A drought frequency formula

Abstract

The important elements of a drought phenomenon are the longest duration and the largest severity for a desired return period. These elements form a basis for designing water storage systems to cope with droughts. At times, a third element, drought intensity, is also used and is defined as the ratio of severity to duration. The commonly available statistics for the causative drought variables such as annual rainfall or runoff sequences are the mean, the coefficient of variation and the lag one serial correlation coefficient, and occasionally some indication of the probability distribution function (pdf) of the sequences. The extremal values of the duration and severity are modelled in the present paper using information on the aforesaid parameters at the truncation level equal to the mean of the drought sequence, which is generally taken as the truncation level in the analysis of droughts. The drought severity has been modelled as the product of the duration and intensity with the assumption of independence between them. An estimate of drought intensity has been realized from the concept of the truncated normal distribution of the standardized form of the drought sequences in the normalized domain. A formula in terms of the extremal severity and the T-year return period has been suggested similar to the flood frequency formulae, commonly cited in hydrological texts.     

Assessment of hydrological droughts for the Yellow River,China, using copulas

Droughts are one of the normal and recurrent climatic phenomena on Earth. However, recurring prolonged droughts have caused far‐reaching and diverse impacts because of water deficits. This study aims to investigate the hydrological droughts of the Yellow River in northern China. Since drought duration and drought severity exhibit significant correlation, a bivariate distribution is used to model the drought duration and severity jointly. However, drought duration and drought severity are often modelled by different distributions; the commonly used bivariate distributions cannot be applied. In this study, a copula is employed to construct the bivariate drought distribution. The copula is a function that links the univariate marginal distributions to form the bivariate distribution. The bivariate return periods are also established to explore the drought characteristics of the historically noticeable droughts. The results show that the return period of the drought that occurred in late 1920s to early 1930s is 105 years. The significant 1997 dry‐up phenomenon that occurred in the downstream Yellow River (resulting from the 1997–1998 drought) only has a return period of 4·4 years and is probably induced by two successive droughts and deteriorated by other factors, such as human activities. Copyright © 2007 John Wiley & Sons, Ltd.     

A multimodel assessment of the climate change effect on the drought severity–duration–frequency relationship

In this study, the patterns of past and future drought occurrences in the Seoul region were analysed using observed historical data from the Seoul weather station located in the Korean Peninsula and four different types of general circulation models (GCMs), namely, GFDL:CM2_1, CONS:ECHO‐G, MRI:CGCM2_3_2 and UKMO:HADGEM1. To analyse statistical properties such as drought frequency duration and return period, the Standardized Precipitation Index was used to derive the severity–duration–frequency (SDF) curve from the drought frequency analysis. In addition, a drought spell analysis was conducted to estimate the frequency and change of drought duration for each drought classification. The results of the analysis suggested a decrease in the frequency of mild droughts and an increase in the frequency of severe and extreme droughts in the future. Furthermore, the average duration of droughts is expected to increase. A comparison of the SDF relationship derived from the observed data with that derived via the GCMs indicated that the drought severity for each return period was reduced as drought duration increased and that the drought severity derived from the GCMs was severer than the severity obtained using the observed data for the same duration and return period. Furthermore, among the four types of GCMs used in this study, the MRI model predicted the most severe future drought for the Seoul region, and the SDF curve derived using the MRI model also resulted in the highest degree of drought severity compared with the other GCMs. Copyright © 2012 John Wiley & Sons, Ltd.     

Space–time modelling of catchment scale drought characteristics

Drought may affect all components of the water cycle and covers commonly a large part of the catchment area. This paper examines drought propagation at the catchment scale using spatially aggregated drought characteristics and illustrates the importance of catchment processes in modifying the drought signal in both time and space. Analysis is conducted using monthly time series covering the period 1961–1997 for the Pang catchment, UK. The time series include observed rainfall and groundwater recharge, head and discharge simulated by physically-based soil water and groundwater models. Drought events derived separately for each unit area and variable are combined to yield catchment scale drought characteristics. The study reveals relatively large differences in the spatial and temporal characteristics of drought for the different variables. Meteorological droughts cover frequently the whole catchment; and they are more numerous and last for a short time (1–2 months). In comparison, droughts in recharge and hydraulic head cover typically a smaller area and last longer (4–5 months). Hydraulic head and groundwater discharge exhibit similar drought characteristics, which can be expected in a groundwater fed catchment. Deficit volume is considered a robust measure of the severity of a drought event over the catchment area for all variables; whereas, duration is less sensitive, particular for rainfall. Spatial variability in drought characteristics for groundwater recharge, head and discharge are primarily controlled by catchment properties. It is recommended not to use drought area separately as a measure of drought severity at the catchment scale, rather it should be used in combination with other drought characteristics like duration and deficit volume.     

Stochastic simulation of droughts. Part 2: regional droughts

In Part 1 we demonstrated the applicability of stochastic models to predicting the characteristics of point drought events within any planning period by means of a case study (Mohan S, Sahoo PK (2007) Hydrological Processes 21 : this issue). In addition, studies on regional droughts are important in the context of regional level planning and evolving management strategies. The small number of drought events from a particular streamflow or rainfall series, when subjected to statistical analysis in order to predict future occurrences, produces results that are not very reliable. To overcome this difficulty, we propose using a long sequence of synthetically generated annual rainfall series at various rain‐gauge stations of a region, and multiyear regional droughts were derived from both historic and generated series. The key parameters for a successful regional multiyear drought study are the critical area ratio and the critical level, and the area affected by the drought can be ascertained using these parameters. The important regional drought parameters were determined and their suitable probability distributions were arrived at by studying a total of nine possible probability models; these models can be used in predicting the longest regional drought duration and the greatest regional drought severity with a given return period. The effect of change of critical parameters on the regional drought parameters is also studied and reported. Copyright © 2007 John Wiley & Sons, Ltd.     

Drought analysis of monthly hydrological sequences: a case study of Canadian rivers

Abstract

Two parameters of importance in hydrological droughts viz. the longest duration, L_T and the largest severity, S_T (in standardized form) over a desired return period, T years, have been analysed for monthly flow sequences of Canadian rivers. An important point in the analysis is that monthly sequences are non-stationary (periodic-stochastic) as against annual flows, which fulfil the conditions of stochastic stationarity. The parameters mean, μ, standard deviation, σ (or coefficient of variation), lag1 serial correlation, ρ, and skewness, γ (which is helpful in identifying the probability distribution function) of annual flow sequences, when used in the analytical relationships, are able to predict expected values of the longest duration, E(L_T ) in years and the largest standardized severity, E(S_T ). For monthly flow sequences, there are 12 sets of these parameters and thus the issue is how to involve these parameters to derive the estimates of E(L_T ) and E(S_T ). Moreover, the truncation level (i.e. the monthly mean value) varies from month to month. The analysis in this paper demonstrates that the drought analysis on an annual basis can be extended to monthly droughts simply by standardizing the flows for each month. Thus, the variable truncation levels corresponding to the mean monthly flows were transformed into one unified truncation level equal to zero. The runs of deficits in the standardized sequences are treated as drought episodes and thus the theory of runs forms an essential tool for analysis. Estimates of the above parameters (denoted as μ_av, σ_av, ρ_av, and γ_av) for use in the analytical relationships were obtained by averaging 12 monthly values for each parameter. The product- and L-moment ratio analyses indicated that the monthly flows in the Canadian rivers fit the gamma probability distribution reasonably well, which resulted in the satisfactory prediction of E(L_T ). However, the prediction of E(S_T ) tended to be more satisfactory with the assumption of a Markovian normal model and the relationship E(S_T ) ≈ E(L_T ) was observed to perform better.     

Study of indices for drought characterization in KBK districts in Orissa (India)

Drought is a temporary, random and regional climatic phenomenon, originating due to lack of precipitation leading to water deficit and causing economic loss. Success in drought alleviation depends on how well droughts are defined and their severity quantified. A quantitative definition identifies the beginning, end, spatial extent and the severity of drought. Among the available indices, no single index is capable of fully describing all the physical characteristics of drought. Therefore, in most cases it is useful and necessary to consider several indices, examine their sensitivity and accuracy, and investigate for correlation among them. In this study, the geographical information system‐based Spatial and Time Series Information Modeling (SPATSIM) and Daily Water Resources Assessment Modeling (DWRAM) software were used for drought analysis on monthly and daily bases respectively and its spatial distribution in both dry and wet years. SPATSIM utilizes standardized precipitation index (SPI), effective drought index (EDI), deciles index and departure from long‐term mean and median; and DWRAM employs only EDI. The analysis of data from the Kalahandi and Nuapada districts of Orissa (India) revealed that (a) droughts in this region occurred with a frequency of once in every 3 to 4 years, (b) droughts occurred in the year when the ratio of annual rainfall to potential evapotranspiration (Pae/PET) was less than 0·6, (c) EDI better represented the droughts in the area than any other index; (d) all SPI, EDI and annual deviation from the mean showed a similar trend of drought severity. The comparison of all indices and results of analysis led to several useful and pragmatic inferences in understanding the drought attributes of the study area. Copyright © 2007 John Wiley & Sons, Ltd.     

Spatial mapping of drought in Zambia using regional frequency analysis

Regional frequency analysis based on L-moments was applied to assess the spatial extent of meteorological droughts in tandem with their return periods in Zambia. Weather station monthly rainfall data were screened to form homogeneous sub-regions-, validated by a homogeneity criterion and fitted by a generalized extreme value distribution using goodness-of-fit test statistics. Predictor equations at regional scale for L-moment ratios and mean annual precipitation were developed to generate spatial maps of meteorological drought recurrences. The 80% of normal rainfall level and two thresholds of 60% and 70% were synonymous with moderate and severe droughts, respectively. Droughts were more severe in the south than in the north of Zambia. The return periods for severe and moderate droughts showed an overlapping pattern in their occurrence at many locations, indicating that in certain years droughts can affect the entire country. The extreme south of Zambia is the most prone to drought.     

Developing a large‐scale water‐balance approach to seasonal forecasting: application to the 2012 drought in Britain

Seasonal hydrological forecasts, or outlooks, can potentially provide water managers with estimates of river flows and water resources for a lead time of several months ahead. An experimental modelling tool for national hydrological outlooks has been developed which combines a hydrological model estimate of sub‐surface water storage across Britain with a range of seasonal rainfall forecasts to provide estimates of area‐wide hydrological conditions up to a few months ahead. The link is made between a deficit in sub‐surface water storage and a requirement for additional rainfall over subsequent months to enable sub‐surface water storage and river flow to return to mean monthly values. The new scheme is assessed over a recent period which includes the termination of the drought that affected much of Britain in the first few months of 2012. An illustration is provided of its use to obtain return‐period estimates of the ‘rainfall required’ to ease drought conditions; these are well in excess of 200 years for several regions of the country, for termination within a month of 1 April 2012, and still exceed 40 years for termination within three months. National maps of sub‐surface water storage anomaly show for the first time the current spatial variability of drought severity. They can also be used to provide an indication of how a drought situation might develop in the next few months given a range of possible future rainfall scenarios. © 2013 CEH/Crown and John Wiley & Sons, Ltd.     

Evaluation, modification, and application of the Effective Drought Index to 200-Year drought climatology of Seoul, Korea

In this paper, progressive methods for assessing drought severity from diverse points of view were conceived. To select a fundamental drought index, the performances of the Effective Drought Index (EDI) and 1-, 3-, 6-, 9-, 12-, and 24-month Standardized Precipitation Indices (SPIs) were compared for drought monitoring data accumulated over 200-year period from 1807 to 2006 for Seoul, Korea. The results confirmed that the EDI was more efficient than the SPIs in assessing both short and long-term droughts.We then proposed the following methods for modifying and supplementing the EDI: (1) CEDI, a corrected EDI that considers the rapid runoff of water resources after heavy rainfall; (2) AEDI, an accumulated EDI that considers the drought severity and duration of individual drought events; and (3) YAEDI, a year-accumulated negative EDI representing annual drought severity. In addition to these indices, to more accurately measure and diagnose droughts, we proposed the utilization of (4) the Available Water Resources Index (AWRI), an existing index that expresses the actual amount of available water.Using the improved methods above, we assessed and summarized important droughts that have occurred in Seoul over the 200 years from 1807 to 2006.     

Prediction of the impact of climate change on drought: an evaluation of six UK catchments using two stochastic approaches

Under future climate scenarios, possible changes of drought patterns pose new challenges for water resources management. For quantifying and qualifying drought characteristics in the UK, the drought severity indices of six catchments are investigated and modelled by two stochastic methods: autoregressive integrated moving average (ARIMA) models and the generalized linear model (GLM) approach. From the ARIMA models, autocorrelation structures are first identified for the drought index series, and the unexplained variance of the series is used to establish empirical relationships between drought and climate variables. Based on the ARIMA results, mean sea level pressure and possibly the North Atlantic Oscillation index are found to be significant climate variables for seasonal drought forecasting. Using the GLM approach, occurrences and amounts of rainfall are simulated with conditioning on climate variables. From the GLM‐simulated rainfall for the 1980s and 2080s, the probabilistic characteristics of the drought severity are derived and assessed. Results indicate that the drought pattern in the 2080s is less certain than for the 1961–1990 period, based on the Shannon entropy, but that droughts are expected to be more clustered and intermittent. The 10th and 50th quantiles of drought are likely higher in the 2080s scenarios, but there is no evidence showing the changes in the 90th quantile extreme droughts. Copyright © 2012 John Wiley & Sons, Ltd.     

水文干旱多变量联合设计及水库影响评估

基于东江流域博罗站月径流数据,采用游程理论提取水文干旱事件.选用Meta-Gaussian Copula函数,统计模拟水文干旱指标的多变量联合分布.采用Kendall联合重现期和最大可能权函数,设计给定联合超越重现期的水文干旱指标组合值,并定量评估水库径流调节作用对水文干旱多变量联合特征的影响.结果表明:东江流域水文干旱历时、强度和峰值的统计优选分布均为韦布尔分布.干旱指标之间具有较高的正相关性,Meta-Gaussian Copula能够很好地模拟水文干旱指标两变量和三变量联合分布.基于任意两个变量联合设计和三变量联合设计,干旱指标设计组合值位于同频位置附近,且同一个干旱指标设计值在不同变量组合之间差别较小.水库径流调节作用对于缓解东江流域水文干旱效果明显,同一组干旱指标的多变量联合超越重现期在水库影响下明显变大.联合超越重现期越小,水库对联合设计值的影响程度越大.根据目前水库运行模式,若要满足河道内最小管理流量目标,联合超越重现期10 a一遇的干旱历时、强度和峰值依然达到了约3.89~4.04月、7.20~7.97亿m3和2.99~3.12亿m3.     

A bivariate pareto model for drought

Univariate Pareto distributions have been so widely used in hydrology. It seems however that bivariate or multivariate Pareto distributions have not yet found applications in hydrology, especially with respect to drought. In this note, a drought application is described by assuming a bivariate Pareto model for the joint distribution drought durations and drought severity in the State of Nebraska. Based on this model, exact distributions are derived for the inter arrival time, magnitude and the proportion of droughts. Estimates of 2, 5, 10, 20, 50 and 100 year return periods are derived for the three variables, drought duration, drought severity and the pairwise combinations: (drought duration, drought severity), (inter arrival time of drought, proportion of drought) and (drought duration, drought magnitude). These return period estimates could have an important role in hydrology, for example, with respect to measures of vegetation water stress for plants in water-controlled ecosystems.     

Propagation of drought through groundwater—a new approach using linear reservoir theory

The effect of drought on groundwater heads and discharge is often complex and poorly understood. Therefore the propagation of a drought from groundwater recharge to discharge and the influence of aquifer characteristics on the propagation was analysed by tracking a drought in recharge through a linear reservoir. The recharge was defined as a sinusoid function with a period of 1 year. The decrease in recharge owing to drought was simulated by multiplying the recharge during 1 year with a drought fraction between 0 and 1, which represents a decrease in the recharge of 100 to 0%, respectively. The droughts were identified using the threshold level approach, with a threshold that is constant in time. For this case analytical formulations were derived, which express the drought duration and deficit in the groundwater discharge in terms of the decrease in recharge, the reservoir coefficient that characterizes aquifer properties and the height of the threshold level. The results showed that the delay in the groundwater system caused a shift of the main part of the decrease in recharge from the high‐flow to the low‐flow period. This resulted in an increase in drought deficit for discharge compared with the drought deficit for recharge. Also the development of multiyear droughts caused an increase in drought deficit. The attenuation in the groundwater system caused a decrease in drought deficit. In most cases the net effect of these processes was an increase of drought deficit as a result of the propagation through groundwater. Only for small droughts the deficit decreased from recharge to discharge. The amount of increase or decrease depends on the reservoir coefficient and the severity of the drought. Under most conditions a maximum in the drought deficit occurred for a reservoir coefficient of around 200 days. Copyright © 2003 John Wiley & Sons, Ltd.     

Stochastic simulation of droughts. Part 1: point droughts

The number of drought events derived from the historic streamflow or rainfall series will be limited and produce results that are not very reliable. This study proposes a drought simulation methodology that uses a long sequence of synthetically generated monthly streamflow/rainfall series, from which it is possible to drive a large sample of drought events and the prediction of drought characteristics will be reliable. The modified Herbst method has been used to identify droughts in the generated streamflow and rainfall series. The drought simulation procedure is illustrated with a case study of the Bhadra reservoir catchment in Karnataka State, India. Monthly droughts were derived from both historic and generated monthly streamflow and rainfall series. The important drought characteristics were determined and the suitable probability distribution for each parameter was arrived at after studying seven different probability models. The use of the probability curves thus derived has been illustrated with examples (referred to in Part 1 as ‘point droughts’). Similarly, the development and application of stochastic models for the prediction of regional drought parameters have been illustrated with examples in the accompanying paper (Part 2: regional droughts). Copyright © 2007 John Wiley & Sons, Ltd.     

Drought hotspot analysis and risk assessment using probabilistic drought monitoring and severity–duration–frequency analysis

Drought hotspot identification requires continuous drought monitoring and spatial risk assessment. The present study analysed drought events in the agriculture‐dominated mid‐Mahanadi River Basin in Odisha, India, using crop water stress as a drought indicator. This drought index incorporated different factors that affect crop water deficit such as the cropping pattern, soil characteristics, and surface soil moisture. The drought monitoring framework utilized a relevance vector machine model‐based classification that provided the uncertainty associated with drought categorization. Using the proposed framework, drought hotspots are identified in the study region and compared with indices based on precipitation and soil moisture. Further, a bivariate copula is employed to model the agricultural drought characteristics and develop the drought severity–duration–frequency (S–D–F) relationships. The drought hotspot maps and S–D–F curves are developed for different locations in the region. These provided useful information on the site‐specific drought patterns and the characteristics of the devastating droughts of 2002 and 2012, characterized by an average drought duration of 7 months at several locations. The site‐specific risk of short‐ and long‐term agricultural droughts are then investigated using the conditional copula. The results suggest that the conditional return periods and the S–D–F curves are valuable tools to assess the spatial variability of drought risk in the region.     

Application of copulas for derivation of drought severity–duration–frequency curves

This study presents copula‐based multivariate probabilistic approach to model severity–duration–frequency (S‐D‐F) relationship of drought events in western Rajasthan, India. Drought occurrences are analysed using standardized precipitation index computed on monthly mean areal precipitation, aggregated at a time scale of 6 months. After testing with a series of probability density functions, the drought variable severity is found to be better represented with log‐normal distribution, whereas duration is well fitted with exponential distribution. Four different classes of bivariate copulas – Archimedean, extreme value, Plackett, and elliptical families are evaluated for modelling joint distribution of drought characteristics. It is observed that the extreme value copula – Gumbel–Hougaard copula – performed better as compared with other classes of copulas, based on results of various statistical tests and upper tail dependence coefficient. The joint distribution obtained from best performing copula is then employed to determine conditional return period and to derive drought severity‐duration‐frequency (S‐D‐F) curves for the study region. The results of the study suggests that the copula method can be used effectively to derive the drought S‐D‐F curves, which can be helpful in planning and adopting suitable drought mitigation strategies in drought‐prone areas. Copyright © 2011 John Wiley & Sons, Ltd.     

Revisiting hydrological drought propagation and recovery considering water quantity and quality

Climate extremes, in particular droughts, are significant driving forces towards riverine ecosystem disturbance. Drought impacts on stream ecosystems include losses that can be either direct (e.g., destruction of habitat for aquatic species) or indirect (e.g., deterioration of water quality, soil quality, and increased chance of wildfires). This paper combines hydrologic drought and water quality changes during droughts and represents a multistage framework to detect and characterize hydrological droughts while considering water quality parameters. This method is applied to 52 streamflow stations in the state of California, USA, over the study period of 1950–2010. The framework is assessed and validated based on two drought events declared by the state in 2002 and 2008. Results show that there are two opposite drought propagation patterns in northern and southern California. In general, northern California indicates more frequent droughts with shorter time to recover. Chronology of drought shows that stations located in southern California have not followed a specific pattern but they experienced longer drought episodes with prolonged drought recovery. When considering water quality, results show that droughts either deteriorate or enhance water systems, depending on the parameter of interest. Undesirable changes (e.g., increased temperature and decreased dissolved oxygen) are observed during droughts. In contrast, decreased turbidity is detected in rivers during drought episodes, which is desirable in water systems. Nevertheless, water quality deteriorates during drought recovery, even after drought termination. Depending on climatic and streamflow characteristics of the watersheds, it was found that it would take nearly 2 months on average for water quality to recover after drought termination.     

Threshold level approach for streamflow drought analysis in the Central Andes of Argentina: a climatological assessment

A streamflow drought climatology was developed over the Central Andes of Argentina, a semi-arid region highly vulnerable to climatic variations, based on the analysis of daily historical streamflow records. A threshold level approach was applied on a daily basis for three different severity levels in order to depict the main characteristics of droughts – number of drought events, mean duration and mean severity – over the period 1957–2014. Based on three annual indices that summarize the frequency of drought events, their duration and severity, we identified the main regional dry periods and the main modes of variability through an empirical decomposition. These modes are linked to La Niña conditions on inter-annual time scales and the Pacific Decadal Oscillation for the decadal variations, showing the influence of the tropical Pacific Ocean in the development of streamflow drought conditions and its relevance for potential predictability of hydroclimatic variations over the region.