Irrigation Notes

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California State University, Fresno, California 93740-0018

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January 1988

Coefficient of Uniformity - What it tells us^*
By David F. Zoldoske and Kenneth H. Solomon

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 ½Xi-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."

Figure 1. Denso-gram

This process transforms the actual catchment values into various intensities of shades. This is done by setting the wettest area to the value of one and displayed as black and setting an area of no water equal to the value of zero and displayed as white. All other values fall between these two points and are given shading values corresponding to their relative position between zero and one. However, while this method may give a "feel" for overall uniformity, it does not provide for a quantitative means to measure uniformity.

A better method might be the sliding window (Figure 2), which is moved systematically over the entire set of water application values found in the sprinkler overlap area. As the window is moved through the overlap area, the values falling in the window are averaged. The average window value is stored as it moves through the sprinkler array. After the window has passed through the entire array, the average window values are sorted and the lowest mean value is identified. This low critical window value is then divided by the total overlap mean, and when multiplied by 100, produces a coefficient in percent which directly addresses the size and magnitude of the critical area.

Figure 2. Sliding Window

It can be stated mathematically as follows:

Za = 100 (M¢/M)

where Z = the sliding window coefficient a = the window size (%)

100 = a constant for percentage

M¢ = the low critical window value

M = the mean overlap value

The window can be any size, but 2, 5, and 10 percent of the overlap area represent values of practical interest. The ability to configure the window to various sizes allows for a sensitivity analysis of the problem area. This gives the irrigation specialist the means to compare changes in overall irrigation efficiencies to specific changes in window size. In other words, the ability to size up the problem.

Once the irrigation specialist has determined the irrigation uniformity of his system, he needs to address the specific amount of water required for quality turf. To help identify the required amount of water to be irrigated, a production function type curve for turf quality can be developed. This curve relates the expected turf quality to the quantity of water applied. Turf quality is rates on a scale of 1 to 10 with 1 indicating very poor and unacceptable and 10 exceeding excellent turf.

Three regions are indicated in Figure 3 (Water Requirement/Turf Quality Curve). Region I is obviously under-irrigation, showing dry or weak turf; and Region III is obviously over-irrigation, typical of water logged or fungus damaged turf and may exceed economic constraints.

Figure 3. Water Requirement/Turf Quality Curve

Region II is adequate turf quality. It would be possible to develop such a curve for each type of turf and each locale.

A coefficient could be developed based on this turf quality curve. The coefficient would use the curve to obtain the "penalty" function to assign penalties to deviations from the mean. Such a coefficient would be roughly analogous to CU, but would avoid the criticisms associated with the first two features of CU mentioned above. Under- and over-watering would not be treated equally, but would be treated as indicated by the turf quality curve and the penalties would not be assigned on a "linear" fashion, but would be assigned in accordance with anticipated effects on quality.

In the above example, the turf manager has selected a turf rating (quality) of 7 or better. In his area, this requires an effective irrigation of between 0.24 and 0.48 inches daily during the summer months. Anything less than 0.24 inches will produce less than desired turf rating and anything more than 0.48 inches will exceed his predetermined economic constraints. The turf manager further decides to use the sliding window to determine if his irrigation system will provide the necessary coverage. Using a window size of 10 percent, the "low" window produces a coefficient of 71 and the "high" window produces a coefficient of 132. When these coefficients are multiplied by the mean application rate of 0.36 inches, the low and the high rates are 0.26 and 0.48 inches, respectively. These two extremes fall within the parameters set forth in Figure 3. Thus, the current irrigation system should

perform satisfactorily with proper water management.

With PC computers becoming a part of turf managers everyday decision-making process, the ability to model irrigation efficiencies and amounts will become increasingly important. This ability will allow for fine tuning of the irrigation system to improve overall turf quality while minimizing the ever increasing costs of power and water.



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