Scottish Road Network Climate Change Study: UKCP09 update Autumn 2011

Appendix A: UKCP09 probability density functions

The following description has been copied from the UKCP09 technical report (Murphy JM, et al, 2009). It gives a clear description and interpretation of the meaning of the probabilistic density functions that underpin the UKCP09 projections.

How are probabilistic projections presented?
Explaining PDFs and CDFs

The provision of probabilistic projections is the major improvement which the UKCP09 brings to users. However, to utilise these appropriately, it is essential that users have a good understanding of what they mean and how they are communicated.

Probabilistic projections assign a probability to different possible climate change outcomes, recognising that (a) we cannot give a single answer and (b) giving a range of possible climate change outcomes is better and can help with making robust adaptation decisions, but would be of limited use if we could not say which outcomes are more or less likely than others.

Within any given range of plausible climate changes, we cannot talk about the absolute probability of climate changing by some exact value - for example a temperature rise of exactly 6.0ºC. Instead we talk about the probability of climate change being less than or greater than a certain value, using the Cumulative Distribution Function (CDF). This is defined as the probability of a climate change being less than a given amount. The climate change at the 50% probability level is that which is as likely as not to be exceeded; it is properly known as the median, but in UKCP09 we refer to it by the more user-friendly name of central estimate. Thus in Figure 1.5 , the CDF (a hypothetical example at a certain location, by a certain future time period, for a given month of the year, under a particular emissions scenario) shows that there is a 10% probability of temperature change being less than about 2.3ºC and a 90% probability of temperature change being less than about 3.6ºC. These statements conventionally concern the probability of change being less than a given threshold, but of course we can turn them around to give the probability of exceeding that threshold. Thus the CDF in Figure 1.5 also shows that there is a 90% probability of temperature change exceeding about 2.3ºC and a 10% probability of temperature change exceeding about 3.6ºC.

Figure 1.5: Top panel, Cummulative distribution function of temperature change for a hypothetical choice of emission scenario, location, time period and month. bottom panel, Corresponding probability density function for this hopothetical case.

The CDF would be useful for those who want to know the probability of climate change being less than some threshold where an impact of interest starts to occur. However, the CDF is not useful for understanding the relative probability of different specific outcomes. The Probability Density Function (PDF, Figure 1.5) is an alternative representation of the same distribution which is a useful visualisation of the relative likelihood of different climate outcomes. For a given value of climate change, the CDF is the area under the PDF to the left of that value of climate change. As the CDF has a maximum value of 100%, the area under the PDF curve cannot be more than 100%.

As probability is represented by the area under a PDF curve, the y-axis in Figure 1.5(b) is referred to as a probability density, with units of "per ºC". However, the PDF can be thought of more simply in relative terms by comparing the ratios of probability density for different outcomes. For instance, as the probability density at 2.9ºC is about 0.7 (per ºC) and the probability density 3.8ºC is about 0.2 (per ºC), then a temperature change of 2.9ºC is about 3.5 times more likely than one of 3.8ºC. Hence, for simplicity, PDF graphs from the User Interface are all labelled relative probability rather than probability density (per ºC).

The hypothetical distribution shown in Figure 1.5 is smooth and almost symmetrical; in practice the UKCP09 distributions vary in shape, dependent on how the effects of uncertain climate system processes combine to produce different projections for different variables, time periods and locations.

It is very important to understand what a probability means in UKCP09. The interpretation of probability generally falls into two broad categories. The first type of probability relates to the expected frequency of occurrence of some outcome, over a large number of independent trials carried out under the same conditions. For example, the chance of getting a 5 (or any other number) when rolling a dice is 1 in 6, that is, a probability of about 17%. This is not the meaning of the probabilities supplied in UKCP09, as there can only be one pathway of future climate. In UKCP09, we use the second type (called Bayesian probability) where probability is a measure of the degree to which a particular level of future climate change is consistent with the information used in the analysis, that is, the evidence. In UKCP09, this information comes from observations and outputs from a number of climate models, all with their associated uncertainties. The methodology which allows us to generate probabilities is based on large numbers (ensembles) of climate model simulations, but adjusted according to how well different simulations fit historical climate observations in order to make them relevant to the real world. The user can give more consideration to climate change outcomes that are more consistent with the evidence, as measured by the probabilities. Hence, Figure 1.5 (top panel) does not say that the temperature rise will be less than 2.3ºC in 10% of future climates, because there will be only one future climate; rather it says that we are 10% certain (based on data, current understanding and chosen methodology) that the temperature rise will be less than 2.3ºC. One important consequence of the definition of probability used in UKCP09 is that the probabilistic projections are themselves uncertain, because they are dependent on the information used and how the methodology is formulated.