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The need to allocate the existing water in a sustainable manner, even with the projected population growth, has made to assess the consumptive use or evapotranspiration (ET), which determines the irrigation demand. As underscored in the literature, Penman-Monteith method which is a combination of aerodynamic and energy balance method is widely used and accepted as the method of estimation of ET. However, the application of Penman-Monteith relies on many climate parameters such as relative humidity, solar radiation, temperature, and wind speed. Therefore, there exists a need to determine the parameters that are most sensitive and correlated with dependent variable ( i.e ., ET), to strengthen the knowledge base. However, the sensitivity of ET using Penman-Monteith is oftentimes estimated using meteorological data from climate stations. Such estimation of sensitivity may vary spatially and thus there exists a need to estimate sensitivity of ET spatially. Thus, in this paper, based on One-AT-A-Time (OAT) method, a spatial sensitivity tool that can geographically encompass all the best available climate datasets to produce ET and its sensitivity at different spatial scales is developed. The spatial tool is developed as a Python toolbox in ArcGIS using Python, an open source programming language, and the ArcPy site-package of ArcGIS. The developed spatial tool is demonstrated using the meteorological data from Automated Weather Data Network in Nebraska in 2010. To summarize the outcome of the sensitivity analysis using OAT method, sensitivity indices are developed for each raster cell. The demonstration of the tool shows that, among the considered parameters, the computed ET using Penman-Monteith is highly sensitive to solar radiation followed by temperature for the state of Nebraska, as depicted by the sensitivity index. The computed sensitivity index of wind speed and the relative humidity are not that significant compared to the sensitivity index of solar radiation and temperature.

Although over 70% of Earth’s surface is covered by water, the amount of freshwater available for appropriation is limited as 97.5% of all water on Earth is saline [

As underscored in the literature [

Based on a 20-yr historical daily climate dataset, in computing ET using FAO56 Penman-Monteith method [

As presented in the previous paragraph, the sensitivity of ET using Penman-Monteith is oftentimes estimated using meteorological data from climate stations. However, such estimation of sensitivity may vary spatially and thus there exists a need to estimate sensitivity of ET spatially. Thus, the objectives of this paper are to a) develop a spatial sensitivity tool that can geographically encompass all the best available climate datasets using Python, an open source programming language supported by a growing user community for its extensive collection of standard and third-party libraries, and the ArcPy site-package of ArcGIS, b) evaluate the spatial sensitivity of ET using Penman-Monteith c) evaluate the spatial sensitivity of ET using aerodynamic method and energy balance method.

The need to manage the available freshwater wisely with ever increasing population and the demand from irrigation has brought ET as one of the critical subject areas to research in the field of hydrology. Over the years, with many research works, numerous methods have been developed to estimate ET. These methods mainly fall under these categories: a) aerodynamic method, b) energy balance method, and c) combination of aerodynamic and energy balance methods [

This method of determining evaporation considers the transport of water vapor by the turbulence of the wind blowing over a natural surface. According to this method, the evaporation (

where M,

where

By substituting Equation (2) in Equation (1),

Considering aerodynamic resistance,

Since

where T is the air temperature in degree Celsius. With further simplification,

As shown in

where

The sensible heat flux defined by Equation (7) is related to Bowen ratio,

Bowen ratio,

surface. By substituting Equation (7) in Equation (6),

Since

As shown in Equation (10), the net radiation (

The net short-wave radiation that is defined by Equation (11) is a function of total extraterrestrial radiation

(

The net long-wave radiation which is in accord with Stefan-Boltzmann’s law of black body radiation is given by Equation (12).

where

As shown in Equation (13), this method of evaporation is obtained by combining the evaporation computed by

aerodynamic (

where

This method is same as the combination method of Penman. However, in this method, similar to

For grass reference crop,

where

where Z_{1}, Z_{2} are measurement heights for levels 1 and 2, respectively. Z_{0} is the reference height where velocity is zero. For open agricultural area, Z_{0} = 0.03. The inner details of the method are presented in

The mathematical model of a dependent variable (

and the derivative of the function are quite complex. On the other hand, in OAT method which the developed tool uses, at a time, one of the parameters is varied while keeping the other parameters at their base or nominal value. Based on how the parameters are varied with respect to the nominal value, many variations of this method have been proposed. To represent the entire spectrum of the parameter variability, it has been recommended to vary each parameter by certain percentiles while keeping the other parameters at their base value [

In the developed spatial sensitivity tool, since there many variables associated in the sensitivity analysis, to summarize the outcome of sensitivity analysis using OAT method, a sensitivity index (

where

ArcGIS provides easy-to-use platform to extend its desktop features by accessing geoprocessing functionalities through programming/scripting languages. Python, an open source programming language supported by a growing user community for its extensive collection of standard and third-party libraries, is one of the scripting languages supported by Environmental Systems Research Institute (Esri). The communication between ArcGIS and Python is through a site-package that is called ArcPy. Using the ArcPy site-package, the customization of desktop features could be in three ways: desktop add-in, standard toolbox, Python toolbox. Since the development of spatial sensitivity tool at grid scale does not involve an event such as dragging a rectangle over a geographical map to define an area of interest, Python toolbox is used to develop the spatial sensitivity tool. The developed python toolbox is basically an ASCII-based file, which contains scripts written using Python scripting language and the ArcPy site-package. As shown in

The graphical user interface of the developed spatial sensitivity tool is shown in

Based on the parameter range (i.e., % of the base value of the parameter at each raster cell) and the number of simulations specified by the user, the interval of sampling is calculated as shown in the variable named “mulRange”, where the Python variables “inSenseHigh”, “inSenseLow”, and “inNumberSimulations” are the maximum value of the parameter, the minimum value of the parameter, and the number of simulations, respectively. The maximum value of the parameter and the minimum value of the parameter are calculated within an if-else block based on the parameter name that is specified by the user and stored by the tool in the Python variable named “inSenseType”.

if(inSenseType==‘Temp’):

inSenseLow=Raster(inRasterTempF)−parameters[

inSenseHigh=Raster(inRasterTempF)+parameters[

elif inSenseType==‘RH’:

inSenseLow=Raster(inRasterRH)−parameters[

inSenseHigh=Raster (inRasterRH)+parameters[

elif inSenseType==‘Wind’:

inSenseLow=Raster (inRasterW)−parameters[

inSenseHigh=Raster (inRasterW)+parameters[

elif inSenseType==‘Solar’:

inSenseLow=Raster (inSolarRadiation)−parameters[

inSenseHigh=Raster (inSolarRadiation)+parameters[

inNumberSimulations=parameters [

mulRange=(inSenseHigh−inSenseLow)/(inNumberSimulations−1)

The below code shows the variables that are used to store the user specified rasters such as the temperature, relative humidity, and wind speed.

inRasterTempC=(Raster(inRasterTempF)−32)/1.8

inRasterRH2=Raster(inRasterRH)

Windspeed=Raster(inRasterW)

inSolarR=Raster(inSolarRadiation)

listRaster=“ ”

As shown below, based on the number of simulations specified by the user, a for-loop is setup. Within the for-loop, the variables that are used to store the user specified rasters such as the temperature, relative humidity, and wind speed, are updated using the Python variable “mulRange” and the iteration number. Since the sensitivity analysis is based on the concept of One-AT-A-Time, within the for-loop, an if-else condition is developed to identify the raster dataset that needs to be modified. For example, if the user specified parameter name is “RH”, the raster dataset associated with relative humidity is modified within the for-loop at each iteration. The other raster datasets (i.e., temperature, wind speed, and solar radiation) are kept at their base values for all the iterations.

for x in range (0, inNumberSimulations):

messages.addWarningMessage (“The iteration number {0} is going on.”.format (x))

if(inSenseType==‘Temp’):

inRasterTempC=(Raster(inRasterTempF)/Raster(inRasterTempF)*(mulRange*x+inSenseLow)−32)/1.8

elif inSenseType==‘RH’:

inRasterRH2=Raster(inRasterRH)/Raster (inRasterRH)*(mulRange*x+inSenseLow)

elif inSenseType==‘Wind’:

Windspeed=Raster(inRasterW)/Raster (inRasterW)*(mulRange*x+inSenseLow)

elif inSenseType==‘Solar’:

InSolarR=Raster(inSolarRadiation)/Raster(inSolarRadiation)*(mulRange*x+inSenseLow)

Having generated the raster dataset for the parameter that is tested in the sensitivity analysis, the ET raster is developed as discussed in Section 2.0. The detailed Python code is outlined by [

PenmanMon.save(outRaster1+str (x))

listRaster=listRaster+outRaster1+str(x)+“;”

As shown in the below code, since the sensitivity index discussed in Section 3.1 requires the maximum and the minimum values of ET at each raster cell, two new raster datasets are created to store this information. The copy_management tool of ArcGIS is used to generate the rasters.

arcpy.Copy_management(outRaster1+str(x), outRaster1+“max”) #Creating a new raster to store the max raster from sense analysis

arcpy.Copy_management(outRaster1+str(x), outRaster1+“min”) #Creating a new raster to store the min raster from sense analysis

The mosaic operation on ET raster datasets as outlined in

arcpy.Mosaic_management(listRaster,outRaster1+“max”,“MAXIMUM”,“FIRST”, “0”, “”, “”, “”, “”)

arcpy.Mosaic_management(listRaster,outRaster1+“min”,“MINIMUM”,“FIRST”, “0”, “”, “”, “”, “”)

SenseRaster=(Raster(outRaster1+“max”)−Raster(outRaster1+“min”))/Raster(outRaster1+“max”)

SenseRaster.save(outRaster1+“SenseIndex”)

As discussed previously, within each iterations, the ET raster datasets are generated to run the mosaic operation to get the maximum and the minimum ET values for the considered number of simulations for each parameter within the given range. After the mosaic operation, the below lines of code with a for-loop are used to delete the ET raster datasets that were used as inputs for the mosaic operation.

for x in range(0, inNumberSimulations):

arcpy.Delete_management(outRaster1+str (x))

The state of Nebraska that lies in both the Great Plains and the Midwestern United States has a total geographical area of 200,520 km^{2}. The total population of the state is 1.8 Million [

The AWDN has around 63 stations to cover the state of Nebraska. The data is available on hourly, daily, and sub-daily basis since 1985. To demonstrate the tool, the daily data in 2010 was downloaded from the online services provided by AWDN. As discussed in Section 2.0, the developed tool requires raster datasets on temperature, relative humidity, wind speed, and solar radiation. Therefore, at first, the average daily data in 2010 was developed based on the daily data of temperature, relative humidity, and wind speed. Using ArcGIS and the spatial locations of the climate stations, the tabular datasets of average daily data in 2010 were transformed to geographical data. Subsequently, the Spatial Analyst Extension of ArcGIS was used to develop the grid level values of temperature, relative humidity, and wind speed at a resolution of 1 km. The Kriging spatial interpolation technique packaged with ArcGIS was used to develop the rasters of required inputs. To ensure that the Krigged data covers the whole state, the extent of the interpolation was set using the state map of Nebraska. The research work carried out by [

The estimated ET using Penman-Monteith method is shown in

Penman-Monteith method is =

ET using Penman-Monteith method is registered in the Eastern part of Nebraska. However, the geographical extent of such high ET is limited to a very small area in contrast to the geographical area with estimated ET using Penman-Monteith method in the range of 0.77 - 0.96, as shown in

Figures 8(a)-(d) show the maps of spatial sensitivity index of ET using Penman-Monteith method for relative humidity, solar radiation, wind, and temperature, respectively. For these maps, as indicated in the legends of the maps, the number of simulations was set to 50, and the variation of the said parameters were set to 25% with respect to the base value for each raster cell. In other words, for each raster cell, the maximum and the minimum are considered as 125% and 75% of the base value for each raster cell, respectively. As can be observed from

temperature, but higher spatial sensitivity index of solar radiation.

Figures 9(a)-(c) show the spatial sensitivity index of ET using energy balance method for relative humidity, solar radiation, and temperature, respectively. Since the energy balance method discussed in Section 2.2 is not influenced by wind speed, the spatial sensitivity index of ET using energy balance method for wind speed is not shown in

Similarly,

method is higher than the spatial sensitivity index of ET using Penman-Monteith and the energy balance method.

To understand the influence of parameter variation (as a % of the base value for each raster cell) on the trend of computed spatial sensitivity index, simulations were carried out for parameter variations of 5%, 10%, 15%, 20%, and 25% of the base value for each raster cell. As previously, the number of simulations was fixed to 50. The outcome of these simulations is placed in

parameter variations increase. In other words, with more variation expressed with the base value, the computed maximum and minimum sensitivity index also increase for all the parameters. The range that is denoted between the minimum and the maximum value of the computed sensitivity index for the selected parameters, too increases with the increased parameter variations. Furthermore, the fitted equations whose strengths are explained through the statistical measure (R^{2}) also reveal that the rate of increase with the increase of parameter variation decreases in the order of solar radiation, temperature, wind speed, and relative humidity.

In this paper, a spatial sensitivity tool that can geographically encompass all the best available climate datasets to produce ET and its sensitivity at different spatial scales is developed. Based on the outcome of this study, the following points are highlighted:

1) Among the considered parameters in the sensitivity analysis of ET using Penman-Monteith, solar radiation registers the highest sensitivity index followed by the temperature for the state of Nebraska. The computed sensitivity index of wind speed and the relative humidity are not that significant compared to the sensitivity index of solar radiation and temperature.

2) Though the local sensitivity analysis such as One-AT-A-Time is the simplest method of sensitive analysis, the combined variability resulting from changing all the parameters simultaneously is precluded. Therefore, extending the developed tool with random sampling methods such as Latin Hypercube could be one of the future research needs.

3) In demonstrating the developed spatial sensitivity tool, the simulations are carried by varying the base value of the parameters, such as the temperature and solar radiation, for each raster cell by a given percentile. The percentile variation of the base value for each raster cell could be based on historical temporal datasets at each raster cell.

4) The developed spatial sensitivity tool is tested for parameters such as temperature, relative humidity, solar radiation, and wind speed. The develop tool can also further be tuned to test the sensitivity of ET with the intermediate or secondary parameters that are derived from the user specified parameters. The developed tool could also be potentially extended to establish multiple regressed equations to predict ET values at each raster cell based on the sensitivity analysis at each raster cell.

The authors would like to thank the Board of Regents, University of Nebraska-Lincoln, Lincoln, for providing the financial support to conduct this research. This research was conducted when author^{*} was a researcher at University of Nebraska-Lincoln, Lincoln.

SivarajahMylevaganam,ChittaranjanRay, (2016) The Spatial Sensitivity Analysis of Evapotranspiration using Penman-Monteith Method at Grid Scale. Journal of Geographic Information System,08,121-136. doi: 10.4236/jgis.2016.81012