Evénements de période sèche en pays semi-aride

On analyse par événement les périodes sèches ou longs événements secs se produisant au cours d'une saison humide en pays semi-aride. Une période sèche se définit comme une série de jours avec pluies quotidiennes inférieures à un seuil donné. Le cas d'espèce de la région de Dodoma en Tanzanie, où l'on observe surtout des précipitations de type convectif illustre la méthodologie. Une analyse conventionnelle des périodes sèches, qui ne fournit pas de relation entre la fréquence et la durée de ces périodes, semble cependant indiquer que les périodes sèches se produisent de façon aléatoire pendant la saison pluvieuse qui est elle-même de longueur aléatoire. L'analyse par événement comprend le nombre d'événements par saison, quisuit approximativement une loi de Poisson, la durée des événements secs, qui est supposée suivre une loi binomiale négative et la durée des événements pluvieux qui est supposée suivre une loi géométrique. On utilise la loi de Pearson III pour estimer les événements secs saisonniers de durées maxima et on compare les résultats obtenus avec ceux de l'analyse par événement, les écarts observés étant expliqués par les différences de conception existant entre ces deux méthodes. La distribution spatiale des événements secs est également analysée, on trouve qu'une forte majorité des événements secs est simultanée sur au moins deux stations, et que près de la moitié des événements secs de durée modérée est simultanée sur trois stations. L'analyse par événement permet de calibrer les modèles de précipitation avec peu de données et de procéder à la génération d'événements synthétiques par simulation.One form of drought is the interruption of the rainy season by a sa called dry spell. Dry spell can be defined as a sequence of dry days including days with less than a threshold value of rainfall.A dry spell, defined a on daily scale, becomes untraceable by statistics using longer than one-day-long equidistant time intervals. If the daily discretization of the rainy season is te be avoided, an intermediate technique is needed.Event-based analysis of the rainfall and dry spell provides This approach. The method is demonstrated with data from the Dodoma Region, situated in the semiarid highlands of Tanzania. The climate is characterized by one rainy tesson from the and of November until the end of April. The occurrence of rainfall is erratic.The average seasonal precipitation is about 600 mm with variations between 450 and over 800 mm. Rainy seasons are separated by an almosl 7 month long dry season.During the rainy season convective type storms prevail. Single storms lasi a few hours, but their occurrence is clearly grouped within the timespan of a few days, separated by the dry spells which are usually much longer.Conventional statistics of dry spells are summarized in tables 1 and 2 using 1.0 mm daily precipitation as the threshold.It is shown that dry spells occur randomly during the rainy season. For the event-based analysis dry spell is detined as a dry event. Dry events are considered as a sequence of dry days separated by rainfall events from each other. Thus the rainy season is detined as a series of rainfall and subsequent dry events. Rainfall events are defined as the uninterupted sequence of rainy days, when at least on one day more than a threshold amount of rainfall has been observed- Rainy days with less Man the threshold depth of precipitation are accounted for the rainfall event if they occur in an uninterrupted sequence. Only isolated subthreshold rainfall will be discarded, and considered as part of a dry event (fig. 2). ln this analysis the threshold value of 5.0 mm/day was seiected.The comparison of tables 1 and 4 shows that the length of the mean maximum dry spell doubles by replacing the 1.0 mm/day threshold by 5.0 mm/day. The sequence of rainfall and dry events is characterized by Dn, m, duration of the mth rainfall of the nth rainy season, and by the inter-event time Zn, m (duration of the dry event) between the end at the preceding and the start of the succeeding rainfall event.In case of convective type storms the series of the subsequent events (either dry or rainfall) could be considered independent, thus their number/season should follow the Poisson distribution.In case of independence of subsequent events, the waiting lime for a new event must follow the exponential pdf.By measuring the waiting time in days the discrets equivalent of the exponential pdf can be used.Since the sequences of convective type storms de not contain purely independent events, the waiting time t follows instead the discrete counterpart of the 2 parameter gamma pdf, the negative binomial pdf. This modified Poisson-modal, Poisson pdf for the number of rainfall events and the negative binomial pdf for the length of the inter-event lime has been applied to describe the rainy season. Table 3 summanzes the parameters r and p for different rein gauges.By focusing on the dry spell event, the duration of the rainfall events Dn, m will in fact be identified as inter-event time. This change of rotes fils the original Poisson model better. Since rainfall events are shorter, their duration follows the geometrical pdf, as theoretically required.The Poisson pdf seems to fit slightly better the number of dry events than the rainfall ones (fig. 5). If has to be pointed out that the event-based definition of the rainy season dues not exactly fit the theoretical precondition, i. e. to have a certain fixed period. Rainy seasons have variable longths, as they are a stochastic fonction of the events themselves. For "modal fitting" the consideration of the core of the rainy season, from January to April would be a better choice. However it would truncate the physical phenomenon wilh the potentiel omission of extrema long dry events. Therefore, in spite of mediocre fitting, the Poisson modal will be used ln analysis.Tables 4 and 5 summarize the statistical characteristics of the dry events for the selected rain gouges for bath the whole, and for the core of the rainy season.Dry events accurring in the core of the rainy season were identified as those ending within the timespan of January -April.The mean lengths of the longest dry spells in the core are less than the corresponding value for the whole season. However, at two stations, Kondoa and Gwandi, in more han 70 % of the season the longest dry spell did occur during the core. This coincidence was only 40 % for the Farkwa rain gauge.For planning purposes, the longest dry spells associated with the varions statistical recurrence periods are derived on the basis of the fitted Pearson III type probability distribution functions (fig. 6, table 6).The event-based analysis, relying on the expected number ot events/season and the negative binomial pdf for the length of the dry events, can also be used to approximate the distribution of the extreme long dry spells. Contrary to the Pearson III distribution fitted to the seasonal extreme values, the negative binomial pdf f (n) determines the probability that a random dry event would last n days.Consequently, the exceedence probability Pe (N), that an extreme long dry event would occur at least once within a given statistical recurrence period of T years must be equal to the reciprocal value of the product λ•T, where λ denotes the expected number of dry eventslyear (season). λ•T specifies the expected number of "trials" needed to observe at least once the extreme duration of N days associated with the return period of T years. The length of this extreme dry spell can then be obtained from the cumulative negative binomial pdf (table 6). The deviations observed for low number of "trials" between the event-based and the extreme seasonal value approach are due to the conceptual difference.Table 7 displays the simultaneous occurrence of dry events at several rain gauges. By using Farkwa as the reference station, table 7 does not account for dry events that might have occurred simultaneously at Gwandi and Kandoa without having been recorded at Farkwa.Except for the very short (1 or 2 days) and the very long dry events (over 30 days), the overwhelming majority of the dry events occurred at least at two stations simultaneously. Furthermore, excluding the 1, 2 or more than 30 day - long events, more than 63 % of the dry events have been observed at all three stations.Two (or three) dry events were only classified as simultaneous if more than hall of the duration of the reference event at Fartwa was overlapped by an uninterrupted dry event at the other station (s).Event-based analysis, even if it is carried out on the bases of a few years of observation, can rely on large number of data points (table 3). While the expected number of events/season is still derived from very few data, this estimate is more reliable than the approximative expected length of the longest seasonal dry spell, since the variability of the former is usually less than that of the latter, for the same data sets (table 8)

A Stochastic Model of Phosphorus Loading from Non-Point Sources

A stochastic model is presented and applied for Lake Balaton, Hungary to estimate the phosphorus (P) loading from non-point sources. Rainfall events cause surface runoff events and erosion events; all three events are random. P is carried by runoff into the lake in two forms: (1) dissolved P and (2) sediment, absorbed or fixed P. P loading is thus considered as a random variable, whose probability density function (pdf) per event is to be estimated. Pdf of seasonal (e.g. annual) loading is determined as the sum of a random number of random events. The annual mass balance of P stored in lake sediment leads to a first order difference equation, the solution of which can be used to predict the expected P available for release at future times The model is applied for the Tetves subwatershed (70 sq. km.) of Lake Balaton. Preliminary results show that during relatively short runoff events about as much P reaches the lake as during the rest of the year and that more sediment P is produced than dissolved P. Since a considerable variance apppears in the annual amounts of P loading, the use of stochastic models to estimate the loading conditions seems to be most appropriate. The stochastic loading model should be incorporated into a broader control model. Elements of such a control model are given in the form of possible P loading reduction measures; also, economic trade-off between these measures is discussed

Electrokinetic-potential fluctuations generated by jet impingement

Reprinted from International journal of heat and mass transfer, v. 7 pages 159-167.CER63LD-JEC-12.Includes bibliographical references.Electrokinetic-potential fluctuations produced by a two-dimensional submerged water jet impinging on a plate have been measured. The potential fluctuations appear to be approximately proportional to the longitudinal-velocity fluctuations ux' in the neighborhood of the boundary. Normalized frequency distributions of potential-fluctuation measurements agree with velocity fluctuation data taken by Klebanoff and Laufer with a hot-wire anemometer at a dimensionless distance y/δ ∼- 10-3 from the wall. Assumptions made concerning the relationship between potential and velocity fluctuations give a possible explanation of the change in the shape of the potential-fluctuation spectrum with the flow velocity and the electrical conductivity of the water. Further analysis is required to establish a definite relationship between electrokinetic-potential fluctuations and velocity fluctuations occurring near a solid boundary

Trade-off Between Cost and Effectiveness of Control of Nutrient Loading into a Water Body

A system consisting of a watershed and a water body is considered, and a methodology is presented for selecting the alternative scheme offering the best compromise between economic activity in the watershed and quality of the water body. The general problem is specified for the system of a watershed and a lake endangered by eutrophication. Both economic activity and eutrophication can be characterized by several criteria. The method is applied to actual data from a subwatershed of Lake Balaton, Hungary, where the economic objective is to minimize the sum of costs and losses for the various control measures and the environmental objective is to minimize the amount of P available for algal growth. Both of these objectives are decomposed into several criteria. The action space consists of six pure strategies, namely, the control of (1) point-source pollution, (2) fertilizer, (3) erosion, (4) land use, (5) runoff control, and (6) sediment yield. These six pure actions lead to the definition of eight mixed alternatives. The phosphorus-loading portion of the model is run repeatedly with different stochastic input sequences to account for hydrologic uncertainty and the corresponding environmental objective is expressed as the probability "uj" that alternative "j" results in the largest decrease of P-loading. Model parameters are estimated using available data or published tables and graphs. Compromise programming is used to find a trade-off (or satisfactum solution) that balances the two conflicting objectives. In order to facilitate further application of the methodology, several points are discussed such as the relationship between the lake and its catchment, the error in stochastic simulation, the consideration of various uncertainties, the effect of snowmelt, and possible coupling with detailed lake eutrophication models. Finally, a step-by-step summary of the methodology is given to facilitate application of the model to other cases. Multicriterion decision-making techniques are briefly reviewed in the appendix so that cases with more than two objectives may also be approached

Optimal Flood Levee Designs by Dynamic Programming

An economic optimal development of a levee system along a river is investigated and a dynamic programming (DP) approach is used to find the optima under various conditions. The system consists of a number of levee reaches or stages. A random input of flood wave is regarded at the upstream point of the system. There are two failure modes considered and, consequently, two parameters of the flood wave (state variables) to trigger failure modes in every stage. Stochastic DP is used since the state transition functions (flood routing along the stages) are random functions. Three methods are discussed. In Method I, the expected value of the objective function is taken first, then DP is used as a numerical technique. In Method II, a fixed design flood is chosen as an input under which both optimum cost and policy is determined. In Method III, the value of the expected optimum objective function is calculated. It is shown that the full power of DP cannot be used if Method I is applied. Future research involves comparing the solutions of the three methods

Une approche floue pour la détermination de la région d'influence d'une station hydrométrique

La notion d'appartenance partielle d'une station hydrométrique à une région hydrologique est modélisée par une fonction d'appartenance obtenue en appliquant les concepts de l'analyse floue. Les stations hydrométriques sont représentées dans des plans dont les axes sont des attributs hydrologiques et/ou physiographiques. Les régions hydrologiques sont considérées comme des sous-ensembles flous. Une méthode d'agrégation par cohérence (Iphigénie) permet d'établir des classes d'équivalence pour la relation floue "il n'y a pas d'incohérence entre les éléments d'une même classe": ce sont des classes d'équivalence qui représentent les régions floues. La fonction d'appartenance dans ce cas est stricte. Par opposition, la seconde méthode de type centres mobiles flous (ISODATA) permet d'attribuer un degré d'appartenance d'une station à une région floue dans l'intervalle [0,1]. Celle-ci reflète le degré d'appartenance de la station à un groupe donné (le nombre de groupes étant préalablement choisi de façon heuristique). Pour le cas traité (réseau hydrométrique tunisien, débits maximums annuels de crue), il s'avère cependant que le caractère flou des stations n'est pas très prononcé. Sur la base des agrégats obtenus par la méthode Iphigénie et des régions floues obtenues par ISODATA, est effectuée une estimation régionale des débits maximums de crue de période de retour 100 ans. Celle-ci est ensuite comparée à l'estimation régionale obtenue par la méthode de la région d'influence ainsi qu'à l'estimation utilisant les seules données du site, sous l'hypothèse que les populations parentes sont des lois Gamma à deux paramètres et Pareto à trois paramètres.The concept of partial membership of a hydrometric station in a hydrologic region is modeled using fuzzy sets theory. Hydrometric stations are represented in spaces of hydrologic (coefficient of variation: CV, coefficient of skewness: CS, and their counterparts based on L- moments: L-CV and L-CS) and/or physiographic attributes (surface of watershed: S, specific flow: Qs=Qmoyen/S, and a shape index: Ic). Two fuzzy clustering methods are considered.First a clustering method by coherence (Iphigénie) is considered. It is based on the principle of transitivity: if two pairs of stations (A,B) and (B,C) are known to be "close" to one another, then it is incoherent to state that A is "far" from C. Using a Euclidean distance, all pairs of stations are sorted from the closest pairs to the farthest. Then, the pairs of stations starting and ending this list are removed and classified respectively as "close" and "far". The process is then continued until an incoherence is detected. Clusters of stations are then determined from the graph of "close" stations. A disadvantage of Iphigénie is that crisp (non fuzzy) membership functions are obtained.A second method of clustering is considered (ISODATA), which consists of minimizing fuzziness of clusters as measured by an objective function, and which can assign any degree of membership between 0 to 1 to a station to reflect its partial membership in a hydrologic region. It is a generalization of the classical method of mobile centers, in which crisp clusters minimizing entropy are obtained. When using Iphigénie, the number of clusters is determined automatically by the method, but for ISODATA it must be determined beforehand.An application of both methods of clustering to the Tunisian hydrometric network (which consists of 39 stations, see Figure 1) is considered, with the objective of obtaining regional estimates of the flood frequency curves. Four planes are considered: P1: (Qs,CV), P2: (CS,CV), P3: (L-CS,L-CV), and P4: (S,Ic), based on a correlation study of the available variables (Table 1).Figures 2, 3a, 4 and 5 show the clusters obtained using Iphigénie for planes P1 through P4. Estimates of skewness (CS) being quite biased and variable for small sample sizes, it was decided to determine the influence of sample size in the clusters obtained for P2. Figure 3b shows the clusters obtained when the network is restricted to the 20 stations of the network for which at least 20 observations of maximum annual flood are available. Fewer clusters are obtained than in Figure 3, but it can be observed that the structure is the same: additional clusters appearing in Figure 3 may be obtained by breaking up certain large clusters of Figure 3b. In Figure 3c, the sample size of each of the 39 stations of the network is plotted in the plane (CS,CV), to see if extreme estimated values of CS and CV were caused by small samples. This does not seem to be the case, since many of the most extreme points correspond to long series.ISODATA was also applied to the network. Based on entropy criteria (Table 2, Figures 6a and 6b), the number of clusters for ISODATA was set to 4. It turns out that the groups obtained using ISODATA are not very fuzzy. The fuzzy groups determined by ISODATA are generally conditioned by only one variable, as shown by Figures 7a-7d, which respectively show the fuzzy clusters obtained for planes P1-P4. Only lines of iso-membership of level 0.9 were plotted to facilitate the analysis. For hydrologic spaces (P2 and P3), it is skewness (CS and L-CS) and for physiographic spaces (P1 and P4) it is surface (Qs and S). Regionalization of the 100-year return period flood is performed based on the homogeneous groups obtained (using an index-flood method), and compared to the well-known region of influence (ROI) approach, both under the hypothesis of a 2-parameter Gamma distribution and a 3-parameter Pareto distribution. For the ROI approach, the threshold corresponding to the size of the ROI of a station is taken to be the distance at which an incoherence first appeared when applying Iphigénie. Correlation of the regional estimate with a local estimation for space P1 is 0.91 for Iphigénie and 0.85 both for ISODATA and the ROI approach. Relative bias of regional estimates of the 100-year flood based on P1 is plotted on Figures 9 (Gamma distribution) and Figure 10 (Pareto distribution). The three methods considered give similar results for a Gamma distribution, but Iphigénie estimates are less biased when a Pareto distribution is used. Thus Iphigénie appears superior, in this case, to ISODATA and ROI. Values of bias and standard error for all four planes are given for Iphigénie in Table 3.Application of an index-flood regionalization approach at ungauged sites requires the estimation of mean flow (also called the flood index) from physiographic attributes. A regression study shows that the best explanatory variables are watershed surface S, the shape index Ic and the average slope of the river. In Figure 8, the observed flood index is plotted against the flood index obtained by regression. The correlation coefficient is 0.93.Iphigénie and ISODATA could also be used in conjunction with other regionalization methods. For example, when using the ROI approach, it is necessary to, quite arbitrarily, determine the ROI threshold. It has been shown that this is a byproduct of the use of Iphigénie. ISODATA is most useful for pattern identification when the data is very fuzzy, unlike the example considered in this paper. But even in the case of the Tunisian network, its application gives indications as to which variables (skewness and surface) are most useful for clustering

Rotation und Vibration in Beispielen zur Methode der direkten Bewegungsteilung

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Selbsttätiges Auswuchten mittels zweier beliebiger Rollen

Zum selbsttätigen Auswuchten des starren Rotors in einer Ebene mittels zweier Kugeln oder Rollen allgemeinerer Gestalt, die sich in einem viskosen Medium unter bestimmten Bedingungen infolge der durch die Zentrifugalkräfte erregten Schwingungen kompensatorisch positionieren, werden für ein ausführlich diskutiertes Modell die Bewegungsgleichungen hergeleitet. Auf analytischem Wege werden für den im Vergleich zu bisherigen Untersuchungen allgemeineren Fall von Kompensationselementen mit ungleichen Zentrifugalkraften unter Voraussetzung vernachlässigbarer Schwingungsdämpfung die möglichen Paare synchroner Phasenwinkel und ihre Stabilität untersucht. Die Ergebnisse der analytischen Untersuchungen einschließlich der für bestimmte Parameterwerte vorhandenen Stabilität einer speziellen Lösung, die im Falle übereinstimmender Zentrifugalkräfte immer instabil ist, werden durch Computer-Simulation bestätigt

Identifying the underlying structure and dynamic interactions in a voting network

We analyse the structure and behaviour of a specific voting network using a dynamic structure-based methodology which draws on Q-Analysis and social network theory. Our empirical focus is on the Eurovision Song Contest over a period of 20 years. For a multicultural contest of this kind, one of the key questions is how the quality of a song is judged and how voting groups emerge. We investigate structures that may identify the winner based purely on the topology of the network. This provides a basic framework to identify what the characteristics associated with becoming a winner are, and may help to establish a homogenous criterion for subjective measures such as quality. Further, we measure the importance of voting cliques, and present a dynamic model based on a changing multidimensional measure of connectivity in order to reveal the formation of emerging community structure within the contest. Finally, we study the dynamic behaviour exhibited by the network in order to understand the clustering of voting preferences and the relationship between local and global properties.Comment: 20 pages, 10 figures, 3 tables, submitted to Physica