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Fig. 10. Probability threshold density for half-hourly rainfall rates (a) and runoffs (b). The different curves correspond to different aggregation step. Concerning rainfall rates, the absolute value of the algebraic tails gp (Eq. (3)) are 5.53, 4.73, 5.66 and 4.20 for decreasing resolution. Concerning runoffs, the slopes are 4.96, 4.70, 4.85 and 4.68 for decreasing resolution.

Figure 10 Probability threshold density for half-hourly rainfall rates (a) and runoffs (b). The different curves correspond to different aggregation step. Concerning rainfall rates, the absolute value of the algebraic tails gp (Eq. (3)) are 5.53, 4.73, 5.66 and 4.20 for decreasing resolution. Concerning runoffs, the slopes are 4.96, 4.70, 4.85 and 4.68 for decreasing resolution.

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Fig. 1. Visualisation of the location and the extend of the three watershed ((a) Aliou, (b) Baget and (c) Fontestorbes),
Fig. 1. Visualisation of the location and the extend of the three watershed ((a) Aliou, (b) Baget and (c) Fontestorbes),
Description of the half-hourly rainfall rates and runoffs of Aliou spring. Note the high number of consecutive data of each series, which constitute an important bank of data for karst hydrologists
Description of the half-hourly rainfall rates and runoffs of Aliou spring. Note the high number of consecutive data of each series, which constitute an important bank of data for karst hydrologists
Fig. 2. Data visualisation: half-hourly rainfall rates measured on Aliou karstic basin under two different periods. Fig. 1a corresponds to the period extending from July 10 1993 to May 24 1995 (32 768 successive data). Fig. 1b corresponds to the period extending from May 25 1995 to February 8 1996 (16 384 consecutive data). One can note the extreme intermittence of the processes at this sampling rate.
Fig. 2. Data visualisation: half-hourly rainfall rates measured on Aliou karstic basin under two different periods. Fig. 1a corresponds to the period extending from July 10 1993 to May 24 1995 (32 768 successive data). Fig. 1b corresponds to the period extending from May 25 1995 to February 8 1996 (16 384 consecutive data). One can note the extreme intermittence of the processes at this sampling rate.
Fig. 3. Data visualisation: half-hourly runoffs measured at the outlet of Aliou karstic basin under eight non-overlapped periods. (Fig. 2a: from May 30 1992 to May 6 1993; Fig. 2b: from February 16 1994 to August 6 1994; Fig. 2c: from July 5 1994 to December 23 1994; Fig. 2d: from August 16 1995 to July 22 1996; Fig. 2e: from November 11 1996 to August | 1997; Fig. 2f: from May 15 1997 to September 29 1997; Fig. 2g: from December 29 1997 to June 17 1998; Fig. 2h: from February 17 1998 to December 31 1999). The length of the series varies between 8192 and 32 768 consecutive data. D. Labat et al. / Journal of Hydrology 256 (2002) 176-195
Fig. 3. Data visualisation: half-hourly runoffs measured at the outlet of Aliou karstic basin under eight non-overlapped periods. (Fig. 2a: from May 30 1992 to May 6 1993; Fig. 2b: from February 16 1994 to August 6 1994; Fig. 2c: from July 5 1994 to December 23 1994; Fig. 2d: from August 16 1995 to July 22 1996; Fig. 2e: from November 11 1996 to August | 1997; Fig. 2f: from May 15 1997 to September 29 1997; Fig. 2g: from December 29 1997 to June 17 1998; Fig. 2h: from February 17 1998 to December 31 1999). The length of the series varies between 8192 and 32 768 consecutive data. D. Labat et al. / Journal of Hydrology 256 (2002) 176-195
Fig. 4. Spectral analysis: variance spectra of daily rainfall rates and runoffs. The frequencies are normalised by the Nyquist frequency. They exhibit several power-law behaviours. The different absolute slopes are (a) 0.68 for rainfall rates, (b) 1.13 for Aliou, (c) 1.89 for Baget runoffs and (d) 2.81 for Fontestorbes runoffs. A cut off scale is visible at around 16 days for all the karstic basins.
Fig. 4. Spectral analysis: variance spectra of daily rainfall rates and runoffs. The frequencies are normalised by the Nyquist frequency. They exhibit several power-law behaviours. The different absolute slopes are (a) 0.68 for rainfall rates, (b) 1.13 for Aliou, (c) 1.89 for Baget runoffs and (d) 2.81 for Fontestorbes runoffs. A cut off scale is visible at around 16 days for all the karstic basins.
Fig. 5. Probability threshold density for daily rainfall rates (a) and runoffs (b—d). The different curves correspond to different aggregation ste from | to 8 days. Concerning rainfall rates (a), the absolute value of the algebraic tails gp (Eq. (3)) are 3.49, 5.63, 5.18 and 5.05 for decreasin: resolutions. Concerning Aliou runoffs (b), the absolute slopes are 4.17, 4.43, 4.6 and 4.58 for decreasing resolutions. Concerning Baget runoff (c), the absolute slopes 5.3, 4.68, 5.22 and 4.8 for decreasing resolutions. Concerning Fontestorbes runoffs (d), the absolute slopes are 4.29 4.78, 4.96 and 5.78 for decreasing resolutions.
Fig. 5. Probability threshold density for daily rainfall rates (a) and runoffs (b—d). The different curves correspond to different aggregation ste from | to 8 days. Concerning rainfall rates (a), the absolute value of the algebraic tails gp (Eq. (3)) are 3.49, 5.63, 5.18 and 5.05 for decreasin: resolutions. Concerning Aliou runoffs (b), the absolute slopes are 4.17, 4.43, 4.6 and 4.58 for decreasing resolutions. Concerning Baget runoff (c), the absolute slopes 5.3, 4.68, 5.22 and 4.8 for decreasing resolutions. Concerning Fontestorbes runoffs (d), the absolute slopes are 4.29 4.78, 4.96 and 5.78 for decreasing resolutions.
Fig. 6. Moment and multifractal analyses: (a) log—log plot of the average gth moments of the daily rainfall rates for different moments (¢q = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits single power-law behaviour for time-scales varying approximately between | and 32 days. (b) Empirical moment scaling function K(q) for daily rainfall rates (0) compared to theoretical one based Eq. (5) and corresponding to universal multifractal models.
Fig. 6. Moment and multifractal analyses: (a) log—log plot of the average gth moments of the daily rainfall rates for different moments (¢q = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits single power-law behaviour for time-scales varying approximately between | and 32 days. (b) Empirical moment scaling function K(q) for daily rainfall rates (0) compared to theoretical one based Eq. (5) and corresponding to universal multifractal models.
D. Labat et al. / Journal of Hydrology 256 (2002) 176-195 Fig. 7. Moment and multifractal analyses: (a) log—log plot of the average gth moments of the daily runoffs for different moments (g¢ = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits single power-law behaviour from time-scales varying approximately between | and 512 days (a — Aliou, b — Baget and c — Fontestorbes). (b) Empirical moment scaling function K(q) for daily runoffs (0) compared to theoretical one based Eq. (5) and corresponding to universal multifractal models (d — Aliou, e — Baget and f — Fontestorbes).
D. Labat et al. / Journal of Hydrology 256 (2002) 176-195 Fig. 7. Moment and multifractal analyses: (a) log—log plot of the average gth moments of the daily runoffs for different moments (g¢ = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits single power-law behaviour from time-scales varying approximately between | and 512 days (a — Aliou, b — Baget and c — Fontestorbes). (b) Empirical moment scaling function K(q) for daily runoffs (0) compared to theoretical one based Eq. (5) and corresponding to universal multifractal models (d — Aliou, e — Baget and f — Fontestorbes).
Fig. 8. Spectral analysis: average variance spectra of half-hourly rainfall rates (a) and runoffs (b). The frequencies are normalised by the Nyquist frequency. They exhibit power-law behaviour and the absolute slopes are B = 0.96 for rainfall rates and B = 3.70 for runoffs.
Fig. 8. Spectral analysis: average variance spectra of half-hourly rainfall rates (a) and runoffs (b). The frequencies are normalised by the Nyquist frequency. They exhibit power-law behaviour and the absolute slopes are B = 0.96 for rainfall rates and B = 3.70 for runoffs.
Multifractal analysis of half-hourly data: time-scale intervals, absolute spectral slope and corresponding multifractal parameters a and C (Eq. (5)) obtained for the different series (the average value of multifractal parameters in parenthesis are obtained without considering Qal3 and Qal4). Note that the average value of both parameters are slightly different between the two scale-processes (small scale and large scale- processes)
Multifractal analysis of half-hourly data: time-scale intervals, absolute spectral slope and corresponding multifractal parameters a and C (Eq. (5)) obtained for the different series (the average value of multifractal parameters in parenthesis are obtained without considering Qal3 and Qal4). Note that the average value of both parameters are slightly different between the two scale-processes (small scale and large scale- processes)
Fig. 9. Visualisation of the aggregation process: from top to bottom, one represents the signal (half-hourly runoffs) and the aggregated signals corresponding to the different sampling rates 2, 8, 128, 2048 and 8192 h.
Fig. 9. Visualisation of the aggregation process: from top to bottom, one represents the signal (half-hourly runoffs) and the aggregated signals corresponding to the different sampling rates 2, 8, 128, 2048 and 8192 h.
Fig. 11. Moment analysis: log—log plot of the average gth moments of the half-hourly rainfall rates for different moments (¢ = 0.15, 0.5, 1, 1.5 2.5, 5). One exhibits a single power-law behaviour for time-scales varying approximately between 30 min and 1024 h.
Fig. 11. Moment analysis: log—log plot of the average gth moments of the half-hourly rainfall rates for different moments (¢ = 0.15, 0.5, 1, 1.5 2.5, 5). One exhibits a single power-law behaviour for time-scales varying approximately between 30 min and 1024 h.
Fig. 12. Moment analysis: log—log plot of the average gth moments of the half-hourly runoffs for different moments (q = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits two different power-law behaviour with a cut-off scale varying between 16 and 64 h.
Fig. 12. Moment analysis: log—log plot of the average gth moments of the half-hourly runoffs for different moments (q = 0.15, 0.5, 1, 1.5, 2.5, 5). One exhibits two different power-law behaviour with a cut-off scale varying between 16 and 64 h.
Fig. 13. Multifractal analysis: empirical moment scaling function K(q) for rainfall rates (0): (a) and (b) corresponds to Pall and Pal2. The convex K(q) function (Eq. (5)) corresponding to universal multifractal models is also plotted on each graph.
Fig. 13. Multifractal analysis: empirical moment scaling function K(q) for rainfall rates (0): (a) and (b) corresponds to Pall and Pal2. The convex K(q) function (Eq. (5)) corresponding to universal multifractal models is also plotted on each graph.
Fig. 14. Multifractal analysis: empirical moment scaling function K(q) for runoffs (0). (a)—(h) correspond to small scale-processes (inferior to the cut-off scale). The convex K(q) function corresponding to universal multifractal models is also plotted on each graph.
Fig. 14. Multifractal analysis: empirical moment scaling function K(q) for runoffs (0). (a)—(h) correspond to small scale-processes (inferior to the cut-off scale). The convex K(q) function corresponding to universal multifractal models is also plotted on each graph.
Fig. 15. Multifractal analysis: empirical moment scaling function K(q) for runoffs (0). (a)—(h) correspond to large scale-processes (inferior tc the cut-off scale). The convex K(q) function corresponding to universal multifractal models is also plotted on each graph.
Fig. 15. Multifractal analysis: empirical moment scaling function K(q) for runoffs (0). (a)—(h) correspond to large scale-processes (inferior tc the cut-off scale). The convex K(q) function corresponding to universal multifractal models is also plotted on each graph.
Related topics:
Stochastic ProcessHydrologyWavelet AnalysisMultifractal AnalysisSimulation MethodsScale dependenceContinuous wavelet transform
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