Composite Goodness of Fit in Reaeration Coeffcient Modeling

Determination of reaeration coefficient is an important factor in surface water quality modeling as it determines the efficiency of the Streeter-Phelps model used for predicting dissolved oxygen deficit of any stream. This study compared the efficiency of Atuwara model with ten other reaeration coefficients models by making use of three data sets obtained from river Atuwara during the prevalent wet and dry seasons using composite goodness of fit test which was developed by quantitatively combining statistical and graphical goodness of fit. The eleven tested models were ranked in order of performance. Results show that the four top ranking models were developed through a process that utilized data from multiple streams while models that were developed from data obtained from the test subject alone performed less competitively. The outcome of the study also suggests that the usual practice of selecting the best model based on statistical analysis alone does not necessarily yield the best result and therefore recommended the incorporation of quantitatively analyzed graphs. The paper concludes that selection of the best performing model among existing reaeration coefficient models using the composite goodness of fit may present a cheaper and better alternative to conventional model development approach.


Introduction
Computation of reaeration coefficient (k 2 ) is an integral part of the process of modeling the dissolved oxygen of any surface water body (Chapman, 1996;Lin & Lee, 2007;Omole, Adewumi, Longe, & Ogbiye, 2012).Several k 2 models have been proposed and their distinguishing factor has been their capacity to predict measured data with minimum error.This has been demonstrated through several publications on reaeration coefficient modeling (Streeter, Wright, & Kehr, 1936;O'Connor & Dobbins, 1958;Owens, Edwards, & Gibbs, 1964, Langbein & Dururn, 1967;Bansal, 1973;Bennet, & Rathburn, 1972;Long, 1984;Baecheler & Lazo, 1999;Jha, Ojha, & Bhatia, 2001;Agunwamba, Maduka, & Ofosaren, 2007;Longe & Omole, 2008;Omole & Longe, 2012) beginning with the pioneering work of Streeter and Phelps (1925) where it was established that dissolved oxygen (DO) content of surface water bodies is used up in breaking down biological and chemical wastes.The quantity of oxygen that would be required to break down these wastes completely was described as biochemical oxygen demand (BOD).Streeter and Phelps (1925) therefore succeeded at providing a mathematical relationship between DO and BOD, thus setting the pace for understanding the process.Subsequent researches proposed different k 2 models, most of which were validated by presenting the regression statistic and probably by comparing it with one other existing model.Although k 2 models are characteristically empirical, models such as O 'Connor & Dobbins (1958), Owens, et al. (1964), Langbein & Dururn, (1967), Bansal, (1973) and, Bennet & Rathburn, (1972) were developed for application in multiple geographical locations.However, since each surface water body is unique, adopting a single k 2 model for modeling multiple water bodies must be done carefully following a robust and objective analysis of multiple models.Furthermore, the development of an empirical k 2 model requires data collection which involves repeated field trips, water sampling, stream geometry measurements, laboratory tests of water samples, and data analysis.This tedious and expensive approach could probably be avoided by use of an alternative approach which involves testing a group of reaeration coefficients models that were previously developed under conditions similar to local conditions.The conventional approach embraces the generation of null hypothesis and the application of statistical analysis of data in drawing inferences because of its quantitative outlook.Error statistics in particular helps to select the model that minimizes error.However, the use of graphs visually demonstrates the comparison between measured and simulated data and thus presents an argument that may agree with or differ from statistical inference.A perfect fit of the two plotted lines therefore show that the simulated data perfectly represents the measured data and the equation of the line simulating data becomes the perfect model.Otherwise, the line that best simulates measured data becomes the preferred model.Therefore, the current study, which may be the first of its kind, proposes a method that combines both statistical and visual inspection of graphs of multiple models using the same data sets.The selected models for this study are Atuwara (Omole & Longe, 2012), Streeter et al. (1936) (which is also known as US Geological Survey equation), O'Connor & Dobbins (1958), Owens et al. (1964), Langbein and Dururn (1967), Bansal (1973), Bennet andRathburn (1972), Long (1984), Baecheler and Lazo (1999), Jha et al., (2001) and, Agunwamba et al. (2007) model which was developed in southern Nigeria using data obtained during the wet season only (Table 1).The Streeter et al. (1936) model was selected because it is the first proposed k 2 model.O' Connor andDobbins (1958), Owens, et al. (1964), Langbein and Dururn, (1967), Bansal, (1973) and, Bennet and Rathburn, (1972) were selected because each of them simulated multiple rivers which possess diverse characteristics such as stream depth and speed.Long (1984) was selected because Texas state has high temperatures during summer which is similar to the tropics.Baecheler and Lazo (1999) was selected because of its gentle slope which is similar to river Atuwara's.Jha et al. (2001) was selected because the climatic conditions in western Uttar Pradesh in India where the model was developed is similar to river Atuwara environ (Yadav et al., 2008).Finally, Agunwamba et al. (2007) was selected because it was developed in Nigeria also.

Materials and Methods
Three data sets obtained from river Atuwara were used for the analysis.The data sets were obtained in March and July 2009 as well as January 2010.The March and January data represented data taken during the dry season while July data represented data during the peak of wet season when there is high dilution of pollutant load.January was the most critical period because of the dry weather flow which is characterized by low stream velocity and discharge.All effluent discharges into the river body at this time have maximum impact because of the low dilution of pollutant concentration.Detailed discussion on how the data sets were obtained and how Atuwara model was developed are fully discussed in Omole and Longe (2012).Selection of the best model from among existing models for river Atuwara, which is the focus of this study, was based on criteria such as availability in literature, the similarity of model parameters, stream geometry, stream speed, the type of climate from which model was developed, and the robustness of analysis that led to the development of the model.Other model specific factors for choosing the test models are summarized in Table 1.

Theoretical Concept
The efficiency of a model can be defined as its ability to adequately predict observed data with minimal error.The best model is therefore deemed as having the best goodness of fit (Berthouex & Brown, 2002;Montgomerry & Runger, 2003;Chatterjee & Hadi, 2006).Goodness of fit can be categorized into two.These are statistical goodness of fit and graphical goodness of fit (Montgomerry & Runger, 2003;Omole, 2011).The former is based on an array of statistically determined error parameters such as estimated variance (standard error), sum of squares of regression (SSR); coefficient of determination (R 2 ); adjusted coefficient of determination (Adj.R 2 ) and root mean square error (RMSE).Furthermore, statistical error parameters such as SE, SSE, SSR, and RMSE whose values are closer to 0 indicate a better fit.Also, models with higher values of statistical parameters such as R 2 and Adjusted R 2 indicate better fit (Runger & Montgomery, 2003).The graphical goodness of fit is based on visual inspection which could be a subjective but nonetheless highly useful tool.This is because a model could have minimum error and still be visually non-predictive (Montgomery & Runger, 2003).In order to compare the predictive capacity of eleven k 2 models, the statistical goodness of fit of each model is determined using the procedure described in the flowchart (Figure 1).Likewise, the best model is expected to have the highest value of coefficient of determination.Therefore, the highest weight is allocated to the model with the highest R 2 or Adjusted R 2 .For the graphical input data, the weights are allocated by inspection.The response trend line that best imitates the measured data trend line is allocated the highest weight.If two models display the same statistical value or trend line, the same values are allocated to them.However, the value of weight that may be allocated to the next model will be m-j, where m is the weight value shared by two or more models and j is the number of models that share the value.Another sensitive part of the composite goodness of fit is the allocation of importance to the statistical and graphical components of the composite goodness of fit (Steps 16-22 of the algorithm).For this study, equal importance was given to them therefore each carried a 50% cumulative weight in the final analysis (Steps 25-26).

Statistical Analysis
Subsequent to the statistical analysis of data, all the pre-selected models were ranked according to how well they minimized error and maximized fit.Models which performed better were allocated the higher scores (Table 2).The ranking was done for each data set and then combined to find an average score which was then converted to percentages in the last row of Table 2. Results show that each data set presented fluctuations in ranking for most of the models but a few of the models had consistently high scores.The fluctuations were expected considering that the different data represented extreme weather conditions.Regardless, the most representative model is expected to have high rankings in both wet and dry seasons.The combined assessment of the three data sets showed that best ranking model is Long (1984) and this is closely followed by Bansal (1973) model, Streeter et al. 1936, and Agunwamba et al. (2007) in that order.Atuwara model ranked eighth in the cumulative statistical analysis.The unique feature of the two top ranking models is the fact that both were developed using data from multiple streams (Table 1).This further buttress the fact that empirical models developed from particular streams may not necessarily be the most representative model for that stream.2007) models, which were developed in the Nigerian environment, performed relatively well in the statistical and graphical analyses, O'Connor and Dobbins displayed more remarkable performance which suggests that it could be safely deployed in DO deficit modeling studies on River Atuwara.The selection of a few models out of the several existing models using the composite goodness of fit approach may provide a cheaper and better alternative than the traditional model development approach.Hence, detailed information on the design of existing and future models should be given prominence in scientific publications so as to aid future researchers in short-listing the most suitable models for use in other environments.Furthermore, the study pointed out that the conventional practice of basing scientific inferences on statistical analyses while relegating graphical analyses to complementary status may not yield the most objective inference.Although, graphical analysis is subjective in approach, this study has proposed a way to assess it quantitatively thus making it a very important tool in the selection of models with the best fit.

Figure 1 .
Figure 1.Flowchart showing the progression of the statistical analysis

Figure 2 .
Figure 2. Plot of individual k 2 models against measured k 2 data gathered in January

Table 2 .
Statistical goodness of fit using January, March and July data Atuwara w O'Connorw Agunwambaw Jha w Streeter w Baechelerw Owens w Bansal w Bennet w Long w Langbein w