Atlas of precipitation extremes for South America and Africa based on depth-duration-frequency relationships in a stochastic weather generator dataset

Anrw Fullhrt , Dvi C.Goorih , Mnru B.Mls , Pulo Trso S.Olivir ,Cristino s Nvs Almi , Josˊ C. Arújo , Sh Burns

a Southwest Watershed Research Center USDA-ARS, 2000 E.Allen Rd., Tucson, AZ, 85719, United States

b Sustainable Agricultural Water Systems Unit USDA-ARS, 239 Hopkins Road, Davis, CA, 95616, United States

c Faculty of Engineering, Architecture and Urbanism and Geography, Universidade Federal de Mato Grosso do Sul, Campo Grande, MS, 79070-900, Brazil

d Water Resources and Environmental Engineering Laboratory, Universidade Federal da Paraíba, Joao Pessoa, PB, 58051-900, Brazil

e Department of Agricultural Engineering, Campus do Pici Fortaleza, Universidade Federal do Cearˊa, Fortaleza, 60020-181, Brazil

Keywords:Africa Depth-duration-frequency Extreme precipitation

A B S T R A C T Information about extreme rainfall is lacking in regions of South America and Africa.This study attempts to fill this scientific gap by use of a gridded parameterization for the stochastic weather generator,CLIGEN,to map depth-duration-frequency (DDF) relationships.Analysis of 500-year point-scale precipitation time series generated at each grid point allowed maps of return period precipitation to be produced for a selection of sixteen durations ranging from 10-min to 1-year and for nine return periods from 2 to 500 years.The generalized extreme value (GEV) probability distribution was fitted for all durations, and given GEV quantiles, an interpolation method was applied to produce maps at 0.1° resolution that better resolve small-scale spatial climate gradients.In addition to uncertainties related to GEV fitting,this study quantifies prediction intervals based on ground validation.This validation was important for identifying biases in CLIGEN, although uncertainties were not always satisfactorily defined due to sampling design and other factors.For daily/multi-day durations, 100 stations with daily observations and ≥50-year records were selected for validation against the 0.1° CLIGEN map series,resulting in a median and average absolute error of 13%and 16%,respectively.For sub-daily durations,prediction errors were larger overall.An analogy using available U.S.data established the degree of bias in CLIGEN for sub-daily durations, and three records in Brazil with high temporal resolutions were used to confirm that applied bias adjustments resulted in error ranges similar to the daily/multi-day cases.This atlas is freely available for study of extreme precipitation.

1.Introduction

Global climate datasets have assisted in frequency analysis of extreme precipitation in places with sparse coverage,such as in regions of South America and Africa (Awadallah & Awadallah, 2013;Br^eda et al.,2022).Typically,global climate datasets have grid-scale,areal precipitation values with lower intensities than would be expected at a single point of measurement (Lau & Behrangi, 2022;Meng et al., 2021; Oliveira & Roca, 2022).Furthermore, frequency analysis requires long record lengths, which may be a limitation of currently available global climate datasets.Stochastic weather generators, such as CLIGEN, can produce point-scale precipitation time series with arbitrarily long record lengths and can represent interannual variability within a stationary climate.CLIGEN stochastically generated weather is commonly used as input to drive several process-based hydrology and soil erosion models, though extreme precipitation in CLIGEN is understudied.Here, we investigate the suitability of CLIGEN for extreme value frequency analysis and establish systematic biases that affect CLIGEN's predictions.This has a secondary outcome of validating model scenarios for extreme events driven by CLIGEN.

Stochastic weather generators have been evaluated across a range of climates, often in the context of application to ungauged basins (Grimaldi et al., 2022).In terms of theoretical framework,CLIGEN is an example of a weather generator that uses ground observations (or estimates) of basic monthly climate statistics in order to define probability distributions, whereas weather generators may be parameterized from various other data sources,including variable fields from climate models(Langousis&Kaleris,2014).CLIGEN's approach is simpler compared to some generators and is effective at representing daily-scale precipitation.For subdaily factors such as peak intensity and storm duration, uncertainty is generally higher in CLIGEN,and other generators(e.g.Park et al., 2021) may produce more realistic hyetographs than that of CLIGEN's idealized hyetograph in which a single peak is assumed and only one major sub-daily input parameter is required.

CLIGEN represents an approach for downscaling grid-scale precipitation data from existing global climate datasets and generating long records that may be used for frequency analysis (Mehan et al.,2017; Wang et al., 2018).CLIGEN has been used in conjunction with soil erosion models such as the Rangeland Hydrology and Erosion Model (RHEM; Hernandez et al., 2017), the Water Erosion Prediction Project model(WEPP;Flanagan et al.,2012),the Soil and Water Assessment Tool(SWAT;Arnold et al.,2012),and the Universal Soil Loss Equation(USLE,Kinnell&Yu,2020).CLIGEN has also been used for hydrological assessment such as in a study of the multi-site Walnut Gulch Experimental Watershed (Zhao et al.,2021).Furthermore,CLIGEN has provided a simplistic means of assessing impacts of climate change by manipulation of its input parameters(Vaghefi&Yu,2016;Zhang,2013).The present study analyzes depth-durationfrequency (DDF) relationships in CLIGEN output by utilizing one of several large-scale CLIGEN parameterization grids available for different regions of the globe (Fullhart et al., 2022a; Wang et al.,2021).The availability of DDF information is important for water resources management, design of storm water infrastructure, and hydrological and soil erosion research.

The objective of this study is to analyze a continental-scale CLIGEN parameterization that allows mapping of spatial DDF relationships and return period precipitation distributions.Using this CLIGEN data,depth maps are produced for a wide range of durations and frequencies.For each individual duration-frequency case,different forms of ground validation are used to assess errors in terms of variance and systematic bias in CLIGEN.Bias adjustments are applied to reduce error in cases with large bias.Uncertainties are determined that are related to fitting annual maxima to the generalized extreme value (GEV) probability distribution and using the GEV distribution to approximate continuous DDF curves,which was done to allow estimation of return period depths for any durationfrequency case of interest.We also discuss implications of the results on CLIGEN applications and how the DDF data may be applied.

2.Methods

2.1.Data inputs and validation datasets

Input precipitation time series were produced using the pointscale stochastic weather generator, CLIGEN v5.3, and a CLIGEN input parameterization grid for South America and Africa at 0.25°resolution (～28 km at the equator).The procedure for this CLIGEN parameterization is described in detail in an open-access article by Fullhart et al.(2022a).This parameterization allows CLIGEN time series to be generated at each point in the grid.To briefly describe how the grid was created, critical input parameters (which are primarily monthly-scale aggregate statistics)were calculated from 20-year time series representing the 21st century.These grid-scale time series were taken from three global climate datasets with differing spatial and temporal resolutions: monthly, daily, and 30-min.These correspond to the TerraClimate, ERA5-Daily, and GPMIMERG datasets, respectively.Monthly accumulation was set directly using the 1/24th degree TerraClimate monthly dataset.On average,daily accumulation in the CLIGEN output will occur on the number of days required for the monthly mean of single-storm accumulation to sum to the TerraClimate monthly accumulation.The mean, standard deviation, and skewness of single-storm accumulation distributions (referred to as MEAN P, SDEV P and SKEW P in CLIGEN) were taken from ERA5-Daily by assuming a maximum of one storm event per day, which is an assumption of CLIGEN.Sub-daily precipitation patterns in CLIGEN are influenced by a parameter defined as the average of monthly maximum 30-min intensity (MX.5P) that was calculated from the 30-min GPMIMERG dataset.In order to downscale the estimated CLIGEN parameters taken from the coarser spatial and temporal resolutions of ERA5 and GPM-IMERG (0.25 and 0.1°, respectively), observed data from ground networks informed statistical downscaling regressions that were trained for each parameter.

A selection of ground records and spatial datasets were used to validate estimated return period precipitation.Records from the NOAA GHCN-Daily network were queried and compared to estimated depths for daily,multi-day,and annual durations.Initially,105 GHCN-Daily stations in Africa and South America were considered that had 50+year daily records and a maximum of one missing day per month.Following removal of five stations to reduce clustering in urban areas and to omit stations with collocated grid cells that mostly cover water bodies, a total of 100 stations were accepted.Ground validation of depths for sub-daily durations was based on three records in Brazil with 13-23 year record lengths and with high temporal resolution measurements of 1-min, 5-min, and 15-min(known high temporal records within Africa were too short in length to be included).The locations of all ground stations are shown in Fig.1.For this selection of stations, annual precipitation ranges from a minimum of 169 mm in South Africa to a maximum of 2,983 mm in French Guiana.Most stations are in Brazil and South Africa because these countries have the longest established gauge networks, though many countries have much greater coverage consisting of shorter records.To address the data limitation caused by the small sample of sub-hourly records,correlations to expected depth values for sub-daily durations were defined based on an analogy to datasets available for the U.S.using n = 100 randomly sampled spatial locations.These correlations defined systematic biases in CLIGEN by assuming known depths may be approximated from DDF relationships given by NOAA ATLAS-14(Perica et al.,2013).Time series taken from two sets of CLIGEN parameterizations were compared to NOAA ATLAS-14:the PRISM CLIgrid dataset(4 km,Daly et al., 2008) and the dense CLIGEN ground observation network of Srivastava et al.(2019).One hundred (n = 100) spatially random locations for the contiguous U.S.were sampled from these datasets(excluding northwestern states with no ATLAS-14 coverage).Given this approach, the three stations in Brazil with high temporal resolution measurements were used to confirm that differences in climate type between the U.S., Africa and South America had minimal impact on the determined systematic bias in CLIGEN.The three records also confirmed that use of determined bias adjustments resulted in improved estimation, since bias in CLIGEN was particuarly large for sub-daily durations.For the 5-min and 15-min records, it was recognized that intensities would tend to show underestimation bias for shorter durations.Therefore,it was deemed necessary to only use these records for longer sub-daily durations.

2.2.Theory of precipitation in CLIGEN

Fig.1.Ground stations for validation with daily records from NOAA-GHCN shown in black (n = 100) and stations with 1, 5, and 15-min records shown in red (n = 3).

CLIGEN is useful for producing long-term climate time series with stationary climate.Since all major precipitation input parameters are monthly statistics, variability arises on monthly-scales (not on seasonal or annual scales,as may be the case in the event of drought or climate cycles).Subsequently,inter-annual variability also arises due to variability within single months.Within a given month, daily precipitation amounts are sampled from a skewed normal distribution parameterized by the mean, standard deviation, and skewness of observed single-storm accumulations.Trace precipitation amounts are not modeled.Variability in the occurrence of precipitation is controlled by a two-state Markov conditional probability chain using transitions of precipitation occurrence between the current day and the following day.By these mechanisms,variability in precipitation arises on timespans ranging from daily to interannual.Baffault et al.(1996) demonstrated that annual sedimentation rates in an erosion model driven by CLIGEN reached convergence at approximately 200 years of simulation.A running average of annual precipitation in CLIGEN also tends to vary minimally after 200 years, even in cases of climates with large precipitation amounts,such as in the example shown in Fig.3.

On the sub-daily time frame,CLIGEN uses a simplified intensitytime storm pattern described by exponential functions partitioned by the time of peak precipitation intensity.The equation set is as follows:

where i(t) is intensity normalized to average storm intensity at normalized time,t(normalized to storm duration),and ipand tpare normalized intensity and time, respectively, at the time of peak intensity (Yu, 2002).A fitting parameter, B, is calculated for each storm.An input parameter defined as the average monthly maximum 30-min intensity (MX.5P), influences peak intensity,storm duration, and a combined factor used in CLIGEN,α0.5, being the ratio of maximum 30-min accumulation to total storm accumulation.Peak storm intensity is given by:

where rpis peak intensity (mm/hr) and R is total accumulation(mm; Yu, 2000).The value of α0.5is sampled from a gamma distribution parameterized by MX.5P.The storm duration, D (h), is given by:

where 3.99 is a dimensionless constant (Yu, 2000).A combination of CLIGEN outputs and use of these equations allows annual maxima for any duration to be determined.Procedures for disaggregating single storms are described in Yu (2002).This procedure first requires that the fitting parameter,B,in Equation(1)be determined by setting up an equation that is solved with Newton iteration.Once B is known, Equation (1) may be used to determine i for any time in the storm duration.

2.3.Depth-duration-frequency analysis of CLIGEN time series

A 500-year time series was generated for each grid point in the parameter grid.The DDF analysis of each time series considered a selection of sixteen durations and nine return periods for a total of 144 duration-frequency cases.The considered sub-daily durations with time units of minutes were as follows:10,15,30, 60,120,180,360,and 720 min.To avoid underestimation bias from using coarse fixed-interval ground data, depths for sub-daily durations were compared to stations with 1-min,5-min,and 15-min resolution records.Daily and multi-day durations had time units of days as follows: 1, 2, 3, 4, 7,10, and 30 days and were calculated using daily records.Additionally,annual(1-year)durations were considered.For each duration,the following return periods were considered:2,5,10,20, 25, 50, 100, 250, and 500 years.The aforementioned stations shown in Fig.1 were used to assess the accuracy of return period depths for each duration.Return period depths were empirically determined from the 500-year CLIGEN time series based on annual maxima ranked by magnitude, i.e., exceedance probabilities of m/n were used where n = 500 and m is an annual maximum ranking(m=1, 2, 3, …, n).As discussed in the following section, fitting the empirical annual maxima from these time series to the Generalized Extreme Value (GEV) distribution allowed for better comparison to the ground data,which were also fitted to the GEV distribution.

Fig.2.Examples of maps from the UCK series (0.1° resolution).

2.4.Generalized extreme value distribution

For each grid point and duration, the GEV probability distribution was fitted to the annual maxima taken from corresponding 500-year CLIGEN time series.The cumulative distribution function of the GEV distribution is as follows:

where p is the precipitation depth (mm), μ and α are the location and scale parameters,respectively,with the same units as p,and k is a non-zero shape parameter (Overeem et al., 2008).The SciPy Python module version 1.7.3 implementation of the maximum likelihood estimation method was used for estimation of μ and α(Virtanen et al., 2020).A simplification was made to hold the k parameter constant at-0.114,as was done in the global frequency analysis of Courty et al.(2019) that considered hourly/multi-hour durations of grid-scale ERA5 precipitation.This assumption may be made because the k parameter has been shown to remain essentialy constant with respect to change in duration and may asymptotically approach a lesser/more negative value with increasing sample size.Courty et al.(2019) found the expected number of statistically significant GEV fits given 95%confidence for annual maxima series with sample size of n = 40.

Fig.3.Running Average and GEV-fit of annual accumulation for a 500-year CLIGEN simulation in the Amazon.

Two sets of uncertainty metrics for GEV fitting were also mapped:the Filliben statistic, which indicates goodness-of-fit to the GEV distribution, and confidence intervals for μ and α parameters.The 95%confidence level was used for reporting statistical significance of both metrics.The Filliben statistic is equivalent to the Pearson correlation coefficient when plotting GEV-estimated versus CLIGENestimated quantiles on a Q-Q plot (Wilks, 2019).In this Q-Q plot,the exceedence probabilities of the empirical CLIGEN depths were given by the Cunnane plotting position, and the GEV-estimated depths were plotted for the corresponding exceedence probabilities.The critical value for the Filliben statistic was found using the formulation by Heo et al.(2008).However, given the large sample size that was used (n = 500), the critical value is extremely strict(～0.998),and a small fraction of tests passed.Therefore,the Filliben statistic is mainly used to understand relative differences in goodness-of-fit.Confidence intervals for μ and α parameters were defined by bootstrap resampling.In this,104replicate annual maxima datasets with sample size n = 500 were produced using sampling with replacement.The GEV distribution was fitted with each replicate resulting in 104parameter estimates for each parameter.The 2.5 and 97.5 percentiles were taken as the 95% confidence intervals for each parameter value.GEV parameter values and their confidence intervals were mapped at 0.25°resolution according to the resolution of the CLIGEN parameterization grid.

2.5.Universal Co-Kriging

The gridded CLIGEN dataset provided point-scale observations at 0.25°resolution,which is roughly equivalent to ～28 km resolution at the equator.Small-scale spatialgradients inclimate may bedifficultto resolve at this resolution,contributing to error.To solve this,univeral co-kriging(UCK)wasused to interpolate the dataset to 0.1°resolution using the ArcMap 10.6.1 (ESRI, 2018) UCK optimization routine for eachdepth-frequencycase,done separately foreach continent.UCK is a means of producing the best linear unbiased prediction for interpolated locations.In UCK,a predicted value is the sum of polynomial trends and random errors that are fitted through known values.Spatial averages of variance are considered in order to minimize residuals (Brus & Heuvelink, 2007).Multiple covariate map layers may be used to inform trend predictions.For this application,1/24th degree resolution TerraClimate layers of 20-year averages of annual precipitation,daily high temperatures,and downward solar radiation were used.For maps of sub-hourly duration return period depths(10,15,and 30-min durations), gridpoints present in a region along the equator were omitted when creating the UCK maps because GEV fits were poor and depths were obviously too low compared to the surroundings.This region was identified by having substantially lower Filliben statistic values.The issue is visually obvious in sub-hourly empirical depth maps where intensities appear nearly constant and have low sensitivty to climate gradients.Omitting gridpoints in this region subsequently resulted in predictions in this region that were informed by only the covariate map layers,and this region should be disregarded because the UCK model was optimized only for regions where grid point values were available.Causes of this issue are discussed in more detail later.Accuracy of interpolated surfaces was assessed using standard prediction error.Since prediction errors for the UCK model were small and only reflected the contribution to error added by the UCK model itself,error surfaces for the UCK map series were created using prediction intervals from ground validation regressions at 95% confidence, which is discussed in the following.Normality testing of residuals was done with the Shapiro-Wilk test to judge whether prediction intervals were satisfactorily defined.

2.6.Bias adjustments and uncertainties determined by regressions to ground data

The degree of systematic bias was determined for each of the 144 considered duration-frequency cases on an individual basis by performing regressions against ground data.For each case,this involved determining the slope and intercept coefficients of ordinary least squares(OLS)linear regressions forthe relationshipbetweenCLIGENderived versus ground-based depths.This assumed that the groundbased depths could be treated as known depths, though they were quantiles from a fitted GEV distribution and have some degree of error.A clear linear relationship was found in most cases,but the trend was not always close to the 1:1 line,which gave motivation to create a bias-adjustedmapseriesusingthedeterminedregressioncoefficients as bias adjustment factors.The OLS regressions against ground data werealsousedtoestablish uncertaintiesbased onprediction intervals for each regression.Regressions for daily/multi-day and sub-daily durations used two different sets of ground data.Depth estimates derived from NOAA GHCN-Daily stations(shown in Fig.1)were used in regressions for all daily/multi-day durations(see Fig.6 for example regressions that are discussed in more detail later).Whereas for all sub-daily duration-frequency cases, data limitations made it necessary to establish regressions using data from the U.S.as an anology in which collocated,spatially randomsampleswere regressed for PRISM CLIgrid versus NOAA Atlas-14 (where the former is taken as the CLIGEN-derived estimate,and the latter is taken as the known ground data).The transferability of these correlations to the climate types present in South America and Africa represents an unquantified source of error.The datasets involved in these correlations were described in more detail in section 2.1.

Validation of sub-daily cases indicated systematic bias in CLIGEN was often the dominant factor over uncertainties in parameterization of CLIGEN inputs, given the wide range of bias that was found for these cases.Bias adjustments were applied based on the slope and intercept coefficients of the OLS regressions corresponding to each duration-frequency case by calculating the observed/known depth (independent X value) from the estimated depth (dependent Y value).Bias-adjusted return depths and uncertainties were mapped starting with the values of the 0.1°UCK map series and prediction intervals were determined for these values from corresponding OLS regressions.Prediction intervals for the unadjusted UCK map series were based on variance along the Y axis.For the bias-adjusted map series, prediction intervals for calculated X values were determined with a parametric procedure that considers the variance of X values and produces interval bands that are measured along the X axis,parallel to Y=0(Sokal&Rohlf,1995).Therefore,use of OLS regressions provided these outcomes:bias adjsutment factors,uncertainties based on prediction intervals,and error distributions as a form of validation.Since the OLS regressions for sub-daily cases were based on U.S.data, it was deemed necessary to include the three high temporal resolution ground stations in different climate regions of Brazil as an independent validation group.Whereas the one hundred daily ground records were used for both development of bias adjustments and validation since these were within the target areas.

2.7.Depth-duration-frequency curves

It was investigated how DDF curves may be constructed by determining scaling regressions for GEV parameters as a function of duration in order to allow estimation of durations other than those for which GEV fitting was done.GEV parameters may follow a linear trend when plotted as a function of duration using log transformation (Courty et al., 2019; Overeem et al., 2008).In this study,linear trends were found when parameter values and durations were plotted with log-log scaling.Regression coefficients were determined for the correlation of log-transformed parameter values versus durations based on ordinary least squares(OLS).For plotting on an arithmetic scale,the following power function was used:

where m and b are slope and intercept coefficients of the OLS regression,respectively,and x is the duration.Using Eq.(5),estimated GEV parameters for a range of durations could then be entered into the GEV quantile equation in order to approximate continuous DDF curves.Minute-scale and daily-scale durations were analyzed separately in terms of GEV parameter scaling factors,such that two sets of DDF curves were produced for a given location.As will be discussed,longer minute-scale durations (e.g.720-min) appeared to diverge from a linear trend in log-log space.Bootstrap resampling was again used with 104replicate annual maxima datasets in order to define uncertainties for the locations of DDF curves that arise from the combination of GEV fitting and fitting of scaling regressions.Confidence intervals for the location of the regression lines for scaling GEV parameters were first found by fitting the replicate datasets to 104regression lines per parameter.The GEV quantile equation then gave 104quantiles for all durations in the range of durations used for plotting the DDF curves.Again,95%confidence levels were used for these uncertainties based on percentiles.

3.Results

3.1.Overview of map output

The DDF atlas created using this methodology is comprised of different map series,each including return period depths for 16 durations and 9 return periods along with associated uncertainties.This atlas is made available online as a dataset in netCDF format(Fullhart et al., 2022b; doi: https://doi.org/10.5281/zenodo.7126441), which can be opened with Geographic Information System software.Table 1 shows the maps produced for each estimation approach, including map data for:(1)empirical return period depths taken directly from the 500-year CLIGEN simulations; (2) GEV-fitted return period depths;(3)return period depths based on interpolation of GEV-fitted data using UCK to interpolate from the original resolution of 0.25°to 0.1°resolution; and (4) bias-corrected return period depths at 0.1°resolution based on analysis of systematic bias in CLIGEN.The empirical data,which represents DDFrelationships according directly to CLIGEN, provides data for assessing the suitability of CLIGEN for DDFanalysis and its representation of other precipitation factors.One issue with the empirical dataset will be discussed that affects the Amazon and equatorial Africa in which simulated sub-hourly intensities appear to be too low for extreme storms.Therefore, unrealistic aspects of how certain climates are represented in CLIGEN can be identified in the empirical dataset,while other issues relating to CLIGEN are apparent in the comparison between empirical and GEVfitted depths.In contrast to the empirical data, the GEV dataset partially solves issues related to CLIGEN that make the GEV dataset more consistent.Moreover, the GEV dataset enables use of the GEV equation to determine quantiles for any duration-frequency case of interest.The UCK maps interpolated to 0.1°resolution based on GEVfitted depths are better suited to delineating small scale climate gradients and may reduce error in regions with strong gradients.The 0.1°resolution may be sufficient to depict climate regions of interest for national spatial extents.Maps for example duration-frequency cases from the UCK series are shown in Fig.2.Maps of bias-adjusted depths at 0.1°resolution are preferred for sub-daily durations because of strong systematic bias that was found by comparison to ground records and other datasets.

The GEV equation marginally improved accuracy compared to the empirical depths.The average Filliben statistic for GEV goodness-offit across all gridpoints and durations was 0.984.In some situations,the Filliben statistic was considerably lower.The bottom panel ofFig.3 illustrates one reason for this, which was a common scenario where empirical depths in CLIGEN appear to unrealistically diverge from the GEV fit for long return periods, such that the empirical depths only increase slightly with longer return periods.This issue relates to the fact that in a typical CLIGEN simulation,the degree of interannual variability tends to affect the running average of annual accumulation for approximately 200 years.As a result, GEV depth estimates for annual accumulations with long return periods were often greater than empirical depths.However, a similar issue affecting estimation for longer return periods occurred with other durations as a result of high precipitation rates.The most obvious instance of this happened for sub-hourly durations in a geographic band along the equator that is particularly wide in South America,and that is narrower across Africa.This is visually apparent in maps for sub-hourly durations in the GEV-fitted map series, and is also reflected in poor GEV goodness-of-fit (i.e.low Filliben statistic values).As a result of this issue,these regions should be disregarded for the three sub-hourly durations.

Table 1 List of map series with their corresponding resolutions, included map types, and uncertainties.The number of maps for each map type are shown in parentheses.

Fig.4.Average error from validation of 100 GHCN-Daily records in South America and Africa for all daily/multi-day durations.

For daily/multi-day durations, averages of absolute error percentages sorted by return period are shown in Fig.4.This analysis suggests that GEV-fitting reduced error for long return periods in daily/multi-day cases.As will be discussed,systematic bias for daily/multi-day durations was not as strong as for sub-daily durations.Average percent bias for daily/multi-day durations sorted by return period(using the same data as Fig.4)ranged between-4%and 4%for empirical depths,and was reduced to-3%and 2%for the GEV depths and to-2%and 3%for UCK.Since bias is low overall for daily/multiday durations, either the GEV or the UCK map series may be used,while the latter may be preferred in areas with strong spatial gradients.In general,a small number of daily/multi-day cases were identified that would warrant bias adjustments given that only four cases had PBIAS values above 10% (see Table A5 for performance metric definitions).Depths of annual durations had considerably higher PBIAS values that suggested underestimation for long return periods.

Additionally,distributions of absolute error for all daily/multi-day durations and return periods were considered on the basis of the different estimation methods used(see Fig.5).For UCK,overall error increases only marginally,contrary to what might be expected given that the sampled depths represent interpolated values.Furthermore in UCK,the largest errors were eliminated from the distributions in Fig.5,reducing the range of error,which may relate to improved accuracy in areas with strong spatial gradients.The same distributions were plotted for the U.S.analogy in Figure A1, which shows distributions that are similar to the UCK outcome.For sub-daily distributions, Figure A2 shows that error is significantly increased.It is important to note that aggregate error statistics and distributions of error depend on the duration-frequency cases that were selected for consideration.

3.2.Ground validation and determination of systematic bias in CLIGEN

3.2.1.Validation of daily/multi-day and yearly durations

For daily/multi-day and yearly durations,the GHCN-Daily stations in Fig.1 were used to validate depths.Reporting of this validation is done for the UCK map series,which is the recommended series for these durations in terms of error reduction and spatial resolution.The ground records for validation had significant GEV fits for 64%of the duration-frequency cases, which is considerably less than the 95%expected using the critical value of the Filliben statistic.Significant GEV fits were commonly found for some durations at a given station but not others.This adds an unquantified uncertainty that could potentially relate to a number of factors.All OLS regressions used to establish bias in CLIGEN were highly significant, but as indicated in Table A1,only 58%of regressions passed the Shapiro-Wilk test.Given the null hypothesis,the interpretation for this outcome is that there was evidence that 42% of regressions are not normally distributed,and therefore, prediction intervals for regressions that show nonnormality are poorly defined and should be considered a minimum error.This outcome is partly due to challenges related to sampling design.ExampleOLSregressionsfortwoduration-frequencycasesare shown in Fig.6 for the 2yr-1day and 100yr-1day cases that are illustrative of the fact that wide prediction interval bands and high variance existed for cases that considered long return periods.The regression for the 100yr-1day case is an example of non-normal residuals that shows clustering of datapoints,which reflects the spatial clustering of ground locations.Statistics for estimation performance of each individual duration-frequency case are given inTable 2.Based on PBIAS values, a pattern was found of overestimation for shorter durations and shorter return periods that trended towards underestimation for longer durations and return periods.Aggregated statistics considering all duration-frequency cases pooled together had median and average absolute errors of 16.7%and 16.4%,respectively,and average PBIAS of 1.2%.

Fig.5.Distributions of absolute error for all daily/multi-day durations and return periods in the validation against 100 NOAA GHCN-Daily ground stations(n=6300 data pairs).The tick marks between 0%and 25%are the medians and averages that remain at approximately 13%and 16%,respectively,for each estimation method.The averages of all PBIAS values for the three estimation methods are within the range of 0%-1%.

Fig.6.Regressions for example duration-frequency cases, 2yr-1day and 100yr-1day.Confidence and prediction bands are shown at significance level of 0.05.

3.2.2.Validation of sub-daily durations

The analogy to U.S.data allowed correlations to be established between CLIGEN-derived return period depths and expected ground values for sub-daily durations.This analysis found signficant bias in CLIGEN,and therefore,bias-adjusted depth maps are recommended for sub-daily durations.The analogy to U.S.data ignores differences in climate types between the U.S., South America, and Africa.Therefore, a limited dataset of three high temporal resolution ground records in Brazil were additionally used to investigate the transferability of the estiblished correlations to South America and Africa.The U.S.analogy used two sets of CLIGEN-derived data that were compared to expected ground values from NOAA ATLAS-14 at 100 spatially random locations.Results comparing the PRISM CLIgrid and NOAA ATLAS-14 spatial datasets are reported in this section, while comparisons are given in the appendix to the nearest available CLIGEN parameterization from the dense ground network of Srivastava et al.(2019), which used ground observations.PRISM CLIgrid and the dataset of Srivastava et al.(2019) gave very similar results, suggesting that bias inherent in CLIGEN was identified.

Metrics of regression performance for the U.S.analogy are given in Table 3.Considerable bias in CLIGEN occurs for these duration cases that tends towards overestimation.The bias adjustments determined from these regressions use the coefficients given in Table A2.The greatest errors tend to occur for return periods of 20 years or less and for durations in the range of 30-min to 120-min.There is also a trend oflongerreturnperiodsshowinglessbias,which subsequentlyresults inabsoluteerror valuesthatgenerally decreasewithincreasingreturn period,counter to what might normally be expected.A similar percentage of regressions(61%,indicated in Table A2)for sub-daily cases passed normality testing compared to the daily/multi-day and yearly cases despite less clustering of the sample design in the U.S.analogy.Tables 4-6 show the estimation accuracy of bias-adjusted depths for the three ground stations in Brazil used to validate sub-daily durations.The records had different temporal resolutions and were therefore only used to validate sub-daily durations long enough to be unaffected by underestimation bias that arises from temporal averaging.Each record is discussed individually in the following.First,aggregate statistics are given for the three records considering all duration-frequency cases pooled together.The average PBIAS was-6.2%,and the median and average absolute percent errors were 11.9%and 14.5%,respectively.Absolute error isbroken downby return period in Fig.7 for the three records,showing that errors are highest for the 500-year return periods.

Table 2 Regression performance statistics for daily-scale and longer durations.The ordering of statistics given for each duration-frequency case is as follows: PBIAS (%), absolute percent error (%), and RMSE (mm).

Table 3 Regression performance statistics for sub-daily durations.The ordering of statistics for each duration-frequency case is as follows: PBIAS (%), absolute percent error (%), and RMSE (mm).

Table 4 Estimation accuracy of bias-adjusted depths against the 1-min record for the Guaraíra Experimental Basin.The ordering of statistics for each duration-frequency case with sample size of n = 1 data pairs is as follows: PBIAS (%) and RMSE (mm).

Table 5 Estimation accuracy of bias-adjusted depths against the 5-min record for the Aiuaba Experimental Basin.The ordering of statistics for each duration-frequency case with sample size of n = 1 data pairs is as follows: PBIAS (%) and RMSE (mm).

Table 6 Estimation accuracy of bias-adjusted depths against the 15-min record for the São Paulo record.The ordering of statistics for each duration-frequency case with sample size of n = 1 data pairs is as follows: PBIAS (%) and RMSE (mm).

Fig.7.Average error from validation of 3 high temporal resolution records in Brazil for sub-daily durations against the bias-adjusted map series.

The three ground records used to validate bias-adjusted sub-daily depths had record lengths ranging from 13 years to 23 years.These records were significantly shorter than the 50-year daily records and containedlongergaps,but thiswasdeemednecessarytovalidate subdaily durations.Short data gaps existed that may be a source of error,but these generally did not occur during periods when the highest rainfall would be expected.Longer gaps affecting entire years meant that continuous periods had to be concatenated, shortening the record lengths.The 1-min,19-year record forthe Guaraíra Experimental Basin(GEB)in Paraíba represents a tropical climate(Coutinho et al.,2014).The reference period is from 2004 through the wet season of 2022.It was used to validate all sub-daily durations, with results showninTable4.Estimateddepthshadthe poorestcomparison tothe GEB record of the three ground records,and particularly highly errors occurred for durations of 30-min or less.This high error could not be confirmed fordurations of 30-min or less using the other two records.Given that GEV fits for the GEB record were significant, suggesting minimal error in ground-derived quantiles, one or more of several possibilities could explain this high error:bias adjustments for these durations were not effective in all cases and tropical climates may require special bias adjustments; or high CLIGEN parameterization error occurs at the GEB location as a result of strong spatial climate gradients, possibly related to localized coastal influence and topographic gradients in the Gramame basin.It is also possible that measurement bias arises due to use of a tipping bucket gauge(0.254 mm/tip),but this cannot explain the large errors that occur in certain cases.The 5-min record for the Aiuaba Experimental Basin(AEB)in Cearˊa represents a semi-arid climate.It was used to validate 30-min through 720-min durations.The reference period is from 2003 to 2018 with a 3-year gap.The continuous record lengths were concatenated to produce a 13-year record.All but one GEV fits were significant.Table 5 shows that errors were generally low,including the 30-min duration,which had greatly reduced error compared to the GEB validation.The 15-min,23-year record for São Paulo represents a subtropical climate.The reference period was from 1997 to 2021 with 2 years of gaps.It was used to validate 120-min and longer durations,with three of four GEV fits being significant.The errors shown in Table 6 were generally in the same range of those found for AEB.

3.3.DDF curves

This section explores the potential of using the created atlas to produce DDF curves.The scaling of GEV parameters is first discussed,followed by the plotting of DDF curves.Fig.8 shows typical outcomes of OLS regression of GEV parameters against duration.Scaling regressions for daily/multi-day durations tend to have smaller residuals than regressions for sub-daily durations in which more non-linearity was evident.An issue consistently occurred for regressions of subdaily durations where the 360-min, and particularly the 720-min durations, diverged from the trend given by the shorter durations,such that 360-min and 720-min GEV parameters were essentially the same.Considering this, the 720-min GEV values were omitted, and the trendline was allowed to extend to 720-min based on the trendline fitted with all shorter durations.Part of the reason that 360-min and 720-min GEV parameter values were often very similar relates to the fact that only a single storm may occur per day in CLIGEN, and therefore,for a given storm,the same 360-min and 720-min depths would be found unless the storm has a particularly long duration.Given the systematic biases found for certain sub-daily cases,it may be more accurate to only include durations with small biases in the scaling regression.The large sample size used for fitting the GEV distribution(n=500)resulted in confidence intervals for individual μ values that were generally very small, resulting in overlapping parameter confidence intervals for parameters and regression bands in Fig.8,while uncertainties for α were generally larger.Example DDF curves are given in Fig.9 for selected return periods.Confidence interval bands for the location of the curve become large for longer return periods and durations.The overestimation bias that affects sub-daily return period depths is evident for the minute-scale DDF curves because the indicated depths at 720 min exceed those at 1 day.The non-linearity and large residuals for the sub-daily parameter scaling regressions may therefore be related to the degree of systematic bias.

4.Discussion

Fig.8.Scaling of location (μ, top lines) and scale (α, bottom lines) for Lagos, Nigeria and São Paulo, Brazil with 95% confidence intervals and bands (overlapping of confidence metrics occurs for location regressions).

Fig.9.DDF curves for Bogotˊa, Colombia, and Kinshasa, D.R.C., with 95% confidence bands reflecting uncertainty of GEV-fitting and GEV parameter scaling.

Our results indicated strong systematic bias in CLIGEN for some duration-frequency cases, making it necesary to use biasadjustments, particularly for sub-daily durations.Given that average bias for daily/multi-day return period depths was small,bias-adjusted depths are only recommended for sub-daily durations.There are similarities in the determined sub-hourly biases to those of the study of Wang et al.(2018) that was done for China with its own representation of climate types.Their study used records with high temporal resolution measurements and found overestimation bias for sub-daily return period depths ranging from factors of ～1.2 to a maximum of ～1.4 going from 10-min to 60-min duration, which then decreased to ～1 (no bias) for 720-min,and stayed close to 1 for longer durations up to 24 h.A similar pattern is evident in PBIAS values in Table 3 and slope coefficients in Table A2 that indicate cases with the largest bias occur for 60-min to 180-min durations.Though the systematic bias is likely a function of climate type,the fact that there was consistency in the ground-validation of bias-adjusted sub-daily depths in differing climate types suggests that climate type may have a relatively small impact on systematic bias in CLIGEN.

The interpolation using UCK resulted in only a marginal increase in error while small-scale climate features were greatly smoothed.For cases with long return periods,this had the additional benefit of smoothing an unrealistic degree of spatial heterogeneity in neighboring grid cells where generated return period storms in adjacent grid cells have dissimilar characteristics to a degree that would not be expected (which is clearly evident in the empirical map series).The UCK interpolation was also able to represent features that are of interest for national spatial extents.Some countries may not have similar coverages of DDF information as provided by the 0.1°UCK and bias-adjusted series, and there are good comparisons in the depicted spatial climate gradients and climate zones to those of existing sources of data.

An example of a country where small-scale climate features are indicated is Ethiopia.The high rainfall depth in the central and northwestern regions (seen in Fig.2) overlaps with high elevation areas consistent with ground observations.The Ethiopian rift valley extending to the southeastern lowlands represents rainfall scarce regions, which were captured by the atlas.Analysis of ～30 years(1980-2012)point data from 55 rain gauges in the Oromia region of Ethiopia showed DDF distributions of 100-year,1-h depths (Tesfaye& Demissie, 2015)with a similar range of precipitation as the atlas.The atlas also captured the general spatial distribution of observed rainfall across Africa.Similar to Ethiopia,the rugged terrain region in eastern Africa shows significant spatial variability.The maps in Fig.2 show other monumental climate regions such as the Sahara Desert as the driest region, the Sahel buffer region above the Congo basin,the high rainfall region in the Congo basin, the variable climate regime in east Africa,and the Kalahari dry regions in Southern Africa.The greatest concentration of rain often happens between July and August in high elevation areas.In these regions, dense rain gauge observation or radar technologies are unavailable to capture the high rainfall variability.Consequently, the rational formula is still widely used for estimating design peak discharges in Africa (Ayalew et al.,2022).In this case, datasets such as the present atlas and corresponding CLIGEN parameterization can provide robust estimates of design rainfall for data scarce regions, and favor use of continuous modeling approaches for design storms,as opposed to event-based empirical models such as the rational formula.

We also noted realistic DDF patterns across Brazil.This country is the largest in spatial extent of South America, having 6 biomes and large climate variation with the highest precipitation amount in the Amazon biome located in the northernportion(>3000 mmyr-1),and the lowest in the Caatinga bioma (semi-arid region) located in the northeast (<800 mm yr-1).High values of precipitation are also observedinthe Atlantic Forest biome(>2500 mmyr-1),located inthe Brazilian coastal area.Intermediate amountsof precipitation(1,000 to 2,500 mm yr-1) are observed in the Cerrado biome located in the centralportionofBrazil(woodlandsandsavanna),thePantanallocated inthe midwestregion(the largesttropical wetlands intheworld),and the Pampa biome,region in southern Brazil(Almagro et al.,2020).In terms of spatial disribution and order of precipitation values, our findings also corroborate with average daily precipitation values presented in Xavier et al., 2022 and Almagro et al., 2021, and with the rainfallerosivitymapofBrazilproposedbyOliveiraetal.,2013.InBrazil,hundreds of daily precipitation data have been collected,organized,and made freely available by the National Water Agency(ANA)and were most recently organized in usefuls datasets such as BR-DWGD(Xavier et al., 2022) and CaBra (Almagro et al., 2021).However, we could not find sub-daily precipitation data or DDF relations with expansivespatialcoverageofBrazil.Forinstance,togeneratesub-daily DDF in Brazil for engineering design,engineers typically desegregate daily rainfall by using tabulated desegregation coefficients.Therefore,the atlas proposed here has the potential to be a useful tool used in hydrological and soil erosion models, soil and water conservation practices,and water resources engineering design.

Applications using data from this atlas should consider several problems that caused certain duration-frequency cases to have large uncertainties or poorly defined uncertainties.Some of the problems were related to sources of bias in CLIGEN.One such issue was the underestimation of sub-hourly intensities along the equator that was identified as a region with low Filliben values.The region appears to largely coincide with the Intertropical Convergence Zone (ITCZ) in which convective rainfall is predominant,mainly due to a zone of low pressure and low-level wind convergence.The generated return period storms in CLIGEN are substantially greater in magnitude in this region than in the surroundings.The poor GEV fitting for sub-hourly durations occurs for these storms due to dampening factors in CLIGEN, where 30-min intensities are reduced for months with large storm accumulations and a large number of rainy days.Subsequently,this region has the lowest α0.5values of any other region.These issues suggest that the scaling of dampening factors for 30-min intensity in CLIGEN could be improved in areas with high precipitation rates.

A considerable number of OLS regressions used for biasadjustments had residuals that failed the Shapiro-Wilk test (successful tests are indicated in Table A1 for daily/multi-day cases and Table A2 for sub-daily cases).Prediction intervals for the regressions that were shown to have non-normal residuals should be considered suspect.Non-normal residuals are largely due to the sampling design of locations used for validation.There was strong spatial clustering of GHCN-Daily stations (see Fig.1), and despite the spatially random sampling of locations in the U.S.analogy,non-normality also affected sub-daily regressions, possibly because of the unequal representation of climate types across the U.S.OLS regression also has an assumption that X values are observed and not subject to error.This assumption is likely violated given the challenges with GEV fitting of ground data and using NOAA ATLAS-14 to approximate observed values.This represents an unquantified source of uncertainty in the established bias adjustments and prediction intervals.

Studies such as this that use ground networks for validation of gridded precipitation may benefit from future research on how sampling bias impacts the resulting uncertainties.The fact that even the OLS regressions for the U.S.data(with locations that were spatially random)had a number of failed Shapiro-Wilk tests seems to suggest that spatial clustring of data(as occurs with the GHCNDaily distribution) is not the only factor for explaning non-normal residuals.The spatial distribution of climate types within the sampled areas may also be a factor, and some means of spatially weighting by climate type may therefore result in better statistical outcomes.Related to this issue, further work could better identify the extent to which error is increased in areas with strong climate gradients within the resolution of the CLIGEN parameterization and how uncertainty is subsequently affected.The inclusion of ground stations for validation in such areas(e.g.,the Guaraíra experimental basin) was done by necessity rather than by a random sampling design, which may lead to bias.

The present analysis supports the finding that daily/multi-day precipitation extremes are well approximated in CLIGEN, while sub-daily extremes tend to be overestimated.This should be considered when using CLIGEN in conjunction with hydrological or soil erosion models for the simulation of extreme events.For soil erosion modeling, overestimation of sub-hourly durations is very likely to lead to overestimation of runoff and sediment yield.In the equation for rainfall erosivity,intensity for a given interval appears as a power term,and 30-min maximum intensity is a multiplicative factor.Therefore, erosion rates are highly sensitive to rainfall intensity.Considering this, the largest erosion events driven by CLIGEN should be considered suspect in any soil erosion modeling application that simulates extreme events.

5.Data availability

The NetCDF formatted atlas is available at https://zenodo.org/record/7126441#.YzZ-LHbMKUk (doi: https://doi.org/10.5281/zenodo.7126441).The map layers are organized by continent and then by the map series shown in Table 1.Additionally, the CLIGEN parameterization used to produce the time series on which the DDF analysis was based is available at https://data.nal.usda.gov/dataset/gridded-20-year-parameterization-stochastic-weather-generatorcligen-south-american-and-african-continents-025-arc-degree

resolution (doi: https://doi.org/10.15482/USDA.ADC/1524754).For interested readers, a gridded CLIGEN parameterization has also become available for the remaining global land masses encompassed bythe latitudinal band of South America and Africa at https://data.nal.usda.gov/dataset/gridded-20-year-parameterization-stochastic-weather-generator-cligen-fill-gaps-coverage-south-40thparallel(doi:https://doi.org/10.15482/USDA.ADC/1528372).

6.Conclusions

This novel methodology for estimation of DDF relationships provides a means of overcoming data limitations, scaling to large spatial extents, and determing return period precipitation for a wide range of durations.The map outputs have sufficient resolution to identify small-scale climate zones that reflect local geographies and climate dynamics.Return period depths for daily/multi-day and yearly durations tended to be better estimated than sub-daily durations.This finding may be useful for assessing the accuracy of extreme events in modeling applications coupled to CLIGEN.Comparisons to ground data provided measures of uncertainty that reflect multiple sources of error, including prediction error.These comparisons identified systematic bias in CLIGEN based on several corroborating lines of evidence.In some cases, determining prediction error from validation against ground networks resulted in poorly defined uncertainties,which warrants further investigation into sampling design and other possible factors.The atlas represents an important step in producing point-scale DDF estimates across large spatial scales in data sparce regions.

Declaration of competing interest

The authors declare no competing or conflicting interests.

Acknowledgements

This research was a contribution from the Long-Term Agroecosystem Research (LTAR) network.LTAR is supported by the United States Department of Agriculture.The Brazilian co-authors are grateful to the Brazilian National Council for Scientific and Technological Development (CNPq)for the PQ fellowships.

Appendix

Table A1 Regression parameters for daily/multi-day durations of UCK versus GHCN-Daily.Regression parameters are listed in the following order:slope coeffciient,intercept,and Pearson rsquared.Cases that passed the Shapiro-Wilk test are marked with an asterisk (*).

Table A2 Regression parameters for sub-daily durations of PRISM CLIgrid versus NOAA Atlas-14.Regression parameters are listed in the following order:slope coeffciient,intercept,and Pearson r-squared.Cases that passed the Shapiro-Wilk normality test are marked with an asterisk (*).

Table A3 Regression performance statistics for sub-daily durations of Srivastava et al.(2019)versus NOAA Atlas-14.The ordering of statistics for each duration-frequency case is given as follows:PBIAS(%),absolute percent error(%),and RMSE (mm).

Table A4 Regression parameters for sub-daily durations of Srivastava et al.(2019)versus NOAA Atlas-14.Regression parameters are listed in the following order:slope coeffciient,intercept,and Pearson r-squared.Cases that passed the Shapiro-Wilk normality test are marked with an asterisk (*).

Table A5 Model performance metrics for observed (O) and predicted (P) values according to Moriasi et al.(2007).

Figure A1.Distributions of absolute error for all daily/multi-day durations and return periods for US analogy data.The tick marks between 0% and 25% are the medians and averages, respectively.

Fig.A2.Distributions of absolute error for all sub-daily durations and return periods for US analogy data.The tick marks between 25% and 50% are the medians and averages,respectively.