Tracing Time-Varying Characteristics of Meteorological Drought through Nonstationary Joint Deficit Index

a. Modified standardized index and its marginal distribution

b. Detection of nonstationarity

The basic definition of nonstationarity implies change in any statistical parameter of the probabilistic distribution over time (Smith 2002). Therefore, it is essential to check for any signature of nonstationarity in the statistical parameters before deciding whether to adopt a stationary/nonstationary model. Changes in the time series alone cannot reflect the actual change in the statistical parameters. Here, the time sliding window (TSW) approach proposed by Vinnarasi and Dhanya (2019) is extended to detect both the short-term changes and the long-term changes in the statistical parameters. Short-term change, also termed as changepoint, occurs when the regime moves from one state to another in a short amount of time, that can be visualized in terms of a step change/sudden jump in a time series. The long-term change (trend) occurs in a relatively long time, and it is reflected as a concave/convex trend in the time series. The steps involved in detecting the nonstationarity in a time series, for a specific rainfall depth and time scale, are outlined below.

1. Using a predefined time sliding window (q), the aggregated rainfall depth for a given time scale $[X_S^m(t)=x_1,\hspace{0.17em}{x}_2,\dots ,x_n]$ is reconstructed into many realizations (R₁, R₂, …, R_n−q) as follows:

   $\begin{array}{c}{R}_1=\{x_1,x_2,\dots ,x_q\}\\ ⋮\\ R_{n-q}=\{x_{n-q},x_{n-q+1},\dots ,x_n\}.\end{array}$(4)
2. Each realization (R₁, R₂, …, R_n−q) is fitted with a two-parameter gamma distribution (Γ) whose parameters are estimated using maximum likelihood approach:

   $R_i\sim \Gamma ({\alpha }_{R_i},{\beta }_{R_i})\hspace{0.17em}\forall \hspace{0.17em}i=1,\hspace{0.17em}2,\dots ,n-q.$(5)
3. Parameter time series (PTS) are formed by using these estimates from each realization. For example,

   $\begin{array}{c}\mathit{\alpha }={\alpha }_{R_1},{\alpha }_{R_2},\dots ,{\alpha }_{R_{n-q}}\\ \mathit{\beta }={\beta }_{R_1},{\beta }_{R_2},\dots ,{\beta }_{R_{n-q}}.\end{array}$(6)

4. Then, the parametric global method is used to detect the changepoint (Killick et al. 2012) in the above mentioned α and β time series (PTS).

5. If a changepoint is detected in one or both PTS (say in shape, α, PTS), then the series is divided into two segments: α₁ denotes the shape PTS segment before the changepoint and α₂ denotes the shape PTS segment after the changepoint.

6. The nonparametric moving block bootstrap Mann–Kendall test is used to detect any significant trend in each segment (when changepoint present) or for the whole PTS (when changepoint is absent). The linear fit estimates for a very short time series will have high uncertainty, and hence, we select the changepoint such that the split time series are at least 35 years long to have reasonable confidence in the estimates.

7. If the parameter shows a significant trend, then it is considered to be nonstationary, and the time-varying parameters are estimated by imposing a linear trend in the following manner (the procedure is shown for shape parameter α; a similar procedure is followed for scale parameter β as well).

   1. In case of changepoint (at t = t₀), with a significant trend in both segments,

      $\mathit{\alpha }=\{\begin{array}{ll}{\mathit{\alpha }}_1=a_{11}+a_{12}t& \forall t=1,2,\dots ,t_0\\ {\mathit{\alpha }}_2=\hspace{0.17em}{a}_{21}+a_{22}t& \forall t=t_0+1,\dots ,T\end{array},$(7)

      where a₁₁ and a₁₂ denote the coefficients of linear relationship for the first segment (t = 1, 2, …, t₀). Similarly, a₂₁ and a₂₂ denote the coefficients of linear relationship for the second segment [t = (t₀ + 1): T].
   2. In case of changepoint (at t = t₀), with a significant trend in only one of the segments (say, the first segment),

      $\mathit{\alpha }=\{\begin{array}{ll}{\mathit{\alpha }}_1=a_{11}+a_{12}t& \forall t=1,2,\dots ,t_0\\ {\mathit{\alpha }}_2=a_{21}& \forall t=t_0+1,\dots ,T\end{array},$(8)

      where a₁₁ and a₁₂ denote the coefficients of linear relationship for the first segment (t = 1, 2, …, t₀) and a₂₁ denotes the time-invariant estimate for the second segment (t = t₀ + 1, …, T). Please note that if the second segment has trend, it will have linear formulation and first segment will have a constant value.
   3. In case of no changepoint and presence of significant trend in the entire time series,

      $\mathit{\alpha }=a_{01}+a_{02}t\forall t=1,2,\dots ,T,$(9)

      where a₀₁ and a₀₂ denote the coefficients of linear relationship for the entire time series (t = 1, 2, …, T).

   4. In case of no changepoint with insignificant trend, the parameter is estimated as a constant value (a₀₀).

Vinnarasi and Dhanya (2019) used Bayesian inference (see supplementary information for the methodology details) to estimate time-varying parameters in Eqs. (7)–(9) (i.e., a₀₁, a₀₂, …, a₂₂). The estimates from Bayesian inference are more robust compared to ordinary least squares estimates in providing uncertainty in the estimates. However, Bayesian inference is computationally intensive and demands large storage space, as the total number of instances would be 144 (cases) × 3930 (grids). Hence, we have used linear fit estimation for all the grids in this study after verifying that the Bayesian and linear fit estimates are reasonably close for selected grids.

c. NJDI

To examine the overall status of a drought, time-varying marginal distributions of aggregated rainfall depths ($u_s^m$) at different time scales of each month are combined through empirical copula to compute the joint cumulative probability. The procedure involved in computing NJDI is described below.

1. Compute the marginal distribution for each month, and for each time scale using time-varying distribution parameters. For instance, the marginal distribution of January month for the 12-month time scale of precipitation is

   $u_{12,t}^{\text{Jan}}=F_{\text{gam}}(R_{12,t}^{\text{Jan}}|{\alpha }_{12,t}^{\text{Jan}},{\beta }_{12,t}^{\text{Jan}}).$(10)
2. The marginal cumulative distribution function of different months for each time scale is reconstructed with respect to the same ending time:

   $u_{12,t}\text{is}\text{the}\text{union}\text{of}{u}_{12,t}^{\text{Jan}},u_{12,t}^{\text{Feb}},\dots ,u_{12,t}^{\text{Dec}}.$(11)

3. Then, the joint cumulative distribution of all the reconstructed marginal distributions is computed using empirical copula (Kao and Govindaraju 2010).

   1. For single variable,

      $P[U_{1,t}\le u_{1,t},U_{2,t}\le u_{2,t},\dots ,U_{d,t}\le u_{12,t}]=q_t.$(12)
   2. For multiple variables, steps (i) to (iii) need to be repeated for the consecutive variables, and then the joint cumulative distribution is computed using

      $P[U_{1,t}\le u_{1,t},\dots ,U_{12,t}\le u_{12,t},U_{13,t}\le {\upsilon }_{1,t},\dots ,U_{d,t}\le {\upsilon }_{12,t}]=q_t.$(13)

4. It may happen that different combinations of (u_1,t, …, u_12,t) may result in same value of q_t. We then use the Kendall distribution function (K_c) (Nelsen et al. 2003; Kao and Govindaraju 2010) to which will enable us to compute a quantifiable index:

   $K_c(q_t)=P[C_{U_{1,\dots ,}{U}_d}(U_1,\dots ,U_d)\le q_t].$(14)

5. Finally, nonstationary standardized index (NSI; refers to drought information at a single time scale, say 6 months) and NJDI (refers to drought information at multiple time scales, say, 1, 2, …, 12 months) are computed by converting the joint cumulative distribution and cumulative distribution of each month, respectively, to the standard normal deviate with zero mean and unit variance. Then, the joint cumulative distribution of all the reconstructed marginal distributions is computed using empirical copula (Kao and Govindaraju 2010):

   1. NSI^mod for 12-month time scale (s = 12) is computed as

      ${\text{SI}}_s^{\text{mod}}={\text{SI}}_{12}^{\text{mod}}=F^{-1}(U_{12,t}).$(15)
   2. NJDI is computed as

      $\text{NJDI}=F^{-1}[K_c(q_t)],$(16)

      where positive NJDI (0.5 < K_c < 1) indicates overall wet condition and negative NJDI denotes overall dry condition (0 < K_c < 0.5) (Kao and Govindaraju 2010). The drought classifications of NSI and NJDI are the same as traditional SI (McKee et al. 1993), as shown in Table 1.

Table 1.

Drought classification based on the SI values.

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a. Extraction of marginals

First, the aggregated rainfall series is extracted for all the months (January, February, …, December) and for 12 time scales (1–12 months), totaling 144 series for each grid, following the procedure given in section 3a, to account for the seasonal variability in India. Since gamma distribution is limited by the presence of zero values, the series having large proportions of zero values are not analyzed which are primarily located in the extremely arid regions located in the western Indian states. The marginal distribution is computed by fitting a two-parameter gamma distribution to each series, and the goodness of fit is estimated using the KS test. There are some aggregated precipitation series (5.6%) which fail the KS test and we use empirical distribution to estimate their probability distribution. The modified SI is extracted for all the aggregated rainfall series. For illustration, the modified SI for a representative grid located at 26°N, 93°E is compared with the conventional SI, as shown in Fig. S2 (see supplementary information).

b. Detection and estimation of nonstationary parameters

Once the marginal distribution of each series is found, the signature of nonstationarity is detected using the TSW approach, as mentioned in section 3b. Here, a 30-yr time sliding window is chosen to slice each series into 83 segments. The size of TSW (30 years) is chosen such that it is large enough for the goodness of fit for selected probability distribution (two-parameter gamma at each segment) (Vinnarasi and Dhanya 2019). Around 0.69% of segments are not considered because they failed to pass the KS test. The PTS showing significant trend is modeled as a linear function of time. We chose linear regression model as it adequately captured the nonstationarity present in the parameter time series without introducing too much complexity in the model and to avoid uncertainty due to the addition of parameters (Luke et al. 2017).

Initially, Bayesian inference utilizing DEMC Metropolis–Hastings algorithm (Cheng et al. 2014; Vrugt et al. 2009; see supplementary information for details about methodology) with five Markov chains is employed to estimate the nonstationary/stationary parameters. The resultant estimates from the Bayesian inference and linear fit estimates are provided in Table 2 for two representative grids. It can be observed that the median of the posterior distribution computed using Bayesian inference and the linear fit estimation are reasonably close (relative difference ≤ 10%). Therefore, linear fit estimation is applied to all the other grids, since Bayesian inference is computationally intensive and demands large storage space, as the total number of instances would be 144 (cases) × 3930 (grids).

Table 2.

Nonstationary parameter statistics for two chosen grids using Bayesian inference and linear fit estimate.

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Any change in the distribution parameter is expected to alter the distribution of the events as shown in Fig. S1; therefore, the changes in shape (α) and scale (β) parameters of the gamma distribution affect the mean (αβ) and variance of the distribution (αβ²). To illustrate this, the parameter series of shape (α) and scale (β) parameters, time sliding window mean, time sliding window variance, and the changepoints for a grid located at 21.5°N, 79.25°E, for the month of September, are shown in Figs. 2a and 2b, respectively. It reveals a decreasing trend in shape parameter and increasing trend in scale parameter before changepoint, which is reflected as no shift in the mean but an increase in variance (i.e., the spread of distribution). However, after the changepoint, both shape and scale parameters follow a decreasing trend that is reflected as a decreasing trend in both the mean and variance. This suggests a decreasing trend in expected annual precipitation with a shrinking uncertainty band (variance), which may result in an increase in dry events. The pattern of changes in mean and variance is also revealed in the scatterplot, in which the events are spread before the changepoint and converge afterward (Figs. 2c–h). Though the changepoint of shape parameter is different from that of the scale parameter (Fig. 2a), the model generated using distinct changepoints could not capture the trend and the spread of rainfall events (Fig. 2g). Hence different alternatives of changepoints were modeled and compared for the following cases: (i) changepoint of mean adopted for both scale and shape parameters (Fig. 2d), (ii) changepoint of variance adopted for both scale and shape parameters (Fig. 2f), (iii) changepoint of shape PTS is adopted for both scale and shape parameters (Fig. 2e), and (iv) changepoint of scale PTS is adopted for both scale and shape parameters (Fig. 2c). Finally, it was observed that the changepoint in mean covers a broader range and captures the divergence and convergence in the spread of data (Fig. 2d). Since the changepoint in scale and shape parameters are close to each other (Fig. 2a), the envelope for changepoints based on variance (Fig. 2f) and scale parameter (Fig. 2e) follows the trend of the mean (Fig. 2d). We have used the changepoint in the TSW mean (Fig. 2d) for deriving NJDI in this study as it also encodes information about short-term changes in TSW variance and shape parameters. The performance of considering the changepoint in the TSW mean is good in the case of gamma distribution because the mean of the gamma distribution comprises an equal contribution of scale and shape parameters. This may be different for different distributions.

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Illustration of the parameter series and the changepoints for a grid located at 21.5°N, 79.25°E for the month of September. (a) Variation of parameters α (blue line) and β (red line) of gamma distribution computed using time sliding window series and their changepoints (solid circles); (b) mean (blue line) and variance (red line) of the gamma distribution and their changepoints (solid circles). Nonstationary modeling of precipitation for the grid: (c) changepoint of α is considered for both α and β; (d) changepoint in the mean is considered as a changepoint for α and β; (e) changepoint of β is considered for both α and β; (f) changepoint in variance is considered for α and β; (g) the respective changepoints of α and β are considered; and (h) without changepoint. The light gray region, dark gray region, and black line represent the area between the 0.05 and 0.95 quantiles, the area between the 0.25 and 0.75 quantiles, and the 0.5 quantile, respectively, while the red dots indicate the observed values.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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Further, the analysis of spatial variations of the nonstationary parameters shows that only less than 10% grids show no long-term or short-term trend in the time series and more than 55% of grids have undergone significant short-term changes. Thus, more than half the grids (55%) show presence of changepoint (short-term change). The majority of grids show a negative long-term trend in the shape parameter and a positive long-term trend in the scale parameter. In the grids showing short-term changes, the majority of the grids reveal a positive trend in shape parameter before changepoint and negative trend after changepoint, while the scale parameter shows a contrary behavior. The trend in the mean depends on the relative rate of change of the scale and shape parameter, and not merely on the sign of change. As an illustration, the spatial variations of the distribution parameters for summer monsoon (June–September) at 1-month time scale are shown in Figs. S3–S6 of the supplementary information, which clearly reveal that majority of grids have undergone changes; especially the shape and scale parameters predominantly show negative and positive trends, respectively.

Further, to emphasize the effect of incorporating time-varying parameters on the distribution of the events, the marginal distribution for two grids located at 10.5°N, 78°E (southern India) and 26.5°N, 72.75°E (northwest) are compared in Fig. 3. Here the parameter series show negative/positive and positive/negative trends for shape/scale parameters, respectively, without any changepoint (Figs. 3a,b). The distributions are computed for the following cases: (i) constant parameter–stationary case (black dotted line), (ii) initial period (1925) of the time-varying parameter estimated using linear trend (black line), and (iii) latter period (1985) of the time-varying parameter estimated using linear trend (gray line). The PDF plots for the above cases are shown in Fig. 3c, which reveals an evident change in the mean and variance of the distribution, in response to the choice of the parameter. It is worthwhile to note that the probability of dry events has increased in the grid which shows negative and positive trends in shape and scale parameters, respectively (right panel), whereas an opposite behavior in the grid which shows positive and negative trends in shape and scale parameter, respectively (left panel), where the probability of the dry events has decreased. This behavior is overlooked in the case of a stationary approach. It can be inferred from Figs. S3 and S4 that dry events are increasing in southern India and the northwest region, especially in recent decades whereas a decrease in the dry events is expected in the northwest region.

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Nonstationarity in the rainfall series of September (1-month time scale) for locations (left) 10.5°N, 78°E and (right) 26.5°N, 72.75°E. (a) Parameter series of α and its trend, (b) parameter series of β and its trend, and (c) PDF of rainfall depth (inset of the right panel is the zoomed portion of PDF of rainfall depth from 0 to 2 mm). Black solid represents the PDF based on 1925 parameter values, gray represents the PDF based on 1985 parameter values, and black dotted represents the PDF based on stationary parameters.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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c. Nonstationary joint deficit index

The joint cumulative probability is computed using empirical copula by combining the marginal cumulative probability of all the time scales. The stationary drought indices, computed using a constant distribution parameter, are compared with the nonstationary drought indices. We initially compare the stationary (${\text{SI}}_s^{\text{mod}}$) and nonstationary (${\text{NSI}}_s^{\text{mod}}$) indices for a single time scale, say, 3 months. For instance, the classification of historical droughts by both ${\text{NSI}}_3^{\text{mod}}$ and ${\text{SI}}_3^{\text{mod}}$, for the month of September, for a grid located at 10.5°N, 78°E (southern India), is depicted in Fig. 4. The rainfall depth of 90–100 mm is classified as D2 (moderate drought) by the traditional approach, whereas in the case of the proposed nonstationary index, the classification for the same amount of rainfall depth is time dependent, i.e., varies from year to year, taking into account the nonstationarity in the precipitation series. The major differences between JDI and NJDI arise due to the use of a time-varying gamma distribution to model the rainfall time series. Similar observation was made by Shiau (2020) for SPI and NSPI that although both time-invariant and time-varying gamma distributions are region-specific the latter is a relative temporal reference that adapts to the trending rainfall series. A lower negative value of NJDI which uses time-varying gamma distribution represents only a more severe degree of dryness but not a greater rainfall deficiency compared to JDI. For example, ${\text{SI}}_3^{\text{mod}}$ identifies the rainfall depth of 93.1 mm in the year 1968 and 98.2 mm in the year 1994 as D2 (moderate drought), while ${\text{NSI}}_3^{\text{mod}}$ identifies them as D3 (severe drought) and D1 (mild drought), respectively. To have a better insight, the 30-yr moving average of the rainfall depth (magenta line) is also overlaid on the plot. For instance, the 30-yr moving-average rainfall for 1968 (1953–82) is 241.1 mm and for 1994 (1979–2008) it is 168.3 mm, respectively, and the rainfall deficit in 1968 (148 mm) is more than that of 1994 (70.1 mm). Therefore, the proposed index classifies 1968 and 1994 as D3 and D1, respectively, considering the nonstationarity in the historical rainfall records, while the traditional approach failed to capture this. Likewise, the deficits in rainfall depth in 1934 and 1952 are 171.9 and 186.9 mm, respectively, which are classified as D4 and D3 by the traditional method, though the deficit is higher in the latter year. Whereas the proposed method classifies both as D4. This further emphasizes the incapability of stationarity-based drought index in reflecting the inherent temporal dynamics in the rainfall series. Similar differences at 6-, 9-, and 12-month time scales are observed between ${\text{SI}}_s^{\text{mod}}$ and ${\text{NSI}}_s^{\text{mod}}$, as shown in Figs. S7–S9 of supplementary information.

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Historical drought classification of 3-month time-scale rainfall depth of September for the period 1902–2013 using the modified (a) NSI and (b) SI at grid location 10.5°N, 78°E.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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Further, it can be noted that ${\text{SI}}_3^{\text{mod}}$ identifies more droughts with higher severity in the latter part of the series and ignores dry events in the initial part of the series, which may be due to the increased number of drier days (lesser rainfall depth than the initial period) in the latter period. This indicates that the drought index based on stationary assumption is biased by the drier events and fails to capture the severity of droughts appropriately, since it uses absolute thresholds to classify the droughts. This is further illustrated by comparing ${\text{NSI}}_3^{\text{mod}}$ and ${\text{SI}}_3^{\text{mod}}$ for another grid located at 26.5°N, 72.75°E, where ${\text{SI}}_3^{\text{mod}}$ identifies fewer severe droughts in the latter period and many severe events in the initial period, as depicted in Fig. 5. This can again be attributed to the concentration of drier events in the initial part, which implies that the traditional method has overestimated the severity of drought in the earlier part and underestimated it in the latter part, whereas the proposed method classifies the drought with better accuracy and harmonious distribution. A similar pattern is observed at 6-, 9-, and 12-month time scales also, as shown in Figs. S10–S12 of supplementary information. This clearly suggests that the proposed nonstationarity-based drought index is capable of resolving the drawback of over/underestimation by the stationary methods by incorporating the dynamic behavior of the rainfall series into modeling.

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Historical drought classification of 3-month time-scale rainfall depth of September for the period 1902–2013 using the modified (a) NSI and (b) SI at grid location 26.5°N, 72.75°E.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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We now compare the stationary (JDI) and nonstationary (NJDI) indices which provide information about drought at considering time scales of 1, 2, …, 12 months. The inferences from comparison of JDI and NJDI reflect the overall drought status of a location (grid). Six historical drought years are chosen, viz. 1918, 1920, 1941, 1965, 1987, and 2002 to compare the spatial distribution of NJDI and JDI (Mishra et al. 2018, 2019; Mishra 2020; Pai et al. 2011). Figures 6a–c show the NJDI and JDI for the drought years 1918, 1920, and 1941, which occurred pre-1950. It can be seen overall that JDI underpredicted the drought in all the three years, except in the northwest region. The severe drought (D4) in the year 1918 due to the El Niño is identified in 6% of grids by NJDI and less than 6% by JDI, which too, are concentrated only in the northwest. Likewise, JDI underestimates the drought in the southern, central, and northeast India for the years 1918 and 1920. This may be due to the fact that drier days are concentrated in the northwest (arid region) than other regions pre-1950, as reported by Vinnarasi and Dhanya (2016). However, a contrary pattern is revealed post-1950, which is depicted in Figs. 6d–f, where JDI overpredicts the drought in southern and northeastern India, which is again due to an increase in the aggregation of drier days in the recent decades in these regions (Vinnarasi and Dhanya 2016). The possible cause for the recent increase in the drier days are due to the increase in diurnal temperature range which in turn influenced by both global (multidecadal oscillation, global warming) and local (urbanization, dewpoint temperature, etc.) factors (Ali and Mishra 2017; Mishra et al. 2022; Vinnarasi et al. 2017; Zhang et al. 2019).

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Spatial comparison of drought classification between NJDI and JDI for (a) September 1918, (b) September 1920, (c) September 1941, (d) September 1965, (e) September 1987, and (f) September 2002.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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As illustrated above for the month of September, similar observations are made for other months also. Figure 7 compares the classification of drought by JDI and NJDI for the years 1918 (early twentieth century) and 2002 (early twenty-first century) for the Indian summer monsoon months of June to October. The severity of 1918 drought is considered to be higher than the 2002 drought as the 1918 drought affected the country in an unforeseen manner (Shewale and Kumar 2005). Shewale and Kumar (2005) also noted that the Indian summer monsoon (June to September) rainfall was almost 25% lower than the long-term average annual rainfall of 880 mm in 1918 resulting in almost 70% of the area under moderate and severe droughts. The JDI and NJDI classify 54% and 60% of area under moderate to extreme drought in 1918 at end of Indian summer monsoon (October) while the 2002 drought covers 27% and 23% of country area as per JDI and NJDI, respectively (Fig. 7 and Fig. S13 in supplementary information). Thus, the drought severity-area estimates from JDI and NJDI are close to previously reported in the literature. The monthly resolution of JDI and NJDI allows use to compare the evolution of the drought as well. The 2002 drought recovered after the month of July (Fig. 7 and Fig. S13) i.e., area under moderate (D2) to extreme (D4) drought decreased from 49% in July to 23% in October (based on NJDI; see Fig. 7 and Fig. S13). The 1918 drought on the other hand maintained the severity from July (42% area under moderate to extreme drought) until the end of Indian summer monsoon in October (59%). NJDI is in closer proximity in estimating the severity of 1918 drought as described in the literature (Shewale and Kumar 2005) than JDI. In the case of 1918, JDI tends to overpredict the drought severity in the northwest and underpredict it in other regions for all the months compared to NJDI. The severity is represented by the classification of drought as D4 category especially in the month of July (Fig. 7c). Whereas, in the case of 2002, JDI tends to underpredict the drought severity in the northwest and overpredict it in other regions with respect to NJDI. These differences are clearly seen in the month of July, which can also be ascribed to the reasons mentioned earlier. Moreover, both JDI and NJDI identified October 1918 as the predominant drought, which also coincides with the major historical drought that occurred in 1918 (Mishra et al. 2019; Mishra 2020).

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Spatial comparison of drought classification between NJDI and JDI for (top two rows) 1918 and (bottom two rows) 2002 in (a) June, (b) July, (c) August, (d) September, and (e) October.

Citation: Journal of Climate 36, 12; 10.1175/JCLI-D-22-0437.1

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Overall, it can be inferred that stationarity-based indices (${\text{SI}}_s^{\text{mod}}$ or JDI) either underpredict or overpredict the severity of the drought compared to historical observations, and the predictions are biased toward aggregated drier events, failing to capture the temporal dynamics in the precipitation. The proposed nonstationary index (NJDI) can capture the time-varying nature of the meteorological drought and can be effectively employed for drought assessment under a changing climate. This study computes the dynamic meteorological drought characteristics using monthly accumulated rainfall data. However, the monthly variation in the meteorological variables cannot capture the intramonthly distribution which produces a skewed distribution of wet days. This may lead to misinterpretation of drought and pose a threat to agriculture. The proposed nonstationary drought index framework can be easily expanded for standardized net precipitation drought index proposed by Singh et al. (2021), which accounts for intramonthly distributions. The proposed methodology can also be employed for studying intramonthly droughts if submonthly precipitation information is available.