Modernizing Probable Maximum Precipitation Estimation (2024)

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3 State of the Science and Recent Advances in Understanding Extreme Precipitation Development of a deep scientific understanding of extreme rainfall is critical for addressing the challenges to estimating PMP in the current and future climate state. Recent advances in the understanding of atmospheric processes at the synoptic to microphysical scales, the accuracy and resolution of numerical weather prediction, and enhancements to observations of precipitation from weather radar provide the foundation for major advances in understanding storms that produce rainfall accumulations with an “extremely low annual exceedance probability.” A greater understanding of the physical processes associated with extreme rainfall can enhance current PMP procedures and guide the development of model-based methods for PMP estimation. Availability of long radar rainfall datasets, in conjunction with surface rainfall measurements, can provide the observational grounding for assessing the climatology of rainfall extremes across the United States. Advanced atmospheric models and computing resources point to the potential for modeling extreme precipitation across the range of spatial scales and durations needed for PMP estimation. Recent studies that link changes in climate to the processes that produce extreme rainfall provide insights into how a changing climate will likely affect the frequency and intensity of extreme precipitation events. Taken together, these advances make it possible to understand and model the impacts of climate change on PMP- magnitude storms and lay the groundwork for model-based approaches for estimating PMP under different scenarios of future climate. The incomplete sampling of extreme rainfall events in storm catalogs precludes the use of extreme value analysis (EVA) for PMP estimation. As opposed to current PMP practice which requires an assumption of an upper bound, EVA methods can accommodate bounded and unbounded distributions. Although the question of bounds on rainfall is unresolved, available evidence does not support the bounded assumption (as discussed below). EVA methods have substantial benefits for the estimation of PMP under a revised definition; when applied to climate model output EVA methods can enable quantitative assessment of uncertainty in PMP estimates. The application of these advances to PMP estimation pose some challenges, but, taken together, they could modernize PMP estimation in a manner that accounts for uncertainty and the impacts of a changing climate. SCIENTIFIC ADVANCES: METEOROLOGY OF EXTREME RAINFALL Storms and Storm Features Producing Extreme Rainfall The previous National Academies study of PMP concluded that “major new research initiatives are needed to improve scientific understanding of extreme rainfall events” (NRC, 1994). The past three decades have seen important advances in the understanding, 36 Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 37 characterization, and prediction of extreme rainfall, many of which are applicable to PMP estimation. Doswell et al. (1996) established a fundamental conceptual framework for understanding and forecasting heavy rain. This “ingredients-based methodology” built on some of the earlier ideas of Showalter and Solot (1942) and is framed around the key concept that large rainfall accumulations result from high rainfall rates that are sustained over a long duration. Doswell et al. expressed this mathematically as 𝑃 = 𝑅𝐷, where P is the accumulated precipitation at a point, 𝑅 is the average rain rate, and D is the duration of the rain. Rain rate R was in turn approximated by the simple equation R = Ewq, where E is the precipitation efficiency, w is the average ascent rate in the saturated updraft, and q is the water vapor mixing ratio at the base of the saturated updraft. This equation implies that high rain rates are the result of ascending moist air (from atmospheric convection, orographic “upslope” flow, frontal uplift, or their combinations) that is efficient at producing cloud and precipitation particles that fall to the ground as rain prior to evaporating. Long-duration rainfall events occur when the size, organization, and motion of the precipitation system promotes the repeated passage of rain cells with high rain rates, when favorable synoptic conditions lead to the repeated passage of storm systems over the same location, or both. The ingredients required for heavy rainfall can be difficult to measure and quantify, however. For example, convective rainfall is very sensitive to small differences in atmospheric water vapor, especially close to the surface (e.g., Schumacher and Peters, 2017). Yet vertical profiles of water vapor are not well observed: twice-daily balloon soundings are insufficient for capturing spatial and temporal variability; observations from aircraft ascents and descents are irregular and inconsistent in space and time; and remotely sensed vertical moisture profiles are promising but have not yet been widely deployed or evaluated operationally. Accurate measurements of updraft speeds are very difficult to obtain, which is discussed in greater depth below. Likewise, precipitation efficiency depends on complex interactions between environmental conditions and cloud microphysical processes that are not fully understood. Thus, although the “ingredients-based methodology” is highly useful conceptually, it is not easy to deploy for quantitative predictions of precipitation. Conclusion 3-1: Shortcomings in observations of water vapor, including its spatial and vertical structure, and water vapor flux limit quantitative estimates of possible upper bounds of heavy precipitation. A wide variety of storm types (Box 3-1) can combine in characteristic ways the necessary ingredients for extreme precipitation, from large-scale weather systems such as tropical cyclones and atmospheric rivers, to mesoscale convective systems, to supercell thunderstorms and even isolated convective cells (e.g., Schumacher, 2017). In some geographic regions, only one or two storm types may be likely to yield extreme precipitation, whereas in other regions a broader range of heavy-rain-producing storm types is possible. The availability of radar observations in the late 1940s rapidly transformed the scientific understanding of storms that produce extreme rainfall (Byers and Braham, 1949). Similar advances have followed from deployment of the Next Generation Weather Radar (NEXRAD) network across the United States in the 1990s, especially through the availability of high-resolution quantitative precipitation estimates (Fulton et al., 1998). For example, Schumacher and Johnson (2005, 2006) summarized the storm types associated with 184 24-hour extreme rainfall events in the central and eastern United States Prepublication copy 

38 Modernizing Probable Maximum Precipitation Estimation based on their characteristics as observed by radar. They found that a large proportion of the events were associated with mesoscale convective systems (MCSs), or collections of individual thunderstorm cells that act in concert, a finding confirmed in a larger dataset by Stevenson and Schumacher (2014). MCSs that produce extreme rainfall are often characterized by “echo training,” whereby individual convective cells repeatedly pass over a location, as if they were train cars lined up along tracks. The findings above used radar observations to build upon previous analyses (e.g., Chappell, 1986; Maddox et al., 1979) that established the importance of mesoscale meteorological processes to extreme rain production on temporal scales of 3-24 hours and spatial scales of tens to hundreds of kilometers (see also Gochis et al., 2015; Hitchens et al., 2013; Moore et al., 2015). BOX 3-1 Storm Types Storm types based on physical characteristics Tropical Cyclone (TC): Encompassing term for hurricanes of various strength designations (i.e., tropical depressions/storms and hurricanes), typhoons, and cyclones that form in the ocean and draw energy from high ocean temperatures and are characterized by large synoptic scale and organized deep convection. Extreme precipitation over land can be caused when landfalling tropical cyclones slow down so that single areas are exposed to the intense rainfall from the TC over an extended period of time. TCs are also capable of producing very heavy short-term rain rates. Extratropical Cyclone: A large (1,000 km or more in spatial scale) weather system at middle or high latitudes with low pressure at the center, typically with warm and cold fronts extending outward from the low-pressure center. Extratropical cyclones occasionally produce extreme precipitation along their fronts; atmospheric rivers (defined below) often develop in association with an extratropical cyclone or a series of them. Extratropical Transition: The process by which TCs transition into extratropical cyclones as they turn poleward. Fronts develop during this transition process, and heavy precipitation often occurs along these fronts. Many extreme rainfall events along the U.S. East Coast have been associated with extratropical transition. Atmospheric River (AR): Narrow plumes of intense horizontal water vapor transport, typically associated with an extratropical cyclone. Extreme precipitation can be produced when ARs encounter topography that forces the large amounts of available moisture to rise and precipitate rapidly. Ordinary Thunderstorms: A single updraft and downdraft; these occur on small spatial scales and generally last less than 1 hour. They serve as the building blocks for larger, organized clusters and lines of storms, including mesoscale convective systems. Mesoscale Convective System (MCS): Collections of thunderstorm cells, often organized into clusters or lines, that move together and produce precipitation over a spatial area of hundreds of km and last up to 24 hours. Extreme precipitation can be caused by multiple cells in a system passing over the same location one after another. continued Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 39 BOX 3-1 continued Supercell Thunderstorm: Rotating storms characterized by strong, persistent updrafts of air. Extreme precipitation can be created through rotating updrafts of higher strength and longer durations than are found in many storm types, producing higher intensity and longer durations of precipitation. Orographic Precipitation: Precipitation that is produced because of interactions of moist air with mountainous terrain that forces the air to ascend. Orographic precipitation associated with synoptic systems preferentially falls on the windward slopes, but warm season orographic precipitation associated with convection may produce more diverse spatial distribution. Storm types based on PMP applications General Storm: A designation for a storm event that produces precipitation on a relatively large scale (>1,300 km2) and long duration (>6 hours), typically associated with a major synoptic (i.e., large scale) weather feature, such as an extratropical cyclone. Local Storm: A designation for a storm event that produces precipitation on a relatively small scale (<1,300 km2, frequently around 500 km2) and short duration (<6 hours, frequently around 1-2 hours). These could include mesoscale convective systems, supercell thunderstorms, orographic precipitation, or even ordinary thunderstorm cells. Tropical Storm: A designation for a storm event that generally aligns with the meteorological definition above of a tropical cyclone. Hybrid Storm: A designation generally associated with storms undergoing extratropical transition, such that they are “hybrids” between tropical and extratropical cyclones. This designation has been used in some PMP studies for the eastern United States. Cool-Season Storm: A special case of the General Storm category that occurs in the cool season (typically November-March). These storms are particularly needed to estimate critical cool season hazards such as rain on snow and extreme floods from atmospheric rivers. Radar observations have also been used to develop storm catalog data for PMP studies. Smith et al. (1996) performed analyses of the 27 June 1995 Rapidan storm using reflectivity and Doppler velocity observations from the Sterling, Virginia, WSR-88D radar. Rainfall fields derived from reflectivity-based rainfall estimates and bucket survey observations have been integrated into recent PMP studies, in which the Rapidan storm controls PMP estimates for time periods less than 6 hours over large regions in the Central Appalachians (AWA, 2015). Storm tracking analyses based on 3-D reflectivity fields illustrate the role of storm size and motion as drivers of Doswell’s ingredients-based formulation of extreme rainfall. Storm tracking analyses also contribute to interpretations of orographic precipitation mechanisms and assessments of storm transposition assumptions in the Central Appalachians. Doppler velocity observations are used with humidity measurements to examine the atmospheric water balance of the Rapidan storm, providing insights into assumptions underlying moisture maximization procedures. Another major advance has been an increase in understanding of the importance of atmospheric rivers (ARs) for extreme rainfall (Ralph and Dettinger, 2011; Zhu and Newell, 1998). ARs are focused plumes of intense water vapor transport, typically associated with an Prepublication copy 

40 Modernizing Probable Maximum Precipitation Estimation extratropical cyclone. They include both the well-studied ARs that collide with the mountain ranges of the western United States (e.g., Ralph et al., 2006), as well as those that originate in the Gulf of Mexico or Atlantic Ocean and provide a favorable environment for extreme precipitation in the eastern United States (e.g., Barros and Kuligowski, 1998; Mahoney et al., 2016; Moore et al., 2012). ARs can enhance all three variables in the rain rate equation. By quickly transporting moisture in a focused area, they both increase the water vapor available to updrafts (increased q) and the precipitation efficiency E. Furthermore, their position near atmospheric frontal zones is often associated with increased synoptic-scale ascent, which can promote individual updrafts and heavy orographic precipitation (increased w). Another important process that has been identified as a potential contributor to PMP-type storms over short durations is updraft rotation, which enhances the updraft speed and duration of individual thunderstorms. Smith et al. (2001) demonstrated that the intense updrafts in rotating thunderstorms, known as supercells, can produce exceptionally high rain rates, in some cases exceeding 300 mm/hour. They hypothesized that many of the most extreme short-term rain events in the contiguous United States are associated with supercells. Nielsen and Schumacher (2018, 2020) confirmed these findings using observations and numerical model experiments, showing that environments with stronger low-level shear result in more storm-scale rotation, stronger low-level updrafts, and more rainfall. However, the interplay between storm dynamics and thermodynamics in the production of extreme rainfall remains an active area of research, and these processes are far from fully understood. In general, updraft speeds (i.e., w in the rain rate equation) remain very difficult to measure or estimate, especially for the intense updrafts in supercells, which are challenging to quantify from both in situ and remotely sensed measurements (e.g., Marinescu et al., 2020). Conclusion 3-2: Interactions between storm dynamics and thermodynamics in extreme rain-producing storms remain difficult to both measure and simulate. The “convective intensity” problem concerns the climatology of extreme rainfall, with supercells representing the high end of convective intensity and convective storms that do not produce lightning representing the low end of convective intensity. Are storms controlling PMP estimates at short durations and small areas concentrated on the low end or the high end of the convective intensity spectrum? The importance of the problem for PMP estimation is tied to the concentration of high-hazard structures (both dams and nuclear plants) in small watersheds (Chapter 2). Cotton et al. (2010) note that “we should not expect that the storm systems producing the largest hailstones [supercells] are also heavy rain producing storms.” Their arguments focus on storm speed and water balance, in particular the low precipitation efficiency observed in some hailstorms. The 28 July 1997 Fort Collins, Colorado, storm (Petersen et al., 1999) is an archetype for “warm rain” storm systems that produce extreme rainfall over short durations and small areas with little or no lightning (see Zipser and Liu, 2021 for climatological context). As noted above, observational evidence and numerical modeling studies show the potential for extreme rainfall at the high end of the convective intensity spectrum (see also Giordano and Fritsch, 1991). Observational evidence from the U.S. Geological Survey (USGS) suggests that storms at the high end of the convective intensity spectrum are principal agents of extreme floods in small watersheds for much of the conterminous United States (Costa, 1987; Crippen and Bue, 1977; J. Smith et al., 2018). In a recent review of the convective intensity Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 41 problem, Zipser and Liu (2021) examine “notable examples of excessive rain events over the United States, with and without intense convection.” They conclude that “evidence now amply supports the prevailing view that intense convection indeed is frequently associated with extreme rainfall rates.” Conclusion 3-3: Storms controlling PMP estimates at short durations and small areas are concentrated at the high end of the convective intensity spectrum for much of the conterminous United States. Continued research is required on how convective intensity modulates rainfall production, especially for supercells and mesoscale convective systems. Although they occur less frequently than some of the other storm types discussed above, the weather systems with perhaps the greatest potential for high-end rainfall accumulations over 1 to 4 days are landfalling tropical cyclones. For example, Kunkel and Champion (2019) showed that 5 of the top 10 rainiest 4-day storms over an area of 50,000 km2 were TCs (Figure 3-1). The largest multi-day rain accumulations are produced by tropical cyclones with slow forward motion, with Hurricane Harvey (2017) along the Gulf of Mexico coast being an exceptional example (e.g., Galarneau and Zeng, 2020; Figure 3-1). Harvey produced more than 1,000 mm of rainfall over a 7-day period in Texas and exceeded earlier PMP estimates at 3-day durations (Kao et al., 2019). Galarneau and Zeng (2020) argued that “the extended period in which deep tropical moisture overlapped with quasigeostrophic forcing for ascent is what set Harvey apart from other rainstorms in 1979–2018.” Li et al. (2020) estimated that Harvey’s maximum rainfall totals would have been much larger if the storm had followed a slower track consistent with earlier forecasts. FIGURE 3-1 Precipitation magnitudes and meteorological causes for the 30 largest 4-day events for an area size of ~50,000 km2. NOTE: Hurricanes Harvey and Florence events are indicated. SOURCE: Kunkel and Champion (2019). Prepublication copy 

42 Modernizing Probable Maximum Precipitation Estimation Topography and Extreme Rainfall Many of the storms with rainfall totals that approach PMP magnitudes occur near topographic features, where enhancement by upslope flow is possible. These include storms along the east side of the Rocky Mountains and Black Hills (e.g., Gochis et al., 2015; Maddox et al., 1978; Petersen et al., 1999), arid/semi-arid regions of the intermountain western United States (Smith et al., 2019), the Balcones Escarpment of Texas (e.g., Nielsen et al., 2016), and both sides of the Appalachian Mountains (e.g., Hicks et al., 2005; Konrad, 2001; Martinaitis et al., 2020; Pontrelli et al., 1999; Smith et al., 1996, 2001), in addition to the storms occurring along the western U.S. mountain ranges associated with ARs as described above. The detailed meteorological processes associated with these events have perhaps received less research attention in recent years, but pioneering studies of major flash floods in the 1970s remain relevant for describing how the ascent of very moist air along sloped terrain can result in rain accumulations that approach PMP. Observing the distribution of heavy precipitation in complex terrain remains a major challenge owing to sharp spatial gradients and radar beam blockage. In fact, it has been suggested that advances in modeling orographic precipitation have outpaced the ability to observe that precipitation, especially for atmospheric rivers in the western United States (Lundquist et al., 2019). The challenges to observing and modeling orographic precipitation are summarized in Banta (1990), Kirshbaum et al. (2018), and Chow et al. (2019); see also Miglietta and Rotunno (2012) and Wilson and Barros (2014). Paleoflood and geomorphic studies have pointed to “hotspots” of extreme flood events in the topographic settings detailed above. Harden et al. (2011) combine paleoflood studies and analyses of major historical storms to support a “hypothesis of distinct differences in flood generation within the central Black Hills.” Similar hypotheses have been proposed for the Front Range of the Rocky Mountains (Jarrett and Costa, 1988), the Central Appalachians (Scott Eaton et al. 2003; Smith et al., 1996), the Balcones Escarpment of Texas (Baker, 1975), and the Colorado Plateau (Webb et al., 1988). The existence of hotspots, as described by these and other studies, raise questions for PMP estimation. What is the scientific rationale for transposing major storm catalog events, such as the June 1972 Black Hills South Dakota storm and the June 1995 Rapidan Virginia storm, if these events are linked to distinctive topographic features in the settings where they occurred? Another meteorological situation that can lead to extremely large rainfall accumulations is the impingement of a landfalling tropical cyclone on elevated terrain. For example, the world record rainfalls at durations of 12, 24, 72, and 96 hours all come from tropical cyclones approaching the steeply sloped terrain on the island of La Reunion in the Indian Ocean (e.g., 4,936 mm in 96 hours in February 2007; Arizona State University, 2023). These world record- setting storms inform understanding of the processes that govern extreme orographic precipitation and may be relevant to assessing the potential for extreme rainfall in U.S. island locations such as Hawaii and Puerto Rico. Although the contiguous United States generally does not have steep terrain near the coastlines where tropical cyclones make landfall, orographic precipitation mechanisms play an important role in amplifying tropical cyclone rainfall in the Appalachian region of the eastern United States (see, e.g., AWA, 2015). Overall, research in recent decades has provided much better characterization of the types of weather systems that are responsible for extreme precipitation. This characterization enables a more informed consideration of storm types in PMP estimates, which is discussed in Chapter 4. These advances have not yet been incorporated into the approaches that are currently in use. Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 43 Conclusion 3-4: Major scientific advances have been made in understanding extreme rainfall since the 1994 National Research Council study of PMP, but they have not translated to major advances in methods for estimating PMP. SCIENTIFIC ADVANCES: RAINFALL DATA Advances and Current State of Radar Observation for Extreme Rainfall Over the past three decades, evolving methods for estimation of rainfall from radar measurements (Berne and Krajewski, 2013; Cifelli et al., 2011; Krajewski and Smith, 2002; Ryzhkov et al., 2005, 2022) have helped to advance PMP estimation (see, e.g., AWA, 2015, 2019). Radar measurements alone are generally not sufficient for accurate estimation of rainfall; surface rainfall measurements are with merged with radar observations in flood forecasting and PMP application. Surface rainfall observations from rain gauges and from bucket surveys play a critical role in developing rainfall fields from radar for PMP estimation (AWA, 2015; Baeck and Smith, 1998; Petersen et al., 1999). Storm catalog rainfall fields at spatial scales as small as 1 km and time scales as short as 5 minutes can be constructed from radar and surface rainfall observations during the current “NEXRAD era,” which began with the initial deployments of WSR-88D radars in 1992. For time periods prior to 2012, radar rainfall estimates are based on power law equations, termed Z-R relationships, relating rainfall rate R to radar reflectivity factor Z. Since the polarimetric upgrade of the NEXRAD radar network in 2012, rainfall estimates are based on polarimetric measurements, including horizontal reflectivity, differential reflectivity, and differential phase shift. Two derived variables, specific differential phase shift (KDP) and specific attenuation (A) play an important role in polarimetric algorithms for rainfall estimation (Ryzhkov et al., 2022). Like the reflectivity-only case, polarimetric algorithms are based on simple estimation equations—often power laws, with empirical parameters that must be specified. There are two paths for developing storm catalog rainfall data during the NEXRAD era. The first begins with the “raw” radar observations (termed NEXRAD Level II data) and computes rainfall fields from radar rainfall algorithms and gauge-radar merging algorithms. Existing storm catalog data have been derived in this fashion using reflectivity-only algorithms (see, e.g., AWA, 2015 and Smith et al., 1996). The second path is based on long-term radar rainfall datasets, developed either as “analysis” fields from operational radar rainfall products or as “reanalysis” rainfall datasets computed after the fact using a standardized algorithm over the entire period of record. “Stage IV” is an analysis dataset produced by compositing operational radar rainfall estimates from radars across the United States and covers the period from 2000 to the present (Nelson et al., 2016). It has a spatial resolution of approximately 4 km and a time resolution of 1 hour. Due to a range of error sources (Nelson et al., 2016), the Stage IV dataset is not suitable for PMP applications without extensive quality control and homogenization. The Multi-Radar Multi-Sensor (MRMS) rainfall estimates developed by the NOAA National Severe Storm Laboratory (Zhang et al., 2016; see also Lengfeld et al., 2020) provide an improved rainfall dataset relative to Stage IV at high temporal and spatial resolution. Archives of operational MRMS analysis products and MRMS reanalysis products are available for limited time periods. The MRMS rainfall products are designed for a broad range of hydrologic applications but are not tailored specifically for extreme rainfall events. Prepublication copy 

44 Modernizing Probable Maximum Precipitation Estimation Errors in estimating rainfall from radar arise from measurement properties—especially the range-dependent sampling of the atmosphere (Berne and Krajewski, 2013); microphysical processes that determine the number, size, and type of hydrometeors (rain, graupel, hail and snow) in a radar sample volume (Krajewski and Smith, 2002; Ryzhkov, 2022); and dynamical processes, especially through the role of updrafts and downdrafts in convective rainfall (Austin, 1987). The role of vertical motion in updrafts and downdrafts imposes fundamental limitations on the accuracy of radar rainfall estimates (Austin, 1987). A key assumption underlying radar rainfall estimation is that hydrometeors (raindrops, graupel and hail) fall at their terminal velocity, which implies that there is no vertical movement of air in the radar sample volume. Convective intensity, as discussed in the previous section, is linked to both microphysical and dynamical sources of errors in radar rainfall estimates. Long lists of empirical Z-R parameters have been tabulated and used for rainfall estimation in differing settings, with convective intensity arguments often invoked to explain the variation in empirical parameters (Battan, 1973; Krajewski and Smith, 2002). Similar issues arise for polarimetric algorithms, with “cold rain” microphysical processes in intense convection creating major challenges to estimation of extreme rainfall (Ryzhkov et al., 2022). Convective intensity is also at the heart of underestimation of extreme rainfall in strong downdrafts. Merging Radar and Surface Rainfall Observations Two general classes of procedures have been used for “merging” radar and surface rainfall observations: mean field bias correction and “local” corrections that exploit the spatial correlation structure of rainfall (Berne and Krajewski, 2013; Krajewski, 1987). Correction of mean field bias in radar rainfall estimates using rain gauge observations has long been recognized as one of the most important tools for improving the accuracy of radar rainfall estimates (Krajewski and Smith, 2002; Steiner et al., 1999). For reflectivity-based rainfall estimates, a mean field bias correction translates to changing the pre-factor in Z-R relationships, thereby providing a data-driven tool for addressing the variability of Z-R parameters noted above. Large bias corrections for PMP magnitude storms have been reported for reflectivity-only rainfall estimates (Baeck and Smith, 1998) and for polarimetric estimates (Smith et al., 2023, 2024). “Conditional bias,” in which errors in radar rainfall estimates systematically underestimate peak rain rates, can be an important factor in extreme rainfall estimation (Ciach et al., 2000). Local corrections have been used to develop rainfall fields for recent PMP studies (AWA, 2018; Parzybok and Tomlinson, 2006) and provide a path for addressing underestimation of peak rainfall. Procedures that provide local corrections impose a heavy burden on determining the accuracy of surface rainfall observations, especially in the region of most extreme rainfall. Enhancements to PMP estimation based on radar will require a concerted effort to obtain high- quality surface rainfall measurements. Availability, accuracy, and sampling properties of extreme surface rainfall measurements point to the importance of radar rainfall estimates based on the full volume-scale radar fields. Conclusion 3-5: Surface rainfall observations should be used in combination with radar observations, from both the polarimetric and reflectivity-only eras, for development of rainfall fields for modernized storm catalogs. Mean field bias algorithms and local Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 45 correction algorithms tailored for PMP application should be developed, standardized, and documented. Conclusion 3-6: Enhancements to radar algorithms for estimating rainfall fields from PMP-magnitude storms, both for reflectivity-based algorithms and for polarimetric-era algorithms, are needed to reduce the dependence on surface rainfall measurements. Standardization and documentation of these algorithms is an important step in assuring transparency in data and methods for estimating PMP. Recommendation 3-1: NOAA should facilitate development of continuous “reanalysis” rainfall datasets covering the NEXRAD era. The reanalysis should build on advances developed by NOAA through the MRMS program and target algorithm structure and parameters for estimation of extreme rainfall. The reanalysis dataset will contribute to identification of storm catalog events and development and evaluation of model-based PMP estimation methods. Radar Capabilities for Climatological Applications Radar rainfall datasets are increasingly used for climatological applications (e.g., Lengfeld et al., 2020; Saltikoff et al., 2019; Smith et al., 2024). With full deployment of radars and implementation of systematic archiving procedures (Droegemeier et al., 2002), near- complete datasets from the NEXRAD network can be developed for the period from 2000 to 2024. Conclusion 3-7: Continuous reanalysis rainfall data and storm catalog data developed from radar observations during the period 2000–2024 can provide a useful observational tool for characterizing current-climate rainfall extremes over much of the United States. Radar-based products provide the ability to assess rainfall across the range of spatial and temporal scales needed for PMP estimation. Radar rainfall estimates are of greatest utility for “local” and TC storm types. Range- dependent sampling by radar typically results in large errors in estimates of stratiform rainfall in which precipitation processes are concentrated in the lowest levels of the atmosphere. This translates to significant errors in estimates of extreme rainfall for many AR episodes along the west coast of the United States. In mountainous terrain, beam blockage and partial beam blockage complicates estimation of radar rainfall. Radar observations are, however, especially valuable for analyses of extreme rainfall in mountainous regions that are not affected by blockage issues. For the Model Evaluation Project (Chapter 5), the long-term radar rainfall dataset will provide a key observational resource for evaluating model simulations. NUMERICAL MODELING AND COMPUTING Numerical modeling of storms and extreme precipitation using models that solve the mathematical equations that describe the dynamical and physical processes of the atmosphere can augment observation-based estimation of PMP, which encounters various limitations related to the sampling and measurement errors noted above. Modeling of storms that might be capable of producing PMP must be done at a fine scale to resolve the intricate, highly transient physical Prepublication copy 

46 Modernizing Probable Maximum Precipitation Estimation and dynamical processes associated with the storms that produce the extreme precipitation. Since the 1970s, two classes of models, large-eddy simulation (LES) and cloud-resolving model (CRM), have been developed to explicitly simulate convective clouds to gain a better understanding of their lifecycle. Designed to explicitly resolve turbulent motions in an inertial subrange, LES (resolutions of ~100 m) is capable of modeling atmospheric convective boundary layers (Bryan et al., 2003), while at resolutions of ~1 km, CRM is more suited for modeling deep convective clouds and associated motions (Guichard and Couvreux, 2017). LES is more often used to simulate shallow convective clouds involving warm-phase microphysical processes; CRM must represent microphysical processes for both warm and ice phases to better simulate deep convective clouds that extend vertically above the freezing level. Besides explicit modeling of clouds and convection, LES and CRM have also been used to inform the development of parameterizations for clouds and convection for large-scale models and to better connect such efforts with local-scale field and aircraft measurements. With advances in computing in the past two decades, LES and CRM models can be used in simulations over relatively large domains, enabling the study of convective cloud ensembles and organized shallow and deep convection spanning tens-to-hundreds of kilometers in scale. At the same time, most regional weather and climate models are now equipped with nonhydrostatic solvers (hence no assumption of hydrostatic balance), allowing them to model atmospheric processes at grid spacings of a few kilometers where deep convection and the mesoscale dynamics of precipitating storm systems are beginning to be explicitly resolved. These models, known as convection-permitting models (CPM) and storm-resolving models (SRM), have played an instrumental role in bridging modeling of individual storms to modeling storms on climate timescales (Prein et al., 2015). Seasonal-to-decadal regional convection-permitting climate simulations have been performed in the past decade covering regional to continental domains, providing important insights on how storms and extreme precipitation may change under global warming (Ban et al., 2015, 2021; Chen et al., 2023; Kendon et al., 2014; Prein et al., 2017b, 2023). For example, comparing regional CPM simulations at 1.5 km grid spacing over the United Kingdom for the present day and future, Kendon et al. (2014) found significant increases in short-duration summer rain exceeding the high thresholds due to changes in the local storm dynamics with warming. However, such trends could be concealed by natural variability, as shown in an ensemble of 12 CPM simulations that capture internal variability (Kendon et al., 2023). CPM simulations covering a much larger domain across the continental United States (CONUS) also showed intensification of hourly precipitation extremes (Prein et al., 2017c). Along with CPM, computationally efficient algorithms for tracking storms such as MCSs, tropical and extratropical cyclones, and ARs in large datasets have expanded our ability to analyze and compare storms in observations and climate simulations (e.g., Feng et al., 2018, 2023; Ullrich et al., 2021). By tracking MCSs in the CONUS simulations, Prein et al. (2017) further showed an increased frequency of intense summertime MCSs, which has also been found through tracking MCS in observations of the past decades (Feng et al., 2016). Identifying precipitation objects in CPM simulations over the western United States, Chen et al. (2023) noted a sharpening of cold season storms (i.e., decreasing area-reduction-factor corresponding to a larger increase in storm peak precipitation intensity than storm area averaged precipitation intensity), particularly for AR-related heavy precipitation events. Tracking ARs in a large ensemble of global climate simulations at low resolution, Huang and Swain (2022) identified a multiweek sequence of AR storms capable of giving rise to a megaflood similar to the “Great Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 47 Flood of 1861–1862” in California. By downscaling the scenario using a regional CPM, they investigated how such an event may unfold in the future. Although the use of regional CPMs in climate research has increased in the recent decade, global CPMs have emerged in the past two decades, thanks to the development of computationally efficient algorithms for nonhydrostatic dynamical solver and mesh generation (e.g., icosahedral and cubed sphere grids) and the availability of large high-performance computing platforms (Satoh et al., 2019). Unlike regional CPMs, which have been employed to produce decadal simulations and projections, global CPMs have mostly been employed in shorter simulations because of their high computational cost. For example, the first intercomparison of global CPM, Dynamics of the Atmospheric general circulation Modeled On Nonhydrostatic Domains (DYAMOND), includes eight models running for 40 days in a boreal summer (Satoh et al., 2019; Stevens et al., 2019). With unstructured grids, global CPMs can be configured for global simulations with convection-permitting modeling limited to regions of interest. Compared to global CPMs, such regionally refined CPM simulations are computationally more affordable, because the computational cost scales with the size of the refined region, and they demonstrate comparable skill to regional CPM simulations for modeling extreme weather events (Z. Liu et al., 2022). To realize the use of global CPMs for climate and Earth system modeling, some efforts have started to couple global CPMs with land and ocean models (Hohenegger et al., 2023), but results have not yet been widely reported. Also importantly, global CPMs must achieve a minimum throughput of 1 simulated year per day (SYPD) to be practically useful for climate production runs. However, all the global CPMs that participated in DYAMOND were Fortran- based codes running on moderate-sized CPU-based supercomputers, with throughput ranging from 0.007 to 0.05 SYPD (Stevens et al., 2019). Because global CPMs must solve the governing equations on billions of grid cells, they are well suited for GPU-based exascale computers built for parallel computations on thousands of nodes. Building global CPMs for efficient and performance portable implementation on exascale computers is a significant technical challenge requiring upgrading of the CPM codes to work with GPU-programming paradigms. As an example, two global CPMs, ICON and SCREAM, have been adapted to GPU-based computers using Fortran/OpenACC and C++/Kokkos, respectively. Because C++/Kokkos allows a single code to run efficiently on a variety of high-performance computing architectures, SCREAM has demonstrated performance portability across CPUs and both NVIDIA and AMD GPUs, achieving a throughput of more than 1 SYPD on Frontier, the first exascale computer on the TOP500 list, and thereby demonstrating the viability of using global CPM for climate simulations (Taylor et al., 2023) (Figure 3-2). Compared to coarser resolution simulations with parameterized deep convection, regional and global CPM simulations have shown clear improvements in their ability to reproduce important observed features. For precipitation, the most notable improvements relate to the diurnal cycle such as the nocturnal peak in regions frequented by MCS, which has been a longstanding challenge for simulations with parameterized deep convection (Feng et al., 2023). Because complex terrain is better resolved by CPM, precipitation in mountainous regions is also noticeably improved compared to lower resolution simulations (e.g., Ban et al., 2014). CPM simulations also show significant improvements in representing the probability density function of daily precipitation rates, reducing the frequent occurrence of drizzles found in lower resolution simulations (Stephens et al., 2010), and producing more realistic intense/extreme precipitation amounts (Kendon et al., 2017; Patricola and Wehner, 2018; Prein et al., 2017a). Prepublication copy 

48 Modernizing Probable Maximum Precipitation Estimation FIGURE 3-2 Left panel: Examples of clouds simulated by SCREAM, a global CPM, at 3.25 km grid spacing and comparison with satellite data. Right panel: Throughput of SCREAM in Simulated Days per day of wall clock time (SDYD) vs. node count on the Frontier (AMD GPUs) and Summit (Nvidia GPUs) demonstrating a throughput of more than 1 SYPD on the exascale Frontier machine. SOURCE: Taylor et al. (2023). There is, however, a tendency for heavy rainfall in CPM to be too intense, because convection is still not fully resolved at kilometer scales. For example, at grid spacings of a few kilometers, entrainment is too weak to mix drier environmental air into the updrafts, leading to overly wide and strong convective updrafts and excessive convective precipitation. Such biases have been found in regional CPMs and consistently across the DYAMOND global CPM simulations (Feng et al., 2023). Despite this similarity, the DYAMOND models simulated diverse frequencies of both deep convection and organized convective systems in the tropics (Feng et al., 2023), suggesting that large uncertainties remain in CPMs because processes such as turbulence and cloud microphysics are parameterized in these models using different formulations. Lastly, although MCSs are much better simulated in CPMs than models with parameterized deep convection, CPMs still exhibit dry biases in MCS precipitation over land regions such as the central United States (Feng et al., 2018; Prein et al., 2017a). Recent studies suggested the need to improve modeling of land surface and coupled land-atmosphere processes to address the dry biases (Barlage et al., 2021; Qin et al., 2023). In summary, advances in numerical modeling and computing have enabled the employment of regional and global CPMs to model intense precipitation of different durations associated with different storm types. Although these advances lay the foundation for model- based estimation of PMP, more efforts are needed to address CPM biases and uncertainties and to improve computational efficiency for more robust estimation of current and future PMP. Dramatic improvements in artificial intelligence (AI)/machine learning (ML) techniques during the past decade offer some promise for further advancing modeling of PMP by improving physics parameterizations used in CPM (e.g., Gentine et al., 2018; Yuval and O’Gorman, 2020) and bias-correcting CPM simulations (Bretherton et al., 2022). AI/ML can also be used to develop emulators of CPMs, which can then be used for effective model calibration and uncertainty quantification (Hourdin et al., 2023) and for production of a much larger ensemble of CPM-like simulations, known as ensemble boosting, at a much lower computational cost (Gibson et al., 2021). Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 49 Conclusion 3-8: Model-based estimation of PMP requires very high-resolution simulations that explicitly represent convection and storm structures that produce extreme precipitation. Advances in regional and global CPMs and high-performance computing have made it feasible to model PMP-magnitude storms on climate timescales. SCIENTIFIC ADVANCES: CLIMATE CHANGE AND EXTREME RAINFALL According to the National Academies of Sciences, Engineering, and Medicine (NASEM; 2016), scientific confidence in climate-driven changes in extreme weather depends on three separate lines of evidence: a clear trend in observations, a clear trend in climate model simulations, and physical understanding of the connection between climate change and extreme trends, with confidence highest if all three lines are present. Fischer and Knutti (2016) argue that extreme precipitation is an example of a scientific prediction of a consequence of climate change that was made first using global climate model output and physical understanding and subsequently was verified by observations as a trend emerged. Because of the high societal importance of extreme precipitation, the relationship between climate change and extreme precipitation remains an active area of research, and improved model simulations, an ever- expanding historical data record, and improved statistical techniques are continually refining our understanding of trends around the globe. Physical Understanding The most direct mechanism by which climate change affects extreme precipitation is through the effect of temperature on saturation specific humidity (Allen and Ingram, 2002; Trenberth, 1999; Trenberth et al., 2003). As discussed in the subsection Meteorology of Extreme Rainfall above, the precipitation rate is directly proportional to the specific humidity when and where the air is saturated. Well-established laws of thermodynamics show that the amount of water vapor in saturated air increases rapidly with temperature. The equation relating water vapor and temperature is known as the Clausius-Clapeyron equation. The proportionality constant itself depends on temperature, but the rate of increase of moisture is approximately 7 percent for each degree Celsius of lower tropospheric temperature increase. Consequently, a baseline expectation for the change in precipitation amount from the strongest storms, assuming 100 percent precipitation efficiency and no change in updraft strength, would be 7 percent per degree of warming. This rate of increase is known as Clausius-Clapeyron scaling, or C-C scaling for short. Considerable research in recent years has been directed toward investigating the dependence of rain rate on local temperatures in the present-day climate, sometimes called “apparent” C-C scaling. However, in model simulations the correlations between temperature fluctuations and precipitation intensity on weather timescales can substantially differ from the effect of long-term warming on precipitation intensity (Bao et al., 2017; Lenderink et al., 2021; Sun et al., 2020; Wang et al., 2017; Zhang et al., 2017). Analyses of extreme precipitation change in observations and models tend to show that average changes have a similar magnitude to C-C scaling, but with larger changes in the tropics and for sub-daily events, and considerable spatial variability, including changes in sign, elsewhere (Förster and Thiele, 2020; Guerreiro et al., 2018; Pall et al., 2007). Prepublication copy 

50 Modernizing Probable Maximum Precipitation Estimation C-C scaling is problematic because it is an incomplete theory of extreme precipitation. It does not specify how extreme the precipitation must be to follow C-C scaling. C-C scaling is not expected to apply to precipitation overall. Surface evaporation is the primary means of net energy transfer from Earth’s surface to the atmosphere, and therefore it is constrained at around 1.5−2.0 percent per degree of warming by changes in the radiative energy transfer from the ground to the lower atmosphere under warming (Allen and Ingram, 2002). Lastly, as discussed in the subsection Meteorology of Extreme Rainfall, the total precipitation in a given interval of time can be characterized as P = EqwD, or precipitation efficiency times total column moisture times mean column vertical motion times storm duration. Expecting C-C to hold everywhere is tantamount to assuming that precipitation efficiency, updraft strength, and storm duration are unaffected by climate change. C-C scaling turns out to be more like a rule of thumb than a constraint, and differences between observed or simulated scaling and C-C scaling are useful for identifying the effects of E, w, and D. In addition, changes in the frequency of events F can be affected by climate change through changes in weather patterns or environmental conditions. These other effects are collectively referred to as the dynamic effects of climate change on extreme precipitation (O’Gorman and Schneider, 2009), while C-C scaling represents the thermodynamic effect of climate change on extreme precipitation. There is no basic theory that states whether the dynamic effect should be positive or negative, nor whether it should be larger or smaller than the thermodynamic effect. Physical principles only help somewhat with these other factors. With tropospheric relative humidity expected to change very little and even decrease over land (Byrne and O’Gorman, 2016; O’Gorman and Muller, 2010), the increasing temperatures of climate change imply an increasing vapor pressure deficit, which could decrease precipitation efficiency, though radiative-convective equilibrium simulations have found the opposite (Lutsko and Cronin, 2018). In addition, increased precipitation through C-C scaling may directly imply increased vertical motion in some circumstances, suggesting that super-CC scaling of extreme precipitation changes is quite reasonable in moist environments such as the tropics and for the most extreme events (Neelin et al., 2022). The absence of a comprehensive physical theory for extreme precipitation changes means that it will be necessary to rely upon historical trends and climate model projections to quantify the impacts of climate change on extreme rainfall at any given location. However, physical understanding justifies the assumption that climate change affects extreme precipitation intensity, most clearly through the C-C effect. Results from Modeling Climate model output provides an opportunity for scientists to test their understanding of the relationship between extreme precipitation and climate change. With climate model simulations, many years of simulated extreme precipitation values can be produced, and cause and effect tested. For example, Kunkel et al. (2013a) found that very extreme precipitable water magnitudes increases over CONUS by 25–42 percent over a century under RCP 8.5, while convergence and vertical motion extremes do not increase nearly as much. One major disadvantage of global climate models, however, is their resolution, which is too coarse to directly simulate thunderstorms or atmospheric moist convection in general. Such precipitation is estimated (parameterized) based on historical observed relationships between environmental conditions and rainfall. Those estimations have limited validity at the extreme end of the precipitation spectrum. Therefore, the Kunkel et al. (2013a) finding regarding convergence Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 51 is probably much more resolution-dependent than their finding regarding precipitable water. Only recently have CPMs been applied to the question of changes in extreme rainfall frequency. In addition, most modeling studies use metrics that are nowhere near as extreme as PMP, usually something like annual maximum 1-day or 5-day precipitation. With the expectation that extreme precipitation is increasing faster than overall precipitation, days with lighter precipitation should be decreasing in frequency. Model simulations generally find that the crossover point is around the 90th percentile of precipitation (Pendergrass and Hartmann, 2014). In other words, only the top 10 percent wettest days are increasing in precipitation intensity in the global average. In addition, studies that have looked at changes in intensity at different return frequencies find that the fractional climate change effect increases as return frequency decreases (Gründemann et al., 2022; Martel et al., 2021; Myhre et al., 2019). Simulated rainfall intensity at durations shorter than 1 day increase faster than daily or multi-day rainfall amounts (Fosser et al., 2020; Martel et al., 2020, 2021; Westra et al., 2014). The thermodynamic effect seems generally to be larger than the dynamic effect over land in midlatitudes, at least for 1-day annual maximum precipitation (Pfahl et al., 2017). The limited number of higher-resolution simulations confirm the trends from larger-scale models but generally tend to show larger trends (Cannon and Innocenti, 2019; Helsen et al., 2020; Kendon et al., 2014; van der Wiel et al., 2016). Some model-based studies have looked specifically at climate change impacts on TCs, generally finding that structural changes lead to an increase in precipitation intensity in addition to thermodynamic enhancements (Gutmann et al., 2018; Liu et al., 2019; Patricola and Wehner, 2018; Reed et al., 2022). MCSs are projected to increase in both intensity (roughly following C- C scaling) and area (Dougherty et al., 2023; Prein et al., 2017c), with the latter possibly leading to greater PMP increases in larger basins. Individual thunderstorms are projected to become rarer and more intense because of simultaneous increases in instability and convective inhibition (Rasmussen et al., 2017). An extensive modeling study of moisture-maximized storms in past and future climates in the southeast United States found that the modeled PMP-magnitude storms exhibited an increase in intensity larger than C-C scaling (Rastogi et al., 2017). Observed Trends in the United States At the individual station level, statistically significant trends in extreme precipitation are relatively rare, because of a small signal-to-noise ratio. However, regional aggregation of station data consistently shows a tendency for increasing 1-day or 2-day extreme precipitation in the central and eastern United States (DeGaetano, 2009; Groisman et al., 2012; Janssen et al., 2014; Kunkel et al., 2013b; Risser et al., 2019a; Westra et al., 2013; Wright et al., 2019; see Figure 3- 3). As with climate model output, the rarer the event, the larger the observed trend (Fischer and Knutti, 2016), both globally and regionally (Sun et al., 2021; Westra et al., 2013). Barbero et al. (2017) found that increases in 1-day annual maxima in the United States have been larger than increases in hourly maxima, contrary to modeling studies. Global changes in the frequency of record-setting daily precipitation are on average close to what would be expected from C-C scaling, except larger in the tropics (Lehmann et al., 2015). Some studies have looked at trends in estimated PMP in the United States by, for example, analyzing PMP over subperiods and constructing a time series of estimates. Although such studies are generally not able to estimate trend values with much precision, the general tendency is for a historical upward trend in PMP to be found (Gu et al., 2022; Lee and Singh, Prepublication copy 

52 Modernizing Probable Maximum Precipitation Estimation 2020). It is also possible to assume that the dynamical effect of climate change is zero and to estimate the PMP trend due to increases in maximum dew point temperature or precipitable water; such studies generally find a positive historical (or projected) trend in moisture, directly implying that PMP magnitudes will increase because of the moisture maximization step even if no new storms are incorporated (Kao et al., 2019; Kunkel et al., 2020; Stratz and Hossain, 2014; Visser et al., 2022). It seems particularly unlikely that dynamical effects will be so negative as to offset thermodynamic effects (Kunkel et al., 2013a). FIGURE 3-3 Observed changes in three measures of extreme precipitation: (a) total precipitation falling on the heaviest 1 percent of days, (b) daily maximum precipitation in a 5-year period, and (c) the annual heaviest daily precipitation amount over 1958–2021. NOTES: The frequency and intensity of heavy precipitation events have increased across much of the United States, particularly the eastern part of CONUS, with implications for flood risk and infrastructure planning. Numbers in black circles depict percent changes at the regional level. Data were not available for the U.S.-Affiliated Pacific Islands and the U.S. Virgin Islands. SOURCE: Figure, caption, and notes from NCA5, Figure 2.8 (https://nca2023.globalchange.gov/chapter/2/). Detection and Attribution Studies Changes in the frequency or intensity of precipitation at very long return intervals are challenging to analyze because of the limited historical record. Most current knowledge of climate change effects on the frequency of extremely rare storms comes from detection and attribution studies of extreme rainfall events. Hurricane Harvey, in August 2017, brought PMP-magnitude precipitation to southeast Texas and has been the focus of several detection and attribution studies. Studies by van Oldenborgh et al. (2017) and Risser and Wehner (2017) analyzed historical trends in extreme precipitation but with different choices for event definition and other analysis aspects. Van Oldenborgh et al. (2017) estimated how much the probability of exceeding about 1043 mm in 3 days had changed, assuming that similar extreme event probabilities apply to the entire northern Gulf of Mexico coastal region. This total was the highest total observed at a long-term Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 53 climate station. Using global mean surface temperature as a covariate, they estimated a roughly 20 percent observed increase over 1880–2017 in the rainfall amount corresponding to the present-day probability of 1043 mm, and an increase in the probability of an event of given intensity over the same period of a factor of four. This estimate leads to a present-day return period of about 9,000 years. One of two Global Circulation Models (GCMs) showed a similar increase, while the other’s increase was about half as large. Risser and Wehner (2017) used gauge data over a smaller area and temporal window but considered average storm-total precipitation over 33,000 km2 or 105,000 km2. They estimated an anthropogenic increase of 20−40 percent in amount and a probability increase of about a factor of 10. However, this analysis excluded gauge data prior to 1950, and any other starting point for the period of record of extreme rainfall yields a smaller estimated anthropogenic increase. Other detection and attribution studies have considered events that were rare but still well below PMP values. Despite differences in methods and geographical setting, they all find increases due to climate change. For example, Tradowsky et al. (2023) finds an increase in the intensity of 1-day point rainfall in West Germany at a return period of about 1,000 years of 22 percent (7–34%) in observations and 5 percent (2–8%) in a model synthesis. Summary Intense precipitation is increasing over the majority of the globe. In the United States, historical trends, model projections, and physical understanding all point to more intense precipitation in the future, with the greatest increases at sub-daily durations and the longest return periods. However, the challenge of relating these trends to PMP is illustrated schematically in Figure 3-4. The portion on the left is from Pendergrass (2018), illustrating that greater return periods (higher percentiles) tend to have larger increases due to climate change. The graph has been extended toward even greater return periods to show those that correspond to PMP in the United States (Caldwell et al., 2011; Schaefer, 2023) and Australia (Nathan and Weinmann, 2019; Nathan et al., 2016). Various annual exceedance probabilities have been estimated for particular PMP values; that range of possible probabilities leads to uncertainty in the climate change effect because (at least at lower return frequencies) the climate change effect increases as the annual exceedance probability decreases (see also Jayaweera et al., 2023). Uncertainty also arises from uncertainty in the climate change effect at easier-to-estimate return frequencies and the need to then extrapolate those values to return intervals consistent with PMP. Extrapolation of the bottom quartile of estimates of the climate change effect yields something in the neighborhood of C-C scaling for PMP, but central and upper estimates for the climate change effect imply enhancements much greater than C-C. In addition, the climate change effect on short-duration storms is thought to be greater than that for daily precipitation. Estimates in the neighborhood of twice C-C scaling seem plausible, but the uncertainty associated with extrapolation over such a large data gap is massive. C-C scaling is a conservative, physically justified assumption, while neglect of climate change entirely is dangerously contrary to the evidence for extreme rainfall in general (e.g., Visser et al., 2022). Prepublication copy 

54 Modernizing Probable Maximum Precipitation Estimation FIGURE 3-4 Illustration of the possible change in intensity of PMP due to climate change, expressed as a percent change per degree of increase of global mean surface temperatures. NOTES: Estimates from shorter return periods are depicted as in Pendergrass (2018), but those provide limited information regarding the appropriate scaling at PMP-like return periods. In addition, evidence indicates that sub-daily extremes are intensifying more rapidly than daily extremes, but the magnitude of that difference is also poorly quantified. As a result, the actual scaling of PMP with climate change is not yet known and is poorly constrained. SOURCE: Adapted from Pendergrass (2018). Figure 3-4 uses global mean surface temperature (GMST) for C-C scaling. This is a common approach (Barbero et al., 2017; Chen et al., 2021; Guerriero et al., 2018; Liang et al., 2023; Jorgensen and Nielsen-Gammon, 2024; Myhre et al., 2019; Pendergrass and Hartmann, 2014; Westra and Sisson, 2011; Westra et al., 2013). Other reference temperatures have also been employed, such as regional mean surface temperature, often in conjunction with regional climate modeling studies (Fujibe, 2013; Förster and Thiele, 2020; Qin et al., 2021; Wood and Ludwig, 2020; Zeder and Fischer, 2020), regional mean surface dew point (Lenderink et al., 2019), local annual mean surface temperature (Bao et al., 2017; Pall et al., 2007), and local surface dew point conditioned on the occurrence of extreme precipitation (Lenderink et al., 2021). The latter approach has the most in common with the traditional PMP estimation procedures and may be most useful for understanding dynamic and thermodynamic contributions to changes of individual storms. GMST is much less directly related to the thermodynamic enhancement of individual storms, but scaling extreme precipitation by GMST has the virtues of (1) GMST being more robustly estimated and projected; (2) empirical and model-based estimates incorporate both dynamic and thermodynamic effects; and (3) the results are easily translated to future climate scenarios. The true C-C scaling factor based on any reference temperature is likely to vary geographically because of changes in weather patterns and differing dynamical changes across different storm types. Conclusion 3-9: The assumption that climate change does not affect extreme rainfall, implicit in traditional stationary analyses, is contrary to multiple lines of evidence. Neglecting climate change generally underestimates both present-day and future risk of extreme rainfall. Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 55 Conclusion 3-10: Clausius-Clapeyron scaling provides a useful means for quantifying changes in extreme rainfall due to warming. A 7 percent per degree scaling using global mean surface temperature is a handy rule of thumb, but it neglects dynamical influences on storm structure and frequency, and those seem likely to further amplify very extreme precipitation, including PMP-magnitude storms, particularly those of short duration. The overall magnitude of amplification is likely to vary with location and storm type. ADVANCES: STATISTICAL METHODS Colloquially, extreme precipitation events are described by “return levels,” for example, a “100-year storm.” Because terms such as 100-year storm can be misinterpreted by the public, and because such terminology is difficult to reconcile with changes in risk due to a changing climate, we choose instead to refer to event magnitudes in terms of their annual exceedance probability (AEP) depth. For a specified rainfall duration 𝑑, and a 𝑝 between 0 and 1, the 𝑝 −AEP precipitation depth is the precipitation magnitude that has a probability 𝑝 of being exceeded in a particular year. Thus, the AEP depth is the 1 − 𝑝 quantile of the distribution of annual maximum precipitation for the duration of interest, which translates to a very high quantile of the overall precipitation distribution of duration 𝑑. For small 𝑝, the data records are often too short to estimate the AEP depth by standard quantile estimation methods; under a stationary climate, the 𝑝 −AEP depth would require well over 1/𝑝 years of precipitation data. The statistical approach to estimating an AEP depth that requires extrapolation into the tail beyond the range of the observed data is based on extreme value analysis (EVA). EVA is now a well-established area of statistics used heavily in climate science, hydrology, and other areas of environmental/earth science to characterize the behavior of extremes. The book An Introduction to Statistical Modeling of Extreme Values (Coles, 2001) serves as a widely used reference in this area. Statistical EVA relies on the fundamental principle of fitting an extreme value distribution using only observations that are extreme, so that inference is not contaminated by data from the bulk of the distribution. The block maxima approach uses the maximum value from each block of data, which in earth/environmental science is often a year (and is also known as the annual maximum series approach). The use of the Generalized Extreme Value (GEV) distribution for modeling block maxima is theoretically justified because it is the appropriate distribution in the hypothetical as the block size goes to infinity. In practice this is interpreted as the block size is “large enough” to justify use of the GEV. The threshold exceedance approach uses values above a carefully chosen threshold, often empirically chosen as a high quantile (and is also known as the partial duration series approach). The theoretical justification for use of the extreme value distribution in this approach is that the distribution is the appropriate distribution in the hypothetical as the threshold goes to infinity. There are two common representations of the distribution in this approach: the generalized Pareto distribution and the closely related point process-based representation. For this approach certain tools can aid in choosing a sufficiently large threshold. Box 3-2 provides characterizations of the GEV and generalized Pareto distributions corresponding to those given in Coles (2001). In both the block maxima and threshold exceedance approaches, one can fit the available extremal data (block maxima or threshold exceedances) and obtain a characterization of the distribution’s tail, which is governed by the three parameters that characterize these distributions. AEP depths are, in turn, simple functions of the three parameters. Prepublication copy 

56 Modernizing Probable Maximum Precipitation Estimation BOX 3-2 Generalized Extreme Value and Generalized Pareto Distributions The cumulative distribution function of the of the Generalized Extreme Value (GEV) distribution is 𝐹 𝑧 = exp − 1 + 𝜉 . (1) The parameter 𝜇 is the location parameter and as it changes the distribution shifts to the left or right. The parameter 𝜎 is the scale parameter, and it stretches or shrinks the distribution similar to a standard deviation. The parameter 𝜉 is the shape parameter. The GEV distribution encompasses the Weibull distribution (𝜉 < 0 ), which has a finite upper bound; the Gumbel distribution (𝜉 = 0 ), which has no upper bound but has a light tail similar to the tail of a normal distribution; and the Fréchet distribution (𝜉 > 0 ), which has no upper bound and a heavy tail. To be precise, the Gumbel distribution’s cumulative distribution function is given by 𝐹 𝑧   =   exp − exp − , which is the limit of Equation 1 as 𝜉 → 0 . The (1 − 𝑝)th quantile, 𝑧 , of the GEV distribution (aka the (1 − 𝑝) ⋅ 100th percentile) is 𝑧 =𝜇− 1 − − log(1 − 𝑝) (2) for 𝜉 ≠ 0 (there is a different equation when 𝜉 = 0 ). When considering block sizes of 1 year (annual maxima), 𝑧 is the precipitation depth with an annual exceedance probability (AEP) of 𝑝. In the case of a bounded distribution, the bound is 𝜇 − . The presence of 𝜉 in the denominator of these expressions helps to explain the sensitivity of AEP depth and upper bound (if it exists) estimates and their uncertainty to the value of the shape parameter. 𝑧 is estimated by substituting estimates for the three parameters into Equation 2 to obtain 𝑧̂ . The sampling variance (the statistical uncertainty) of 𝑧̂ can be derived from the functional relationship in Equation 2 and the variance-covariance matrix for the estimates of the three parameters (when using maximum likelihood estimation, this is estimated by the inverse of the information matrix). This first- order approximation for the sampling variance is known as the delta method. Other methods such as bootstrapping or profile likelihood are also available and have been shown to have improved coverage performance. Considering the threshold exceedance approach, the Generalized Pareto (GP) distribution has cumulative distribution function, 𝜉 (𝑧 − 𝑢) 𝐹 (𝑧|𝑧 > 𝑢) = 1 − 1 + ,  𝜎 for values 𝑧 > 𝑢 above a threshold 𝑢 , where 𝜎 =  𝜎 + 𝜉 (𝑢 − 𝜇), and 𝜇 , 𝜎 , and 𝜉 are as in the GEV distribution. A convenient alternative representation of the threshold exceedance model uses a Poisson process representation to derive the probability density function (not shown) for threshold exceedance observations as a function of the GEV parameters, 𝜇, 𝜎, 𝜉 . Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 57 The fundamental behavior of the tail of the distribution (and therefore AEP depths for very small probabilities) is determined by the shape parameter 𝜉, also known as the tail index. Using the representation (parameterization) of the distributions given in Coles (2001), the distribution is unbounded when the shape parameter is non-negative. Note that this parametrization is not universal, and in the hydrology literature 𝜉 is sometimes replaced by −𝜅 implying the distribution is unbounded when 𝜅 is not positive. Figure 3-5 shows how the upper bound and AEP depth are affected by changing the value of the shape parameter while holding the location and scale parameters fixed. For the most negative shape parameter values, the upper bound is not very different from small-𝑝 AEP depths, but it increases quickly, tending to infinity, as the shape parameter approaches zero. AEP depths increase with the shape parameter, but not as quickly. Under stationarity (i.e., assuming the parameters do not change over time), these models can be fit to observations (either block maxima or values over the threshold) using a variety of statistical fitting techniques, including maximum likelihood, L-moments, and Bayesian methods. To account for variation over time, it is common to represent the parameters (particularly the location parameter) as regression-style functions of time or proxies for time such as global mean temperature or CO2 concentration, and to estimate the parameters using maximum likelihood or Bayesian methods. In such analyses, AEP depth estimates change with time or the proxy variable. FIGURE 3-5 Relationships of the upper bound (black curve) and of precipitation depths corresponding to extreme AEPs (green, blue, and red curves for return periods of 104, 105, and 106 years, respectively) to the shape parameter of the extreme value distribution. The upper bound exists only for negative values of the shape parameter. NOTE: The location and scale parameter values are based on a GEV fit to GHCN daily precipitation data for Berkeley, California, but the qualitative results (i.e., the curve behavior) are similar for parameters from GEV fits for other U.S. locations. Prepublication copy 

58 Modernizing Probable Maximum Precipitation Estimation Uncertainty for the parameter estimates and estimated AEP depth can be characterized using a variety of standard statistical approaches, including likelihood-based, moment-based, and Bayesian approaches. In particular, for maximum likelihood, standard statistical theory enables approximation of the sampling distribution of the AEP depth estimator and therefore computation of a confidence interval (see Box 3-2). These approaches can also be used determine the sample size needed to achieve a chosen sampling variance (or equivalently the length of a confidence interval, as presented in Chapter 5). As is typical when quantifying statistical uncertainty, uncertainty associated with parameter and AEP depth estimates decreases as the sample size increases. By its nature, EVA is often limited by relatively short data records, resulting in a relatively small dataset of extreme values. In particular, the shape parameter that governs the fundamental tail behavior is often found to have large uncertainty (e.g., see Martins and Stedinger 2000 for hydrologic examples). The amount of uncertainty associated with AEP depth can be uncomfortably large and grows as extrapolation moves further into the tail (estimates more extreme AEP depths). One possibility for increasing the amount of information is to borrow strength from nearby locations. Borrowing strength has a long history in the study of extreme precipitation and can be done via many methods such as regional frequency analysis, hierarchical Bayesian methods, distance-weighted local likelihood, or by directly smoothing return values estimated at individual locations. By borrowing strength across locations, uncertainty associated with parameter estimates, and in turn AEP depth estimates, are reduced. NOAA Atlas 14, the national precipitation frequency estimates currently in use, uses regional frequency analysis to combine data from regions determined to be homogeneous. Various approaches for borrowing strength across locations are being considered for use in the development of NOAA Atlas 15. The extreme value methods described thus far are essentially univariate in that they aim to describe the tail of an individual variable. Even the aforementioned methods that borrow strength across multiple locations do so to better estimate the univariate distribution’s parameters. The more advanced topic of analyzing the dependence of extremes for different variables, either reflecting different physical variables (de Haan and de Ronde, 1998; Heffernan and Tawn, 2004; Zscheischler et al., 2020) or the same variable at different spatial locations (Davison et al., 2012, Huser and Wadsworth, 2022), has seen an explosion of interest over the past couple of decades. Questions arising from the compound effects of coincident extremes require knowledge of dependence in the multivariate tail. Multivariate, spatial, and time series models have been developed to characterize extremal dependence and can be used to quantify risk of compound extreme events. Models for multivariate extremes are often computationally challenging, and the development of computationally tractable multivariate models is a continuing focus of extremes research. If beginning with a locationwise characterization of extreme precipitation, PMP estimates for a spatial area (e.g., basin) encompassing multiple locations would require accounting for spatial dependence in the extremes. Similarly, PMP estimates for an aggregated temporal duration could require modeling of temporal dependence. However, with access to complete space-time fields (as recommended in the long-term model-based approach to estimating PMP), precipitation data can be aggregated to the space-time resolution of interest (e.g., 3-day precipitation over a basin of interest) and standard univariate EVA methods described above can be applied; the dependence is captured in the space-time fields and propagates into the aggregated statistics. Multivariate methods could still be relevant for describing and understanding the dependence of extreme precipitation at different spatial and temporal Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 59 resolutions. Practitioners could also use multivariate extremes methods in conjuncture with the space-time field data to quantify risk of specific compound events not captured in PMP estimates. In EVA, because the focus is on return values for precipitation over a given time duration, the notion of an individual event (a storm) is not directly relevant, and standard analysis would include precipitation from all types of events. To include data only from specific storm types, the standard EVA must be modified to account for the probability of the storm type occurring, in addition to modeling the distribution of precipitation given the storm type. However, mixtures of storm types do create challenges for standard use of EVA. Fitting extreme value distributions using block maxima or threshold exceedances is justified based on asymptotic arguments and therefore assumes long blocks or large thresholds to achieve unbiased statistical estimation. When shorter blocks (e.g., annual maxima) or smaller thresholds are used, as is often needed with short observational records, the data being fit likely represent a mixture distribution across different types that cannot be well represented by a single extreme value distribution (e.g., Ben Alaya et al., 2020; Morrison and Smith, 2002; Villarini and Smith, 2010). A different type of mixture occurs in arid lands. Annual maximum rainfall and flood records in arid and semi-arid regions of the western United States exhibit mixtures in which some years have large events, but most years have effectively no events (J. Smith et al., 2018; Wang, 1990). Conclusion 3-11: Extreme Value Analysis (EVA) is a well-developed branch of statistics specifically aimed at quantifying the magnitude of very rare events. Applying EVA to estimate PMP-relevant precipitation depths has specific challenges: the precipitation observational record is relatively short for estimating AEP depths associated with PMP- relevant probabilities, and data arising from mixtures of storm types or in arid regions can require specific consideration to avoid statistical bias. Precipitation frequency approaches that are not grounded in EVA have been proposed for PMP applications. Hershfield introduced an influential method for estimating extreme rainfall accumulations that is based on moment-based frequency analyses (Hershfield, 1961; see also Hershfield, 1965 and Koutsoyiannis, 1999). Douglas and Barros (2003) introduced multifractal methods for precipitation frequency analysis and applied them to daily and monthly rainfall series. Paired with methods for precipitation frequency analysis, the authors introduced Fractal Maximum Precipitation as a potential replacement for PMP. Multifractal methods provide innovative insights into the problem of PMP estimation, but they do not have the mature statistical foundations of EVA methods. PMP AS AN UPPER BOUND? The philosophy for engineering design developed by the Miami Conservancy (Chapter 2) was grounded in the assumption that rainfall and flood magnitudes cannot exceed an intrinsic physical upper limit (Miami Conservancy, 1916; Morgan, 1917). This view was expressed by the preeminent hydrologist of the era, Robert Horton, in a 1927 letter to the editor of Engineering News Record: “It is not difficult to show from sound meteorological reasoning, and aside from any statistical proof, that there is a natural limitation to rain intensity for any given duration” (Horton Archive, see Vimal and Singh, 2022). Prepublication copy 

60 Modernizing Probable Maximum Precipitation Estimation For short-duration small-area rainfall, Horton’s arguments for limiting rates of rainfall initially centered on updraft velocity, updraft size, and water vapor content (Horton, 1919). Subsequent studies expanded the conceptual formulations to address downdraft properties, microphysical processes, and storm rotation (Horton, 1948a, 1948b, 1949; these studies were all published posthumously). Horton recognized that the size of downdrafts and rainfall distribution in downdrafts are important determinants of rainfall extremes for small time and space scales (Horton 1948a, 1948b). He also concluded that size sorting of hydrometeors in updrafts and downdrafts plays an important role in dictating raindrop size distributions and the distribution of rainfall rates in downdrafts (Horton, 1948b). Horton also examined the role of rotational motion as a significant component of hailstorms that produce extreme short-duration rainfall (Horton, 1949). Horton never produced a comprehensive theory for bounds on rainfall, although research on the topic continued intermittently until his death in 1945. Much of Horton’s research on extreme rainfall was unpublished when he died (Horton, 1948a, 1948b, 1949), reflecting the incomplete picture that had emerged on the question of bounds. Fundamental problems, such as specifying maximum updraft velocities remain unsolved (see Horton, 1949, Marinescu et al., 2020, and discussion above). Little research has been carried out on limiting rates of precipitation subsequent to Horton’s studies. Conclusion 3-12: A first-principles theory has not emerged to support the existence and characterize the magnitudes of upper bounds on precipitation. To some hydrologists and hydrometeorologists, compelling arguments for the existence of bounds on rainfall and flood peaks can be based on empirical evidence provided by “envelope curves” relating maximum rainfall observations to duration (Figure 3-6; Jennings, 1950; Shands, 1947) and maximum flood peaks to drainage area (e.g., Costa, 1987; Crippen, 1982; Enzel et al., 1993). Jennings’ summary of world record rainfall observations stimulated research on scaling laws relating rainfall to duration, with results pointing to maximum rainfall scaling with the square root of duration. These curves typically show world record (or near record) point rainfall accumulations for various durations. For example, in NRC (1994), their Figure 1 shows an envelope curve with the equation R = 16.4 D0.48, where R is in inches and D in hours, as an apparent upper bound. However, since this time, the purported “upper bound” has been exceeded by large amounts. For example, the February 2007 rain event at La Réunion, a small, mountainous island in the Indian Ocean, exceeded this estimate by 500 to more than 1,000 mm over 3–6-day durations (Figure 3-6). This event indicates that the past empirical estimates of upper bounds are not necessarily well supported. Recent studies have taken a different perspective on record values, focusing on exceedance probabilities of envelope curves (Castellarin et al., 2005; Vogel et al., 2007). Envelope curve analyses provide insights into bounds on rainfall but have not provided tools that are used for specifying bounds on point rainfall as a function of duration for particular locations or regions. Statistical Nature of an Upper Bound Extensive statistical research has addressed the question of bounds on rainfall and floods, partly motivating the development of extreme value analysis (Gumbel, 1941). EVA has been applied to precipitation data from around the world in a large number of analyses in recent Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 61 FIGURE 3-6 Envelope curves (linear and log scales), with world record point rainfall measurements with respect to duration. NOTES: The blue line shows the function R = 16.4 D0.48 (with R in inches and D in hours; converted to mm) that was shown in Figure 1 of NRC (1994). The February 2007 observations at La Réunion that exceeded this line are highlighted in both panels. SOURCES: Data from https://www.weather.gov/owp/hdsc_record_precip and Jennings (1950). Prepublication copy 

62 Modernizing Probable Maximum Precipitation Estimation decades, with fitting usually done to individual station data. In many cases the estimated shape parameter is non-negative, suggesting unbounded precipitation distributions (Cavanaugh et al., 2015; Papalexiou and Koutsoyiannis, 2013; Papalexiou et al., 2018 and references therein; Serinaldi and Kilsby, 2014), as also seen in Figure 3-7. Analyses of more than 5,000 flood records in the United States also point in the unbounded, heavy-tailed direction (J. Smith et al., 2018). Koutsoyiannis (1999) advocates for abandoning PMP as an upper bound and replacing it with a very high quantile estimate obtained via EVA methods. There are caveats (limited sample sizes, statistical assumptions needed to carry out EVA) to the EVA of rainfall and floods, but there is little statistical evidence supporting the bounded assumption. Horton (1919) assumed that over time, large rainfall observations would pile up close to the bound, providing a natural path for statistical estimation of the bound, in line with modern extreme value statistics for bounded distributions. However, subsequent analyses of rainfall observations have not supported that path. For example, Hurricane Harvey, which approached or exceeded some multi-day PMP estimates, could have produced much larger precipitation totals if the storm had followed a slightly different path (Li et al., 2020). Given that PMP is often interpreted as a depth that cannot be exceeded, it is difficult to reconcile the current PMP definition with extreme value analyses that tend to indicate that precipitation is unbounded and difficult to empirically estimate an upper bound for use as a PMP estimate. Conclusion 3-13: Statistical evidence does not support the assumption that precipitation is bounded; the evidence points to unbounded, heavy-tailed distributions. From a statistical perspective, even if an upper limit exists, use of the upper bound of a distribution as the quantity of interest has critical shortcomings relative to use of an AEP depth for a very small probability, as illustrated in Figure 3-5. First, even if the upper bound exists in principle, the quantity cannot be estimated empirically when the distribution is estimated to be unbounded. Second, in the case of a bounded tail, as the shape parameter increases toward zero, the upper bound becomes much larger than depths for even very extreme AEPs, such as for an AEP of 10 . The estimate of the upper bound in this situation will also likely be highly sensitive to the exact data used and the statistical estimation procedure chosen. Finally, as the shape parameter becomes more negative, the case where it is most practical to use the upper bound, the upper bound is very similar to depths for very extreme AEPs. Climate Change Global, regional, and local temperature increases are occurring, which results in increased moisture-holding capacity of the atmosphere through the Clausius-Clapeyron relation. The concept of a physical upper limit to rainfall must be referenced to a particular climate state under the presence of climate change. The current PMP definition does not address climate change and thereby neglects the increase in atmospheric water vapor due to climate change, which can lead to an increase in PMP (e.g., Kunkel et al., 2013b) (see section on Implications of Climate Change for PMP in Chapter 4 for further details). Recent studies and summaries on extreme event rainfalls (e.g., Risser and Wehner, 2017; van Oldenborgh et al., 2017) and PMP (Visser et al., 2022) suggest that extreme rainfall magnitudes are increasing and PMP estimates will increase in the future. Prepublication copy 

State of the Science and Recent Advances in Understanding Extreme Precipitation 63 FIGURE 3-7 Distribution of shape parameter estimates from fitting individual station- and season- specific GEV distributions to GHCN daily precipitation data from stations in the contiguous United States. NOTES: Estimated shape parameter values of zero or more correspond to unbounded distributions. The spread in the estimates reflects both statistical sampling uncertainty in the estimates for a specific location and season and variability in the true parameter values across locations and seasons. SOURCE: Plot is based on parameter estimates from Risser et al. (2019b), provided to the committee. Prepublication copy