11. Confidence Intervals for Flood Return Level Estimates assuming ...

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232 11 Flood Level Confidence IntervalsEmpirical0 50 100 150Return Level0 50 100 15020 40 60 80 100Model0.2 1.0 5.0 20.0 100.0Return PeriodFig. 11.8. Result of the Ml-estimation of the Gumbel parameters for the Vilsbiburg yearly maximaseries compared to the empirical maxima series in a quantile plot (left panel) and return level plot(right panel).the variance cycle. As we find also indication for a weekly component in theperiodogram, we subtract this component analogously. The Box-Cox transformationas well as the seasonal filters have been suggested by Hipel andMcLeod [11.28] for hydrological time series. Although the proposed estimatesof the periodic components in mean and variance are consistent, especially theestimates of the annual cycle exhibits a large variance. Several techniques suchas STL [11.11] are advocated to obtain smoother estimates. Studying thoseperiodic components is, however, not in the focus of this paper.Figure 11.9 shows a comparison of the transformed and mean adjusteddaily run-off record to a Gaussian distribution in form of a density plot (leftpanel) and a plot of empirical versus theoretical quantiles (right panel). Thetransformed distribution is much less skewed than the original one. Differencesto a Gaussian are mainly found in the tails.We now fit Farima[p, d, q] models of various orders with 0 ≤ q ≤ 5, 0 ≤ p ≤6 and q ≤ p and compare the Hic of the different models in Fig. 11.10(left). Thesmallest value for the Hic is obtained for the Farima[3, d, 0] process, which isthus chosen to model the autocorrelation of the daily run-off. The parametersestimated for this process with their asymptotic standard deviation are: d =0.439 ± 0.016, φ 1 = 0.415 ± 0.017, φ 2 = −0.043 ± 0.007, φ 3 = 0.028 ± 0.008and ση 2 = 0.2205. Using the goodness-of-fit test proposed by Beran [11.4], weobtain a p-value of p = 0.015. The model thus cannot be rejected on a 1%level of significance. The result of this fit is shown in the spectral domain inFig. 11.10(right).Modelling the ACF of the Maxima Series. Using this model, a longseries is generated with N long = 100N data . This simulated series is partiallyback-transformed: the overall mean and the seasonal cycles in mean and vari-
Henning W. Rust et al. 233Density0.0 0.2 0.4Sample Quantiles−4 −2 0 2 4−4 −2 0 2 4Quantile−4 −2 0 2 4Theoretical QuantilesFig. 11.9. Comparison of the Box-Cox transformed, deseasonalised and mean adjusted run-offvalues to a Gaussian distribution. Histogram (left, grey) with a density estimate (solid line) andquantile plot (right).HIC30050 30065 30080[0,0][1,0][1,1][2,0][2,1][2,2][3,0][3,1][3,2][4,0][4,1][4,2][4,3][4,4][5,0][5,1][5,2][5,3][6,0][6,1][6,2][6,3][6,5]Spectrum1e−06 1e−02 1e+025e−05 1e−03 5e−02FARIMA Model Order [p,q]Best HIC : FARIMA[3,d,0]FrequencyFARIMA[3,d,0]Fig. 11.10. Model selection for the Box-Cox transformed and deseasonalised daily run-off withmean removed. The spectral density of the model with smallest Hic is shown in double logarithmicrepresentation together with the periodogram (grey) of the empirical series (right panel).ance are added. Note, that the Box-Cox transform is not inverted in this step.From the resulting record, we extract the annual maxima series. Figure 11.2(left panel) shows the Acf of the original maxima series (solid) and comparesit to the Acf of a section of the same length cut out of the maximaseries gained from the simulated run (dotted). The original series has beenBox-Cox transformed as well to achieve a comparable situation. The autocorrelationbasically fluctuates within the 95% significance level (dashed) for awhite noise. However, the full maxima series from the simulated run of 6 200data points exhibits prominent autocorrelation (Fig. 11.2, right panel). Thisindicates that, although an existing autocorrelation structure is not necessarily
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