This Resource serves to explain and contain the methodology, R codes, and results of the United States Geological Survey (USGS) irrigation freshwater withdrawals and irrigated acreages analysis for my thesis. For more information, see my thesis at the USU Digital Commons.

Statewide USGS estimates of total, ground, and surface withdrawal estimates are available one year in every five years for the period between 1950 and 2015. Statewide USGS estimates of total irrigated acres are available one year in every five years for the period between 1960 and 2015. USGS county-level irrigation withdrawal and acreage estimates, including sprinkler and flood irrigated acres, are available for the period between 1985 and 2015. Spatial delineations in Utah for this analysis include statewide, northern, and southern divisions (roughly splitting the state into two halves), the Great and Colorado River basins, and all 29 counties. Sub-state correlation analyses were only possible using USGS county-level data for the period between 1985 and 2015.
USGS irrigation withdrawal data were compiled for statewide and county-levels for all years available in Utah. Statewide irrigation estimates data were made available in the USGS national water use reports. County-level irrigation data for Utah were available from the USGS National Water Information System: Web Interface.

USGS irrigation data in Utah are limited. At most, 14 points of data can be analyzed. Using a density function, these data do not satisfy normal distribution assumptions. These data also have tied values (i.e., subsequent estimates in the analysis that are identical). Kendall’s Tau-b statistic handles situations such as these robustly. Kendall’s Tau-b is a non-parametric statistic that measures the correlation between ranked pairs, via their number of concordant and discordant pairs. Being a non-parametric test, values can be continuous, such as irrigation withdrawal estimates, or ordinal, such as the water-year quintile rank of freshwater availability key indicators discussed in the other resources connected with this resource.

If X and Y represent two datasets of interest, for instance total irrigation withdrawals and sprinkler irrigated acres, concordance and discordance are defined as follows:

Given the pairs (X_0,Y_0) and (X_1,Y_1 ):
Concordant pair= (Y_1-Y_0)/(X_1-X_0 )>0
Discordant pair= (Y_1-Y_0)/(X_1-X_0 )<0
Tied pair= (Y_1-Y_0)/(X_1-X_0 )=0

These equations reveal how well the pair of data follow each other, indicating a whether a significant relationship exists. The result of Kendall’s Tau-b test is a number between -1 and 1. A value of 1 means there is perfect positive association, or agreement i.e., when the pair are sorted in descending order according to one parameter, their ranks, or order, are mirrored. Conversely, a value of -1 shows perfect negative correlation, or inversion, i.e., as one increases or decreases, the other does the opposite.
The formula in R software for Kendall’s Tau-b (Yao 2021) is:
τ_b= (n_c-n_d)/(√(N_1 ) × √(N_2 ))
where
n_c=number of concordant pairs
n_c=number of discordant pairs
N_1=number of data pairs not tied in a target feature
N_2=number of data pairs not tied in the other target feature
To test the significance of Kendall Tau-b results, where ‘n’ is the number of observations, the following formula can be used to obtain a Z-score that can be referred to the normal distribution:

Z= (3τ_b*√(n(n-1)))/√(2(2n+5))

Once a Z-score is obtained, it can then be mapped to a p-value from the normal distribution to check for statistical significance. A Z-score of ≥1.96 denotes that with 95% confidence a statistically significant relationship exists. Using a Z-score of 1.96, along with 14, 12 and 7 as the numbers of observations, τ_b is equal to 0.39, 0.43 and 0.62 respectively. These thresholds were chosen as indicating that, with 95% confidence, a significant relationship exists between the pair of parameters.

For sub-state analyses’ spatial delineations, Konieczki and Heilman (2004), Ramsey et al. (2009), and Wallace et al. (2012) were used as references to break up the Great Basin and Colorado River basins according to counties in Utah (see thesis for more information). Northern and southern area counties were split according to visual inspection, each representing roughly half of the state (see thesis for maps of these areas).

The R programming software (R version 4.1.0 (2021-05-18) -- "Camp Pontanezen") was used to analyze statewide and county-level USGS irrigation data for Utah, utilizing Kendall’s Tau-b test to discover significance of relationships between parameters. R software produces matrices of Kendall Tau-b results. Using the R ‘corrplot’ package, the lower half of results matrices were plotted.