Flood Mathematical Models That Are Used In Flood Modelling - Statistics Project Example

Flood Modelling Your name Name of Assignment 25thFebruary , 2013 Outline I Introduction II base flow and direct runoff A). Justification of choice of method for separation and lose evaluation B). Synthetic IUH conceptual models III Goodness of fit between the observed predicted flat flows. IV The reliability of models for predicting future flood river flows V The future improvements to the models VI Conclusion References FLOOD MODELLING Introduction Flooding can be very devastating if proper prediction is not carried out using modelling techniques that are accurate. The models adopted in flood prediction should be able to accurately predict the volume of water that will be flowing during heavy rainfall, determine base run off as well as direct run off. To begin with determining peak runoff rates as well as runoff volumes is essential in the models that are used simulation of flooding. This is information can also be used in designing water and soil conservation in flooding areas. The base flow and direct runoff information is usually recorded from the area of study and calculated from automatic recorders that are put there. This paper is going to study flood mathematical models that are used in flood modelling 3.1 base flow and direct runoff Using the data provided, the observed data event has been separated base flow and direct runoff separately. This has been done to provide losses associated to the rainy water. In determining the base flow, it is assumed that the gradient does not affect the base of river. This is comparable to dry season when the flow rate is different. If we look at the initial SDM one can note that a small change leads to increase in flow rate per second. When the base flow model is applied to stimulate an event, the total flow is not known until after the base flow has been simulated and added to the direct runoff hydrograph. Rainfall/runoff model treats each catchment as a single unit, allowing some of the model parameters to be evaluated from physical catchment data. Direct runoff model shows the SDM as well as the actual evapotranspiration and the runoff components such as interflow, base flow and overland flow. This shown below From the figures above the discharge shows that cease to a high flow was a common occurrence period under study and not month that does not high flow rate. This is supported the figure below for the month of January 1972 Justification of choice of method for separation and lose evaluation SDM contribution to flood flow is not at the same rateas surface runoff thus it is separately analysed which is done as a requirements in hydrograph analysis. The groundwater recession curve is a characteristic of the particular catchment in which it was recorded, and some part of this curve will be a constituent of the total hydrograph. If individual storm recession curves are fitted together to form a composite curve, the result is a master recession curve.The recession curve has a continuous discharge recordcovering event 2291.It represents the base flow contribution, after surface runoff hasceased, at as many different stages as possible The base flow sections on the hydrograph are then plotted to a log Qvertical scale against a linear time scale; so that the lowest point on therecession curves are tangential to a common line. Log of data refers to a statistical technique which is used to level out the fluctuations or variations in data. After making adjustments in the data for the variation effects, it is relatively easy to sort out changes caused by non-seasonal factors in the data. If these adjustments are not made, then identifying the differences caused by non-seasonal factors becomes nearly impossible. These adjustments are intended to even out ups and downs reflected in the data related to a data. It is possible to attempt to relate the runoff component of the record of discharge against time and the fraction of the rainfallhyetograph (called the net or effective rainfall) that produced it.In order to estimate the net (effective) rainfall from the total it is necessary toseparate out the quantity of rainfall that becomes evapotranspiration, infiltration and surface detention in pools. In view of the difficulties inobtaining separate measurements of these components they are oftenreferred to collectively as losses. Hence: Net precipitation = total precipitation – losses (2) Where the volume of net precipitation = the total volume of runoff. Estimation of the losses for a storm on a particular catchment is usually basedon the analysis of existing rainfall and record of discharge against time. Synthetic IUH conceptual models IUH conceptual models provide an understanding of the flow dynamics along the length of the catchment area. From the Matlab we note that RMSE store for the models vary and have a mean score of 0.00659 for the observed flat event. I carried out various models that were to help in improving the accuracy of the results that I obtained. The optimum RMSE score was 0.00469. The following table shows RMSE score values using Matlab models model description RMSE score Model 1 Single Linear reservoir 0.0083 Model2 Scalene triangle peak occurring at 1/3 of the time base 0.006 Model 3 Cascade of two linear reservoirs 0.0088 Mode 9 A cascade of N equal linear reservoirs 0.0031 Model 10 A cascade of two unequal linear reservoirs 0.0085 Model 12 Scalene triangle 0.0089 Model 20 A cascade of three unequal linear reservoirs 0.0025 Average 0.00659 Comparing the catchment size and the period showed that events 2282 had the largest RMSE followed by event 2287. This leads to the formation of a successful model which helped to create a relationship between runoff and rainfall. Forecasters should use practicestake into account uncertaintyeven though they are complex to use and compute, some expression of forecast uncertaintyis clearly better than none. As an absolute minimum, even if no formal uncertainty analysis is carried out, brusquepredicts should include by a cautionary statement regarding the uncertainties associated with the forecasts. 3.2 Goodness of fit between the observed predicted flat flows. In order to have accurate results of goodness -of -fit in multiple flood events like in our case it was necessary to record rainfall for every event in millimetres and the predicted value. Any event that is selected in estimating the model should show tendencies of flooding. In my case the event that i selected from eleven observed events had a peek flow above the median. This gave me results that was less and certain and required less calibration. SUMMARY OUTPUT Regression Statistics Multiple R 0.036893 R Square 0.001361 Adjusted R Square -0.02563 Standard Error 90.57574 Observations 39 ANOVA   df SS MS F Significance F Regression 1 413.7141 413.7141 0.050429 0.823555 Residual 37 303546.7 8203.964 Total 38 303960.4         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 79.23751 27.02041 2.932506 0.005739 24.48897 133.9861 24.48897 133.9861 X Variable 1 -0.43217 1.924498 -0.22456 0.823555 -4.33157 3.467232 -4.33157 3.467232 Residual analysis is very important in a regression model. Given a set of data, a regression model is built to find out to what extent the value of dependent variable can be predicted by independent variable. For a given set of data, a regression model is built in such a way that the deviation of predicted values from the actual values is minimized. A residual analysis helps in finding out to what extent this deviation has been reduced by the regression model. If the value of residual sum of square after running a regression on a data set is found to be low, then it can be said that the model is efficient enough in predicting the value of the independent variable given the value of dependent variable For the purpose of assessing the goodness of fit of the regression model implemented above two measures can be evaluated including coefficient of determination and lack-of-fit sum of squares. From the results above it could be indicated that the value of coefficient of determination R-square is 0.043 which implies that the regression model is able to predict 4.3% of the total variations observed. This is quite low value which means that the regression model implemented in this exercise is strong and has a low level of goodness of fit. The reason due to which the line is termed as best fit is that the line minimizes the SSE (Sum of the Squared Errors of prediction) and therefore the line formed after minimizing the sum of squares is likely to be more correct as compared to other methods. The error (Y – Y’) shown in this regression method is the difference found between the value of the dependent variable determined through regression (Y’) and the actual value (Y). Due to this, it is comparatively easy to identify the trends between the dependent and independent variables by means of the line passing through the points representing the standard errors. Other benefits associated with the application of linear regression are that, The method enables the determination of trends by providing a single slope; The data fit to the line is not biased, if it is considered that the deviations are randomly distributed with respect to the trend; The best fit allows the minimization of errors; and There is consistency in the trends shown by the line and it is therefore possible that whenever linear regression is applied over similar values, the results will always be same (Grob, 2003). Regression Statistics Multiple R 0.050305 R Square 0.002531 Adjusted R Square -0.02372 Standard Error 58.6174 Observations 40 ANOVA   df SS MS F Significance F Regression 1 331.2518 331.2518 0.096406 0.757882 Residual 38 130568 3435.999 Total 39 130899.2         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 75.90058 20.14352 3.767989 0.000558 35.12214 116.679 35.12214 116.679 2283 -0.43047 1.386409 -0.31049 0.757882 -3.23711 2.376167 -3.23711 2.376167 The slope parameter is -0.43 and the intercept parameter is75.9. The estimated regression equation for the data is Y = 75.9 - 0.43047 X. The slope parameter is the average increment in the dependent variable when the dependent variable increases by one unit. Hence it suggests that when the number of time by one unit, the flow rate decreases by 0.43 on average. The intercept parameter is the value of the dependent variable in the absence of independent variable. It suggests that the flow rate will be 75.9m3/s. Another important measure in regression analyses is the coefficient of determination. According to Freund, E.f. (1999), “The proportion of the total variability of the responses explained by the model is called the coefficient of determination, denote R2.” R2 is expressed in percentage. It indicates the goodness of the fit of the model. Also it is simply the square of the correlation coefficient. R2for the data is 0.002531, which indicates that nearly 0.25% of the variability is explained by the linear model. It accounts a moderate fit of the model to the data. A percentage of 0.2531% will indicate a good fit. The regression line is incorporated in the scatter plot and it is displayed as follows. All the points are very close to the regression line and there is no outlier. The goodness of the model can be observed from graphs like residual plots. The residual plot given in the appendix does not show any pattern and it indicates that the samples are random and no assignable causes are present. Hence we can assure the model for goodness of fit. There are certain specifications that are characteristic of having a catchment area that has influence within the model. This is because factors are assumed to impact the conditional variation while spill overs in the runoff equation are permitted at the same time. 3.3 The reliability of models for predicting future flood river flows In estimating future flood river flows for catchment areas, models play an important role. However, it should be noted that most factors play an important role in determining the amount of flooding in this catchment areas. This makes the calculation of catchment runoff complex due to the fact that lead-time may be difficult to estimate, the data used may be obsolete because of its historical nature, boundary conditions may change and government policies if the river flows across nations. The behaviour of flood river flows is usually influenced by factors such as availability of vegetation in one side of catchment area, the usage of land near the catchment area, urbanisation, availability of snow near the catchment area, changes in temperature and usage of water resources. The factors affect rain runoff models which are data driven and the data is usually historical when some of the parameters are different from the period being predicted. If care is not taken, the obtained results may not be accurate. The reliability of models for predicting future flood river flows is measured using its ability to simulate the behaviour of the system to use. The systems will reflect the accuracy and consistency with which the output from the model simulates the corresponding observed output. Theaccuracy of the individual parameter values may not be a requirement to reasonable performance of the model in simulating floods, reflecting the equifinality phenomenon (Knapton, 2006). The inexactness of parameter values will have negative implicationof the results and its interpretation especially when it comes to catchment behaviour such as predicting the effects of changes in climate. Logically, the objective function adopted to calibrate a model should be directly reflected in the primary model efficiency index and clearly that index should directly reflect the primary requirement(s) of the modelling task (Clewett, Clarkson, et al. 2003). standard Nash-Sutcliffe model efficiency index R2 is designed to show goodness of fit of the model output to the system output over the whole calibration or verification range of the data, without emphasis on the flow magnitudes and peak flow matching at which errors occur (Doherty, 2003). The hydrodynamic model which usually relies on time function is used to calculate flow conditions and is useful when the river has many tributaries and subsidiaries flowing either across countries or across areas where many parameters are affected by these changes. These models are used to determine the amount of water that is expected in a river at a specific time. These models are usually shown in graphs in form of rating curves. The lead time for the model usually does not take into consideration snow melt at the time as well as other parameters. Increasing lead-time will affect the accuracy of the models. This means forecasting river flows during flooding will be affected by other conditions which are difficult to forecast thus making simulations and calculations have less accuracy. They also rely on historical data which does not take into consideration the changes in weather conditions. The models usually use calibration in modelling process because of the difficulty of measuring the parameter that is used with the intention of ensuring that the process maintains some accuracy. In predicting future flood river flows, is challenging when it comes to environmental changes since it is not easy to predict the coursethe climate takes. The models can estimate or quantify the impact of climate or land use in predicting flood river flows but it is inaccurate or uncertain. Another problem with models, they use poorly measured catchments which ends up giving wrong or inaccurate results. Barriers affecting the ability of gauging catchment areas should be understood and eliminated to improve the data results. There should be also proper strategies that will help reduce errors during analysis or help in diagnosing the error as well as show the uncertainty in the data used. There is also erroneous representation of relationship between the runoff and the controlling input variables such as assuming linearity when it isnot linear system. One may also assume that snow melt has no effect while it has. This will lead to erroneous model parameters that give poor results. 3.4 The future improvements to the models. The models for estimating flood river flows are important and they need to improve for proper accuracy. In order to have accurate results, reservoirs which stores large amounts of water needs to be well defined and any form of flow which is not common from year to year should be always ignored. The models that are employed should have their methodology changed or introduce new method of prediction which combines modelling ideas, current and historical data as well as remote sensed data. The Autoregressive Conditional Heteroscedastic (ARCH) and Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model ARCH models will be employed in measurement of many variance situations in this research (co-integration tests) (Bollerslev 1986, p. 311). Co-integration Tests can used, according to Johnsen’s co-integration test is used to test the relationship between the two variables. Given the availability of other tests like Engle and Granger (1987) and Nelson &Daniel (1991), Johansen’s co-integration test (1991) is preferred since of its ability to consider all variables and tests of co-integrating relationship. If the runoff and Flow rate are considered as belonging to a dynamic linear model then their behaviour will be assumed stationary. Johansen’s co-integration tests show that only linear transformation of a number co-integrating factors (vectors) can be stationary (Bollerslev, Kroner and Chou 1992, p. 59). For example xt and yt are said to be co-integrated if there exists a parameter α such that Ut=Yt-axt Nelson &Daniel (1991) asserts that if one assumes that xt and yt are integrated processes, and there is a linear combination integrated then both variables are said to co-integrated. That is, if the variables of a dynamic linear model are co-integrated, fluctuations are considered stationary (Gonzalo and Granger, 1995, p. 27). To apply Johansen co-integration test in the tests for rainfall and floods, one needs to begin by constructing a vector autoregressive model with k lags. Yt=B1+Yt-1+ B2+Yt-2+……..+ Bk+Yt-k+Ut This correctly corresponds to Markov Switching (MS) models, represented as shown in the equation below. whereyt-i is output at time t-i for i=0, 1,2,.. Co-integration can easily be tested by examining the ranks of a matrices obtained by getting their characteristic roots that are not zero. When it is zero then all variables are considered non-stationary hence no co-integrating factors and If >1, there must be more than 1 co-integrating vectors. Generally Johansen’s co-integration test as shown above indicates that only linear combinations of a number co-integrating factors may be stationary and that this behaviour of co-integration is expected to exist (Gonzalo and Granger 1995, p. 30). Another model to be used is Vector Autoregressive Modelsince it isused for forecasting systems of interrelated time series and for analyzing the dynamic impact of random fluctuations on given factors. A combination of simultaneous equation models and univariate time series models form the vector autoregressive model are treated as a function of lagged values of all the endogenous variables in the system. That is to say, current values depend on previous values of all variables and error terms. Variables in a VAR model are taken as endogenous and no restrictions are imposed. Least square method is used for each equation with assumption of all of the components being stationary. ARCH and GARCH models are non-linear which are very useful in forecasting flooding data. Portmanteau and specific tests are used to detect nonlinear pattern in the data and find applicable models. Portmanteau tests detect the non-linearity in the data and specific tests identify specific types of non-linear structure. Autoregressive Conditional HeteroscedasticModel is used to model and forecast conditional variances in the factors being tested. The model specificationsare the variance of the dependant variable depends upon the past values of internal variables and external variables. For basic ARCH model, conditional variance of a shock at time t is a function of the squares of past shocks:. (Where, h is the variance and  is a “shock,” “news,” or “error”). Since the conditional variance needs to be nonnegative, the conditions have to be met. If 1 = 0, then the conditional variance is constant and is conditionally homoscedastic. Generalized Autoregressive Conditional HeteroscedasticModel allows current conditional variance to be dependent upon lagged conditional variance and lagged squared of errors. GARCH (1, 1): . Where other terms are explained as follows: The variance (ht) is a function of an intercept (), a shock from the prior period () and the variance from last period (). This model is known as conditional variance model, GARCH (1, 1). With this model one can do a one-period-ahead estimation using the past information. GARCH model maximum technique is employed. This works by identifying a log-likelihood function and maximizing the values of the parameters (Zakoian 1994, p. 931). Maximum likelihood method is utilized in both linear and non-linear models with reliable results. Conclusion Direct runoff and Base flow hydrographs developed in this paper have no much difference with other mathematically developed models although this was related to a specific case. These models separates base flow rate and direct runoff and plots graphs for each for comparability. In the model used in this case a plot was made on marking direct runoff contribution on the curve while base flow separation line drawn. The use of hydrographs and the data provided on Nash model provided information that is easy to extrapolated flood plain on base flow and direct runoff. Other models have explored as future possible models for improving flood modelling. The explored models can also use base flow as well as direct runoff to generate hydrograph. The Autoregressive Conditional Heteroscedastic (ARCH) and Generalized Autoregressive Conditional Heteroscedastic (GARCH) Model are expected to be more accurate as they incorporate many parameters as compared to Nash model although it is considered versatile and accurate. References Bollerslev, T 1986, Generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, vol. 31, 307-27. Bollerslev, T, Kroner, K & Chou, RY 1992, ARCH modeling in finance: a review of the theory and empirical evidence, Journal of Econometrics, vol. 52, 5–59. Chadwick, A., Morfett, J. and Borthwick, M. 2004. Hydraulics in Civil and Environmental Engineering. London: E & F N Spon. Clewett, J. F., N. M. Clarkson, et al. 2003. RainmanStreamFlow (version 4.3): A comprehensive climate and stream flow analysis package on CD to assess seasonal forecasts and manage climatic risk. Department of Primary Industries, Queensland. Connell Wagner Pty Ltd, 2001. Flood investigation of the communities of Beswick, Mataranka, Djilkminggan and Elsey – Study Report, unpublished. Dooge, J. C. I. and O'Kane, J. P. 2003.Deterministic Methods in Systems Hydrology.Balkema, Lisse. Dooge, J. C. I. 1977. Problems and methods of rainfall-runoff modeling. Ciriani, T.A., Maione, U. and Wallis, J.R. (Eds) Mathematical Models for Surface Water Hydrology. Proceedings of the Workshop held at the IBM Scientific Center, Pisa, Italy. Wiley, London.pp 71-108. Engle, RF & Granger, CWJ 1987, ‘Co-integration and error correction: representation, estimation and testing’, Econometrica vol. 55, pp. 251-76. Gonzalo, J & Granger, CWJ 1995.‘Estimation of common long memory components in co-integrated systems’, Journal of Business and Economics Statistics, vol. 13, no. 1, pp. 27-35. Institute of Hydrology, 1999.Flood Estimation Handbook. Institute of Hydrology, Wallingford, UK. Knapton, A. 2006.Regional Groundwater Modelling of the Cambrian Limestone Aquifer System of the Wiso Basin, Georgina Basin and Daly Basin. Alice Springs, NTG Dept. Natural Resources, Environment and The Arts. Knapton, A. 2009.Development of a Surface Water Model of the Roper River using MIKE11.Alice Springs, NTG, Department of Natural Resources, Environment, the Arts and Sport. Nelson, DB 1991, ‘Conditional heteroskedasticity in asset returns: a new approach’, Econometrica,vol. 59, pp. 347–370. URS 2008.Integrated hydrologic modeling of the Daly River catchment and Development of a Water Resource Monitoring Strategy. Darwin, NT. Wilson, E M. 1990. Engineering Hydrology. London: MacMillan. Zakoian, JM 1994, ‘Threshold heteroskedastic models’, Journal of Economic Dynamics and Control, vol. 18, pp. 931–955. Read More