Recurrence interval of drought events through stochastic analysis of rainfall and streamflow data

Transcription

1 Hydrological Sciences - Journal des Sciences Hydrologiques, 30, 2,6/1985 INTRODUCTION Recurrence interval of drought events through stochastic analysis of rainfall and streamflow data R. SRIKANTHAN & T. A. McMAHON Department of Civil Engineering, University of Melbourne, Parkville, Victoria, Australia 3052 ABSTRACT The efficient management of an agricultural system or the operation of a water supply scheme requires knowledge about critical droughts, such as their length, severity and frequency of occurrence. Several theoretical distributions are fitted to drought events of up to one year's duration and the average recurrence intervals calculated. A methodology based on stochastically generated data is proposed and used to obtain the average recurrence intervals of events of duration longer than one year. These methods are applied to five river basins in Victoria. The accuracy of the method is dependent on the adequacy of the generating model. Intervalle de recurrence des sécheresses étudié par 1'analyse stochastique des données des précipitations et des débits des rivières RESUME La gestion efficace d'un système d'aménagement agricole et l'exploitation d'installations pour l'approvisionnement en eau exige la connaissance des données suivantes concernant les sécheresses les plus critiques: leur longueur, leur sévérité et la fréquence de leur occurrence. Plusieurs distributions théoriques sont ajustées à des sécheresses durant jusqu'à un an et les intervalles moyens de récurrence sont calculés. Une méthodologie basée sur des données simulées par un processus stochastique est proposée et utilisée en vue d'obtenir l'intervalle moyen de récurrence pour des événements dont la durée dépasse un an. Les méthodes sont appliquées à cinq rivières dans l'etat de Victoria. L'exactitude des résultats de la méthode dépend de l'exactitude du modèle de génération des données. A knowledge of the recurrence interval of droughts is essential to manage an agricultural system or to operate a water supply scheme efficiently. Calculation of recurrence intervals of extreme events of duration up to one year is relatively straightforward in the sense that the method using an appropriate distribution for the annual series is established. However, there is no known analytical procedure available at present to calculate the recurrence interval of hydrological events of duration longer than one year. Even 197

2 198 R. Srikanthan & T.A. McMahon though several empirical procedures (Stall, 1962; Jeppson, 1967) have been proposed, there are problems associated with them. The objective of this paper is to examine procedures for estimating the recurrence intervals of low rainfall and low streamflows that have been observed in the past or that may occur in the future. The inflow data for five major reservoirs in Victoria, Australia, and the average rainfall data for the corresponding catchments are utilized in the analysis. REVIEW OF EXISTING METHODS OF DROUGHT FREQUENCY ESTIMATION Droughts of one year duration or less When the duration of an event is less than or equal to one year, an annual series can be formed with the values of interest. A theoretical distribution can be fitted to the annual series and the probability of occurrence (or recurrence interval) for various events may be obtained from the fitted distribution. But what is the appropriate distribution? Using annual minimum daily and 7-day low flows for 34 streams, Matalas (1963) concluded that the extreme value type III and Pearson type III distributions fitted the data equally well and recommended their use with maximum likelihood estimates of the distribution parameters, rather than moment estimates. Based on an analysis of 37 streams, Joseph (1970) concluded that the gamma distribution was the best of the five distributions tested. Pattison (1966) used a two-parameter gamma distribution to calculate the return period of the lowest 12 months of rainfall on record for a number of sites in the Hunter Valley, New South Wales, Australia. Droughts of more than one year's duration Stall (1962) converted monthly flow data to running totals for durations under consideration. These running totals of, say, n months duration for the entire period of record (N years) were inspected and the lowest one was selected. All the running totals that overlapped to include any of the n months within this lowest period were excluded from further consideration. The remaining running totals were inspected to locate the second most severe n-month low flow. Again, overlapping running totals were excluded. The procedure was repeated until the sequence of running totals was exhausted. The recurrence interval of the series of events was computed from: in which T is the recurrence interval in years and m refers to the rank of the low flow events. After obtaining the sequence of running totals from monthly data, Jeppson (1967) obtained a frequency distribution by ranking the sequence from low to high and assigning a probability (%) to each event given by:

3 Recurrence interval of drought events 199 (2) N n + 1 where m is the rank of the event and N n is the total number of running totals (N n = 12N - n + 1). The recurrence interval for this overlapping series is computed by: *, -» ls^f tj1 To estimate the dependency of the events selected by this procedure, Jeppson eliminated all events on both sides of the lowest event (as Stall did) that contained any of the same monthly data, and then selected the next lowest event from the remaining data. After continuing this process until no events remained, he obtained the probability of the independent sequence from the equation: m p = (4) N n - n + 1 Jeppson did not convert the probability to recurrence interval for this series. The above methods always relate the length of historical record to the recurrence interval of the lowest observed flow event, irrespective of its magnitude. Although it is true that the sample probability of the rank 1 event from an N year record is constant, the population estimate depends also on the magnitude of the event. The procedures of Stall (1962) and Jeppson (1967) do not recognize this fact. An analytical method using the gamma distribution has been proposed by the writers (McMahon & Srikanthan, 1982). However, the difficulty with this method is that the d-year events correspond to non-overlapping sequences and as such should not be used to calculate the probability of exceedance of historical droughts. Furthermore, the relationship between probability of exceedance and recurrence interval is not clear. These aspects are being taken up in another paper nearing completion by the writers. PROPOSED METHOD The method proposed in this paper uses stochastically generated data to obtain the recurrence interval of observed extreme events of duration greater than one year. A long record of generated data (in this case, 000 years) is divided into a number of replicates, each of length equal to N years. From each replicate the rank 1 d-year total (that is, the lowest d-year total) is obtained and these are averaged. The average value gives the magnitude of the event having a recurrence interval of T years given by: T=Jl±JKj2 0.6 v ' This plotting procedure is based on the algorithm recommended by Cunnane (1978) and confirmed by the writers (Srikanthan & McMahon, 1981) for a single distribution-free formula and takes into account the bias in sample estimates of recurrence interval. By varying N from to 500 years, an empirical curve of drought

4 200 R. Srikanthan & T.A. McMahon events vs. recurrence interval may be obtained. The empirical curve can be used to determine the recurrence interval of any observed event of d-year duration. This procedure is repeated by varying d to cover all durations under consideration, resulting in a set of curves of magnitude vs. recurrence interval. It is of interest to note that Wright & Walsh (1983) have adopted a similar approach of using synthetic data in their recent study of drought frequency characteristics in New South Wales. However, in their study, two separate sets of empirical curves were presented, one for the negative run lengths and the other for the deficit sums with respect to a given base value. As a result, the deficit sums do not have an associated run length and vice versa. The use of rainfall or inflow totals of specific durations allows totals to be associated with durations and these are independent of any base values. APPLICATION OF PROBABILITY AND STOCHASTIC SIMULATION METHODS For each catchment, average catchment rainfalls were calculated on a daily basis using the Thiessen method. When rainfall data at one or more stations were not available, data from the remaining available stations were used with appropriate modified Thiessen weights. Consequently, the resulting rainfall series may not be fully homogeneous as one would expect from a single-site data set. Daily values were summed to provide monthly totals. The inflows to the various reservoirs were not gauged completely, but were estimated from some gauged streams and from reservoir level variations. Hence the inflow data are subject to estimation errors. The annual characteristics of the rainfall and inflow data are given in Table 1. Procedure for events of duration one year and less Using monthly rainfall and inflow data, sequences of annual minimum 6-month and 9-month totals were formed. For these two sets of data plus the annual series, parameters of gamma, Pearson type III and Table 1 Annual characteristics of rainfall and inflow data Strearngauging station location Length of Coefficient Coefficient Catchment Latitude Longitude record Mean of of Lag one area, (km 2 ) 'S 'E Variable (years) (mm) variation skewness auto-corr. Yarra (337) Eildon (3900) Glenmaggie (1900) Eppalock (2350) Rocklands (1450) ,

5 Recurrence interval of drought events 201 log Pearson type III distributions were calculated. Based on these, recurrence intervals of selected low rainfall and inflow events were estimated. However, only the log Pearson type III distribution gave results consistent with the historical information for all cases except the annual rainfall for Rocklands. The reason for this anomaly is that the annual rainfall data for Rocklands is normally distributed and the use of the log Pearson type III distribution is inappropriate. Consequently, a normal distribution was used for this case. Results from 6-month and annual minimum events are presented in Table 2. Table 2 Recurrence interval of lowest rainfall and inflow totals in the historical record (years) Station Yarra Eildon Gleni naggie Eppa lock Rockl; ands Variable R 1 R 1 R I R I R I Length of record Duration Method 6 months 1 year 2 years 5 years years LP.Ill data f / 60 ) 250 j > t R = I =, tnorme il dlstributi on. Proposed method for events of duration longer than one year A first order Markov model was used to generate 000 years of annual rainfalls and inflows for the five catchments. A normal distribution was used for the annual rainfalls of Eppalock and Rocklands, while the Wilson-Hilferty transformation (Wilson & Hilferty, 1931) was used to take into account the skewness in the remaining three rainfall data sets. Since all the inflow data exhibited skewness, the Wilson-Hilferty transformation was initially used for all five inflow data sets. Because the generated inflow sequences for Eildon, Eppalock and Rocklands contained a number of zero values (as a result of replacing negative values with zeros during generation) and a comparison of the frequency distributions of the historical and generated data showed unsatisfactory fits, the Box-Cox transformation (Box & Cox, 1964) was tested in place of the Wilson-Hilferty transformation. However, data generation was not possible for Eppalock and Rocklands because negative values occurred in the transformed domain and these values cannot be raised to a real power. Their occurrence could have been avoided by using a positive location parameter, but this would have produced negative values in the absolute domain as a result of subtracting the location parameter from the generated values. Hence, the Box-Cox transformation was used only for the inflows of Yarra, Eildon and Glenmaggie. For Eppalock and Rocklands, a logarithmic transformation was first applied to the data and the resulting skew in the log domain was modelled through the Wilson-Hilferty

6 202 R, Srikanthan &T.A. McMahon transformation. To validate the generated data, the sequences were broken into a number of replicates of length equal to the corresponding historical record. Annual parameters were estimated from each replicate and averaged. These are given in Table 3, along with the equivalent historical parameters. The frequency distributions of the generated and historical data were also compared. Table 3 and frequency plots (not presented) indicated that the generation models were operating satisfactorily. In this study, a short memory model was used because both the rainfall and inflow series did not show clearly any long term persistence. The average Hurst coefficient estimated by Hurst's estimator (Hurst, 1951) was 0.66 for both rainfall and inflow. Lengths of runs obtained from historical and generated data were not compared in this study. If the data clearly indicate long term persistence, then one should use a long memory model such as a fractional Gaussian noise or a broken line model. The generated annual data were then partitioned into a number of replicates of length N, varying from to 500 years. From each replicate, the smallest 1-, 2-, 5- and -year totals were obtained and averaged. The average values were plotted against the recurrence interval (that is, (N + 0.2)/0.6) and were used to estimate the recurrence interval of selected events of one year or Table 3 Comparison of historical and generated annual parameters Standard Coefficent Lag one Mean deviation of auto- Model (mm) (mm) skewness correlation Yarra Eildon Glenmaggie Eppalock Rocklands

7 Recurrence interval of drought events 203 longer duration. As an example, Fig.l shows the empirical plots of low rainfall and low inflow for durations of one to years against recurrence interval for the Eppalock catchment. RESULTS Frequency of severest droughts on record From the historical rainfall and inflow data, the annual minimum 6-month and 1-, 2-, 5- and -year totals were obtained. Recurrence intervals of events up to one year's duration were estimated using the log Pearson type III distribution, and for events of one year's duration and longer, the recurrence interval was read off from the empirical curves, like Fig.l, developed from the generated data. The estimated recurrence intervals are presented in Table 2. Duration E E S E ~a 'a tr Recurrence Interval (years) Fig. 1 Recurrence interval of low rainfall and streamflow events for Eppalock catchment. An intermediate check on the results can be obtained for events of one year's duration because estimates of recurrence interval of the severest one year drought were obtained using both the analytical method (log Pearson III distribution) and the empirical procedure (replicates of generated data). Good agreement between the two results is noted in Table 2.

8 204 R. Srikanthan & T.A. McMahon Table 4 Recurrence interval of low rainfall and inflow totals observed within the last five years (years) Duration Yarra Eildon Glenmaggie Eppalock Rocklands R I R I R I R I R I 6 months 9 months 1 year 2 years 3 years 5 years < < < < < R =. I =. Recurrence interval of 1982/1983 drought To estimate the recurrence interval of the 1982/1983 drought, a particularly severe event over much of Australia, its duration and severity were first assessed by examining the behaviour diagrams of the stored water for the five reservoirs and the corresponding rainfalls and inflows on a monthly basis over the past few years. For the five catchments, it appears that the recent drought began in April-May 1982 and continued for 12 months. In order to examine the recurrence interval of the drought in relation to longer and shorter ones over the past few years, the annual minimum 6- and 9-month and minimum 1-, 2-, 3- and 5-year rainfall and inflow totals were extracted from the historical records and the corresponding recurrence intervals calculated using the analytical and empirical approaches outlined earlier. The results are presented in Table 4. DISCUSSION OF RESULTS It is observed in Table 2 that the average recurrence intervals of the rainfall for the worst drought on record tend to be slightly larger than those for the corresponding inflow events. Furthermore, the results suggest that the recurrence intervals are not inconsistent with the length of the historical data. One would expect some rank 1 values (that is, the extreme events) to have recurrence intervals less than the length of the historical record and others greater than the length of the historical record. Such a picture is observed. Recurrence intervals in Table 4 clearly indicate the 12-month drought (1982/1983) was a rare event. The mean of the recurrence intervals of rainfall over the five catchments was about 65 years and that of inflow was about 120 years. Nevertheless, the likelihood or probability of an event of this magnitude occurring again in the next decade is between 8% and 15%. Limitations The validity of the proposed method depends on the choice of an

9 Recurrence interval of drought events 205 appropriate data generating model. The use of an incorrect model makes the results meaningless. It should be noted that the results from the proposed method are correct only to the same degree as the generating model is correct. CONCLUSIONS Based on five river basins in Victoria, the paper illustrates how stochastic data generation can be used as a tool of analysis to estimate the probability of extreme events. Estimation of frequencies of droughts of one year's duration and less is straightforward. But techniques adopted hitherto to estimate the recurrence interval of long droughts have been unsatisfactory in that the recurrence interval is directly related to the length of record and parts of the data are excluded in determining the recurrence interval of less severe events. The proposed method of using stochastic data generation to determine the frequency of occurrence of extreme droughts of length greater than one year overcomes this and other inadequacies, provided a suitable model is adopted. ACKNOWLEDGEMENT The research reported here was funded by the Australian Research Grants Scheme. The data were provided by the Bureau of Meteorology, the Melbourne and Metropolitan Board of Works and the State Rivers and Water Supply Commission of Victoria. REFERENCES Box, G.E.P. & Cox, D.R. (1964) An analysis of transformations. J. Roy. Stat. Soc. B26, Cunnane, C. (1978) Unbiased plotting positions - a review. J. Hydrol. 37, Hurst, H.E. (1951) Long term storage capacity of reservoirs. Trans. ASCE 116, Jeppson, R.W. (1967) Frequency analysis and probable storage requirements by frequency mass curve methods. Utah Water Research Laboratory Report, Utah State University, Logan, Utah, USA. Joseph, E.S. (1970) Probability distribution of annual droughts. J. Irrig. Drain. Div. ASCE 96, IR4, McMahon, T.A. & Srikanthan, R. (1982) Probability of extreme low flows of various durations. In: Optimal Allocation of Water Resources (Proc. Exeter Symp., July 1982), IAHS Publ. no Matalas, N.C. (1963) Probability distribution of low flows. Statistical Studies in Hydrology. USGS Prof. Pap. 434-A. Srikanthan, R. & McMahon, T.A. (1981) Log Pearson III distribution - an empirically derived plotting position. J. Hydrol. 52, Stall, J.B. (1962) Reservoir mass analysis by low flow series. J. Sanit. Engng Div. ASCE 88 (SA5), 21-. Wilson, E.B. & Hilferty, M.M. (1931) Distribution of chi-square.

10 206 R. Srikanthan & T.A. McMahon Proc. Nat. Acad. Sci. USA 17, Wright, G.L. & Walsh, M.A. (1983) An investigation of drought frequency characteristics in New South Wales. In: Hydrology and Water Resources Symposium (Hobart), Institution of Engineers Australia, Nat. Conf. Publ. no. 83/13. Received 24 February 1984; accepted 1 November 1984.