The GLUE procedure recognizes the equivalence of different sets of parameters in the calibration of models. It is based upon running a model with different sets of parameter values chosen randomly from the specified spatial distributions. Many papers have applied this methodology and emphasize the effects of the likelihood measure in the whole applying process [9] [10] . The term “likelihood” was used in a general sense, as a fuzzy, belief, or possible measure of how well the model conforms to the observed behavior of the system [4] , yet not in the restricted sense of maximum likelihood theory which is developed under specific assumptions of zero mean, normally distributed errors [11] [12] . Moreover, it is subjective to choose a suitable threshold for the likelihood measure to identify the behavior of the model. In the past studies, usually the threshold was chosen subjectively on the scale of some summary goodness of fit index [13] [14] [15] . And the GLUE method is used well in uncertainty research [16] [17] .

Here in the application of GLUE methodology, for each set of parameters, whether the model is behavioral or not is determined by the likelihood value on a basis of comparing predicted and observed responses.

The requirements of the GLUE procedure are given as follows:

1) A formal definition of a likelihood measure. At this stage it is worth noting that a formal definition is required but the choice of a likelihood measure will be inherently subjective.

2) An appropriate definition of the initial range and the distributions of the parameters to be considered for a particular model.

3) Definition of a feasible threshold value.

The NASH model [18] [19] [20] [21] is a conceptual hydrological concentration model developed by Nash, J.E., and it is widely used in the watershed concentration simulation [22] [23] . In the model, the research basin is divided into a series of identical reservoirs, and the reallocation of the net rainfall in the catchment is assimilated to be an adjustment of the reservoirs. So the instantaneous unit hydrograph (IUH) can be deduced as Eq. (1):

where is the y-coordinate of instantaneous unit hydrograph; is the Gamma function; reflects the regulation and storage capacity of the basin, and it could be the number of the reservoirs, termed shape parameter; is the storage-discharge parameter of the reservoirs, termed scale parameter. The Nash model with its clear conception and simple structure has been used extensively in flood forecasting [24] [25] .

According to Eq. (1), when the net rainfall is given, the flow hydrograph at the basin outlet can be deduced as

To avoid the input-output uncertainty and the model structure uncertainty, the concept of “ideal data” is proposed to do the research of model parameters uncertainty.

On the basis of the physical interpretation of the hydrological model and the characteristics of the research catchment, the model parameter spaces can be determined based on the prior information. Choose one set of parameters randomly in the parameter spaces as the “true parameter values”. First assume the input data has no error, then the input is called “ideal input”, and the “ideal output” can be calculated by using the “true parameter values” in the model.

For NASH model, the input is net rainfall and output is flow at the basin outlet.

The GLUE methodology has been first applied to the real data (observed discharge) research with likelihood measure L1. In this study the choice of a model efficiency rejection criterion (<0) has been included, which assure all of the possible modeling results can be contained.

Scatter plots of parameters and based on the likelihood measure L1are reported in Figure 1 and Figure 2 which show the effects of using different flood events with real input-output data. It can be seen that good and poor simulations are available virtually throughout the parameter ranges, and the effects of different flood events for parameters and are seen clearly. It can be inferred that there is uncertainty of input-output data or others simultaneously to influence the distributions of the scatter plots.

The results shown in Figure 1 and Figure 2 suggest that the parameter response surfaces are very complex and for different data that is for different events they vary showing the uncertainty of data.

It is difficult in rainfall-runoff modeling to bracket all the discharge observations (or other predicted variables) within 90% of the calculated confidence bounds, as seen in Figure 3, the discharge is often outside of the 90% confidence bounds, which furthermore proves the data limitations as well as the model structure limitations.

Figures 4-6 show the distribution of errors, , , computed considering the four flood events under real data research, referring to peak discharge, , discharge 10 hours after peak time, , discharge 10 hours before peak time, , respectively. It is easy to see from the figures that the dis- tributions are non-Gaussian and vary in shape through time under the existence of parameter uncertainty, data uncertainty and model structure uncertainty.

Then the GLUE methodology has been applied in parameter uncertainty research by proposing the “ideal data” to avoid the influence of other uncertainty factors. Following the above generation steps of the “ideal data”, one set of parameters is chosen as the “true parameter values”:, then the model is determined. With the ideal input of net rainfall data and the confirmed NASH model, the output data of the model will be obtained to be the ideal output data. So there is no uncertainty for the input data, output data and

model structure, under which situation the NASH parameter uncertainty is simulated by GLUE methodology. Here the choice of a high model efficiency rejection criterion (<0.90) has been included, since this is one way of trying to estimate the relationship of the two parameters.

Figure 7 and Figure 8 are scatter plots of efficiency results for parameters and under ideal data. It is easy to see that each parameter has very similar efficiency under different flood events. By Comparing Figure 7 and Figure 8 with Figure 1 and Figure 2, it also can be inferred that there are no other uncertainty factors affecting the parameters uncertainty research.

Comparing Figure 9 to Figure 3, it is easy to prove that all of the ideal discharges in NASH modeling are bracketed within the 90% calculated uncertainty

bounds, as shown in Figure 9. And for different flood events with ideal input data, the results are consistent, which demonstrate that there are no other uncertainty factors except the model parameters and assure the research accuracy of the model parameter uncertainty and correlation estimation. Obviously, the ideal data can assure the accuracy of the parameter uncertainty research.

Figure 10 shows the response surface of parameters and with goodness of fit represented as contours and suggests the obvious correlation and uncertainty of the two parameters.

Figure 11 gives the scatter plots of parameters and with likelihood

value greater than 0.98 trying to find the ridge line, from which we can assume that the distributions of the scatter plots can be matched by the function as below:

where is equal to 3.82.

For the ideal data analysis, the uncertainty assessment refers only to the parameters, so it is accurate to analyze the error distributions of the discharges as the parameter errors. Same as the real data research, Figures 12-14 show for

each flood event the error distributions of three discharge values, which demonstrate how distributions are non-Gaussian and vary in shape through time under the single existence of parameters uncertainty.

Figure 15 is the research result under idea data based on the likelihood L2 with parameter N = 20. For the four flood events, all of the “real parameters of model” are obtained in the figure.