Appendix B

History of PMP

This Appendix provides historical context for conceptual models that informed the development of PMP in the United States, and for the development and evolution of PMP definitions, expanding on the overview given in Chapter 2.

CONCEPTUAL MODELS FOR EXTREME RAINFALL AND PMP

Conceptual models have played a central role in providing a basis for a “theory” of physical limits to rainfall (PMP) and the associated evolution of PMP definitions. The conceptual model for PMP over areas unaffected by terrain influence (Figure B-1) was based on an idealized model of a convective cell (thunderstorm model), which is described and illustrated in numerous publications from the 1940s (Bernard, 1944; Paulhus and Gilman, 1953; Showalter and Solot, 1942; USWB, 1943b, 1947a). This conceptual model (Figure B-1) assumes that the convective cell is the most efficient at producing precipitation and that there is a physical limit to the depth of precipitable water in the column. Critical variables are the depth of the inflow column, the height to which this column is lifted (nearly the tropopause), and the difference in moisture between the inflow and outflow columns (Showalter and Solot, 1942). Other physical “limits” in this model were the rate at which wind can transport water vapor over a basin and the fraction of water vapor that can be converted to surface precipitation (NRC, 1994). Notably, the roots of this model were also used for application to forecasting using mass storage and vertical velocity equations described in Showalter (1944a, b). At that time, there was a tight connection between personnel developing PMP concepts and models for forecasting precipitation magnitudes. This convective cell conceptual model was later investigated in Hydrometeorological Report (HMR) 23 for possible refinement using three vertical layers (USWB, 1947b), but no changes were made. This model was used to provide generalized PMP estimates in the eastern United States (HMRs 23, 33,

and 51) through 1978. It was also used, and continues to be used, in many statewide and regional PMP studies with no modification.

Since its development in the early 1940s, no changes have been made to this conceptual PMP model that is still used in practice today (WMO, 2009). The connection between developers of PMP models and methods, and those from the forecasting community, generally ceased in the early 1970s. The basic equation to estimate PMP from an observed storm rainfall depth using moisture maximization (described in the Moisture Maximization sections in Chapters 2 and 4) has not changed since the 1940s. This model does not reflect modern atmospheric science knowledge in convection (e.g., Houze, 2004; Schumacher and Rasmussen, 2020), 50 years of advances in forecasting precipitation, and current practices in understanding and estimating extreme rainfall magnitudes that are described in Chapter 3.

Orographic Precipitation

In regions with prominent orography (most of the western United States), conceptual PMP models were developed in the early 1940s to account for orographic precipitation in California. These models were based on ideas from Bjerknes on dynamics of air currents ascending over a mountain barrier (USWB, 1943b). Various models were developed, applied, and refined to estimate PMP over the Sacramento River basin (HMR 3; USWB, 1943b), Los Angeles River Basin (HMR 21B; USWB, 1945), and San Joaquin Basin (HMR 24; USWB, 1947c). As shown in Figure B-2, these 2D models included wind, pressure, and moisture flow over a ridge.

These orographic precipitation models were further developed and improved over time in the 1950s and 1960s for generalized PMP estimates over California and the Pacific Northwest with pressure layers shown in Figure B-3 (USWB, 1966). In the 1970s and 1980s, conceptual orographic models for PMP were further developed for the Southwest (HMR 49) and the Rocky Mountains, summarized by Hansen (1987).

Investigations stagnated in 1980s on exploring, researching, and testing numerical weather prediction (NWP) models for use in estimating PMP, especially in orographic areas, to replace the models used in HMR 36 and HMR 43. Attempts were made in the early 1980s using a steady-state orographic model (Rhea, 1978), but that approach could not reproduce the June 1964 Gibson Dam storm (Hansen et al., 1988). The storm separation method (SSM) was a conceptual model that separated convergence and orographic rainfall to estimate PMP (Hansen et al., 1988). A key model assumption in the SSM was that “non-orographic precipitation is directly proportional to the effectiveness of atmospheric forcing and inversely proportional to the effectiveness of the orographic forcing mechanisms” and that this precipitation can be transposed in the domain (Hansen et al., 1988). The SSM was later used (1990–1999) to revise generalized PMP estimates in the Pacific Northwest (HMR 57) and California (HMR 59), with little to no improvement to the orographic methods.

Uncertainty of PMP Estimates and Changes over Time

The current PMP definitions convey a concept of “exact” physical (deterministic) magnitude and do not clearly convey that these quantities are estimated with uncertainty, or the fact that PMP estimates change over time (can increase or decrease) (Salas et al., 2020). In fact, significant changes in PMP estimates over time have occurred.

WMO (1986) discusses accuracy and confidence limits, invoking the use of meteorological judgment:

There is no objective way of assessing the accuracy of the magnitude of PMP estimates derived by the procedures described here or by any other known procedures, Judgement of meteorologists, based on meteorological principles and storm experience, is most important.

The delineation of lower and upper limits to PMP estimates is somewhat analogous to the confidence bands used in statistics. It would be convenient if a confidence band could be placed about a PMP estimate in an objective manner, similar to the standard statistical method, but this is not possible because PMP is not estimated by formal

statistical procedures. This limitation, however, does not invalidate the concept of a confidence band about the estimate, but it means that such limits must be based in considerable measure on judgement, as is the PMP estimate itself. WMO (2009) briefly discusses accuracy, as follows:

The accuracy of PMP/PMF estimation rests on the quantity and quality of data on extraordinary storms and floods and the depth of analysis and study. Nonetheless, it is impossible to give precise values for PMP and PMF. As yet, there are no methods to quantitatively assess the accuracy of PMP and PMF. Presently, it is most important to analyze, compare and harmonize results of PMP/PMF from multiple perspectives.

This statement is essentially the same as WMO (1986).

Generalized PMP estimates in the eastern United States have increased by 10 to 30 percent from 1947 to 1978 (England et al., 2011; NRC, 1985, pp. 47–48). In contrast, generalized PMP estimates in the Rocky Mountain region decreased by 10 to greater than 40 percent at high elevations from HMR 55 to HMR 55A (Figure B-4 below;

Hansen et al., 1988); other recent statewide studies show similar decreases compared to HMR 51 estimates. Likewise, recent PMP estimates for the states of Colorado and New Mexico decreased by up to 62 percent from HMR 55A (Table B-1 below; AWA, 2018). PMP revised estimates in HMRs 57 and 59 are highly variable for individual watersheds, ranging from –63 percent to +63 percent for watersheds in the Pacific Northwest (Table B-2; Hansen et al., 1994) and California (Table B-3; Corrigan et al., 1999). Understanding and quantifying this variability and changes over time (potential increases and decreases) should be reflected in a modern definition of PMP.

PMP DEFINITIONS

The U.S. Weather Bureau developed PMP definitions in the late 1930s and early 1940s, building on concepts and procedures introduced by the Miami Conservancy District (Showalter and Solot, 1942; see also Myers, 1967 and Chapter 2). The U.S. Weather Bureau (USWB), U.S. Army Corps of Engineers (USACE), and U.S. Bureau of Reclamation (USBR) collaborated in development of PMP concepts, definitions, and

estimation procedures, which played a central role in design and construction of large dams in the United States by USACE (Hathaway, 1939a, b) and USBR (Billington et al., 2005; USWB, 1947b). These concepts and definitions were later adopted by other federal agencies and states for use in dam design, construction, and safety programs (e.g., Leopold and Maddock, 1954; USWB, 1960) and for use in designing nuclear facilities (USNRC, 1977).

Evolving challenges and perspectives on risk, uncertainty, and physical limits to extreme rainfall informed and are reflected in PMP definitions. An important part of this history is that the definition of PMP changed over time, but the concept of upper bounds on rainfall have been a fundamental element of the evolving definitions from the 1930s to the present. Hydrometeorologists from the USWB (and later the National Weather Service [NWS]) subsequently refined both PMP definitions and concepts as they conducted studies for specific watersheds and dams (e.g., USWB, 1939) and then developed generalized PMP estimates to cover large areas (USWB, 1947b). The NWS hydrometeorologists later wrote PMP definitions and methods in guidance documents for the world through the World Meteorological Organization (WMO; 1973, 1986). PMP definitions were also published by the American Meteorological Society (AMS, 1959, 2022). The PMP definition and methods in the WMO guidance document were updated in 2009 to reflect experience and practices in China (WMO, 2009).

Current PMP Definitions

Three PMP definitions are in current use as reflected in PMP reports, textbooks, manuals, and guidance documents: HMR 52 (Hansen et al., 1982), WMO (1986), and WMO (2009). In the United States, PMP definitions are from NWS HMRs. The WMO definitions are presented and reviewed here to encompass an international perspective.

HMR 52 (Hansen et al., 1982)

The most widely used definition of PMP is from HMR 52 (Hansen et al., 1982):

Probable Maximum Precipitation (PMP). Theoretically the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of the year.

Hansen et al. (1982, p. 2) note the following regarding this definition: “This definition is a 1982 revision to that used previously (American Meteorological Society [AMS] 1959) and results from mutual agreement among the National Weather Service, the U.S. Army Corps of Engineers, and the Bureau of Reclamation.” The definition of PMP was revised in HMR 52 to focus on the fact that PMP should reflect storm area rather than watershed area, which reflected the practice of providing generalized PMP estimates (e.g., Schreiner and Riedel, 1978). They also defined three important and related terms: “PMP storm pattern,” “storm-centered area-averaged PMP,” and “drainage-averaged PMP” based on the computation methods in HMR 52. Notably, this definition of PMP is used in both WMO (1986) and AMS (2022).

WMO (1986)

WMO (1986) provides the following definition of PMP:

Precipitation associated with the uppermost limits is known as the probable maximum precipitation (PMP), which is currently defined (Hansen et al.,1982) as theoretically the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of year. Such is the conceptual definition of PMP. This definition is a description of the upper limit of precipitation potential that is storm centred, i.e., related to the center of the precipitation pattern of the storm irrespective of the configuration of the boundaries of a particular basin.

WMO (1986) was an update and revision to WMO (1973) (see Historical PMP Definitions section below for details on WMO [1973]). The principal author J.F. Miller worked extensively on precipitation frequency and PMP at NWS. WMO (1986) provided conceptual and operational definitions of PMP, with important terms and concepts on probable maximum storm, accuracy, and confidence limits. The definitions were nearly the same as those in WMO (1973), but with two important changes; (1) the use of the HMR 52 PMP definition, noting that it is “currently defined as” and (2) the critical operational definition was revised to include “with virtually no risk of being exceeded.”

Along with the definition, WMO (1986) included statements that PMP values may change with new knowledge, and that climate trends are not considered, as follows. “The values derived as PMP under these definitions are subject to change as knowledge of the physics of atmospheric processes increases. They are also subject to change with long-term climatic variations, such as would result from changes in solar radiation intensity. Climatic trends, however, progress so slowly that their influence on PMP is small compared to other uncertainties in estimating these extreme values. Climatic trends are therefore, not considered when preparing PMP estimates.”

WMO (1986) also provided an “operational definition” of PMP:

In addition to the conceptual definition of PMP, an operational definition may be considered as consisting of the steps followed by hydrometeorologists in arriving at

the answers supplied to engineers or hydrologists for hydrological design purposes. Whatever the philosophical objections to the concept, the operational definition leads to answers that have been examined thoroughly by competent meteorologists, engineers, and hydrologists and judged as meeting the requirements of a design criterion with virtually no risk of being exceeded.

WMO (1986) also defined probable maximum storm:

The term probable maximum storm (PMS), has been used to refer to any maximized, observed or hypothetical storm that is equal to PMP for durations and area sizes critical for developing the probable maximum flood (PMF) for a basin. The term has also been applied to a hypothetical storm that would produce PMP for all durations at the total basin area and somewhat lesser values for smaller areas within the basin. … PMP for various durations and sizes of area within a specific basin is usually determined by several types of storms.

This definition conflates the definition of PMP (storm area) with that of the watershed area, and what is a maximum.

WMO (2009)

WMO (2009) provides three definitions of PMP with slight variations: Summary page xxiii:

Probable maximum precipitation (PMP) is defined as the greatest depth of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year, with no allowance made for long-term climatic trends.

Section 1.1:

PMP is the theoretical maximum precipitation for a given duration under modern meteorological conditions. Such a precipitation is likely to happen over a design watershed, or a storm area of a given size, at a certain time of year.

Glossary, p. 243:

Probable Maximum Precipitation (PMP) Theoretically, the greatest precipitation for a given duration that is physically possible over a given watershed area or size of storm area at a particular geographic location at a certain time of year, under modern meteorological conditions.

The slight variations between the three WMO (2009) definitions can lead to some confusion. The first explicitly excludes climate trends, whereas the second and third state “under modern meteorological conditions” without defining what that means. The second includes the concept of “theoretical maximum” whereas the first and third refer to the greatest precipitation meteorologically or physically possible. The first and third

could produce the heaviest, meteorologically-possible precipitation over a specific area for all durations within a storm. A distinction between precipitation and storm is now generally recognized. The probable maximum precipitation, or PMP, as now generally known, for a specific area for various durations is usually determined by several types of storms. For example, the PMP for an area under 100 sq. mi. and for durations less than 6 hours is very likely to be realized from thunderstorms, but general storms are more likely to provide the limiting precipitation values for longer durations.

HMR 43

HMR 43 (USWB, 1966) provided generalized PMP estimates for the Pacific Northwest. Notably, PMP estimates were made as the sum of convergence precipitation and orographic precipitation. An orographic precipitation model was used following HMR 36. A definition of PMP is stated in Chapter 1 as follows, noting “rainfall that approaches the upper limit.” There are several important statements on PMP estimation that follow the definition, including wind, how much to maximize, and the use of judgments (emphasis added).

PMP over a watershed is the depth of rainfall that approaches the upper limit of what the atmosphere can produce. In mountainous regions it is derived in part by physical methods, in that maximum winds and moisture are input to an orographic storage equation that makes use of several principles of airflow to compute precipitation due to lift by mountain slopes. Involved in the procedure is maximizing storms of record for moisture and indirectly for wind. How much and which storms to maximize is all-important to the upper limit. Much of this report is devoted to answering these questions. It is easily shown that if storm transposition were unlimited and if maximum values of winds, moisture and other variables of storms were combined, the results would be unrealistic. Limited transposition and combination of near maximum values of a variable with high but not necessarily highest values of other variables require making judgments at several steps in the procedure. Such judgments are influenced through study of record storms. Additional guidelines come from results of other PMP studies and statistical analyses of extremes in observed variables.

HMR 51

HMR 51 (Schreiner and Riedel, 1978) provided updates to generalized PMP estimates for the eastern United States, along with estimates for larger drainage areas. HMR 51 provided two definitions of PMP from AMS (1959) and WMO (1973), with notes about estimation and judgment, as follows (emphasis added).

PMP is defined as the theoretically greatest depth of precipitation for a given duration that is physically possible over a particular drainage area at a certain time of year (AMS, 1959). In consideration of our limited knowledge of the complicated processes and interrelationships in storms, PMP values are identified as estimates.

Another definition of PMP more operational in concept is ‘the steps followed by hydrometeorologists in arriving at the answers supplied to engineers for hydrological design purposes’ (WMO, 1973). This definition leads to answers deemed adequate by competent meteorologists and engineers and judged as meeting the requirements of a design criterion.

HMR 52

HMR 52 (Hansen et al., 1982) was created as a supplement to HMR 51. The report established procedures to apply PMP estimates found in HMR 51 to watersheds. The definition of PMP was revised in HMR 52 to focus on the fact that PMP reflects storm area rather than watershed area. This revised definition is listed below, along with definitions for storm pattern and area averages (emphasis added).

Probable Maximum Precipitation (PMP). Theoretically the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of the year. (This definition is a 1982 revision to that used previously (American Meteorological Society 1959) and results from mutual agreement among the National Weather Service, the U.S. Army Corps of Engineers, and the Bureau of Reclamation.)

PMP Storm Pattern. The isohyetal pattern that encloses the PMP area plus the isohyets of residual precipitation outside the PMP portion of the pattern.

Storm-centered area-averaged PMP. The values obtained from HMR No. 51 corresponding to the area of the PMP portion of the PMP storm pattern. In this report all references to PMP estimates or to incremental PMP infer storm-area averaged PMP.

Drainage-averaged PMP. After the PMP storm pattern has been distributed across a specific drainage and the computational procedure of this report applied, we obtain drainage-averaged PMP estimates. These values include that portion of the PMP storm pattern that occur over the drainage, both PMP and residual.

HMR 55A

HMR 55A provided generalized PMP estimates for an area between the 103rd meridian and the Continental Divide. It retained the PMP definition from HMR 52 and provided definitions for generalized and individualized estimates.

Generalized. When used as an adjective to modify names such as PMP or estimates or charts, is to be taken in the sense of “comprehensive,” i.e., pertaining to all things belonging to a group or category. Thus, a generalized PMP map for a specific area and duration defines PMP for all points in the region; no location is excluded.

Individualized. As applied to drainage estimates, indicates studies for specific drainages that include considerations for possible local influences. In the sense of applications to

specific basins, it is commonly implied that information obtained from a generalized study will be processed and result in specific drainage-averaged values.

HMR 57

HMR 57 (Hansen et al., 1994) provided updated generalized PMP estimates from HMR 43. It retained the PMP definition from HMR 52. Some relevant definitions and philosophy for the estimates are as follows, from HMR 57 section 1.2.

The definition of PMP was changed in 1982 (Hansen et al., 1988) to read, “theoretically, the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of the year.” This change to the definition used previously (American Meteorological Society, 1959), and in HMR 43, resulted from mutual agreement among the NWS, the U. S. Army Corps of Engineers (COE), and the Bureau of Reclamation (USBR) among others. The new definition stresses the independence of atmospheric control over precipitation from that relative to a particular drainage area mentioned in the earlier definition.

The foundation of PMP estimation lies in observations of rainfall amounts as observed in major storms. PMP studies deal with the potential rainfall that may be produced from the coincidence of an optimum set of atmospheric conditions and circumstances. It is important to realize that the PMP is a theoretical value that represents a limiting precipitation amount for a particular duration and area, and as such is not a quantity that is expected to be observed. Because of this concept, the PMP in this report as others should always be regarded as an estimate. Recent NWS PMP reports (Schreiner and Riedel, 1978; Hansen et al., 1988) have described the procedures used to derive PMP estimates, based on observed storm rainfall maxima and atmospheric knowledge.

Two important atmospheric conditions that are considered in most PMP studies are the moisture content and the efficiency with which a storm converts moisture into precipitation. A procedure known as moisture maximization is used to approximate the highest moisture potential in storms. It is also recognized that records of observed storm rainfalls are relatively short, generally less than 100 years. One means to improve the adequacy of the storm sample has been to apply a procedure of storm transposition. By increasing the storm sample at a location through transposition, it is assumed that at least one storm in the sample has contained maximum efficiency. This assumption is necessary because not all aspects of the physical processes resulting in the most extreme rainfall are known. PMP estimates are the result of envelopment and smoothing of a number of moisture maximized, transposed storm rainfall amounts. This report will discuss these procedures as applied to Pacific Northwest storms.

The concept of PMP as an upper limit often evokes concerns that the procedure combines maximized quantities to reach a level that cannot reasonably be expected to occur. It will be noted in this study, as in past NWS studies, that this is not the case. While moisture is indeed maximized, numerous other factors are involved at a lesser level to effectively control unreasonable compounding of extremes.

Terrain plays an important role in precipitation and can act both to enhance as well as reduce (shelter) observed rainfall. It is well known that storms that move slowly or become stalled, or reoccur over a specific location result in more precipitation falling in a particular rain gage than do rapidly moving storms. Thus, orographic effects from storm-terrain interactions to the extent that they trigger moisture release or block storm movement, play an important role in PMP studies. The Pacific Northwest has some of the most complex terrain features in the country and makes this region a difficult, although interesting, challenge for study.

HMR 59

HMR 59 (Corrigan et al., 1999) provided updated generalized PMP estimates from HMR 36. It retained the PMP definition from HMR 52 (as did HMR 55A). Some relevant definitions and philosophy for the estimates are as follows, from HMR 59 section 1.3.

The PMP definition used for this report was given in HMR 55A (1988) as ‘theoretically, the greatest depth of precipitation for a given duration that is physically possible over a given storm area at a particular geographical location at a certain time of the year.’ This is slightly different from the previous definition (American Meteorological Society 1959), which was used in HMR 36. The HMR 36 definition stressed that the estimate was for a particular drainage area. The current definition is more generalized, and emphasizes the control the atmosphere has over a broad geographic region. At the same time, the techniques from this report provide estimates of PMP for specific basins.

The PMP storm for a region is considered the upper limit of precipitation. Moisture maximization, storm transposition, and envelopment are tools that provide estimates of the upper limits of precipitation for a region from intense storms. However, the remaining procedures used to develop a PMP design storm do not maximize the other factors involved in the estimation of these potential storms. Moisture is maximized, but other factors are allowed to act in a lesser manner, so that an unreasonable compounding of extremes does not occur. These procedures produce a PMP design storm. For orographic regions, only that portion of the precipitation that can be considered non-orographic is transposed. No attempt is made to transpose the orographic components of a storm.

World Meteorological Organization Manuals

WMO first produced a manual for estimating PMP in 1973. It was revised in 1986 and 2009 with slight changes in definitions. The 1973 and 1986 manuals were written by current or recent employees of the NWS Office of Hydrology, implying endorsement by NWS.

WMO 1973

WMO (1973) provided conceptual and operational definitions of PMP as listed below. The conceptual definition is from AMS (1959). Important terms and concepts on

than can general storms, but they are relatively short-lived, and individual storms cover relatively small areas. General storms, although they often include thunderstorms, produce less intense rainfall on the average, but their longer life and greater areal coverage result in greater rainfall amounts for durations of about 6 hours and longer, and for large areas.

Normally, it would appear illogical to assume that PMP for all durations and sizes of area could be realized from one storm, but this is not necessarily so. PMP for small basins may be, and is often assumed to be, obtainable from a single storm. In such cases, PMP and PMS are synonymous, but this is not always so. PMP values for all ranges of duration and sizes of area in a basin are always understood to represent limiting rainfall amounts without regard to storm type. In other words, PMP values envelop the probable maximum amounts that might be realized from any type of storm that could produce heavy precipitation over the basin. PMS, on the other hand, may refer to any maximized observed or hypothetical storm that is equal to PMP for at least one duration and size of area. The term has been applied also to a hypothetical storm that would produce PMP for all durations at the total basin area and somewhat lesser values for smaller areas within the basin.

Accuracy of PMP estimates

That the procedures described here for deriving estimates of PMP yield results to the nearest millimeter or tenth of an inch should not be taken as an indication of the degree of accuracy of the estimates. There is no objective way of assessing the general level of PMP estimates derived by the procedures described here or by any other known procedures. Judgment based on meteorology and experience is most important. Obviously, estimates subsequently exceeded by observed storm rainfall were too low. There is no way, however, that an estimate can be labelled with certainty as being too low or too high at the time it is mode. Their accuracy may be assessed, however, by consideration of the following factors: (1) excess of estimated PMP over the maximum observed rainfall values for the project basin and surrounding region; (2) number and severity of record storms; (3) limitations on storm transposition in the region; (4) number, character, and interrelationship of maximizing steps; (5) reliability of any model used for relating rainfall to other meteorological variables; and (6) probability of occurrence of the individual meteorological variables used in such models, with care being taken to avoid excessive compounding of probabilities of rare events.

Subsequent chapters show that various steps in the procedures require meteorological judgment. Consequently, the resulting estimates can be conservative or liberal depending on decisions affecting the degree of maximization used in their derivation. Thus, in effect, lower and upper limits to PMP can be estimated, although only one set of values is usually derived.

Confidence bands

The delineation of lower and upper limits to PMP is somewhat analogous to the confidence bands used in statistical work. It would be nice if a confidence band could be placed about a PMP estimate in an objective manner, similar to the standard statistical method, but this is not possible because PMP is not estimated by formal statistical

methods. This limitation, however, does not invalidate the concept of a confidence band, but it means that its limits must be based in considerable measure on judgment, as is the PMP estimate itself. Factors influencing such judgment are the same as those for assessing the general level of PMP listed in the preceding paragraph.

WMO 2009

WMO (2009) was an update and revision to WMO (1986), with revisions that reflect experience since 1986 in the United States, Australia, and India, with a focus on direct watershed estimates of PMP in China. The PMP definition was changed to indicate that PMP could represent a storm area or a design watershed, and eliminates the operational PMP definition. This version recognizes PMP is an estimate of a physical upper limit and makes simple statements about its accuracy. The discussion on confidence bands in WMO (1986) was eliminated. WMO (2009) provides three slightly different definitions of PMP. The first one includes “meteorologically possible” and a statement about climate trends; the second one includes the “theoretical maximum” concept and “modern meteorological conditions,” without mentioning climate trends. The second definition is close to the third definition that is listed in the glossary. The term “modern meteorological conditions” is not specifically defined.

Definition of PMP (Summary page xxiii)

Probable maximum precipitation (PMP) is defined as the greatest depth of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year, with no allowance made for long-term climatic trends.

Definition of PMP (Section 1.1)

PMP is the theoretical maximum precipitation for a given duration under modern meteorological conditions. Such a precipitation is likely to happen over a design watershed, or a storm area of a given size, at a certain time of year. Under disadvantageous conditions, PMP could be converted into PMF – the theoretical maximum flood. This is necessary information for the design of a given project in the targeted watershed.

Definition of PMP (Glossary, p. 243)

Probable Maximum Precipitation (PMP) Theoretically, the greatest precipitation for a given duration that is physically possible over a given watershed area or size of storm area at a particular geographic location at a certain time of year, under modern meteorological conditions.

1.4.1 Basic knowledge

Storms, and their associated floods, have physical upper limits, which are referred to as PMP and PMF. It should be noted that due to the physical complexity of the

phenomena and limitations in data and the meteorological and hydrological sciences, only approximations are currently available for the upper limits of storms and their associated floods.

1.6 ACCURACY OF PMP/PMF ESTIMATION

The accuracy of PMP/PMF estimation rests on the quantity and quality of data on extraordinary storms and floods and the depth of analysis and study. Nonetheless, it is impossible to give precise values for PMP and PMF. As yet, there are no methods to quantitatively assess the accuracy of PMP and PMF. Presently, it is most important to analyse, compare and harmonize results of PMP/PMF from multiple perspectives. This task is called a consistency check in the United States (Hydrometeorological Reports 55A, 57 and 59: Hansen and others, 1988; Hansen and others, 1994; and Corrigan and others, 1998) and is termed a rationality check in China (section 7.2.7 of the manual). Results are quality controlled through such a comparison. When evaluating various PMP estimates, there are some other aspects to consider:

1. the amount by which the estimated PMP exceeds the maximum observed rainfall values for the surrounding meteorologically homogeneous region;
2. the frequency and severity of recorded storms that have occurred in the region;
3. limitations on storm transposition in the region;
4. the number of times and character of maximization, and correlations between them;
5. the reliability of relations between rainfalls and other meteorological variables in the model;
6. occurrence probabilities of individual meteorological variables in the model, though excessive combination of rare occurrences should be avoided.

Although the procedures described here produce PMP estimates to the nearest millimetre or tenth of an inch, this should not be used to indicate the degree of accuracy.

Federal Agency and Industry Guidelines

NRC 1985 Safety of Dams Flood and Earthquake Criteria

NRC (1985, pp. 56-57) contained the following finding relevant to modernizing PMP and definitions:

The committee has found general agreement in the following observations regarding current spillway capacity criteria:

• Interpretations of data from past storms and storm model concepts are required to make estimates of PMP.
• As shown by past experience, PMP estimates can change as more data become available; thus, the PMP estimate cannot be regarded as a fixed criterion, but confidence in the estimates should rise with successive PMP estimates for a given locality.

Swain et al. (2006) restate the PMP definition from HMR 52 and note PMP estimates from the HMRs are used:

The PMP, as defined by these three agencies at that time, is “theoretically, the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of the year.” PMP must always be termed as an estimate because there is no direct means of computing and evaluating the accuracy of the results. Since the mid-1980s, Reclamation has considered that the series of HMRs prepared and updated by the National Weather Service provide the best estimates of PMP potential within the limits of each report.

USACE 1991 Engineering Regulation 1110-8-2 Inflow Design Floods for Dams and Reservoirs

The USACE (1991) regulation utilizes the PMP definition from HMR 52.

NRC NUREG/CR-7046 and ANSI/ANS 2.8-2019

Prasad et al. (2011) summarized the design flood estimation at nuclear power plants for the Nuclear Regulatory Commission; they cite the WMO (1986) PMP definition in the glossary of their report. Prasad et al. (2011) acknowledge the idea of risk for design purposes as follows.

In the past, NRC [U.S. Nuclear Regulatory Commission] adopted the concept of a “probable maximum event,” for estimating design bases. The probable maximum event, which is determined by accounting for the physical limits of the natural phenomenon, is the event that is considered to be the most severe reasonably possible at the location of interest and is thought to exceed the severity of all historically observed events. For example, a probable maximum flood (PMF) is the hypothetical flood generated in the drainage area by a probable maximum precipitation (PMP) event. … The PMP is assumed to be a theoretical maximum and its estimation uses no associated probability distribution. In standard practice, estimating the PMF from the PMP involves some subjectivity and also uses no probabilistic basis.

More recently, probabilistic methods have also gained acceptance for determining design-basis events. The advantage of probabilistic methods is that an estimate of the probability-of exceedance of the selected design basis can be made. This capability enables clear articulation of the level of risk that an SSC important to safety encounters during its operation. The emphasis, therefore, is not on determining the worst-case scenario as a basis for design, but to state the level of risk a chosen design would face.

The American Nuclear Society (ANS, 2019) in its revision to ANSI/ANS 2.8 rescinded the use of PMP and Probable Maximum Flood (PMF) as a design flood standard, replacing it with a probabilistic flood hazard evaluation. Relevant excerpts for modernizing PMP are as follows.

This standard differs from its predecessor in the following areas:

• The applicability of the standard extends to all nuclear facilities, not just power reactors.
• Probabilistic assessment: This standard replaces the prescriptive “probable maximum” approach for establishing design flood hazards with a probabilistic approach for analyzing the frequency and magnitude of flood hazards. Thus, this standard focuses on the performance of a probabilistic flood hazard assessment and development of site probabilistic hazard frequency curves. An integral part of this process is the treatment of uncertainty.

UK PMP/PMF Improvements 2021 Draft

We propose this working definition of the present-day PMP: The greatest depth of precipitation for a given duration that is meteorologically possible under contemporary climatic conditions for a catchment at a particular time of year.

American Meteorological Society

AMS first provided a definition of PMP (AMS, 1959) that is based principally on HMR 33. A subsequent revision (AMS, 2022) reflects the HMR 52 theoretical definition that focuses on storm area. Both are shown below.

Probable maximum precipitation - (Also called maximum possible precipitation.) The theoretically greatest depth of precipitation for a given duration that is physically possible over a particular drainage area at a certain time of year. In practice, this is derived over flat terrain by storm transposition and moisture adjustment to observed storm patterns. (AMS, 1959)

Probable maximum precipitation - [Also called maximum probable precipitation, maximum possible precipitation (rare).] Theoretically, the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of year. (AMS, 2022)

Hydrology Texts and Statewide PMP Reports

Handbook of Applied Hydrology – 1964

From Gilman (1964, pp. 9-62 to 9-64)

… For these reasons, design engineers have asked meteorologists for estimates of the probable maximum precipitation as the basis for design of such spillways.

The estimates represent the best judgment of the meteorologists of the realistic upper limit of precipitation that can occur. Many meteorologists have thought that there is no upper limit on precipitation amount - that any given amount can conceivably occur. Such a view is not realistic mathematically or physically speaking, since it is certainly

possible to put an upper bound on the precipitation that can occur. And if it is possible to fix an upper bound, there must exits a least upper bound, which might be called the possible maximum, or probable maximum, precipitation (PMP).

Advantages of the procedure are several. It provides empirical or statistical controls. The values are directly related to the largest that have occurred. The experience of a basin is extended through transposition. The use of actual storms for patterns ensures realism in that nature’s integrations are used rather than hard-to-justify synthetic ones. The overcompounding of probabilities is minimized.

Several features of the results obtained are worthy of note. The highest estimates of PMP often exceed the greatest value of observed precipitation in certain basins by only a small percent. In other basins they may be several times as great as the maximum observed. The greatest maximizing process for a given basin is storm transposition. If a precipitation value several times as large as any over a given problem basin has been observed over a nearby basin, then it is considered that the observed isohyets in the actual storm can be transferred, or transposed, so as to indicate the maximum amount over the problem basin. … The principal place this difficulty appears is in the rugged mountainous areas such as the West Coast of the United States. The estimation of PMP in mountainous areas is usually considered to be much more uncertain than in large homogeneous areas.

Wiesner 1970

From Wiesner (1970, p. 186)

Definition of probable maximum precipitation, PMP. It is recognized that there is a physical upper limit to the amount of precipitation that can fall over a specified area in a given time (Bernard, 1944). This upper limit has become known as the Probable Maximum precipitation, PMP, and is more precisely defined as, that depth of precipitation, which, for a given area and duration, can be reached, but not exceeded under known meteorological conditions.

Chow Maidment Mays 1988

From Chow et al. (1988, p. 418)

The concept of an estimated limiting value is implicit in the commonly used probable maximum precipitation (PMP) and the corresponding probable maximum flood (PMF). The probable maximum precipitation is defined by the WMO (1983) as a “quantity of precipitation that is close to the physical upper limit for a given duration over a particular basin.

Viessman et al. 1989 Introduction to Hydrology, 3rd Edition

Viessman et al. (1989, p. 372) provide an unusual PMP definition; it includes the term “reasonable” and notes a low probability of occurrence.

The PMP is defined as the reasonable maximization of the meteorological factors that operate to produce a maximum storm. The PMP has a low, but unknown, probability of occurrence. It is neither the maximum observed depth at the design location or region nor a value that is completely immune to exceedance.

McCuen 1989 Hydrologic Analysis and Design

McCuen (1989, p. 600) cites the PMP definition from HMR 52.

Handbook of Hydrology—Maidment 1993

Smith (1993, p. 3.33) utilized the PMP definition from WMO (1986) (same as HMR 52). He notes the following, relevant for modernizing PMP, including the concept of “very low risk of exceedance.”

For design of high-hazard structures such as spillways on large dams it is necessary to use precipitation values with very low risk of exceedance. Ideally a hydrologist would like to choose design storms for which there is no risk of exceedance. A theoretical problem that has plagued the search for such a storm is determining whether there is indeed an upper limit on rainfall amount. The conclusion of Gilman in 1964 that the existence of an upper limit on rainfall amount is both mathematically and physically realistic remains valid. The spatial and temporal context of the upper bound on rainfall amount is incorporated into the definition of probable maximum precipitation (PMP) which is defined by the World Meteorological Organization as ‘theoretically, the greatest depth of precipitation for a given duration that is physically possible over a given size storm area at a particular geographical location at a certain time of year.’ A more troublesome problem than ascertaining whether an upper bound exists is determining what it is.

Viessman and Lewis 2002 Introduction to Hydrology, 5th Edition

Viessman and Lewis (2002, pp. 551–552) present a definition that retains the “reasonable” term and provide two other definitions, including one with the notion of a “low probability of occurrence.”

The PMP is generally defined as the reasonable maximization of the meteorological factors that operate to produce a maximum storm for any given duration and areal extent. Other definitions have been proposed, including:

1. The PMP is the maximum amount and duration of precipitation that can be expected to occur in a drainage basin.
2. The PMP is the flood that may be expected from the most severe combination of critical meteorologic and hydrologic conditions that are reasonably possible in the region. The PMP has a low, but unknown, probability of occurrence. It is neither the maximum observed depth at the design location or region nor a value that is completely immune to exceedance.

Shuttleworth 2012

Shuttleworth (2012, pp. 207–208) provided a definition of PMP along with relevant discussion (emphasis added).

A further measure of extreme precipitation for a region that might be helpful in infrastructure design is the concept of probable maximum precipitation (PMP). Although the name implies PMP is a statistical measure, it is largely a physical estimate of what might be the greatest possible precipitation given a certain set of extreme atmospheric conditions. PMP is a hypothetical concept which is defined as “the analytically estimated greatest possible depth of precipitation that is physically possible and reasonably characteristic over a geographical region at a certain time of year”. PMP is usually defined with respect to a given area, often a drainage basin, and includes estimates of the inflow of moisture over the basin and the maximum likely amount of that moisture which could be precipitated.

The name total precipitable water (W) is inaccurate because not all of the water in the atmosphere can be precipitated by any known mechanism. Consequently, in addition to depending on W, the calculation of PMP needs to recognize and make allowance for realistic restrictions on the rate of convergence of water vapor towards a storm and the maximum effect of vertical motion within a storm. One approach used to estimate PMP is to adopt (and if necessary transpose from elsewhere) models of real extreme storms to estimate these additional restrictions, but then to index these to local extreme values of W. However, the assumptions and generalizations made when adopting the storm model approach are such that a sometimes preferred technique involves the use of actual storm occurrences, which are then ‘maximized’ to become an extreme storm for the area using the highest observed surface dew points and most extreme morphological conditions. … In regions with topography the estimation of PMP is much more difficult.

Bedient 2019

From Bedient (2019, p. 191):

When it is not possible to reduce the risk to a desired level by designing for a high (but hypothetical) return period event, an alternative is to design for the probable maximum flood, which is the flood that results from the probable maximum precipitation (PMP) event. The PMP is the highest precipitation likely to occur under known meteorological conditions (Smith, 1993; Mays, 2001) and has been computed for most areas by the National Weather Service Hydrometeorological Design Studies Center.

Handbook of Applied Hydrology Singh 2017

Mukhopadhyay and Kappel (2017) rely on the HMR 52 PMP definition; they also include brief statements on estimation, probability, and climate change.

Probable Maximum Precipitation (PMP) is defined as the theoretically greatest depth of precipitation for a given duration that is physically possible over a given size storm

area and reasonably characteristic over a given geographic location at a certain time of the year (Hansen et al., 1982; Hansen, 1987). PMP is only an analytical estimate representing a theoretical upper limit and therefore cannot be exact. Generally, the probability of occurrence of a PMP is not given in its estimation. Furthermore, due to the large uncertainty in regards to the effects of climate change, any potential effect of climate change on the estimation of a PMP is generally not evaluated (WMO, 2009).

Statewide PMP Studies

Statewide PMP studies, such as for Nebraska (Tomlinson et al., 2008), Texas (AWA, 2016), and North Dakota (AWA, 2021), generally use the HMR 52 PMP definition.