Background



Velocity Distributions in “Natural” Stream Channels

÷A Second Look÷

John L. Kelliher

University of New Hampshire, Fluvial Hydrology

5/10/00

Introduction

Because so much of quantitative fluvial hydrology entails calculations involving the velocity of a given flow, it is of great interest to gain an ability to estimate what that velocity will be at any particular time in any specified reach. Velocity is used in the calculation of discharge through a single cross-section or a reach or river, as well as in the estimation of the total discharge for the entire basin. It is integral for the estimation of Manning’s “n”, Chezy’s “C”, energy slope, and other friction factors that make up the resistance of a given system to a flow. It is also used to estimate stream power, which is related to bed form development, and mass transport issues such as entrainment and deposition. Yet another use would be in the estimation of the energy coefficient (α). For these reasons and others, we would like to be able to estimate the mean velocity of a flow.

Historically, this has been accomplished through several methods. Widely utilized are the velocity profile method first described by Prandtl (1926) and later by Von Karman (1930) (Dingman, 1984). The USGS currently uses the six-tenths method for shallow flows (Y < 0.75 m), whereby the average velocity is assumed to exist at a depth of 0.6 Y, where Y is the depth of the flow at the site of measurement. For deeper flows (Y > 0.75 m) the USGS uses the Two-Tenths and Eight-Tenths Method described by Dingman (1994). Velocity can also be estimated using the Surface Velocity or Float Method. It can be back calculated through dilution gaging, flumes, and weirs, and slope-area measurements discussed in Dingman (1994). Each of these methods involves some degree of inaccuracy due to errors involved in the measurement or observation of velocity. These methods can also be dangerous at high flows due to increased velocity, turbulence, and debris. Miscalculations can also result from erroneous assumptions about the resistance of the reach or that of constant discharge through the reach if velocity is back solved through Manning’s equation or Chezy’s equation.

The main thrust of this research is to estimate a probability distribution for velocity measurements of natural stream channels. If we know which probability distribution best describes a sample population, we can use the quantile function, f(x), to obtain quantile estimates of the sample population from which we will obtain our mean velocity. Central to this project are L-moments, which are a statistical method of examining data sets that exhibit large skew, as is commonly found in hydrologic data. This process will allow us to estimate the probability distribution from which a sample of velocity is drawn, without having to “get our feet wet” so to speak. If we know the probability distribution of a sample population, we gain many insights into the data, and begin to be able to make certain inferences about the data. These insights include information about the shape, location, scale, and dispersion of the distribution.

Statistical Background

Hosking and Wallis (1997) have described probability distributions as the relative frequency that each sample of a given magnitude occurs in a sample population. In our example, these samples are velocity measurements. Therefore, x.50 is equivalent to the median velocity; 50% of the measurements will be greater than x.50, 50% will be less. Another way that we can quantify probability distributions is through the use of non-exceedence probabilities:

F (xi) = Pr [X ( xi], [Eqn. 1]

or the probability that X is exceeded by xi. Here F(xi) is the cumulative distribution function (cdf) of X and is a monotonically increasing function for 0 ( F(xi) ( 1. Thus, if we know F(xi), we also know every possible value of X. We can also estimate the values of xi through the use of the probability density function (pdf) of X. The pdf can be shown to be the derivative of the cdf and can also be plotted as a curve. The area under this curve and bounded by the X-axis is equal to one, and we can then find the probability that A ( xi ( B by subtracting F(A) from F(B), that is, through an application of the fundamental theorem of calculus (Walpole, 1982).

Since we most often do not know every possible value of X, we are forced to estimate these values for a sample population. This is often accomplished through the use of quantile estimators. For example, the median quantile estimator of a sample population, x-hat.50 is the value of xi that is described by:

Pr{X < xi} ( .50 ( Pr{X ( xi}. [Eqn. 2]

By accurately describing the quantiles of the probability distribution, we are describing how X is distributed, and we can begin to understand some important things about X. For instance, x-hat.50, or the median value of X, illustrates the central tendency of the distribution. We can also express the dispersion of the distribution as x-hat.75 minus x-hat.25 as described by Dingman (1994).

The Method of Moments has been traditionally used to describe probability distributions. This method involves the determination of the sum of the differences between xi and the mean value of X, exponentiated to various powers. Hosking and Wallis (1997) describe this as bringing the quantile function to higher powers. For example, the variance of a distribution (the second moment of the distribution) is:

[pic], [Eqn. 3]

We can clearly see that the difference between xi and the mean (() is brought to the second power. Similarly, the third and fourth order moments (skew and kurtosis) are brought to the third and fourth powers, respectively. Therefore extreme values of xi will yield even more extreme estimates of the higher order moments.

This leads us to a critical problem in using the Method of Moments to evaluate hydrologic data. Because hydrologic data are often highly skewed, with many extreme values, the Method of Moments often renders inaccurate estimates of the higher order statistics. Since probability distributions of a random sample can be estimated by plotting the coefficient of skewness against the coefficient of kurtosis on a moment-ratio diagram, these deviations from the true values of skew and kurtosis can lead to invalid estimates of the probability distribution.

Hosking (1990) and Hosking and Wallis (1997) have suggested a solution to this problem. They propose to describe a probability distribution of a sample population through the use of L-moments. L-moments differ from traditional moments in that they do not involve the exponentiation of the deviation from the mean that was so problematic in the use of hydrologic data with the Method of Moments. L-moments instead use linear combinations of the probability-weighted moments (see Greenwood et al. (1979) and Hosking and Wallis (1997) for a complete description). This means that inferences that are based on the L-statistics (L-mean, L-variance, L-skewness, and L-kurtosis) have less bias and should also be more reliable estimators of the shape of the curve that is the probability distribution of a given sample.

The use of L-moments for the determination of probability distributions of hydrologic data is becoming more common, because of their robust behaviors with regard to the highly skewed data often encountered in hydrologic samples. Vogel and Fennesey (1993) investigated stream discharge in Massachusetts using L-moments, and found that L-moments could be used to discern from which probability distribution the streamflow data arose from. In their paper, they compare L-moments with the traditionally used Method of Moments and concluded that the Method of Moments provides little information about a sample’s probability distribution. Vogel et al. (1993b) showed L-moments could be used to estimate stream discharge in Australia. They found that the continent receives annual precipitation in a bimodal manner. They divided their total population into two subsets based on this bimodal hetereogeneity. Their analyses showed that drainage basins that received the bulk of their annual precipitation in the winter could be modeled by a Generalized Extreme Value (GEV) distribution, and that basins that obtained the bulk of their precipitation in the summer were better approximated with a Generalized Pareto distribution (GPA). Other efforts have been made by Hosking and Wallis (1987a) and by Hosking (1990) to describe how the scientific community could benefit through the use of L-moments and that L-moments should be expressly used when analyzing hydrologic data in particular.

Ben-Zvi and Azmon (1997) used L-moment theory to describe extreme discharges in 68 basins in Israel. While they were able to find that the L-moment method can provide reasonable estimates of a sample’s probability distribution, additional statistical analyses needed to be conducted due to the subjectivity of choosing the “best” distribution. They showed that the Anderson-Darling test could be used to assess how well a chosen distribution could re-create a sample population. Future research in this direction could easily be applied in the estimation of velocity distributions in natural stream channels, or in the estimation of flow duration curves.

Van Gelder and Neykov (1998) used the Method of L-Moments in a study that attempted to characterize flood events in the Netherlands with regional frequency analysis. Their report stresses the importance of accurately identifying and defining homogeneity of a region. Following the examples of Hosking and Wallis (1997), they used a discordancy measure to distinguish heterogeneity. From this close examination of homogeneity, their conclusions were that a completely homogeneous region was very difficult to obtain for the Netherlands because of the varied geography and ocean influences. However, they also inferred that regional frequency analysis (and the use of L-moments) was more accurate than single-site statistical surveys.

Method

Dingman (1989) showed that the distribution of streamflow velocity data could be estimated from the Power Law distribution. He explained that knowledge of the probability distribution of discharge velocities would allow scientists to make estimates of the energy coefficient, which could then be used to make better estimations of reach resistance. Haire (1995) collected cross-sectional streamflow velocity data and compared Dingman’s (1989) approach with the Prandtl-Von Karman Universal Velocity Distribution Law and a Maximum Entropy model. She showed that the Power Law could account for over 80% of the variation between observed and predicted streamflow velocities at the 95% significance level, whereas the Prandtl-Von Karman method could only account for approximately 70% of the variation. She concluded that none of these statistical distributions could accurately describe the velocity data at the 95% significance level, but that some could be used with discretion as the 90% significance level.

This study will re-examine the data collected by Haire. She obtained 27 velocity distribution sample measurements from 20 streams in New York. Haire generated 22 additional velocity measurements through flume experiments at the Cold Regions Research and Engineering Laboratory (CRREL) in Hanover, NH. She also included velocity measurements from previously cited work (Haire, 1995) to obtain a sample population of N = 81. This study will focus solely on those cross-sections with positive velocity measurements. Therefore, our sample size is somewhat decreased to N = 39, with 26 samples taken from the natural cross-sections, and 13 from the flume studies.

This study will use the Method of L-Moments to estimate from which probability distribution these velocity measurements are drawn. This assumes that the entire data set is homogeneous, that is that each sample shares the same probability distribution, but this assumption will be shown to have been erroneous. If each of the samples shares a similar probability distribution, we can say that these data compose a homogeneous subset of the original data set. The size of this subset is limited by the size of the total population and zero. Clearly, we would like to include all the sample data from the population into this homogeneous subset, but this is often not the case due to heterogeneity of the population. Monte Carlo simulations will be used to attempt to describe the degree of homogeneity (or heterogeneity) of the region in a visual manner. Knowing the probability distribution will enable us to obtain quantile function values of velocity, from which the mean velocity can be estimated.

Estimates of the L-skew and L-kurtosis for each of the 39 samples were calculated from the formulas derived by Hosking (1990). These values were plotted on an L-moment ratio diagram (Fig. 1) from which we will attempt to determine which of the possible probability distributions is most likely to be representative of the sample population. L-moment diagrams are similar to customarily used moment-ratio diagrams. The L-moment-ratio diagram plots the coefficient of L-skewness against the coefficient of L-kurtosis. As each probability distribution has a unique plot on this type of diagram, the distribution that most closely approximates the coefficients of L-skewness and L-kurtosis for each sample in the population is taken to be the probability distribution from which the entire population is drawn from. To re-iterate, knowledge of the probability distribution of the sample population provides us with a means to estimate the various quantile measures for the distribution.

Ideally, we would like the data (the coefficients of L-skewness and L-kurtosis) to plot such that they fall directly on one of the theoretical distributions on the L-moment-ratio diagram. In some situations (see Vogel and Fennessey, 1993), the data very closely approximates a particular distribution and the choice is simple. However, this is not always the case. Van Gelder and Neykov (1998) and Sankarasubramanian and Srinivsan (1999) found that their data did not plot in a quasi-linear manner, and in fact, looked more like a cloud of data points.

This is the case as well with Haire’s (1995) data set. The L-statistics show increased variation and thus plot as more of a cloud type feature (Fig. 1). This poses some problems because it appears that several distributions could be used as potential probability distribution estimates. These include the GPA, GEV, and Pearson Log-normal (PE3) distributions. Because of previous familiarity with the GPA distribution, this paper will examine how well it can estimate streamflow velocities. We will then perform a Monte Carlo simulation to assess how close to the GPA our observed data really is. One possible reason for this variation is due to the likelihood that these sample data sets do not form a homogeneous subset. It would not take much imagination to see several subsets in this data as indicated in Figure 1.

Even though several of these cross-sections have L-skew and L-kurtosis values that indicate that they would probably be better modeled using a different distribution, we will attempt to model them using the Pareto distribution. This ambiguity is probably indicative of a non-homogeneous "region", which would make the results of this "regional frequency analysis" quasi-invalid, but still more statistically sound than (and therefore preferable to) the Method of Moments (Van Gelder and Neykov, 1998). The statistical analysis of the other distributions that may fit this data is beyond the scope of this project, but certainly would be worth pursuing as future research.

The Pareto Distribution

Hosking (1990) describes the generalized Pareto distribution as:

[pic] [Eqn 5]

where y = -κ-1log{1-κ(x-ξ)/α} when κ ( 0,

or y = (x-ξ)/α when κ = 0,

f(x) is the probability density function,

ξ, α, and κ are parameters related to the location (mean), scale (variance), and shape (skew) of the data respectively.

The quantile estimators for the GPA distribution are likewise found as:

x(F) = ξ + (α / κ)( 1 − ( 1 − F )κ ) when κ ( 0 [Eqn. 6]

or

x(F) = ξ + α log (1-F) when κ = 0, [Eqn. 7]

where x(F) is the quantile function of the cumulative distribution,

and F is the probability that the actual value of velocity is at most x, or the non-exceedence probability.

The three parameters involved in the above equations are estimated by linear combinations of the L-moments calculated from the observed values of each of the samples in the population. The value of kappa can be estimated for each sample in the population by a linear combination of the L-moment ratios that were used to determine the probability distribution of the sample population. We can then estimate the other parameters, ξ and α, through linear combinations of kappa and second L-moment, the same scaling factor (λ2) that was used to determine the coefficients of L-skewness and L-kurtosis. Since ξ is a measure of location, it represents a lower bound for the estimation of velocity. For example, if we were to eliminate any sample that had observed negative velocities, we would expect that ξ would be 0. Hosking and Wallis (1997) provide a thorough discussion on the derivation of these parameter values.

Once these parameters are known for each cross-section, it becomes quite trivial to calculate the quantile estimators of the actual streamflow velocity data. The trick then is to determine how κ, α, and ξ are related to the cross-sectional geometry or other factors, so that they can be estimated for ungaged and unmeasured reaches. Once this is accomplished, we will have a method or estimating streamflow velocity in a cross-section literally without having to get our feet wet.

Monte Carlo Simulations

Now that we have chosen (albeit qualitatively) the Pareto as the distribution that best describes our data, we need to ensure that this is the correct choice. The test that we propose involves the use of the parameter values that were generated from the L-moment analyses just discussed for each cross-section. For each cross-section, we will use the Pareto distribution quantile equation (Eqn. 6) in a Monte Carlo experiment that will attempt to re-create our observed data. 99 synthetic velocity distributions were generated for each sample cross-section for which we have velocity distribution data. Then each of these simulation distributions was run through a spreadsheet that calculated the L-moments and parameters (α, κ, ξ) of that simulation of velocity measurements.

Because we know that these simulations follow a Pareto distribution, we know as well that the calculated L-skew and L-kurtosis values for these simulations should plot on the Pareto distribution on an L-moment ratio diagram. If we then plot our observed L-skew and L-kurtosis values on this graph as well, we can get an idea of how reasonable our assumption was about our choice of probability distribution. That is to say, if our observed values fall within the cloud generated by a random Monte Carlo simulation of the Pareto distribution, we can assume that the probability is high that we have made the right choice. Likewise, if our observed values do not fall within this cloud, we know that the Pareto distribution probably does not describe our data very well.

This test has been performed for the 26 selected natural cross-sectional velocity distributions. The results of these efforts are included in Appendix 1. These graphs show that the Monte Carlo simulations tended to fall on or about the GPA line as expected. The grouping of these 100 simulations varies widely, owing to the variation in the sample size. Larger sample sizes tended to be more closely grouped than smaller sample sizes, showing perhaps that larger sample sizes result in less bias. Also plotted on these diagrams are the L-skew and L-kurtosis values from the observed velocity distributions. If the observed velocity distributions were indeed distributed in an exact Pareto manner, we would expect the plot of L-skew and L-kurtosis values estimated from the L-moments to fall on the line of the GPA.

In fact, many of these observed velocity distribution L-skew/L-kurtosis values fall either within the cloud of Monte Carlo generated values, or near to it. This is encouraging because this means that we have at hand a simple method of checking the validity of assuming a homogeneous sample in regional frequency analysis. The CRREL flume study data (Appendix 2) also show these characteristics. One flume cross-section (T31600ML) showed anomalous behavior in that the observed L-skew and L-kurtosis values did not plot inside or near the cloud of Monte Carlo generated values. This cross-section had a moderate number of sampled observations (N = 100), but had the largest discharge (1852.96 GPM), high slope (= 0.010), and a relatively low roughness (mesh), which perhaps contributed to this anomalous behavior. Because this anomaly is so blatant, we may choose not to include this sample due to its “heterogeneous” behavior. Other samples where the observed L-moments plotted either outside the cloud or on its periphery should be further scrutinized for heterogeneous characteristics. These heterogeneous characteristics may quite possibly show up in the regression analysis of the channel geometry to estimate the parameters of the Pareto distribution.

Multiple Linear Regression Analysis

It was hypothesized that alpha, kappa, and xi are related to channel geometry and/or other factors. One method of relating these parameters to channel geometry is through the application of multiple linear regression. The average values of the Pareto distribution parameters (α, κ, and ξ) that were calculated from the Monte Carlo simulation were regressed against channel geometry information for each CRREL cross-section using a step-backward approach. Haire’s data (1995) is listed for these selected cross-sections in Table 1. Again, only samples with non-negative velocities were used, so N = 13 samples. We did not use Manning’s n or average velocity as regressors because velocity is implicit in their calculation and this would introduce multicollinearity into our model.

Step-backward regression analysis is a process whereby regressors are eliminated in a sequential manner until the model contains only statistically significant regressors. It is up to the modeler to specify what significance level they wish to use. In our example we originally used a 90% significance level. The results of these regression analyses are included in Appendix 3. We can see from the studentized residuals that several of the residuals are potentially outliers. Studentized residuals are distributed such that they have a zero mean and a unit standard deviation. Therefore, we can use the empirical rule to estimate how likely that observation could occur by chance. A studentized residual of greater than about 2.5 is usually indicative of an outlier. Although none of the studentized residuals in our analyses have values greater than 2.5, several have residuals that approach 2.0, and these have been highlighted in yellow. These observations with large studentized residuals may be outliers, but more statistical work would need to be done to prove this, perhaps by using SAS or some other statistical package to analyze leverage and /or influence of the observation. A thorough discussion of these statistical tests is found in Freund and Wilson (1998).

We can see from the multiple linear regression analysis at the 90% significance level that all three parameters could be regressed against various combinations of channel geometry. This is proven by examination of the F-test values for each regression analysis. Since the “significance F” values are less than 0.1, we know that each of these regression analyses are significant at the 90% level. Once we know that the regression is valid, we can obtain estimates of the values for each parameter at the 90% significance level by using the coefficients of each regressor in a linear manner:

κ.90 = 6.871 + 43.8(AREA) – 52.3(H.RADIUS) –146.7(DEPTH) + .001(Q) [Eqn. 8],

α.90 = 104.4 – 259.9(H.RADIUS) + .054(Q) [Eqn. 9],

ξ.90 = 18.816 – 10.74(AREA) + .018(Q) [Eqn. 10].

Appendix 4 shows the results of these analyses in application. These graphs plot the observed velocities against the predicted velocities, a one-to-one line for the observed values, and an average velocity measurement taken from the observed data. We used the estimated parameter values to calculate a Pareto distribution of simulated velocities. We then compared how these parameters fared at predicting velocity distributions by comparing the two plots.

We can immediately see that the fits are fair at best. Interestingly, the velocity distributions generated through the estimated parameters for higher discharges seem to fit best (See plots for T3 1600 AH, T3 900 ML, T3 1600 TH, T3 1600 ML). The plots for the smaller discharges don’t appear to fit very well. The results for this were surprising because of the previous discussion on Monte Carlo simulations and graphing of the L-moment diagrams in Appendix 2. The only “stream” that was expected to not do well was T3 1600 ML because it’s observed L-moments plotted outside the cloud of numerically simulated L-moments.

In many cases, the general slope of the distributions generated by the regression relationships is the same as that of the observed distribution, but there appears to be a translation of the curve. By translation, we mean that the curve appears to need to be shifted left or right to achieve the best fit. At this point the reason for this is unknown. However, the fit is quite good in several of the cross-sections.

With these conundrums in mind, we opted for a recalculation of the regression for the parameter estimations at the 95% significance level. Perhaps with these “stricter” stipulations, we could obtain more rigorous estimations of the parameters.

Unfortunately, at the 95% significance level, we were unable to find any combination of regressors through the step-backward method that was statistically significant for Kappa or Alpha. Examination of the regression analyses (Appendix 5) shows us that the regression for Xi is significant at the 95% level (F-test), and this provides us with a more rigorous method of estimating this parameter. The equation for Xi at the 95% significance level is:

[pic], [Eqn. 10]

which is the same equation that we generated at the 90% significance level. Therefore, we propose to use the equations generated from the multiple linear regression model at the 90% significance level.

Table 2 further re-iterates the fact that this model doesn’t predict observed velocities for all the streams, particularly at low flows. Indeed, it predicted velocities that were up to 400 times the observed velocities. However, the model does perform well for larger discharges, achieving a lowest % change of about 3.4. The average % change between the observed and predicted velocities was on the order of 97.4 %. The average percent change for the largest discharges (T3 1600 XX) was much better (about 14.3 %).

We find that for moderate discharges (T3 900 XX), the average percent deviation was about 42 %, but for lower discharges (T3 100 XX), this percentage is significantly larger (328 %).

Future Studies

Clearly, this study was shown that velocity distributions are difficult to model. One of the key tenets of regional frequency analysis is that the data sample must form a homogeneous region (Hosking and Wallis, 1998). When the data is heterogeneous, as is probably the case in this example, the results of the analyses can be quite spurious. This study has sparked much interest in this researcher, and the employment of the Monte Carlo experimentation was very interesting. Future research could include examination of some of the other probability distributions that appeared to fit the cloud of L-moments, such as the LN3, PE3, or segregating the sample into homogeneous “regions”. These regions need not be in a spatial sense, but more as a sense of the characteristics of the distribution of the data. We can quickly examine this by looking at Appendix 6, which is essentially Figure 1 without the natural stream samples. This L-moment diagram shows us how varied the CRREL data set is, which is a good indication that this data set is not homogeneous. We can clearly see that the data doesn’t fit the Pareto very well, although some points are close to the line. The average values of the L-moments for this data set fall into a probabilistic "no man's land," one that is not really modeled by any one distribution. Perhaps the Wakeby five parameter or the Kappa distribution would do well with these data because these distributions plot as an area on this type of diagram, and this might fit our cloud better.

In particular, the three points near the apex of the GPA parabola appear to fit very well. These points correspond to samples T3 900 BL, T3 1600 AH, and T3 1600 AL. However, even though these observations should be distributed in a quasi-Pareto manner, examination of these three samples in Appendix 4 shows that only predicted values T3 1600 AH appears to fit the observed data well.

One interesting thing to note about this is that when we compare the graph of Appendix 6 with the individual graphs in Appendix 2, we can get an idea of the scale of the graphs. A point that appears to be "close" to the Pareto in the graphs of Appendix 2 may in reality be quite far from the Pareto distribution, and in fact may well be modeled better with another distribution. T3 1600 ML, which represents our "best fit" for the average velocity (% change = 3.4) falls not on the Pareto distribution, but almost directly on the LN3 plot. The more one looks at this graph, the more one realizes that these data have serious homogeneity issues…for example, T3 900 BL almost falls directly on the GPA line, yet still has a % error of about 75%.

For these reasons among others, we recommend that future research be directed at the proper identification of homogeneity, because this seems to be a central issue in the use of L-moments. Mis-identification of a probability distribution yields erroneous estimations of its parameters, which in turn generates incorrect estimates of its parameters. If we have values for the parameters that are not correct, we obviously cannot use regression analysis to relate these parameters to the channel geometry or what have you. Another avenue that will probably be worth exploring would be in trying to find other factors that could be correlated with the parameter estimates of the chosen distribution. These would have to be measured in the field, but if we can increase the R2-value for the regression equations, then we will also increase our ability to accurately predict natural velocity distributions in natural stream channels. A final suggestion would be to investigate the possibility of a non-linear regression model. The estimation of velocity, although crucial to much of quantitative hydrology will have to wait until we have a better way of estimating the probability distributions of a given sample population, and the parameters of such a probability distribution.

References:

Ben-Zvi, A. and B. Azmon. 1997. Joint Use of L-moment Diagram and goodness-of-fit Test: A Case Study of Diverse Series. Journal of Hydrology, 198: 245-259.

Chang, H. H. 1998. Fluvial Processes in River Engineering. Krieger Publishing Company, Malabar, FL.

Dingman, S. L. 1984. Fluvial Hydrology. W.H. Freeman and Company. New York.

Dingman, S. L. 1989. Probability Distribution of Velocity in Natural Channel Cross-Sections. Water Resources Research, 25(3), pp. 509-518.

Dingman, S. L. 1994. Physical Hydrology. Prentice Hall. Upper Saddle River, NJ.

Freund, R. J., and W. J. Wilson. Regression Analysis: Statistical Modeling of a Response Variable. Academic Press. Boston, MA.

Greenwood, J. A. et al. 1979. Probability Weighted Moments: Definition and Relation to Parameters of Several Distributions Expressable in Inverse Form. Water Resources Research, 15(5), pp 1049-1054.

Haire, P. K. 1995. “The Ability of Various Models to Describe Velocity Distibutions in Arbitrarily Selected Natural Stream Channels.” Unpublished Masters Thesis, UNH, Durham, NH.

Hosking, J. R. M. and J. R. Wallis. 1987. Parameters and Quantile Estimation for the Generalized Pareto Distribution. Tectnometrics, 29: 339-349.

Hosking, J. R. M. and J. R. Wallis. 1997. Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press. New York.

Sankarasubramanian, A. and K. Srinivasan. 1999. Investigation and Comparison of Sampling Properties of L-Moments and Conventional Moments. Journal of Hydrology, 218, 13-34.

Van Gelder, P. and N. M. Neykov. 1998. Regional Frequency Analysis of Extreme Water Levels Along the Dutch Coast using L-Moments: A Preliminary Study. Delft University of Technology, Delft, Netherlands.

Vogel, R. M. and T. A. McMahon, and F. H. S. Chiew. 1993. Floodflow frequency model selection in Australia. Journal of Hydrology, 146: 421-449.

Vogel, R. M. and N. M. Fennessey. 1993. L-moment Diagrams should replace product moment diagrams. Water Resources Research, 29: 1745-1752.

Walpole, R. E. 1982. Introduction to Statistics. MacMillan Publishing Company, Inc. New York.

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Table 1. Channel geometry of CRREL simulations (Haire, 1995).

Figure 1. L-Moment ratio diagram of selected data set of Haire (1995). Note scatter results in ambiguous determination of a probability distribution. Average L-skew and L-kurtosis are plotted for both the selected data set (light blue) and the total data set (royal blue). Note also possible subsets of homogeneous data.

Table 2. Comparison of model estimates of average velocities with observed velocities.

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