Keywords: k-means fuzzy clustering, catchment characteristics, calibration, validation, low flow.

In this paper a new approach is introduced whereby the location and scale parameters of the Extreme Value Type 1 (EV1) distribution of the annual minimum, 3-, 7-, 10-, 15-, and 30-day sustained low flow series are estimated from catchment characteristics. Two different models representing this approach have been constructed and their performance was tested using data from 55 flow measuring stations in the Shannon River Basin (Ireland).

The first model is a linear relationship relating each of the EV1 parameters with various combinations of five catchment characteristics as follows: (Sc1) area and rainfall; (Sc2) area, rainfall, mean elevation, and mean slope; (Sc3) area, rainfall, mean elevation, mean slope, and soil type; (Sc4) area, rainfall, mean elevation, mean slope, and geology; and (Sc5) area, rainfall, mean elevation, mean slope, and aquifer category. The optimum values of the EV1 parameters for the five low flows series in each station were obtained by fitting the EV1 distribution to each of the five low flow series using the method of probability weighted moments [1].

The second model applies the k-means fuzzy clustering method [2] to divide the flow measuring stations into two homogenous groups each of which is represented by a separate linear relationship similar in form to that in the first model. Each station was assigned a weight proportional to its closeness to each group, as calculated with an exponential membership function. A separate linear model relating the EV1 parameters to catchment characteristics is fitted to each group. Then the final predicted values of the EV1 parameters were calculated as a weighted (according to its membership function) average of the two linear models.

The parameters of the two models were calibrated using both the unconstrained and the constrained least squares methods. The calibrated models are validated by applying them to 20 stations not used in the calibration, i.e. the split sample method. In general the validation results highlighted the value of the new approach and indicated that:

1. The best estimate of the EV1 location parameter was obtained from the catchment variables (Sc3: area, rainfall, mean elevation, mean slope, and soil type) using the unconstrained simple model, and
2. The best estimate of the EV1 scale parameter was obtained with the catchment variables (Sc2: area, rainfall, mean elevation, and mean slope) using the constrained fuzzy clustering model, indicating that it is a more difficult parameter to estimate.

1

J.A. Greenwood, J.M. Landwehr, N.C. Matalas, and J.R. Wallis, "Probability weighted moments - Definition and relation to parameters of distributions expressible in inverse form", Water Resources Research, 15(5), 1049-1054, 1979. doi:10.1029/WR015i005p01049

2

J.A. Hartigan, and M.A. Wong, "Algorithm AS 139: A k-means clustering algorithm", Applied Statistics, 28, 100-108, 1979. doi:10.2307/2346830

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