MULTIFRACTAL MODELING OF THE BLAVET RIVER DISCHARGES AT GUERLEDAN MODELADUR MULTIFRAKTEL SKORVOÙ AR BLAVEZH E GWERLEDAN MODELISATION MULTIFRACTALE DES DEBITS DU BLAVET A GUERLEDAN P. Hubert (1), I. Tchiguirinskaia (2), H. Bendjoudi (3), D. Schertzer (4), S. Lovejoy (5) (1) UMR Sisyphe, CIG, Ecole des Mines, Paris, [email protected] (2) UMR Sisyphe, LGA, Université Paris VI, [email protected] (3) UMR Sisyphe, LGA, Université Paris VI, [email protected] (4) Météo-France and LMM, Université Paris VI, [email protected] (5) Physics Department, McGill University, Montreal, [email protected] Diverrañ Mont en-dro kas ar stêr zo anavezet eveit bezañ bet muzuliet abaoe hanter-kant bloavezh da vihanañ [Hurst, 1951]. Nevez zo eo bet kavet eo multifraktel rummadoù skorvoù ar stêr e meur a vuzuliadenn etre div sizhun ha seizh miz da vihanañ [Tessier et al., 1996; Pandey et al., 1998; Labat et al., 2002]. Diouzh ar muzulioù-se e voe priziet ar parametroù multifraktel hollvedel (H, C1, α) [Schertzer and Lovejoy, 1987] kenkoulz hag ezponant war var (qD) al lezenn a wana gwirheñvelder an dasparzh. An nevesañ eo ivez urzh diforc'hañ ar prantadoù stadegel, o klotañ gant urzh multifraktel kentañ ur prantad treuziñ [Schertzer and Lovejoy, 1992] ha gant dont war-wel strukturioù war var emaozet [Schertzer et al., 1993, Chigirinskaya et al., 1994]. Ar parametroù-se a hañval bezañ sichennet en tu all d'ar muzulioù, met ivez bezañ ur seurt ment diazad bras (eus un nebeud km² betek milionoù a km²). Er studiadenn-mañ e tielfennomp an dastumadoù pemdeziek graet etre 1939 ha 1999 diwar skorvoù ar Blavezh e Gwerledan. Evidomp ez eo qD = 3 urzh war var al lezenn a wana gwirheñvelder an dasparzh. Diouzh un tu all, diwar un dielfennañ spektrel o tiskouez o deus skorvoù ar stêr un ingalded kreñv diouzh ar c'houlzoù, e arguzennomp diwar-benn kemer kement-mañ e kont. Evel just, kementmañ a dorr amzer (stadegel) digemmusted an treuzkas, ma talc'her kont anezhi en dielfennañ multifraktel boas. Hervez an hent-se e arguzennomp penaos ober gant ur modeladur multifraktel ingal hag eeun-tre [Tchiguirinskaia, 2002] diazezet war lammoù stokastek lieskementiñ gant unanoù adaozet. Dont a ra dimp diouzh un tu muioc'h a briziadurioù eus ar parametr hollvedel ha diskouez e adkrou ar modeladur kas ur stêr ekeñver koulzoù muzuliet, met ivez e tegas kemmoù e reizhiadoù muzuliañ evit amzer hiroc'h. Abstract River flow phenomena have been known to be scaling for at least fifty years [Hurst, 1951]. More recently, river discharge series have been found to be multifractal over a

range of scales spanning at least from 2 weeks to 7 months [Tessier et al., 1996; Pandey et al., 1998; Labat et al., 2002]. On this range of scales the universal multifractal parameters ( H,C1 ,α ) [Schertzer and Lovejoy, 1987] were estimated as well as the critical exponent (q D ) of the power-law fall-off of the probability distribution. The latter is also the order of the divergence of the statistical moments, which corresponds to a first order multifractal phase transition [Schertzer and Lovejoy, 1992] and the appearance of selforganized critical structures [Schertzer et al., 1993, Chigirinskaya et al., 1994]. These parameter estimates seem not only to be robust over this range of time scales, but also over a wide range of basin sizes (from a few km2 up to millions of km2). In the present study, we analyse daily records from 1939 to 1999 of the Blavet river discharges at Guerledan (Brittany). We estimate as q D = 3 the critical order of the powerlaw fall-off of the probability distribution. On the other hand, since a spectral analysis shows that the river discharges have a strong seasonal periodicity, we discuss how to take it into account. Indeed, it breaks the (statistical) time translation invariance, which is implicitly assumed in the usual multifractal analysis. In this perspective, we discuss the applicability of a rather simple seasonal periodic multifractal model [Tchiguirinskaia et al., 2002] based on stochastic multiplicative cascades with re-ordered singularities. We obtain on the one hand more reliable estimates of the universal parameter and show that this model adequately reproduces a river flow not only at seasonal time scales, but also changes in scaling regimes for longer time scales. Résumé Le caractère scalant du débit des rivières est reconnu depuis au moins une cinquantaine d’années [Hurst, 1951]. Plus récemment, on a pu montrer que les séries de débits étaient multifractales sur une gamme d’échelles allant au moins de 2 semaines à 7 mois [Tessier et al., 1996; Pandey et al., 1998; Labat et al., 2002]. Sur cette intervalle, les paramètres multifractals universels ( H,C1 ,α ) [Schertzer and Lovejoy, 1987] de même que l’exposant critique (q D ) de décroissance algébrique de la distribution de probabilité ont pu être estimés. Ce dernier paramètre est aussi l’ordre de divergence des moments statistiques, qui correspond à une transition de phase multifractale de première espèce [Schertzer and Lovejoy, 1992] et à l’apparition de structures auto-organisées [Schertzer et al., 1993, Chigirinskaya et al., 1994]. Ces paramètres semblent non seulement robustes sur la gamme d’échelles sur laquelle ils ont été estimés, mais se retrouvent aussi sur une large gamme de tailles de bassins (de quelques km2 à plusieurs millions de km2). Dans la présente étude, nous avons analysé les débits journaliers du Blavet à Guerledan (Côtes d’Armor) de 1939 à 1999. Nous avons estimé à q D = 3 l’exposant critique (q D ) de décroissance algébrique de la distribution de probabilité. Par ailleurs, comme l’analyse spectrale montre que les débits présentent une forte périodicité saisonnière, nous avons cherché à en tenir compte. Cette périodicité rompt en effet l’invariance (statistique) d’échelle au cours du temps, hypothèse implicite des analyses multifractales habituelles. Dans cette perspective, nous avons proposé l’application un modèle multifractal périodique simple [Tchiguirinskaia et al., 2002] basé sur des cascades stochastiques multiplicatives dont on ordonne les singularités. Nous obtenons ainsi une estimation plus fiable des paramètres universels et nous avons pu montrer que ce modèle reproduit correctement le débit de la rivière à toutes les échelles de temps et rend compte du changement de régime de la scalance observé pour les grandes échelles de temps.

b) a) Figure 1a,b: 52 years time series (01/01/1948-31/12/1999) of daily discharges at Guerledan (a) and annual maxima of these daily discharges (b).

Figure 2: Spectra (from top to bottom) of annual maxima of daily discharges and of daily discharges themselves (log-log plot).

Spectral analysis of Blavet discharges at Guerledan The scaling range may be assessed with the help of classical spectral analysis in determining the frequency bands where the spectrum displays a power-law. When studying the spectra of thirty French rivers, [Tessier et al., 1996] observed the existence of a scale break that appears roughly between 16 and 30 days. The ensemble averaged spectral exponent was estimated as β=1.3 for the 1 to 16 days regime and β=0.52 for the 1 month to 30 years regime of river runoff. The authors pointed out that the period associated with the break was not significantly correlated neither with the size of basin nor with the geology. Therefore, they argued that this 16-day period was associated to the atmospheric synoptic maximum [Kolesnikova and Monin, 1965], which is the typical lifetime of planetary scale atmospheric structures. As a consequence, the multifractal analysis was performed over two distinct frequency bands, respectively the high frequencies of 1-16 days using daily data and the low frequencies of 1-360 months using monthly averages of daily records. [Pandey et al., 1998] obtained similar results for daily river flow data from 19 river basins of varying watershed areas in the continental USA. For most of the rivers, the authors observed a break in the scaling regime, which was associated to half of the atmospheric synoptic maximum. The spectral exponent for low frequency region was estimated as β=0.72 and the multifractal analysis was only performed for the time-scales longer than 8 days. Although the basin areas varied over nearly six orders of magnitude, the scaling results were independent of the basin size and geology. In a recent study [Labat et al., 2002] of

three karstic springs located in the French Pyrenées Mountains, a change of scaling behavior was noted on the spectra and was also attributed to the synoptic maximum. In spite of it, a unique multifractal analysis was performed over about 11 years of daily records that displays a convincing unique multifractal regime over the range from 1 to 512 days. It is important to note that no upper time scale (in particular of 512 days) was found for the lower frequency scaling regime months [Tessier et al., 1996; Pandey et al., 1998; Labat et al., 2002]. On the other hand, the influence of annual cycle was disregarded in all three studies.

Figure 3: A log-log plot of the DTM as a function of the scale ratio λ for q = 1.5 and various values of η . The straight lines indicate scaling of moments of filtered daily discharges over time-scales from 1 to 512 days.

Figure 4: A log-log plot of empirical scaling function K(q,η) vs. η for (from top to bottom) q =2.0; 1.5 and 0.8. The straight lines have a unique slope which corresponds to α=1.62. For each q, the intersection of such curve with the axis Logη=0 gives the corresponding LogK(q). Then the universal multifractal expression for K(q) is used to compute C1=0.14.

The time series of daily discharges of Blavet river at Guerledan as well as of their annual maxima are displayed on Fig.1 for the period from 1939 to 1999. It is notable that this Celtic river visually exhibits much stronger intermittent spikes than those observed on other river discharges. Furthermore, the intermittency of the annual maxima and that of the full time-series are rather the same. The corresponding spectra are presented in Fig.2. The spectral exponent of daily discharges may be estimated as β=1.3 for time-scales shorter than a year. Since spectral analysis decomposes the statistics according to frequencies, the seasonal periodicity of Blavet discharges corresponds to a prominent annual spectral spike. Before this annual spike, the spectral slope is rather flat and the spectrum exponent β is close to 0.3. The annual maxima have the same spectral exponent. In contrast to months [Tessier et al., 1996; Pandey et al., 1998; Labat et al., 2002], we do not observe any peculiarity of the spectrum behavior on time-scales of the order of two weeks. In the following section we will proceed to a more involved scaling analysis of daily discharges keeping in mind three main remarks based on the spectral results: • since no unique power-low exponent was found for the entire spectrum, we may face similar changes in scaling regimes within multifractal analysis; • the annual spike may pose significant problems for a multifractal analysis performed in the physical space, whereas it does not in the Fourier space; • since β=1.3 for high frequencies, the discharge statistics can not be directly in agreement with the statistics of a multifractal field produced by a multiplicative

cascade (which yields β