This document discusses using genetic programming to determine the most effective parameters that influence scour depth at seawalls. Previous studies have developed equations to predict scour depth but they considered a limited number of parameters and had uncertainties. The genetic programming approach was able to evolve models that identify the most significant parameters and omit non-significant ones. The results indicated that relative scour depth is most affected by the reflection coefficient and water depth at the toe, while the influence of sediment grain size is negligible.

1. Determination of the most effective parameters on scour depth at seawalls using Genetic Programming (GP)
Keywords: Coastal structures; breaking waves; Genetic programming; Scour depth; Regression-based equations
Abstract:
Scour at coastal structures is one of the major problems that may lead to their failure. Therefore, predicting accurate scour depth at coastal structures is important. Extensive laboratory studies have been conducted predicting the maximum scour depth. These studies have developed their formulas using the limited set of effective input parameters. Using irrelevant parameters can lead to complex models that are very difficult to interpret and implement as compared to the models evolved with the most significant parameters. Sensitivity analysis can be used for determination of the most important input parameters. In this study, the genetic programming (GP) built-in power in determination of the most significant parameters and omitting the non-significant parameters during the models evolution has been used for sensitivity analysis. In order to verify the GP models performance in determination of the most significant parameters, the Liong et al. (2002) [1] proposed approach has been utilized. The results indicated that the relative scour depth at seawalls is mostly affected by the reflection coefficient and the water depth at the toe. Moreover, the influence of mean diameters of bed sediments is negligible.
Introduction:
Coastal structures such as breakwaters and seawalls are built to protect the shore from the force of the waves and provide calm water condition for loading, unloading and repairing the ships. Scour, one of the significant problems for the coastal structures stability according to Oumeraci (1994) [2] and Lilly and Hughes (1993) [3] reports, results from the interaction between structure, bed sediments and incident waves. An understanding of this process is very important for the optimum design of breakwaters. Being so complicated, it is so difficult to establish a general empirical model to predict the ultimate scour depth. However, many investigators had conducted laboratory studies on this subject. In addition to these studies, different methods such as numerical methods have been developed for this purpose. Empirical investigators such as Herbich et al. (1981) [4], Xie (1985) [5], Fowler (1992) [6], Sumer and Fredsoe (2000) [7], Sutherland (2006) [8], Lee and Mizutani (2008) [9] and Ching-Piao Tsai et al. (2009) [10], among others, produced empirical equations for predicting maximum scour depth in front of the toe of coastal structures. These formulas are simple and fast ones. However, their low accuracy and limited application are their problems.Since most of the seawalls are under the action of breaking waves, understanding the mechanism of interaction between breaking waves and these structures, and also developing an applicable and comprehensive model with the most significant input parameters for prediction the maximum scour depth can be very useful for designing purposes. The main drawback of empirical formulas is that their
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2. formula did not contain all of the effective parameters. These empirical investigations have
proposed formulas that only consider the effect of one or two parameters. Therefore, introducing
the most significant input parameters can lead to the models that are accurate, comprehensive
and applicable for various conditions.
Fowler (1992) [6] conducted experimental investigation into scour at the toe of vertical
seawall under the action of broken waves. He found that the relative scour depth (Smax/H0),
where S is the maximum scour depth and H0 is the deep-water wave height, increases with the
increase in (htoe/L0) where htoe is the water depth at the toe of the structure and L0 is the deep-water
wavelength. He also indicated that on the average, regular wave-induced scour depths
exceeded scour depths associated with comparable irregular waves. Furthermore, he developed
an empirical formula for prediction of maximum scour depth at seawalls using the irregular
wave test results as below:
22.72 0.25
0 0
max  
L
h
H
S toe (1)
Fowler (1992) [6] formula has developed based on laboratory tests with constant structure
configuration and bed slope, one kind of bed sediments and limited relative water depth at the
toe for hhtoe/L0≤0.015. Hence, they did not consider the effects of relative water depth for the ranges
ofhhtoe/L0≥0.015,bed slope, structure configuration and bed sediments properties in his formula.
Sutherland et al. (2006) [8] conducted experimental investigation into scour at the toe of the
sloped seawall under the action of breaking waves. They deduced that scour depth is highly
dependent on the form of wave breaking prior to the seawall. Moreover, they pointed out when
waves plunge directly onto the wall generate jets of water that may penetrate to the seabed and
cause a local scour hole just adjacent to the seawall. Therefore, they introduced serf similarity
parameter as the most significant one.They suggested the following formula for predicting scour
depth at seawalls as a function of Iribarren number:
1.3 0.169
0
max  Ir 
H
S
(2)
Where:
0
0
tan
L
H
Ir

 (3)
Although Sutherland et al. (2006) [8] investigated the effect of bed slope (β), wall slope, and
relative water depth, but their proposed formula did not contain the sediments properties,
structure configuration and water depth at the toe on the maximum scour depth.
Ching-Piao Tsai et al. (2009) [10] conducted experimental studies of toe scour of seawall on a
steep seabed with slope of 1:5 under the action of breaking waves. Their experimental results
indicated that the maximum depth of toe scour increased as steepness of theincoming wave
increased, but an increase in the water depth at the toe makes it decrease. Moreover, they
introduced the breaker type in front of the seawall as another significant parameter in scour
depth. They deduced that the scour depth due to a plunging breaker is larger than that of a
spilling breaker or non-breaking wave in front of the seawall.They developed two empirical
equations of the relative scour depth in terms of the relative water depth at the toe and Iribarren
number as below:
0.92 0.18
0
max  Ir 
H
S
(3)
And:

3. 1.35
0
0
max
2
0.231
 


 



L
h
Sinh
H
S
toe 
(4)
Similar to Sutherland et al. (2006) [8], Tsai et al. (2009) [10] proposed formulas for predicting of
maximum scour depth did not contain all of the effective parameters. Besides of limited
applicability, these regression-based equations, obtained from laboratory experimental data using
non-dimensional governing parameters, have large uncertainties that inevitably increase the
safety factor and construction cost. Hence, a transparent and comprehensible model with the
most effective input parameters for prediction of maximum scour depth can be very useful. The
purpose of this study is introducing the most effective parameters on the scour process at the
seawalls and their relative importance using genetic programming.
Genetic programming:
GP is a branch of the genetic algorithm belonging to the family of evolutionary algorithms, first
proposed by Koza (1992) [11]. The GP is similar to Genetic Algorithm (GA); however, it
employs a “parse tree” structure for the search of its solutions, whereas the GA employs bite
strips. The technique is truly a “bottom up” process, as there is no assumption made on the
structure of the relationship between the independent and dependent variables but an appropriate
relationship is identified for any given data sets. The relationship can be logical statements; or it
is normally a mathematical expression, which may be in some familiar mathematical format; or
it may assemble mathematical functions in a completely unfamiliar format. The GP
implementation of relationships has two components: (i) a parse tree, which is a functional set of
basic operators emulating the role of RNA and (ii) the actual components of the functions and
their parameters (referred to as the terminal set), which emulate the role of proteins or
chromosomes in biological systems. When these two components work hand in hand, only then
efficient emulation of evolutionary processes become possible. The relationship between the
independent and dependent variables are often referred to as the “model”, the “program” or the
“solution” but whatever the terminology ,the identified relationship in a particular GP modeling
is continually evolving and never fixed [12]. GP finds the best solution for a problem by
implementing following steps [11]:
i) Initialize a population of individuals known as chromosomes at random.
ii) The fitness value of each model is evaluated using the values of independent and dependent
variables with respect to a target value.
iii) Select the fittest individuals for being modified. There are various selection methods
including:
(i) Ranking, in which individual models are ranked andselected according to their fitness
value and
(ii) Selection by tournament, in which the population is regarded as a “gene pool” of
models and a certain number of models are picked up randomly and are then compared
according to their fitness; a set number of winners are picked based on their fitness
values.
iv) Modify a selected individual with a relatively high fitness using a genetic operator. Applying
operators like crossover and mutation to the winners, “children” or “offspring” are produced, in
which crossovers are responsible for maintaining identical features from one generation to
another but mutation causes random changes in the parse tree, although data mutation is also
possible. This completes the operations at the initial generation.
v) Repeat steps 2-4 until a termination criterion is met. In this case, the best solution is printed.
There are now various software applications for implementing GP models and Fig.1 presents a
typical implementation procedure. This study was carried out by using Genexpro software

4. application. The modeling algorithms of GeneXproTools are based on Gene Expression
Programming (GEP), an extremely fast and powerful learning algorithm. This application
provides tools to post-process the identified model and refine it if necessary.
In this study, the GP was used for determining the relative importance of input parameters
which can affect the maximum scour depth at coastal structures under the action of breaking
waves. Based on previous studies, the mathematical form of such a relation can be shown as
follows:
 


 


 b
toe b b Ir
H
d
L
H
L
h
L
h
f Cr
H
S
, , , , ,
0
50
0 0 0 0
max (5)
Where Smax/H0is the relative maximum scour depth at the seawall,Cr is the reflection coefficient,
hb/L0 is the relative breaking depth, htoe/L0is the relative water depth at the toe, Hb/L0 isthe
relative breaking wave height, Irbis the breaking serf similarity parameter which determines the
type of wave breaking, and d50/H0 is the normalized mean diameter of bed sediments.
Fig.1 Flowchart of Genetic Programming[11]
The used data sets:
A combination of Fowler (1992) [6], Sutherland et al. (2006) [8] and Ching-Piao Tsai et al.
(2009) [10] data sets was used in order to develop the models with genetic programming (GP).
These data sets contain experimental results of the maximum scour depth at coastal structures
under the action of breaking waves. Overall, 41 data sets were extracted from these literatures.70
percentage of data used for training and the remaining 30% were used for testing. Table.1 shows
the ranges of the ranges of different parameters of the datasets.

5. Table 1) Range of different train and test data used for the prediction of scour depth
GP modeling:
As discussed in previous sections, in order to evolve a model with GP, the function set and the Characteristics of the employed Genetic Programming must be introduced. In this study, a function set, composed of the operators that have been used in previous investigations, used to achieve the best model evolved by genetic programming.Table.2 shows the Characteristics of employed Genetic Programming.
Table 2) Characteristics of employed Genetic Programming.
For quantitative evaluation of the models performance, different statistical measures including the determination coefficient (R2), root mean square error (RMSE), BIAS and Scatter index (SI) were calculated as below:
parameters
Train range
Test range
minimum
Average
Maximum
Cr
0.260-0.576
0.270-0.576
0.260
0.397
0.576
hb/L0
0.012-0.094
0.012-0.094
0.012
0.0500
0.094
htoe/L0
0.006-0.150
0.006-0.150
0.006
0.0587
0.150
Hb/L0
0.009-0.087
0.009-0.087
0.009
0.0474
0.087
Irb
0.001-4.761
0.001-4.761
0.001
1.269
4.761
d50/L0
0.0004-0.0016
0.0004-.0016
0.0004
0.0006
0.0016
Smax/L0
0.017-0.90
0.031-0.814
0.017
0.3816
0.900
Setting of parameter
Description of parameter
parameter
+,−,×,÷,√풙,풆풙 ,풙ퟐ,√풙ퟑ ,퐭퐚퐧풙 ,퐬퐢퐧퐡풙
Function set
P1
30
Number of chromosomes
P2
8
Head size
P3
3
Number of genes
P4
Addition
Linking function
P5
RMSE
Fitness function
P6
0.044
Mutation rate
P7
0.3
One-point and two-point recombination
P8
0.1
Gene transposition
P9
2
Constants per gene
P10
-10 to 10
Range of constants
P11

6.  

  
  

i i
i
i
i
X X Yi Y
X X Yi Y
R
2 2
2
( ) ( )
(( ) ( ))
(6)
N
X Y
RMSE i
 i  i

2 (( )
(7)
N
X Y
BIAS i
 i  i

( )
(8)
X
RMSE
SI  (9)
Where Xi is the observed parameter and Yi is the corresponding simulated parameter, X is the
mean value of the observed parameters and Y is the mean value of the predicted parameters and
N is the number of measurements.
Applications and results:
For sensitivity analysis, the exclusion of parameters separately and developing a model with the
other ones is wildly utilized by many studies (e.g. Azamathulla et al. (2010) [13]). One of the
major capabilities of GP is its inherent power in determination of significant variables and
gradually omitting the input variables that do not contribute beneficially in models accuracy
(Jayawardena et al. (2005) [14]).Using this capability, the GP best models evolved by several
runs can be used to assess the relative significance of each input parameter on predicted scour
depth. The aggregated selection times of each input parameter in all of the GP best models
specifies the significance of that parameter. Results of evolving various GP models given in
Tables. 3, 4 indicated that Smax/H0 is mostly affected by Cr. Moreover, the next important
parameters are hb/L0, htoe/L0, Irb, and Hb/L0, respectively.
Table 3) Number of input variable selections in 30 GP runs.
Input variable
Number of
selections
Cr 110
hb/L0 53
htoe/L0 47
Hb/L0 13
Irb 40
d50/L0 4

7. In order to verify the GP models performance in determination of the most significant input
parameters, the Liong et al. (2002) [1] proposed approach has been utilized. According to their
approach, only one input variable varies while the others are constant. Moreover, they proposed
a variation of ±15%, ±15%, ±15% for each input parameter in each stage. The influence of this
change on Smax/H0is measured in terms of Average Percentage Change (APC) calculated as:
   
 
i
N
i org
org
S
S S
N
APC 







 

1 max
100 max max mod
(10)
Where (Smax/H0)org is the GP predicted relative scour depth with original values of the input
variables, and (Smax/H0)mod is the modified GP predicted relative scour depth due to variation of a
particular variable and N is the number of data points. The process is repeated for all of the input
variables. The results given in Table. 4 indicate the order of parameters’ significance and the
sensitivity of (Smax/H0) to each of the parameters and conform with the results of evolved GP
models (Table. 3). Moreover, the results of this table confirm the mentioned results, given in
5thsection, about the variation trend of relative scour depth against input parameters. For
instance, the (Smax/H0)decreases by increasing Cr, moreover, decreasing of (Smax/H0) with an
increase in htoe/L0 indicates that the majority of data set, used for evolving models, have htoe/L0
larger than 0.015.
Table 4) Average Percentage Change (APC) in relative scour depth due to variation of variable values
Summary and conclusions:
In this study, the application of GP to determine the most significant input parameters on the
relative scour depth due to the breaking waves were investigated. Firstly, based on the previous
laboratory studies of Fowler (1992), Sutherland et al. (2006), and Tsai et al. (2009), the
reflection coefficient, the relative water depth at the toe, the water depth at the breaking point,
the breaking wave serf similarity parameter, the normalized mean diameter of bed sediments,
and the breaking wave height were introduced as the effective parameters on the scour at
seawalls. Therefore, using these irrelevant or non-significant parameters can results in the
complicated models that are very difficult to interpret. Theresults of sensitivity analysis by GP
indicated that the reflection coefficient and the relative water depth at the breaking point are the
Significance Order
Percentage Change in Variable (Eq. 5)
Considered
variable
-15% -10% -5% +5% +10% +15%
Cr -10.399 -6.485 -3.076 2.897 5.765 8.763 1
Irb 2.312 1.532 0.762 -0.753 -1.498 -2.234 4
Hb/L0 -0.457 -0.288 -0.136 0.123 0.236 0.338 5
hb/L0 -3.889 -2.548 -1.253 1.214 2.392 3.537 2
htoe/L0 -3.144 -2.109 -1.061 1.075 2.165 3.269 3
d50/L0 -0.035 -0.023 -0.012 0.016 0.019 0.029 6

8. most significant parameters in the broken wave-induced scour modeling. Moreover, the results of the GP evolved models were verified with the Liong et al. (2002) proposed approach. The importance order can be used in the future for developing optimum models predicting the maximum scour depth at seawalls.

9. References:
[1] Liong, S.Y., Gautam, T.R., Khu, S.T., Babaovic, V., Muttil, N., 2002. Genetic programming: A new paradigm in rainfall-runoff modeling. Journal of American Water Resources Association 38 (3), 219-231.
[2] Oumeraci H., 1994. Review and analysis of vertical breakwater failures-lessons learned. Coastal Engineering, special issue, vertical breakwaters 22, 3-29.
[3] Lillycrop, W.J., Hughes, S.A., 1993. Scour hole problems experienced by the corps of engineers: data presentation and summary. Miscellaneous papers. CERC-93-2. U.S. Army Engineers Waterways Experiment Station. Coastal Engineering Research Center, Vicksburg, MS.
[4] Herbich, J., Ko, B., 1981. Scour of sand beaches in front of seawalls. In: Proc. Coastal Eng. ASCE, pp.622–643, Chap. 40.
[5] Xie, S.L., 1981. Scouring patterns in front of vertical breakwaters and their influence on the stability of the foundations of the breakwaters. Department of Civil Engineering, Delft University of Technology, Delft, the Netherlands, 61 pp.
[6] Fowler, J.E., 1992.Scour problems and methods for prediction of maximum scour at vertical seawalls. Technical Report CERC- 92-16. U.S. Corps of Engineers. Coastal Engineering Research Center, Department of the Army Waterways Experiment Station, Vicksburg, Mississippi.
[7] Sumer, B.M., Fredsoe, J., 2000. Experimental study of 2D scour and its protection at a rubble- mound breakwater. Coastal engineering 40, 59-87.
[8] Sutherland, J., Obhrai.C., Whitehouse, R.J.S., Pearace, A.M.C., 2006. Laboratory tests of scour at a seawall. In: Proceedings, 3rd international conference on scour and erosion (CD-ROM).
[9] Kwang-Ho, Lee., Norimi, Mizutani.,2008. Experimental study on scour occurring at a vertical impermeable submerged breakwater. Applied Ocean Research 30 (2008) 92_99.
[10] Tsai, C.P., Chen, H.B., You, S.S., 2009. Toe scour of seawall on a steep seabed by breaking waves. Journal of Waterway, Port, Coastal, and Ocean engineering 135 (2), 61-68.
[11] J. R. Koza, Genetic Programming, The MIT Press, Cambridge, Massachusetts, 1992.
[12] Ghorbani, Mohammad Ali., Khatibi, Rahman., Aytek,Ali., Makarynskyy, Oleg., Shiri, Jalal , 2010. Sea water level forecasting using genetic programming and comparing the performance with Artificial Neural Networks. Computers & Geosciences 620–627.
[13] Aytek, A., Kisi, O., 2008. A genetic programming approach to suspend sediment modeling. Journal of Hydrology 351, 288-298.
[14] Jayawardena, A.W., Muttil, N., Fernando, T.M.K.G., 2005. Rainfall-Runoff modeling using genetic programming. MODSIM 2005 International Congress on Modeling and Simulation: Advances and Applications for Management and Decision Making, Modeling and Simulation Society of Australia and New Zealand, Melbourne, Australia, 1841-1847.