The Weihe Riveristhe largest tributary of the Yellow River and thebasincovers a drainage area of 0.135× 10⁶km². The river originates from the Niaoshu Mountains, goes through the provinces of Gansu, Ningxia and Shaanxi, and runs into the Yellow River at the TongguanGorge(Fig. 1). The basin extends from 33° 50′ N to 37° 18′ N and from 104° 00′ Eto 110° 20′ E. The elevation within the basin ranges from 304 to 3836 ma.s.l. with a west-east decline. The annualmeantemperature ranges between 3.7° C and 13.9° C, and the mean annual precipitation ranges between 400 and 550mm. The precipitation within the basin is spatially uneven and exhibits a declining trend from southeast to northwest with most of the annual precipitation occurring between July and October. The land uses within the basin include farmlands (46%), pastures (37%), and forests (16%).

Fig. 1

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	Fig. 1 Location of the Weihe River Basin and the distribution of rain gauge stations.Xiji, Changwu and Huashan are three selected stations representing the upper, middle, and lowerreachesofthebasin, respectively.

2.2.1 Satellite precipitation data

The TMPA 3B42 post-real-timeestimatesused in this study were produced in four consecutive steps: (1) the precipitation estimates derived from available Low Earth Orbit (LEO) microwave (MW) radiometers were calibrated using the TRMM combined instrument (TCI) dataset; (2) all of the estimates at 3 h intervals were then merged; (3) the gaps in the analyses were filled using Geosynchronous Earth Orbit (GEO) infrared data that were regionally calibrated to the merged MW product; and (4)a month-to-month bias-adjustment with the global precipitation climatology project (GPCP)data was employed (Huffman et al., 2007). During recent years, the TMPA algorithm has undergone some upgrades to cope with new sensor-resulted data expansion (Yong et al., 2012). The latest upgrade includes the release of the Version-7 of the TMPA satellite precipitation products. It should be mentioned that there are four major upgrades of the TMPA Version-7 over original Version-6. First, the special sensor microwave imager sounder (SSMIS) data on the defence meteorological satellite program (DMSP) F-16 satellites were incorporated into TMPA data to improve the accuracy of microwave precipitation estimates. Second, the TMPA Version-7 used the new global precipitation climatology centre (GPCC) “ full” gauge analysis. Third, the non-uniformlyprocessed advanced microwave sounding unit (AMSU) record in Version-6 was uniformly processed in Version-7. Fourth, the TRMM microwave imager (TMI) and the special sensor microwave imager (SSMI) input data were upgraded to the GPROF2010 algorithm (Huffman and Bolvin, 2013). In this study, we adopted the latest post-real-time research TMPA 3B42V7 satellite precipitation products. The spatial resolution of the TMPA 3B42V7 used in this study is 0.25° and the temporal resolution is 3 h. The 3-h satellite precipitation products were accumulated to form the monthly precipitation products for drought monitoring.

2.2.2 Gauged precipitation data

We collectedthe precipitation data from 15 meteorological stations for the period of 1961-2013 from the Climatic Data Center, National Meteorological Information Center, China Meteorological Administration, including Xi’ an, Tianshui, and Pingliang, which are the stations included in the GPCC gauge analysis. So we only adopted the data from the rest 12 meteorological stations (which are completely independent from those used inthe TMPA post-real-time retrieval system) as the benchmark data. These data have been widely used for different studies previously and the quality has been approved to be reliable (Zuo et al., 2014).In addition, three stations (i.e., Xiji, Changwu and Huashan) were selected for graphicpresentationsof the precipitation in the upper, middle and lower reaches of the basin (Fig. 1), respectively. The inverse distance weighting (IDW) method was first used to interpolate the rain-gauge precipitation onto 0.0625-deggrids and the 0.0625-deg data were then aggregated to 0.25-deg grids to match the spatial resolution of TMPA data (Bartier and Keller, 1996). The topographic data were obtained from the U.S. Geological Survey (USGS) 30 arc-second global digital elevation model (GTOPO30) (https://lta.cr.usgs.gov/GTOPO30).

The SPI was first proposed by McKee et al. (1993) to estimate the drought conditions in Colorado. The SPI value for any location can be calculated based on the long-term precipitation record for a designatedtimeperiod. The proposal was based on an assumption that theprecipitation series fit a gamma distribution pattern. The probability density function of precipitation can be expressed as follows:

$g(x)=\frac{1}{{{\beta }^{\alpha }}\Gamma (\alpha )}{{x}^{\alpha -1}}{{e}^{{-x}/{\beta }\; }}, $x> 0.(1)

Wherexdenotes precipitation value; Γ (α ) is gamma function; α and β are shape parameter and scale parameter, respectively.

Then, the cumulative probability of precipitation for the given month and time scale for the considered station is computed as:

$G(x)=\int_{0}^{x}{g(x)dx}=\frac{1}{{{\beta }^{\alpha }}\Gamma (\alpha )}\int_{0}^{x}{{{x}^{\alpha -1}}}{{e}^{{-x}/{\beta }\; }}dx$.(2)

For locations where observations of zero precipitation occur, the fitting of a gamma distribution becomes problematic since it is not defined for zero. In this case, the cumulative probability H(x) becomes:

$H(x)=q+(1-q)G(x)$.(3)

Where, q is the probability of zero precipitation calculated from the frequency of observations of zero.

When $0< H(x)\le 0.5$,

$SPI=-\left( t-\frac{{{c}_{0}}+{{c}_{1}}t+{{c}_{2}}{{t}^{2}}}{1+{{d}_{1}}t+{{d}_{2}}{{t}^{2}}+{{d}_{3}}{{t}^{3}}} \right)$, (4)

$t=\sqrt{\ln \left( \frac{1}{{{\left( H(x) \right)}^{2}}} \right)}$.(5)

When $0.5< H(x)\le 1$,

$SPI=t-\frac{{{c}_{0}}+{{c}_{1}}t+{{c}_{2}}{{t}^{2}}}{1+{{d}_{1}}t+{{d}_{2}}{{t}^{2}}+{{d}_{3}}{{t}^{3}}}$, (6)

$t=\sqrt{\ln \left( \frac{1}{{{\left( 1-H(x) \right)}^{2}}} \right)}$.(7)

Where c₀, c₁, c₂, d₁, d₂ and d₃are calculating parameters, and their values are:c₀=2.515517, c₁=0.802853, c₂=0.010328, d₁=1.432788, d₂=0.189269, and d₃=0.001308; t is the function of H(x).

The fundamental strength of SPI is that it can be calculated for a variety of time scales (i.e., 1-, 3-, 6-, 9- and 12-month). This versatility allows SPI to monitor short-term water supplies, such as seasonal soil moisture availability; and also to monitor long-term water resources, such as groundwater supplies, streamflow amounts, and lake levels (McKee et al., 1993; Mishra and Singh, 2010). Since its initiation, the SPI has been successfully and widely applied in drought monitoring around the world (Hayes et al., 1999; Zhang et al., 2009, Li et al., 2013a). SPI index was divided into 7 classes to indicateeitherdryconditions or wet conditions (Table 1).

Table 1

Table 1

Table 1 Drought and wet classification based on the SPI (Standardized Precipitation Index) index values SPI Classes for dryness SPI Classes for wetness 0.00--0.99 Normal (N) 0.00-0.99 Normal (N) -1.00--1.49 Moderately dry (MD) 1.00-1.49 Moderately wet (MW) -1.50--1.99 Severely dry (SD) 1.50-1.99 Severely wet (SW) ≤ -2.00 Extremely dry (ED) ≥ 2.00 Extremely wet (EW)

Table 1 Drought and wet classification based on the SPI (Standardized Precipitation Index) index values

Mean errors (ME), relative bias (BIAS), relative absolute bias (ABIAS), and correlation coefficient (CC) were employed to compare the monthly TMPA satellite precipitation data with rain gauge observations for evaluating the acceptability of the TMPA data in monitoring drought conditions. The ME means the average difference between the satellite-based precipitation estimates and rain gauge observations. The BIAS describes the systematic bias of the satellite-based precipitation estimates, whereas the ABIAS represents the absolute bias of the satellite-based precipitationestimates.The CC indicates the agreement between the satellite-based precipitation estimates and rain gauge observations. The formulas for ME, BIAS, ABIASand CC are given by following equations.

$ME=\frac{1}{n}\sum\limits_{i=1}^{n}{({{S}_{i}}-{{G}_{i}})}$.(8)

$BIAS=\frac{\sum^{n}_{i=1}(S_i-G_1)}{\sum^{n}_{i=1}(G_i)}\times 100%$(9)

$ABIAS=\frac{\sum\limits_{i=1}^{n}{\left| {{S}_{i}}-{{G}_{i}} \right|}}{\sum\limits_{i=1}^{n}{{{G}_{i}}}}\times 100%$. (10)

$CC=\frac{\sum\limits_{i=1}^{n}{({{G}_{i}}-\overline{G})({{S}_{i}}-\overline{S})}}{\sqrt{\sum\limits_{i=1}^{n}{{{({{G}_{i}}-\overline{G})}^{2}}\sqrt{\sum\limits_{i=1}^{n}{{{({{S}_{i}}-\overline{S})}^{2}}}}}}}$. (11)

Where n is the total sets of rain gauge data or satellite precipitation data; S_iandG_i are the i^thvalues of the satellite-based precipitation data and rain gauge observatio.ns, respectively; and $\overline{S}$.and $\overline{G}$ are the mean values of the satellite-based precipitation data and rain gauge observatio.ns, respectively.

In addition, the validation statistical indices of CC and the categorical statistical indices of critical success index (CSI) were adopted to measure the correspondence between the satellite-based drought estimates and rain gauge-based drought estimates. The CC shows the agreement between the satellite-based SPI and rain gauge-based SPI. The CSI represents the proportion of the times of drought events (as well as flood events) that were correctly diagnosed by the TMPA satellite-based precipitation data (Zeng et al., 2012). The drought events (as well as flood events) are defined in the Table 1. The perfect value of CSI is1.0. The formula of CSI is given by the following equation:

$CSI=\frac{{{t}_{H}}}{{{t}_{H}}+{{t}_{M}}+{{t}_{F}}}$. (12)

Where H, M, and F are different cases; H, observed drought events that were also correctly diagnosed by satellite-based precipitation data; M, observed drought events that were not diagnosed; F, drought events that were diagnosed but not observed; and t_H, t_Mandt_F are the times of occurrence of the corresponding case (see details in Ebert et al. (2007)).

The statistical indices (i.e. ME, BIAS, ABIAS and CC) were first calculated to compare the TMPA 3B42V7 precipitation data with rain gauge observations at both station and basin scales. Table 2 includes the statistical indices of TMPA 3B42V7 and rain gauge precipitation data at monthly time scale. Table 2 shows that the ME ranges from-4.39to 4.95 mm; the BIAS ranges from-7.09% to 11.72%; the ABIAS ranges from20.09%to 32.96%; and the CC ranges from0.90to 0.95. It can be seen thatthe differences between TMPA 3B42V7 data and rain gauge observations are generally small and TMPA 3B42V7 data are thus acceptable for drought monitoring. But, the ABIAS is still too large and the distribution of the TMPA 3B42V7 at monthly or sub-monthly time scale thus still needs to be improved. Generally speaking, the TMPA 3B42V7 slightly overestimated the precipitation with exceptions at following four stations: Baoji, Wugong, Luochuan, and Huashan. Figure 2 is the scatter plots of three representative stations (i.e., Xiji, Changwu and Huashan) and the basin average at monthly time scale. It can be seen that linear relationships between the TMPA data and rain gauge observations are rather robust and the CC is 0.91 (R²=0.82) at Xiji, 0.92 (R²=0.85) at Changwu, and 0.95 (R²=0.91) at Huashan. It is quite notable that CC is as high as 0.98 (R²=0.97) at basin scale (i.e., basin average), demonstrating that the TMPA 3B42V7 precipitation data are acceptable formonitoring drought conditions in the Weihe River basin.

Table 2

Table 2

Table 2 Comparison of statistical indices between TMPA 3B42V7 estimates and rain gauge observations at monthly time scale Station ME (mm) BIAS (%) ABIAS (%) CC Station ME (mm) BIAS (%) ABIAS (%) CC Huajialing 1.06 2.73 22.20 0.95 Wugong -1.41 -2.81 27.95 0.90 Xiji 3.72 11.72 32.96 0.91 Wuqi 3.30 8.51 27.64 0.94 Guyuan 3.77 10.52 31.65 0.91 Tongchuan 0.58 1.13 26.65 0.91 Baoji -1.24 -2.29 25.70 0.91 Luochuan -0.41 -0.79 26.26 0.93 Huanxian 4.11 11.67 29.99 0.93 Huashan -4.39 -7.09 20.09 0.95 Xifengzhen 4.95 10.88 24.37 0.95 Basin average 2.40 5.40 11.44 0.98 Changwu 4.70 9.64 26.53 0.92 Note: ME, mean errors; BIAS, relative bias; ABIAS, relative absolute bias; CC, correlation coefficient.

Table 2 Comparison of statistical indices between TMPA 3B42V7 estimates and rain gauge observations at monthly time scale

Fig. 2

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	Fig. 2 Precipitation scatter plots of three representative stations (a-c) and the basin average (d). TMPA, TMPA 3B42V7 precipitation estimates at monthly time scale. Gaugerepresentsrain gauge observations at monthly time scale.

Figure 3a shows the spatially-distributed patterns of the rain-gauged mean annual precipitation and Figure 3bshows the spatial distribution of TMPA-estimated precipitation. The spatial similarity between these two patterns (Figs. 3a and b) is rather high with more details expressed in Figure 3b (i.e., TMPA-estimated precipitation). We speculate that Figure 3b is a more realisticspatial expression of the annual precipitation than Figure 3a. Our speculation is based on following three facts. First, Figure 3a was interpolated based on a limited number of rain gauge observations. Second, Figure 3b was estimated based on sufficient data points (TMPA 3B42V7 cells). Third, the average CC between TMPA 3B42V7 data and rain gauge observations at those 12 participating stations was as high as 0.98.

Fig. 3

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	Fig. 3 Spatial distributions of mean annual precipitation based on rain gauge observations (a) and TMPA 3B42V7 estimates from 1998 to 2013(b)

The TMPA 3B42V7 data became available only from January 1998 onward and its limited time lengthis a major restriction for fitting a long-term probability distribution function (Naumann et al., 2012; Li et al., 2013b). Thus, we investigated the robustness of the short-term data (1998-2013)forfitting a long-term probability distribution function. We firstly analyzed the coincidence of several statistical indices, such as monthly mean (Mean), maximal (Max), minimal (Min) and standard deviation (Std), between a long-term (1961-2013) and a short-term (1998-2013) rain-gauge observations at three representative rain-gauge stations (Xiji, Changwu and Huashan)andalso at basin scale (i.e., basin average). Table 3 shows that the long-term Meanwas 34.05 mm at Xiji, 48.62 mmatChangwu, 67.60 mm atHuashan, and 45.16 mm at basin scale. The short-term Meanwas 31.73 mm at Xiji, 48.77 mmatChangwu, 61.93 mm atHuashan, and 44.38mm at basin scale. It means that the short-term monthly means are all not significantly different from the long-term monthly means. It should be stressed that other statistical indices (e.g., Max, Min andStd) are also similar between the long-term (1961-2013) and the short-term (1998-2013) data series. Table 3 also shows that the short-term (1998-2013) rain-gauged monthly means at the three representative stations and also at the basin scale (i.e., basin average) are all notsignificantly different from the short-term TMPA-inferred monthly means. And, other statistical indices (e.g., Max, Min, andStd) are also quite similar between the rain-gauged monthly means and the TMPA-inferred monthly means for the data overlapping period (1998-2013). Figure 4 is the comparisons of monthly mean precipitation data series between the short-term (1998-2013) and the long-term (1961-2013) and also between rain-gauge observations and TMPA 3B42V7 estimates. Figure 4 shows that the short-term TMPA 3B42V7 estimates are generally similar toboth the short-term and the long-term rain-gauge observations at the three representative stations. In comparison, the TMPA 3B42V7 estimates are almost identical with both the short-term and the long-term rain-gauge observations. Our results are quite consistent with those of Li et al. (2013b). The aforementioned similarities suggest that a short-term data series can be used for fitting a long-term probability distribution function.

Fig. 4

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	Fig. 4 Comparisons of monthly mean precipitation between the short-term (1998-2013) and the long-term (1961-2013) data series and also between rain gauge observations and TMPA 3B42V7 estimates

Table 3

Table 3

Table 3 Comparison of statistical indices between short-term (1998-2013) and long-term (1961-2013) data series Station Dataset Mean(mm/m) Max(mm/m) Min(mm/m) Std(mm/m) Xiji Gauge 1961-2013 34.05 226.40 0.00 39.15 Gauge 1998-2013 31.73 200.20 0.00 35.24 TMPA 1998-2013 35.45 212.17 0.00 37.31 Changwu Gauge 1961-2013 48.62 312.00 0.00 51.37 Gauge 1998-2013 48.77 312.00 0.00 53.32 TMPA 1998-2013 53.47 276.87 0.00 57.25 Huashan Gauge 1961-2013 67.60 328.70 0.00 61.92 Gauge 1998-2013 61.93 328.70 0.00 61.11 TMPA 1998-2013 57.54 309.20 0.00 58.28 Basin average Gauge 1961-2013 45.16 227.17 0.00 43.89 Gauge 1998-2013 44.38 227.17 0.09 44.48 TMPA 1998-2013 46.78 240.03 0.16 46.48

Table 3 Comparison of statistical indices between short-term (1998-2013) and long-term (1961-2013) data series

The SPI values at different time scales were computed for each grid (0.25° × 0.25° ) in the Weihe River Basin using the monthly TMPA precipitation data from January 1998 to December 2013.

And, the rain gauge observations were also used to calculate the SPI. A positive SPI means that the precipitation over a designated period was higher than thatover the whole time series, while a negative SPI indicates that the precipitation over a designated period waslower than thatover the whole time series.

At 1-month scale, the CC between TMPA-basedSPI and raingauge-based SPIis 0.64 (R²=0.41) in Xiji, 0.78 (R²=0.61) at Changwu, 0.86 (R²=0.74) at Huashan and 0.92 (R²=0.86) at basin scale (Table 4; Fig. 5). Generally speaking, the TMPA-based SPI has good consistency with the raingauge based-SPI. Figure 6 shows the occurrence frequency (%) of the different dry and wet classes expressed both by TMPA-inferred SPI₁ and by rain-gauged SPI₁. The normal class (i.e., -0.99≤ SPI₁≤ 0.99) had the largest share, accounting for 60%-70%, and the extremely dry class (ED)and extremely wet class (EW) had the lowest frequency, being less than 5%. The other classesincluding moderately dry (MD), moderately wet (MW), severely dry (SD) and severely wet (SW) varied between 3% and 15%. Again, the high degree of temporal coincidence between the TMPA-inferred SPI₁ and the rain-gauged SPI₁means that the TMPA 3B42V7 data can be used to determine the dry and wet classes in the Weihe River Basin.

Table 4

Table 4

Table 4 Validation of different time scales TMPA-based SPI Station SPI1 SPI3 SPI6 SPI9 SPI12 CC CSI (%) CC CSI (%) CC CSI (%) CC CSI (%) CC CSI (%) Huajialing 0.78 68.23 0.81 66.15 0.85 61.98 0.86 62.50 0.88 61.98 Xiji 0.64 60.94 0.64 60.42 0.70 59.90 0.72 63.54 0.70 59.38 Guyuan 0.70 65.63 0.64 63.54 0.74 66.67 0.74 69.27 0.70 65.10 Baoji 0.75 69.27 0.74 67.71 0.82 73.44 0.85 69.27 0.87 71.35 Huanxian 0.74 66.15 0.80 64.58 0.84 61.46 0.80 55.73 0.78 54.17 Xifengzhen 0.82 68.75 0.81 65.63 0.88 71.35 0.91 78.65 0.93 80.21 Changwu 0.80 61.98 0.76 69.27 0.77 65.10 0.73 66.67 0.75 70.83 Wugong 0.82 76.56 0.82 65.10 0.85 72.92 0.88 69.27 0.90 70.31 Wuqi 0.75 66.15 0.81 71.35 0.89 67.71 0.90 61.98 0.90 58.85 Tongchuan 0.81 68.23 0.81 67.19 0.82 59.90 0.79 61.46 0.81 59.90 Luochuan 0.82 62.50 0.81 67.19 0.81 61.98 0.79 60.42 0.82 59.38 Huashan 0.86 73.44 0.87 67.19 0.87 69.27 0.88 65.63 0.89 65.10 Basin Average 0.92 80.73 0.92 78.13 0.96 85.42 0.96 85.94 0.97 86.46 Note: CC, correlation coefficient; CSI, critical success index.

Table 4 Validation of different time scales TMPA-based SPI

Fig. 5

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	Fig. 5 SPI1scatterplotsof three representative stations (a-c) and the basin average (d). TMPA-SPI1, TMPA-inferred SPI1estimates; Gauge-SPI1, rain gauged SPI1.

Fig. 6

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	Fig. 6 Occurrence frequencies (%) of the different dry and wet classes based both on rain gauged SPI1 and TMPA-inferred SPI1.EW, extremely wet; SW, severely wet; MW, moderatelywet; N, normal; MD, moderately dry; SD, severely dry; ED, extremely dry.

For comparing the results in different time scales, we computed the statistical indices at 1-, 3-, 6-, 9- and 12-month scales (Table 4).Table 4 shows that the CC between the rain-gauged SPI₁ and the TMPA-inferred SPI₁ranged from 0.64 to 0.86 and the CSI rangedfrom 60.94% to 76.56%, indicating that the TMPA-based SPIestimationis an acceptable tool for drought monitoring in the Weihe River Basin. Furthermore, the results at 3-, 6-, 9- and 12-month scales are similar with the result of 1-month scale, again meaning that the TMPA-based SPIestimationis an acceptable tool for drought monitoringat different time scales in the Weihe River Basin. Figure 7 shows the inter-annual variability of SPI based on TMPA 3B42V7 estimates and rain gauge observations at different time scales at basin scale (i.e., basin average). It shows that the TMPA 3B42V7-based SPI matches the rain gauge-based SPI very well at all time scales (1-, 3-, 6-, 9- and 12-month scales), again confirming that the TMPA 3B42V7 data havegreatpotentials for the drought monitoring in the Weihe River Basin and probably also in other areas.

Fig. 7

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	Fig. 7 Comparison of SPI time series calculated both from rain gauge observations and from TMPA 3B42V7 estimates at 3-month scale(a), 6-month scale(b), 9-month scale(c), and 12-month scale(d)

To further evaluate the reliability of using the TMPA 3B42V7 data to monitor drought conditions, we targeted two typical drought events as validation cases: one occurred in the period spanning from March to August 2000 and the other occurred in the period spanning from October 2012 to March 2013. The validation demonstrated that TMPA 3B42V7-inferred SPI captured the drought events very well in both cases(Table 5). In both cases, the TMPA 3B42V7 told almost exactly the same stories as the rain gauge did, again further confirming that the TMPA 3B42V7 data havegreatpotentials for the drought monitoring in the Weihe River Basin and probably also in other areas.

Table 5

Table 5

Table 5 Validation of SPI1 for two typical drought events: one occurred in the period spanning from March to August 2000, and the other occurred in the period spanning from October 2012 to March 2013 Period Description of the drought event Gauge-SPI1 TMPA-SPI1 Classification Gauge TMPA Mar 2000 The drought affected a wide area with not only severe drought but also long drought period. -0.80 -0.87 Normal Normal Apr 2000 -0.35 -0.52 Normal Normal May 2000 -1.21 -1.38 Moderately dry Moderately dry Jun 2000 1.89 1.90 Severely wet Severely wet Jul 2000 -1.66 -2.08 Severely dry Extremely dry Aug 2000 0.24 0.49 Normal Normal Oct 2012 The precipitation has been 50% below normal value, and this change leads to a severe drought in the next spring. -1.79 -1.61 Severely dry Severely dry Nov 2012 0.02 0.11 Normal Normal Dec 2012 0.21 -0.08 Normal Normal Jan 2013 -0.70 -0.85 Normal Normal Feb 2013 0.39 0.40 Normal Normal Mar 2013 -1.60 -1.87 Severely dry Severely dry

Table 5 Validation of SPI1 for two typical drought events: one occurred in the period spanning from March to August 2000, and the other occurred in the period spanning from October 2012 to March 2013