A novel generalized combinative procedure for Multi-Scalar standardized drought Indices-The long average weighted joint aggregative criterion

Figures & data

Fig. 1. Flowchart of the proposed criterion.

Fig. 2. Flowchart of the LAWJADC equation.

Table 1. Drought classification criterion.

Range of Index Values	Class
$\mathit{SDI}\ge 2$	Extreme Wet (EW)
$1.5\le \mathit{SDI}<2$	Severe Wet (SW)
$1\le \mathit{SDI}<1.5$	Moderate Wet (MW)
$-1\le \mathit{SDI}<1$	Normal Dry (ND) (Near Normal)
$-1>\mathit{SDI}\ge -1.5$	Moderate Dry (MD)
$-1.5>\mathit{SDI}\ge -2$	Severe Dry (SD)
$\mathit{SDI}<-2$	Extreme Dry (ED)

Fig. 3. Geographical locations of selected meteorological stations of Pakistan.

Table 2. Climatology of the selected stations.

Station	Months	Precipitation		Temperature (max)		Temperature (min)
		Mean	Kurtosis	Mean	Kurtosis	Mean	Kurtosis
Astore	1	40.079	0.994	3.033	−0.616	−7.157	−0.165
	2	45.545	0.779	4.418	0.805	−5.185	0.610
	3	73.862	1.558	9.097	−0.344	−0.812	0.106
	4	79.423	0.641	15.472	0.202	4.083	−0.191
	5	65.636	0.667	20.207	−0.172	7.784	0.089
	6	24.8	−0.772	24.647	−0.274	11.419	1.348
	7	25.266	2.243	27.123	−0.625	14.505	0.287
	8	25.838	0.544	26.769	−0.449	14.497	0.204
	9	25.551	15.810	23.251	−0.951	10.216	−0.271
	10	20.923	2.655	17.834	0.776	4.632	0.809
	11	17.192	2.153	11.631	0.103	−0.283	−0.559
	12	27.681	8.709	5.734	−0.111	−4.397	−0.559
Bunji	1	6.233	4.481	10.041	0.300	0.136	−0.602
	2	9.592	8.260	12.776	−0.057	2.892	0.548
	3	14.102	7.409	18.474	−0.545	7.786	−0.674
	4	24.956	2.342	24.467	−0.199	12.269	−0.380
	5	27.679	1.936	28.697	0.006	15.316	−0.152
	6	10.6	0.926	33.32	−0.313	19.2	1.050
	7	17.875	3.600	35.386	0.366	22.603	−0.210
	8	22.434	3.233	34.91	−0.091	21.733	0.429
	9	14.575	5.695	31.455	−0.245	16.674	1.986
	10	7.436	24.935	25.678	−0.092	10.592	−0.094
	11	2.941	9.026	18.751	−0.711	4.696	2.095
	12	4.945	12.467	12.179	0.427	1.156	0.007
Gupis	1	7.403	2.344	4.681	−0.426	−5.639	−0.561
	2	12.136	21.059	7.021	0.577	−3.281	0.463
	3	15.86	1.759	12.625	−0.346	1.776	−0.712
	4	40.606	11.641	19.044	−0.521	7.104	−0.675
	5	26.961	0.921	23.787	−0.008	10.939	−0.180
	6	17.859	3.845	28.856	0.064	14.972	0.024
	7	16.091	4.370	31.705	−0.348	17.587	0.491
	8	24.767	9.035	30.546	−0.480	16.763	0.099
	9	13.698	13.381	26.521	−0.116	12.458	0.040
	10	8.051	2.392	20.555	−0.179	6.545	−0.840
	11	2.483	1.166	14.23	0.153	1.046	−0.911
	12	4.475	10.181	6.932	0.500	−3.454	−0.202

Table 3. BIC for Scale 1 SPI, SPEI and SPTI for Astore Bunji and Gupis.

	Astore			Bunji			Gupis
Distribution	SPI	SPEI	SPTI	SPI	SPEI	SPTI	SPI	SPEI	SPTI
2P Beta	−735.45	−565.00	−287.79	−722.44	−1019.68	−125.12	−399.44	−775.08	−254.43
3P Weibull	−1036.51	−700.54	−483.52	−1030.98	−1178.06	−188.38	−777.57	−910.74	−370.55
4P Beta	−1031.38	−700.28	−473.37	−1020.69	−1210.97	−181.74	−788.69	−823.15	−374.24
Arcsine	−853.92	−643.79	−274.88	−791.05	−1089.24	7.71	−577.40	−817.37	−151.87
Burr	−777.82	−656.97	−332.22	−753.84	−1152.29	158.50	−537.17	−905.52	−21.11
Cauchy	−906.22	−673.83	−357.62	−790.41	−1147.13	−2.03	−640.15	−884.69	−235.77
Chi	−772.18	−571.38	−253.75	−747.42	−1024.70	−89.65	−535.49	−779.77	−188.78
Chi-Square	−778.82	−570.07	−450.78	−771.56	−1024.59	−49.17	−549.29	−779.77	−146.95
Cosine	−827.71	−706.01	−182.80	−582.27	−1175.57	165.44	−409.73	−880.80	13.08
Curvilinear Trapezoidal	−770.35	−590.81	−369.28	−561.78	−1128.83	−40.20	−399.24	−809.67	−158.55
Exponential	−992.52	−569.99	−398.97	−727.50	−1024.70	30.67	−596.82	−779.77	−206.77
F-	−735.45	−565.75	−459.87	−863.89	−1019.68	2.54	−623.87	−775.23	−124.81
Gamma	−1024.97	−573.48	−482.01	−879.99	−1019.80	−116.91	−667.73	−775.23	−294.28
Generalized Extreme Value	−959.45	−700.89	−450.09	−886.80	−1216.78	−88.95	−744.99	−943.97	−338.99
Generalized normal	−974.59	−699.88	−472.53	−929.73	−1208.10	−124.81	−759.83	−938.39	−355.54
Gumbel	−942.14	−692.28	−382.34	−754.06	−1159.45	29.41	−607.56	−881.41	−205.28
Inverse Chi-Square	−847.29	−573.96	−260.44	−849.03	−1024.70	74.59	−615.75	−779.84	−126.69
Inverse Gamma	−894.39	−575.70	−393.13	−864.79	−1019.80	−75.36	−737.28	−775.30	−333.93
Inverse Gaussian	−848.87	−565.00	−355.22	−790.50	−1019.68	−12.65	−740.25	−775.08	−334.88
Johnson SB	−971.05	−640.08	−475.88	−925.95	−1248.44	−122.60	−457.52	−977.62	−351.09
Johnson SU	−969.88	−695.65	−467.89	−924.84	−1203.21	−120.21	−755.59	−936.90	−351.09
Laplace	−914.20	−683.10	−377.36	−756.94	−1150.62	10.91	−630.07	−886.00	−238.05
Logistic	−920.09	−697.50	−356.29	−752.10	−1166.09	37.57	−596.22	−895.15	−187.98
Log-normal	−970.31	−574.42	−474.91	−882.30	−1019.80	−105.32	−729.92	−775.30	−338.98
Normal	−917.65	−702.40	−348.30	−747.38	−1170.79	46.87	−585.58	−898.11	−172.80
Rayleigh	−934.02	−700.69	−365.81	−748.60	−1166.60	42.39	−591.38	−888.30	−182.47
Scaled/shifted t-	−915.53	−698.17	−359.42	−814.16	−1165.90	−11.77	−642.21	−893.57	−237.55
Skewed-normal	−951.45	−700.79	−378.12	−744.10	−1218.27	44.30	−590.96	−937.90	−185.69
Trapezoidal	−940.42	−710.05	−354.12	−712.78	−1224.00	80.19	−570.94	−946.94	−151.71
Triangular	−945.13	−685.96	−358.76	−664.59	−1228.23	76.39	−557.57	−947.69	−148.56
Uniform	−741.92	−664.32	−220.71	−622.63	−1084.39	121.33	−478.66	−884.52	−37.54
von Mises	−752.43	−568.01	−355.83	−752.74	−1019.80	46.82	−531.25	−775.31	−173.35

Fig. 4. Probability distribution at scale one of SPI, SPEI and SPTI at Astore, Bunji and Gupis.

Table 4. Characteristics of some important distributions.

Sr. No	Distributions	Probability function	CDF	Range
1	3P Weibull	$P(x)=\frac{\alpha }{\beta }{\left(\frac{x-\gamma }{\beta }\right)}^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[-{\left(\frac{x-\gamma }{\beta }\right)}^{\alpha }\right]$	F$(x)=1-\mathrm{}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[-{\left(\frac{x-\gamma }{\beta }\right)}^{\alpha }\right]$	$\gamma \le x<+\text{∞}$
2	Gamma	$f(x)=\frac{{\left(x\right)}^{\alpha -1}}{{\beta }^{\alpha }\Gamma (\alpha )}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{-\left(x\right)}{\beta }\right)$	$F(x)=\frac{\Gamma (\alpha ,\frac{x}{\beta })}{\Gamma (\alpha )}$	$x>0$
3	Normal	$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{-1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2\right]$	$F(x)=\frac{1}{2}\left[1+\text{erf}\hspace{0.33em}\left(\frac{x-\mu }{\sigma \sqrt{2}}\right)\right]$	$-\mathrm{\infty }<x<+\mathrm{\infty }$
4	Cosine	$f(x)=\frac{1}{2s}\left[1+\left.\text{cos}\left(\frac{x-\mu }{s}\pi \right)\right]\right.$	$F(x)=\frac{1}{2}\left[1+\left.\frac{x-\mu }{s}+\frac{1}{\pi }\text{sin}\left(\frac{x-\mu }{s}\pi \right)\right]\right.$	$\mathrm{\mu }-s\le x\le \mathrm{\mu }+s$
5	Skewed normal	$f(x)=\frac{2}{\omega \sqrt{2\pi }}\text{exp}\hspace{0.17em}\left(-\left(\frac{x-\mathrm{\xi }}{2{\omega }^2}\right)\right){\int }_{-\mathrm{\infty }}^{\alpha \left(\frac{x-\mathrm{\xi }}{\omega }\right)}\frac{1}{\sqrt{2\pi }}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\left(\frac{t^2}{2}\right)\right)dt$	F$(x)=\varphi \left(\frac{x-\mathrm{\xi }}{2\omega }\right)-2T\left(\frac{x-\mathrm{\xi }}{2\omega },\alpha \right)$ $T\left(h,\alpha \right)\mathrm{}is\mathrm{}\mathit{Owe}{n}^{\mathrm{\prime }}s\mathrm{}\mathit{Function}$	$-\mathrm{\infty }<x<+\mathrm{\infty }$
6	Trapezoidal	$f(x)=\left\{\begin{array}{l}\frac{2}{d+c-a-b}\frac{x-a}{b-a}\mathit{for}a\le x<b\\ \frac{2}{d+c-a-b}\mathit{for}b\le x<c\\ \frac{2}{d+c-a-b}\frac{d-x}{d-c}\mathit{for}c\le x\le d\end{array}\right\}$	$F\left(x\right)=\left\{\begin{array}{l}\frac{1}{d+c-a-b}\frac{1}{b-a}{\left(x-a\right)}^2\mathit{for}a\le x<b\\ \frac{1}{d+c-a-b}\left(2x-a-b\right)\mathit{for}b\le x<c\\ 1-\frac{1}{d+c-a-b}\frac{1}{d-c}{\left(d-x\right)}^2\mathit{for}c\le x\le d\end{array}\right\}$	$a\le x\le d$
7	Triangular	$f(x)=\left\{\begin{array}{l}\frac{2\left(x-a\right)}{\left(b-a\right)\left(c-a\right)}\mathit{for}a\le x<c\\ \frac{2}{b-a}\mathit{for}x=c\\ \frac{2\left(b-x\right)}{\left(b-a\right)\left(b-c\right)}\mathit{for}c<x\le b\end{array}\right\}$	F$(x)=\left(\begin{array}{l}\frac{(x-a)²}{\left(b-a\right)\left(c-a\right)}\mathit{for}a\le x<c\\ 1-\frac{(b-x)²}{\left(b-a\right)\left(b-c\right)}\mathit{for}x=c\\ 1\mathit{for}c<x\le b\end{array}\right)$	$a\le x\le b$
8	Johnson SU	$f\left(x\right)=\frac{\delta }{\lambda \sqrt{2\pi }\mathrm{}\sqrt{z^2+1}}\mathrm{e}\mathrm{xp}\left(-\frac{1}{2}{\left(\mathrm{\gamma }+\mathrm{\delta }\ln \left(z+\sqrt{z^2+1}\right)\right)}^2\right)$$z=\frac{x-\mathrm{\xi }}{\lambda }$	$F\left(x\right)=\varphi \left(\mathrm{\gamma }+\mathrm{\delta }\ln \left(z+\sqrt{z^2+1}\right)\right)$$\varphi \mathrm{}is\mathrm{}\mathit{Laplace}\mathrm{}\mathit{integral}$	$-\mathrm{\infty }<x<+\mathrm{\infty }$
9	Johnson SB	$f\left(x\right)=\frac{\delta }{\lambda \sqrt{2\pi }\mathrm{}z\left(1-z\right)}\mathrm{e}\mathrm{xp}\left(-\frac{1}{2}{\left(\mathrm{\gamma }+\mathrm{\delta }\ln \left(\frac{z}{1-z}\right)\right)}^2\right)$	$F\left(x\right)=\varphi \left(\mathrm{\gamma }+\mathrm{\delta }\ln \left(\frac{z}{1-z}\right)\right)$$\varphi \mathrm{}is\mathrm{}\mathit{Laplace}\mathrm{}\mathit{integral}$	$\mathrm{\xi }\le x\le \mathrm{\xi }+\mathrm{\lambda }$
10	Log normal	$f(x)=\frac{\hspace{0.17em}\text{exp}\hspace{0.17em}\left[\frac{-1}{2}{\left(\frac{In(x)-\mu }{\sigma }\right)}^2\right]}{\left(x\right)\sigma \sqrt{2\pi }}$	$f\left(x\right)=\frac{1}{2}+\frac{\mathit{erf}\left[\frac{In(x)-\mu }{\sqrt{2}\sigma }\right]}{2}$	$0<x<+\mathrm{\infty }$
11	Laplace	$f\left(x\right)=\frac{1}{2b}+\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{\left|x-\mu \right|}{b}\right)$	$F\left(x\right)=\left\{\begin{array}{l}\frac{1}{2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{x-u}{b}\right)\mathrm{}\mathit{for}\mathrm{}x\le u\mathrm{}\\ 1-\frac{1}{2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{x-u}{b}\right)\mathit{for}x\ge u\end{array}\right\}$	$x\in R$
12	Logistic	$f\left(x\right)=\frac{\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{x-\mu }{s}\right)}{s{\left(1+\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{x-\mu }{s}\right)\right)}^2}$	$F(x)=\frac{1}{1+\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{x-\mu }{s}\right)}$	$-\mathrm{\infty }<x<+\mathrm{\infty }$
13	Gumbel	$f\left(x\right)=\frac{1}{\beta }\hspace{0.17em}\text{exp}\hspace{0.17em}\left[-(z+\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-z\right))\right]$$z=\left(\frac{x-\mu }{\beta }\right)$	$\text{exp}\hspace{0.17em}\left(-\hspace{0.17em}\text{exp}\hspace{0.17em}\left(-\frac{x-\mu }{\beta }\right)\right)$	$x\in R$
14	Rayleigh	$f\left(x\right)=\frac{x}{{\sigma }^2}\mathrm{e}\mathrm{xp}\left(\frac{{-x}^2}{{(2\sigma }^2)}\right)$	$F\left(x\right)=1-\exp \left(\frac{{-x}^2}{\left({2\sigma }^2\right)}\right)$	$0<x<+\mathrm{\infty }$
15	4P Beta	$f(x)=\mathrm{}\frac{{(x-\mathit{min})}^{\alpha -1}{(\mathit{max}-x)}^{\beta -1}}{\mathit{B}\left(\alpha ,\mathrm{}\beta \right){(\mathit{max}-\mathit{min})}^{\alpha +\beta -1}}$	$F(x)=\mathrm{}\frac{\mathit{B}\left(\frac{x}{\alpha },\mathrm{}\beta \right)}{\mathit{B}\left(\alpha ,\mathrm{}\beta \right)}$	$\mathit{min}<x<\mathit{max}$

Table 5. BIC for all scale of SPI, SPEI and SPTI indicators at Astore, Bunji and Gupis stations.

Astor				Bunji		Gupis
Index	Scales	Distribution	BIC	Distribution	BIC	Distribution	BIC
For SPI	1	3P Weibull	−1036.51	3P Weibull	−1030.98	4P Beta	−788.69
	3	Gamma	−1279.05	Gamma	−824.87	Gamma	−1264.95
	6	Gamma	−892.67	Skewed-normal	−1162.15	Gumbel	−1305.35
	9	Gamma	−896.09	Normal	−649.05	Johnson SU	−1518.96
	12	Cosine	−913.33	Laplace	−688.10	Johnson SU	−937.61
	24	Skewed-normal	−1294.42	Laplace	−843.74	Scaled/Shifted t	−1407.98
For SPEI	1	Trapezoidal	−710.05	Johnson SB	−1248.44	Johnson SB	−977.62
	3	Trapezoidal	−941.49	Johnson SB	−1323.78	Johnson SB	−1098.61
	6	Trapezoidal	−1440.21	Johnson SB	−1094.44	Trapezoidal	−1482.59
	9	Triangular	−1405.81	Trapezoidal	−976.73	Trapezoidal	−1513.12
	12	Trapezoidal	−1052.48	Logistic	−1158.15	Laplace	−1063.66
	24	Johnson SU	−1471.48	Gumbel	−1508.11	Laplace	−1474.12
For SPTI	1	3P Weilbull	−483.52	3P Weibull	−188.38	4P Beta	−374.24
	3	Johnson SU	−542.87	Gumbel	−190.69	Chi-Square	−432.18
	6	Log-normal	−721.00	Skewed-normal	−300.34	Johnson SU	−410.04
	9	Gamma	−725.22	Rayleigh	−380.83	Johnson SU	−463.21
	12	Triangular	−702.87	Laplace	−411.86	Johnson SU	−466.57
	24	Laplace	−564.00	Trapezoidal	−304.73	Scaled/Shifted t	−629.46

Fig. 5. Graphical representation of TPM of SPI, SPEI and SPTI at scale 1 in Astore, Bunji and Gupis.

Table 6. Steady-states probabilities of various drought classes.

			ED	EW	MD	MW	ND	SD	SW	Sum
Astore		SPI	NA	0.02309	0.09254	0.09583	0.67654	0.06949	0.04252	1.0
	Scale-1	SPEI	NA	0.03013	0.16716	0.08661	0.66113	0.02135	0.03361	1.0
		SPTI	NA	0.01952	0.07995	0.11872	0.67887	0.05339	0.04954	1.0
		SPI	NA	0.02674	0.09626	0.07487	0.76292	NA	0.03922	1.0
	Scale-3	SPEI	NA	0.01756	0.16475	0.11997	0.63395	0.03761	0.02616	1.0
		SPTI	NA	0.01575	0.13215	0.10907	0.65843	0.01267	0.07193	1.0
		SPI	0.01905	0.02137	0.10311	0.10330	0.66614	0.04428	0.04275	1.0
	Scale-6	SPEI	NA	0.02151	0.15233	0.11470	0.63620	0.03226	0.04301	1.0
		SPTI	NA	0.01603	0.13401	0.09798	0.65198	0.04300	0.05700	1.0
		SPI	NA	0.07027	0.02162	0.08468	0.77117	NA	0.05225	1.0
	Scale-9	SPEI	0.04573	0.00178	0.08788	0.15350	0.62473	0.05424	0.03213	1.0
		SPTI	0.00727	0.01791	0.10650	0.08598	0.67892	0.04968	0.05374	1.0
		SPI	0.01131	0.01974	0.10654	0.11128	0.66767	0.05654	0.02692	1.0
	Scale-12	SPEI	0.02434	0.02316	0.09480	0.09800	0.68597	0.05771	0.01604	1.0
		SPTI	NA	0.02136	0.13068	0.09610	0.64246	0.06491	0.04449	1.0
		SPI	0.01111	0.03704	0.07037	0.05741	0.73889	0.07593	0.00926	1.0
	Scale-24	SPEI	0.01667	NA	0.12778	0.10556	0.66111	0.06481	0.02407	1.0
		SPTI	NA	NA	0.12407	0.11111	0.64074	0.07407	0.05000	1.0
		SPI	NA	0.02131	0.28419	0.07460	0.57549	NA	0.04441	1.0
	Scale-1	SPEI	NA	0.00534	0.16201	0.08852	0.68365	0.04629	0.01419	1.0
		SPTI	NA	0.01243	0.28419	0.09414	0.55773	NA	0.05151	1.0
		SPI	0.01248	0.01070	0.05526	0.07665	0.77897	0.01604	0.04991	1.0
	Scale-3	SPEI	NA	0.00713	0.16934	0.14617	0.63815	0.03209	0.00713	1.0
		SPTI	NA	0.02674	0.09626	0.07487	0.76292	NA	0.03922	1.0
		SPI	NA	0.03943	0.09140	0.05376	0.77240	NA	0.04301	1.0
	Scale-6	SPEI	NA	0.00713	0.16934	0.14617	0.63815	0.03209	0.00713	1.0
		SPTI	NA	0.02674	0.09626	0.07487	0.76292	NA	0.03922	1.0
Gupis		SPI	NA	0.06126	0.04144	0.08288	0.74955	NA	0.06486	1.0
	Scale-9	SPEI	0.01622	0.01622	0.11892	0.08649	0.68829	0.04505	0.02883	1.0
		SPTI	NA	0.07027	0.02162	0.08468	0.77117	NA	0.05225	1.0
		SPI	NA	0.06703	0.04891	0.07246	0.74457	NA	0.06703	1.0
	Scale-12	SPEI	NA	0.03623	0.10326	0.10507	0.69565	0.00543	0.05435	1.0
		SPTI	0.03986	0.00362	0.07428	0.10870	0.69746	0.01812	0.05797	1.0
		SPI	0.03704	NA	0.05741	0.13148	0.65926	0.04444	0.07037	1.0
	Scale-24	SPEI	NA	0.02963	0.10926	0.11111	0.67037	NA	0.07963	1.0
		SPTI	NA	0.04074	0.05741	0.08333	0.73704	NA	0.08148	1.0
		SPI	NA	0.01422	0.20121	0.08695	0.64801	NA	0.04961	1.0
	Scale-1	SPEI	NA	0.00706	0.18527	0.17128	0.59370	0.02673	0.01597	1.0
		SPTI	NA	0.01424	0.20472	0.07275	0.66037	NA	0.04793	1.0
		SPI	NA	0.02674	0.09626	0.07487	0.76292	NA	0.03922	1.0
	Scale-3	SPEI	NA	0.01756	0.16475	0.11997	0.63395	0.03761	0.02616	1.0
		SPTI	NA	0.01575	0.13215	0.10907	0.65843	0.01267	0.07193	1.0
		SPI	0.01905	0.02137	0.10311	0.10330	0.66614	0.04428	0.04275	1.0
	Scale-6	SPEI	NA	0.02151	0.15233	0.11470	0.63620	0.03226	0.04301	1.0
		SPTI	NA	0.01603	0.13401	0.09798	0.65198	0.04300	0.05700	1.0
Bunji		SPI	NA	0.07027	0.02162	0.08468	0.77117	NA	0.05225	1.0
	Scale-9	SPEI	0.04573	0.00178	0.08788	0.15350	0.62473	0.05424	0.03213	1.0
		SPTI	0.00727	0.01791	0.10650	0.08598	0.67892	0.04968	0.05374	1.0
		SPI	0.01131	0.01974	0.10654	0.11128	0.66767	0.05654	0.02692	1.0
	Scale-12	SPEI	0.02434	0.02316	0.09480	0.09800	0.68597	0.05771	0.01604	1.0
		SPTI	NA	0.02136	0.13068	0.09610	0.64246	0.06491	0.04449	1.0
		SPI	NA	NA	0.14815	0.12778	0.61111	0.04815	0.06481	1.0
	Scale-24	SPEI	0.05185	NA	0.06111	0.18704	0.64630	0.04815	0.00556	0.9
		SPTI	NA	NA	0.14444	0.13519	0.61111	0.02778	0.08148	1.0

Fig. 6. Temporal plot of SPI, SPEI, SPTI, PWJADI and LAWJADC for Astore.

Fig. 7. Temporal plot of SPI, SPEI, SPTI, PWJADI and LAWJADC for Bunji.

Fig. 8. Temporal plot of SPI, SPEI, SPTI, PWJADI and LAWJADC for Gupis.

Fig. 9. Bar plot counts of drought categories observed in SPI, SPEI, SPTI, PWJADI and LAWJADC for Astore.

Fig. 10. Bar plot counts of drought categories observed in SPI, SPEI, SPTI, PWJADI and LAWJADC for Bunji.

Fig. 11. Bar plot counts of drought categories observed in SPI, SPEI, SPTI, PWJADI and LAWJADC for Gupis.

Table 7. Kappa values showing association among various drought indices and criterion.

			Kappa					P-value
Station	Scale	Method	SPI	SPEI	SPTI	PWJADI	LAWJADC	SPI	SPEI	SPTI	PWJADI	LAWJADC
	Scale-1	LAWJADC	0.2850	0.0176	0.2220	0.8250	1.0000	0.0000	0.6030	0.0000	0.0000	0.0000
Astore		PWJADI	0.2970	0.0970	0.2430	1.0000	0.8250	0.0000	0.0048	0.0000	0.0000	0.0000
	Scale-3	LAWJADC	0.2650	0.1060	0.1090	0.4800	1.0000	0.0000	0.0002	0.0002	0.0000	0.0000
		PWJADI	0.4840	0.0381	0.1630	1.0000	0.4800	0.0000	0.2920	0.0000	0.0000	0.0000
	Scale-6	LAWJADC	0.3900	0.1730	0.2790	0.5560	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		PWJADI	0.3210	0.4060	0.3870	1.0000	0.5560	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-9	LAWJADC	0.2180	0.0263	0.0615	0.6320	1.0000	0.0000	0.2580	0.0216	0.0000	0.0000
		PWJADI	0.4320	0.0189	0.1590	1.0000	0.6320	0.0000	0.5570	0.0000	0.0000	0.0000
	Scale-12	LAWJADC	0.1850	0.0029	0.4960	0.3920	1.0000	0.0000	0.9400	0.0000	0.0000	0.0000
		PWJADI	0.2960	0.3630	0.5310	1.0000	0.3920	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-24	LAWJADC	0.1730	0.3250	0.3140	0.0683	1.0000	0.0000	0.0000	0.0000	0.0880	0.0000
		PWJADI	0.4500	0.0638	0.0515	1.0000	−0.0683	0.0000	0.0717	0.2100	0.0000	0.0880
	Scale-1	LAWJADC	0.3430	0.1190	0.2850	0.9040	1.0000	0.0000	0.0002	0.0000	0.0000	0.0000
Bunji		PWJADI	0.2980	0.1710	0.2990	1.0000	0.9040	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-3	LAWJADC	0.1160	0.0689	0.2330	0.6160	1.0000	0.0000	0.0047	0.0000	0.0000	0.0000
		PWJADI	0.1530	0.1530	0.3640	1.0000	0.6160	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-6	LAWJADC	0.0727	0.1300	0.1840	0.1730	1.0000	0.0064	0.0000	0.0000	0.0000	0.0000
		PWJADI	0.5990	0.0363	0.6270	1.0000	0.1730	0.0000	0.3820	0.0000	0.0000	0.0000
	Scale-9	LAWJADC	0.2240	0.2060	0.2670	0.5780	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		PWJADI	0.2930	0.2540	0.2950	1.0000	0.5780	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-12	LAWJADC	0.4890	0.5080	0.5670	0.6750	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		PWJADI	0.5380	0.5680	0.7560	1.0000	0.6750	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-24	LAWJADC	0.2200	0.6010	0.2770	0.7300	1.0000	0.0000	0.0000	0.0000	0.0000	0.0000
		PWJADI	0.3920	0.5520	0.4070	1.0000	0.7300	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-1	LAWJADC	0.2060	0.1280	0.2250	0.9120	1.0000	0.0000	0.0004	0.0000	0.0000	0.0000
Gupis		PWJADI	0.2120	0.1830	0.2450	1.0000	0.9120	0.0000	0.0000	0.0000	0.0000	0.0000
	Scale-3	LAWJADC	0.1420	−0.0109	0.2040	0.6590	1.0000	0.0000	0.6680	0.0000	0.0000	0.0000
		PWJADI	0.2000	0.0145	0.4010	1.0000	0.6590	0.0000	0.6530	0.0000	0.0000	0.0000
	Scale-6	LAWJADC	0.3340	0.0581	0.2880	0.6560	1.0000	0.0000	0.0478	0.0000	0.0000	0.0000
		PWJADI	0.5500	0.0739	0.0739	1.0000	0.6560	0.0000	0.0461	0.0461	0.0000	0.0000
	Scale-9	LAWJADC	0.4020	0.1390	0.4960	0.5850	1.0000	0.0000	0.0001	0.0000	0.0000	0.0000
		PWJADI	0.6030	0.1490	0.7340	1.0000	0.5850	0.0000	0.0002	0.0000	0.0000	0.0000
	Scale-12	LAWJADC	0.7520	0.3060	0.0715	0.8780	1.0000	0.0000	0.0000	0.0782	0.0000	0.0000
		PWJADI	0.7100	0.3790	0.0264	1.0000	0.8780	0.0000	0.0000	0.5200	0.0000	0.0000
	Scale-24	LAWJADC	0.5860	0.1750	0.0809	0.8770	1.0000	0.0000	0.0000	0.0400	0.0000	0.0000
	Scale-24	PWJADI	0.5430	0.1930	0.0658	1.0000	0.8770	0.0000	0.0000	0.1140	0.0000	0.0000