**3.2 Methodology**

We propose to develop a simple drought index called the evapotranspiration Water Deficit Drought Index (EWDI), which is derived from precipitation, meteorological data, and vegetation information. EWDI uses the structure of SPI with the monthly difference between ETW and ET. This value represents water deficit using the complementary relationship. The complementary relationship to estimate ET was addressed in the previous sections and a nonparametric approach to calculating the probability-based drought index will be addressed in this sector.

### *3.2.1 EWDI formulation*

With a known ET value, the difference between ETW and ET for the month *i* is calculating using Eq. (14):

$$D\_i = ET\mathbf{W}\_i - ET\_i \tag{14}$$

Given the monthly time series of *Di*, EWDI uses a nonparametric approach in which empirically derived probabilities are obtained through an inverse normal approximation [20] because this probabilities approach allows a consistent comparison between EWDI against other standardized indices [21, 22].

The probability distribution function of the *Di*, according to the Tukey distribution, is given by Eq. (15):

$$P(D\_i) = \frac{i - 0.33}{n + 0.33} \tag{15}$$

where *P*(*Di*) is the empirical probability of *Di* which is aggregated across the period of interest. In this study, we used 12-month duration for accumulating *Di* because 9- to 12-month time scale is the most useful in estimating the extreme drought conditions [23]. For example, to calculate a 12-month EWDI in December, *Di* is summed over the period from January to December. *i* is the rank of the aggregated *Di* in the historical time series (*i* = 1 is the maximum *Di*), and *n* is the number of observations in the series being ranked. EWDI then can be easily derived following the classical approximation of [20] as shown in Eq. (16):

oroxx formation of [20] as shown in Eq. (16):

$$EWDI = W - \frac{C\_0 + C\_1W + C\_2W^2}{1 + d\_1W + d\_2W^2 + d\_3W^3} \tag{16}$$

**87**

**Table 3.**

*EWDI.*

*All indices data from 2001 to 2015 were collected.*

*An Advanced Evapotranspiration Method and Application*

\_\_\_\_\_\_\_\_\_\_

If *P*(*Di*) >0.5, replace *P*(*Di*) with [1 − *P*(*Di*)] and the sign of EWDI is reversed. The constants are *<sup>C</sup>*<sup>0</sup> = 2.515517, *C*<sup>1</sup> = 0.802853, *C*<sup>2</sup> = 0.010328, *d*<sup>1</sup> = 1.432788, *d*<sup>2</sup> = 0.189269, and *d*<sup>3</sup> = 0.001308. The average value of EWDI is 0, and the standard deviation is 1. A zero

equal to the median value, positive value indicates drought, and negative is wet condition. Hereafter, drought index EWDI estimated from the modified GG [5] is called EWDI-MOD. Similarly, drought index EWDI estimated from GG-NDVI [24] is

Required meteorological data to calculate both ET values (modified GG or GG-NDVI) are air temperature, precipitation, elevation (pressure), net radiation, wind speed, and NDVI. Net radiation was estimated using the equations suggested by [25]. Air temperature and precipitation data are from the PRISM (Parameterelevation Regressions on Independent Slopes Model) climate group (available at http://prism.oregonstate.edu/) at 4-km resolution for the period 2000–2015 covering the CONUS. Wind speed was collected from the Climate Monitoring at NOAA's National Centers for Environmental Information (available at https://www.ncdc. noaa.gov/societal-impacts/wind/). Monthly NDVI data required for the GG-NDVI method are from the NASA Earth Observations (NEO, available at http://neo.sci.

To assess the capability of EWDI, we used USDM to compare the differences between the two indices during the evolution of drought through time and space. USDM is derived from measurements of climatic, hydrologic, soil conditions, and regional expert comments [26]. USDM is not a forecast instead it assesses the current drought conditions. USDM divides drought severity into five classes: abnormally dry (D0), moderate drought (D1), severe drought (D2), extreme drought (D3), and exceptional drought (D4). All drought indices used in this study were converted to USDM classes as presented in **Table 3**. Additionally, we compared EWDI against PDSI and SPI which were retrieved from the WestWide Drought Tracker (WWDT, available at http://www.wrcc.dri.edu/wwdt/about.html). USDM data from 2000 to 2015 were collected from the USDM website (http://

**Drought condition USDM PSDI SPI EWDI** Abnormally dry D0 −1.0 −0.5 −0.5 Moderate drought D1 −2.0 −0.8 −0.8 Severe drought D2 −3.0 −1.3 −1.3 Extreme drought D3 −4.0 −1.6 −1.6 Exceptional drought D4 −5.0 > −2.0 > −2.0 >

*Drought classes of USDM and corresponding threshold value for classifying drought with PDSI, SPI, and* 

−2 ln*P*(*Di*) for *P*(*Di*) ≤ 0.5 (17)

accumulated over the aggregation period in the year of interest is

*DOI: http://dx.doi.org/10.5772/intechopen.81047*

*W* = √

where

EWDI means that *Di*

called EWDI-NDVI.

gsfc.nasa.gov/).

**3.3 Data**

where

*Advanced Evapotranspiration Methods and Applications*

**3.2 Methodology**

*3.2.1 EWDI formulation*

calculating using Eq. (14):

tion, is given by Eq. (15):

(SEDI, [19]) was developed by using actual ET based on [3] and a structure of the SPI. They estimated ET using the modified GG model of Anayah and Kaluarachchi [5] and ETW minus ET to measure drought conditions. As a result, the spatial patterns of the SEDI were consistent with the PDSI and SPI over the contiguous United States (CONUS), and this index could roughly identify vegetative droughts such as a Vegetation Health Index (VHI). Although the results of SEDI demonstrated that the use of actual ET can provide a reliable measure for drought monitor, it would have been much more useful if the authors addressed the precipitation and used the accurate ET model. Taking these limitations into account, this chapter has focused on developing a drought index with an advanced ET model including precipitation and remote sending vegetation information. The specific object is to evaluate the applicability of the proposed drought index over the CONUS by comparing it with US Drought Monitor (USDM) which is most widely used tool in the United States.

We propose to develop a simple drought index called the evapotranspiration Water Deficit Drought Index (EWDI), which is derived from precipitation, meteorological data, and vegetation information. EWDI uses the structure of SPI with the monthly difference between ETW and ET. This value represents water deficit using the complementary relationship. The complementary relationship to estimate ET was addressed in the previous sections and a nonparametric approach to calculating

With a known ET value, the difference between ETW and ET for the month *i* is

*Di* = *ETWi* − *ETi* (14)

Given the monthly time series of *Di*, EWDI uses a nonparametric approach in which empirically derived probabilities are obtained through an inverse normal approximation [20] because this probabilities approach allows a consistent com-

The probability distribution function of the *Di*, according to the Tukey distribu-

where *P*(*Di*) is the empirical probability of *Di* which is aggregated across the period

9- to 12-month time scale is the most useful in estimating the extreme drought condi-

over the period from January to December. *i* is the rank of the aggregated *Di* in the historical time series (*i* = 1 is the maximum *Di*), and *n* is the number of observations in the series being ranked. EWDI then can be easily derived following the classical

*<sup>n</sup>* <sup>+</sup> 0.33 (15)

\_\_*\_\_\_\_\_\_\_\_\_\_\_*\_\_\_ 1 <sup>+</sup> *<sup>d</sup>*1*<sup>W</sup>* <sup>+</sup> *<sup>d</sup>*2*W*<sup>2</sup> <sup>+</sup> *<sup>d</sup>*3*W*<sup>3</sup> (16)

because

is summed

the probability-based drought index will be addressed in this sector.

parison between EWDI against other standardized indices [21, 22].

of interest. In this study, we used 12-month duration for accumulating *Di*

tions [23]. For example, to calculate a 12-month EWDI in December, *Di*

*EWDI* <sup>=</sup> *<sup>W</sup>* <sup>−</sup> *<sup>C</sup>*<sup>0</sup> <sup>+</sup> *<sup>C</sup>*1*<sup>W</sup>* <sup>+</sup> *<sup>C</sup>*2*W*<sup>2</sup>

*P*(*Di*) = \_\_\_\_\_\_ *<sup>i</sup>* <sup>−</sup> 0.33

approximation of [20] as shown in Eq. (16):

**86**

$$\mathcal{W} = \sqrt{-2\ln P(D\_i)} \text{ for } P(D\_i) \le 0.5 \tag{17}$$

If *P*(*Di*) >0.5, replace *P*(*Di*) with [1 − *P*(*Di*)] and the sign of EWDI is reversed. The constants are *<sup>C</sup>*<sup>0</sup> = 2.515517, *C*<sup>1</sup> = 0.802853, *C*<sup>2</sup> = 0.010328, *d*<sup>1</sup> = 1.432788, *d*<sup>2</sup> = 0.189269, and *d*<sup>3</sup> = 0.001308. The average value of EWDI is 0, and the standard deviation is 1. A zero EWDI means that *Di* accumulated over the aggregation period in the year of interest is equal to the median value, positive value indicates drought, and negative is wet condition.

Hereafter, drought index EWDI estimated from the modified GG [5] is called EWDI-MOD. Similarly, drought index EWDI estimated from GG-NDVI [24] is called EWDI-NDVI.
