Every dam safety file contains a number that tries to describe the greatest rainfall physically possible over the catchment, called probable maximum precipitation (PMP). However, a publication I came across recently when working on a PMP for a project in central Africa, concluded that the quantity this number claims to describe has never been demonstrated to exist. Usually, if not working on a specific topic for a while, I tend to “update” myself a bit to see what changed since the last time I worked on it, and this project made me dig a bit more on PMP.

Modernizing Probable Maximum Precipitation Estimation (NASEM, 2024) was commissioned by NOAA and published in June 2024. It is a substantial re-examination of the PMP concept, and it draws on the considerable body of research on extreme rainfall. It is not an easy read but highly recommended to everyone who works in this space. Whilst it is focused mostly on the USA, many conclusions, criticisms and recommendations apply throughout because the scientific critique is directed at the method rather than at the geography.

The aim of this article is to summarise the most interesting points because there are many. Among its conclusions is that the very definition of PMP is deficient and requires replacement.

The new definition describes PMP as the depth of precipitation for a particular duration, location and areal extent, such as a drainage basin, with an extremely low annual probability of being exceeded, for a specified climate period.

Note that there is no reference to a maximum, to a physical upper limit, or to the greatest depth physically possible, something most practitioners carry in their heads which has appeared in some form in every definition the profession has used since the 1930s.

Two new elements appear in the updated definition: the first is an exceedance probability, so that PMP becomes a depth that can be exceeded rather than a ceiling beyond which nothing lies and the second is a specified climate period, which makes PMP a quantity defined for a stated climate and therefore one that changes as the climate does.

Under the previous, deterministic definition, PMP was a property of the atmosphere waiting to be discovered; under the revised definition it is a quantile of a distribution, estimated from data, carrying the uncertainty that estimation implies.

These points are interesting because they go directly against the World Meteorological Organisation,  international authority on this topic.

Departure from WMO No. 1045

Where no national guidance exists, practice rests principally on the WMO Manual on Estimation of Probable Maximum Precipitation, third edition (WMO, 2009). The revised definition departs from it in three key respects.

1. The upper-bound framing is retained throughout WMO No. 1045, whose definitions describe the greatest depth meteorologically possible and the theoretical maximum precipitation under modern meteorological conditions. It basically implies the ceiling.
2. Stationarity is likewise explicit. The manual states that climatic trends progress slowly enough that their influence on PMP is small relative to other uncertainties in estimating these values, and that trends are therefore not considered. There is no climate change in the definition.
3. The third departure is more interesting. The 1986 edition of the manual carried both a conceptual and an operational definition of PMP, the first describing PMP as a physical limit and the second describing it as the value the accepted procedures produce, together with a discussion of confidence bands; the 2009 edition removed the operational definition and dropped the confidence-band discussion. On the one hand, the industry is moving to risk-informed decision making, on the other the international standard discarded the two elements that would have supported either development.

This does not make WMO No. 1045 a poor document and it remains the practical basis for a great deal of competent work. Hydrologists and practitioners should nonetheless be clear that this is the standard the industry is moving away from.

Theoretical and operational PMP

If asking what PMP is, there are two answers in practice. The conceptual answer is that it is the physical upper limit of rainfall for a given duration and area. The procedural answer is that it is whatever the accepted method returns. Both cannot hold for a single quantity.

A confidence interval requires a fixed quantity that the calculation is trying to estimate.

The conceptual definition names one that cannot be estimated from a storm catalogue, which is not a sample in any conventional sense; the operational definition makes the calculation and its target the same thing, leaving nothing for the interval to be about.

Perturbing the inputs and reporting the spread remains possible and useful, but a sensitivity range is not a statement of statistical confidence.

This is why PMP estimates have arrived as single numbers for as long as they have been produced.

Australian Rainfall and Runoff, one of the great guidelines in this field and readable even for beginners, has included AEP estimates for PMP since 1987, on the stated reasoning that operational PMP estimates are not theoretical upper limits and may conceivably be exceeded. This way practitioners are absolved from finding the upper bound and allowed to be, well, practical.

The case against the upper bound

A greatest possible rainfall closes the design problem, which is convenient for designer, regulator and owner alike. The idea also carries an intuitive physical justification, in that the atmosphere holds a finite quantity of water and there must therefore be a limit to how much of it can fall.

However, as per the report, no first-principles derivation has been produced that establishes an upper bound on precipitation or characterises its magnitude. The argument from atmospheric water content establishes that any given storm delivers a finite depth, which is not disputed. It does not locate the limit, and it does not constrain the distribution, since a bounded and an unbounded distribution both yield a finite depth at every exceedance probability.

Unbounded here does not mean that some rainfall depth is infinite, which never happens. It means there is no depth above which the probability is exactly zero. Observed rainfall is consistent with unbounded, heavy-tailed distributions. The practical meaning of a heavy tail is that the depths which must be treated as possible continue to grow as rarer events are considered, rather than converging on a ceiling at which the curve flattens and yields a maximum.

Finally, if rainfall depth has no demonstrable upper bound then the largest depth worth designing for is not a maximum but a depth sufficiently rare that the residual risk of exceeding it is accepted.

That is what the revised definition states. And this is engineering language, which I welcome.

Implications

The PMP evaluation is becoming a risk-informed decision, just like everything else in engineering.

What changes is the meaning behind the PMP number. It is a very rare quantile, produced by a chosen procedure, for a climate that is not the one the structure will likely experience over its design life.

The question this raises is, if PMP is a quantile, which quantile is it?

---

References

1. NASEM (2024) Modernizing Probable Maximum Precipitation Estimation. Washington DC: National Academies Press. doi:10.17226/27460.
2. Nathan, R. and Weinmann, E. (2019) Australian Rainfall and Runoff, Book 8: Estimation of Very Rare to Extreme Floods. Geoscience Australia.
3. NRC (1994) Estimating Bounds on Extreme Precipitation Events: A Brief Assessment. Washington DC: National Academies Press.
4. WMO (2009) Manual on Estimation of Probable Maximum Precipitation (PMP), WMO-No. 1045. Geneva: World Meteorological Organization.