1 ψ 2 ), each model tree's leaf represents a linear regression model (Y ¼ ζ 0 þ ζ 1 ψ 1 þ ζ 2 ψ 2 ). M5 algorithm was originally proposed by Quinlan (1992) and further developed by Wang and Witten (1997) to so-called M50 algorithm. The M50 algorithm operates based on partitioning the input space into many subspaces and fitting a linear regression model (LRM) in each of the sub-spaces by using if-then rules. M50 builds a regression tree by recursive splitting based on treating the standard deviation (σ ) of the class values that reach a node as a measure of the error at the node. The attribute that maximizes the expected error reduction is selected for splitting at the node. Quinlan (1992) proposed the standard deviation reduction (σ R ) as a measure of the error at each node (Eq. (13)).
σ R ¼ σ ðT Þ
X jT i j i
jT j
σ ðT i Þ
ð13Þ
where T is the set of instances that reach the node, Ti are the sets resulted from splitting the node according to a given attribute and split value (Wang and Witten, 1997). If the output values of all instances that reach the node have slight variation or when few instances remain, the splitting procedure will automatically terminate. The splitting process may frequently impose too much accuracy on the training data set and hence may over-fit the training data which could result in poor performance of MT when running for unseen data (Bhattacharya and Solomatine, 2005). In this study a pruning technique is employed to overcome the problem of over-training the dataset. Pruning technique basically merges some of the lower sub-trees into one node. Pruning the MT can cause sharp discontinuities between the adjacent linear
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
models in the leaves of the pruned tree. Smoothing procedures has been considered to compensate for the discontinuities of pruned tree and improve the accuracy of MT predictions. In the present paper, capability of M50 model tree algorithm for prediction of the wave runup height is explored. 3.2. Statistical error parameters To evaluate the performance of M50 MT, statistical measures are utilized. In this paper, Correlation Coefficient (R2 ), Scattering Index (SI), bias and RMSE are calculated for wave runup predictions proposed by M50 MT. Eqs (14)–(17) present the statistical error parameters, which have been considered in this study. Bias ¼
N X
1 p oi Ni¼1 i
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X 2 p oi RMSE ¼ t Ni¼1 i SI ¼
RMSE Om
R2 ¼
100
ð14Þ
ð15Þ
ð16Þ
PN
i ¼ 1 ðOi Om Þðpi pm Þ 0:5 P 0:5 N 2 2 i ¼ 1 ðOi Om Þ i ¼ 1 ðpi pm Þ
PN
ð17Þ
Oi represents the observed value, P i is the predicted value, N is the number of observation data, O m is the mean observation value and P m represents the mean prediction value.
4. Data set description The wave runup is predicted by application of M50 MT algorithm. MTs were trained and tested using six effective parameters in the wave runup including surf similarity parameter (ξ), slope (cot α), bed permeability (SP), relative wave height (H/h), wave momentum flux (m) and wave energy spectrum (SS). To achieve the highest possible accuracy and best configuration of input parameters atrial & error procedure utilized. The data set implemented in this study consists of 451 runup data including monochromatic and irregular wave conditions. The irregular wave runup data were collected from Ahrens (1981), Mase (1989) and Van der Meer and Stam (1992) laboratory measurements. The runup data for monochromatic waves are obtained from Hughes (2004). Irregular wave runups on smooth permeable and impermeable slopes have been studied using Ahrens (1981). For the irregular wave runup on gentle impermeable slopes, Mase (1989) laboratory records with the slope range of 1/5–1/30 and Van der Meer and Stam (1992) runup data with slopes range of 1/ 1.5–1/4 have been utilized in the modelling procedures. For the regular wave runup on smooth, impermeable slopes with the slope range 2/3 r tanα r1/30, Hughes (2004) dataset have been utilized which include existing monochromatic wave runup measurements by Granthem (1953) and Saville (1955). The input variables and data range used for the development of M50 MTs are shown in Table 2. Permeability of slope material is a geomorphological parameter which could play a major role in determining the wave runup height. Beaches and structures made from porous materials (e.g., gravel beach) could results in higher kinetic energy dissipation and therefore decreasing runup height. This paper employed Van der Meer and Stam (1992) methodology to present permeability as a quantitative parameter (SP) based on the beach materials. Fig. 4
79
Table 2 Modelling parameters & range. Parameter Surf similarity parameter Permeability Bed slope Relative wave height Wave momentum flux Wave energy spectrum Relative wave runup Relative wave runup
ξop SP cot α H/h m SS Ru2%/H Ru2%/h
Variable type
Data range
Input Input Input Input Input Input Output Output
0.13–18.21 0.1–0.6 1–30 0.06–1.48 0.01–3.43 0.00–1.00 0.46–4.75 0.02–1.2
shows the permeability conditions that were considered for the data of this study. The effect of wave spectral shape (SS) on the wave runup is evaluated with M50 MTs as a quantitative input. Narrow band spectrum is assumed to be 1; Pierson-Moskowitzis 0.5 and wideband spectrum is assigned to 0. Although Iribarren number comprises the bed slope parameter (cot α), the wave runup height is a function of both Iribarren number and slope angle due to the interactions between individual runup bores (Bakhtyar et al., 2008). When a wave hits coastal structures or reaches the shoreline and transforms its kinetic energy into potential by moving up the beach/structure face, before it complete the runup cycle, the next wave will hit the shoreline. For the second wave two possible scenarios exist, either the wave overtakes the first wave during its uprush or collides with the first wave during the backwash. Such interactions between the incident and reflected waves always exist. Therefore, maximum wave runup is not corresponding to the uprush of the highest wave in the train. This behaviour is directly influenced by the foreshore slope. For the mild slopes the water film have longer time to uprush and backwash, however, in the case of steeper slope the runup time for each wave is more limited (Bakhtyar et al., 2008). Hence, for the irregular wave climate with complex wave interactions the effect of slope in wave runup prediction is significant. Ahrens (1981) and Van der Meer and Stam (1992) have studied relatively sharp slopes and Mase (1989) recorded the runup data for mild slopes (Table 3). Therefore, the data used in this paper cover wide range of slopes and both ξ and cotα parameters were considered separately in the modelling procedures to account for the effect of wave interactions on the wave runup.
5. Evaluation of M50 model trees predictions M50 MT algorithm is used for predicting the wave runup height. 451 laboratory runup data have been gathered and nondimensionalized for the purpose of developing model trees. Wave runup predicted based on dimensionless 2% wave runup. Two sets of MTs were developed to predict the relative wave runup based on significant wave height (Ru2%/H) and water depth (Ru2%/h). Selection of appropriate combinations of inputs which produce the best prediction, carried out by a trial and error procedure, as preferred in most MT applications (Bhattacharya and Solomatine, 2005; Kumar and Singh, 2010 and Samadi et al., 2014). Many M50 MTs were constructed with the 6 effective variables (Section 4) to achieve the best wave runup predictions. The input parameters are arranged according to their degree of influence in the wave runup phenomena. The relation between the input parameters and relative runup is non-linear; however, M50 MT algorithm is only capable of producing linear relationship between the input and output parameters. In order to overcome this limitation, all models were developed using log (inputs) and log
80
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
i- Impermeable Slope
ii- Permeable Slope
Sp = 0.1
Sp = 0.4
iii- Permeable Slope
iv- Homogeneous Slope
Sp = 0.5
S p = 0.6
Fig. 4. Permeability conditions based on bed materials.
4
M50 MTs
MT 1 R=H MT 2 R=H MT 3 R=H MT 4 R=H MT 5 R=H MT 6 R=H MT 7 R=H
Input parameters
ξ ξ, ξ, ξ, ξ, ξ, ξ,
SP cot α, H/h SP,cot α,Ss SP, H/h, Ss SP,cot α,Ss,m SP, cot α, H/h,Ss, m
Statistical Measures R2
RMSE
SI (%)
0.82 0.85 0.84 0.88 0.86 0.89 0.92
0.18 0.17 0.16 0.12 0.14 0.11 0.09
14 12.6 11.1 10.2 12.4 7.6 6.3
(outputs) and therefore a nonlinear prediction in the form of outp ut ¼ γ input 1 ψ 1 input 2 ψ 2 … input N ψ N was achieved. All model trees are constructed and trained with the same randomly chosen 70% training data. An algorithm has been developed to choose the training and test data set randomly. Efforts have been made to prevent from over-training the dataset by adopting pruning and smoothing procedures (Section 3). The trained M50 models are tested by the 30% testing dataset. None of the test dataset employed in training phase. Performance of M50 model trees are evaluated and compared with the measured runup by use of statistical error measures (Section 3.2). 5.1. Relative wave runup (Ru2% /H)
Predicted relative wave runup Ru2%/Hmo
Table 3 Input parameters and statistical measures for Ru2%/H model trees.
3
2
1 Input parameters: ξ, SP, cotα, H/h, Ss, m
0
3 1 2 Measured relative wave runup Ru2%/Hmo
Fig. 5. Measured relative wave runup versus predicted values for MT 5 R=H .
Table 4 Input parameters and statistical measures for Ru2%/h model trees. M50 MTs
Input parameters
0
451 experimental data was utilized for developing M5 MTs, 70% of the data were used for training and 30% applied for the testing set. Several models were developed and examined to achieve the best possible accuracy and extract meaningful relations between the input parameters and relative runup (Ru2%/H). 7 different combinations of input variables were tested with M50 MT algorithm. Accuracy of proposed models is evaluated with statistical error measures. The laboratory runup measurements are used as the benchmark for evaluating the performance of the proposed relations by M50 MTs. Table 3 presents the structure of proposed model trees and their performance. The results show that M50 MTs employs hydrodynamic parameters (ξ, H/h) as the primary contributor in wave runup prediction relations. The results demonstrated that including geomorphological parameters such as slope of the beach (cot α) and permeability of bed materials (SP) could improve the predictions of M50 MTs. The model with the input of both hydrodynamics and geomorphological parameters (MT 7 R=H ) demonstrates the best prediction performance with R2, RMSE & SI of 0.92, 0.09 and 6.3, respectively. Fig. 5 indicates the scatter diagram of observed and predicted Ru2%/H values by M50 algorithm for MT 7 R=H . 5.2. Modelling based on Ru2%/h The relationship between relative wave runup (Ru2%/h), hydrodynamic and geomorphological parameters is investigated
4
MT 1 R=h MT 2 R=h MT 3 R=h MT 4 R=h MT 5 R=h
ξ ξ, ξ, ξ, ξ,
SP cot α SP, cot α, Ss SP, cot α, Ss, m
Statistical Measures R2
RMSE
SI(%)
0.88 0.90 0.91 0.94 0.97
0.16 0.14 0.10 0.08 0.05
11.9 10.4 8.7 6.9 5.3
using M50 MT algorithm and laboratory data described in Section 4. MTs were developed, using randomly data split for training and testing set. 316 data points were included in the training dataset (70% data) and the rest of the data was employed for testing procedures (135 data points). M50 MTs input and performance are presented in Table 4. The results show the accuracy of M50 MTs improves by increasing number of input parameters in the model. MT 5 R=h achieved the best prediction accuracy amongst the tested models with R2 of 0.97 and scattering index 5.3. Fig. 6 presents the scattered diagram of measured and predicted relative runup (Ru2%/h) for the case of MT 5 R=h . The results confirm good agreements between the runup measurements and predicted values from M50 MTs relations. The advantage of model tree algorithms on other soft computing methods such as ANN, is its ability to produce meaningful relation between the input and output parameters. MT 5 R=h proposed number of rules to predict the relative wave runup (Ru2%/h) based on the data described in Section 4 (Table 5). The proposed rules by
S. Abolfathi et al. / Ocean Engineering 112 (2016) 76–81
Predicted relative wave runup Ru2%/h
4
3
2
1 Input parameters: ξop, Sp, cotα , Ss , m
0
1 2 3 Measured relative wave runup Ru2%/h
4
Fig. 6. Measured relative wave runup versus predicted values for MT 5 R=h . Table 5 M50 MTs prediction rules for relative wave runup (Ru2%/h). M50 MT proposed formulae for relative runup (Ru2%/h) prediction
Valid range
MT 5 R=h
1:55ξ 1:06 cotα 0:23 Sp 0:05 m 0:37
ξ r 0:4
0:4ξ 1:16 cotα0:09 Sp 0:05 m 0:24
0.4 o ξ r 0.64
0:44ξ 1:38 cotα 0:16 Sp 0:05 m 0:43
0.64o ξ r 1.2 & cotα r 4.47 0.64o ξ r 1.2 & cotα 44.47 1.2 o ξ r 3.4, cotα o 4.47 & SP o 0.22 ξ 4 3.4 & SP o 0.22
4:42ξ 1:22 cotα 0:87 Sp 0:03 m0:13
2:38ξ 1:33 cotα 0:41 Sp 0:03 m0:09 0:12ξ 0:32 cotα0:54 Sp 1:09 m0:15 0:71ξ 0:15 cotα 0:08 Sp 0:23 m0:16
develop model trees. To achieve accurate relative wave runup (Ru2%/H & Ru2%/h) prediction, 12 model trees were developed in total. Different combinations of 6 input parameters was tested and analysed to find the best prediction. Model trees were construction and trained with the same randomly chosen training data. 316 data were used in training set and 135 data employed for the testing set. To evaluate the performance of developed M50 MTs statistical error parameters is used. The M50 MTs with 5 input parameters (ξ, SP, cotα, Ss, m) and the output of relative runup as a function of water depth (Ru2%/h) yields the best result out of all developed models. The results demonstrate the promising ability of M50 MT algorithm in dealing with complex multi parameters input–output systems with non-linear relationship between the input and output. It is also evident that M50 MT algorithm has great ability to learn from data and predict the runup with better accuracy than existing regression models. Unlike soft computing approaches such as ANN, M50 MT algorithm does not require many trial and error procedures to build the model and therefore it is a time-efficient approach. Also, in contrast with ANN approach, M50 MT algorithm is capable of producing meaningful and simple relations between the input and output parameters. Therefore, M50 MT rules could provide some information regarding the importance of each parameter in prediction of wave runup height.
References
Model
0:38ξ 1:49 cotα 0:13 Sp 0:05 m0:25
81
ξ 4 1.21, SP 4 0.22 & m r 0.12 ξ 4 1.21, SP 4 0.22 & m40.12
M50 MTs algorithm are meaningful from engineering sense and in good agreement with the existing empirical relations. The proposed rules for MT 5 R=h (Table 5) and statistical measures (Table 4) show that M50 MT algorithm is capable of predicting wave runup with high precision without any understanding of the complex physical processes associated with runup phenomena.
6. Concluding remarks Wave runup height is the consequence of numerous complex multifaceted time and space-dependent hydrodynamic and geomorphological parameters. The available formulae for prediction of the wave runup are solely based on statistical regression of the experimental data. Empirical approaches based on regression models have drawbacks mainly due to linearity, normality and variance constancy. In this study, M50 model tree algorithm was used to predict the maximum wave runup height based on hydrodynamics and geomorphological parameters obtained from laboratory measurements. 451 experimental data was utilized to
Ahrens, J.P., 1981. Irregular Wave Run up on Smooth Slopes. 1981. Coastal Engineering Research Center, Ft. Belvoir (VA), p. 57, CETA No. 81-17. U.S. Army Corps of Engineers. Bakhtyar, R., Yeganeh-Bakhtiary, A., Ghaheri, A., 2008. Application of neuro-fuzzy approach in the prediction of run-up in swash zone. Appl. Ocean. Res. 30, 17–27. Battjes, J.A., 1974. Surf similarity. In: Proceedings of the 14th International Coastal Engineering Conference, Vol 1, American Society of Civil Engineers, pp. 466– 479. Bhattacharya, B., Solomatine, D.P., 2005. Neural networks and M5 model trees in modelling water level–discharge relationship. Neuro Comput. 63, 381–396. Granthem, K.N., 1953. A Model Study of Wave Run-up on Sloping Structures. 3. Institute of Engineering Research, University of California, Berkeley, California, Technical Report, Series. Hughes, S.A., 2004. Estimation of wave run-up on smooth, impermeable slopes using the wave momentum flux parameter. Coast. Eng. 51, 1085–1094. Hunt, I.A., 1959. Design of Seawalls and Breakwaters. Journal of the Waterway, Port, Coastal, and Ocean Division, 85. ASCE, pp. 123–152. Kumar, K.S., Singh, M.V.P., 2010. Estimation of mean annual flood in Indian catchments using Backpropagation neural network & M5 model tree. J. water Resour. Manag. (24), 2007–2019. Mase, H., 1989. Random wave runup height on gentle slope. J. Waterw., Port., Coast., Ocean. Eng. 115 (5), 649–661. Quinlan, J.R., 1992. Learning with Continuous Classes. World Scientific, pp. 343–348 Proceedings of AI'92 (Adams and Sterling. Samadi, M., Jabbari, E., Aza,athulla, H.M., 2014. J. Neural Comput. Appl. 24, 357–366. Saville Jr., T., 1955. Laboratory Data on Wave Run up and Overtopping on Shore Structure, ,. Beach Erosion Board, U.S. Army Corps of Engineers, Washington DC, Technical Memorandum No. 64. Solomatine, D.P., Dulal, K.N., 2003. Model trees as an alternative to neural networks in rainfall-runoff modeling. J. Hydro Sci. 48 (3), 399–411. Solomatine, D.P., Siek., M.B., 2006. Modular learning models in forecasting natural phenomena. Neural Netw. J. 19 (2), 215–224. U.S. Army Corps of Engineers, 2002. Coastal Engineering Manual. Engineer Manual 1110-2-1100. U.S. Army Corps of Engineers, Washington, D.C., in 6 volumes. Van der Meer, J.W., Stam, C.-J.M., 1992. Wave runup on smooth and rock slopes of coastal structures. J. Waterw., Port., Coast., Ocean. Eng. 118 (5), 534–550. Wang, Y., Witten, I.H., 1997. Induction of model trees for predicting continuous lasses, Proceedings of the Poster Papers of the European Conference on Machine Learning. University of Economics, Faculty of Informatics and Statistics, Prague. Wassing, F., 1957. Model investigation on wave run-up carried out in the Netherlands during the past twenty years. In: Proceedings of the 6th International Coastal Engineering Conference, American Society Civil Engineers, pp. 700–714.