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.     

Predicting return periods of hydrological droughts using the Pearson 3 distribution: a case from rivers in the Canadian prairies

Abstract

The standardized series of monthly and weekly flow sequences, referred to as standardized hydrological index (SHI) series, from five rivers in the Canadian prairies were subjected to return period (T_r) analysis of drought length (L). The SHI series were truncated at drought probability levels q ranging from 0.5 to 0.05 with the intention of deducing drought events and corresponding drought lengths. The values of L were fitted to the Pearson 3, the gamma (2-parameter), the exponential (1-parameter), the Weibull 3 and the Weibull (2-parameter) probability density functions (pdfs). A priori assignment of one week or one month for the location parameter in the Pearson 3 pdf proved logical and also facilitated the rapid estimation of other parameters using either the method of moments or the method of maximum likelihood. The Pearson 3 turns out to be the most suitable pdf to describe and to estimate return periods of drought lengths. At the monthly and weekly time scales, it was inferred that the sample size (T, months or weeks) of SHI series could be treated equivalent to the return period of the largest recorded drought length. At the annual time scale, however, the sample size (T, years) should be modified using either the Hazen or the Gringorten plotting position formula to reflect the actual return period of the largest recorded drought length in years.

Editor D. Koutsoyiannis; Associate editor E. Gargouri     

Combined analysis of precipitation and water deficit for drought hazard assessment

Abstract

The combined analysis of precipitation and water scarcity was done with the use of the Standardized Precipitation Index (SPI) and the Standardized Runoff Index (SRI), developed as a monthly, two-variable SPI-SRI indicator to identify different classes of hydrometeorological conditions. Stochastic analysis of a long-term time series (1966–2005) of monthly SPI-SRI indicator values was performed using a first-order Markov chain model. This provided characteristics of regional features of drought formation, evolution and persistence, as well as tools for statistical long-term drought hazard prediction. The study was carried out on two subbasins of the Odra River (Poland) of different orography and land use: the mountainous Nysa K?odzka basin and the lowland, agricultural Prosna basin. Classification obtained with the SPI-SRI indicator was compared with the output from the NIZOWKA model that provided identification of hydrological drought events including drought duration and deficit volume. Severe and long-duration droughts corresponded to SPI-SRI Class 3 (dry meteorological and dry hydrological), while severe but short-term droughts (lasting less than 30 days) corresponded to SPI-SRI Class 4 (wet meteorological and dry hydrological). The results confirm that, in Poland, meteorologically dry conditions often shift to hydrologically dry conditions within the same month, droughts rarely last longer than 2 months and two separate drought events can be observed within the same year.     

Analysis of streamflow droughts using fixed and variable thresholds

Hydrological drought analysis is very important in the design of hydrotechnical projects and water resources management and planning. In this study, a methodology is proposed for the analysis of streamflow droughts using the threshold level approach. The method has been applied to Yermasoyia semiarid basin in Cyprus based on 30‐year daily discharge data. Severity was defined as the accumulated water deficit volume occurring during a drought event, in respect with a target threshold. Fixed and variable thresholds (seasonal, monthly, and daily) were employed to derive the drought characteristics. The threshold levels were determined based on the Q₅₀ percentiles of flow extracted from the corresponding flow duration curves for each threshold. The aim is to investigate the sensitivity of these thresholds in the estimation of maximum drought severities for various return periods and the derivation of severity–duration–frequency curves. The block maxima and the peaks over threshold approaches were used to perform the extreme value analysis. Three pooling procedures (moving average, interevent time criterion, and interevent time and volume criterion) were employed to remove the dependent and minor droughts. The application showed that the interevent time and volume criterion is the most unbiased pooling method. Therefore, it was selected to estimate the drought characteristics. The results of this study indicate that monthly and daily variable thresholds are able to capture abnormal drought events that occur during the whole hydrological year whereas the other two, only the severe ones. They are also more sensitive in the estimation of maximum drought severities and the derivation of the curves because they incorporate better the effect of drought durations.     

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.     

Analytical procedures for weekly hydrological droughts: a case of Canadian rivers

Abstract

Two entities of importance in hydrological droughts, viz. the longest duration, L_T , and the largest magnitude, M_T (in standardized terms) over a desired time period (which could also correspond to a specific return period) T, have been analysed for weekly flow sequences of Canadian rivers. Analysis has been carried out in terms of week-by-week standardized values of flow sequences, designated as SHI (standardized hydrological index). The SHI sequence is truncated at the median level for identification and evaluation of expected values of the above random variables, E(L_T ) and E(M_T ). SHI sequences tended to be strongly autocorrelated and are modelled as autoregressive order-1, order-2 or autoregressive moving average order-1,1. The drought model built on the theorem of extremes of random numbers of random variables was found to be less satisfactory for the prediction of E(L_T ) and E(M_T ) on a weekly basis. However, the model has worked well on a monthly (weakly Markovian) and an annual (random) basis. An alternative procedure based on a second-order Markov chain model provided satisfactory prediction of E(L_T ). Parameters such as the mean, standard deviation (or coefficient of variation), and lag-1 serial correlation of the original weekly flow sequences (obeying a gamma probability distribution function) were used to estimate the simple and first-order drought probabilities through closed-form equations. Second-order probabilities have been estimated based on the original flow sequences as well as SHI sequences, utilizing a counting method. The E(M_T ) can be predicted as a product of drought intensity (which obeys the truncated normal distribution) and E(L_T ) (which is based on a mixture of first- and second-order Markov chains).

Citation Sharma, T. C. & Panu, U. S. (2010) Analytical procedures for weekly hydrological droughts: a case of Canadian rivers. Hydrol. Sci. J. 55(1), 79–92.     

Spatio-temporal drought analysis in the Trakya region,Turkey

Abstract

Since droughts are natural phenomena, their occurrence cannot be predicted with certainty and thus it must be treated as a random variable. Once drought duration and magnitude have been found objectively, it is possible to plan for the transport of water in known quantities to drought-stricken areas either from alternative water resources or from water stored during wet periods. The summation of deficits over a particular period is referred to as the drought magnitude. Drought intensity is the ratio of drought magnitude to its duration. These drought properties at different truncation levels provide significant hydrological and hydrometeorological design quantities. In this study, the run analysis and z-score are used for determining drought properties of given hydrological series. In addition, kriging is used as a spatial drought analysis for mapping. This study is applied to precipitation records for Istanbul, Edirne, Tekirdag and Kirklareli in the Trakya region, Turkey and then the drought period, magnitude and standardized precipitation index (SPI) values are presented to depict the relationships between drought duration and magnitude.     

An analysis of non-normal Markovian extremal droughts

In many arid and semi-arid environments of the world, years of extended droughts are not uncommon. The occurrence of a drought can be reflected by the deficiency of the rainfall or stream flow sequences below the long-term mean value, which is generally taken as the truncation level for the identification of the droughts. The commonly available statistics for the above processes are mean, coefficient of variation and the lag-one serial correlation coefficient, and at times some indication of the probability distribution function (pdf) of the sequences. The important elements of a drought phenomenon are the longest duration and the largest severity for a desired return period, which form a basis for designing facilities to meet exigencies arising as a result of droughts. The sequences of drought variable, such as annual rainfall or stream flow, may follow normal, log-normal or gamma distributions, and may evolve in a Markovian fashion and are bound to influence extremal values of the duration and severity. The effect of the aforesaid statistical parameters on the extremal drought durations and severity have been analysed in the present paper. A formula in terms of the extremal severity and the return period ‘T’ in years has been suggested in parallel to the flood frequency formula, commonly cited in the hydrological texts. © 1998 John Wiley & Sons, Ltd.     

Uncertainty and variability in bivariate modeling of hydrological droughts

There are two kinds of uncertainty factors in modeling the bivariate distribution of hydrological droughts: the alteration of predefined critical ratios for pooling droughts and excluding minor droughts and the temporal variability of univariate and/or bivariate characteristics of droughts due to the impact of human activities. Daily flow data covering a period of 56 hydrological years from two gauging stations from a humid region in South China are used. The influences of alterations of threshold values of flow and critical ratios of pooling droughts and excluding minor droughts on drought properties are analyzed. Six conventional univariate models and three Archimedean copulas are employed to fit the marginal and joint distributions of drought properties, the Kolmogorov–Smirnov and Anderson–Darling methods are used for testing the goodness-of-fit of the univariate model, and the Cramer-von Mises method based on Rosenblatt’s transform is applied for the test of the bivariate model. The change point analysis of the copula parameter of bivariate distribution of droughts is first made. Results demonstrate that both the statistical characteristics of each drought property and their bivariate joint distributions are sensitive to the critical ratio of excluding minor droughts. A model can be selected to fit the marginal distribution for drought deficit volume or maximum deficit, but it is not determined for drought duration with the lower ratios of the pooling and excluding droughts. The statistical uncertainty of drought duration makes the modeling of bivariate joint distribution of drought duration and deficit volume or of drought duration and maximum deficit undermined. Change points significantly occurred in the period from the late 1970s to the middle 1980s for a single drought property and the copula parameter of their joint distribution due to the impact of human activities. The difference between two subseries separated by the change point is remarkable in the magnitudes of drought properties and the joint return periods. A copula function can be selected to optimally fit the bivariate distribution, provided that the critical ratios of pooling and excluding droughts are great enough such as the optimal value of 0.4 in the case study. It is valuable that the modeling and designing of the bivariate joint correlation and distribution of drought properties can be performed on the subseries separated by the change point of the copula parameter.     

A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies

Abstract

A hydrological drought magnitude (M _T ) expressed in standardized terms is predicted on annual, monthly and weekly time scales for a sampling period of T years in streamflow data from the Canadian prairies. The drought episodes are considered to follow the Poisson law of probability and, when coupled with the gamma probability distribution function (pdf) of drought magnitude (M) in the extreme number theorem, culminate in a relationship capable of evaluating the expected value, E(M _T ). The parameters of the underlying pdf of M are determined based on the assumption that the drought intensity follows a truncated normal pdf. The E(M _T ) can be evaluated using only standard deviation (σ), lag-1 autocorrelation (ρ) of the standardized hydrological index (SHI) sequence, and a weighting parameter Φ (ranging from 0 to 1) to account for the extreme drought duration (L _T ), as well as the mean drought duration (L_m ), in a characteristic drought length (L_c ). The SHI is treated as standard normal variate, equivalent to the commonly-used standardized precipitation index. A closed-form relationship can be used for the estimation of first-order conditional probabilities, which can also be estimated from historical streamflow records. For all rivers, at the annual time scale, the value of Φ was found equal to 0.5, but it tends to vary (in the range 0 to 1) from river to river at monthly and weekly time scales. However, for a particular river, the Φ value was nearly constant at monthly and weekly time scales. The proposed method estimates E(M _T ) satisfactorily comparable to the observed counterpart. At the annual time scale, the assumption of a normal pdf for drought magnitude tends to yield results in close proximity to that of a gamma pdf. The M _T , when transformed into deficit-volume, can form a basis for designing water storage facilities and for planning water management strategies during drought periods.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2013. A semi-empirical method for predicting hydrological drought magnitudes in the Canadian prairies. Hydrological Sciences Journal, 58 (3), 549–569.     

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.     

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.     

Estimation of regional meteorological and hydrological drought characteristics: a case study for Denmark

Information on regional drought characteristics provides critical information for adequate water resource management. This study introduces a method to calculate the probability of a specific area to be affected by a drought of a given severity and demonstrates its potential for calculating both meteorological and hydrological drought characteristics. The method is demonstrated using Denmark as a case study. The calculation procedure was applied to monthly precipitation and streamflow series separately, which were linearly transformed by the Empirical Orthogonal Functions (EOF) method. Denmark was divided into 260 grid-cells of 14×17 km, and the monthly mean and the EOF-weight coefficients were interpolated by kriging. The frequency distributions of the first two (streamflow) or three (precipitation) amplitude functions were then derived. By performing Monte Carlo simulations, amplitude functions corresponding to 1000 years of data were generated. Based on these simulated functions as well as interpolated mean and weight coefficients, long time series of precipitation and streamflow were simulated for each grid-cell. The probability distribution functions of the area covered by a drought and the drought deficit volumes were then derived and combined to produce drought severity-area-frequency curves. These curves allowed an estimation of the probability of an area of a certain extent to have a drought of a given severity, and thereby return periods could be assigned to historical drought events. A comparison of drought characteristics showed that streamflow droughts are less homogeneous over the region, less frequent and last for longer time periods than precipitation droughts.     

On the generation of drought events using an alternating renewal-reward model

This research study focused on the hypothesis that extreme drought and high streamflow events come from different independent populations with different probability distributions which need to be studied separately, rather than considering the streamflow population as a whole. The inability of traditional streamflow generator models to consistently reproduce the frequency of occurrence of severe droughts observed in the historical record has been questioned by many researchers. Our study focused on the development of astochastic event generator model which would be capable of doing so. This was accomplished in a two-step process by first generating the drought event, and then deriving the streamflows which comprised that event. The model considered for this analysis was an alternating renewal-reward procedure that cycles between eventon andoff times, and is representative of drought or high streamflow event duration. The reward gained while the event ison oroff represents drought severity or high streamflow surplus. Geometric and gamma distributions were considered for drought duration and deficit respectively. Model validation was performed using calculated required capacities from the sequent peak algorithm.     

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.     

Prediction of hydrological drought durations based on Markov chains: case of the Canadian prairies

Abstract

Hydrological drought durations (lengths) in the Canadian prairies were modelled using the standardized hydrological index (SHI) sequences derived from the streamflow series at annual, monthly and weekly time scales. The rivers chosen for the study present high levels of persistence (as indicated by values exceeding 0.95 for lag-1 autocorrelation in weekly SHI sequences), because they encompass large catchment areas (2210–119 000 km²) and traverse, or originate in, lakes. For such rivers, Markov chain models were found to be simple and efficient tools for predicting the drought duration (year, month, or week) based on annual, monthly and weekly SHI sequences. The prediction of drought durations was accomplished at threshold levels corresponding to median flow (Q50) (drought probability, q?=?0.5) to Q95 (drought probability, q?=?0.05) exceedence levels in the SHI sequences. The first-order Markov chain or the random model was found to be acceptable for the prediction of annual drought lengths, based on the Hazen plotting position formula for exceedence probability, because of the small sample size of annual streamflows. On monthly and weekly time scales, the second-order Markov chain model was found to be satisfactory using the Weibull plotting position formula for exceedence probability. The crucial element in modelling drought lengths is the reliable estimation of parameters (conditional probabilities) of the first- and second-order persistence, which were estimated using the notions implicit in the discrete autoregressive moving average class of models. The variance of drought durations is of particular significance, because it plays a crucial role in the accurate estimation of persistence parameters. Although, the counting method of the estimation of persistence parameters was found to be unsatisfactory, it proved useful in setting the initial values and also in subsequent adjustment of the variance-based estimates of persistence parameters. At low threshold levels corresponding to q < 0.20, even the first-order Markov chain can be construed as a satisfactory model for predicting drought durations based on monthly and weekly SHI sequences.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2012. Prediction of hydrological drought durations based on Markov chains in the Canadian prairies. Hydrological Sciences Journal, 57 (4), 705–722.     

Hydrological identification of drought

The problem of drought identification is considered from a primarily hydrological viewpoint. The problems related to the definition, identification and prediction of drought have not yet been solved. Although rainfall data are analysed as the main indicator and characteristic of drought, other characteristics resulting from a rainfall deficit are also important. A time unit of one month was selected as the most suitable for analysis. Special attention was paid to the selection of truncation levels with respect to their influence on the results of drought identification. Three methods for drought identification were applied to a series of monthly rainfall data in Osijek from January 1882 to December 1990: (1) run analysis; (2) a discrete Markov process; and (3) the percentile method. Although the results of these three methods are encouraging, different methods yielded similar results. Some drawbacks of the application of distribution curves are discussed. Drought events should be identified using a number of different procedures.     

Observed advantage for negative binomial over Poisson distribution in partial duration series

The goodness of fit of the negative binomial and the Poisson distributions to partial duration series of runoff events is tested. The data have been recorded by eight hydrometric stations located on ephemeral rivers in Isreal. For each station, a number of threshold discharges are considered, by that series of nested subsamples are formed. Owing to size limitations, a Chi-square test is conducted on samples associated with low to moderate threshold discharges. Positive results, at a 5% significance level, are obtained in 30 out of the 53 tests of the Poisson distribution, and in 22 out of the 28 tests of the negative binomial distribution. The fit of the Poisson distribution to samples of conventionally recommended sizes (of 2 to 3 events per year) is found positive for five rivers and negative for the three other rivers The fit of the negative binomial distribution to these samples is found positive for six rivers, inconclusive for one river and short of data for the eighth river. Mixed results are obtained as the threshold level is raised. Therefore, no direct extrapolation is possible to samples associated with high thresholds.An indirect extrapolation is drawn through a comparison of the actual properties of the samples with those expected under a perfect fit of the distribution functions. Ranges of such properties are defined with respect to the properties of the tested samples and to the test results. The actual properties of nine of the eleven samples associated with high thresholds (i.e. mean number of events <-0.1year ^–1) are found within these ranges. This provides a hint for a probable good fit of either distribution, and particularly the negative binomial, to the occurrence frequency of high events.     

Observed advantage for negative binomial over Poisson distribution in partial duration series

The goodness of fit of the negative binomial and the Poisson distributions to partial duration series of runoff events is tested. The data have been recorded by eight hydrometric stations located on ephemeral rivers in Isreal. For each station, a number of threshold discharges are considered, by that series of nested subsamples are formed. Owing to size limitations, a Chi-square test is conducted on samples associated with low to moderate threshold discharges. Positive results, at a 5% significance level, are obtained in 30 out of the 53 tests of the Poisson distribution, and in 22 out of the 28 tests of the negative binomial distribution. The fit of the Poisson distribution to samples of conventionally recommended sizes (of 2 to 3 events per year) is found positive for five rivers and negative for the three other rivers The fit of the negative binomial distribution to these samples is found positive for six rivers, inconclusive for one river and short of data for the eighth river. Mixed results are obtained as the threshold level is raised. Therefore, no direct extrapolation is possible to samples associated with high thresholds.An indirect extrapolation is drawn through a comparison of the actual properties of the samples with those expected under a perfect fit of the distribution functions. Ranges of such properties are defined with respect to the properties of the tested samples and to the test results. The actual properties of nine of the eleven samples associated with high thresholds (i.e. mean number of events <-0.1year ^–1) are found within these ranges. This provides a hint for a probable good fit of either distribution, and particularly the negative binomial, to the occurrence frequency of high events.