The hydrology of areas of low precipitation - L'hydrologie des régions à faibles précipitations (Proceedings of the Canberra Symposium, December 1979; Actes du Colloque de Canberra, décembre 1979): IAHS-AISH Publ. no. 128.
Assessment of some streamflow simulation models used in an irrigation scheme design study
I. SIMMERS University of Waikato, HamUton, New Zealand
Abstract. This paper investigates streamflow data requirements for irrigation scheme design in the low rainfall Maniototo Plains area of Central Otago, New Zealand. The data collection network is theoretically acceptable for most purposes, but the measured discharge values do not allow estimation of all the needed basin and sub-basin parameters at the required precision level. Record synthesis is necessary in order to increase the amount of information contained in the present streamflow data series. Techniques chosen for time series extension and spatial extrapolation are linear, curvilinear and multiple regression, water balance, and conceptual models. Discussed are: theoretical aspects which concern precision of population parameter estimates from blended data; standard errors of prediction for synthetic records; and the relative simulation and prediction abilities of models chosen. The analyses showed that increased information could be obtained from the blended records, though the study objective was not fully realized. In some instances, record extension resulted in a decrease of statistical information. For design of an irrigation scheme to proceed now, the chosen error criterion is judged too stringent for the data. If the calculated errors cannot be accepted, all streamflow records from the area will require extension by additional observations.
Evaluation de quelques modèles d'écoulement simulé utilisés dans une étude pour le projet d'un système d'irrigation Résumé. Cette communication étudie les besoins en données d'écoulement pour le projet d'un système d'irrigation dans les plaines de Maniototo de l'Otago Central en Nouvelle-Zélande, où les précipitations sont peu abondantes. Le réseau qui collecte les données est acceptable en théorie pour la plupart des cas, mais les valeurs de débits observées ne permettent pas d'apprécier avec la précision nécessaire tous les paramètres du bassin total et des bassins partiels. La synthèse des relevés est essentielle pour augmenter le volume d'information contenu dans la série actuelle des données d'écoulement. Nous avons choisi pour l'extension des séries dans le temps et l'espace les techniques suivantes: régressions linéaires, curvilignes, multiples, le bilan hydrique, et les modèles conceptuels. Nous discutons: les aspects théoriques qui concernent la précision des calculs des paramètres de populations basés sur les données reconstituées; les écarts-types des prédictions basées sur les relevés reconstitués et les capacités relatives des modèles choisis pour la simulation et la prédiction. Les analyses ont montré qu'on pouvait obtenir une augmentation de renseigne ments à partir des relevés reconstitués bien que l'étude n'ait pas atteint tous ses objectifs. Dans certains cas, l'extrapolation des relevés a eu pour résultat une diminution de l'information statis tique. Nous jugeons que le critère d'erreur-type choisi est trop rigoureux pour les données, pour que l'on puisse procéder actuellement à l'exécution du projet d'un système d'irrigation. Si les erreurs calculées sont inacceptables, il faudra majorer le volume des relevés d'écoulement de la région au moyen d'observations supplémentaires.
WATER RESOURCE ASSESSMENT TECHNIQUES Data precision requirements To plan a data collection or synthesis programme it is vital to first establish clearly the purpose for which the data are to be used, and the degree of precision of the infor mation at a particular confidence level that will be adequate. Purpose will determine the required data precision. Precision requirements will in turn dictate the minimum record lengths of each variable necessary to estimate population parameters at a point, and for climatological data, the network density needed for parameter estimation over an area. Determination of these areal mean values also requires a statement on whether the requirements are for daily, monthly, or long-term data.
169 170 I. Simmers It is necessary to understand the relationship between precision and the data required to obtain it if a sampling programme is to be established at a point or over an area on a sound scientific basis. The relationships are well established for normal and lognormal distributions, and are outlined by Montgomery and Hart (1971). A numerical statement as to what data precision is required for water resource system design may be different for each parameter, and may refer to any number of variables over whatever time period. Parameters commonly chosen are the mean, sample standard deviation, coefficient of variation and extreme values. The suggested approach for any study is to use the existing data to establish statistically the network design and period of sampling which would be needed to estimate parameters of the population with stated precision at a stated level of confidence. This will determine the need for additional measured or synthesized information.
Determination of yield parameters in ungauged areas The translation of streamflow parameters from one basin to another is a prediction problem, and is usually based on relationships between flow characteristics and several variables which describe basin and climatic characteristics. Several techniques are in common use for spatial extrapolation of hydrological data, and for time series exten sion either when records are missing or when measured information is insufficiently precise. These are linear, curvilinear and multiple regression, water balance and con ceptual models. A necessary condition for sound prediction is good model fit. However, the ultimate value of any model is determined not only by goodness of fit, but also by the degree of precision with which flow parameters are estimated in other basins, irrespective of the number of physiographic and climatic variables chosen. If the estimates do not accord with some previously stated allowable error criteria, determined by the purpose for which the data are required, then they must be considered to have little value.
Precision of new parameter estimates Theoretical aspects of whether increased information in the statistical sense is added by time extension and spatial extrapolation of records are described in detail by Matalas and Langbein (1962), Fiering (1963) and by Frost and Clarke (1973). How ever, to summarize, statistical techniques may be used to examine the validity of using linear correlation analysis to augment data for any variable, when concurrent and additional data are available at a nearby station. Results from the theory of sampling give the variance of the first two statistical moments for a record of length m, taken to include only measured data. If correlation estimates are used to provide n2 addi tional values, the result is a combined record of nx measured values and n2 estimated values. If the variance of a parameter computed from this blended record exceeds that computed from the record of size nx alone, the combined record provides a less precise estimate of the parameter. However, if the variance is less than that computed from the original record alone, the correlation technique has provided a more precise estimate. A method to calculate the effective or net period of a combined short-term and extended record is given by Clark and Bruce (1966). Regression theory also affords a general solution for correlation of either similar or dissimilar variables (e.g. precipitation versus discharge), with application to linear, curvilinear, or multiple regression models. The estimated standard error in prediction by linear regression of the population mean j', given x, is shown by Snedecor and Cochran (1969) to be 1 1/2 2 Sixt_x=Sy.x -+C„(x-(x» (1) in Assessment of some streamflow simulation models 171 with (n — 2) degrees of freedom. Sy. x is the standard error of estimate from regres 2 sion; n is the number of items used in the correlation; Cn is l/(« — 1) (Sx) , where Sx is the standard deviation of the x values. For x = Ql ^12 C13" X — (x)~ a/2 2 2 3 3 2 2 x{(x-(x),x - The degrees of freedom are (n — 3) for xh x2 only in equation (4). For 95 per cent confidence limits, equations (1) to (4) must be multiplied by the appropriate value of t005 given in standard statistical tables. In equations (1) to (3) for 2 3 first, second and third order polynomials, Sy.x, x , x is given by SbiJJCn, where Sbx is the standard error of the by coefficient. EXAMPLE FROM THE UPPER TAIERI RIVER, OTAGO Basin description A surface water resources analysis completed by Simmers (1975a) for the Upper Taieri River, affords a practical example of the theoretical principles discussed here. The area studied is outlined in Fig. 1, covers 738 km2 and lies 64 km northwest of Dunedin. Much has been written on the general climatic character of Central Otago and for a detailed description reference should be made to Lister and Hargreaves (1965). The region is one of extremes — there is a wide range of annual and diurnal temperatures, a high likelihood of frost, and high potential évapotranspiration rates in the warmer months. Further, summer concentration of an only modest rainfall accentuates the dryness of Central Otago, since a large proportion of the rain comes when temperature, insolation and evaporation are all high. Significant moisture deficiencies are thus pro duced within the lowland areas and give an air of uncertainty to pastoral farming. Irrigation of the area has been investigated since 1906, and has caused considerable controversy. Two major proposals have been considered in recent years. The first was presented in 1961, and proposed the irrigation of 16 600 ha solely by gravity feed. The second scheme, in 1964, involved a pumping extension to the earlier gravity pro posal, with a consequent increase in the irrigated area to 26 100 ha. Both these schemes rely on the construction of a storage dam below Paerau Bridge (Fig. 1), with inundation of about 3800 ha of the Styx basin lowlands. More recent proposals, however, appear to favour either a 'run-of-river' approach to irrigation, or the use of upland reservoir storage. 172 I. Simmers FIGURE 1. Upper Taieri basin, Central Otago. The water resources project completed by Simmers in 1975 was initiated to con sider data requirements and analyses for the planning of these various schemes. To plan for both the 'run-of-river' and single storage reservoir irrigation proposals, streamflow information was required at three sites on the main river above the Maniototo Plain, and on each of the principal upper tributaries. The 95 per cent confidence level was adopted for required data precision. Allowable standard errors of population para meter estimates from measured and synthesized data were stated to be 10 per cent for streamflow and precipitation, and five per cent for temperature. Streamflow parameter estimates from the measured data Continuous streamflow measurements were available for only the Taieri River at Paerau and Patearoa—Paerau Bridges and the Loganburn. Parameters considered in the original analyses of these data were the monthly, seasonal and annual means, sample standard deviations, coefficients of variation and extreme values. However, for this Assessment of some streamflow simulation models 173 *n \o o\ •* NO CS ^H ir> T-H -^ lO oô o *A o c- o o ^o -i d r4 ^ «n l"~ ON in oo co t> co p o es co vûHOcn Ot CO t-. \q o O Tf r-f CO —î d vô \o CO CO CO CS \£> \£> 0\ HOOIC- c- o o co t-; in o> o r- co c •ata c'a .3 o 00HO 00 .-H |£ o\ cei oj S ° t-" O CT\ M3 HOHO\ CO & CO CO o in VO rH 60 X, O^ • o ^-< o o c-- SOONO .2 ^2 g.g to O w CO OO a « "^ -ct co CD io VO t-j IfHO o't d >n "t '6do\d 12 ^£j T—l C\ •s "* ° d £ : cs KO CO CO <1> Q5 0j r <« ^ s >> ta -x C3 s« to S, S.; II bo es , - a^ S co co < tt, 2 to to < O J" fa fa ^ Paerau Bridge record extension — comparison of models Results from the simple and multiple regression analyses are summarized in Tables 2 and 3. Values for the simple rainfall—runoff regression models demonstrate that in only three months does the blended record give estimates of the long-term mean which are any more precise than those calculated from the 30 year measured record. Standard errors of up to 31 per cent were shown for October and November, and only in the winter months was the value less than 10 per cent. The maximum gain in effective record length was calculated to be 11 years. Record extension by multiple regression again did not allow parameter estimation at precision levels which are markedly greater than for the measured series alone. However, percentage standard errors of the long-term means were reduced overall compared with the simple regression analyses, and the high values previously calcu lated for October and November decreased to less than 12 per cent. Standard errors of yield parameters derived by water balance or conceptual models cannot be calculated directly (Clarke, 1973). However, in this study, parameter preci sion limits may be qualitatively deduced by comparing calculated indices of model efficiency with values derived for the simple and multiple regression models which have Assessment of some streamflow simulation models 175 CO O Tt OO C-^ CN c- O iq 00 00 CN >, •a -3" CO o g CD f": co cOU I oi o oi o CO 00 & SO CO •* •a CO O 1 •a o CD S g •r> so SO *o o 1—( in io to o TI M3 O CS rj- co co o C/5 3 U-4 O OS SO 00 r-; es ana l 3 J$ o CO O c~ ssio n :a l t o U") OO SO 3 gre : CO O 00* en ord ; CD ° CO r» CD 21. 6 o CD *^g M) CS 00 t-4 a. o S 00 CS Tf c*-i O CO 00 ifro m measure d r d erro r sure d t-; (Tabl e s a per c O p oj rt •5 o> cs o CO s 3 ^ CD •s g combine d J3 C3 COS art g ° o « 3 g0 UIO I R g >.Q 'O CS O O CD ÏÏS •-=* g O ^—s rt ctl -H--H- fi CD cJ T3 -ç> -~- ccj iea n •o CM ^~s Dfl O E Ti oa ft (« , i s 3 0 y e "—M » >> Correspond s T3 Estimate d f n eighte d valu e Derive d fror c %>+*• s lende d mea n a. a. CD TZÎ S*. ^ Cd en COCO 4— -H- & K» eu H H o orre s « C enve i y>( T CD a. a. Q m H 3 coco 3 Co «» 176 I. Simmers a known error. Indices chosen were Fm2 and R2 as defined by Nash and Sutcliffe (1970) and by Clarke (1973). That is, Fm2 = (lln)7fl(y1-y)2 (5) and R 2 = [var (y) - Fm2]/var (y) (6) yl and y are the computed and observed concurrent values, n is the number of evenly spaced values in the record; var(y) is the initial variance given by (1/n) S'ifj1 - (y))2, where (y) = {IIn) 2"j>. Table 4 summarizes results for all four models using a 10 year test period. In each case, except for the water balance models, the first five-year inter val was used to test model prediction ability. The remaining five-year period was used in model development and hence provided the basis for tests of simulation ability. For the water balance models both five-year intervals could be used to test 'goodness of reconstruction', since neither period contributed data for initial model development. Comparison of Table 4 with results from the simple and multiple regression analyses shows that the water balance model is unreliable overall when used as a runoff predic tion method for the Paerau Bridge station. Although the version with a 70 per cent carryover factor is the most efficient of all the models considered for yield prediction in the first test period, this performance is not maintained for the second five-year interval. More consistent results are shown by the multiple regression models, though with slightly lower efficiency values. TABLE 4. Comparison of residual variance and index of efficiency calculations for Paerau Bridge yield models Simple Water balance models: rainfall Multipl e (50% (70% runoff regression carryover carryover Conceptual models models factor) factor) model Fm as ai percentage of measured mean yield for: Pred.f Sim4 Pred. Sim. Pred. Sim. Pred. Sim. Pred. Sim. January 47.4 52.9 43.7 40.6 52.7 96.5 41.7 98.5 50.2 108 February 69.2 93.5 76.5 76.2 78.7 170 94.7 221 106 205 March 50.2 47.3 52.2 42.3 96.8 73.2 108 74.2 98.8 83.4 April 46.2 52.2 41.7 36.6 41.1 86.0 22.5 48.7 41.4 33.4 May 53.9 37.5 45.4 41.4 57.7 97.0 50.6 54.1 56.6 39.8 June 47.0 46.6 44.4 36.9 31.6 188 35.3 166 33.4 89.7 July 50.7 28.7 51.8 28.9 25.5 94.1 32.4 89.1 45.1 56.7 August 26.4 26.2 28.8 25.5 50.7 90.0 24.6 55.8 22.1 19.2 September 40.1 40.4 20.6 47.9 38.9 70.5 28.5 54.9 49.3 68.1 October 20.9 47.0 19.5 50.2 25.0 51.3 19.0 39.8 35.0 42.9 November 35.8 44.9 40.5 24.7 25.5 58.5 27.0 54.1 39.6 23.7 December 31.7 68.1 54.9 54.4 42.5 56.1 55.0 48.9 46.1 33.0 Year 20.3 23.7 22.0 22.1 16.7 35.0 15.8 33.6 20.0 20.2 Monthly dataFm2 0.360 0.346 0.316 0.334 0.372 1.138 0.288 0.705 0.460 0.529 Monthly datai?2 0.507 0.587 0.567 0.602 0.490 0.000 0.605 0.160 0.370 0.369 Annual dataFm2 11.991 11.657 14.025 10.115 8.069 25.298 7.263 23.378 11.596 8.468 Annual datai?2 0.005 0.221 0.000 0.324 0.330 0.000 0.397 0.000 0.038 0.434 t Initial five-year period to test 'goodness of reconstruction' (prediction). $ Second five-year period to test 'goodness of fit' (simulation). Assessment of some streamflow simulation models 177 The conceptual model used is also shown to be generally unsatisfactory for Upper Taieri runoff prediction purposes. Although improved simulation and prediction ability is shown for the annual data, the monthly results are no better than those derived by the alternative simpler methods. It is concluded from these results that little information was gained by the analyses. The addition of synthesized data to the measured Paerau Bridge record still does not allow population parameters to be determined for all months with the required degree of precision. None of the models chosen for Paerau Bridge record extension allow parameter estimation to markedly greater precision levels than for the measured series alone. Tributary and other main channel station record extensions Streamflow synthesis for the five remaining Upper Taieri stations (Patearoa-Paerau Bridge, Upper Styx Valley Bridge, Loganburn, Serpentine Creek and Styx Creek) was achieved by way of simple linear regression analysis, with the measured Paerau Bridge data as independent variables. Effective record extensions were calculated to vary from nine to 21 years and marked improvements in mean discharge estimate precision were shown by the extended Patearoa-Paerau Bridge and Loganburn records when com pared with the results in Table 1. Mean annual discharge was estimated to within five per cent for both stations, and for the Loganburn only four months had standard errors which exceeded 10 per cent (two months in the case of Patearoa-Paerau Bridge). Prediction standard errors for Upper Styx Bridge mean discharge estimates were shown to be less than eight per cent. The Styx and Serpentine Creek values were higher and generally in excess of 10 per cent. However, the Paerau Bridge discharges used in these sub-basin regressions are not error free, since the {y) data listed in Table 1 represent /x*, with errors as given by SE(y). As such, if the mean discharges calculated by simple regression for the remain ing Upper Taieri stations are also to be considered as estimates of M* , rather than (y) with the independent variable assumed error free, then the standard errors will exceed the above values. For a rigorous determination of errors in such situations reference may be made to Madansky (1959) and Carlson et al. (1966). As an approximation for this study, the (y) data in Table 1 were maximized and minimized in accordance with their listed errors, a range of new discharges calculated and the standard errors re computed. Results of this analysis are given in Table 5. TABLE 5. Long-term mean discharge standard errors where the Paerau Bridge data are not assumed to be error free Values oîSny(xy as a percentage of mean discharges for: Patearoa- Upper Styx Paerau Valley Serpentine Styx Bridge Bridge Loganburn Creek Creek January 15.5 17.6 18.9 34.4 35.0 February 19.5 22.4 19.8 46.6 43.6 March 24.0 17.5 38.9 33.4 34.0 April 30.3 14.9 37.0 24.3 25.5 May 17.7 15.7 14.4 22.3 21.6 June 19.2 13.5 27.7 20.3 20.7 July 12.2 12.9 13.3 18.6 19.2 August 14.1 13.0 20.4 17.4 16.8 September 12.6 15.4 12.7 19.4 18.0 October 17.6 15.6 17.0 19.5 17.7 November 14.2 14.4 15.4 19.5 18.1 December 19.1 15.8 15.9 23.8 23.8 Year 9.63 10.7 9.68 16.3 16.6 178 I. Simmers The data show that only the long-term mean annual discharges for the Patearoa— Paerau Bridge and Loganburn stations may now be estimated with a standard error of less than 10 per cent. However, comparison of Tables 5 and 1 shows that for these two stations, more precise estimates of mean discharge may still be made from the com bined record than from the measured series alone. Standard errors calculated for the Paerau Bridge record extension must also be con sidered as minimum values, since the rainfall data used in the simple and multiple regressions were not error free. Structural function analysis shows that although the established gauges are within the bounds of density and location to satisfy the stated error criterion of project design for annual data, the results are less conclusive for monthly values (Simmers, 1975b). In no month does the present network allow mean areal rainfall estimation to within 20 per cent of actual at the 95 per cent confidence level. Further, in only five months is it possible to estimate values with standard errors of less than 10 per cent, no matter how dense the network. Implications for irrigation scheme design The implications of these results are far reaching when considering use of the available measured data for design of the proposed Upper Taieri irrigation scheme. If synthetic streamflow records are derived by one of the many basin models available which use precipitation input, the streamflow output cannot be more precise than the original mean rainfall input. The Upper Taieri analyses showed that increased information could be obtained from the blended records, though the study objective was not realized. For design of any irrigation scheme to proceed now, therefore, the chosen error criteria must be judged too stringent for these data. If the calculated errors cannot be accepted, all precipitation and streamflow records from the area will require extension by additional observations. CONCLUSIONS There is no doubt of the need for continuing hydrological data collection in New Zealand. However, several aspects need to be stressed at this stage, in order to avoid perpetuation of any ill-conceived and wasteful programmes. It is clear that any sampling network must serve the purpose for which the measured data are to be used. The network should be a flexible system capable of change as demanded by different hydrological problems. It should also be operated for a finite time, with the length of record determined by the purposes for which data are required and hence the necessary degree of precision. If records are short, the statistics may be poor estimates of the population parameters. Ultimate value for engineering design can only be achieved when the available measured data are considered in terms of the degree of precision to which the parameters are estimated, and comparison of these values with the required precision levels. For ungauged basins, the hydrologist has relied on judgement or on empirical relations with doubtful accuracy. Because the basin is ungauged, the reliability of these approaches is rarely checked (Linsley, 1976). The ultimate value of any model used for data translation is determined by the degree of precision with which flow parameters are estimated in other basins. If the estimates do not accord with a stated allowable error, determined by the purpose for which the data are required, they are of little value. It is also not unknown in hydrology to have mathematical models formulated with only a notional or garbled idea of their purpose. More often than not in these circumstances, the choice of model is inappropriate for the ostensible objectives. This account demonstrates the relative merits of some of these water resource assess ment techniques. The experience gained illustrates that planners need to be more critically aware of limitations in the data they use. Assessment of some streamflow simulation models 179 Acknowledgements. Background work for this study formed the basis of a University of Waikato doctoral project. Initial phases were carried out under the auspices of a New Zealand Ministry of Works and Development hydrological research programme - the use of their facilities and equip ment, and those of the University of Otago, are thus acknowledged. Many people were associated with the project, and thanks are due in particular to staff of the Ministry of Works and Develop ment, Dunedin, and of the Otago and Waikato Universities. REFERENCES Carlson, F. D., Sobel, E. and Watson, G. S. (1966) Linear relationships between variables affected by errors. Biometrics 22, no. 2, 252-267. Cislerova, M. and Hutchinson, P. (1973) The redesign of the rain gauge network of Zambia. Meteorological Notes, series A, no' 11, Meteorological Department, Lusaka, Zambia. Clark, R. H. and Bruce, J. P. (1966) Extending hydrological data. Proceedings, Hydrological Symposium no. 5, McGill University, Canada. Clarke, R. T. 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