Water Stress Estimation from NDVI-Ts Plot and the Wet Environment Evapotranspiration

In this work we present a new simple index to estimate water stress (WS) for different types of surfaces, from remotely sensed data. We derive a WS index, named WSIEw, modifying the Water Deficit Index (WDI) proposed by Moran et al. by using the wet environment evapotranspiration (Ew) instead of the potential evapotranspiration (Epot) concept. Jiang and Islam model was used to simulate actual evapotranspiration (ET) and Priestley and Taylor equation to estimate Ew. The WSIEw results were compared to ground observations of ET, precipitation (PP), soil temperature (Tsoil) and soil moisture (SM) in the Southern Great Plains-EEUU. Preliminary results suggest the method is sensitive to the water status of different surfaces. However, the WSIEw would range from 0 to 0.7, having a value of 0.4 for a dry surface with 5% of SM. The methodology is operationally simple and easy to implement since it requires only information from remote sensors.


Introduction
The future of food depends largely on the water availability and strategic planning of the water resources.Therefore, to aid farmers to optimize the water uses is critical to maximize food production for the World.Nowadays vegetation WS indexes are extensively used in assisting farmer to maximize the crop yield when optimizing the irrigation system [1].The need to monitor large areas motivated the development of WS indexes based on remotely sensed data.Indeed, the advent of the thermal infrared sensors allowed surface temperature (T s ) to be monitored and related to vegetation water deficits [2][3][4][5].The correlation between surface temperature and water stress is based on the assumption that as a crop transpires, the leaves cool the air below them and the air temperature (T a ) drops [6].Examples of successful water stress indexes are the Critical Temperature Variability (CTV) [5,7], the Crop Water Stress Index (CWSI) [3] and the Water Deficit Index (WDI) [8], among others.
Moran et al. [8] provided the theoretical derivation of the WDI based on the relative evaporation, defined as the ratio between actual evapotranspiration (ET) and poten-tial evapotranspiration (E pot ).One of the main assumptions of the WDI is that soil and vegetation exchange energy and they cannot be analyzed separately [8].This concept is particularly interesting for remote sensing applications where mix pixels, i.e. soil + vegetation, provide a single signal in every electromagnetic wavelength.The calculation is based on the vegetation index (VI) and T s space, introduced by Price [9].The most common VI used is the Normalized Difference Vegetation Index (NDVI) [10][11][12].The IV-T s space has also been used to estimate ET [13][14][15].
In this work we propose a new form of the WDI based on Jiang and Islam [16], and Priestley and Taylor [17] methods.The theoretical background provided by Moran et al. [8] is consistent to Jiang and Islam [16] methodology to estimate ET, therefore they are linkable in a single index called WSI Ew .

Methodology
The background methodologies are the WDI published by Moran et al. [8] and Jiang and Islam [16] method.In this session both methods are briefly presented before we present the rationale behind the new index.

The WDI Index
Moran et al. [8] discussed the validity of the CWSI theory for partially vegetated areas.The derivation of their new index is based on the interpretation of the trapezoidal shaped IV-(T s -T a ) plot, where T a is the air temperature.
Figure 1 exemplifies the borders of the trapezoid.The segment 1 -3 represents the cold edge, with well watered conditions and vegetation cover ranging from bare soil to fully vegetated areas (NDVI ranging from 0 -1).The segment 2 -4, the warm edge, represents dry surfaces with vegetation varying from bare soil to full canopies.The segment 1 -2 represents fully vegetated areas.Point 1 corresponds to well watered vegetation.Point 2 characterizes dry vegetation.The segment 3 -4 represents bare soil [18].Given the value of T s -T a at any point in the trapezoid (i.e. point B in Figure 1), the evaporation can be obtained by from the distances AB and BC.
The WDI considers two important assumptions associated to the relationship between IV and the difference T s -T a .First, the authors assume that the difference T s -T a is linearly related to percentage of vegetated area and the canopy and soil temperatures.Another important statement made by the authors is that given a certain net energy (R n ), the temperature of the foliage and soil are linearly related to the transpiration and evaporation respectively.Therefore, the variations in T s -T a would be associated to ET.Thus, for a partially vegetated area, where (ET/E pot ) is the relative evaporation.E pot is associated to a surface with unlimited water supply.The authors computed it with Penman-Monteith´s equation [19] assuming the vegetation resistance (r cp ) closes to zero, without being canceled out.Moran et al. [8] asserted that the VI has to be sensitive to the canopy variations and insensitive to spectral changes in soil background.Hence, the soil-adjusted vegetation index (SAVI) was the selected VI, and the linear edges of the SAVI-T s space were adjusted to determine ET/E pot .Details of the WDI are provided in Moran et al. [8].

Jiang and Islam Method
The Jiang-Islam's interpretation of the NDVI-T s relationship provides the basis to estimate ET by modifying Priestley and Taylor's equation [17].Jiang and Islam introduced a coefficient to account for unsaturated areas, which replaced the original Priestley and Taylor's coefficient (α).The resulting modified equation is, where  is Jiang-Islam's parameter, Δ isthe slope of the saturation vapor pressure curve, γ is the psychrometric constant, R n is the net radiation at the surface level = and G is soil heat flux.
The parameter  varies from zero, for a dry bare soil surface, to α for a saturated or well vegetated surface, i.e. it becomes equal to Priestley and Taylor's equation.This parameter  is calculated by a simple two-step linear interpolation between the sides of the NDVI-T s triangle, as shown in Figure 2.
Jiang and Islam interpreted the upper edge, with high temperatures and low values of , as the minimum value of ET for each class of NDVI, while the cold edge, associated with low T s and maximum values of , represents maximum ET rate.Therefore, the value of  vary within the limits of the triangle.Thus, NDVI-T s plot is applied to derive  by using the normalized temperature, max max min where T max and T min are the maximum and minimum T s for a given vegetation class and T i is the radiometric temperature for a given pixel.
In practice, the value of T max is the temperature obtained extrapolating the upper edge to intersect the T s axis (Figure 2) for a NDVI = 0, while T min is obtained as the average T s of those pixels identified as water, i.e.
with NDVI < 0. A full description of  calculation can be found in Jiang and Islam [16].

The New Water Status Index (WSIEw)
Moran et al. [8] related the WS with ET/E pot (see Equation (1)), where E pot was defined from the Penman-Monteith's equation [19] suggesting that the R n , T a , vapor pressure deficit (VPD) and the soil moisture (SM) are the main controlling factors for the stomata opening [2,4,8].One of the main assumption of this model is that r cp tends to zero.This E pot definition is closer to a wet environment evapotranspiration (E w ) concept than to a truly potential concept, where the energy is maximum, the surrounding air is dry while the surface is saturated [20].Hence, for a given atmospheric condition and unlimited water supply, the soil+ plant complex will evapotranspire at its maximum rate according to the available energy.In this case, the maximum rate of ET is given by E w [17,[21][22][23].
The vegetation stress is mainly caused by deficit of moisture at the root zone.In the absence of watering, the moisture content in the root zone will be reduced as a result of crop intake.In turn, water stress causes the closure of the stoma of the plants and hence a reduction in the transpiration rate.Then, the ratio ET/E w is a good indicator of evapotranspiration deficit [24].Thus, E w can replace the E pot in Equation (1), and the WDI (see Equation (1)) can be written as, In this new form of WDI, ET can be replaced by the Equation (2) [16] and E w by the Priestley-Taylor equation.The new index, WSI Ew in terms of the parameters  and  is: replacing  by equation 3, the WSI Ew becomes, max min max min max min where T i is the radiometric temperature for a given pixel and T max and T min are the Jiang and Islam's parameters.
Operational robust methods could be achieved with purely remotely sensed data, as we propose here.In this case, T a and the VPD are not explicitly required to compute the new index.The WSI Ew does not require the understanding of the crop type biophysical functions under specific climates.The WSI Ew could be calculated from data recorded from current satellite missions, such as NOAA series, EOS-Terra and EOS-Aqua.In this work we applied to WSI Ew to the Southern Great Plains (SPG), using MODIS images and comparing the results with observations.

Study Area
The SGP region of US is a flat terrain, heterogeneous land cover with seasonal variation in temperature and humidity.It extends over the State of Oklahoma and southern part of Kansas, running from longitude 95.5˚W to 99.5˚W and from latitude 34.5˚N to 38.5˚N.
This region has relatively extensive and well distributed coverage of ground stations, maintained by the Atmospheric Radiation Measurement (ARM) program.The stations are widely distributed over the whole domain (Figure 3).E8 and E22 are located in a grazed rangeland region, E4 in an ungrazed rangeland area, E13 is positioned in a region with pasture and wheat, E7, E9, E15, E20 and E27 are located in pastures.E18 and E19 are in ungrazed pasture area, E12 is located in a native prairie, E10 is in alfalfa, E16 is in wheat region and E2 is in grass region.

Data and Images
The Energy Balance Bowen Ratio (EBBR) system compute 30-min estimates of the sensible and latent heat vertical fluxes at the local scale.Flux estimates are calculated from observations of net radiation, soil surface heat flux, and the vertical gradients of temperature and relative humidity.The instruments and measurement applications are well established and have been used for validation purposes in many studies [22,23,25,26].Further information about the ARM EBBR data and methodology is available at http://www.arm.gov.
Daytime MODIS-Aqua images for nine days in years 2009, 2010 and 2011 in spring and summer with at least 82% of the study area free of clouds were selected.Table 1 summarizes the image information including date, day of the year, satellite overpass time and image quality.The product MYD02 and MYD11 were used in this work.MYD02 provides corrected radiance, reflectance and geolocations for 36 bands and MYD11 provides T s images on a daily base [23,29,30].

Preprocessing
The MODIS images were georeferenced from the Latitude and Longitude associate to each pixel.The study area was pulled out of each image and geographically projected in a grid of 445 columns by 445 rows, with The NDVI-T s spaces were plotted to obtain Jiang-Islam's parameters (T max and T min ) and then obtain the WSI Ew by Equation (6).Then, the WSI Ew was validated with field data and contrasted with variables that are commonly associated to water stress.
Given that the WS is not directly measurable, we computed the WS with Equation (4) using ET ground observations (ET obs ) and Pristley and Taylor's equation as follows, where is the net radiation observed at the bowen ratio stations, is the ground data of soil head flux,  and  were computed from observed air temperature and atmospheric pressure.

WS Results
The regional statistic of WSI Ew , i.e. maximum, mean and standard deviation, for each of the days are shown in Table 2.
The regional minimum is always equal to 0.0, since Jiang and Islam's method [16] requires free water pixels to estimate T min .The minimum temperature represents ET = E w , i.e. no-stress condition; thus, no-stress pixels are always present in this methodology.The regional maximum ranges from 0.50 and 0.72.The mean values of WSI Ew vary from 0.14 to 0.42.For all the study days, the standard deviation is lower than 0.12 suggesting little regional dispersion around the mean of WSI Ew .Wang et al. [12] analyzed the phenological cycle of winter wheat under irrigation in the North China Plain.The authors associated CWSI values lower than 0.34 to no stress conditions.Kar and Kumar [31] analyzed the CWSI in peanut crops under irrigation.They found CWSI values between 0.61 and 0.63 just before the application of irrigation.Hence, the regional WSI Ew obtained here are comparables with those published by other authors.How-ever further analysis is needed to characterize the Results of WSI Ew and WS obs were compared.The bias was obtained as ∑(WS obs − WSI Ew )/n and RMSE as (∑(WS obs − WSI Ew ) 2 /n) 0.5 , where n is the number of observations.These statistics show bias of 0.05 and RMSE of 0.120, which represents approximately 24% of the mean WS (assuming a mean value of 0.5).
It is not evident how to validate the stress indexes from the literature revised here.For instance, Colaizzi et al. [32] compared the WDI and the soil water deficit index (SWDI).They found values of RMSE lower than 0,143 (29% of the mean) and bias lower than 0.112, consistent with the results presented here.In both cases, the RMSE are of the same order indicating that the method's errors are about 30% of the mean.
The comparison between WDI and WSI Ew remarks the differences between both models, with a bias of 0.28 and RMSE of 0.27 (54%).These results could be due to the separation between the bulk of the pixels and the dry border in the trapezoid (see Figure 1).These differences would be related to the differences in the E w and E pot concepts already explained in Section 2.3.
The WSI Ew results were also compared with records of different variables that index water stress.Figure 4 presents the WSI Ew versus observed SM (at 5 cm below the surface).
There is a well defined inverse relationship between the SM and the WSI Ew (Figure 4) with a coefficient of determination (R 2 ) equal to 0.52.The dissimilar spatial resolution of both data sets may be the cause of the dispersion observed in Figure 4. Certainly the WSI Ew is calculated as a result of the signal from 1 km × 1 km pixel while the ancillary data are representative of few meters around the point station [30].Nevertheless, values of SM of 5%, characteristic of low moisture in most soils, correspond to WSI Ew of about 0.45 while a SM of 30%, i.e. a well watered surface, could be associated to a WSI Ew of 0.2.
Figure 5 displays WSI Ew versus T soil (at 5 cm below the surface).These relationship showed a R 2 = 0.72.It should be noted that  is a dimensionless temperature, what may explain the relationship with the soil temperature presented in Figure 5. Patel et al. [33] examined the potential of using canopy-air temperature difference (T c -T a ) for assessing the crop water status.The authors correlated T c -T a with SM (at 15 cm) with a R 2 = 0.59.Fensholt and Sandholt [34] found a R 2 = 0.48 when comparing the Shortwave Infrared Water Stress Index (SIWSI) with soil moisture observed in situ.
The inverse relationship between observed ET obs and calculated WSI Ew presented in Figure 6 is noteworthy, given that the ET records are independent to ET J-I /E w.In general, the WSI Ew seems to capture the surface water stress condition well.The WSI Ew was compared with the rainfall accumulated five days prior to the analyzed dates (PP).The comparison is shown in Figure 7 where the decrease of WS when rainfall increases is observed (R 2 = 0.62).As the surface gets humid more water is available for ET and the ratio ET/E pot tends to one.Thus, the stress decreases with the precipitation increase.Wang et al. [12] studied the wheat in the plain of China.They associated the wheat low stress to high rainfall events.The authors concluded that a good wheat harvest was obtained for CWSI lower than 0.34, representing a not-stress condition.
Figure 8 shows the comparison of WSI Ew with soil moisture and PP for station E12.The SM observed at 5 cm is supplied by PP and represent the roots water availability [34], thus any reduction of PP, causes a decrease in the SM and it is expected to cause an increase of the WSI Ew .
Finally, Figure 9 shows the WSI Ew map for the SGP for the May 4 th 2011, where we observed the stress would decreases from West to East.In general, the mountain areas present WSI Ew of 0.7 and the prairies at the East show values of WSI Ew about 0.3.The black areas are cloudy masked pixel [16,30].

Sensitivity Analysis
T max is the main parameter of the index presented in this paper, thus the sensitivity of the WSI Ew to variation of T max is analyzed here.
Different methods to draw the warm and cold edge of the triangle [10,11] would render different values of T max in Jiang-Islam method.Thus, we apply the First Order Analysis to estimate the effect of T max on WSI Ew variance.
The results of applying Equation ( 9) with differents T min , T max , T i statistic and assuming that max varies from 1% to 15%, suggested that

Discussion
Several methods have been developed to determine the WS from observations of the T s [2][3][4][5].The index proposed by Moran et al. [8], based on the relative evaporation concept, is widely applied all over the world [18,32,35].In this work we presented a modification of WDI proposed by Moran et al. [8].The new index replaces the E pot used in the original formulation for the E w concept.
Given that E w  E pot [20], the ratio ET/E pot is smaller than ET/E w , yielding two WS indexes that point out different causes of vegetation stress.The WSI Ew would only consider the stress due to water shortness in the root zone while the WDI would also reflect the stoma closure cause by the atmospheric and radiation effects.
The WDI required field information (observations of R n , G, wind speed, T a ) to establish the end-points that define the trapezoid (Figure 1(a)).If such information was not available, it would be necessary to guess them from each analyzed image; i.e. indentifying groups of pixels that represent the natural conditions of the endpoints.Guessing the extreme points creates uncertainty in the interpretations of the diagrams [35].The WSI Ew also requires determination of the triangle borders, however in this case, the source of error would be mainly associated with definition of T max .
The WSI Ew is another "thermal index" with the form of a normalized temperature, i.e.

   
. T max is the temperature for a hypothetical dry bare soil pixel and T min is the temperature for a saturated pixel.
It must be noted that the T max is not the maximum thermal infrared of the image or maximum canopy temperature; on contrast, it is a "hypothetical" maximum temperature.It is usually found in the literature thermal stress indices calculated as a normalized T s , the difference between the methods is given the extreme temperatures estimation.For instance, Galleguillos et al. [18] used the WDI to derive the daily ET.The authors assumed that T a is constant in the study area and they estimated the maximum and minimum T s from the energy balance.Wang et al. [35] made used of the WDI to estimate soil moisture.These authors explored the T s -EVI (Enhanced Vegetation Index) space to determine the minimum and maximum values of T s .The relationship between the WSI Ew and SM (R 2 = 0.52) provided interesting information.In Figure 3(a) we observe that a soil with 5% of moisture matches a WSI Ew of about 0.45.On the other extreme, a SM of 30% agrees with a WSI Ew of 0.1.In other words, the WSI Ew would not get close to 1 although the SM indicates a dry surface.
We compare the WSI Ew with the rainfall accumulated five days prior to the analyzed dates, observing that the WS decreases as precipitation increases, however there is no sufficient analysis about irrigation fields and precipitation spatial distribution in this work to deepen the conclusions about the WSI Ew range of variation.
The advantage of this new formulation is that there is no need of crop-field observations, yet overestimated T max values might yield an underestimated water stress condition.Thus, a T max significantly larger than the actual dry soil temperature, would not index the stress with a WSI Ew value close to 1, as it would be expected by the end-users.In general, the WSI Ew shows significant correlations with water stress indicators.
The WSI Ew is applicable to different satellite missions and requires minor image processing.The new index can be very useful for end users who require quick and easy methods of application.

Figure 2 .
Figure 2. NDVI-T s triangle space with upper and lower bonds and Jiang-Islam's parameters for date 04/05/2011.

Figure 3 .
Figure 3.Southern Great Plains and ground station locations.
was compared with the WS obs and WDI obtained from NDVI-(T s -T a ) space.Finally, in order to analyze the applicability of WSI Ew we compared it with different variables associated to the stress.

Figure 8 .
Figure 8. WSI Ew vs SM and PP for station E12.

Figure 9 .
Figure 9. WSI Ew map for the SGP for the May 4th 2011.

Table 2 . regional statistics WSI Ew (values maximum, aver- age and standard deviation).
WSI Ew valid range.
5% to 30% approximately.These results indicate that variations up 10% of T max would cause errors of about 15% in WSI Ew .