The most widely accepted measure of irrigation uniformity in the turf industry is JE Christiansen's uniformity coefficient (CU). Developed before the advent of the computer, Christiansen's CU can be calculated employing only simple arithmetic procedures. Stated in formula form, CU is given by:
CU = 100 (1-D/M)
D = (1/n) å ½ Xi-M ½
M = (1/n) å Xi
Where: CU = Christiansen's Coefficient of Uniformity (%)
D = Average Absolute Deviation From the Mean
M = Mean Application
Xi = Individual Application Amounts
n = Number of Individual Application Amounts
and the two parallel vertical bars in the definition of D imply "absolute value." The absolute value of a deviation considers only its magnitude, not its sign. Thus, for a mean application of 10 (M = 10), individual application amounts of 8 and 12 (Xi = 8 and Xi = 12) both contribute absolute deviations of 2 to the determination of D.
There are three important features of the CU formula that should be recognized and considered when interpreting CU values. The first is that due to the absolute value used in determining D, CU treats over- and under-watering (relative to the mean value, M) equally. D may be thought of as an average "penalty" function - it assigns a penalty to each catchment of individual application amount. The penalty assigned to application amounts are the same equally above and below the mean.
Second, the computation of D assigns penalties in what is mathematically called a "linear" fashion. This means that the penalty assigned to each catchment is in direct proportion to the amount by which it deviates from the mean. Again, for a mean application of 10, individual catchments of 8 and 14 are "penalized" 2 and 4 units, respectively. Note that the 14 is penalized twice as much as the 8, since its deviation from the mean is twice as large.
The third feature of CU is that it is an average measurement. By comparing the average absolute deviation (D) to the mean application (M), CU indicates on average how uniform the sprinkler pattern is. It can give no indication of how bad a particular localized area might be, or how large that critical area might be.
There is no question that CU has been a valuable tool in the design and evaluation of sprinkler irrigation systems. But the three features of CU noted above have caused some to discount the significance of CU. "Over- and under-watering should be treated the same," they say. "Large deviations from the mean are far more significant than small ones. The penalty should be more proportionate to its size," suggest others. Still others state, "The average conditions are of no concern to me, I need to know how bad things are in the critical area."
Christiansen's CU has also been criticized unfairly, it seems to me as follows: It is possible for two, very different sprinkler application patterns to result in the same CU. While this observation is true, it is not really fair to criticize CU for this "defect." This potential exists for any and all coefficients that have been or could be invented. This is an unavoidable consequence of trying to represent a whole array of values (all individual application amounts, Xi) by a single indicator value. This "defect" is a trade-off that is necessary if we are to have the convenience of referring to a single performance indicator.
In spite of these criticisms, and in spite of the development of computers, elegant statistical analyses, and numerous other formulas for uniformity measure, CU is still the single most used yardstick for water uniformity. A few fundamentally different approaches are discussed below.
One method which emphasizes the under-watered area and looks at the critical regions is the "distribution uniformity," or DU. This method sorts all data points in the overlap area and ranks them from low to high, with the mean value for the lowest 25 percent (low quarter) divided by the mean value for the entire area. However, this method does not take into account the location of the water values or any benefit which might be derived from water values immediately adjacent to the low values.
A non-quantitative way to look at the overlap area is to have it graphically displayed using a shading technique ( Figure 1 ), or "denso-gram."