Gep For Scour Depth Downstream Of Sills Biology Essay


Local scour is an important issue in environmental science and engineering in order to prevent degradation of river bed and safe the stability of grade control structures-stilling basins, aprons, ski-jump bucket spillways, bed sills, weirs, check dams, etc. This study presents Gene-Expression Programming (GEP) which is an extension to Genetic Programming (GP) as an alternative approach to predict scour depth downstream of sills. Published data were compiled from the literature for the scour depth downstream of sills. The proposed GEP approach produce satisfactory results compared (R2=0.967 and RMSE =0.088) to existing predictors for scour depth.

Local scour modelling is an important issue in environmental engineering in order to prevent degradation of river bed and safe the stability of grade control structures (Laucelli and Giustolisi, 2010). In an estuary or a river, a sill may be the initial foundation or the lower part of structure that has to be constructed on a bed of alluvial material. The bed in the direct neighborhood of hydraulic structure is generally protected against current, waves, and eddies (Hoffman and Verheij, 1997). The length of the bed protection depends on the permissible scour depth. Local scour is the erosion of bed surface and the hydraulic structures due to the impact effect of flowing water. Grade-control structures are built in order to prevent excessive channel-bed degradation in alluvial channels. However, local scour downstream of grade-control structures occurs due to erosive action of the weir overflow and this action may undermine these structures (Bormann and Julien, 1991). Hydraulic grade-control structures have been widely used to increase slope stability and control scour in mountain streams (Chinnarasri and Kositgittiwong, 2008). They are built across the rivers in low-stability areas, or in areas that have to be adjusted from steeper slopes to less severe slopes (Gaudio, et al 2000; Lenzi et al., 2002; and Marion et al., 2004).

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Most of the previous researchers focused on local scouring at isolated drop structures by free jets through experimental studies (Volkart et al 1973 and Whittakar, 1987) Summaries of research for the problem of single, isolated drop structures can be found in Lenzi et al. (2002) Owing to the complexity of flow characteristics, such as flow depth, sills spacing, height of water jet and time evolution, much less is known about the case of a staircase-like sequence of grade control structures (Gaudio and Marion, 2003; Lenzi et al., 2003; and Lenzi and Comiti, 2003). The principle of grade-control structures is to decrease bed slope by dividing it into partitions. Initial steep bed slope is scoured greatly, but when there are grade-control structures, longitudinal channel slope is decreased to a lower value called an ultimate slope, representing a dynamic equilibrium between bed scouring and aggradation (Lenzi et al., 2003; and Lenzi and Comiti, 2003).

The processes of degradation stop and bed profile are stabilised. Under the same flow and sediment rates, the bed slopes between sills are found to be less severe than without sills: a result which corresponds to the case of clear water. Less severe slopes, in comparison with the initial slope, show the potential degradation prevented by the sills (Martin-Vide and Andreatt, 2006; Marion et al., 2006).

The parameters concerned with the flow and local scouring downstream of bed sills may consist of critical specific energy (Hs), maximum depth of the scour hole at the equilibrium condition (ys), initial bed slope So, equilibrium bed slope (Seq), sill spacing (L), median sediment size D50, density of water ρw, submerged density of sediment ρs, sorting index (SI) and acceleration due to gravity (g), as shown in Fig. 1 (Chinnarasri and Kositgittiwong, 2008). The effect of sediment sorting can be described by a reference size, D50, and a geometric standard deviation, σg, of the particles. The sorting index

was proposed by (Chinnarasri and Kositgittiwong, 2008), and morphological jump a= (S0-Seq)L which equivalent to head drop (Gaudio et al, 2000). Scour can be expressed as


A dimensional analysis Eq. (1) can be reduced to a set of six non-dimensional parameters, it gives


where ∆= (ρs-ρw)/ρw is the relative submerged density of sediment. Lenzi et al. (2002) carried out local scouring studies in high gradient streams where the initial bed slopes were 0.0785 m/m, 0.1145 m/m and 0.1480 m/m, respectively. They found that the maximum scour depth on low- and high-gradient streams can be expressed with the non-linear equation as (valid for 0.16≤a/∆D95≤ 1.15)

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During the last two decades, researchers were primarily using soft computing techniques for controlled laboratory data, and the results were demonstrated to be significantly better than those from conventional statistical methods (Giustolisi, 2004; Azmathulla et al., 2010). Use of artificial neural networks (ANN) to predict the scour around and downstream of hydraulic structures, was reported by Azmathullah et al. (2005). However, using ANNs as a mere black-box to reproduce an input-output sequence well does not help in advancing the scientific understanding of hydraulic processes so not attempted in the present study. Recently, gene-expression programming (GEP) has attracted attention in the prediction of hydraulic characteristics; yet its use for hydraulic applications is limited, and needs further exploration. This study presents a new soft computing GEP as alternative tool for estimating scour downstream of sills.

Overview of GEP

GEP, which is an extension of GP (Koza, 1992), is a search technique that involves computer programs (e.g., mathematical expressions, decision trees, polynomial constructs, and logical expressions). GEP computer programs are all encoded in linear chromosomes, which are then expressed or translated into expression trees (ETs). ETs are sophisticated computer programs that have usually evolved to solve a particular problem and are selected according to their fitness at solving that problem.

GEP is a full-fledged genotype/phenotype system, with the genotype totally separated from the phenotype, whereas in GP, genotype and phenotype are mixed together in a simple replicator system. As a result, the full-fledged genotype/phenotype system of GEP surpasses the old GP system by a factor of 100-60,000 (Ferreira 2001a, b).

Initially, the chromosomes of each individual in the population are generated randomly. Then, the chromosomes are expressed, and each individual is evaluated based on a fitness function and selected to reproduce with modification, leaving progeny with new traits. The individuals in the new generation are, in their turn, subjected to some developmental processes, such as expression of the genomes, confrontation of the selection environment, and reproduction with modification. These processes are repeated for a predefined number of generations or until a solution is achieved (Ferreira 2001a, b). The functionality of each genetic operator included in GEP system has been explained by Guven and Aytek (2009).

Derivation of Froude Number based on GEP

In this section, the sediment load is modeled using the GEP approach. Initially, the "training set" is selected from the entire data set, and the rest is used as the "testing set". Once the training set is selected, one could say that the learning environment of the system is defined. The modeling also includes five major steps to prepare to use GEP. The first is to choose the fitness function. For this problem, the fitness, fi, of an individual program, i, is measured by:


where M is the range of selection, C(i,j) is the value returned by the individual chromosome i for fitness case j (out of Ct fitness cases) and Tj is the target value for fitness case j. If |C(i,j) - Tj| (the precision) ≦ 0.01, then the precision is 0, and fi = fmax = CtM. In this case, M = 100 is used; therefore, fmax = 1000. The advantage of this kind of fitness function is that the system can find the optimal solution by itself.

Secondly, the set of terminals T and the set of functions F are chosen to create the chromosomes. In this problem, the terminal set consists of single independent variable, i.e., T = {h}. The choice of the appropriate function set is not so clear; however, a good guess is helpful if it includes all the necessary functions. In this study, four basic arithmetic operators (+, -, *, /) and some basic mathematical functions (√) are utilized.

The third major step is to choose the chromosomal architecture, i.e., the length of the head and the number of genes. We initially used single gene and two head lengths and increased the number of genes and heads one at a time during each run while we monitored the training and testing performances of each model. We observed that more than two genes more and a head length greater than 8 did not significantly improve the training and testing performance of GEP models. Thus, the head length, lh = 8, and two genes per chromosome are employed for each GEP model in this study.

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The fourth major step is to choose the linking function. In this study, addition and multiplication operators are used as linking functions, and it is observed that linking the sub-ETs by addition gives better fitness (Eq. 4) values. The fifth and final step is to choose the set of genetic operators that cause variation and their rates. A combination of all genetic operators (mutation, transposition and crossover) is used for this purpose (Table 2).

Table 3 compares the GEP model with one of the independent parameters removed in each case and any independent parameter from the input set that yielded larger RMSE and lower R2 values also removed. These five independent parameters affect ; thus, the functional relationship given in Eq. (1) is used for the GEP model in this study. The GEP approach resulted in a highly nonlinear relationship between and the input parameters, and the GEP model had the highest accuracy and the lowest error (Table 3).

The GEP model was calibrated with 105 input-target pairs of collected data (Table 1). Among the 105 data sets, 25 (25%) were reserved for validation (testing), and the remaining 80 sets were used to calibrate the GEP model.

The best individual in each generation has 30 chromosomes has and a fitness 555.6 for . The explicit formulations of GEP for are given in Eq. (5), and the corresponding expression trees are shown in Fig. 4.



Training and testing results of GEP modeling

The performance of GEP in training and testing sets is evaluated in terms of four common statistical measures such as R2 (coefficient of determination), RMSE (root mean square error), MAE (mean average error) and d (average absolute deviation) which are expressed as follows:



where denotes the target values of , while and denotes the observed and averaged observed values of , respectively, and N is the number of data points. The range of variation of collected data for this study, and its parameters are shown in Table 1. The functional set and operational parameters used in the present GEP modeling are listed in Table 2.

Results and discussion

The results of the GEP model and Chinnarasri and Kositgittiwong (2008) equation are computed using the collected data set and are compared with the measured data. It is observed that, the GEP has good result and there are considerable errors in comparison with the measured data. This indicates the poor performance of empirical equation proposed by Chinnarasri and Kositgittiwong (2008) for the prediction of scour downstream of sills. From Figure 2 it is clear that there is substantial scatter between observed and predicted relative scour depth. The GEP model predicted fairly accurate and comparable (R2=0.967 and RMSE =0.088) with the previous researchers. With the advancements in computer hardware and software, the application of soft-computing tools should not pose problems in even routine applications. The advantage of the GEP technique is that it is easy to deal with physical prior knowledge perhaps because it works in a similar way as humans, especially when applied to field data from the rivers, to perform scientific discovery, as in this work.


A gene-expression programming approach is used to derive a new expression for the prediction of scour downstream of sills. The proposed equation can be used to estimate scour depth for Mountain Rivers for various bed slopes. Performance of the GEP expression is carried out by comparing its predictions with the published data (R2=0.967 and RMSE =0.088). The comparison shows that the new expression has the least root mean square error and the highest coefficient of determination. The expression is found to be particularly suitable for bed slopes where predictions are very close to the measured scour depth