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<\/a>\nIntroduction<\/h2>\n
Numerous extraordinary flood events that occurred in Middle Europe during the last twenty years seem to prove the IPCC\u2019s\u00a0climate change predictions of an increase of extreme hydrologic events in terms of frequency and intensity. Exemplarily, the return periods of the Labe (in German \u201eElbe\u201c) floods at gauge Dresden (Saxony) were 100 to 200 years in 2002 and 50 to 100 years in 2013 In many cases, the occurrence of inhomogeneities (structural, biological, soil\u00ad\u2011mechanical or other) in a\u00a0levee section triggers a\u00a0failure process (fig. 1<\/em>). As conventional stability computations that are usually conducted on basis of two\u00ad\u2011dimensional, vertical finite\u00ad\u2011element\u00ad\u2011models do not incorporate the occurrence of such discontinuities, their suitability for reliability assessment of levees is questionable. To bypass some of these facts, the approach presented in herein is based on the collection and statistical evaluation of data from failed and non\u00ad\u2011failed levee sections in order to predict a\u00a0levee\u2019s\u00a0failure probability and to identify the main drivers for the failure. So far, the application of the approach is limited to (quasi-) homogeneous levees that had been in operation for many decades (\u201eold levees\u201c) and do often not meet current design standards.<\/p>\n Whenever a\u00a0levee fails during a\u00a0flood event the question about the reasons for failure arises. In most cases, the failure process itself was neither observed (by eye witnesses) nor measured (e.g. saturation or deformation process). Moreover, the local conditions of the levee at the point of failure are unclear as there usually is a\u00a0larger time gap between the date of survey and the time of failure. Thus, the only way to investigate the possible sources of the failure is to collect all data available for the levee section within a\u00a0post event analysis. Aerial photographs and laserscanning data, taken from a\u00a0satellite, a\u00a0plane\/helicopter or nowadays also from unmanned aerial vehicles (UAV) can be an important source of information, e.g. in order to identify the levee\u2019s\u00a0geometric\u00a0properties, if the levee section was overtopped or if tree growth was present\u00a0at the breach. Other conditions, such as the inner structure of the levee or the subsoil conditions, remain widely unknown, and can only be estimated by\u00a0setting the properties equal to the levee sections adjacent to the breach. There are numerous examples where institutions or researchers stored the levee failure data in appropriate tables or databases [1, 3]. In the Czech Republic, levee failures during the flood 1997 were investigated in the Morava and Odra catchment [4, 5]. In many cases the data was used for basic descriptive statistics only and simple univariate statistical analysis was conducted. To the author\u2019s\u00a0knowledge, Uno\u00a0et\u00a0al. [6] were the first to analyse the failure data in a\u00a0more sophisticated manner by utilizing data of failed and non\u00ad\u2011failed levee sections in Japan for multivariate, statistical analysis by means of the logistic regression method.<\/p>\n The method of logistic regression allows an exploratory data analysis, aiming for the identification of correlations between several parameters within a\u00a0system. Although it has been widely used in the field of medicine, social science or economics it is not very common in engineering (table 1<\/em>).<\/p>\n The application of logistic regression is preferable, whenever dependencies between several factors and observed events are undoubted, but the processes leading to the observation are uncertain. Regression models generally aim for the prediction of the outcome of a\u00a0dependent variable (response variable), Y, given a\u00a0set of influencing factors, Xi<\/sub> (equation 1). Although not compulsory, logistic regression is often used when the response variable has binary realisations (dichotomous variable), e.g. \u201efailure\u201c or \u201enon\u00ad\u2011failure\u201c [7].<\/p>\n A\u00a0specific characteristic of logistic regression models is the ability to combine variables of different scale levels. Metric\u00ad\u2011scaled (cardinal) parameters are commonly used in engineering. Non\u00ad\u2011metric (categorical) parameters can be subdivided in nominal and ordinal variables (table 2<\/em>).<\/p>\n Whenever regression parameters are linked linearly, we refer to generalised linear regression models. The link between the combined regression parameters X and the response variable Y is defined by a\u00a0link function. In the case of logistic regression the logistic function is used (equation 2, fig. 2<\/em>).<\/p>\n In this respect, P(z) represents the probability that the binary response variable Y has the realisation of y = 1. The logistic function is given in equation (2), belongs to the group of sigmoid functions. Since the functional values can be interpreted as probabilities in the range of 0 < P(z) < 1 its usage is popular in probability theory. Therefore neither certain, P(z) = 1, nor impossible, P(z) = 0, events will be predicted. Using this approach a\u00a0logistic regression model computes the occurrence probability of an event P(z)y=1<\/sub>, depending on the value of the parameter z, which is often called \u201elogit\u201c. The logit z\u00a0is an index that combines all regression parameters Xi<\/sub> by means of a\u00a0linear sum. The logits of a\u00a0data sample with n observations are calculated by using equation (3).<\/p>\n with:<\/p>\n Thus, the logit of the n\u00ad\u2011th observation derives from:<\/p>\n
\nwith another major flood event in 2006. The European floods in 2002 with an overall damage of approximately 15 billion EUR (of which 9 billion EUR only in Germany) were probably the main drivers for the preparation and implementation of the EU Floods Directive in November 2007 requiring all member states to develop flood risk management plans for endangered regions until the end of 2015. These plans are incorporating technical as well as non\u00ad\u2011technical measures. Regarding the technical measures, flood levees continue to be the most important flood mitigation structures. In this context, recent flood events\u00a0revealed that long stretches of existing river levees need to be resurveyed, strengthened or reconstructed, since levees broke at many locations. In most cases, the higher discharges and water levels respectively exceeded the design conditions. However, a\u00a0survey of levee failures during the 2002 flood in Saxony [1]\u00a0showed, that levees also broke at sections that were considered safe according to the results of conventional levee stability computations. On the other hand, levees resisted at spots where such computations predicted failure for the given conditions. The reason for the aforementioned phenomena is the uncertainty deriving from [2]:<\/span><\/p>\n\n
<\/a>\nFig. 1. Levee failure at the Mulde River (Saxony) during the flood 2002 (photo: Ott, 2002)
\nObr.\u00a01. Selh\u00e1n\u00ed hr\u00e1ze na \u0159ece Mulda (Sasko) b\u011bhem povodn\u00ed v\u00a0roce 2002 (fotografie: Ott, 2002)<\/h6>\nMethodology<\/h2>\n
Levee failure databases<\/h3>\n
Logistic regression<\/h3>\n
Table 1. Sample applications of logistic regression
\nTabulka 1. Uk\u00e1zka aplikace logistick\u00e9 regrese<\/h5>\n<\/a>\n<\/a>\nTable 2. Scale levels
\nTabulka 2. \u00darovn\u011b m\u011b\u0159en\u00ed<\/h5>\n<\/a>\n<\/a>\n<\/a>\n<\/a>\n<\/a>\n