Empirical Model and Validation of Bar Formation in Sand Bed Channel

Both present experimental and previous historical data were used to develop the empirical model. Generally, the aim of this study to establish an empirical model of bar formation from the present experimental data and selected historical data. Statistical techniques using Multiple Linear Regression Analysis were employed for empirical model development and the selection of bar formation parameters were done based on the works of previous investigators. Validation of the newly developed empirical model was done by using a different set of historical data from selected laboratory studies. Model development involved selection of parameters through review of established models, dimensional analysis to check on the homogeneity of the model and statistical analysis. Derived empirical model has been validated using a different set of data from previous studies. Analysis confirmed that the empirical model derived using linear regression technique depicts the highest discrepancy ratio accuracy of 90% with s d D and D B as the most significant parameters that promote bar height formation.


Introduction
Bars formation are present in most sand and gravel bed rivers and shows strong influence on flow and sediment transport processes and consequently on channel morphology [1] and as a fundamental sedimentary feature of braided rivers [2].It is being observed in a field study that simple lobate unit bars occur with distinct downstream margins that may have avalanche faces, especially in the sand bed channel.Many data have been collected on the occurrence of bars in laboratory study to develop the empirical criteria for alternate bar formations and predictors for the equilibrium length and height of bars [3] [5].Theoretically, a large number of linear studies that seek the conditions for incipient bar formation and linear growth rate under steady flow conditions has been developed.
Generally, the aim of this study to establish an empirical model of bar formation from the present experimental data and selected historical data.Statistical techniques using Multiple Linear Regression Analysis were employed for empirical model development and the selection of bar formation parameters were done based on the works of previous investigators.Validation of the newly developed empirical model was done by using a different set of historical data from selected laboratory studies.

Multiple linear regression analysis
In multiple linear correlation and regression, the additional independent variables denoted by X 1 , X 2 …X k. help to better explain or predict the dependent variables (Y).The general linear model (First order multiple regression model) are as follows; or where; Y = dependent variable X 1 ,X 2 ,X 3 , …, X K = dependent variables E 0 /A = intercept, the value of Y when all the X's are zero E j /B j = amount by which Y changes when that particular X j increase by one unit, with the values of all other independent variables held constant.
H =random error term If the multiple linear regression is estimated using sample data, the estimated regression equation is written as where Ŷ is the estimated value for Y.The assumptions of the multiple linear regression method are [6].(a) The mean of the probability distribution of H is zero, that is E (H) = 0 (b) The errors must be normally distributed and have a constant variance, V H 2 (c) The independent variables are not linearly related, but they can have a nonlinear relationship.High correlation among independent variables would result to the existence of multicollinearity.(d) There is no linear association between the random error term, H and each independent variable.

Methodology
Selections of the bar formation parameters are based on the review done by previous researcher and through dimensionless analysis.In this analysis, six dimensionless independent parameters namely ratio of grain size ), ( RS There were a total of ten experimental works regarding bar height formation, including the present study that gives a total of 189 data.These data were being classified into two groups; data that collected during the experimental works and data of independent parameters that computed using the collected data.Selection of data to be use in model development and model validation were depend on the range of flow rates.For the data of present study, the recorded flow rates were the initial flow rates imposed to the channel while the recorded data for velocity, channel width and flow depth were for final condition.The grains size and slope of the channel were assumed constant for all experimental run.Velocity recorded is below the critical velocity, as this is due to resistance of bed configuration of bar formation at the final measurement.The data were tabulated in Table 1.analysis through graphs representation is used to confirm the trends and accuracy of the prediction.The accuracy can be evaluated from the distribution pattern of the data.Homoscedastic scatter with high positive correlation signify good agreement of the predicted to observed values.

Table 1. Range of hydraulics and sediment characteristics data used in the analyses
Based on these three elements, linear equation is seemed to perform better than nonlinear models for all sets of data from the present study, Kinoshita, 1961;Yoshino, 1967;Fujita, 1980;Ikeda, 1984;Lanzoni, 2000a. [7]- [9], [3], [10] and W* excluded.The prediction accuracy for this linear regression model is 90% on data from the present experimental study and previous researchers.Summary of the performance of derived empirical models using linear and nonlinear regression method with the respective value of R 2 and discrepancy ratio are shown in Table 2.

Validation of empirical models
An attempt was also made to validate all the derived models for other sets of data from previous studies [5; 11; 12; and 13] and the results are shown in Table 3. Analysis confirms that newly derived empirical model prediction gives satisfactory performance when validated on data from previous researchers.The results from [5] indicates the most accurate prediction as the value of discrepancy ratio of 100%.This is evident from the distribution pattern of the data of each researcher that lies within the acceptable limit as shown in Fig. 1.Non-linear equation gives significant contribution that shows the highest R 2 value of 0.5 as compared to linear regression method.However, there is clear indication that distribution pattern of data in which the scatter plot shown in Fig. 1 and analyzed on their correlations with the dependent variable, ratio of bar height to grain size ( linear and nonlinear regression technique used to investigate the relationship between the dependent variable and independent variables.The statistical technique was employed in the analysis, namely multiple linear regression.The new empirical model will be developed using data from the present experimental study and selected data from previous researchers.The newly developed empirical model will be validated using a different set of data from previous investigators.The procedure involved in the analyses are as follows; (a) Selection of the data from the previous researchers.(b) Derivation of the empirical models through a various combination of the variables using multiple linear regression techniques.(c) Validation of the newly developed empirical model using selected data from previous researchers.
.The derived model using linear regression have established two dimensionless parameters namely shows well distribution pattern for linear Equation as compared to nonlinear.Thus linear equation derived using linear regression technique is the best derived empirical model with 90% accuracy with significant parameters that promote bar height formation.This indicates the findings of the present study corroborate the previous work done by[3].

Figure 1 .
Figure 1.Graph of validation using linear regression method.

Figure 2 . 5 Conclusion
Figure 2. Graph of validation using non-linear regression method.

Table 2 .
Performance of derived models in this study

Table 3 .
Validation of derived models on data by previous studies