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AUC Geographica AUC Geographica 9 ACTA UNIVERSITATIS CAROLINAE AUCAUC GEOGRAPHICAGEOGRAPHICA 47 2/2012 CHARLES UNIVERSITY IN PRAGUE • KAROLINUM PRESS • 2012 AUC_Geographica_02_2012_47_2592.indd 1 19.11.12 14:29 © Charles University in Prague – Karolinum Press, 2012 ISSN 0300-5402 AUC_Geographica_02_2012_47_2592.indd 2 19.11.12 14:29 CONTENTS ARTICLES RADEK DušEK, RENATA POPELKOVÁ: Theoretical view of theS hannon index in the evaluation of landscape diversity . 5 MICHAL JENÍček, HANA BEITLEROVÁ, MARTIN HASA, DANA KučerOVÁ, HANA PEVNÁ, SLÁVEK PODZIMEK: Modelling snow accumulation and snowmelt runoff – present approaches and results . 15 JAN Klimeš: Geomorphology and natural hazards of the selected glacial valleys, Cordillera Blanca, Peru . 25 PAVEL NOVÁK, PETR Fučík, IVAN NOVOTNÝ, TOMÁš HEJDUK, DANIEL ŽíŽALA: An integrated approach for management of agricultural non-point pollution sources in the Czech Republic . 33 BLANKA MARKOVÁ, TOMÁš BORUTA: TheP otential of Cultural Events in the Peripheral Rural Jesenicko Region . 45 LUKÁš VÍT, JAN D . BLÁHA: A Study of the User Friendliness of Temporal Legends in Animated Maps . 53 REPORTS JAN KALVODA, JAROSLAV KADLEC: Summary of papers by Czech participants at the XVIIIth INQUA Congress in Bern . 63 AUC_Geographica_02_2012_47_2592.indd 3 19.11.12 14:29 AUC_Geographica_02_2012_47_2592.indd 4 19.11.12 14:29 AUC Geographica 5 THeoretical VIEW OF THE SHANNON INDEX IN THE evaluation OF landscape DIVERSITY RADEK DušEK, RENATA POPELKOVÁ University of Ostrava, Faculty of Science, Department of Physical Geography and Geoecology ABstract Shannon’s diversity index is frequently used in the determination of landscape diversity. Its indisputable advantage is a possibility to obtain numeric values that can subsequently be easily compared. However, accurate evaluation of landscape diversity from obtained results is rather complicated. The aim of the article is (i) to take a closer look at the theoretical origin of the formula that stems from the principles of the calculation of information entropy and (ii) to draw attention to several issues connected to the Shannon index application in landscape diversity assessment. Numeric value of the Shannon’s index depends on applied logarithm base that is not precisely specified by the formula. Presenting the resulting Shannon index value without stating the logarithm base is not very suitable. Nevertheless, a bigger problem is the dependence of the resulting Shannon’s diversity index value on two parameters, namely the number of studied categories and evenness of spatial distribu- tion of individual categories. The resulting value may be identical for different types of the division of the study area.T herefore, the number of categories and the evenness of spatial distribution need to be taken into consideration in the very assessment of the Shannon index result. The number of categories could also be presented along with the resulting Shannon’s index value. A major drawback of the Shannon index is its inability to express spatial distribution of patches within the area; it only presents the total extent of each category. Out of existing mod- ifications of the index that try to take spatial distribution into consideration, the most convenient is the coefficient of the distance between the extent of identical and different categories. Based on arguments deriving from theoretical basis of the Shannon index formula and its practical application, a new view of landscape diversity maximum is presented. The application of the Shannon index disregards the fact that the original relation required for entropy calculation presupposes independence of the existing state (e.g. land cover categories in case of landscape assessment). With regard to the fact that commonly defined categories of patches are independent; the index calculation should make use of the relation considering conditional probabilities of the occurrence of a certain category. Key words: Shannon index, entropy, landscape diversity, maximum diversity 1. Introduction which is defined as the ratio of the area that originated due to mining to the area representing original cultural Landscape assessment represents a complex activity . landscape . The coefficient represents the transformation Landscape can be described from the point of quality and process of original cultural landscape into mining land- quantity . Quality description of landscape mosaic focus- scape (Mulková 2007) . es particularly on its content . On the base of qualitative The use of landscape metrics for the assessment of characteristics all structures of landscape mosaic are landscape structure and evolution is, however, connected ranged into individual categories, e .g . land cover cate- with many problems . The metrics cannot always be ap- gories . Quantitative approach deals with quantitative as- plied to all data . Special attention thus needs to be paid to sessment, i .e . possibilities to measure and calculate vari- the interpretation of quantitative data . ous values of the landscape structure . Quantitative land- Landscape can be characterised by diversity scape characteristics are determined by means of land- expressing the extent of heterogeneity and variety of scape metrics describing landscape structure and evolu- landscape structure . In ecology, landscape diversity in- tion . A benefit of quantitative values consists in obtain- dices are, for example, the Simpson’s diversity index, ing exact numeric data on the landscape structure that which is particularly sensitive to species richness, and can be compared, e .g . various years within one locality or the Shannon index, which is sensitive to rare species (Fa- various localities in individual years (Popelková 2009) . rina 2006) . Shannon’s and Simpson’s diversity indices can The number of landscape metrics is high . For exam- also be applied to landscape (UMass Landscape Ecology ple, McGarigal and Marks (1995) present 100 land- Lab 2012) . scape metrics, many of which are mutually dependent If landscape diversity needs to be assessed, it is Shan- (Cushman et al . 2008) . Existing metrics are often modi- non’s diversity index that is used most frequently, as sta- fied and completely new metrics occur as well . One of ted by McGarigal and Marks (1995) . The aim of the article them is a newly created coefficient of mining-based is to draw attention to difficulties related to the applica- landscape transformation (Mulková, Popelková 2008), tion of this landscape assessment index . To be quoted as: Dušek, R. – Popelková, R. (2012): Theoretical View of the Shannon Index in the Evaluation of Landscape Diversity AUC Geographica, 47, No. 2, pp. 5–13 AUC_Geographica_02_2012_47_2592.indd 5 19.11.12 14:29 6 AUC Geographica 2. Theoretic background where k is any constant – the issues of the constant are dealt with in the section ‘Logarithm base’ . The equation 2.1 Origin expresses the information extent within one message . If we want to express the average information extent for one The author of Shannon’s diversity index equation is symbol of the message, then we get Claude Elwood Shannon (1916–2001), an American 1 H = = k log s . (2) electronic engineer and mathematician, known as the n father of information theory . The equation, which was published in A Mathematical Theory of Communication This relation is valid in the case of equal probability in 1948 (Shannon 1948), was derived within informa- (frequency) of the occurrence of individual symbols in one tion theory and the quantity that it expresses was named message . If the symbols appear with dissimilar probabi- entropy . Shannon adopted this denomination from Boltz- lity pi ∈ (p1, p2 … ps), where 0 ≤ pi ≤ 1, then after mak- mann’s thermodynamic entropy . Although there is formal ing modifications and using Stirling’s formula (Shannon congruence of the relations within information and ther- 1948) we get the relation expressing the amount of infor- modynamic entropy, it took many years until their mu- mation for one symbol of a message tual relation was proved . At present, the relation is ap- 1 s plied in many scientific fields; among others, in geogra- H = = −k pi log pi . (3) n ∑i=1 phy and biology for diversity determination . As stated above, the quantity H was denominated en- 2.2 Information and entropy tropy (information entropy) . The relation (3) presents the features of entropy: In order to understand the importance of the relation 1 . entropy only depends on probabilities (not on that is used in connection with landscape diversity, its ori- values of symbols in a message), gin within the information theory needs to be shown . 2 . entropy is invariant to the sequence of symbols, Basic concepts: 3 . for a concrete s, maximum entropy occurs for – alphabet is a set of symbols – the number of symbols s, p1 = p2 = … = ps = 1/s, i e. for steady representation of e .g . for the alphabet {a, b, c, d, e} is s = 5 . symbols, – message is a sequence of symbols, e .g . ‘babaccdbeae’, 4 . minimum entropy occurs for s = 1, then p1 = p = 1 of the length n . In the given example n = 11 . and p log p = 1 ∙ log 1 = 1 ∙ 0 = 0 → H = 0 . The number of possible messages N of the length n over the alphabet of symbols s is calculated as a variation 2.3 Application for landscape diversity with repetition: N = sn . The original equation contained probabilities of phe- In the above case N = 511 = 48,828,125 possible mes- nomena (symbols) that in landscape assessment were sages . substituted with proportional representation of areas We look for a function that can express information
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