Development of an upper envelope curve of frequency factors for Probable Maximum Precipitation (PMP) estimation in tropical regions under convective regimes: a deterministic approach

Martins, Vitor Matheus Passos; Pinto, Eber José de Andrade

doi:10.1590/2318-0331.312620250147

ABSTRACT

This study revisits and adapts the Hershfield Statistical Method to estimate the point Probable Maximum Precipitation (PMP) for a tropical climate region in northern Brazil, focusing on durations from 24 hours to 30 days. To improve the representativeness of PMP estimates, regional upper envelopes of frequency factors ( $K_{m}$ ) were developed based on 52 rainfall stations, capturing the distinct climatic behavior of convective systems typical of equatorial regions. The results indicate a decreasing trend in frequency factor ( $K_{m}$ ) values as rainfall duration increases, consistent with the predominance of short-duration, high-intensity events in the region. Additionally, local frequency factors ( $K_{l o c a l}$ ) exhibited significant variability and positive skewness, reflecting the spatial heterogeneity of extreme precipitation events. This approach can support hydrological risk assessments and infrastructure projects in tropical environments, specifically in part of the state of Pará, Brazil. This article is the first part of a two-part study on frequency factors for PMP estimation.

Keywords:
Probable Maximum Precipitation; Frequency factor; Upper envelope; Extreme rainfall; Tropical climate

RESUMO

Este estudo revisita e adapta o Método Estatístico de Hershfield para estimar a Precipitação Máxima Provável (PMP) pontual para uma região de clima tropical no Norte do Brasil, com foco em durações de 24 horas a 30 dias. Para melhorar a representatividade das estimativas de PMP, foram desenvolvidas envoltórias regionais dos fatores de frequência ( $K_{m}$ ) com base em 52 estações pluviométricas, capturando o comportamento climático distinto de sistemas convectivos típicos de regiões equatoriais. Os resultados indicam uma tendência decrescente nos valores de $K_{m}$ à medida que a duração da chuva aumenta, o que é consistente com a predominância de eventos de curta duração e alta intensidade na região. Além disso, os fatores de frequência locais ( $K_{l o c a l}$ ) apresentaram variabilidade significativa e assimetria positiva, refletindo a heterogeneidade espacial de eventos extremos de precipitação. Essa abordagem pode dar suporte em avaliações de risco hidrológico e projetos de infraestrutura em ambientes tropicais, especificamente em parte do estado do Pará, no Brasil. Este artigo constitui a primeira parte de um estudo em duas etapas sobre fatores de frequência para estimativa de PMP.

Palavras-chave:
Precipitação Máxima Provável; Fator de frequência; Envoltória; Precipitação extrema; Clima tropical

INTRODUCTION

Estimating the Probable Maximum Precipitation (PMP) is a critical task for hydrological safety assessments and infrastructure design, particularly for large dams and high-risk hydraulic structures. Several technical and regulatory frameworks explicitly recommend the consideration of PMP or the associated Probable Maximum Flood (PMF) in safety evaluations (Centrais Elétricas Brasileiras, 2003; Szymanski & Davies, 2004; International Commission on Large Dams, 2011, 2022; CBDB as cited in Pinheiro, 2011; Agência Nacional de Águas e Saneamento Básico, 2016; Associação Brasileira de Normas Técnicas, 2017; Agência Nacional de Mineração, 2022; Salgado et al., 2025).

According to the World Meteorological Organization (World Meteorological Organization, 2009) PMP is defined as the maximum theoretical precipitation that can occur over a given region, during a specific duration, and under particular meteorological conditions. It represents an upper-bound estimate intended to support conservative design and risk mitigation strategies.

The conceptual foundation of PMP is rooted in the early recognition that extreme hydrometeorological processes are bounded by physical constraints. Horton (1936) argued that storms and floods within a given basin must be subject to upper physical limits, establishing a conceptual basis for subsequent theoretical developments. Later discussions refined this understanding, emphasizing that the principal difficulty of PMP lies not in its conceptual validity, but in its quantification, which is inherently constrained by observational data, methodological assumptions, and epistemic uncertainty (Dooge, 1986; Berod et al., 1992).

Historically, two main approaches have been employed for PMP estimation: a physical approach and a statistical approach. The physical approach is based on storm maximization techniques, moisture maximization, and atmospheric analyses aimed at approximating a physically plausible upper limit. The statistical approach, in contrast, extrapolates extreme precipitation records using frequency analysis and envelope curves of frequency factors.

Within the statistical framework, the Hershfield Statistical Method (Hershfield, 1961a, 1961b, 1965) became one of the most widely adopted deterministic/empirical procedures for PMP estimation. Developed using precipitation records from the United States, its formulation assumes the applicability of an upper envelope of frequency factors ( $K_{m}$ ) derived predominantly from temperate and cold-climate stations. This envelope has frequently been applied, often without regional recalibration, as a reference for diverse climatic contexts.

Although PMP has traditionally been interpreted as a physical upper bound, subsequent studies demonstrated that statistical implementations implicitly correspond to extremely low but finite probabilities of exceedance. Revisiting Hershfield’s formulation, Koutsoyiannis (1999) showed that the fixed value $K_{m} = 15$ for 24-hour PMP estimation is statistically equivalent to a quantile associated with a return period on the order of 60,000 years under a Generalized Extreme Value (GEV) distribution. Similar interpretations were reinforced by Pinto (2001). Later, World Meteorological Organization (2009) and Salas et al. (2014) explicitly distinguished between the theoretical definition of PMP as a physical limit and its operational interpretation as a methodological construct for estimating extremely rare events. More recently, NASEM (National Academies of Sciences Engineering and Medicine, 2024) formalized the probabilistic interpretation of PMP and highlighted the need to reconcile deterministic formulations with non-stationary climate conditions.

Empirical quantifications of this implicit probability have produced return periods spanning several orders of magnitude. In the United States, National Research Council (1994) suggested that statistical PMP estimates may correspond to return periods ranging from $10^{5}$ to $10^{9}$ years, depending on regional and methodological assumptions. Nathan & Weinmann (2001, as cited in Fernandes, 2009) reported values on the order of $10^{4}$ to $10^{7}$ years, while Ball et al. (2019) obtained recurrence intervals equal to or exceeding 2,000 years, illustrating the strong methodological dependency of such estimates. More recently, Kappel et al. (2024) estimated median return periods between $10^{6}$ to $10^{7}$ years for 24 and 72 hours PMP in the Wasatch–Uinta Mountains region (USA), with possible ranges extending from $10^{5}$ to $10^{9}$ years. In South America, Sugai (1989) reported average return periods of approximately $1.4 \times 10^{5}$ years for 24-hour PMP in southern Brazil. These studies collectively demonstrate that the probabilistic interpretation of PMP is neither recent nor geographically restricted, but rather reflects methodological assumptions, sample length, and climatic characteristics.

Subsequent investigations in different climatic regions revealed that the direct transposition of Hershfield’s Upper Envelope or WMO-recommended correction factors may lead to inconsistencies outside their original development context. Rezacova et al. (2005) reported substantial overestimation of daily PMP values in the Czech Republic when applying WMO procedures, prompting regional methodological adjustments. Similarly, Casas-Castillo et al. (2018), using 258 stations across the Iberian Peninsula, derived a regional envelope curve for 24-hour PMP estimation, reinforcing the necessity of climatic adaptation in statistical PMP methods.

In Brazil, several studies have questioned the indiscriminate application of Hershfield’s original envelope, particularly in tropical and subtropical environments characterized by intense convective activity (Sugai, 1989; Burger, 2014; Cavalcanti et al., 2018; Conceição & Fontes, 2019; Silva Neto et al., 2020; Barbosa et al., 2023). These investigations consistently identified frequency factors lower than those proposed by Hershfield, reinforcing the importance of a regional approach.

However, although regional adaptations have been proposed in temperate and Mediterranean climates, studies explicitly focused on genuinely tropical, equatorial, hot, and humid regions — such as northern Brazil — remain limited. Unlike extra-tropical regimes, equatorial environments are dominated by persistent deep convection, high precipitable water content, weak thermal seasonality, and strong moisture recycling, factors that may influence the statistical behavior of annual precipitation maxima and, consequently, the estimation of frequency factors.

One notable tropical application is the regional adaptation proposed by Sarkar & Maity (2020) in India, where frequency factors were reassessed under monsoon-dominated convective conditions. Their approach provides an important methodological reference for tropical environments and will be evaluated in this study.

The objective of this study is therefore to reassess and adapt the Hershfield Statistical Method to support the estimation of frequency factors in a genuinely tropical, equatorial, hot, and humid region of northern Brazil characterized by intense convective activity. The analysis is based on historical series of annual maximum precipitation for durations from 24 hours to 30 days.

MATERIAL AND METHODS

Study area and dataset

Figure 1 presents the global Köppen–Geiger climate classification according to Kottek et al. (2006, as cited in Nascimento et al., 2017), based on data from 1951 to 2000 and a spatial resolution of 0.5° × 0.5°. The map highlights the marked climatic contrast between the United States — which served as the basis for Hershfield’s original formulation and is predominantly characterized by temperate and snow climates — and northern Brazil, which is classified as equatorial, with persistently high temperatures and humidity. Additionally, hemispheric climatic asymmetries reinforce the need for region-specific analyses.

Figure 1
Köppen–Geiger climate classification for the whole world.

Given this climatic divergence, a study area was selected in the northern portion of the state of Pará, Brazil, characterized by strong convective activity and classified as having a tropical/equatorial climate (Instituto Brasileiro de Geografia e Estatística, 2002; Alvares et al., 2013). Beyond its meteorological relevance, the region has strategic socio-economic importance. It comprises major infrastructure such as refineries, reservoirs, hydroelectric and mining facilities, as well as export ports, particularly in the municipalities of Belém (the state capital) and Barcarena, a prominent industrial and logistical hub.

To construct a Regional Upper Envelope of frequency factors suitable for this climate, 52 rain gauge stations (listed in Table 1) were selected based on convective climatology criteria. The selected area was defined by a 200 km radius centered at latitude -1.5058° and longitude -48.6271° (SIRGAS 2000), covering data from 1949 to 2023, with an average record length of approximately 34 years.

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Table 1
Basic characteristics of the 52 rain gauge stations. The hydrological year considered was from January to December. ^[1]Climate Classification based in Alvares et al. (2013) and Instituto Brasileiro de Geografia e Estatística (2002). ^[2]Geographical coordinates in SIRGAS 2000.

Some rainfall stations have record lengths shorter than 20 years. These shorter series were retained to enhance spatial coverage in areas with limited gauge density, while acknowledging the higher uncertainty associated with limited sample size.

The durations analyzed were: 24 hours, 3 days, 5 days, 10 days, 15 days, and 30 days. In addition, the research was performed using daily rainfall data for each station. A factor of 1.14 was applied to daily rainfall to obtain 24-hour rainfall depths (Weiss, 1964, as cited in World Meteorological Organization, 2009).

The study area lies entirely within the Amazon biome (Instituto Brasileiro de Geografia e Estatística, 2019), primarily in the Tocantins–Araguaia Hydrographic Basin, bordering the Amazon Basin and the Western Northeast Atlantic Basin (Instituto Brasileiro de Geografia e Estatística, 2021).

From a climatic perspective, the region is strongly influenced by large-scale atmospheric circulation patterns, including the Hadley and Walker cells (Ferreira & Mello, 2005; Liberto, 2014). The Intertropical Convergence Zone (ITCZ) plays a central role in seasonal rainfall modulation, while Mesoscale Convective Complexes (MCCs) are responsible for high-intensity precipitation events, especially in the North and Northeast of Brazil. These convective systems, typically covering areas between 120,000 and 150,000 km², are frequent during summer and autumn and significantly influence the regional precipitation regime (Maddox, 1980; Machado et al., 1994; Salio et al., 2007; Lyra et al., 2020). Machado et al. (1994) confirmed MCCs in South America with radii of approximately 240 km, reinforcing the relevance of convective phenomena in the precipitation regime of the study area.

According to Lima et al. (2025), the study area is predominantly located in zone 2 of homogeneous climate risk. The authors mention that zone 2 is a region “directly influenced by the ITCZ and, in its coastal part, by the Easterly Wave Disturbances. There is an out-of-phase relationship with the ONI, PDO, and AMO climate indices, influencing its interannual (ONI) and decadal/multidecadal (PDO/AMO) variability”. Lima et al. (2025) divided the Brazilian territory into 7 zones to facilitate the design of a climate risk management strategy.

Figure 2 and Figure 3 illustrate the spatial distribution of the 52 rain gauge stations and provide a geographic and climatic characterization of the region, which is dominantly tropical, warm, and humid. For more detailed descriptions of the regional climatic system and its variability, see Martins (2024).

Figure 2
Spatial distribution of the 52 rain gauge stations used in the study.

Figure 3
(a) Köppen-Geiger climate characterization (Alvares et al., 2013) in the region. The image also shows the mean annual rainfall isohyets, as well as the 52 rainfall stations considered in the study; (b) Instituto Brasileiro de Geografia e Estatística (2002) climate characterization of the study region; and (c) hypsometric map of the region with hydrography.

Table 1 summarizes the main attributes of each station, including coordinates (SIRGAS 2000), climate classification, elevation, and data availability. The hydrological year was defined from January to December.

Theoretical basis

Hershfield (1961a, 1961b, 1965), building upon the statistical formulation proposed by Chow (1951), proposed that the Probable Maximum Precipitation (PMP) could be represented by a theoretical upper limit, i.e., a precipitation value that would not be exceeded under given climatic conditions. Based on this assumption, he proposed the following formulation in Equation (1):

X_{P M P} = {\bar{X}}_{n} + K_{m} * S_{n}

(1)

where $X_{P M P}$ is the estimate of the Probable Maximum Precipitation (PMP) for a given location and duration; ${\bar{X}}_{n}$ is the mean of the series of n values of maximum annual rainfall, for a given duration; $S_{n}$ is the corresponding standard deviation; $K_{m}$ (or simply $K)$ is the hydrological frequency (or recurrence) factor.

Adapting Equation (1) and determining $X_{P M P} = X_{m a x}$ , Hershfield calculated the frequency factors for each of the rainfall series from the stations studied (also known as local frequency factors). It should also be noted that the maximum value was excluded when calculating the mean and standard deviation.

K_{l o c a l} = \frac{X_{m a x} - {\bar{X}}_{n - 1}}{S_{n - 1}}

(2)

where $K_{l o c a l}$ is the local frequency factor; $X_{m a x}$ is the maximum value of the time series; ${\bar{X}}_{n - 1}$ is the mean of the series excluding the value of maximum annual rainfall ( $X_{m a x}$ ); $S_{n - 1}$ is the standard deviation of the series excluding the value of maximum annual rainfall ( $X_{m a x}$ ).

After computing local factors for multiple stations, Hershfield constructed a general upper envelope of $K_{l o c a l}$ values as a function of the mean annual maximum precipitation, allowing the derivation of a generalized upper-bound $K_{m}$ beyond the original dataset. This procedure is illustrated in Figure 4. To account for sampling variability and the influence of extreme values, additional adjustments were introduced to the estimation of the mean and standard deviation as functions of sample size and dispersion characteristics (Figure 5).

Figure 4

K_{m}

as a function of mean maximum annual rainfall.

Figure 5
Adjustment of the mean and standard deviation based on the maximum value observed and the sample size.

It is important to note that the empirical basis for Hershfield's Upper Envelope was derived predominantly from rainfall stations located in temperate and cold climates, primarily in the United States, with records available up to the early 1960s. Consequently, the original envelope does not explicitly incorporate more recent precipitation records nor climatic regimes dominated by persistent deep convection, such as tropical and equatorial regions.

Reboita et al. (2010) emphasize that South America, especially Brazil, experiences high precipitation rates influenced by the complex interaction of upper-level and lower-level atmospheric systems. Such climatic characteristics may affect the statistical properties of annual precipitation maxima and, consequently, the representativeness of the generalized $K_{m}$ upper envelope when directly transferred to different climatic contexts.

The estimation of a representative $K_{m}$ value is sensitive to record length, dispersion, and climatic heterogeneity relative to the stations used to construct the original envelope. Overestimation of $K_{m}$ leads to inflated PMP values and potentially overconservative design parameters, whereas underestimation may reduce safety margins by understating the hydrometeorological hazard (World Meteorological Organization, 2009). Consequently, careful regional assessment of frequency factors is required when applying the Hershfield methodology outside its original development context.

Methodological framework

In general, a simplified flowchart summarizing the statistical PMP estimation procedure is presented in Figure 6.

Figure 6
Simplified flowchart of the statistical procedure for PMP estimation.

Sarkar & Maity (2020) proposed a modification to Hershfield’s methodology in which the Regional Upper Envelope is defined by two segments: (i) a constant plateau corresponding to the maximum observed $K_{l o c a l}$ , extending over the range of lower adjusted mean values; and (ii) an exponential decay segment beyond a transition mean, preserving the decreasing tendency of the original Hershfield envelope (see Figure 7).

Figure 7
Relationship between mean annual maximum daily rainfall, the original Hershfield Upper Envelope, and the modified envelope proposed by Sarkar & Maity (2020).

The steps to estimate the PMP, using the statistical methodology presented by Sarkar & Maity (2020), are as follows:

Statistical characterization of annual maxima: For each analyzed duration, the mean ( ${\overset{ˉ}{X}}_{n}$ ) and standard deviation ( $S_{n}$ ) of the annual maximum precipitation series were computed for all selected stations.
Adjustment for sample size and outliers: The mean and standard deviation were adjusted to account for the influence of extreme values and limited record length, following the adjustment procedures originally proposed by Hershfield (1961b) (see Figure 5). The resulting statistics are referred to as adjusted mean and adjusted standard deviation.
Computation of local frequency factors and regional envelope construction: Local frequency factors ( $K_{l o c a l}$ ) were computed using Equation (2). A Regional Upper Envelope was then constructed according to the formulation proposed by Sarkar & Maity (2020), consisting of:

A constant plateau corresponding to the maximum $K_{l o c a l}$ observed;
Followed by an exponential decay function as a function of the adjusted mean.

The upper envelope was considered valid within the interval defined by the minimum and maximum adjusted means of the regional dataset. In summary, the plateau extends from the minimum adjusted mean to the transition mean ${\bar{X}}^{t}_{a d j}$ , which is determined through numerical fitting of Equation (3). After that, exponential decay occurs.

PMP estimation: The PMP was finally estimated using Equation (1), with $K_{m}$ obtained from the Regional Upper Envelope defined in Step 3.

Equation (3) shows the formulation of Sarkar & Maity (2020) for the mathematical adjustment of regional upper envelopes:

K_{m} = {\begin{matrix} K_{p l a t e a u}, i f {\bar{X}}_{a d j_m i n} \leq {\bar{X}}_{a d j} \leq {\bar{X}}^{t}_{a d j}, \\ K_{p l a t e a u} * e^{- α * ({\bar{X}}_{a d j} - {\bar{X}}^{t}_{a d j})}, i f {\bar{X}}_{a d j} > {\bar{X}}^{t}_{a d j} \end{matrix}

(3)

where: $K_{m}$ is the regional frequency factor; $K_{p l a t e a u}$ is the maximum observed $K_{l o c a l}$ defining the constant plateau segment of the upper envelope; $α$ is the regional upper envelope adjustment coefficient, obtained through numerical optimization procedures; ${\bar{X}}_{a d j}$ is the adjusted mean of the annual maximum precipitation series, for a given duration; ${\bar{X}}^{t}_{a d j}$ is the adjusted mean corresponding to the transition point between the constant plateau and the exponential decay segment; and ${\bar{X}}_{a d j_m i n}$ is the adjusted mean among all stations for the analyzed duration.

Brief discussions on Probable Maximum Precipitation (PMP) and associated return periods (RPs) were also included in this study. The following probability distributions were evaluated using parameter estimation by the method of L-moments: Exponential (exp), Gamma (gam), Generalized Pareto (gpa), Generalized Logistic (glo), GEV (gev), Gumbel (gum), Log-Normal (ln2), Log-Normal-3P (ln3), Log-Pearson-III (lp3) and Pearson-III (pe3).

The goodness-of-fit between the theoretical and empirical distributions was assessed through the non-parametric Kolmogorov–Smirnov test at a 5% significance level, complemented by the Root Mean Square Error (RMSE) between theoretical and empirical quantiles, along with visual inspection of the graphical fit. The procedures are detailed in Hosking & Wallis (1997) and Naghettini & Pinto (2007). It is important to note that all statistical adjustments were performed using daily rainfall data for each station.

RESULTS AND DISCUSSIONS

Figure 8 displays the Regional Upper Envelopes constructed for all six durations analyzed, as described in the Materials and Methods section. The results suggest a negative correlation between the magnitude of the mean annual maximum rainfall and $K_{m}$ , both within individual durations and across different durations. This inverse relationship was also identified by Burger (2014) in the southern region of Brazil.

Figure 8
Regional

K_{m}

Upper Envelopes as a function of mean maximum annual rainfall from 24 hours to 30 days. Yellow points represent

K_{l o c a l}

values at rainfall stations.

The introduction of a constant plateau in the regional upper envelope, following Sarkar & Maity (2020), may modify the implicit exceedance probability associated with PMP estimates, particularly for lower mean rainfall values. While this structure limits unrealistically large frequency factors in low-mean regimes, it may produce PMP values lower than those derived from Hershfield’s Upper Envelope. Consequently, the resulting PMP estimates should be interpreted as regionally calibrated upper-bound estimates rather than absolute physical limits. Any implications regarding safety levels must therefore be evaluated within the broader framework of engineering criteria and regulatory standards applicable to critical infrastructure.

It is important to emphasize that the adoption of a plateau-based envelope is not motivated by economic considerations or by the intention to reduce design values. Rather, the objective is to avoid statistical inflation of frequency factors when the classical Hershfield envelope is extrapolated beyond the climatic conditions under which it was originally derived. In tropical regimes characterized by different precipitation dynamics, regional recalibration seeks to improve climatic representativeness while preserving the conservative intent inherent to PMP estimation.

Thus, the proposed regional envelope should be understood as a data-driven adaptation grounded in the statistical structure of the analyzed dataset. Its application requires professional judgment and must be integrated with existing safety regulations and risk management practices, particularly in contexts involving high-consequence hydraulic infrastructure.

Figure 9 presents boxplots illustrating the variation in magnitude of both the 52 values of $K_{l o c a l}$ and the resulting $K_{m}$ across rainfall durations ranging from 24 hours to 30 days, accompanied by supporting statistical tables. Analyzing the behavior of $K_{m}$ , a clear decreasing trend is observed as the rainfall duration increases.

Figure 9
Boxplots of the

K_{m}

and

K_{l o c a l}

values for different durations.

For all durations, the distribution of $K_{l o c a l}$ exhibits positive skewness, a typical feature of extreme-value-related statistics. A notable feature of the dataset analyzed is the progressive reduction in the standard deviation of $K_{m}$ values as duration increases, particularly evident for the 10-, 15-, and 30-day periods. This behavior may be associated with the convective-dominated rainfall regime of the region. Short-duration extremes are often controlled by isolated high-intensity convective events, which can generate substantial inter-station variability. In contrast, longer accumulation periods integrate the effects of multiple rainfall systems (e.g., ITCZ influence and mesoscale convective activity), leading to greater spatial coherence and reduced dispersion of regional frequency factors.

Additionally, the coefficient of variation for $K_{l o c a l}$ also shows a consistent decreasing pattern, suggesting reduced relative dispersion of the parameter as the duration increases.

Table 2 presents a synthesis of contributions from various authors, both in Brazil and globally, regarding the estimation of frequency factors. The discrepancies between Hershfield’s values and those observed in tropical regions such as Brazil are evident. Brazilian studies tend to yield lower $K_{l o c a l}$ _s, reflecting updated data and regional climatic differences. This suggests that constructing a regional upper envelope could lead to more realistic and technically sound PMP estimates.

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Table 2
Summary of contributions relating to the frequency factors in Brazil and worldwide.

In general, Barbosa et al. (2023), in a nationwide study across Brazil, identified a maximum $K_{l o c a l}$ of approximately 9.6 for the 24-hour duration, very close to the value obtained in the present study (9.46). In southern Brazil, Sugai (1989) reported a maximum $K_{l o c a l}$ of 9.0 for the same duration, within a humid subtropical climate. Similarly, Desa et al. (2001), in Selangor, Malaysia — a region characterized by a hot and humid equatorial/tropical climate (Af in the Köppen–Geiger classification) — found a maximum $K_{l o c a l}$ of 8.7.

Some studies, such as Chavan & Srinivas (2015), adopt a simplified approach by applying a single $K_{m}$ value (typically the regional maximum) to estimate the deterministic statistical PMP across a set of stations. However, in the context of this study, such generalization is inadequate. Given the convective nature of rainfall systems in the region, which are highly localized and intense, it is necessary to preserve the association between the station-specific adjusted mean precipitation values and their corresponding $K_{m}$ derived from the Upper Envelope. The joint variability of mean precipitation, standard deviation, and frequency factors reflects the intrinsic spatial structure of extreme rainfall in the region and should therefore be explicitly maintained in regional analyses.

Figure 10 presents the statistical attributes of PMP estimates based on the regional upper envelope and Equation (1) using boxplots and supporting statistical tables. The medians increased from 384 mm (24-h) to 1,194 mm (30 days), following the expected behavior of accumulated precipitation. The coefficient of variation remains low throughout the durations (between 0.15 and 0.35), and the skewness is moderate, indicating a statistically robust and stable result.

Figure 10
Boxplots of the point PMP estimates by Regional Upper Envelope for different durations.

The relative consistency in PMP values, particularly for longer durations, suggests that the envelope performs well in attenuating local anomalies while preserving regional trends, fulfilling the methodological expectations of deterministic point PMP estimation.

Figure 11 illustrates the spatial distribution of the 24-hour PMP, estimated using Hershfield’s Upper Envelope and the Regional Upper Envelope. Overlaid on the map are the corresponding $K_{m}$ values from both methods, the $K_{l o c a l}$ values, and the highest observed 24-hour rainfall at each station.

Figure 11
Spatial distribution of the 24-hour PMP using Hershfield’s method and Regional Upper Envelope method. Frequency factors are represented by scaled symbols, and the highest 24-hour rainfall records at each station are also indicated.

To spatially interpolate the PMP estimates, the Inverse Distance Weighting (IDW) method was applied with a power parameter $p = 2$ , based on the evaluation by Ly et al. (2013). These authors reviewed several interpolation methods for hydrological variables and found IDW to be suitable for daily precipitation data.

PMP values for 24-hour estimated via the Hershfield method are consistently higher and more homogeneous, with $K_{m}$ ranging from 13.24 to 16.22 and PMP exceeding 850 mm in several locations (ranging from 440.0 to 959.1 mm).

In contrast, the regional method produces more spatially diverse and climatically coherent PMP values, with values predominantly ranging from 300 to 600 mm and $K_{m}$ values between 2.0 and 9.5. The resulting spatial pattern exhibits a more gradual transition across the domain, which aligns more closely with known regional rainfall mechanisms, including the influence of mesoscale convective systems and the Intertropical Convergence Zone (ITCZ) (Salio et al., 2007; Reboita et al., 2010; Lyra et al., 2020; Martins, 2024).

On average, the PMP values estimated by the two methods differed by 60%, with a minimum difference of 43% and a maximum of 135%. Furthermore, the $K_{l o c a l}$ values are directly influenced by the precipitation records at each station. The highest $K_{l o c a l}$ observed of 9.46 was associated with the station that recorded the highest historical 24-hour rainfall within the study area.

The consistent spatial patterns observed across all four panels of Figure 11 underscore both methodological distinctions and physically meaningful regional contrasts. The spatial distribution of higher PMP estimates is not random; rather, it delineates a coherent NE–SW-oriented corridor. This corridor encompasses stations in historically convection-prone zones, such as the Salgado coastal region (northeastern Pará, including Marapanim, Magalhães Barata, and Maracanã), the Pará River estuary (the Belém Metropolitan Area, Barcarena, and Santo Antônio do Tauá), and the confluence area of the Pará and Tocantins rivers (Muaná, Limoeiro do Ajuru, and Ponta de Pedras).

This observed pattern is consistent with the dominant atmospheric circulation features over northern Brazil. The northeasterly trade winds, which strike the coast at an average angle of 30° relative to the equator, constitute the primary vector for moisture influx from the Atlantic. This oceanic supply is augmented by local evapotranspiration from the Amazon. The region's weather is further modulated by the large-scale Hadley and Walker circulations and the seasonal southward migration of the ITCZ, which collectively create a favorable environment for deep convection. Within this context, MCCs can serve as the triggering mechanism for the most extreme events. Additionally, a local physiographic factor may amplify this effect: the natural channel of the Pará River and Marajó Bay, oriented NE–SW in alignment with the trade winds, may facilitate persistent moisture advection. This water body likely functions as a preferential moisture transport corridor: its smooth surface reduces friction and provides a continuous local source of humidity, enhancing convection in adjacent areas.

Therefore, the concentration of high PMP estimates in these areas directly reflects the local climatology of extremes. The annual maximum precipitation series for these stations exhibit a higher frequency and greater magnitude of extreme events. This is corroborated by mean annual rainfall totals ranging from 2,500 to 3,100 mm, with a notable maximum in the Belém Metropolitan Area (Companhia de Pesquisas de Recursos Minerais, 2011). The region's inherent vulnerability to extreme precipitation is further supported by its climatic classification. As detailed by IBGE (Instituto Brasileiro de Geografia e Estatística, 2002), the Belém Metropolitan Area, Barcarena, and their surroundings form the core of a Hot/Super-humid Equatorial climate ( $r a d i u s ~ 25 k m$ ), distinguished by the complete absence of a dry season. This core is itself surrounded by a broader belt (~50 km radius) of a related but slightly drier subtype: Hot/Super-humid Equatorial sub-dry. This nested climatic structure highlights the area's persistent and intense humidity, rendering it inherently susceptible to the extreme events captured by the PMP analysis.

All point PMP estimation results, as well as the corresponding frequency factor values for all rainfall stations and durations, are available in the following public repository: https://doi.org/10.5281/zenodo.15103909.

Regarding the evaluation of return periods (RP), different probabilistic distributions exhibited varying performance across stations, as summarized in Table 3. The results highlight the statistical heterogeneity of rainfall behavior within the study area, with no single distribution consistently providing the best fit for all stations.

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Table 3
Number of rainfall stations associated with each of the probabilistic distributions included in the study.

Although several three-parameter distributions achieved satisfactory goodness-of-fit according to the Kolmogorov–Smirnov test and RMSE criteria, their application to high return periods revealed limitations associated with instability or upper-bound constraints imposed by the shape parameter. Such behavior is problematic in PMP estimation, which requires stable extrapolation to very low exceedance probabilities.

Therefore, the final selection was guided not solely by goodness-of-fit ranking, but by theoretical suitability for high-return-period extrapolation. The Gumbel distribution was adopted because it provided adequate fit at all stations, demonstrated low RMSE values and presents an unbounded upper tail, ensuring greater numerical stability in extreme quantile estimation. Its application in large-scale extreme rainfall analyses, such as those reported by Fernandes Juste et al. (2023), further supports its suitability for this purpose.

The Equations (4), (5) and (6) show the cumulative probability function of Gumbel, and the $α$ and $β$ parameters estimated by the L-moments method.

F_{X} (x) = \exp [- \exp (- \frac{x - β}{α})], w i t h - \infty < x, β < \infty a n d α > 0

(4)

α = \frac{l_{2}}{\ln 2}

(5)

β = l_{1} - 0.5772 * α

(6)

Lastly, Figure 12 presents the return periods (RP) associated with the PMP estimates for all rainfall stations and durations, displayed as boxplots.

Figure 12
Boxplots of RP for different durations.

There is significant variability in the RP values, as indicated by the high standard deviations. For instance, for the 24-hour duration, RP estimates ranged from a minimum on the order of 100 years to values exceeding 1 million years. The median and mean RP values for 24 hours were approximately 140,000 and 300,000 years, respectively. This is consistent with the values reported by Sugai (1989) in southern Brazil, who obtained similar magnitudes using the Gumbel and Exponential distributions.

In terms of average or median values across the analyzed durations, RP values ranged from $10^{3}$ to $10^{5}$ years, characterizing the events as extreme. All the references cited in this study, such as National Research Council (1994), Nathan & Weinmann (2001, apud Fernandes, 2009), Ball et al. (2019), and Kappel et al. (2024), partially cover this range.

Overall, there was a decreasing trend in return periods magnitudes with increasing duration. This behavior is consistent with the rainfall dynamics of tropical convective systems, which are more likely to generate extreme events in short periods. This further supports the notion that return periods associated with PMP estimates should not be viewed as immutable values; rather, they represent statistical constructs contingent upon both the selected frequency factor and the specific distribution applied to the dataset.

Regarding skewness, values are elevated for durations between 24 hours and 5 days (1.53 to 1.59), reflecting the influence of localized extremes in a few stations.

Finally, this study constitutes the deterministic component of a broader research framework. A complementary study (Martins & Pinto, 2026) extends this analysis through Monte Carlo simulation and extreme value theory, providing probabilistic insights into point-specific PMP estimates.

CONCLUSIONS

This study demonstrates that the traditional Hershfield envelope — originally derived from temperate and cold climates in the Northern Hemisphere (Kottek et al., 2006, as cited in Nascimento et al., 2017) — is not directly transferable to hot, humid, equatorial regions such as northern Brazil. The regional envelope developed here captures distinct meteorological mechanisms of rainfall generation, climatic regulators, and geographical positioning, yielding PMP estimates that are both spatially coherent and physically plausible. These findings underscore that regionalization is not merely a statistical exercise, but a climatic necessity.

A key methodological insight is the consistent decreasing trend of $K_{m}$ values of the Regional Upper Envelope with increasing rainfall duration, a pattern likely generalizable to other tropical regions governed by convective systems. The high variability and positive skewness of $K_{l o c a l}$ further reinforce the spatial heterogeneity of extreme rainfall in such environments. Notably, the highest $K_{l o c a l}$ (9.46) corresponded to the station with the most extreme 24-hour rainfall record.

The adoption of a plateau-based regional envelope improves climatic representativeness but may alter the implicit exceedance probability of PMP estimates relative to the Hershfield formulation. This trade-off between regional specificity and conservative design must be carefully weighed against applicable safety margins and regulatory frameworks for critical infrastructure. Return period estimates, while confirming the extremity of the events (ranging from $10^{3}$ to $10^{5}$ years), exhibited sensitivity to the chosen probabilistic distribution — reinforcing the need for distributional assumptions to be physically and statistically justified.

Despite these sensitivities, the regional upper envelope methodology proved consistent and appropriate for representing the climatological and statistical structure of the study area. It offers a transferable framework for point PMP estimation in data-scarce tropical regions, where convective processes dominate and where the direct application of Hershfield's original parameters may lead to overestimation or misrepresentation of risk.

Future research may integrate probabilistic methods or non-stationary climate modeling to refine PMP estimates under evolving climatic conditions. A companion study (Martins & Pinto, 2026, under revision) advances this agenda by associating frequency factors with probabilities of non-exceedance through Monte Carlo simulation and extreme value theory, thereby bridging the gap between deterministic and probabilistic PMP frameworks.

ACKNOWLEDGEMENTS

The authors are grateful to the Postgraduate Program in Sanitation, Environment and Water Resources of the Universidade Federal de Minas Gerais (UFMG), where this research was developed. The authors also thank the reviewer and the editorial board for their constructive comments, which contributed to improving the quality of the manuscript.

DATA AVAILABILITY STATEMENT

Research data is available in a repository: https://doi.org/10.5281/zenodo.15103909

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Edited by

Editor-in-Chief:
Adilson Pinheiro

Associated Editor:
Carlos Henrique Ribeiro Lima

Publication Dates

Publication in this collection
10 July 2026
Date of issue
2026

History

Received
07 Sept 2025
Reviewed
28 Feb 2026
Accepted
10 Mar 2026

This is an Open Access article distributed under the terms of the Creative Commons Attribution license (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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» https://doi.org/10.5194/nhess-18-3109-2018

[2] Agência Nacional de Águas e Saneamento Básico – ANA. (2016). Manual do empreendedor sobre segurança de barragens – Volume V: diretrizes para a elaboração de projeto de barragens Brasília, DF: Ministério do Meio Ambiente e Mudança do Clima.

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» https://doi.org/10.1127/0941-2948/2013/0507

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[6] Ball, J., Babister, M., Nathan, R., Weeks, W., Weinmann, E., Retallick, M., & Testoni, I. (2019). Australian rainfall and runoff: a guide to flood estimation Canberra: Commonwealth of Australia.

[7] Barbosa, A. G., Silva, A. F. R., & Costa, V. A. F. (2023). Estimativa espacial da Precipitação Máxima Provável com duração de 24 horas para o território brasileiro. In XXV Simpósio Brasileiro de Recursos Hídricos (pp. 10). Porto Alegre: ABRHidro.

[8] Berod, D., Devred, D., Laglaine, V., Chaix, O., Altinakar, M., & Delley, P. (1992). Calcul des crues extrèmes par dés methodes deterministes du type pluie maximale probable (PMP)/crue maximale probable (PMF): application au cas de la Suisse Lausanne: École Polytechnique Fédérale de Lausanne.

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[10] Casas-Castillo, M. C., Rodríguez-Solà, R., Navarro, X., Russo, B., Lastra, A., González, P., & Redaño, A. (2018). On the consideration of scaling properties of extreme rainfall in Madrid (Spain) for developing a generalized intensity-duration-frequency equation and assessing probable maximum precipitation estimates. Theoretical and Applied Climatology, 131(1-2), 573-580. https://doi.org/10.1007/s00704-016-1998-0
» https://doi.org/10.1007/s00704-016-1998-0

[11] Cavalcanti, R. L. C., Silva, C. H. D. S., Jesus, E. S. D., Filho, M. C., & Stošić, T. (2018). Estudo da Precipitação Máxima Provável e tempo de retorno em Recife (PE). In III Congresso Internacional das Ciências Agrárias – COINTER, João Pessoa. Recife: Instituto IDV. https://doi.org/10.31692/2526-7701.IIICOINTERPDVAGRO.2018.00125
» https://doi.org/10.31692/2526-7701.IIICOINTERPDVAGRO.2018.00125

[12] Chavan, S. R., & Srinivas, V. V. (2015). Probable maximum precipitation estimation for catchments in Mahanadi river basin. Aquatic Procedia, 4, 892-899. https://doi.org/10.1016/j.aqpro.2015.02.112
» https://doi.org/10.1016/j.aqpro.2015.02.112

[13] Centrais Elétricas Brasileiras – Eletrobras. (2003). Critérios de projeto civil de usinas hidrelétricas Rio de Janeiro: Eletrobras.

[14] Chow, V. T. (1951). A general formula for hydrologic frequency analysis. Transactions - American Geophysical Union, 32(2), 231-237. https://doi.org/10.1029/TR032i002p00231
» https://doi.org/10.1029/TR032i002p00231

[15] Conceição, M. P. D. A. S., & Fontes, A. S. (2019). Risco hidrológico associado à Precipitação Máxima Provável em Barragens do semiárido baiano. In XXIII Simpósio Brasileiro de Recursos Hídricos Porto Alegre: ABRHidro.

[16] Companhia de Pesquisas de Recursos Minerais – CPRM. (2011). Levantamento da geodiversidade: projeto atlas pluviométrico do Brasil: isoietas anuais médias: período 1977 a 2006 Brasília: CPRM.

[17] Desa, M. N., Noriah, A. B., & Rakhecha, P. R. (2001). Probable maximum precipitation for 24 h duration over southeast Asian monsoon region - Selangor, Malaysia. Atmospheric Research, 58(1), 41-54. https://doi.org/10.1016/S0169-8095(01)00070-9
» https://doi.org/10.1016/S0169-8095(01)00070-9

[18] Desa, M. N., & Rakhecha, P. R. (2007). Probable maximum precipitation for 24-h duration over an equatorial region: Part 2-Johor, Malaysia. Atmospheric Research, 84(1), 84-90. https://doi.org/10.1016/j.atmosres.2006.06.005
» https://doi.org/10.1016/j.atmosres.2006.06.005

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[38] Lima, C. E. S., Souza Filho, F. A., & Rocha, R. V. (2025). Homogeneous climate risk zones in Brazil. RBRH, 30, e36. https://doi.org/10.1590/2318-0331.302520240102
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[39] Ly, S., Charles, C., & Degre, A. (2013). Different methods for spatial interpolation of rainfall data for operational hydrology and hydrological modeling at watershed scale: a review. Biotechnologie, Agronomie, Société et Environnement, 17(2), 392-406.

[40] Lyra, M. J. A., Fedorova, N., Levit, V., & Freitas, I. G. F. (2020). Características dos complexos convectivos de mesoescala no Nordeste Brasileiro. Revista Brasileira de Meteorologia, 35(spe), 727-734. https://doi.org/10.1590/0102-7786355000001
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[41] Machado, L. A. T., Guedes, R. L., Silveira, J. M. B., Waltz, R. C., & Alves, M. A. S. (1994). Ciclo de vida de sistemas convectivos. InAnais do VIII Congresso Brasileiro de Meteorologia e II Congresso Latino-Ibérico de Meteorologia(Vol. 2, pp. 323-326). Rio de Janeiro: SBMET.

[42] Maddox, R. A. (1980). Mesoscale convective complexes. Bulletin of the American Meteorological Society, 61(11), 1374-1387. https://doi.org/10.1175/1520-0477(1980)061<1374:MCC>2.0.CO;2
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[43] Martins, V. M. P. (2024). Emprego da simulação de Monte Carlo na estimativa do fator de frequência adotado no cálculo da Precipitação Máxima Provável (PMP) pontual em uma região de clima equatorial sujeita a atuação convectiva (Dissertação de mestrado). Universidade Federal de Minas Gerais, Belo Horizonte.

[44] Martins, V. M. P., & Pinto, E. J. A. (2026). Derivation of frequency factors for Probable Maximum Precipitation (PMP) estimation through stochastic simulation of annual precipitation maxima in tropical convective regions. Revista Brasileira de Recursos Hídricos, 31, e22. https://doi.org/10.1590/2318-0331.312620250148.

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[47] National Academies of Sciences Engineering and Medicine – NASEM. (2024). Modernizing Probable Maximum Precipitation Estimation Washington, DC: The National Academies Press. https://doi.org/10.17226/27460
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Station	Code	Operator	Climate Classif. ^[1]	Geographical Coordinates ^[2]		Average Altim. (m)	Data Range	Consol. Hydrol. Years	Dist. to Centroid (km)
Station	Code	Operator	Climate Classif. ^[1]	Lat.	Long.	Average Altim. (m)	Data Range	Consol. Hydrol. Years	Dist. to Centroid (km)
ABAETETUBA	00148010	CPRM	Af/EQW-2	-1.750°	-48.867°	9	1980 to 2022	39	38
ACAMPAMENTO MENDES JUNIOR	00247008	CPRM	Af/EQW-3	-2.343°	-47.847°	45	1995 to 2023	21	125
ACARÁ	00148009	CPRM	Af/EQW-2	-1.961°	-48.206°	26	1980 to 2022	40	67
ANAJÁS	00049009	CPRM	Am/EQW-3	-0985°	-49.939°	3	1999 to 2022	23	159
BADAJÓS	00247000	CPRM	Af/EQW-3	-2.513°	-47.768°	10	1972 to 2022	47	145
BAIÃO	00249003	CPRM	Af/EQW-3	-2.785°	-49.669°	37	1984 to 2022	34	183
BELÉM	00148001	EMBRAPA/ IPAGRO	Af/EQW-1	-1.450°	-48.500°	15	1961 to 2014	28	15
BELÉM	00148002	INMET	Af/EQW-1	-1.435°	-48438°	7	1949 to 2019	67	22
BELÉM (INMET)	82191	INMET	Af/EQW-1	-1.436°	-48.437°	7	1961 to 2022	58	22
BELÉM (INMET)	A201	INMET	Af/EQW-1	-1.411°	-48.440°	21	2003 to 2023	20	23
BOM JARDIM	00147006	ANA	Am/EQW-3	-1.567°	-47.100°	38	1972 to 1989	16	169
BUJARU	00148017	CPRM	Af/EQW-2	-1.521°	-48.046°	10	1994 to 2022	28	63
CAMETÁ	00249001	INMET	Af/EQW-3	-2.250°	-49.500°	6	1975 to 1998	23	127
CAMETÁ (INMET)	82263	INMET	Af/EQW-3	-2.240°	-49.500°	10	1970 to 2023	50	127
CAPANEMA	00147002	CPRM	Am/EQW-4	-1.204°	-47.178°	25	1966 to 2022	55	164
CASTANHAL	00147007	CPRM	Am/EQW-3	-1.298°	-47.939°	48	1972 to 2022	47	79
CASTANHAL (INMET)	A202	INMET	Am/EQW-3	-1.301°	-47.948°	47	2003 to 2023	20	78
COLÔNIA SANTO ANTÔNIO	00147011	CPRM	Am/EQW-3	-1.655°	-47.491°	37	1982 to 2022	40	126
CURUÇÁ	00047003	CPRM	Am/EQW-4	-0.738°	-47.854°	17	1981 to 2022	34	121
FAZENDA JAUARÁ	00247003	CPRM	Am/EQW-3	-2.036°	-47.751°	12	1982 to 2023	37	112
FAZENDA MARINGA	00348001	CPRM	Am/EQW-3	-3.154°	-48.086°	61	1982 to 2022	35	191
FAZENDA SÃO RAIMUNDO	00247004	CPRM	Am/EQW-3	-2.426°	-47.516°	70	1982 to 2022	35	159
FAZENDA URUCURE	00248001	CPRM	Af/EQW-3	-2.448°	-48.579°	31	1982 to 2022	38	103
IGARAPÉ-AÇU	00147010	CPRM	Am/EQW-3	-1.133°	-47.628°	31	1980 to 2022	39	118
IGARAPÉ-MIRI	00148018	CPRM	Af/EQW-3	-1.971°	-48.934°	12	1995 to 2023	25	61
JOANA COELI	00149004	CPRM	Af/EQW-3	-1.898°	-49.382°	5	1994 to 2023	25	95
MAGALHÃES BARATA	00047007	CPRM	Am/EQW-4	-0.794°	-47.600°	24	1995 to 2022	25	139
MARUDA	00047005	CPRM	Am/EQW-4	-0.634°	-47.658°	17	1989 to 2022	29	145
MOSQUEIRO	00148012	CPRM	Am/EQW-3	-1.094°	-48.399°	14	1982 to 2022	39	53
MUANÁ	00149006	CPRM	Af/EQW-3	-1.536°	-49.232°	6	2004 to 2023	17	68
NOVA TIMBOTEUA	00147019	CPRM	Am/EQW-3	-1.211°	-47.386°	44	1995 to 2022	26	141
OEIRAS DO PARÁ	00149003	CPRM	Af/EQW-3	-2.002°	-49.864°	6	1982 to 2022	38	149
OUREM	00147016	CPRM	Am/EQW-3	-1.552°	-47.117°	15	1988 to 2022	31	167
PA-256 / RIO ACARA-MIRIM	00248006	CPRM	Am/EQW-3	-2.782°	-48.352°	25	1990 to 2022	32	144
PA-256 / RIO CAPIM	00247006	CPRM	Am/EQW-3	-2.948°	-47.814°	31	1989 to 2022	31	182
PRIMAVERA	00047004	CPRM	Am/EQW-4	-0.929°	-47.099°	16	1982 to 2022	39	181
SALINÓPOLIS	00047002	CPRM	Am/EQW-4	-0.623°	-47.354°	26	1977 to 2022	39	172
SANTA CRUZ DO ARARI	00049011	CPRM	Am/EQW-3	-1.016°	-48.960°	6	1999 to 2023	23	68
SANTA ISABEL DO PARÁ	00148003	CPRM	Am/EQW-2	-1.296°	-48.171°	34	1972 to 2022	49	55
SANTA LUZIA DO PARÁ	00146011	CPRM	Am/EQW-4	-1.516°	-46.906°	76	1994 to 2022	27	190
SANTA MARIA DO PARÁ	00147018	CPRM	Am/EQW-3	-1.357°	-47.574°	40	1990 to 2022	29	117
SÃO DOMINGOS DO CAPIM	00147008	CPRM	Am/EQW-3	-1.678°	-47.770°	5	1975 to 2023	45	96
SÃO JOÃO DE PIRABAS	00047006	CPRM	Am/EQW-4	-0.775°	-47.178°	27	1989 to 2022	31	180
SÃO SEBASTIÃO DA BOA VISTA	00149005	CPRM	Af/EQW-3	-1.717°	-49.517°	10	2004 to 2022	18	102
SOURE (ILHA DE MARAJÓ)	00048000	INMET	Am/EQW-4	-0.728°	-48.516°	9	1961 to 1998	33	89
TAILÂNDIA	00348002	CPRM	Am/EQW-3	-2.950°	-48.967°	49	1994 to 2022	26	164
TERRA ALTA	00147017	CPRM	Am/EQW-3	-1.039°	-47.913°	34	1991 to 2022	31	95
TOMÉ-AÇU	00248003	CPRM	Af/EQW-3	-2.421°	-48149°	12	1984 to 2022	36	113
VIGIA	00048006	CPRM	Am/EQW-4	-0.868°	-48.110°	15	1982 to 2022	39	92
VILA DO CARMO	00249004	CPRM	Af/EQW-3	-2.450°	-49.438°	15	1994 to 2023	27	138
VILA DO CONDE	00148011	CPRM	Af/EQW-2	-1.566°	-48.767°	17	1980 to 2022	38	17
VILA ELIM	00249002	CPRM	Af/EQW-3	-2.804°	-49.378°	9	1980 to 2023	41	166

Authors	Data information	Contribution
Barbosa et al. (2023)	270 INMET stations.	They designed a Regional Upper Envelope of frequency factors for the whole of Brazil, with a maximum $K_{l o c a l}$ of around 9.6 and spatialized the PMP estimates for the whole of the country.
Silva Neto et al. (2020)	10 rainfall stations in Tocantins with data from 1989 to 1999.	Hershfield $K_{m}$ for 24 hours = 16.64. On the other hand, the maximum $K_{l o c a l}$ was 4.81.
Sarkar & Maity (2020)	110 years of observed daily data from 4,951 points on 0.25 x 0.25° grids provided by the India Meteorological Department (IMD).	An Upper Envelope of frequency factors was proposed, consisting of a constant plateau and an exponential curve.
Conceição & Fontes (2019)	4 rainfall stations in the semi-arid region of Bahia, with data from 1942 to 2018.	$K_{l o c a l}$ values ranging from 2.43 to 4.57, different from Hershfield's Original Upper Envelope with $K_{m}$ between 15.95 to 16.46.
Cavalcanti et al. (2018)	Database covering the years 1998 to 2012 from the Recife/PE rainfall station.	$K_{l o c a l}$ equal to 4.9.
Afzali-Gorouh et al. (2018)	6 rainfall stations and 1 meteorological station in the Qareh-Su basin, located in Iran.	$K_{l o c a l}$ values varying between 2.9 and 5.3, different from Hershfield's Original Upper Envelope with $K_{m}$ between 17.2 to 18.0.
Lan et al. (2017)	65 rainfall stations in the Hong Kong region with data from 1985 to 2010.	$K_{l o c a l}$ values varying between 1.79 and 5.46.
Casas-Castillo et al. (2018)	258 rainfall stations in Spain, with data between 1951 to 2012	An Upper Envelope of frequency factors was proposed, consisting of an exponential curve with a maximum value of 19.6. However, the maximum $K_{l o c a l}$ found was 10.4
Burger (2014)	50 conventional rainfall stations and 37 pluviographs in the state of Paraná.	Elaboration of a regression equation for Hershfield's Original Upper Envelope; Paraná Regional Envelope; synthetic simulations of frequency factors with AR(1) models.
Desa & Rakhecha (2007)	39 rainfall stations in Johor, Malaysia, most of which have records of more than 40 years.	Most of the local $K_{l o c a l}$ values were below 6.0, and the highest value recorded as a $K_{l o c a l}$ factor was 7.5, a long way from the maximum values of Hershfield's Original Upper Envelope ( $K_{m}$ = 20). Desa et al. (2003, apud Desa & Rakhecha, 2007) determined a maximum $K_{l o c a l}$ of 7.4 for the Malaysian province of Kelantan.
Rezacova et al. (2005)	849 rainfall station in the Czech Republic, with an average length of 33 years.	A constant frequency factor assumption (11.54) was used throughout the Czech territory, which should be considered a first approximation.
Desa et al. (2001)	33 rainfall stations in Selangor, Malaysia, with records of more than 50 years, for the most part.	Most of the local $K_{l o c a l}$ values were below 5.0, and the highest value recorded as a $K_{l o c a l}$ was 8.7, a long way from the maximum values of Hershfield's Original Upper Envelope ( $K_{m}$ = 20).
Koutsoyiannis (1999)	He investigated the 2,645 rainfall stations included in Hershfield's studies.	It is possible to associate a probability of exceedance, using return period of 60,000 years and GEV probabilistic distribution, for PMP_s that employ a fixed $K_{m}$ equal to 15. Pinto (2001) and Koutsoyiannis (1999) showed that Hershfield's Original Method (Hershfield, 1961a, 1961b and 1965) does not exactly estimate a physical limit for the PMP, but rather associates a probability of exceedance with it.
Sugai (1989)	130 rainfall stations in southern Brazil, totaling 4,150 years of observation.	Maximum local $K_{l o c a l}$ equal to 9.0.

Development of an upper envelope curve of frequency factors for Probable Maximum Precipitation (PMP) estimation in tropical regions under convective regimes: a deterministic approach

Desenvolvimento de uma curva envoltória de fatores de frequência para estimativa da Precipitação Máxima Provável (PMP) em regiões tropicais sob regimes convectivos: uma abordagem determinística

ABSTRACT

RESUMO

INTRODUCTION

MATERIAL AND METHODS

Study area and dataset

Theoretical basis

Methodological framework

RESULTS AND DISCUSSIONS

CONCLUSIONS

ACKNOWLEDGEMENTS

DATA AVAILABILITY STATEMENT

REFERENCES

Edited by

Publication Dates

History