**Title:** *Probability Distribution for***Author:** M D Zaidman**Author:** V Keller**Author:** A R Young**Author:** Environment Agency**Document Type:** Monograph**Annotation:** Environment Agency Project ID:EAPRJOUT_1265, Representation ID: 425, Object ID: 2475**Abstract:**

Probability distributions describing the occurrence of D-day annual minima flow events have been determined for twenty-five British rivers having long, stable and natural flow records. In each case a pre-determined set of rules and criteria were applied to the flow record to derive time series of D-day annual minima and to ensure that these were both stationary and independent. So that low flow events of different duration could be examined a range of values were used for D including 1, 7, 30, 60, 90, 180 and 365 days. For each time series, estimates of the non-exceedance probabilities corresponding to the annual minima were derived using the Gringorten Plotting Position Formula. These were then used to build the observed probability curve. It is important to note that the shape of this curve is influenced by the number and rank order of the annual minima derived for each series, as well as the range of annual minima observed. A representative probability distribution should describe the frequency of occurrence of events beyond the observed range. However, as observed data usually represents only the central portion of the true probability distribution of annual minima, the shape of the tails of the distribution must be inferred from theory or experience. Extreme Value Theory suggests that the frequency behaviour of annual minima will follow that of a Generalised Extreme Value distribution. However Pearson Type III and Generalised Logistic distributions are commonly used for this type of analysis. Using the L-Moment method of parametric estimation, the parameters of each of these three candidate distributions were determined based on the flows and probabilities of the observed data. A fourth distribution, the Generalised Pareto, was also investigated. Although this distribution was not theoretically suitable for describing extreme events such as annual minima, it was included to provide a control. In order to differentiate the most representative of the four parameterised distributions, their descriptive and prescriptive characteristics were considered. Goodness-of-fit tests and root mean square errors (RMSE) were used to quantify the ability of the modelled curve to match the observed data. The results indicated that there was little difference between the performances of the different distributions, and therefore that with the best goodness-of-fit and lowest RMSE values was considered, in a descriptive sense, the most representative. The analysis showed that no one distribution best represents the frequency behaviour of annual minima. However, for annual minima of short duration (values of D in the range 1 to 30 days) the low flow frequency curves in permeable, high storage catchments tend to be best described using the Generalised Logistic or Generalised Extreme Value distributions. In contrast those for the low storage catchments tend to be best described by the Pearson Type-III or Generalised Extreme R and D TECHNICAL REPORT W6-064/TR2 ii Value distributions. Where the annual minimum is of long duration, with values of D above 90 days, many annual minima series were best described by the Generalised Pareto distribution. The shape of the flow-return period relationships derived was examined to qualify the predictive ability of each distribution, i.e. to determine whether the tails of the distribution provide sensible estimates of return period for given annual minima and vice versa. In general sensible estimates are obtained for annual minima of short duration (D=1 or D=7 days) where the prescribed flow is less than 10% of the mean flow, only where catchments are impermeable. As annual minima of longer duration are considered, or as the catchment type is permeable sensible estimates are obtained only for higher prescribed flows. The methodology was also applied to streamflow records shorter than 20 years in length. However, as it is unrealistic to expect to identify a representative probability distribution where there are few observed data points, a slightly different approach was adopted. Properties of the probability distribution for each short-record were inferred from that of an analogue site, having similar low flow behaviour and catchment characteristics but a much longer flow record. Two variations of this approach were considered. Firstly analogue catchments were assumed to have similar probability distributions (provided that the flow values were suitably standardised by, for example, expressing as a ratio of the mean annual minimum value). Hence the probability curve of the long-record analogue was re-scaled by the short-record mean flow to provide an estimate of that for the short record catchment. Although this method proved successful for certain catchments, for others it provided poor results. In the second approach the true probability of non-exceedance for the annual minima occurring in a particular year within a short record was assumed to be equivalent to the probability of the annual minima occurring in the same year of the long record. This method generally provided fairly accurate predictions of the probability distribution, but requires the flow records for the short-record catchment and its long-record analogue being wholly coincident. R and D TECHNICAL REPORT W6-064/TR2 iii CONTENTS FOREWORD i EXECUTIVE SUMMARY ii LIST OF SYMBOLS vii LIST OF ABBREVIATIONS vii 1. 1.1 1.2 1.3**Publisher:** Environment Agency**Subject Keywords:** Low flow; Annual minima; Flow-duration-frequency models.**Extent:** 306**Permalink:** http://www.environmentdata.org/archive/ealit:4736

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