Selection of Truncation Level for Partial Duration Series Analysis: Applications in Canada
Accurate forecasting of flood flows is required for the efficient design and construction of hydraulic structures in rivers as well as for the effective management of water resources. Underestimation of flood flows can result in tragic consequences while overdesigned structures are expensive. One method for estimating flood flows is the partial duration series analysis. In this approach a truncation level defining the intermittent time series is chosen. All flows above this level (exceedances) are analyzed by assuming the time of occurrence of these floods to represent a Poisson distribution. In addition, exceedances are considered to be independent random variables identically distributed over a one-year time interval. The selection of the truncation level is somewhat problematic and not very well defined in the literature. This study presents a truncation level estimation technique based on a series of regional regression equations for 9 distinct regions in Canada, and documents their graphical derivation. According to previous research, the truncation level can be obtained in two ways. First, it is selected in accordance with physical criteria such as the overflowing of a river, the critical flow for flooding of a crop, etc. The second method of truncation level selection is primarily mathematical such as satisfying the analytical fit or the assumptions of the model. In the present study, the truncation level was selected based on the mathematical approach using a graphical technique and using the Chi-square test. The graphical truncation level selection technique is based on the equality of the mean and variance of the Poisson distribution. Given this property of the Poisson distribution, one can study the mean-to-variance ratio as a function of truncation level. Truncation levels were selected graphically from 238 gauging stations across Canada. The Chi-square at a level of significance of 5 % was used for their validation. It was observed, upon examination of the mean-to-variance ratio, that selection of the truncation level was easier in the eastern and western regions of Canada where the ratio varied very little around unity. In the prairies and northern regions, a small variation in the truncation level led to a large variation in the ratio, making the selection of a level more difficult. Following the selection of a truncation level for each hydrometric station, homogeneous regions were selected based on previous countrywide hydrological studies. The regression analysis was then carried out to explain this truncation level using several variables (including bath physiographic variables and streamflow characteristics). Physiographic variables included area of drainage basin (km2); area of lakes and swamps (km2) ; area of forests (km2) ; mean elevation (m) ; slope of drainage basin (%) ; slope of principal watercourse (m/km, %) ; length of principal water course (km) ; area controlled by lakes and swamps (km2) ; and drainage density (km/km2). The Streamflow characteristics included the mean annual flow and mean annual flood. The mean annual flood was included because previous research has shown a strong correlation between the truncation level and the two-year flood estimated by a log-Pearson type III distribution function. For ease of application, the mean annual flood was used as a low return flood estimate as it represents the two-year flood calculated by a normal distribution function. Low return floods of any given distribution function are demonstrated to be similar in magnitude. Multiple regression was carried out using original data and logarithmically transformed data. Equations were derived by selecting one variable, two variables, and so on until all explanatory variables were accounted for in a single equation. Among the selected parameters, the mean annual flood was the best parameter to explain the truncation level for all of the regions across Canada. The coefficient of determination, R2, of the equations calculated using mean annual flood varied between 0.847 and 0.987 from original data, and varied between 0.824 and 0.988 for logarithmically transformed data. In practice, the regional equation can eliminate the graphical estimation technique and a first truncation level can be obtained. This truncation level can be used in the partial duration series analysis with the different tests involved. If this first estimated level does not meet all the statistical tests, a second usually higher level is selected. It is important to note that when the selected level meets the statistical test requirements, the estimation of QT should be relatively insensitive to levels near the selected truncation level.