An Optimized System for the Classification of Meteorological Drought Intensity with Applications in Drought Frequency Analysis
By Vogt, Jürgen V | |
Proquest LLC |
ABSTRACT
The adequacy of meteorological drought intensity threshold levels based on deviations of monthly precipitation totals from normal climatological conditions is reconsidered. The motivation for this study is the observation that reference classification systems are fixed for all climatological regions, and threshold levels have been proposed without regard for the statistical distribution of accumulated precipitation in space and time. This misrepresentation of precipitation variability may lead to erroneous estimates of meteorological drought onset in specific areas where natural breaks in the cumulative distribution of monthly precipitation do not fit the generalized classification systems. In this study, a new optimized classification system based on the nonparametric ''Fisher-Jenks'' algorithm is proposed for the estimation of meteorological drought intensity threshold levels from monthly precipitation totals. The optimized classification system is compared using the tabular accuracy index (TAI) to three fixed classification systems that are proposed in the literature and widely applied in the operational setting. An assessment of drought intensity classifications with optimized and fixed threshold levels shows that 1) six optimized categories most accurately divide precipitation totals into the most appropriate drought intensities, 2) optimized thresholds always give considerably improved drought intensity category allocations over fixed thresholds with the same number of categories, and 3) fixed thresholds underestimate the drought onset. An analysis of monthly and long-term drought frequency for
(
1. Introduction
Drought originates from a temporary water deficit that results in an inability to meet the demands of human activities and the environment (Smakhtin and Schipper 2008). This deficit may be related to a lack of precip- itation, soil moisture, streamflow, or any combination of the three taking place at the same time. Following Dracup et al. (1980) and Wilhite and Glantz (1985), meteorological (lack of precipitation), agricultural (de- cline in soil moisture), hydrological (low streamflow), and socioeconomic droughts are often distinguished. The last form may be considered a consequence of the other physically based types of drought and is associated with a failure of water resources systems to meet the demands for an economic activity (Wilhite and Glantz 1985; Heim 2002; Keyantash and Dracup 2002). Drought types represent different stages of the same process, re- flecting the impacts on different water use sectors as a function of the timing, duration, and amount of a pre- cipitation deficit. Thus, if one is interested in determining the cause of drought events, attention should be focused on meteorological (precipitation) drought (Dracup et al. 1980). Indeed, while the occurrence and progression of drought conditions also depend on other variables such as temperature, wind speed, evapotranspiration, and soil moisture, the precipitation deficit is the main driver of any drought event (Heim 2002; Smakhtin and Schipper 2008; Kumar et al. 2009; Vicente-Serrano et al. 2010; Mishra and Singh 2011). The longer and the more spa- tially extensive the lack of precipitation, the more likely different types of droughts will occur. In this paper we, therefore, concentrate on a methodology to define ap- propriate threshold levels for classifying precipitation deficits into different categories of meteorological drought intensities.
To improve drought mitigation, different indicators are used to trigger a drought response (
The magnitude of negative SPI values corresponds to percentiles p(x) of a probability distribution that are frequently used as threshold levels (triggers) to classify meteorological drought intensity (e.g., McKee et al. 1993;
Although the SPI classification system proposed by McKee et al. (1993) provides a consistent and replica- ble standard for drought intensity classification using threshold levels easily understood by nonexperts, its values have been questioned, for example, by
In the context of the U.S. Drought Monitor (USDM), the SPI is one of six objective indicators (see Table 2 in Svoboda et al. 2002) that are used as a starting point in the elaboration of the weekly Drought Monitor product (Svoboda et al. 2002;
Given the wide use of the SPI and the importance of the threshold levels' selection for correctly characterizing meteorological drought intensities, we propose a new methodological approach to derive such thresholds in an optimized way. The methodology is based on the Fisher- Jenks optimal classification algorithm and the threshold levels are determined as precipitation percentiles on the basis of the empirical cumulative distribution function (ECDF) of historical precipitation totals for each season and geographic location. Our study aims to provide a blueprint for a new way to estimate the threshold levels of meteorological drought intensity classification systems with a different number of categories, even in dry climates or those with a distinct dry season where zero values are common. As a benchmark, we compare the proposed approach with the aforementioned classification systems of drought intensity based on precipitation totals and evaluate its ability for monthly and long-term drought frequency estimation. While the methodology has been tested for different accumulation periods, in this paper we present the results for 1-month accumulations only. The method is the same for other accumulation periods and the results are comparable.
2. Data and methods
This section describes the precipitation dataset used for validating and testing the proposed classification approach and to compare reference classification systems, the com- putation method of drought intensity from reference classification systems, the formulation of the Fisher-Jenks optimal approach for meteorological drought intensity classification, and the standard measure used to assess the accuracy of drought intensity classification systems.
a. Precipitation data: The GPCC Full Data Reanalysis Version 6.0
The analysis of drought intensity classification sys- tems is performed with monthly precipitation totals from the Full Data Reanalysis Monthly Product Version 6.0 of the Global Precipitation Climatology Centre (GPCC). The GPCC was established in 1989 by request of WMO and provides a global gridded analysis of monthly precipitation over land from operational in situ rain gauges based on the Global Telecommunication System and historic precipitation data measured at global stations. The data supplied from 190 worldwide national weather services to the GPCC are regarded as a primary data source, comprising observed monthly totals from 10 700 to more than 47 000 stations since 1901. The monthly gridded datasets are spatially interpolated with a spherical adaptation of the robust Shepard's empirical weighting method (
The data used cover the whole
b. Reference classification systems
The classification systems of meteorological drought intensity proposed by McKee et al. (1993),
Independently of the classification system used, Guttman (1999) and Ntale and Gan (2003) pointed out that different drought intensity categories will be ob- tained for an observed precipitation amount if different parametric probability distributions are fitted to the precipitation time series. In this context, the Weibull plotting position (Weibull 1939) has been shown to be a suitable alternative for estimating unbiased non- exceedance probabilities for any statistical distribution (Maidment 1993; Ntale and Gan 2003; Makkonen 2006). This method avoids completely the parametric distribu- tion hypotheses and therewith the associated problems. Since this plotting-position formula only uses data ranks, it is nonparametric and has attractive properties, the most obvious being its ability to handle disproportionate out- liers (Ntale and Gan 2003; Makkonen 2006). Although the Weibull plotting position has a discrete nature, it is comparable to a parametric distribution fitted to the precipitation data if the time series set is reasonably long, say about 100 yr or more (Guttman 1999; Ntale and Gan 2003;
To classify monthly precipitation observations into ref- erence drought intensity categories through the Weibull plotting position, the long-term records of precipitation are first converted to percentiles by developing ECDFs. In this approach, historical precipitation totals for each geo- graphic location g are stratified by month m,1# m # 12. Then, the data within the 12 monthly sets of N long-term annual records are ranked from the lowest to the highest precipitation total. Estimators of the nonexceedance probabilities p(xg,m,n) are then calculated according to the Weibull plotting-position formula (Weibull 1939):
... (1)
where x 5 monthly precipitation total, g 5 geographic location, m 5 month (1 # m # 12), n 5 year in the time series (1 # n # N), i 5 rank of the order statistic, and N 5 number of years with monthly precipitation in the time series.
Given a monthly precipitation total xg,m,n whose group membership is unknown, it is mapped into ref- erence drought intensity category j if
... (2)
where t j 5 precipitation percentile corresponding to the fixed threshold level for drought intensity category j. The upper bound of a category is established by tj and the lower bound by t j21. Note that t 0 5 0 and tk 5 0.5 (i.e., median of historical precipitation) for all classifi- cation systems presented in Fig. 1, and tj represents the maximum probability of occurrence for drought in- tensity category j.
c. The nonparametric classification system
A shortcoming of the reference classification systems presented in Fig. 1 is that the drought intensity threshold levels for each category are fixed for all geographic lo- cations and months. This standardization has been con- sidered to be an advantage, as the deficit of precipitation can be unambiguously computed and compared over space and time (e.g., Hayes et al. 1999; Lloyd-Hughes and Saunders 2002; Vicente-Serrano et al. 2004). However, none of the fixed classification systems was specifically designed to attend the geographical variability of pre- cipitation and its impacts on local drought conditions. For example, if the probability of zero precipitation exists, that is, p(0) . 0, then the maximum drought intensity will be bounded by p(0). Indeed, if p(0) 5 0.11, then the prob- ability of ''severe'' or more intense drought conditions cannot be depicted by any of the reference classification systems (see Fig. 1), and a zero precipitation value will be categorized as a ''moderate'' drought event. Our solution is to empirically classify precipitation observations into homogeneous groups and identify the particular per- centiles that correspond to the various drought intensity threshold levels t 5 ftj: j 5 1, ..., k categoriesg at each geographic location and time of the year.
Cluster analysis is one of a number of multivariate techniques that can be used to accomplish classification of precipitation regimes for different purposes (e.g., Guttman 1993; Gong and Richman 1995). Here, drought intensity categories (or clusters) can be defined as rela- tive constellations of contiguous low-precipitation ob- servations that are more similar to each other than observations in different clusters (Wilks 2005). Clus- tering involves the grouping of similar observations that exhibit two main properties: external isolation and in- ternal cohesion (Cormack 1971). External isolation re- quires that observations within one cluster be widely separated from observations in another cluster. Internal cohesion requires that observations within the same cluster be similar to one another. Cluster analysis can be performed hierarchically or nonhierarchically. For the classification of precipitation regimes for different pur- poses, it has been shown that nonhierarchical methods outperform hierarchical methods (Gong and Richman 1995). In this context, the nonhierarchical Fisher-Jenks classification algorithm (Fisher 1958; Jenks and Caspall 1971) has a mathematical foundation that guarantees an optimal solution to external isolation and internal co- hesion (
The main advantage of the Fisher-Jenks optimal al- gorithm over the three fixed classification systems is the ability to accommodate the temporal and geographical variations of observed precipitation amounts during the computation of drought intensity threshold levels. Be- cause the threshold levels corresponding to different drought intensity categories are not fixed a priori, then for every time period and geographic location there is a set of threshold levels that optimize the partition of the precipitation observations into some specified k number of drought intensity categories. In applying the Fisher- Jenks optimal algorithm to meteorological drought in- tensity classification, we aim to estimate the drought intensity threshold levels ^t 5 ftj : j 5 1, ..., k categoriesg that minimize the sum of absolute precipitation de- viations about the median of drought intensity cate- gories for each geographic location g and month m, 1 # m # 12, as
... (3)
where Nj 5 total number of years with precipitation observations below the median classified in category j, xn,j 5 precipitation observation in category j and year n, and x~j 5 median of precipitation observations in cate- gory j. Note that, similar to the three reference classifi- cation systems, the median of historical precipitation is the best guess of unknown normal climatological pre- cipitation conditions for each location and month. In agreement, the classification of drought intensity is based only on precipitation observations below the median.
d. Accuracy assessment of meteorological drought intensity classification
Since each classification system communicates a dif- ferent concept of drought intensity and respective probability of occurrence, it is apparent that these concepts cannot be equally valid. The accuracy of drought intensity classification systems, that is, their ability to correctly measure the intensity of a drought event is, for example, performed on the basis of qual- itative biophysical impacts on vegetation vigor or quantitative impacts on crop yield, energy production, public water supply, and other economic activities (e.g., Gouveia et al. 2009; Caccamo et al. 2011; Logar and
... (4)
where k 5 number of drought intensity categories, N 5 number of years with precipitation observations below the median, Nj 5 number of years with precipitation observations below the median in class j, xn 5 pre- cipitation observation in year n, xn,j 5 precipitation ob- servation in category j and year n, xj 5 mean of precipitation observations in category j,andx 5 mean of all precipitation observations.
TAI values range from 0 to 1, with 0 representing the lowest accuracy and 1 representing the highest accuracy. A TAI value of 0 will result only when no grouping (i.e., one class) has been applied to the data observations; a TAI value of 1 occurs only when the values of data ob- servations in each category are identical (i.e., the variance is zero within each class). For example, a six-class system can only achieve a TAI value of 1 if there are exactly six distinct values in a numerical sample of observations.
The GPCC grid cells that for a given month have fewer than five nonzero historical precipitation records below the median are not used for the accuracy assess- ment of drought intensity classification systems on that month, the reason being that the number of distinct precipitation records is insufficient to estimate the drought intensity threshold levels tj for the six-class classification systems and to compute the respective TAI values. The removal of these grid cells ensures that the accuracy assessment of all classification systems is based on the same sample set and comparisons between TAI values are statistically sound for each month.
3. Results and discussion
In this section, we present a comparison between the reference classification systems of meteorological drought intensities proposed by McKee et al. (1993),
a. The optimization of drought intensity categories with the Fisher-Jenks algorithm
In the case of clustering the GPCC precipitation data into two drought categories, the single optimized threshold ^t 2 divides between the sample observations that are closer to the observed minimum historical pre- cipitation value and those that are closer to the estimated median precipitation value for that geographic location. Since the minimum historical precipitation value repre- sents the driest conditions observed for that location and the estimated median represents the respective expected normal climatological conditions, then the two categories that are identified in the data can be conceptually nomi- nated as drought and nondrought. The estimated threshold level ^t 2 conceivably represents the precip- itation percentile that optimally divides the precipitation observations into those two conditions and marks the effective onset of a meteorological drought at this level of generalization. If the precipitation data below the median are uniformly distributed, then the single optimized threshold ^t2 allocates the same number of observations in the drought and nondrought categories. However, it is often the case that no clear uniform structure is observed in the data, and the positioning of the optimized threshold level t^2 depends on the shape of the left tail of the ECDF fitted to the whole precipitation time series.
An analysis of the GPCC data reveals that three typical situations represent the optimized partition of the ordered precipitation time series below the median into two generalized drought categories (Fig. 2): in Fig 2a, the precipitation distribution is free of ''isolated'' groups of observations near its minimum or median values, and the two-class threshold is a robust and resistant measure of central partition that optimally discriminates between drought and nondrought events for the location in a given month; in Fig 2b, the pre- cipitation distribution has a disrupted single set of low-precipitation records isolated in the left tail, and the two-class threshold discriminates these ''extreme'' drought intensity events for the location; in Fig 2c, the precipitation distribution has a disrupted single set of isolated precipitation records near the median, and the two-class threshold discriminates these ''moderate'' drought intensity events for the location.
To avoid the semantic complexity in the drought in- tensity classification of samples represented by Figs. 2b,c, the number of categories can be increased. When a clas- sification system with four categories (i.e., ''extreme,'' ''severe,'' ''moderate,'' and ''no drought'') is used to group the precipitation data, the positioning of the two additional threshold levels depends on the percentile rank of the single threshold level ^t2, as shown in Fig. 2. Statistical analysis of the cases introduced in Fig. 2 reveals that 97.8 6 1.2% ( p , 0.05) of the geographical points in the study area match the category in Fig. 2a each month. Consequently, the ''severe'' drought threshold level in the four-class drought intensity system can be used as a generalized onset boundary that divides between effec- tive drought and nondrought states for the region, while the two extra thresholds are further used to identify particular regimes within those two states, namely near- normal and extremely low-precipitation conditions. Moreover, the optimized four-class system reduces the bias in the positioning of the generalized two-class threshold level according to the arrangement of ex- treme observations and eases the operational monitoring of drought by better depicting the evolution of its con- ditions through the inclusion of more drought intensity categories. The ability for improved monitoring of me- teorological drought conditions can be further enhanced with the use of a six-class drought system (i.e., ''excep- tional,'' ''extreme,'' ''severe,'' ''moderate,'' ''abnormally dry,'' and ''no drought''). In this case, the ''extreme'' and ''moderate'' drought intensity threshold levels move to- ward the ''severe'' drought threshold to find the optimal positioning in the new system. The two new threshold levels are added with the designations of ''abnormally dry'' and ''exceptional'' drought to identify limit pre- cipitation conditions for the region.
Figure 3 shows the monthly frequency distributions of TAI in the study area for the classification of precipitation data in two, four, and six drought clusters. It can be seen from the monthly frequency distributions that the rate of improvement in the classification accuracy has an expo- nential behavior and decreases beyond four drought categories (i.e., ''extreme,'' ''severe,'' ''moderate,'' and ''no drought''). Indeed, there is an average monthly gain in the classification accuracy of 60% from two to four clusters, but only of 10% from four to six clusters. Given that 98.7 6 0.4% (p , 0.05) of the geographic points in the study area attained more than 80% TAI per month with the six-class drought system, we are confident that the use of six optimized groups to classify precipitation observations into drought intensities is sufficient for the study area and precipitation data used.
b. Comparison of reference classification systems
Figure 4a summarizes the differences between the mean monthly overall accuracy (as measured by the TAI) attained with the reference classification systems for the whole study area. The results give an estimate of the ac- curacy gain from the first to the second classification system. The comparison between the reference systems with four drought intensity categories (Fig. 1), that is, McKee et al. (1993) and
As expected, the experiments also show that the six- class drought intensity classification system proposed by Svoboda et al. (2002) (Fig. 1) performs significantly better (p , 0.01; Fig. 4a) than the four-class drought intensity classification system introduced by
c. Comparison between reference and optimized classification systems
Figure 4b shows the differences between the mean monthly overall accuracy of drought intensity classifi- cations performed with the Fisher-Jenks classification algorithm and the reference classification systems pro- posed by
Overall, the results presented in Figs. 4 and 5 show that there is a specific set of threshold levels that optimize the grouping of precipitation data within a particular number of drought intensity categories for each month and geographic location. The optimized grouping can be obtained by tuning the positioning of the class bound- aries with respect to a numerical minimization of clas- sification error, as provided in Eq. (2). Indeed, our tests reveal that neither the classification system proposed by
d. Evaluation of optimized threshold levels
Let us finally compare the precipitation percentiles corresponding to the optimized drought intensity thresh- old levels t 5 ftj: j 5 1, ... , k categoriesg with the equivalent boundaries in the reference classification sys- tems (as presented in Fig. 1). Figure 6 presents the em- pirical PDFs of the drought intensity threshold levels t optimized with four and six clusters in the study area for representative months of different seasons. Although the PDFs converge asymptotically to the empirical modes presented in Table 1, it is also true that the variance of the optimized threshold levels is effectively too large to use fixed thresholds. Naturally, this implies that the threshold levels for splitting drought intensity categories are better estimated on the basis of individual charac- teristics of precipitation distribution along the number line than on fixed boundaries for each geographic loca- tion. It is confirmed from the results detailed in Fig. 6 that a classification system with a priori defined and fixed threshold levels cannot split precipitation observations into drought intensity categories with the same accuracy in space and time.
The main concern with the reference classification systems is that the threshold levels underestimate on average the onset of mid-intensity drought events. For example, the probability of occurrence of ''severe'' drought is defined in
Nevertheless, it is important to highlight that the modes of the empirical PDFs for the precipitation percentiles corresponding to the optimized threshold levels of ''ex- ceptional'' and ''extreme'' drought categories (Fig. 6 and Table 1) are comparable to, respectively, the Svoboda et al. (2002) and
4. Drought frequency analysis
The potential applications of the proposed classifica- tion approach are numerous and varied. In this paper, we present a case study on the optimized estimation of meteorological drought intensity threshold levels for adaptive drought frequency computation. When the concept of frequency is applied to drought-related var- iables, it indicates the number of times that a drought intensity class is triggered in a given geographic location over a period of time (
... (5)
where j 5 1, ... , k drought intensity categories, g 5 geographiclocation,p015Pftn11#jjtn.jg"j51,...,k, n 2 N, p11 5 Pftn11 # j j tn # jg"j 5 1, ..., k, n 2 N, and N 5 total number of months in the precipita- tion time series. The maximum likelihood estimators (MLEs) of p01 and p11 are simply computed from the number of monthly transitions between the respective drought categories during the analysis period as, for example,
... (6)
where n01 is the number of monthly transitions between tn . j and tn11 # j,andn0 is the total number of monthly transitions between t n . j and another data point in the time series.
Figure 7 shows the geographic distribution of the precipitation percentiles corresponding to the optimized threshold levels t j (i.e., monthly frequency) of ''moder- ate,'' ''severe,'' and ''extreme'' drought from a four-class system. The months depicted are representative of dif- ferent seasons and the maps displayed for the study area are informative in several ways. It can be seen that there is a spatial variation of the frequency for each season and drought intensity class, indicating that there is a unique percentile threshold that depends on the specific clima- tological characteristics of the geographic location in that month. Because of their standardization and the fixed thresholds, the reference classification systems cannot depict this geographic and temporal variability in the meteorological drought frequency, because all locations will have the same frequency at a given drought intensity category for all months.
The results of long-term drought frequency (p) esti- mated with the two-class threshold level and the ''se- vere'' drought intensity threshold levels of both four- and six-class classification systems are compared in Fig. 8.We have obtained corresponding spatial patterns in the three maps, demonstrating that there is a match between the generalized two-class threshold level and the inter- mediate drought intensity threshold levels of ''severe'' drought in the four- and six-class systems. This outcome supports the hypothesis that this mid-drought intensity threshold is a transverse and robust benchmark that can be used as a reference for detecting the effective drought onset in any of the optimized classification systems. In- deed, Figs. 8a-c consistently show a match between the geographic pattern of ''severe'' meteorological drought frequency in
The additional threshold levels located above or be- low the precipitation percentile that corresponds to the mid-drought intensity boundary in any of the optimized classification systems show the transitions between different phases of the drought process that can be of interest during the preparedness phase of risk manage- ment. For example, Fig. 9 shows the spatial distribution of the long-term drought (p) threshold levels optimized for the six-class drought intensity system. The maps in- dicate that the whole eastern and northeastern region of
5. Conclusions
As pointed out throughout this paper, there is a need to improve the positioning of meteorological drought intensity threshold levels based on precipitation per- centiles, since a misrepresentation leads to a bias in the estimation of drought intensity categories based on precipitation amounts. Thus, the main objective of this work was to formulate, validate, and test an approach for the optimized classification of meteorological drought intensity threshold levels from precipitation data that can better represent the frequency of occurrence and the intensity of this complex hazard in space and time. As a benchmark, we compare it with some of the reference classification systems commonly presented in drought literature and used operationally.
The use of the Fisher-Jenks classification algorithm was found to accurately estimate the threshold levels that better depict the spatial and temporal variably of meteorological drought intensity in regions with differ- ent climate conditions, including those with a distinct dry season where zero values are common. Using an independent approach for determining meteorological drought intensity ensures that variations in climate are correctly accounted for so that its frequency is accu- rately computed. The goal of the method is to develop an ordinal classification of precipitation, based entirely on the positioning of observations along the number line, in which total within-class variance is minimized. To evaluate the goodness-of-fit of the classification ap- proach, we used a statistical measure, namely the tabular accuracy index. Classification results over
The comparison between the ''optimal'' system and three reference drought systems, namely those proposed by McKee et al. (1993),
Considering the optimized classification of drought intensity threshold levels, some geographical variations in the monthly frequency of meteorological drought events are obtained throughout the study area in
So far, experience shows that the position where the meteorological drought thresholds lie is fixed and/or entirely the analyst's decision. Our method guarantees an adapted threshold level classification that follows the natural breaks in the precipitation data distribution and avoids a bias in the estimation of meteorological drought onset. The proposed classification approach can then be used as a basis for the accurate estimation of the return period of meteorological droughts with different lengths, as well as of their expected inter- arrival time. Thus, future work will focus on the as- sessment of historic changes in meteorological drought over the last century, as well as on the study of its fre- quency and severity in the future as a result of climate change.
Acknowledgments. This research received support from the EUROCLIMA regional cooperation program between the
REFERENCES
Allen, S. K., and Coauthors, 2012: Summary for policymakers. Managing the Risks of Extreme Events and Disasters to Ad- vance Climate Change Adaptation,
Caccamo, G.,
Carrão,H.,G.Sepulcre,S.Horion,andP.Barbosa,2013:Amulti- temporal and non-parametric approach for assessing the impacts of drought on vegetation greenness: A case study for
Cormack, R. M., 1971: A review of classification.
Cromley, R., and
Dracup, J. A.,
Fisher, W. D., 1958: On grouping for maximum homogeneity.
Gong, X., and
Gouveia, C.,
Guttman, N. B., 1993: The use of L-moments in the determination of regional precipitation climates. J. Climate, 6, 2309-2325, doi:10.1175/1520-0442(1993)006,2309:TUOLMI.2.0.CO;2.
-, 1999: Accepting the standardized precipitation index: A calculation algorithm.
Hayes, M. J.,
Heim, R. R., 2002: A review of twentieth-century drought indices used in
Hofer,B.,H.Carrão, and
Jenks, G. F., and
Keyantash, J., and
Kottek,M.,J.Grieser,C.Beck,B.Rudolf,andF.Rubel,2006: World map of the Köppen-Geiger climate classifica- tion updated. Meteor. Z., 15, 259-263, doi:10.1127/ 0941-2948/2006/0130.
Kumar, M. N.,
Llano, M. P., W. Vargas, and G. Naumann, 2012: Climate vari- ability in areas of the world with high production of soya beans and corn: Its relationship to crop yields. Meteor. Appl., 19, 385-396, doi:10.1002/met.270.
Lloyd-Hughes, B., and
Logar, I., and
Magrin, G.,
Maidment, D. R., 1993: Handbook of Hydrology.
Makkonen, L., 2006: Plotting positions in extreme value anal- ysis.
McKee, T. B., N. J. Doeskin, and
Meyer, V., and Coauthors, 2013: Review article: Assessing the costs of natural hazards-State of the art and knowledge gaps. Nat. Hazards Earth Syst. Sci., 13, 1351-1373, doi:10.5194/ nhess-13-1351-2013.
Mishra, A. K., and
-, -, and
Morid, S., V. Smakhtin, and
Naumann, G.,
Ntale, H. K., and
Panu, U., and T. Sharma, 2002: Challenges in drought research: Some perspectives and future directions. Hydrol. Sci. J., 47, 19-30, doi:10.1080/02626660209493019.
Paulo, A. A.,
Quiring, S. M., 2009: Developing objective operational definitions for monitoring drought.
Santos, J. F.,
Sepulcre-Canto, G., S. Horion, A. Singleton, H. Carrão, and
Smakhtin, V. U., and
-, and
Svoboda, M., and Coauthors, 2002: The Drought Monitor. Bull. Amer. Meteor. Soc., 83, 1181-1190.
-,
Traun, C., and
Trenberth, K., and
Vargas, W. M., G. Naumann, and
Vicente-Serrano, S. M., 2006: Spatial and temporal analysis of droughts in the
-,J.C.González-
-, S. Beguería, and J. I. López-Moreno, 2010: A multiscalar drought index sensitive to global warming: The standardized precipitation evapotranspiration index. J. Climate, 23, 1696- 1718, doi:10.1175/2009JCLI2909.1.
Weibull, W., 1939: The Phenomenon of Rupture in Solids. Hand- lingar, Vol. 153, Ingeniörs Vetenskaps Akademien, 45 pp.
Wilhite, D. A., and
Wilks, D. S., 2005: Statistical Methods in the Atmospheric Sciences: An Introduction. 2nd ed.
Wu, H.,
Yadav, S.,
Zhao, M., and
HUGO CARRÃO,ANDREW SINGLETON,GUSTAVO NAUMANN,PAULO BARBOSA, AND JÜRGEN V. VOGT
Climate Risk Management Unit,
(Manuscript received
Corresponding author address:
E-mail: [email protected]
Copyright: | (c) 2014 American Meteorological Society |
Wordcount: | 10313 |
Verification of European Subseasonal Wind Speed Forecasts
Advisor News
Annuity News
Health/Employee Benefits News
Life Insurance News