# Section 2 Annual maximums

## 2.1 Introduction

In this document I will show how to use floodnetRfa to perform at-site frequency analysis using annual maximum discharge. First we will extract the annual maximums from the daily discharge data of the Saint-John River at Fort Kent (NB). Note that incomplet year with missing observations will be removed. In total 88 annual maximums are extracted and showed in a histogram.

library(floodnetRfa)

anData <- ExtractAmax(flow ~ date, flowStJohn, tol = 365)

npDens <- density(anData$flow) hist(anData$flow,
freq = FALSE,
main = '',
xlab = 'Discharge',
ylim = c(0, max(npDens$y))) lines(npDens, col = 2, lwd = 2) In stationary situation, the annual maximums are assumed to be independant and identically distributed. Risk is then measure in terms of return period that characterizes the average time separating two events of the same magnitude. In practice this is equivalent to calculate the probability $$p = 1-1/T$$ of a fitted distribution. The test of Mann-Kendall is frequently performed to verify if the data contains a significant trend that would invalidate the assumption of stationarity. The present data have a p-value of 0.21, which does not suggest the present of a trend. plot(flow~date, anData) MKendall(anData$flow)
##
##
## Mann-Kendall Test for trend
## S:  346
## p-value:  0.2136

## 2.2 Estimation of the flood quantiles

According to extreme value theory, as the number of annual maximums increase, their distribution converge to Generalized Extreme Value (GEV) distribution

$$$F(x) = \exp\left\{ - \left[ 1 - \kappa \left(\frac{x-\xi}{\alpha} \right) \right]^{1/\kappa} \right\} \tag{2.1}$$$

The GEV distribution in Equation (2.1) can be fitted using the FitAmax function. The example below shows how the parameter are estimated using the maximum likelihood method. See for instance (Coles 2001). A brief summary of the fitted model is reported, including the estimated parameter, their standard deviation and the sample L-moments.

fit <- FitAmax(anData\$flow, 'gev', method = 'mle')

print(fit)
##
## At-site frequency analysis
##
## Distribution: gev
## AIC: 1411
## Method: mle
## Estimate:
##        xi     alpha     kappa
## 2101.3761  668.9518    0.1667
##
## Std.err:
##       xi    alpha    kappa
## 78.39437 54.31311  0.06214
##
## Lmoments:
##     l1  lcv     lsk    lkt
## 1 2390 0.17 0.05924 0.1399

The flood quantile of the GEV distribution can be obtained from the formula

$x_T = \mu + \frac{\alpha}{\xi}\left[ 1+\log(1/T)^\kappa \right] .$ These predicted value are computed using the predict function. In the example below show the flood quantile for return period 2, 5, 10, 20, 50 and 100. The standard deviation of the flood quantiles is estimated using the Delta method that assume that the estimated parameter follow a Normal distribution.

yhat <- predict(fit, se = TRUE, ci = 'delta')
Table 2.1: Predicted return levels
pred se lower upper
2339.216 82.44124 2177.634 2500.798
2989.156 99.81724 2793.518 3184.794
3356.637 117.79148 3125.770 3587.504
3668.464 144.52611 3385.198 3951.730
4020.327 193.47341 3641.126 4399.528
4250.404 238.46296 3783.025 4717.783

## 2.3 Verification of the model

The return level plot in 2.2 provide a visual assessment of the fitted distribution by comparing the sample and the predicted flood quantiles. The graphic below shows a good agreement between the two.

plot(fit, ci = TRUE)

Another diagnostic to ensure that the GEV distribution is appropriate is the Anderson-Darling test. The p-value superior to 0.05 indicates that the hypothesis of a GEV cannot be rejected.

## Time consuming when estimated by 'mle'
GofTest(fit, nsim = 500)
##
## Goodness-of-fit test for annual maxima
##
## Test = Anderson-Darling
## Distribution = gev
## statistic : 0.266
## p-value : 0.534

### References

Coles, S. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer Verlag.