Idealized Testing: An Example using Quantile Mapping

1. Idealized Testing: An Example using Quantile Mapping Joe Barsugli

2. What is Quantile Mapping? • A bias correction technique that corrects for the whole distribution of values, not just the mean • Has been used in weather forecasting to adjust forecast model output • Used in the BCSD and BCCA downscaling method • We will focus on the “BC” in these methods

3. BC One GCM; and Bias Corrected data using quantile mapping GCM 2050 2100 1950 2000 36 GCM runs from 16 GCMs ; A2 emissions scenario GCM BC When run through a hydrologic model the amounts shown here lead to approximately 1 billion cubic meters of additional flow at Lee’s Ferry in the Bias Corrected Runs

4. The percent change in the future forthe bias corrected data is modified from, and can even differ in sign from, the GCMs. This is not true everywhere, and depends on the details of the probability distributions. Q: Can we use idealized testing to better understand what features of the probability distribution contribute to this effect?

5. Assume that precipitation follow a Weibull Distribution (this has been assumed in many publications for daily precipitation going back to at least Dan Wilks’s work in the late 1980’s) Quantile mapping formulas are solvable with pencil and paper for the Weibull distribution. If we assume that the Past and Future GCM data, as well as the past observed data are all from Weibull distributions with different parameters, then we can compute the bias corrected future data, which is also a Weibull distribution. We can then see if the same “wettening effect” happens, and under what combination of parameters. We will be looking at distribution like this one

6. BC change -8.6% GCM change -10.4%