2.3. Drought Indices
We considered five drought indices based on two categories of meteorological (SPI, SPEI, and PDSI) and agricultural drought (SMDI and ETDI) in order to comprehensively assess future changes in long-term drought under different degrees of global warming.
Figure 2 provides the flow chart of the modeling process.
The SPI allows monitoring and analysis of meteorological drought through the transformation of aggregated rainfall data at different time scales [
34]. SPI is a statistical monthly indicator (e.g., 3, 6, 12, and 24 months) that is fitted to a long-term precipitation time series using a gamma distribution function, as given by the following equation [
35,
36]:
where,
is monthly precipitation, α and β are shape and scale parameters of the probability distribution, respectively. The SPI time series gives positive or negative precipitation anomalies as the basis for defining drought events [
34] (
Table 4).
- b.
Standardized precipitation evaporation index (SPEI)
SPEI is a meteorological drought index, which takes into account the difference between monthly precipitation and reference evapotranspiration demand of the atmosphere [
37]. Several studies confirm that this index is suitable for drought assessment under the effect of climate change as it captures the influence of potential evapotranspiration (PET) on drought severity [
28,
38,
39,
40,
41]. SPEI is computed by simple subtraction as given in following equation:
where,
P is the monthly precipitation (mm) and PET is the potential evapotranspiration (mm) for
months. Vicente-Serrno et al. [
37], proposed a method to estimate
values using the default three-parameter log-logistic distribution function instead of a two-parameter gamma distribution function needed for SPI [
42]. The probability density function (pdf) (
f(
x)) of the three-parameter log-log distribution variable is defined by Equation (3):
where,
α,
β, and
y are the scale, shape, and origin parameters, respectively, which are obtained using the L-moment procedure for
D values in the range (
y >
D > ∞). Therefore, the PDF of the
D series is given by the following equation:
SPEI values were calculated as the standardized values of
F(
x) following Abramotitz and Stegun. [
43] and details by Vicente-Serrano et al. [
37] using the following equation:
where,
for
and
is the exceeding probability determined by the
value,
If
, then
P is used to replace by 1 −
P and the sign of the resultant SPEI is reversed. The constants are
,
,
,
,
, and
. The definition of a drought event using the SPEI is similar to SPI (
Table 4).
- c.
Self-Calibrated Palmer Drought Severity Index (scPDSI)
The Self-Calibrated Palmer Drought Severity Index (scPDSI) calibrates the PDSI for the location of interest based on water demand and supply instead of simply variation in rainfall (Wells et al., 2004 [
44]). This index employs three factors: rainfall, temperature, and locally available soil water capacity (SWC). The variables of evapotranspiration (ET), recharge (R), runoff (RO), potential recharge (PR), potential evapotranspiration (PET), loss (L), potential runoff (PR), and potential loss (PL) are estimated for AWC. These four variables are weighted according to the regional climate. The weighting factors,
α,
β,
γ, and
δ, are water-balanced coefficients calculated using the following equations:
where,
E,
R,
RO, and
L represent evaporation, soils recharge, runoff, and water loss of the soil layer, and their potential values are
PE,
PR,
PRO,
PL, respectively, so that the variables rely on the soil AWC.
where,
p is the amount of precipitation required to maintain a normal soil moisture level for a particular month under consideration.
Following Palmer’s algorithm [
8,
44,
45], the moisture departure (
d) was derived from the difference between one month of rainfall (
p) under normal conditions and evapotranspiration estimation (
ET). The monthly moisture anomaly index (
Z index) for a given location and time period indicates the degree of dryness or wetness without considering recent rainfall trends:
where,
K is the climate coefficient characteristic of the location:
The only difference between the PDSI and scPDSI is the replacement of the empirically derived climatic characteristic (
K) and duration factors (0.897 and 1/3), with values calculated automatically using historical climatic data for the study area. Monthly scPDSI values were estimated for all of the meteorological stations in the SSRB using the homogenized monthly precipitation and PET (Penman method) for the historical and projected periods. This drought index classification, ranging from negative to positive values, is shown in
Table 4.
- d.
Soil Moisture Deficit Index (SMDI).
The SMDI is based on the soil water content (SWC) anomaly during a given time step (i.e., week, month). The long-term SWC for each month was obtained by taking the median, maximum, and minimum values from 10 RCMs during the historical and projection periods. Following Narasimhan and Srinivasan [
46], the median was chosen over the mean as a measure of “normal” SWC because the median is more stable and is not influenced by outliers. SMDI values for 30-year historical and future periods were obtained by the following set of equations:
where,
is soil water deficit (%),
is the mean monthly SWC in the soil profile (mm),
is the long-term median SWC (mm),
and
are the long-term minimum and maximum SWC (mm),
is the year, and
is the month. SMDI ranges from −2 to +2, with negative values referring to drought. In the current study, the Soil and Water Assessment Tool (SWAT) was used as a hydrologic model to simulate SWC [
28].
- e.
Evapotranspiration Deficit Index (ETDI)
The ETDI was calculated using a procedure similar to the one explained above for
SMDI; however, it is based on the water stress anomaly relative to its long-term average. In this study, the monthly SWAT model’s output of actual evapotranspiration (
AET) and potential evapotranspiration (
PET) was achieved using the median, maximum, and minimum values from the historical and projected 10 RCM simulations. The monthly water stress ratio is calculated using Equation (12):
where
WS is the monthly water stress ratio,
PET is the monthly potential evapotranspiration, and
AET is the monthly actual evapotranspiration. Next, monthly long-term
ETDI is calculated using the following equations:
where,
is monthly water stress anomaly,
,
, and
are median, minimum, and maximum water stress (mm), respectively. The thresholds of agriculture drought resulting from
SMDI and
ETDI are shown in
Table 5.
The indices described above enable the assessment of drought duration, severity, and intensity [
47]. Drought duration (m) refers to the number of months between the starting (included) and ending (not included) months. Drought severity (
) is the absolute value of the sum of index values during a drought event [
48,
49]. We used the three indices SPI, SPEI, and scPDSI and the corresponding drought thresholds to determine duration and severity. Drought duration is the period during which the SPI, SPEI, and scPDSI are continuously negative, starting from the threshold values of −1 and ending with positive values. Drought severity is the cumulative SPI, SPEI, and scPDSI values over the drought duration, while intensity is the ratio of drought severity to duration as given below.
where
e is a drought event;
j is a month;
is the SPI, SPEI, and scPDSI values in month
j; and
m and
are the duration and severity of a drought even, respectively.
2.4. Hydrologic Modeling
A novel aspect of this study was the use of the spatially distributed Soil and Water Assessment Tool (SWAT) model for developing the agricultural drought index data. SWAT is a model of catchment hydrology. Sub-basins are delineated based on land use as well as soil type and slope, all of which are discretized into hydrologic response units (HRUs). SWAT is a temporally continuous, semi-distributed, and physically based model that uses daily, monthly, and yearly time steps to simulate physical processes of climate, soil moisture, plant growth, nutrients, pesticides, bacteria and pathogens, and soil management [
50]. Hydrologic cycle in SWAT was based on the following form of the water balance equation:
where
is the final soil water content (mm),
is the initial soil water content on day
i (mm),
t is the time (days),
is the daily precipitation
i (mm),
is the daily surface runoff (mm),
is the daily evapotranspiration
i (mm water),
is the amount of water entering the vadose zone from the soil profile bottom layer on day
i (mm), and
is the amount of return flow on day
i (mm). A total of 4873 HRUs were delineated by defining thresholds of 2% for land use and 5% for soil type. Moreover, Penman–Monteith method was used to estimate the potential evaporation. In this study, we applied multi-step calibration that included streamflow [
50] and soil moisture by measuring data for each variable [
51,
52]. The SWAT-CUP and the Sequential Uncertainty Fitting (SUFI-2) program [
53] were used to conduct sensitivity, calibration, and uncertainty analyses of the model. Monthly simulated and observed flows during the calibration period (1993–2005) and validation period (2006–2013) were used by three statistics to evaluate the model performance: the Nash–Sutcliffe efficiency (
NSE) [
54], the percent bias (
PBIAS) [
55], and the coefficient of correlation (
r) [
56] according to the following set of equations:
where,
and
are the mean monthly simulated and observed discharge,
is observed discharge on the ith day,
is the simulated monthly discharge,
n is the total number of months, and
is the average observed monthly streamflow. According to Zare et al. [
26], SWAT modeling has limitations when applied to cold regions, where streamflow is predominantly generated by melting snow during spring. Streamflow is underestimated due to an abrupt increase during the thawing time. Likewise, rain-on-snow (ROS) events occasionally occur in the SSRB during winter and are frequently observed in spring. These events create uncertainty for model performances in regional watersheds because runoff from snowmelt is not homogeneous at the watershed scale [
57]. Therefore, SWAT source code needs to be modified for cold regions. We used a combination of the physically based soil temperature module (not empirical) along with an energy budget equation for snowmelt for ROS events. In essence, the modified version of SWAT is capable to simulate freeze–thaw cycles and variation of frozen water content in soils. As such, this module closely simulates streamflow and soil temperature in cold weather. Detailed information regarding SWAT modification, including functions and variables, was given in Zare et al. [
28].