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1 Hydrological Sciences Journal ISSN: (Print) (Online) Journal homepage: Identification of homogeneous regional classes for flood frequency analysis in the light of regional taxonomy / Identification de classes régionales homogènes pour l'analyse fréquentielle des crues à la lumière d'une taxinomie régionale Lubomír Solín To cite this article: Lubomír Solín (2005) Identification of homogeneous regional classes for flood frequency analysis in the light of regional taxonomy / Identification de classes régionales homogènes pour l'analyse fréquentielle des crues à la lumière d'une taxinomie régionale, Hydrological Sciences Journal, 50:6, -1118, DOI: /hysj To link to this article: Published online: 15 Dec Submit your article to this journal Article views: 358 Citing articles: 3 View citing articles Full Terms & Conditions of access and use can be found at

2 Hydrological Sciences Journal des Sciences Hydrologiques, 50(6) December Identification of homogeneous regional classes for flood frequency analysis in the light of regional taxonomy UBOMÍR SOLÍN Institute of Geography, Slovak Academy of Sciences, Štefánikova 49, Bratislava, Slovakia Abstract The paper analyses delineation of hydrological regional classes in the light of regional taxonomy. A brief review of terminological and methodological aspects of regional taxonomy is outlined. The analysis of identification of hydrological regional classes from the point of view of the definition of the basic spatial unit, formulation of the regional taxonomic problem and evaluation of the hydrological response of the physical regional classes is then followed. A more detailed delineation of physical regional classes and a marked separation concerning their hydrological response are achieved if the basic spatial unit is defined as a small basin. Formulation of a hydrological regionalization or regional typification by means of problems defined in regional taxonomy can remove ambiguous and inconsistent features in identifying regional classes. The physical regional classes formed for the purpose of regional flood frequency analysis are considered as regional also from the hydrological point of view only if they satisfy both conditions of intra-class similarity and of inter-class dissimilarity regarding the hydrological attributes. Key words hydrological heterogeneity; hydrological homogeneity; region; regional flood frequency analysis; regional taxonomy; regional type Identification de classes régionales homogènes pour l analyse fréquentielle des crues à la lumière d une taxinomie régionale Résumé L article analyse la délimitation de classes hydrologiques régionales à la lumière d une taxinomie régionale. Une brève revue des aspects terminologiques et méthodologiques de la taxinomie régionale est présentée. Puis le processus d identification des classes hydrologiques régionales est analysé du point de vue de la définition de l unité spatiale de base, de la formulation du problème taxinomique régional et de l évaluation de la réponse hydrologique des classes physiques régionales. Une délimitation plus détaillée des classes physiques régionales et une séparation nette du point de vue de leur réponse hydrologique sont obtenues si l unité spatiale de base est définie comme un petit bassin versant. La formulation d une régionalisation ou d une typologie régionale hydrologiques, via la définition des problèmes en termes de taxinomie régionale, peut éliminer les ambiguïtés et les incohérences lors de l identification des classes régionales. Les classes physiques régionales, identifiées dans le but de l analyse fréquentielle régionale des crues, sont également considérées comme régionales du point de vue hydrologique seulement si elles satisfont à la fois aux deux conditions de la similarité intra-classe et de la dissimilarité inter-classes en termes de caractéristiques hydrologiques. Mots clefs hétérogénéité hydrologique; homogénéité hydrologique; région; analyse fréquentielle régionale des crues; taxinomie régionale; type régional INTRODUCTION The estimation of T-year return flood by the equation ( ) µ ( ) Q T i F i q F = (1) which was introduced by Dalrymple (1960) as the index flood method, is a widely used regional method of flood estimation for ungauged basins, or those with relatively Open for discussion until 1 June 2006

3 1106 ubomír Solín short gauging histories. The symbol Q T ( ) i F is the estimated T-year return flood for the ith basin of regional class; µ i is the index flood, expressed usually by arithmetical means of maximum annual discharges, and q(f) is the dimensionless quantile function of the regional frequency distribution, which is the same for all basins of the regional class. Cunnane (1988), GREHYS (1996) and Hosking & Wallis (1997) presented summarized surveys of other methods of regional flood frequency analysis. The index flood procedure includes three key problems: the delineation of hydrological homogeneous regional classes, the estimation of the regional distribution function and the estimation of the index flood. In the paper, emphasis is laid on analysis of identification of hydrological regional classes for the purpose of regional flood frequency analysis. The issue of estimation of regional distribution function is thoroughly analysed, for example, by Greis & Wood (1981), Hosking et al. (1985), Lettenmaier et al. (1987), Chowdhury et al. (1991) and Hosking & Wallis (1997). The estimation of index flood for ungauged basins is normally made by regression equations based on the physical attributes of the basins. On the basis of the general assertion that similar physical properties of territory produce a similar hydrological response, hydrologically homogeneous regional classes initially coincided with physical geographical regions. The advantage of this approach was that the regional division of the study territory was exhaustive. However, additional tests of geographical regions showed that in many cases regions displayed traits of a marked hydrological heterogeneity (Wiltshire, 1986a). Consequently, spatially contiguous geographical regions were gradually abandoned and clustering of basins into groups according to hydrological or physical basin characteristics started to apply (Mosley, 1981; Gottschalk, 1985; Wiltshire, 1985, 1986b; Acreman & Sinclair; 1986). This approach was gradually accepted and it dominates now (e.g. Burn, 1988, 1990a,b; Nathan & McMahon, 1990; Zrinji & Burn, 1994; Solín & Pola ik, 1994; Hall & Minns, 1999; Burn & Goel, 2000; Ouarda et al., 2001; Solín, 2002; Kohnová & Szolgay, 2000, 2002; Jingyi & Hall, 2004; Chang & Burn, 2003; Lim & Lye, 2003). The nature of regional classes delimited in this way was not that of spatial contiguity in the geographical space anymore and they were characterized by a higher degree of homogeneity as far as the chosen hydrological attributes were concerned. As far as identification of regional classes for the purpose of regional flood frequency analysis is concerned, attention is focused on application of different clustering algorithms (logical division, clustering analysis, multivariate statistical analysis, fuzzy logic, artificial neural network) producing homogeneous groups of basins from the hydrological point of view. However, regional division of a territory also involves additional and equally important aspects, such as definition of the basic spatial unit that is the subject of clustering, definition of the character of regional classes in terms of subject and terminology, systemization of regional classes related to solved problems, evaluation of the regional classes and assessment of optimal regional class number covering the whole study area. Identification of regional classes is the key issue of geographical research. It was in geography where the basic terminological and methodological problems of regional division were formulated and their solutions were also developed. The geographical approach to the regional division is presented in hydrology in a very reduced form, however. For example Acreman & Sinclair (1986) and others only see it as an

4 Identification of homogeneous regional classes for flood frequency analysis 1107 application of geographical regions. Hydrologists incorrectly conceive geographical regional classes only in the sense of a region, that is a spatially contiguous and closed area delimited on the basis of administrative or physiographic boundaries (Wiltshire, 1986a). The reduction of geographical regionalization to the process of interpolation of the location-specific data across a spatially contiguous area, the products of which are isoline maps (e.g. McKay, 1976; Acreman & Wiltshire, 1989) is also erroneous. It limits the interpretation of the term geographical to the location exclusively, and it is presumed that location vicinity predestines attribute similarity. The aim of this paper is to outline in a brief way some terminological and methodological aspects of identification of regional classes as described in geographical literature and then to analyse delineation of hydrologically homogeneous regions for regional flood frequency analysis from the geographical point of view. This reflection may help to remove some ambiguous and inconsistent features in identifying hydrological regional classes. BASIC CONCEPTS, DEFINITIONS AND PROBLEMS OF REGIONAL TAXONOMY The traditional approach to the regional division of the landscape developed in physical geography is based on clearly visible regions in geographical space, which are a priori given (cf. Grigg, 1965; McDonald, 1966), or on the delimitation of the individual landscape spheres into relatively homogeneous areas concerning certain significant attributes (Armand, 1975). However, in geography, as an alternative to this approach, a general consensus was reached that the regional division of territory represents spatial forms of a general classification system where the function of classified object belongs to the place and not to the individual. Bunge (1962) and Grigg (1965) first developed this idea. Clustering of individuals into a system of classes based on certain criteria then corresponds to clustering of places into the system of area classes regions with regard to the established criteria. The stress upon the fact that regionalization is the spatial variant of general classification eventually led to the introduction of the term regional taxonomy (Spence & Taylor, 1970). The basic concepts and theoretical and methodological principles of regional taxonomy were further elaborated by Fischer (1978, 1987) and a thorough review of regional taxonomy was presented by Bezák (1993, 1996). The basic problem of regional taxonomy is defined as follows: Let us presume that the set B consisting of n basic spatial units B 1, B 2,, B n, which are characterized by the set of predicates A 1, A 2,, A p, is given. The task of regional taxonomy is to find such a partition of the set B to non-empty classes R 1, R 2,, R k (while it holds 1 k n ), which convenes to certain objective function defined in relation to the set p of predicates. (Bezák, 1993). In regional taxonomy, the study territory is divided into basic spatial units in an exhaustive way and each unit is characterized by a set of attributes or relationships (common term predicate). The objective function relates to the intra-class similarity and/or inter-class dissimilarity of basic spatial units and intra-class and/or inter-class links between basic spatial units, respectively. The area classes fulfilling the criterion of inner homogeneity and/or outer mutual heterogeneity in relation to the set of

5 1108 ubomír Solín Fig. Classification of regional taxonomic problems according to Fischer (1987). attributes or links, as related to relationships of spatial units, assume the character of regional classes (regional taxa). Fischer (1987) identified four basic groups of regional taxonomic problems in relation to regional classes (Fig. 1): (a) regional classes with the character of regional type or region, (b) formal (homogeneous) or functional regional classes, (c) hierarchic or non-hierarchic regional classes, and (d) mutually overlapping (non-disjoint) or not overlapping (disjoint) regional classes. The classes possessing the character of regional type or region are delimited with regard to their spatial contiguity. Basic spatial units of regional type, in contrast to the region, do not occupy contiguous areas in geographical space. The process of identification of spatially contiguous regions is then referred to as regionalization and the process of identifying regional types is called regional typification. Formal (homogeneous) regional classes express similarity of basic spatial units concerning the set of attributes, while delimitation of functional regional classes is based on relationships between pairs of basic spatial units. The distinction between hierarchic and nonhierarchic regional classes expresses whether the delimited regional classes are or are not arranged into the hierarchic system. The result of hierarchic regional taxonomy is a hierarchically arranged sequence of regional classes, whilst the result of the nonhierarchic regional taxonomy is a mere establishment of the number of regional classes, which is considered optimal from a certain point of view. In the case of nondisjoint regional classes, there is at least one basic spatial unit, which is covered by two or more regional classes, while in the case of disjoint regional classes, each basic spatial unit is under one regional class only. So, according to character of regional classes, 16 different regional taxonomic problems can be identified. According to Fischer (1987), the regional taxonomic process consists of several phases within which some subjective decisions have to be made. The first phase contains the decision about specification of regional taxonomic problems, delimitation of the basic spatial units and choice of their attributes important from the point of view of the regional taxonomic process. The second phase concerns data transformation. The spatial units must be comparable in terms of chosen attributes. Therefore, it is necessary to make some corrections connected with the size of regional units and to standardize them in the case of different measuring units. Further, it is necessary to consider orthogonal transformation of attributes by component or factor analysis in case of their mutual dependence.

6 Identification of homogeneous regional classes for flood frequency analysis 1109 The third phase deals with the choice of similarity measures in order to express the similarities or dissimilarities of basic spatial units regarding the set of chosen attributes. The choice of similarity measure is influenced by the type of the attributes and, additionally, by the specific properties of measure. In the case of metric attributes, a similarity measure from the group of Minkowski metrics (e.g. Euclidean distance) or Mahalanobis metric should be used. On the other hand, the similarity measure of nonmetric attributes can be expressed by a product moment correlation, for example. The fourth phase contains the choice of method for clustering the basic spatial units into regional classes. It should be stressed that application of different similarity measures and clustering methods to the same data set normally leads to different regional taxonomic results. The fifth phase includes evaluation of the regional taxonomic results obtained. The process focuses on evaluation of the regional taxonomic system as a whole, evaluation of regional classes and evaluation of the significance of regional taxonomic results. ANALYSIS OF DELINEATION OF HYDROLOGICAL HOMOGENEOUS REGIONAL CLASSES As a result of the limited number of gauged basins, regionalization or regional typification from the hydrological point of view acquired some specific features: knowledge of the regional hydrological structure of the study territory is induced on regional structure identified in a sample of gauged basins; hydrological homogeneous regional classes are delineated on the basis of the physical rather than the hydrological attributes of basins; the hydrological responses of the identified physical regional classes are evaluated; the additional inclusion of basins, which were not subjects of the regional taxonomic process, into physical regional classes identified in the framework of the sample of gauged basins is necessary. In the case that homogeneous regional classes are identified directly from hydrological characteristics, the additional physical justification of location of the individual hydrological regional classes is carried out, too. Irrespective of the clustering algorithm, the necessity to manage these specific features of hydrological regional typification or regionalization is met. Therefore some considerations on definition of hydrological basic spatial units, regional taxonomic problems solved in hydrology, evaluation of the hydrological response of physical regional classes and an exhaustive character of hydrological homogeneous regional classes from the point of view of regional taxonomy are presented in the following part of this paper. Hydrological basic spatial units fulfilling the function of classified objects Generally, the objects of regional taxonomy basic spatial units should be spatially continuous, not overlapping, be comparable in size and they should completely cover the study territory (Bezák, 1993). Delimitation of the basic spatial unit fulfilling the criterion of spatial continuity in hydrology is comparatively unequivocal it is the basin. However, less obvious is its area. A set containing basins with an area ranging

7 1110 ubomír Solín from 1 2 km 2 to more than 2000 km 2 (for instance, Wiltshire, 1986a) inspires certain doubts concerning their mutual comparability as regards surface area. Constitution of a sample of basins as a combination of large, medium and small basins is questionable. In fact, a set of small basins (e.g. basins with areas below 250 km 2 ) makes it possible to achieve a more detailed delimitation of physical regional classes and a marked separation concerning their hydrological response than does a set of large basins. The predictive function for ungauged basins is enhanced in the case of hydrologically clearly separated regional classes. In the case of large rivers, which traverse a wide range of climates, lithologies, vegetation and soil types, Mosley (1981) pointed out that it is doubtful that any regional scheme could be of much practical utility, except for very small basins, other than treating separately each portion of a catchment that falls within more than one region. As gauging is not available for all basins covering the study territory, the sample of gauged basin should fulfil the requirement of representativeness. This means that selected basins should represent the physical diversity of the basin population reasonably well. Only in such a case, will the regional classes identified in the sample reflect the regional classes in population. The basins should not overlap. Overlapping of a sample of gauged basins is very frequently mainly due to nested basins. The values of upstream gauging profiles affect the hydrological values of those situated downstream. The mutual dependence of hydrological values can markedly influence the results of testing the hydrological response of physical regional classes by the methods of inductive statistics, which are sensitive to infringement of the independence of analysed data. The incidence of mutual dependence of hydrological data in the consequence of nested basins should therefore be eliminated by the correct constitution of a sample of basins. An exhaustive coverage of the study area by basic spatial units (small basins) is the inevitable step for inclusion of ungauged basins into physical regional classes identified in a sample of gauged basins and for the cartographic presentation of regional taxonomic results. The division of Slovakia into more than 5000 small basins by Solín & Grešková (1999) can be quoted as an example. Identification of hydrological regional classes: hydrological or physical basin characteristics One of the basic principles of regional taxonomy states that regionalization or regional typification should be based directly on characteristics of classified objects that agree with the declared aim of regionalization or regional typification and not on attributes, which presumably influence the spatial variability of the differentiating characteristics (Grigg, 1965). The delineation of hydrological homogeneous regional classes should then be based on the hydrological characteristics of basins. However, hydrological regional structure can be directly identified on the basis of hydrological characteristics only within the set of gauged basins. Therefore, inclusion of the remaining basins into the delimited hydrological regional classes poses a problem. This is the reason why the delimitation of hydrological regional classes based on the physical attributes of basins, which are accessible for the basin population of the study territory, is preferred. The GIS technology, digital model of relief, interpretation of land cover from satellite images and digital form of analogue thematic maps provide the opportunity to remove

8 Identification of homogeneous regional classes for flood frequency analysis 1111 the limitations which caused that only the most readily available physical basin could be used. Overlaying of digital map layers in the GIS makes it possible to create a much more precise and extensive database of the physical characteristics of basins than before (cf. Solín et al., 2000). For delineation of physical regional classes, only those physical basin characteristics are important which decisively influence the spatial variability of hydrological characteristics. In such a case, the dependence between the hydrological characteristic and the physical characteristics is the causal and not only correlated dependence. The estimation of the nature of dependence leans on the logical judgment. An exact expression of the weight of the individual physical characteristics is reached if the methods of mathematical statistics, such as partial regression analysis, stepwise regression or factor/component analysis, are applied (e.g. Nathan & McMahon, 1990). In many cases, using a simple graph of scatter plot is very useful. The area of a basin is often considered an important physical attribute (e.g. Jingyi & Hall, 2004). However, if the rule is met that the basic spatial units of regional taxonomy are of similar size, the effect of basin area on hydrological characteristic is eliminated. Therefore, its use as an important factor for clustering is questionable. The basin area is an important scaling factor, but it is not a regional variable. The indispensable step when using physical characteristics is the assessment of the hydrological response of the delimited physical regional classes. Application of different combinations of physical characteristics produces a different hydrological response. At first, conventional moments of maximum annual discharges were used to express the hydrological response (e.g. Mosley, 1981; Wiltshire, 1985, 1986b). Using the parameters of a particular distribution function (e.g. Acreman & Sinclair, 1986) is not recommended, because the same hydrological data can fit several distribution curves. However, the assessment of hydrological response is now carried out by means of L-moments introduced by Hosking (1990). Compared to conventional moments, the L-moments offer several advantages. They are less affected by extreme observations and less biased, have natural bounds r < 1, and have a greater discriminant power to identify the distribution from which the sample was drawn (Hosking & Wallis, 1997). If the characteristics used for regional division are in agreement with the declared aim of regional typification or regionalization, then, by creating classes according to similarity of attributes of basic spatial units, the clustering algorithms simultaneously cause the basic spatial units among the regional classes to be different. This is the reason why it suffices, if classes comply (with regard to the attributes of basic spatial units), either with the condition of intra-class similarity, or with that of inter-class dissimilarity, to be considered regional classes. However, physical regional classes formed for the purpose of regional flood frequency analysis are considered as regional also from the hydrological point of view only if they comply with both the condition of intra-class similarity and that of inter-class dissimilarity regarding the hydrological response. Types (character) of hydrological regional class In hydrology, the concept of spatially contiguous physical regions has been abandoned, but it has not been coped with physical or hydrological regional classes that are not spatially contiguous in terms of subject and terminology. The regional taxonomic

9 1112 ubomír Solín problem is still formulated as the identification of hydrologically homogeneous regions. As has already been mentioned, the classes in regional taxonomy, as far as their contiguity or non-contiguity in geographical space is concerned, are termed regions or regional types, respectively. However, the use of the term region for both spatially contiguous and spatially non-contiguous regional classes is widespread among hydrologists. Traditionally, a region in flood frequency analysis is defined as an area within which the statistical distribution of standardized annual maximum floods is broadly similar in all of the drainage basins (Acreman & Wiltshire, 1989). This loose definition is a source of imprecision and ambiguities regarding the consistence of theoretical and methodological principles. The currently used methods of clustering normally produce regional types. Identification of regions requires the condition of contiguity to be included in the regionalizing algorithm directly, or after the delineation of regional types it is necessary to make additional modification concerning the geographical location of regional types. So, clustering of several regional types together produces a region with a certain pattern of arrangement of the regional types, for which the condition of inner attribute homogeneity is then not a priority. A certain attempt to distinguish by terminology the spatially non-contiguous from contiguous regional classes has been done by Nathan & McMahon (1990), Jingyi & Hall (2004) by using the term subregions for regional classes composed of basins not contiguous in geographical space. With regard to predicates the character of regional classes for the purposes of frequency analysis is exclusively that of formal (homogeneous) regional classes that are identified on the basis of hydrological or physical attributes of basins. Regional classes with occurrence of the same unit in several regional classes have the character of non-disjoint classes. Acreman & Wiltshire (1989) inappropriately denoted the fact that a basin can be classified into several regional classes according to its physical attribute, the regionalization without regions. This approach was later elaborated by Burn (1990a,b), who denoted it the region of influence (ROI). The idea that a proper region can be created for each basin as formed from basins, which possess similar physical or hydrological characteristics, is crucial for this approach. The inclusion of a station into a ROI for a site is made on the basis of an a priori determined threshold value of the Euclidian similarity measure. Using distance to centroid of groups (Wiltshire, 1986a), discriminant analysis (Wiltshire, 1986b) or fuzzy c-method (Hall & Minns, 1999; Jingyi & Hall, 2004) is another way of classifying a basin into several regional classes. Fractional membership then served as a weight to make more accurate the estimation of dimensionless hydrological quintiles for a basin. However, the results achieved by Jingyi & Hall (2004) implied that the use of the membership levels obtained by the application of the fuzzy c-means algorithm as weights in determination of the at-site growth curves did not render the expected results, because the differences between the weighted and unweighted growth curves of the identified regions were relatively small. In disjoint regional classes arranged into either hierarchic or non-hierarchic classes, each basic spatial unit is under one regional class only. They are implicitly delimited by traditional methods, such as the logical division and numerical classification, or application of new clustering algorithms based on artificial neural networks or fuzzy logic (cf. Hall & Minns, 1999; Jingyi & Hall, 2004). In the latter, however, hard (crisp) partition must be used. In connection with disjoint regional classes, it is

10 Identification of homogeneous regional classes for flood frequency analysis 1113 important to set the optimal number of physical regional classes that are to be delimited within the sample of basins. If the logical division is applied, the number of classes depends on the selection of differentiating values of physical characteristics producing different basin groups regarding the hydrological response. If the hierarchic clustering procedures are used, the number of classes depends on the proper location of the vertical line cutting dendogram. As regards non-hierarchic procedures, the number of delimited classes must be set a priori, and in the case of artificial neural networks, the optimal number of regional classes is set automatically when the unsupervised learning scheme is used. The application of an artificial neural network, after Hall & Minns (1999) and Jingyi & Hall (2004), objectifies the clustering into classes, as the classification process takes place free of subjective decisions that are unavoidable in the application of other methods. However, the results of testing hydrological response of physical regional classes are vital for setting their optimum number. The physical regional classes must satisfy with regard to hydrological response both the condition of intra-class similarity and that of inter-class dissimilarity. Evaluation of hydrological response of physical regional classes The physical regional types delimited for flood frequency analysis are considered homogeneous from the hydrological point of view if the between-site dispersion of the sample L-moments ratios for a group of basins under regional types is not larger than expected of a homogeneous regional class. Chowdhury et al. (1991), using the sample L-moments (L-Cv, L-Cs), expressed the homogeneity level of physical regional classes by χ (R) 2. But Hosking & Wallis (1997) stress that it is reasonable to concentrate on L-Cv because the between-site variation in L-Cv has a much larger effect than the variation in L-skewness or L-kurtosis on the variance of the final estimates of all but the most extreme quantiles. The homogeneity of regional class as regards the variation at site L-Cv is then expressed by the statistics of the H heterogeneity (cf. Hosking & Wallis, 1993, 1997). If H is less than 1, the regional unit is considered acceptably homogeneous ; if H is in the interval 1 H 2 it is considered heterogeneous, and if H is greater than 2, the regional unit is considered definitely heterogeneous. Many authors applied the rate of H heterogeneity for expression of the homogeneity level. However, Hosking & Wallis (1997) pointed to the not entirely justified strict interpretation of inner homogeneity testing results of regional classes. As shown by the results of simulation in even heterogeneous regional classes (with the precision of T- year quantiles estimation less by 20 40% compared to homogeneous regional classes), the estimate of quantiles by regional frequency curves was still more precise than that made by the at-site frequency curves. The satisfaction of the condition of hydrological homogeneity for setting the optimal number of physical regional classes as regards hydrological response does not suffice. The physical regional classes must satisfy, as regards to hydrological response, the condition of inter-class dissimilarity, too. For instance the H value of three physical regional types delimited by the method of Kohonen network (Jingyi & Hall, 2004) is in the interval, which allows them to be considered homogeneous. But the physical regional types produce the following regional L-Cv values: , and , respectively. The differences between them are not large. The question arises whether

11 1114 ubomír Solín the physical regional types as regards the hydrological response are distinctly different inter se in spite of hydrological homogeneity. If they are not, it is not justified to delimit them as independent regional classes. Whether or not substantial differences exist between physical regional types from the point of view of flood frequency behaviour can be tested using the analysis of variance model (ANOVA), wherein differences in the regional values of hydrological characteristics reflect the essential physical regional type effects. For example, Wiltshire (1986a) and Acreman & Sinclair (1986) applied the F test to evaluate the hydrological response of physical regional classes expressed by the coefficient of variation of maximum annual runoff. In the case of F statistics, the null hypothesis on equality of the expected mean regional values of the hydrological characteristic is tested. If the null hypothesis is rejected, one can conclude that the dependence exists between the physical regional classes and the regional values of hydrological characteristics, but cannot draw conclusions on the significance of the differences between the individual physical regional types as regards the regional hydrological response. Evaluation of hydrological differences between the individual physical regional classes can be based on application of the method of pairwise comparison (cf. Neter et al., 1985; Hsu, 1996). Pairwise comparison tests the null hypotheses that the differences of expected regional hydrological values between physical regional types is equal to zero. The testing criterion is the interval of the expected difference values between mean regional values. If the interval does not contain the value 0, the null hypotheses are rejected and the physical regional classes are considered hydrologically different. Applying this approach, Solín (2002) has evaluated the hydrological response of physical regional types delineated by different methods. Table 1 shows the regional values of L-Cv of delineated physical regional types and in Table 2 are the results of the F test and pairwise comparison related to differences in regional L-Cv values. At-site L-Cv moments were calculated using the mean annual maximum runoff for the period. Four hydrological regional types identified in the sample of 156 small gauged basins by logical division (LD-P+A) on the basis of differentiation of values of mean annual precipitation and altitude of basins (Table 3) are considered to be the optimum for use in regional flood frequency analysis. The exhausting character of regional typification The last step in the regional-taxonomic process in hydrology is the inclusion of the remaining basins in the study territory into the regional classes identified in the framework of the gauged basins sample. If the regional classes are delineated on the set of hydrological attributes, the exhausting character of the regional taxonomic results can be ensured by means of linkage of the location of regional classes to general physical regions or geo-ecological regional types. The linkage between the delimited hydrological regional classes and traditional physical regions by means of a simple visual comparison is not very consistent. Hydrological regions of New Zealand delineated by Mosley (1981) directly on the basis of mean annual specific maximum discharge and the coefficient of variability of maximum annual discharges by hierarchical cluster analysis were compared to the regions of climatic regime, topography or geology. In the South Island, four hydrological clusters broadly coincide with

12 Identification of homogeneous regional classes for flood frequency analysis 1115 Table Regional value of sample L-Cv and the number of basins in the regional types delineated by different methods. Method of regional typification Regional types L-Cv (number of basins) I II III IV HC-average linkage (68) (69) (12) (7) HC-group average (13) (74) (50) (19) NHC-K means (51) (54) (44) (7) LD-A (11) (87) (52) (6) LD-P (24) (82) (35) (6) LD- P+A (12) (60) (34) (50) HC: hierarchical cluster analysis; NHC: non-hierachichal cluster analysis; LD: logical division; P: mean annual basin precipitation; A: mean altitude of basin. Table 2 Results of the tests of hydrological consequences. Method of regional F test Non significant differences typification between regional types HC-average linkage II-III, II-IV, III-IV HC-group average III-IV NHC-K means III-IV LD-A III-IV LD-P III-IV LD-P+A Table 3 Physical regional types: differentiation values of mean annual precipitation and mean altitude of basins. Physical regional types P (mm) A (m a.s.l.) I II III 950 >700 IV >950 >700 climatic regions. However, a quite different situation was found in the North Island, where the agreement between the hydrological cluster on the one side and some physical regions on the other was not displayed. This approach represents a certain return to hydrological regional division based on physical regions. Wiltshire (1986b) applied a more exact comparison. Allocation of basins to ten clusters, which were derived from flood statistics by non-hierarchic cluster analysis, was compared to allocation of basins into ten clusters obtained by discriminant analysis on the basis of basin characteristics. A good correspondence was found in the case of six clusters. The poor performance of the remaining four clusters reflects the ever-present difficulty of relating flow statistics to basin characteristics (Wiltshire, 1986b). If identification of hydrological regional classes is based on physical basin attributes, the exhausting character of regionalization or regional typification can be achieved by classifying the rest of the basins into regional classes according to their basin attributes. This approach is conditional on the existence of a database of physical characteristics for the whole basin population of the studied territory. The discriminant

13 1116 ubomír Solín Fig. 2 Slovakia: physical regional types of small basins for regional flood frequency analysis. analysis can be used to include basins into regional types delineated by application of the numerical method. Using some logical principles, the basins are classified into physical regional classes on the basis of the value of a physical attribute. For example, more than 5000 small basins of Slovakia was classified into four physical regional classes, which were delineated in the framework of sample of basins by the method of logical division (LD-P+A) on the basis of the mean annual total of precipitation and the mean sea-level altitude of the basin (Fig. 2). The mean annual basin precipitation was calculated from rasters of mean annual precipitation derived from isohyet map by the method of regularized spline with tension (Mitášová & Mitáš, 1993). The mean sea-level altitude of each basin was determined on the basis of the digital relief model (Hofierka et al., 1998). CONCLUSION Identification of hydrologically homogeneous regional classes is analysed from the point of view of regional taxonomy. Regional taxonomy represents a spatial form of the general classification system, where the function of classified objects belongs to the place instead of the individual. The first part of the paper presents a brief outline of the basic principles and the individual steps of the regional-taxonomic process. In the second part, delineation of hydrological regional classes is analysed from the following points of view: definition of the basic spatial unit, which is the subject of the regional taxonomic process; classification of the regional taxonomic problems; evaluation of the obtained regional taxonomic results; and exhaustive regional division of the study territory. Specific features of the regional taxonomic process in hydrology were also analysed, e.g. identification of hydrological regional classes in a sample of basins; delineation of hydrological regional classes on the basis of the physical rather than the hydrological basin attributes; the need to evaluate the hydrological response of

14 Identification of homogeneous regional classes for flood frequency analysis 1117 physical regional classes; and the additional inclusion of the remaining basins into the physical regional classes identified in the framework of the sample of gauged basins. The basic spatial unit of regional taxonomic process in hydrology is the basin. The paper highlights the fact that basins should be comparable in terms of their size, they should not overlap and the selected set of gauged basins should represent the whole population of basins. The predictive function for ungauged basins is enhanced in the case of hydrologically clearly separated physical regional classes. This is the reason why it is proper to define the basic spatial unit from the point of view of area size as small basins, having areas below 250 km 2 approximately. In fact, a set of small basins makes it possible to achieve a more detailed delimitation of physical regional classes and a marked separation concerning their hydrological response. In hydrology, the concept of spatially contiguous physical regions has been abandoned, but it has not coped with physical or hydrological regional classes that are not spatially contiguous in terms of subject and terminology. The classification of regional taxonomic problems in the sense of region regional type, formal (homogeneous) functional, hierarchic non-hierarchic, disjoint non-disjoint, can facilitate a more precise formulation of regional taxonomic problems in hydrology. Testing of the hydrological response of the delineated physical regional classes is an indispensable step in delineation of hydrological regional classes on the basis of physical characteristics of basins. But the setting of the optimal number of physical regional classes requires more than satisfaction of the condition of hydrological homogeneity. The classes must also satisfy the condition of mutual dissimilarity (heterogeneity). Application of the F test as the testing criterion is only the first step. It must be followed by pairwise comparison. Only physical regional classes that differ from each other and simultaneously preserve a certain degree of inner homogeneity from hydrological point of view represent the optimal regional division of the territory for the purposes of regional flood frequency analysis. Acknowledgement This article has been written under the 2/3085 Project financially supported by the VEGA Grant Agency of the Slovak Academy of Sciences. REFERENCES Acreman, M. C. & Sinclair, C. D. (1986) Classification of drainage basins according to their physical characteristics; an application for flood frequency analysis in Scotland. J. Hydrol. 84, Acreman, M. C. & Wiltshire, S. (1989) The regions are dead; long live the regions. Method of identifying and dispensing with regions for flood frequency analysis. In: FRIENDS in Hydrology (ed. by T. Roald, K. Nordseth & K. A. Hassel) (Proc. Bolkesjø Symp., April 1989), IAHS Publ. 187, IAHS Press, Wallingford, UK. Armand, D. L. (1975) Nauka o Landschafte. Mysel, Moskva, Russia. Bezák, A. (1993) Problémy a metódy regionálnej taxonómie (Problems and methods of regional taxonomy, in Slovak). Geographia Slovaca 3, Bezák, A. (1996) Regional taxonomy: a review of problems and methods. Acta Faculatis Rerum Naturalium Universitatis Comenanae, Geographica 28, Bunge, W. (1962) Theoretical geography. Lund Studies in Geography, Series C, General and Mathematical Geography 1 C.W.K. Gleerup, Lund, Sweden. Burn, D. H. (1988) Delineation of groups for regional flood frequency analysis. J. Hydrol. 04, Burn, D. H. (1990a) An appraisal of the region of influence approach to flood frequency analysis. Hydrol. Sci. J. 35(2), Burn, D. H. (1990b) Evaluation of regional flood frequency analysis with a region of influence approach. Water Resour. Res. 26(10), Burn, D. H. & Goel, N. K. (2000) The formation of groups for regional flood frequency analysis. Hydrol. Sci. J. 45(1), Chang Shu & Burn, D. H. (2003) Spatial patterns of homogeneous pooling groups for flood frequency analysis. Hydrol. Sci. J. 48(4),

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