For an LP3 distribution, remember to always use the common (base 10) logarithm values for the calculations. For a GEV distribution, remember to use the real space values for the calculations.
When skew is equal to zero, the LP3 distribution simplifies to a log-normal distribution which means that the logarithm of flow has a normal distribution. This is the reason for plotting LP3 frequency curves on a logarithmic scale for flow and a normal probability scale (or z scale) for AEP. The concept is called linearization. When skew is equal to zero, the frequency curve plots as a straight line. The frequency curve will be approximately linear for nominal skew values on the order of -0.5 to +0.5.
When skew is equal to zero, the mean (μ) of the frequency curve is equal to the common logarithm of flow at an AEP equal to 0.5 which is equivalent to a value of z equal to 0. The mean (μ) is equivalent to the intercept b in y=mx+b. On a log-normal frequency curve plot, y=Log(Q), x=z, and b=μ. The value of z is calculated for the standard normal distribution using the Microsoft Excel function z=NORM.S.INV(1-AEP). The standard deviation (σ) of the frequency curve on a log-normal plot is equivalent to the slope m in y=mx+b where m=σ. The equation for a log-normal frequency curve is commonly written as Log(Q) = μ + zσ which is the equation for a line written in the form y=b+xm.
The equation for an LP3 frequency curve is commonly written as Log(Q) = μ + Kσ where K depends on the skew parameter (γ) and z. The value of z is a function of the AEP where z = NORM.S.INV(1-AEP). The K parameter gives the frequency curve its curved shape because of the way K changes with AEP. When skew is equal to zero, K is equal to z at every AEP resulting in a linear frequency curve on the log-normal plot. When skew is positive, K is greater than z and increases at a faster rate than z as the AEP gets smaller. This gives the frequency curve a "concave up" shape. The opposite happens when skew is negative giving the frequency curve a "concave down" shape. Figure 2 shows an example of K values for different values of skew and AEP. Figure 3 shows how K changes relative to z for different values of skew.
The value of K can be found within Appendix 3 of see B17B or estimated directly. The workbooks provided with this study guide include a user defined function that can calculate K for an LP3 frequency curve for a given value of skew (γ) and z:
fndAdjustedWilsonHilfertyK ( γ, z )
Exercise
Graphically estimate the mean and standard deviation for the LP3 frequency curve shown in Figure 4 using the concept of linearization. Select any two points on the curve and calculate the slope (the standard deviation) and the intercept (the mean). The skew is zero for this frequency curve. Using the estimate of mean and standard deviation, estimate the flow (Q) at the 0.002 AEP (500-year) using the equation Log(Q) = μ + zσ.
Question 2. Compute values for the parameters shown in Table 3.
