Low Flow Analysis Using Filter Generated Series For Lake Victoria Basin

Low Flow Analysis Using Filter Generated Series For Lake Victoria Basin Julius Kabubi 1, Francis Mutua 2, Patrick Willems 3, R.J. Mngodo 4 1. Kenya Meteorological Department, Institute for Meteorological Training and Research (IMTR) Nairobi, P. O. Box 56392-2 City Square, Nairobi. Email: juliuskabubi@yahoo.com 2. Department of Meteorology, University of Nairobi, P. O. Box 3197 Nairobi Email: francismutua23@yahoo.com 3. K. U. Leuven Hydraulics Laboratory, Kasteelpark Arenberg 4 B-31 Heverlee, Belgium. Email: Patrick.Willems@bwk.Kuleuven.ac.be 4. Water Resources Department, Ministry of Water & Livestock Dev., P.O. Box 3566 Dar Es Salaam.Email: raymngodo@yahoo.com Abstract The Nzoia is one of the main Kenyan rivers, which drain into Lake Victoria. It is also perennial with a total drainage area of about 12,8 Km 2. The lower parts of the basins of this river, a few kilometres just before it enters the lake is characterised by frequent episodes of floods, which often cover expansive inhabited areas especially when the upper areas of the basin receive intense rainfall amounts for significant period of time. Theses flood periods are often punctuated with long periods of low flow or drought. In water engineering applications, extremes are often analysed for time series. Those extremes have to be extracted first as peak-over-threshold values or annual maxima/minima from the time series in a preliminary study. For the former, they can take the form of instantaneous, aggregated or averaged values in fixed time duration or cumulative values in events such as storm volumes. As the extremes in an extreme value analysis have to be independent. An independency criterion is used in the extraction process. In river flood applications, for instance, the U.S. Water Resources Council (1982) considers consecutive peak floods as independent if the intervening time exceeds a critical time and if an intervening discharge drops below a critical flow. Such independency criterion influences the number of extremes and also the interpretation of the return period of an extreme event. As a well-considered interpretatio n is needed in most applications, the independency criterion is a subjective choice. It does often not totally agree with statistical independency, which is meant in the theory. The subjective criterion, however, reaches very often a large physical independency. This large independency then guarantees the existence of an extreme value distribution or GPD distribution. When large return periods are estimated in the application, the influence of the choice of the independency criterion becomes less important. 1

In this paper a new criterion (Partrick. Willems, 24) which, identifies all independent low flow values is utilised. This criterion employs an objective method of Peak Over Threshold (POT) method to first identify both the independent maximum flow data points in a given set of input flow data. The POT maxima are subsequently used to locate the independent flow minima. The method has been found to work successfully on perennial rivers such as the Nzoia River within the lake Victoria basin. The extracted flow minima data is then analysed for the tail behaviour by fitting it to the three basic Generalised Pareto Distributions (GPDs): the Exponential, Pareto and Weibull distributions. A threshold value in the low flow series is identified as the value corresponding to the minimum mean squared error (MSE) of each of the distributions. With this threshold, a technique based on the regression in the quartile -quartile plots (QQR method) is used to discriminate between the distributions after applying it on the transformed data (1/Q). All the three river gauging stations on the Nzoia river exhibited a normal tail case and a high correlation in the Q-Q plots, which supports the use of the exponential distribution for this catchment. Paper prepared for presentation to the FRIEND/NILE conference12th to 14th November 25 in Cairo, Egypt 1. Introduction 2

Low flow studies have tended to receive little attention from researchers at the Lake Victoria basin simply because the basin is more prone to flooding that it is to drought. In view of the importance of this lake as a shared resource, knowledge of low flow conditions are absolutely necessary if conflict is to be avoided during the times of low water levels. In addition, low flow characteristics of a river are essential for a successful planning, control and management of water for various uses such as domestic, industrial, irrigation and even for pollution control In most hydrologic research particularly those involving rainfall-runoff, modelling, the one single source of errors is in the flow separation into its various components of base flow, interflow and overland flow. This uncertainty complicates outputs of models especially when extreme flow events are to be evaluated. In practice three models are used for studies of extreme events such as low flows. These are the annual minimum, the low flow threshold and the time series analysis. One minimum value is selected on the assumption that each year is independent from each other climatologically. The low flow series selected in this way is limited in data length and hence the results are not able to produce a good and lengthy forecast for future return periods. When a large return period is required for hydrological design purposes, this model is lacking. Researchers switch on to the use of stochastically generated random series of appreciable length using the statistical properties of the observed short duration series. The second method for selecting a low flow series uses the low flow threshold criteria. In this method a minimum value is set and any other value below it is selected for analysis. The danger here is whether the statistical requirement of independency is satisfied and when the river is not a perennial and hence recording zero flows. The procedure to select such a series while ascertaining high degree of independency has been developed using a mathematical filter and forms the basis of this paper. 2. Study objectives The main objective of this study is to generate an independent low flow series from a given time series of input flow data using WETSPRO and ECQ computer codes and use the generated series to identify the most suitable probability distribution for analysing the extreme low flows for the lake Victoria basin. Consequently, the identified distribution is used to evaluate the low flow (Q) frequency (1/T) relationship for the same area. Location of Nzoia catchment in relation to Lake basin N W E S N zoia_ rive r s.shp La ke bas in.s hp Figure1a: Relative position of the study area the Nzoia catchment 3

3. Study Data Daily flow discharges for the Nzoia River Gauging Station (RGS) number 1DD1 is initially selected for this study. The data starts from 1 st February 1963 and ends on 31 st of May 1999. This brings to a total of 13,269 continuous daily observations. Small gaps that were originally in the data series were filled up via linear interpolation using the HYDATA software. The gauge is located at o 22 2 N and 34 o 29 15 E. The contributing upstream area at the gauge location is 3916 Square Kilometres and with a mean flow of 62 cumecs. Two more stations, 1BA1 and 1EE1 upstream and downstream of this location are also filtered for sub flow components and analysed for low flow frequency behaviour. These gages have got upstream catchment areas of 262 and 11829 square kilometres respectively. The later is near the flood plains just before the river enters into the lake. The low flow is finally standardized with the catchment areas and for the ease of comparison. Low flow filtering procedures of these two stations are not displayed in this paper but the out puts have been used while drawing the conclusions for this research. The Nzoia river catchment and its relative position to the Lake Victoria basin and the country at large is shown in Figure 1 above. A second RGS with different flow conditions is considered from Tanzanian part of the lake, the Simiyu River at Ndagalu, RGS number 5D1, The Simiyu catchment of area 532 km 2 is located between 33 o 15 35 o E and 2 o 3 3 o 3 N as shown on the map below (Fig. 1b). The Simiyu River drains from the Serengeti National Park plains to the Lake Victoria. Data for this station were analyzed for data consistence and reliability. The Simiyu River is highly seasonal and occasionally the flow falls down to zero. This introduces another dimension of complexity due to the presence of the zeros in the low flows. Figure1b: The Simiyu catchment 4. Water Engineering Time Series Processing (WETSPRO): Selection of Independent low flow series 4

This tool contains a series of Microsoft Excel macros for analysing extreme flow data. The beauty and uniqueness of this code lies in the procedure it uses to ascertain independence of successive low flow peaks using a set threshold value. The method employs the principal of linear reservoir whereby the input series (qt) of daily discharge data is filtered and decomposed into its various components such as subsurface flow or interflow, overland flow and base flow. The linear reservoir model can be seen as a low pass filter of the form: b t = αb t 1 + 1 α q t -------------------------------------------Equation (1) ( ) ( ) ( ) ( ) Where; b(t) = the outflow discharge α = Exp (-1/k) k =Recession coefficient Chapman (1991) advanced this linear reservoir filter by incorporating the idea of frequency of flow com ponents for a given time t to take care of exponential recessions. WETSPRO is based on this Recursive digital filter that Chapman developed. A Peak-Over-Threshold (POT) model for extreme value analysis is used to identify the independent maximum flow data points in a given set of daily flow data. The POT maxima are the subsequently used to locate the independent flow minima. The flow minima data is filtered and then fitted to the Generalized Pareto Distributions (GPDs) to establish its extreme value index which is critical in shaping the tail of the distribution, Three categories can be identified for less than Zero, Zero and greater than Zero values of the extreme value index for a given RGS. Researchers in KU Leuven developed a procedure and prepared an EXCEL based computer macro called ECQ for computing quantiles which are then plotted for ease of visual comparison in the identification of the best of the GPD s. Low -flow frequency analysis for higher aggregation levels are also possible. For simplicity, only a 1-day independent minimum low flow series is extracted and subjected to further statistical analysis in this paper. 5. Sub flow filtering The decomposition of the input flow time series using the filter parameters w and w 1 is shown in Figure 2. The accuracy of this parameter is very critical in determining the filter results and it varies for each river gauging station. Input flow q(t) Interflow = w 1 (quick flow) Base flow = w(q(t)) Quick flow = (1-w)(q(t)) Overland flow = (1-w 1 ) (Quick Figure 2: Decomposition of the input flow time series 5

Guided by the basic statistics of the time series of flow at the test station, the WETSPRO filter parameters were optimised and set as follows; (i) Initial flow = 2 cumecs for base flow and interflow (ii) Recession coefficient of 12 days for base flow, 7 days for interflow and 2 days for overland flow (iii) The w and w 1 -parameters, which apportions the input series into the various sub flows, are optimised to.73 for the base flow and.4 for the interflow. 1 Discharge (Cumecs) 1 6 65 7 75 8 85 9 Number of time steps Time series Filtered baseflow Figure 3: WETSPRO filtered base flow for Nzoia 1DD1 Slope recession constant baseflow It is important to note that, once the base flow and inte rflow are optimally filtered, the overland flow can then be calculated as the difference between the sum of the two and the input series. Figure 3 above shows the filtered base flow using a uniform w-parameter for the entire series. Note that the left slanting lines represents approximately the recession coefficient k; assumed constant for the entire duration. Theoretically, it is known that the w-parameter and the recession coefficient are dependent on both storm and catchment characteristics. A more realistic sub filtering technique would be to use seasonally varying filter parameters. This approached was used in this analysis and found to improve the results. However, more work is required on the seasonal variation of w-parameter and hence the results on its applications are not displayed in this paper. Based on the assumptions of a constant w and k parameters, the filter results showing the decomposition of the total input series into base flow and interflow is shown in Figure 4 below. 6

3 Time series Filtered interflow Filtered baseflow Discharge (Cumecs) 2 1 135 185 1135 1185 Number of time steps Figure 4: Sub flows filter results for Nzoia 1DD1 Filter results after using seasonally average w-parameter shows that, there is some improvement in the base flow separation when the seasonally weighted w-parameter is applied as can be inferred from figure 4b below. 5. Independent Low Flow Periods Selection After decomposition of the input series into its various components using the appropriate sub filtering parameters, the next step is to isolate the independent low flow series. This is done by using the Peak Over Threshold (POT) method first to isolate the independent quick flow periods. These periods are consequently used to locate the independent slow periods corresponding to the slow flow values, which are required for further analysis. For this station, the selection parameters were found to be 2 days for the independency period with a set minimum peak of 2 cumecs. With these parameters, 186 independent quick flow periods were used to isolate 36 independent slow flow values. Figures 5a and 5b below shows the POT selection procedure. After every rainfall hydrograph, the program is adjusted interactively until on a single event is selected. In this manner, only values very close to base flow values are selected. These are by definition, independent or nearly independent low flow values. 7

4 Time series POT values indep. quick flow periods POT values indep. slow flow periods Hydrograph separation slow flow 3 Discharge (Cumecs) 2 1 6 65 7 75 8 85 9 Number of time steps Figure 5a: Low flow selection using POT for independent quick and slow flow periods for Nzoia 1DD1 For purposes of comparison of the WETSPRO isolated low flows and the annual minimum flows, a hydrological year for this basin is set to run from 1 st November to 31 st October of the proceeding year. During this period, the minimum flow value (taken as the annual minimum) is extracted and plotted together with the filtered low flow series. The results show a very close resemblanc e for the two methods as is clearly shown on Figure 5b below. 4 Time series POT values indep. quick flow periods POT values indep. slow flow periods Hydrograph separation slow flow annual minima 3 Discharge (Cumecs) 2 1 6 65 7 75 8 85 9 Number of time steps Figure 5b: Comparison of WETSPRO Selected low flow with annual minimum series 6. Data transformation Analysis of our test station produced thirty-six filtered independent low flow values, which coincidentally corresponds to the total number of years of the input series. These values are 8

transformed into 1/Q and ranked for the purpose of fitting them into a Quartile Quartile (Q- Q) plots for the exponential and Pareto type of probability distributions together with their respective slope of the Q-Q plots and their ranked threshold. An extreme value index, which is critical in shaping the tail of the distribution, is also plotted against the threshold rank together with the resultant MSE for each ranked value. The results of these plots are then used to determine the behaviour of the distributional tail and hence for the selection of the appropriate distribution. Literature on these procedures is contained in WETSPRO manual (Willems, 24) in which its associated computer code ECQ is outlined. The tails of the distributions are classified as either Normal corresponding to Exponential type of distributions with an extreme value index γ ~ or a heavy for Pareto type of distributions with γ >. A third case of a light tail with γ < is rarely encountered in Hydrology and hence not discussed in this paper. For the Nzoia, the ECQ produced a Normal type of tail corresponding to the Exponential distribution. Figure 6 shows the slope of Q-Q plot approaching zero and get to the minimum square error at a threshold of 24. This threshold is used to evaluate the distribution parameters (γ (location) and β (scale)) for exponential distribution..12.1 slope MSE.25.2.8.15 slope.6.4.1 MSE.2.5 5 1 15 2 25 3 35 threshold rank Figure 6: Slope of the Exponential distribution derived from the Q-Q plot for Nzoia RGS 1DD1 A case of the normal tailed distribution 24 In the following Figure 7, after truncating the low flow series at the threshold value obtained from Figure 6, a linear fit is done on the transformed data for the exponential distribution. The more linear the fit the higher the correlation coefficient and hence the more suitable is the selected distribution. The cumulative distribution function (CDF) for the exponential distribution is given as: γ x F ( β Where the variable x is the transformed (1/Q) low flow series x) = 1 exp( ) --------------------------------------------------------Equation 2. 9

.35.3.25 1/observations.2.15.1.5 observations extreme value distribution optimal threshold.5 1 1.5 2 2.5 3 3.5 4 -ln(exceedance probability) Figure 7: The exponential distribution Q-Q plot after 1/Q transformation of the filtered low flows 7. Calculation of the return pe riods The main aim of every flow frequency analysis is to come up with a discharge (Q) and return period (T) usually referred to as Q-T relationship for the purpose of forecasting future events of various magnitudes. The final stage of this analysis there fore involves the calculation of the return periods for various low flow magnitudes using the parameters of the exponential distribution. To do this, the transformed (1/Q) independent low flow series used in the categorization of the distribution is then re-transformed back to original low flow series. Using the exponential distribution parameters generated by the ECQ, the data is fitted to this distribution and extrapolated for 1 years return period. Figure 8a shows the low flow forecasts for the Nzoia River at 1DD1. This is the low flow return period relationship, which is very critical for water resources planning and management. 7 6 5 empirical data calibrated distribution Discharge [m 3 /s] 4 3 2 1.1 1 1 1 1 Return period [years] Figure 8a: The Q-T relationship for Nzoia 1DD1 1

Similar results of Q -T relationship is also done for the other two river gauging stations upstream and downstream of our study station respectively. These results are shown in Figures 8b and 8c below. 1.2 1 empirical data calibrated distribution.8 Discharge [m 3 /s].6.4.2.1 1 1 1 1 Return period [years] Figure 8b: The Q-T relationship for Nzoia 1BA1 upstream 8 7 empirical data 6 calibrated distribution 5 Discharge [m 3 /s] 4 3 2 1.1 1 1 1 1 Return period [years] Figure 8c: The Q-T relationship for Nzoia 1EE1 downstream There is a good agreement in the results of 1BA1 and our test station 1DD1 in the low flow projections for the future forecasts. The rather poor fit for the low flows in this station 11

(1EE1) is associated with its location being very close to the flood plains and hence the many channelling at this region affects the actual data collected. It is suspected that, the rating curve from which this data were extracted have not been updated for a long time for this RGS. Finally, the return periods at each of the two RGS s with a good fit are standardized with the upstream catchment area, which contributes to the flow of each gage. This standardization enables comparison of low flow behaviour across the entire catchment. The curves are in agreement especially for higher return period forecasts..5.45.4.35 empirical data 1BA1 calibrated distribution 1BA1 empirical data 1DD1 calibrated distribution 1DD1 Discharge [m 3 /s] / A.3.25.2.15.1.5.1 1 1 1 1 Return period [years] Figure 9: The Q-T standardised curves for low flows for the test station and its upstream station As for the case of Simiyu river, the relationship between the dry spells and return period was investigated since this river frequently dries up, which means this is not a perennial river. Perhaps it is more meaningful to analyse the dry spells in such rivers instead of the return period of an event Q. This relationship is shown in Figure 1 below. For seasonal rivers there is an upper limit to the dry spell. 12

9. Conclusions Figure 1: The Dry spells -T relationship for Simiyu at Ndagalu 5D1 The new criterion for isolating independent or nearly independent low flow series after locating the independent high flow periods is undoubtedly an objective way of attaining a statistical requirement of independency and randomness in time series analysis. The lengthening of the data series particularly for the discharges is of great importance for extreme value forecasts. However, it was observed that, for rivers with substantially high flows with well define climatic seasons, the number of independent values of low flow tends to converge to that annual minimum series (number of years considered). This was the case for the two down stream RGS for the catchment. However, for the independent high flow values, the criterion is able to increase the number of high flow values. It was observed that, both the w-parameter for both base flow and interflow together with the chosen length of independent period plays a key role in apportioning the various subcomponents of the input series. More work is needed in improving the selection of this parameter probably from the statistical characteristics of the input series. A goodness of fit criteria should also be included to test the suitability of a distribution ounces isolated as the best by the Q-Q plots in the ECQ code. For the Nzoia catchment, it was found to produce normal tailed distributions for all the three stations tested. This indicates that, exponential distribution is more suited for low flow studies for this catchment. However, more trials with other catchments within the basin need to be carried out before any generalization can be made. Standardized curves for the return period discharge were performed for the two stations, which showed a good low flow fit by the flow at each RGS with the upstream catchment areas. The results for these two stations are encouraging and perhaps mark the first steps in 13

Regionalization studies. Clearly shown from the Q-T forecasts for all the three river gauging stations, the Nzoia River is dependable upon the low flow regulation and has some room for more water resources development projects. It more meaningful to consider the dry spells for non perennial rivers to get an idea on the uncertainty of the length dry periods. This procedure of flow filtering undoubtedly paves the way of being a very important tool for decomposition of a rainfall input series into its sub-components for use in a Rainfall Runoff modelling. 1. Acknowledgements This paper was prepared based on the research activities of the FRIEND/Nile Project which is funded by the Flemish Government of Belgium through the Flanders-UNESCO Science Trust Fund cooperation and executed by UNESCO Cairo Office. The authors would like to express their great appreciation to the Flemish Government of Belgium, the Flemish experts and universities for their financial and technical support to the project. The authors are indebted to UNESCO Cairo Office, the FRIE ND/Nile Project management team, overall coordinator, thematic coordinators, themes researchers and the implementing institutes in the Nile countries for the successful execution and smooth implementation of the project. Thanks are also due to UNESCO Offic es in Nairobi, Dar Es Salaam and Addis Ababa for their efforts to facilitate the implementation of the FRIEND/Nile activities. 11. References Condie R. & Cheng R. (1982), Low Flow Frequency Analysis, Program LOFLOW, Inland waters Directorate, Environme nt Canada Cunnane C. (1989), Statistical distributions for flood frequency analysis: WMO operational Hydrology report No. 33 Maciej R. (24), Development, use and applications of HYDROSPECT data analysis system for detection of changes in Hydrological time series for use in WCP-Water and National hydrological Series (WCASP-65, WMO/TD-No. 124) Mahe l G. (22), Climatic and anthropogenic impacts on the flow regime of the Nakambe River in Burkina Faso: IAHS Publication no.274: Bridging the gap between research and practice ( Page 69) Paul J, & Harvey D. (1993), Consolidated Frequency Analysis (CFA) version 3.1 Reference manual U.S. Water Resources Council (1982) Hydrology Subcommittee, Guidelines for Determining Flood Flow Frequency, Hydrology Subcom mittee Bulletin 17B, with editorial corrections, Interagency Advisory Committee on Water Data, U.S. Geological Survey, 28 pp., 1982. Svensson C et.al (24) Trends in Floods and low flow hydrological time series, World Climate programme (WCASP-66, WMO/TD-No.1241) 14

Willems P. (24), Water Engineering Time Series PROcessing tool (WETSPRO) users manual 15