Theoretical Drought Classification Method (Hao et al., 2016) PDF

Advances in Water Resources 92 (2016) 240–247 Contents lists available at ScienceDirect Advances in Water Resourc...

Advances in Water Resources 92 (2016) 240–247 Contents lists available at ScienceDirect Advances in Water Resources journal homepage: www.elsevier.com/locate/advwatres A theoretical drought classification method for the multivariate drought index based on distribution properties of standardized drought indices Zengchao Hao a,∗, Fanghua Hao a, Vijay P. Singh b, Youlong Xia c, Wei Ouyang a, Xinyi Shen d a Green Development Institute, School of Environment, Beijing Normal University, Beijing, 100875, China b Department of Biological and Agricultural Engineering and Zachry Department of Civil Engineering, Texas A&M University, College Station, TX 77843-2117, USA c I.M. System Group at Environmental Modeling Center, National Center for Environmental Prediction, College Park, Maryland, USA d Department of Civil and Environmental Engineering, University of Connecticut, Storrs, CT, USA a r t i c l e i n f o a b s t r a c t Article history: Drought indices have been commonly used to characterize different properties of drought and the need Received 18 November 2015 to combine multiple drought indices for accurate drought monitoring has been well recognized. Based on Revised 15 April 2016 linear combinations of multiple drought indices, a variety of multivariate drought indices have recently Accepted 16 April 2016 been developed for comprehensive drought monitoring to integrate drought information from various Available online 19 April 2016 sources. For operational drought management, it is generally required to determine thresholds of drought Keywords: severity for drought classification to trigger a mitigation response during a drought event to aid stake- Drought holders and policy makers in decision making. Though the classification of drought categories based on Drought classification the univariate drought indices has been well studied, drought classification method for the multivariate Drought category drought index has been less explored mainly due to the lack of information about its distribution prop- Linearly combined drought index erty. In this study, a theoretical drought classification method is proposed for the multivariate drought index, based on a linear combination of multiple indices. Based on the distribution property of the stan- dardized drought index, a theoretical distribution of the linear combined index (LDI) is derived, which can be used for classifying drought with the percentile approach. Application of the proposed method for drought classification of LDI, based on standardized precipitation index (SPI), standardized soil moisture index (SSI), and standardized runoff index (SRI) is illustrated with climate division data from California, United States. Results from comparison with the empirical methods show a satisfactory performance of the proposed method for drought classification. © 2016 Elsevier Ltd. All rights reserved. 1. Introduction across space and time (Ellis et al., 2010; Hao and Singh, 2013; Mishra and Singh, 2010). Many of these drought indices are gener- The devastating effects of drought and the potential increase in ally based on the anomaly, percentile (e.g., soil moisture percentile, frequency and severity due to climate change have led to exten- SMP), or the standardized indices (e.g., SPI) of various hydrocli- sive studies for better understanding, monitoring and prediction of matic variables. The derivation of SPI applies to a large number drought. Drought indices are the key components for monitoring of drought indices, such as the Standardized Precipitation Evapo- drought, based on meteorological or hydrological variables. A large transpiration Index (SPEI), Standardized Soil Moisture Index (SSI), number of individual drought indicators, such as Standardized Pre- Standardized Runoff Index (SRI) (Hao et al., 2014; Mo, 2011; Shukla cipitation Index (SPI) (McKee et al., 1993) that is commonly used and Wood, 2008; Vicente-Serrano et al., 2010), based on the dis- for meteorological drought monitoring, have been developed in the tribution of various hydro-climatic variables. A variety of proba- past few decades for characterizing different aspects of drought bility distributions (e.g., gamma, generalized extreme value (GEV), generalized logistic distribution, and beta distributions) have been ∗ used to fit monthly observations of different hydroclimatic vari- Corresponding author. Tel.: +861058801971. E-mail address: [email protected] (Z. Hao). ables for the computation of drought indices (Guttman, 1999; http://dx.doi.org/10.1016/j.advwatres.2016.04.010 0309-1708/© 2016 Elsevier Ltd. All rights reserved.  Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 241 McKee et al., 1993; Sheffield et al., 2004; Stagge et al., 2015; 2. Method Vicente-Serrano et al., 2010). Meanwhile, a suite of multivariate drought indices (MDI) or 2.1. Normal quantile transformation products to integrate drought information from multiple sources have been developed in the past decades (Hao and Singh, 2015; The normal quantile transformation (NQT) (or normal score, in- Kao and Govindaraju, 2010; Vicente-Serrano et al., 2010; Ziese et verse normal) has been commonly used to transform the variable al., 2014), such as the U.S. Drought Monitor (USDM) (Svoboda et of interest to the Gaussian variable in order to make it more treat- al., 2002), Aggregate Drought Index (ADI) (Keyantash and Dracup, able with statistical models (Bogner et al., 2012; Krzysztofowicz, 2004), and Multivariate Standardized Drought Index (MSDI) (Hao 1997; Montanari and Brath, 2004). The general steps in the NQT in- et al., 2014). For example, the linearly combined drought index clude the computation of cumulative probabilities of observations (LDI) is among the commonly used multivariate drought indices for and transformation into normal realizations by applying the stan- drought monitoring and combines different drought indices in a dard normal (or Gaussian) distribution. For a random variable Z, linear manner with associated weights to each index (Mo and Let- the cumulative distribution function (CDF) F(Z) can be expressed tenmaier, 2014; Xia et al., 2014a). These multivariate indices have as been compared with other commonly used drought indices to as- F (Z ) = P (Z ≤ z ) (1) sess their performance for drought monitoring due to the lack of the ground truth of drought observations. However, since multi- The normal quantile transformation of Z (denoted as NQT(Z)) variate drought indices are developed with different methods and takes the form (Bogner et al., 2012; Krzysztofowicz, 1997; Monta- may not be directly comparable with univariate indices (e.g., many nari and Brath, 2004) indices are not developed based on a percentile approach or stan- NQT (Z ) = N −1 [F (Z )] (2) dardized in the same way), it is important to accurately process and evaluate these indices before implementing them for opera- where N is the standard normal distribution with zero mean and tional drought management. unit standard deviation. This transformation ensures that the dis- Drought management of many regional drought plans requires tribution of NQT(Z) is Gaussian, regardless of the original distri- the determination of a drought threshold to trigger a response bution form of Z, which fulfills the underlying normal distribution (e.g., limiting water use) during the drought event (Quiring, 2009; assumption that is intrinsic to many statistical models (Kelly and Steinemann, 2003). Based on a suitable definition of the threshold Krzysztofowicz, 1997; Hao and Singh, 2016). of drought severity, drought condition can be classified into several The concept of deriving SPI has been used to derive the stan- drought categories to aid stakeholders and policy makers to imple- dardized drought index (SDI) based on other hydroclimatic vari- ment drought management measures. The USDM has been widely ables, such as soil moisture, runoff, snowmelt, ground water and used for various applications for both drought research and appli- precipitation minus potential evapotranspiration, for drought char- cation purposes, which classifies the drought into five major cate- acterizations (Hao and Singh, 2015). Due to the standardized na- gories, including abnormally dry (D0), moderate drought (D1), se- ture, SPI (and SDI) values for different time scales can be compared vere drought (D2), extreme drought (D3), and exceptional drought at different locations and seasons, which is a desirable property of (D4), from the least intense to the most intense (Svoboda et al., drought indicators for drought assessment. With the introduction 2002). The classification is based on a percentile approach (e.g., of NQT in Eq. (2), it becomes clear that the computation of SPI es- the D1 drought condition corresponds to the 20th percentile) and sentially adopts the NQT procedure. Thus, the SDI, such as SPI, SSI, has been used in several studies to depict the drought condition SRI and SPEI, possesses the properties of the NQT variables. in the drought category form (Goodrich and Ellis, 2006; Steine- mann, 2003; Steinemann et al., 2015). For example, Mo and Letten- 2.2. Distribution of linear combination of normal random variables maier (2014) developed the grand mean drought index to combine SPI, SMP and SRI with equal weight, which is then remapped into Suppose X is a random vector (n × 1) of the multivariate nor- a uniform distribution that enables the classification of drought mal distribution (MVN). Let a random variable Y be composed of X severity into different drought categories based on the percentile with the weight bT = [b1 ,b2 ,..,bn ]: approach. Other methods have also been developed for drought Y = bT X (3) classification to aid drought management. For example, Mallya et al. (2014) proposed a probabilistic drought classification method Then, the random variable Y is a univariate normal random based on SPI using a gamma mixture model. However, most of the variable with mean uy and variance σ 2 y as (Wilks, 2011) previous studies are based on the univariate drought index and the  uy = bT ux σ 2 y = bT b (4) drought classification for the multivariate drought index has been x less explored, which is even more challenging, since the distribu- tion forms of these drought indices are not explicit or clear. where ux and  x are the mean and covariance matrix of the ran- The objective of this study therefore is to develop a theoret- dom vector X. Eq. (4) states that the linear combination of the ical method for objective drought classification of the multivari- MVN vector X is also normal. ate drought index with focus on the linearly combined drought Since each SDI is essentially the NQT variable, which is a stan- indicator (LDI). By deriving the theoretical distribution of LDI, the dard normal random variable, an assumption is made here that the proposed method enables the classification of drought severity of joint distribution of the SDI is a multivariate normal distribution LDI into drought categories based on the percentile approach. The (note that a vector of normally distributed random variables does proposed method is tested with climate division data from Cali- not imply that the vector has a joint normal distribution). Based fornia in the United States and results show the validity of the on this assumption, the linear combination of SDIs is also normally proposed method for drought classification. This paper proceeds distributed with mean and variance expressed in Eq. (4). as follows. The method to derive the distribution of multivariate 2.3. Distribution of linearly combined drought index drought indices is introduced in Section 2. The case study to illus- trate the application of the proposed drought classification method As a general form, the LDI can be expressed as is presented in Section 3, followed by conclusions and discussion in Sections 4 and 5. LDI = α1 SDI1 + α2 SDI2 +... + αn SDIn (5) 242 Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 where SDIs represent various forms of the standardized drought where P is the probability; n is the length of observed data and i index, such as SPI, SSI, SRI, and SPEI; and α 1 ,…,α n are the weights. is the rank of the observed values. When drought indices are computed in a different way from SDI, The drought classification can then be achieved through the such as PDSI, they can be transferred to the SDI by fitting a distri- empirical distribution of LDI (or other indices) based on the per- bution followed by the inverse of the standard normal distribution centile recommended by the USDM and the standardization can (Goodrich and Ellis, 20 06; Quiring, 20 09), which essentially is the also be obtained through the NQT in Eq. (2). In this study, the em- NQT procedure. pirical drought classification was used as a benchmark for com- The main purpose is to explore the distribution property of parison of theoretical drought classification. The advantage of em- LDI to facilitate drought classification based on the percentile ap- pirical method is that it does not rely on the assumed parametric proach. Since SDI in Eq. (5) is essentially normally distributed, the distribution form. distribution of LDI can then be derived from the distribution prop- erty based on the assumption of multivariate normal distribution, 3. Case study as shown in Section 2.2. From Eq. (4), LDI is a normal random vari- able with mean uLDI and variance σ 2 LDI. Specifically, by denoting 3.1. Data αT = [α 1 , α 2 ,…,α n ], the mean of LDI can be expressed as uLDI = α1 u1 + α2 u2 +... + αn un (6) Monthly precipitation, soil moisture and surface runoff data for the period 1932–2011 from climate divisions in California were where u1 ,u2 ,…un denoted the mean values of SDIs which are gen- used for the case study to illustrate the application of the proposed erally close to 0. method. The locations of different climate divisions are shown The covariance matrix of LDI in Eq. (5) can be expressed as in Fig. 1. A one-layer leaky bucket hydrological model (Huang et ⎛ ⎞ al., 1996; Van den Dool et al., 2003) was used to calculate soil cov(SDI1 , SDI1 ) cov(SDI1 , SDI2 )... cov(SDI1 , SDIn ) moisture, evaporation and runoff with observed monthly precipi- ⎜cov(SDI2 , SDI1 ) cov(SDI2 , SDI2 ) ··· cov(SDI2 , SDIn )⎟ LDI =⎝ ⎠ tation and temperature over 344 climate divisions from National ··· ··· ··· ··· Climatic Data Center (NCDC) as forcing variables (assuming a soil cov(SDIn , SDI1 ) cov(SDIn , SDI2 ) ··· cov(SDIn , SDIn ) column depth of 1600 mm). These climate division data were ob- (7) tained from the Climate Prediction Center (CPC), National Oceanic Thus, one can obtain the distribution function of LDI from Eq. and Atmospheric Administration (NOAA) (ftp://ftp.cpc.ncep.noaa. (4), which can be expressed as gov/wd51yf/us). The monthly data were used for the computation of differ- LDI ∼ N (uLDI , α T LDI α ) (8) ent drought indices, including 6-month SPI, 1-month SSI, and 3- where αT denotes the weights associated with different SDIs that month SRI representing meteorological, agricultural and hydrolog- can be determined empirically or through optimization approaches ical droughts, respectively. In this study, the Weibull plotting po- (Hao and Singh, 2015; Mo and Lettenmaier, 2014; Xia et al., 2014b). sition formula was used to compute these drought indices, which were then used to obtain the linearly combined drought index to evaluate the performance for drought classification. The proposed 2.4. Drought classification method was also tested based on data from other climate divisions and similar results were found (not shown). Since the distribution property of LDI is known from Eq. (8), the drought classification can be achieved based on the percentile rec- ommended by the USDM for the drought category D0 (20 to 30), 3.2. Comparison of univariate and multivariate drought indices D1 (10 to 20), D2 (5 to 10), D3 (2 to 5), and D4 (≤ 2) or thresh- old of the index. Specifically, for SPI (or SDI), these percentiles cor- Three drought indices, SPI, SSI and SRI, were computed based respond to the threshold values T = [–0.5, –0.8, –1.3, –1.6, and – on monthly data from 1932 to 2011 (80 years) for Climate Di- 2](Svoboda et al., 2002). For LDI, the threshold of index value for visions 5 (denoted as CD 5) and Climate Divisions 7 (denoted each drought category can be obtained from the theoretical distri- as CD 7), which are San Joaquin Drainage and Southeast Desert bution in Eq. (8), based on the specified percentile. Specifically, for Basin divisions, in California. A plot of these indices for the pe- a specified percentile p0 (e.g., p0 = 2% corresponds to the thresh- riod 1998–2011 is shown in Fig. 2. Generally, all three drought old separating D3 and D4 drought categories), the corresponding indices showed historical drought conditions in California, such quantile x0 can be expressed as as the 2002 drought and the 20 07–20 09 drought (with indices lower than -0.8). However, these three drought indices performed x0 = N −1 ( p0 ; uLDI , σ 2 LDI ) (9) differently in depicting the onset, severity, and end of drought where N−1 is the inverse normal distribution function. condition, which have been highlighted in a few studies (Hao et Alongside the proposed theoretical method for drought classifi- al., 2014; Mo, 2011; Shukla and Wood, 2008). The differences in cation of LDI, an empirical method for the classification of multi- characterizing drought conditions are understandable, since gen- variate drought indices can be developed by empirically estimating erally drought originates from precipitation deficit (meteorological the distribution (and the percentile) of LDI. A suite of plotting posi- drought), which then leads to the depletion of soil moisture (or tion formulas, such as Weibull, Gringorten, Gumbel, Harris, Hazen, agricultural drought) with the onset generally lagging that of me- Beard, California and others, can be used for the empirical prob- teorological drought, and the deficiency of streamflow or ground- ability estimation (Cook, 2011; Fuglem et al., 2013; Gringorten, water resulting in the hydrological drought (Entekhabi et al., 1996; 1963; Makkonen, 2006), especially when the sample size is rela- Hao and Singh, 2015; Heim, 2002). For example, for the drought tively large. The Weibull plotting position formula has been com- condition during 2008 in CD 5, meteorological drought (SPI) gen- monly used to empirically estimate probability and was also used erally precedes agricultural drought (SSI) and hydrological drought to estimate the distribution of LDI in this study. Specifically, the (SRI). This also shows that an individual index is generally not suf- Weibull plotting position formula is expressed as ficient to characterize complicated drought conditions for all sea- sons and regions, and thus accurate drought monitoring requires i P= (10) the integration of multiple drought related variables or indicators. n+1  Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 243 43 42 1 41 3 40 2 39 38 Latitude 37 5 36 4 35 7 34 6 33 32 -125 -124 -123 -122 -121 -120 -119 -118 -117 -116 -115 -114 Longitude Fig. 1. Locations of climate divisions in California, USA. (a) Division 5, CA SPI SSI SRI LDI Drought Indices 2 0 -2 1998 2000 2002 2004 2006 2008 2010 (b) Division 7,CA Drought Indices 2 0 -2 1998 2000 2002 2004 2006 2008 2010 Fig. 2. Plot of monthly drought indices SPI, SSI, SRI, and LDI (between January 1998 and December 2011) for climate divisions 5 and 7 in California (CA), USA. In this study, a linear combination of three drought indices SPI, 1.01, –1.44, and –1.36, respectively, and LDI0 was –1.27. Basically, SSI and SRI was used to integrate drought information from me- LDI consolidates drought information with the average of the three teorological, agricultural and hydrological droughts to illustrate the drought indices. Notice that different drought indices convey dif- application of the proposed drought classification method. Specif- ferent aspects of drought condition and thus LDI is not meant to ically, the linearly combined drought index in Eq. (5) for this case be superior to the univariate drought index. In the operational was defined with SDI1 = SPI, SDI2 = SSI and SDI3 = SRI. Following drought management, LDI can be used along with other drought Mo and Lettenmaier (2014), here weights of the three drought in- indices to characterize drought. dices were assigned to be equal (α 1 = α 2 = α 3 = 1/3). The LDI val- ues for the period 1998–2011in CD 5 and CD 7 in California are 3.3. Drought classification also shown in Fig. 2, in which the LDI clearly shows historical drought events in California, such as the 20 07–20 09. For exam- Here the LDIs for October in CD 5 and July in CD 7 in Califor- ple, for October 2008 in CD 5, values of SPI, SSI and SRI were – nia were used for illustrating the drought classification results. The 244 Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 r=0.60 r=0.26 r=0.85 4 4 4 Division 5, CA 2 2 2 SRI SRI SSI 0 0 0 -2 -2 -2 -4 -4 -4 -5 0 5 -5 0 5 -5 0 5 SPI SPI SSI r=0.94 r=0.78 r=0.86 4 4 4 Division 7, CA 2 2 2 SRI SRI SSI 0 0 0 -2 -4 -2 -2 -5 0 5 -5 0 5 -5 0 5 SPI SPI SSI Fig. 3. Scatter plots and correlation coefficients (r) of three drought indices SPI, SSI, and SRI for October in climate division 5 and July in climate division 7 in California, USA. scatter plots of pairs of drought indices SPI, SSI and SRI for July percentile was estimated as –1.33, which differed significantly from and October (sample size n = 80) in these two climate divisions the threshold for the SPI recommended by USDM (–1.6), since the are shown in Fig. 3, along with the Pearson correlation coefficient distribution of LDI was not standard normal. For July in CD 7, due (r) to measure the stochastic dependence between two drought in- to the relative closeness of the standard deviation (0.91) to unity, dices. The SPI-SSI and SSI-SRI were highly correlated in the specific the threshold values for drought classification were relatively sim- month of the two climate divisions. However, the correlation be- ilar to those recommended by USDM for the SPI. tween SPI-SRI for October in CD 5 was rather weak, while that for To evaluate the overall performance of the proposed method for July in CD 7 was relatively strong. For drought indices of October drought classification, the percentages of different drought cate- in CD 5 (July in CD 7) in California, the mean and standard devi- gories for the whole period from 1932 to 2011 (80 years) for all cli- ation of LDI estimated from Eqs. (6) and (7) were 0 (0) and 0.81 mate divisions (totally 7) in California were obtained, as shown in (0.91). These results indicate that the variance of LDI depended on Fig. 5 with boxplots. The central mark of the box is the median of the stochastic dependence of drought indices SPI, SSI and SRI and drought percentages with edges of the box representing 25th and different degrees of correlations (or covariance) would result in dif- 75th percentiles and whiskers of the box representing the maxi- ferent values of the variance of LDI (may differ from the unity). mum and minimum values of the percentages. Note that the num- Thus, the distribution of LDI may not be standard normal and Eq. ber of drought categories estimated from the empirical method, (9) can be used to aid the classification in this case. which depended on the rank of the data, for different months The theoretical cumulative distribution function (CDF) of LDI (and climate divisions) was the same and thus the percentages (sample size n = 80) for the two months is shown in Fig. 4(a) and of drought categories from the empirical method were generally (b), respectively, along with the empirical distribution based on the the same for all climate divisions. Overall, the drought percent- Weibull plotting position formula. It can be seen that the theoret- age from the empirical method fell within the boxplot, indicating ical CDF was close to the empirical CDF, implying that the theo- a satisfactory performance of the drought classification from the retical distribution in Eq. (8) was generally valid. The Kolmogorov– proposed method. For example, the median of the drought per- Smirnov (K–S) test (Bloomfield and Marchant, 2013) was used for centage for the D3 drought category from the proposed method the goodness-of-fit of the theoretical distribution with a sample was 2.8%, which was close to the value of 2.5% estimated from the size of 80 and results indicated that the null hypothesis (i.e., data empirical method. These results showed that the proposed method come from the normal distribution) could not be rejected at the based on the theoretical distribution of linearly combined drought 5% significance level, which confirmed the validity of the theoreti- indices through comparison with the empirical method performed cal distribution. We also assessed the validity of the distribution of relatively well for drought classification. LDI in all climate divisions in California for different months and record length (n = 30 and 50) based on the Kolmogorov–Smirnov 4. Discussions (K–S) test and it was found that the theoretical distribution was valid for almost all cases. The theoretical method mainly applies to the multivariate After the validation of the distribution property of LDI, the drought index based on the linear combination while the em- drought classification for each month can then be achieved by pirical method can be used for any multivariate drought index computing the associated threshold of drought indices based on based on empirical estimation of the percentile (or probability). the percentiles suggested by USDM from Eq. (9). For October in Several multivariate drought indices, such as MSDI and ADI, have CD 5, for example, the threshold value corresponding to the 5th been developed in recent years with various methods (but not the  Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 245 (a) Division 5, CA 1 CDF 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 LDI (b) Division 7, CA 1 CDF 0.5 Theoretical Empirical 0 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 LDI Fig. 4. Theoretical and empirical distributions of LDI for October in climate division 5 and July in climate division 7 in California, USA. 0.7 0.6 0.5 Percentage 0.4 0.3 0.2 0.1 0 ND D0 D1 D2 D3 D4 Drought Category Fig. 5. Comparison of the percentages of different drought categories from the theoretical method (shown with boxplot) and empirical method (shown with point) for all climate divisions in California, USA. linear combination) and the distribution property is generally un- Though MSDI is standardized with the normal distribution, it clear. In this case, the empirical method can be used to obtain the is generally not normally distributed since the distribution of the percentile (or standardization) of the index to facilitate drought joint percentile in Eq. (11) is not uniform within [0 1]. To illustrate classification. The MSDI, which has been proposed to incorporate this point, the MSDI based on three indices SPI, SSI and SRI for two drought information from multiple sources based on the joint dis- months (August and October) from 1932 to 2011 for CD7 in Califor- tribution (Hao et al., 2014), was used in this section for illustrative nia is shown in Fig. 6. It can be seen that generally more values of purposes. The trivariate MSDI based on three random variables X1 , MSDI falls below 0. For example, the percentage of the MSDI val- X2 and X3 can be expressed as ues below 0 and -0.5 for October is 75% and 54%, respectively. In addition, none of the two samples passed the Kolmogorov–Smirnov MSDI = φ −1 [P (X1 ≤ x1 , X2 ≤ x2 , X3 ≤ x3 )] (11) normality test (sample size n = 80). The transformed MSDI (termed where ϕ is the standard normal distribution; and P(X1 ≤ x1 , X2 ≤ x2 as MSDIt ) based on NQT with the empirical method for the two and X3 ≤ x3 ) is the joint percentile (or probability). months is also shown in Fig. 6. Due to NQT, the distribution of 246 Z. Hao et al. / Advances in Water Resources 92 (2016) 240–247 August 4 MSDI MSDIt 2 Index 0 -2 -4 1932 1947 1962 1977 1992 2007 October 4 2 Index 0 -2 -4 1932 1947 1962 1977 1992 2007 Fig. 6. Comparison of the trivariate MSDI and its transformation (MSDIt ) based on the empirical distribution for August and October for the period 1932–2011 (sample size n = 80) for climate division 7 in California, USA. MSDIt for the two months passed the Kolmogorov–Smirnov (K–S) in this case. In the past decade, various univariate and multivariate test (sample size n = 80). The difference between MSDI and MSDIt drought indices have been assessed and compared with each other would result in the difference in drought classification. For ex- (and USDM) for drought monitoring across space and time. Due ample, the percentage of MSDIt values below 0 and –0.5 for Oc- to differences in univariate and multivariate indices, it is recom- tober was 50% and 30%, respectively. As such, the MSDIt can be mended to evaluate the multivariate drought indices by transform- employed for the drought classification and monitoring in a sim- ing them into categories or percentiles with the proposed theoreti- ilar way as other standardized drought indices such as SPI and cal or empirical method to ensure the statistical consistency across SPEI. Detailed comparison of the performance of MSDIt for drought space and time for operational drought management. monitoring is beyond the scope of this study and will be carried out in the future. Acknowledgments 5. Conclusions The authors are grateful to the Editor and anonymous review- ers for their valuable comments and suggestions. This work was From an application standpoint, it is important to define a supported by Youth Scholars Program of Beijing Normal Univer- threshold for drought indices to trigger a response for drought sity (Grant No. 2015NT02) and also partly supported by the China management. In this study, a statistical method for the objective Three Gorges Corporation (Project No. 0799557). drought classification of LDI is proposed by deriving its theoreti- cal distribution function to classify drought categories based on a References percentile approach. 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