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Drought is a complex natural hazard. Its several adverse impacts are prevailing in almost all climatic zones around the world. In this regards, drought monitoring and forecasting play a vital role in making drought mitigation policies. Therefore, several drought monitoring tools based on probabilistic models had been developed for precise and accurate inferences of drought severity and its effects. However, risk of inaccurate determination of drought classes always exists in probabilistic models. To overcome this issue, we proposed a new system based Probabilistic Weighted Joint Aggregative Drought Index (PWJADI) criterion for three multi-scalar drought indices, namely Standardized Precipitation Index (SPI), Standardized Precipitation Temperature Index (SPTI), and Standardized Precipitation Evapotranspiration Index (SPEI) at one-month time scale. By the basic assumption of the Markov chain, the PWJADI is based on the temporal switched weights that are propagated from the transition probability matrix of each temporal classification of drought index. Application of the proposed method is made for three meteorological stations of Pakistan. We found that our proposed model has ability to restructure the drought classes by capturing and bending the information from the historical behaviour of each drought class. Consequently, to make accurate and precise drought mitigation policies, the proposed method may integrate into effective drought monitoring systems.

Drought is a complex natural hazard that impacts on human activities by devastating natural system, water supply, socioeconomic and ecosystems (Heim,

However, continuous and accurate drought monitoring is useful for making drought mitigation policies. From an operational point of view, drought characterization allows early warning drought risk analysis (Kogan,

A drought index based on a single climatic variable (Vicente-Serrano et al.,

Contrary to PDSI, McKee et al. (

The primary issue in making advance drought management and mitigation policies is the selection of appropriate drought index for accurate and precise drought monitoring and forecasting. However, the uncertainty in the estimation of drought classes always exists in probabilistic models of SPI, SPEI and SPTI. Further the selection of probability distribution for each indicator is purely subjective in nature. The choice of appropriate drought index depends on several factors such as availability of climatic data, type of drought and the periodic need of water in agricultural sectors (Wilhite and Buchanan-Smith,

On the same rationale of SPI, SPEI is one of the most popular drought indices due to its multivariate capability for drought assessment. This multivariate quality of SPEI makes it more acceptable and superior than SPI, as SPI uses only one climatic variable (i.e. precipitation only). Still, the primary issue in SPEI is the existence of undefined values in a temporal series of low temperature regions/month (Quiring,

Besides the use of influential climatic factors in the estimation procedures of drought indices (SPI, SPEI and SPTI), statistical procedure is vital because of reasons for accurate and the precise determination of drought classification. First, Blain (

In this paper, we aimed to develop a new criterion – the PWJADI to overcome the uncertainty in the accurate determination of drought classes. The PWJADI has capability to give joint decision on the classification of the region under study based on various drought indices. The developed model is on the drought classification of SPI, SPEI and SPTI at one month time scale. Description of the proposed method and its application are given in

In this section, we discussed the methodological structure of the proposed drought classification criterion termed as PWJADI.

McKee et al. (

Vicente-Serrano et al. (

However, empirical analysis of this research is based on Hargreaves equation (Hargreaves and Samani,

Ali et al. (

A stochastic process, or random process

Thus, in the current scenario, time series data on drought classes determined by SPI, SPEI and SPTI for a single station can be considered as a sequence of drought classes and formulated it in a discrete Markov model. Each Markov chain is characterized by a transition probability matrix that represents the probability of transition from one drought state to another drought state. By the assumption of first order Markov chain, this research assume that, given the present month drought classes, the future drought classes/class at particular month/station are conditionally independent (Sanusi et al.,

Let

Further, the transient behaviours of all drought classes are presented by the following transition probability matrix.

In this article, extremely wet (EW), very wet (VW), moderate wet (MW), normal drought (ND), moderate drought (MD), severe drought (SD), and extreme drought (SD) classes are considered as the states of the discrete stochastic process (i.e.

Drought classification criteria of SPI, SPEI and SPTI.

Here, we proposed a new criterion termed as; the PWJADI to utilise the varying methodologies of commonly used three drought indices (SPI, SPEI and SPTI). As these drought indices have uniform mathematical structure and classification states, therefore the use of these indices to report joint characterization is reasonable. Following three steps are involved in the proposed framework (see

Flow chart of the proposed framework.

The first step is straightforward. In this step historical quantitative data on SPI, SPEI and SPTI for single station is classified according to the basic drought classification criteria (McKee et al.,

In the second step, separate transition probability matrices are calculated from each temporal classified series. Here, following the work of Paulo et al. (

In the third step, a new criterion of decision aggregation is suggested in such a way that in each time step (i.e. each month), each drought class of SPI, SPEI and SPTI receives transient probabilities as a weight. Among the three drought indices, succeeding drought category which receives maximum weight from the list of switching probabilities is declared as a joint determined drought class. The basic idea of using transient probabilities is the assumption that next drought class is somehow having a great chance of dependence on the previous drought class.

Following section described the development procedure of the PWJADI.

Let _{1}, _{2,} …, _{n}_{1}, _{2}, …, _{n}_{1}, _{2}, …, _{n}

Hence, to account the effect of the transient behaviour of each drought class in each index, drought class receiving maximum value of switching probability is declared as a joint aggregative drought class. Here, switching probabilities are considered as a weight in the candidacy of respective drought class. Therefore, for our proposed framework, separate transition probability matrices are required to obtain the probability of moving from one drought class to another under the Markov chain framework. Following matrices show the mathematical structure for the selection of switching probabilities from one drought class to another drought class for each index.
_{ij}_{ij}_{ij}

In the next step, weights in terms of switching probability that belong to particular classes of particular index, are arranged in chronological order. These weights will help us to determine next month drought class by accounting the effect of historical transient behaviour of drought classes in each index. In

Those drought categories which have maximum switching probabilities from corresponding transition probability matrices of each drought index is considered as a joint aggregative drought class. However, this process will be repeated to all the three temporal vectors of drought classes and weights for the generalization purpose, and to generate historical time series of newly generated index.

Algorithm evaluation of temporal drought class using PWJADI criterion.

In _{c/c–1}_{c/c–1}_{c/c–1}_{ij}

To evaluate the framework of PWJADI for defining drought classes, we first estimated a time series data on SPI, SPEI and SPTI at one month time scale for three meteorological stations Astor (Latitude: 35.367, Longitude: 74.850, Elevation: 2600m) , Chilas (Latitude: 35.43, Longitude: 74.083, Elevation: 950m) and Islamabad (Latitude: 33.738, Longitude: 73.084, Elevation: 540m) located in different climatic regions of Pakistan.

Geographical locations of the selected stations.

Temporal behaviour of rainfall, minimum temperature, maximum temperature.

Climatology of the selected stations (Monthly data length: 1955–2017).

Note: Avg. (Average), Std. (Standard deviation), and Sk. (Skewness); Rainfall and temperature are measured in millimetre and centigrade, respectively.

In the estimation procedure, several probability distributions are fitted to check their appropriateness on the respective time series of each index. In current research, Kolmogorov–Smirnov, chi-squared and Anderson–Darling tests were used to check the goodness of fit at the most commonly used level of significance 0.05 by using Easyfit software (Schittkowski,

(a) Selected probability distribution for SPI, SPEI and SPTI in Astor. (b) selected probability distribution for SPI, SPEI and SPTI in Chillas. (c) selected probability distribution for SPI, SPEI and SPTI in Islamabad.

BIC of various distributions.

In the

Q–Q plots of Astor station. A- for precipitation, B- for deficient index (D), and C-for DAI index.

Pairwise correlation between quantitative values.

In our experimental results, we found a negative correlation (0–.07) between classified values of SPI and SPEI (see

Joint aggregation analysis of PWJADI criterion with SPI, SPEI and SPTI for Astor stations. The correlation among each categorical observed classes is present in top left panel. The scatter plot of historical accumulated drought classes are displayed in top right panel and bottom panel for SPI, SPEI and SPTI, respectively.

Further, in terms of accumulated drought classes in their corresponding historical series, each drought index has significantly different behaviour. Although there is significant correlation between SPTI and SPEI, however, a significant drop in the agreement between SPTI and SPI that was found when quantitative values were transformed into qualitative drought classes. To observe the disagreement among the temporal behaviour of drought classes, correlation among each indices is presented in

To accumulate the joint effect of all the drought indices by considering their memory effect, outcomes associated with the proposed criterion of new drought indices show positive correlation among SPI, SPEI and SPTI. Selection of drought classes, which have maximum transient probability among the available vector of three classes reduced the incorrect determination drought classes.

Further, to observe the in depth behaviour of PWJADI with other selected indices, scatter plots of qualitative drought classes are presented in

To assign weights, we prepared separate transition probability matrices for each index. In this research, the assumption of first order Markov chain are validated on small segments of the total of time data. We used chi-squire test by “

Temporal representation of switching weights for historical observed Drought classes of SPI, SPEI and SPTI at Astor station.

Bar plot of Drought Categories at Astor station: where 1, 2, 3, 4, 5, 6, and 7 represent ED, EW, MD, MW, ND, SD, and SW drought categories, respectively.

Histograms of SPI, SPEI, SPTI and PWJADI for Astor.

Transition probability matrix for Astor station.

Shapiro–Wilk normality test.

In this study, we introduced a new joint aggregative criterion for assessing accurate drought classes by using SPI, SPEI and SPTI drought indices. We found that aggregate decisions based on three drought indices (SPI, SPEI and SPTI) can be useful for accurate and precise drought monitoring. We concluded from the analysis of three meteorological stations as follows:

The choice of appropriate probability distribution for each drought indicator increase its efficiency for exact drought category.

Although there are positive correlations among each quantitative value of SPI, SPEI and SPTI, but it does not guarantee that in a particular month each drought indicator produces same drought class.

The transient memories as a weight help to reduce the error rate of inaccurate drought class.

Utilization of more than one drought index for drought monitoring, the proposed model can be considered for making reliable drought mitigation policies.

Further, the inferences and numerical computations can be generalized for other time scales and other drought indices such as RDI.

However, the limitation of the proposed methods is not to consider the nonstationary behaviour of Markov chain on whole data length. Moreover, in the computations, the study assumed each Markov chain as first-order Markov process.

Authors are very grateful to the Deanship of Scientific Research at King Khalid University, Kingdom of Saudi Arabia for their administrative and technical support and for funding this work through research groups program under the project number RGP-1/103/40.

No potential conflict of interest was reported by the authors. The manuscript is prepared by using secondary data and authors have not received any financial support. The authors of manuscript certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.

The manuscript is prepared in accordance with the ethical standards of the responsible committee on human experimentation and with the latest (2008) version of Helsinki Declaration of 1975.