Changes in regional daily precipitation intensity and spatial structure from global reanalyses

We conducted an analysis of hydrological cycle variations across 13 regions of varying sizes distributed across different continents. The analysis is based on five reanalysis datasets of daily precipitation, all produced by the European Centre on Medium‐Range Weather Forecasts (ECMWF): ERA5 high‐resolution, ERA5 ensemble, CERA‐20C, ERA20‐C and ERA20‐CM. We examined several climate indicators, including the daily mean precipitation, the 75th and 99th percentiles, the precipitation area fraction and the area fractions with precipitations exceeding 10 and 20 mm. We evaluated the ability of the reanalyses to capture precipitation at specific spatial scales using scale‐separation diagnostics based on 2D wavelet decomposition. The climatological energy spectra of precipitation derived from the analysis describe the scales that each reanalysis can accurately reproduce, serving as a unique signature for each dataset. We compared the spatial scales that were comparable across the different reanalyses and examined the temporal trends of energy on those scales. The results indicate that the hydrological cycle is undergoing changes in all regions, with some variations observed across different regions. Common features include an increase in intense precipitation events and a decrease in the corresponding spatial extent. The ensemble of ERA5 reanalyses exhibited the smallest effective resolution, as determined by the scale‐separation method, and displayed more pronounced trends compared to other reanalyses. Notably, an acceleration of changes is evident in the last 20 years. However, Central Asia may be an exception, showing relatively less noticeable changes in the hydrological cycle.

seasonal time scales, is estimated to be around 7% per 1 C of warming due to the increased capacity of the atmosphere to hold water.Recent studies, such as the one by Rodell and Li (2023), which analysed observations from the GRACE satellite, and the study by Rohde (2023), based on satellite data, confirm the findings of the IPCC report and observe an increase in climate extremes and the intensification of the water cycle due to climate change.
The global hydrological cycle plays a crucial role in Earth's climate system, influencing both energy flow and the circulation of water.It is intertwined with the greenhouse effect, and the two cannot be reduced to individual components (Benestad, 2016).Recent analysis also indicates an increase in the total mass of water falling on Earth's surface from 1950 to 2020, and the link between global daily water mass, precipitation surface area, and statistics of local mean precipitation intensity and wet-day frequency is documented by Benestad et al., in preparation.These parameters are important for understanding the probability of local extreme rainfall events (Benestad et al., 2019;Benestad et al., 2020).
In our previous work (Benestad et al., 2022), daily global total rainfall, global mean precipitation intensity and global surface area were used as climate indicators to study the global hydrological cycle.The study, based on reanalysis data from 1950 to 2020, found a decrease in the global area of daily precipitation as a percentage of the global area, while total global rainfall increased over time.Scale-separation methods using wavelet decomposition were employed to analyse variations in the spatial structure of precipitation.Other studies, such as Chen et al. (2023), have also examined changes in the spatial structure of precipitation, demonstrating the intensification of cold-season storms over the western United States under a warming climate using high-resolution regional simulations.
Reanalysis datasets are valuable for studying the global hydrological cycle as they incorporate diverse observations from various sources to create daily precipitation fields.While reanalyses provide temporal consistency, the variations in observational datasets over time can impact the obtained precipitation trends.To address this, the present study utilizes five different reanalysis datasets, enabling better interpretation and extraction of the climate signal.Alternatively, output from global climate model simulations, such as the CMIP ensembles (Eyring et al., 2016), can be analysed without data assimilation.
This study builds upon the previous work by Benestad et al. (2022), focusing on 13 specific overlapping regions to investigate regional variations in the hydrological cycle.We employ a scale-separation diagnostic based on 2D wavelet decomposition to assess changes in the spatial structure of precipitation within each region.To account for the deformation of precipitation areas when moving across each region, a resampling and regridding procedure has been implemented for accurate comparison and application of wavelet decomposition.
The document is organized as follows.Section 2 provides a description of the reanalyses used.The domains, along with the resampling and regridding procedure, are explained in section 3. Section 4 outlines the wavelet representation of a precipitation field.Section 5 presents the results, including the time series of climate indicators and the key findings from the scale-separation diagnostics.Finally, in the conclusions, we summarize the changes in daily precipitation across all regions.

| DATA
We utilized five reanalysis datasets generated by the European Centre for Medium-Range Weather Forecasts (ECMWF).The geographical reference system for all reanalyses is based on latitude and longitude adapted to a spherical Earth.To obtain daily precipitation data, we downloaded the precipitation fields at subdaily resolutions, such as hourly and 6-hourly intervals.The daily precipitation values were then computed by summing the subdaily fields corresponding to each day.
The availability periods for the different reanalyses are different.However, for the purpose of comparing all five reanalyses, we focused on the common period of 1959-2010, which corresponds to 52 years of daily precipitation fields.
As the usage of ERA5 grows over time, the advantages and disadvantages of this reanalysis become more apparent.Bandhauer et al. (2022) compared ERA5 daily precipitation fields with observational gridded datasets over Europe and found a reasonable agreement between ERA5 and the observational datasets.However, ERA5 was found to overestimate mean precipitation in all regions, which was attributed to an overestimation of the number of wet days.Lavers et al. (2022) recommended the use of ERA5 for extratropical precipitation monitoring based on their analysis of monthly precipitation.They noted that ERA5 errors tend to increase in tropical regions.In our study, our main focus is on detecting variations within each individual reanalysis and comparing those variations among the reanalyses.The technical characteristics of all reanalyses used in our study are listed in detail in Hersbach et al. (2020, table 1).
As previously mentioned, all the reanalyses considered in our research were produced at ECMWF and were generated using various cycles of the ECMWF Integrated Forecasting System (IFS).Consequently, when examining the differences between these reanalyses, we primarily attribute them to differences in model resolution, parameterizations and variations in assimilated observational networks.We operate under the assumption that the fundamental model dynamics remain relatively consistent across different versions of the IFS.
A natural expansion of our study, which is not presented here, would involve conducting similar analyses using precipitation data from sources such as MERRA-2 (Gelaro et al., 2017) and JRA-55 (Kobayashi et al., 2015) reanalyses.This would enable us to evaluate the sensitivity of the climate indicators we have employed to variations in the numerical models used for reanalysis generation.

| ERA5 and ERA5-EnDA
ERA5, as described in Hersbach et al. (2020), is the latest global reanalysis produced by ECMWF on behalf of the Copernicus Climate Change Service (C3S).We obtained the hourly precipitation fields from 1959 to 2022 from the Climate Data Store (CDS) using the product "ERA5 hourly data on single levels from 1940 to present" (DOI: https:// doi.org/10.24381/cds.adbb2d47),accessed on April 29, 2023.
We utilized both the high-resolution fields, referred to as ERA5, and the coarse resolution 10-member ensemble, referred to as ERA5-EnDA.ERA5 reanalyses were generated using the IFS cycle 41r2, which was operational in 2016, and the 4D-Var data assimilation utilized 12-h windows.The data assimilation process incorporates a substantial amount of observations, which vary significantly over time.
For ERA5, we downloaded the hourly fields on a regular grid with a resolution of 0.25 in both the zonal and meridional directions, resulting in a grid size of 1440 by 721 points.According to the ERA5 user guide, the data resolution corresponds to approximately 31 km.
Regarding ERA5-EnDA, the grid spacing used is 0.5 , resulting in a grid size of 720 by 361 points.This grid spacing corresponds to a resolution of approximately 63 km.

| CERA-20C
CERA-20C, as described in Laloyaux et al. (2018), is a climate reanalysis dataset covering the period from 1901 to 2010.It is generated using a coupled data assimilation system called CERA, which assimilates various observations including surface pressure, marine wind observations, and ocean temperature and salinity profiles.
In our study, we have utilized the 10-member ensemble of CERA-20C, and the dataset has been downloaded from the Meteorological Archival and Retrieval System (MARS).CERA-20C was produced using the ECMWF IFS cycle 41r2 (implemented on March 8, 2016).
The grid spacing of CERA-20C corresponds to approximately 125 km.For our analysis, we downloaded the dataset on a regular grid with a spacing of 0.5 , resulting in a grid size of 720 by 361 points in both the zonal and meridional directions.

| ERA-20C and ERA-20CM
ERA-20C, as described in Poli et al. (2016), is a deterministic climate reanalysis dataset covering the period from 1901 to 2010.It has a grid spacing of approximately 125 km.On the other hand, ERA-20CM, as described in Hersbach et al. (2015), is a 10-member ensemble climate reanalysis dataset covering the same period as ERA-20C.It also has a grid spacing of approximately 125 km.ERA-20C and ERA-20CM are based on the IFS cycle 38r1 (implemented on June 19, 2012).
In both ERA-20C and ERA-20CM, sea-surface temperature and sea-ice cover are prescribed using an ensemble of realizations from HadISST2 (Titchner & Rayner, 2014).The forcing terms in the model radiation scheme follow the recommendations of the Coupled Model Intercomparison Project (CMIP5) (Taylor et al., 2012).
ERA-20C assimilates observations of surface pressure and surface marine winds only.On the other hand, ERA-20CM does not include the assimilation of observations.Therefore, ERA-20CM is expected to have less accuracy and precision in reproducing the actual precipitation fields compared to the other reanalyses.However, ERA-20CM is not sensitive to variations in the number of observations, allowing it to serve as a reference for assessing the impact of observation variations on the results.
For both datasets, we have downloaded the data from MARS on a regular grid with a spacing of 0.5 , resulting in a grid size of 720 by 361 points in both the zonal and meridional directions.

| DOMAINS
The study primarily focuses on 13 spatial domains or regions, as depicted in Figure 1.These domains are loosely based on those defined within the CORDEX framework (The COordinated Regional Downscaling EXperiment, www.cordex.org,last access: November 11, 2023), with the aim of encompassing a substantial portion of the Earth's landmass, where the majority of the global population resides.
It is worth noting that we have adjusted the regions to ensure that they are perfectly square in our analysis.Consequently, the domains shown in Figure 1 are transformed into square domains during the pre-processing steps, as described in section 3.1.
The characteristics of all domains are summarized in Table 1, which also provides references for the regridding procedure and the wavelet decomposition discussed in sections 3.1 and 4, respectively.
The "target grid" serves as the basis for our analysis.In the table, we specify the side length of each square domain (abbreviated as "Ext.").These domains are relevant for capturing atmospheric phenomena across different scales.
In the following section, we will adopt the atmospheric scale definitions originally introduced by Orlanski (1975) and also cited by Thunis and Bornstein (1996).These definitions delineate various scales within the atmosphere based on specific characteristics.The general circulation and synoptic cyclones fall into the macroscale category.These phenomena are distinguished by their relatively long lifespan, typically exceeding 1 week, and horizontal length scales that extend beyond 2000 km.Fronts and hurricanes, on the other hand, are categorized within the meso-α scale.They exhibit a lifespan ranging from 1 day to up to 1 week, with maximum horizontal length scales reaching approximately 2000 km.Phenomena with a lifespan of 1 day, such as thunderstorm groups, are classified under the meso-β scale.This scale is characterized by horizontal length scales ranging from 20 km to approximately 200 km.Single thunderstorms are categorized as meso-γ scale events since they typically endure for about 1 h, and their spatial extent falls within the range of 2-20 km.
The selected domain sizes consistently range from above 2000 km to below 10,000 km.This choice ensures their suitability for capturing the multiday evolution of atmospheric phenomena spanning from the meso-β to the macroscales.
In term of expected behaviour of precipitation patterns over the 13 regions, we can refer to chapter 8 of IPCC AR6 (IPCC, 2023b) and IPCC AR6 regional fact sheets (https://www.ipcc.ch/report/ar6/wg1/resources/factsheets, last access: November 11, 2023).For example, According to IPCC AR6 ch.8, there is likely an overall increase in annual mean precipitation over midlatitude land areas in the Northern Hemisphere, with medium confidence noted after 1951.Moreover, there is evidence of a more pronounced increase since the 1980s.Concurrently, global warming has intensified droughts in several regions.An increased contrast between wet and dry areas in the Tropics and subtropics have been observed.This indicates increased rainfall in wet regions and slight decreases in dry regions from 1988 to 2019.For a review of wetting and drying trends trends under climate change, readers are directed to Zaitchik et al. (2023).A comprehensive review of regional precipitation trends can be found in IPCC AR6 ch.8, section 8.3.1.3.It is worth noting that within each of the 13 selected regions, multiple subregional precipitation trends may exist, making our analysis a summary of a more intricate subregional situation.For an exploration of the causes and mechanisms behind large-scale responses of the water cycle to current climate change, we recommend referring to Allan et al. (2020).They highlight that "Rapid adjustments to forcings, cooling effects from scattering aerosols, and observational uncertainty can explain the current challenges in detecting observed global precipitation responses.However, these responses are expected to become more detectable and accelerate as global warming increases and aerosol forcing diminishes".

| Pre-processing of the gridded fields
The pre-processing steps are conducted to mitigate the impact of coordinate reference system distortions on the final results, specifically in relation to the shape of precipitation patterns.The original grids are based on geographical coordinate systems, and since we are working with global datasets that have different grid spacing, we transform the data to a Lambert Azimuthal Equal-Area (LAEA) projected coordinate system.Additionally, we work with square domains to ensure consistency and facilitate analysis.
The choice of LAEA is motivated by its accurate representation of area across the sphere, which is crucial for assessing variations in the areal extent of precipitation events.In contrast, the geographical coordinate system introduces distortions in shape and area due to map projection, making it challenging to interpret the results.By using LAEA, we can accurately capture variations in precipitation area without confounding effects from systematic shifts in precipitation system trajectories along the meridional direction.
The transformation of precipitation fields from their original grids to the target LAEA grids is carried out through a two-step procedure.Figure 2 provides an example for the daily precipitation totals on July 14, 2021, which experienced extreme flooding in parts of Belgium, Germany and surrounding countries.The NEU domain is shown as an illustration.
T A B L E 1 The regional grid specifications apply to CERA-20C, ERA-20C and ERA-20CM datasets.Note: The original data was downloaded on a global grid with a spacing of 0.5 in both the northing and easting directions.The target grid for these datasets always consists of 1024 × 1024 grid points ("Ext."represents the extension of the domain, which is a square-shaped area; we provide the length for only one dimension in km).For further clarification, the definitions of the symbols used in the column headers can be found in sections 3.1 and 4. The distances are expressed in km.
In the first step (from the left panel to the middle panel of Figure 2), the precipitation fields are reprojected onto an intermediate LAEA grid that is coarser than the target grid.When reprojecting the ERA5 grid points from the geographical coordinate system to the LAEA reference system, the spacing between points changes, and the reprojected points no longer form a regular grid.The transformed ERA5 grid points are represented by the red points in Figure 3.The goal of resampling on the regular LAEA intermediate grid (represented by the filled blue points in Figure 3) is to ensure a similar resolution of the precipitation field across the domain.The spacing of the intermediate grid, denoted as N x × N y grid points, is determined independently for the easting and northing directions.The spacing values, δ x and δ y , represent the grid spacing along the easting and northing directions, respectively.For each row, the average distance among ERA5 grid points on the same row in the original grid is computed once converted to the LAEA reference system.The 99th percentile of these average distances is taken as the grid spacing in the zonal direction of the intermediate grid.The same procedure is applied to columns to determine the grid spacing in the meridional direction.
The second step of the transformation procedure (from the middle panel to the right panel of Figure 2) involves bilinear interpolation from the intermediate grid to the target grid.The target grid is a dyadic grid, chosen for convenience in the discrete wavelet transformation and to obtain a smoother field that better represents the spatial patterns reconstructed by the model, as compared to the nearest neighbour resampling performed in the first step.
It is important to note that without the resampling on the intermediate grid, the precipitation field on the target grid would be reconstructed from grid points that are closer to each other in some regions of the domain compared to others.This would introduce smaller-scale details in certain regions that may not be present elsewhere, depending on the number of grid points involved in describing a precipitation system moving across the domain over time.The first step ensures a more uniform size for the smaller-scale details across the domain and over time, allowing for correct assessment of variations in the shape and amplitude of precipitation events that are independent of the specific locations of events over the domain.

| REPRESENTATION OF PRECIPITATION FIELDS BY MEANS OF 2D WAVELET DECOMPOSITION
The wavelet transform is a space-frequency decomposition method that represents a daily precipitation field as a hierarchy of gridded fields.We apply a 2D Haar discrete wavelet transform on dyadic grids, where the grid size is 2 n by 2 n , and n is a positive integer.The grid spacing in both the northing and easting directions is denoted as δ, which varies across different domains (Table 1).
The wavelet transform decomposes the precipitation field into an approximation field and detail fields at different spatial scales.The approximation field, obtained by averaging the original grid cell values in a coarser grid cell, represents the precipitation field at a coarser  field on the 0-grid represents the average daily precipitation over the entire domain.
The energy of the detail fields on the m-grid, associated with the spatial scale 2 n − m ð Þ Á δ, is computed as the average variance of the three detail fields normalized by the number of original grid points in a cell of the m-grid.This energy quantifies the variations in precipitation intensity at different scales.The energy of the approximation field is the mean value of the square of the daily precipitation at each grid point over the domain.The energy is expressed in units of mm 2 .
To facilitate interpretation and comparison of results, the energies can be presented as percentages of the total energy.The total energy for detail fields is the sum of the energies of the three detail fields at all available spatial scales.For approximations, the total energy includes the energies of the detail fields as well as the energy of the approximation field.
Figures 4 and 5 provide examples of the wavelet decomposition of ERA5 daily precipitation fields.The left panel shows the daily precipitation map, while the right panel displays the energy spectrum.The x-axis represents the spatial scale associated with the detail fields, and the y-axis represents the energy.The thick blue line represents the energy of the detail coefficients, and the blue dot represents the energy associated with the approximation field at the largest scale, which corresponds to the domain extension.
In the winter example depicted in Figure 4, the energy peak of the detail coefficients is observed in the macroscale, specifically at a scale of 2518 km, which is associated with a synoptic cyclone.This scale roughly corresponds to the size of the square region covering a quarter of the domain, characterized by stratiform precipitation in the northwest corner.Conversely, in the summer example shown in Figure 5, the prominent feature is the occurrence of heavy precipitation over Central Europe.The energy peak in this case is found in the mesoscale, at a scale of 315 km, which approximately represents the size of the most intense part of the precipitation events across the entire domain.
It is important to note that the total energy associated with the winter case is higher than that of the summer case.However, during summer, there is more energy associated with smaller scales (e.g., 315 km) compared to the winter case.Additionally, the energies associated with the meso-γ and smaller scales (smaller than 39 km) are relatively small in both cases, as ERA5 does not accurately capture precipitation patterns at those scales.

| Regional statistics of daily precipitation
For each reanalysis and for each region, we have calculated several climate indices.Figure 6 shows three statistics calculated over Africa, taking into account grid points where the daily precipitation was greater or equal to 1 mm. Figure 7 shows the statistics of the daily area fractions of precipitation, representing the percentage of the domain where precipitation exceeds a predefined threshold.In this case, we have chosen three thresholds: 1, 10 and 20 mm.
Referring to Figure 6, the top row presents the 5-year running average of the annual mean 99th percentile of daily precipitation.This means that for each day of a given year, we calculated the 99th percentile (aggregating spatially), considering only grid points with values equal to or greater than 1 mm.Then, these values were averaged over 1 year (aggregating temporally), resulting in one value per year.Each point on the lines depicted in the graph represents the average of five values: the value corresponding to that specific year, the values of the two preceding years, and the values of the two subsequent years.We employ the 5-year running average to filter out short-term variability and focus more on the interannual variability over multiple years.It is worth noting that, due to the use of a 5-year running average operator, the last value for the reanalyses corresponds to 2 years before the nominal end of the datasets (e.g., 2008 for all reanalyses ending in 2010).Similarly, the middle and bottom rows of Figure 6 display the 75th percentile and the mean value of daily precipitation, respectively.The structure of Figure 7 is analogous to Figure 6, as previously described.
In both figures, the left panels (a), (c) and (e) show the original statistics for each reanalysis.The right panels (b), (d) and (f) show the statistics of the relative anomalies compared to the mean value of the 1961-1990 period.To calculate these relative anomalies, we first compute the deviations between the original statistics and their mean value from 1961 to 1990 for each reanalysis separately.Then, these deviations are divided by the mean value of the 1961-1990 period.We chose the 30-year normal period of 1961-1990 since all reanalyses are available for this period.If a climate index does not exhibit a significant temporal trend, the line on the graph will fluctuate around zero.Systematic and gradual deviations from the zero line indicate the presence of a positive or negative trend.
For reanalyses that provide ensembles, such as ERA20-CM, CERA-20C and ERA5-EnDA, the ensemble spread is represented by a dashed region, while the tick line represents the median of the ensemble.It is important to note that ERA20-CM shows the largest spread among its members for all computed climate indices.
Normalizing the statistics with respect to the 30-year average allows for more meaningful comparisons between reanalyses, especially when examining their temporal trends.This approach assumes that a quantitative comparison of their absolute values would primarily highlight systematic differences.For example, let us consider Figure 7 and the comparison between panel (a) and panel (b).The differences between the reanalyses result in more recent reanalyses consistently yielding higher values for the area fraction with a threshold of 20 mm.This information is crucial for intercomparison.When we analyse the relative anomalies in panel (b), we gain additional insights.For instance, the trends of the relative anomalies are more similar among reanalyses compared to the trends of the absolute values.Furthermore, ERA20-C exhibits both the smallest area fraction and the fastest increasing trend over time.Additionally, when considering the relative anomalies, both ERA5 reanalyses show temporal trends that are quite similar to those of the other reanalyses, whereas it is not as evident when examining the absolute values.
The same results presented in Figures 6 and 7 for Africa have been obtained for all domains, and they are documented in Supporting Information.

| Climatological energy spectra as indicators of reanalyses effective resolutions
In section 4, we introduced the concept of the energy spectrum for individual daily precipitation fields.Now, we consider the mean energy spectrum of daily precipitation over the period from 1959 to 2010, which we refer to as the climatological energy spectrum.
Figure 8 presents the energy spectrum for Africa, both in terms of absolute energies and energy percentages (normalized by the total energy of the detail coefficients).The layout of the energy spectra in Figure 8 is the same as that of Figure 4, with the x-axis representing the spatial scale and the y-axis representing the energy.
The absolute energy spectra (thick lines in Figure 8a) provide a ranking of the reanalyses based on the magnitude of their respective detail fields.Reanalyses with higher energy values are capable of capturing more features at that spatial scale.The energy percentages (thick lines in Figure 8b) are related to the effective resolution (Grasso, 2000) of the reanalyses.Reanalyses with a faster decay of energy on smaller scales indicate a coarser effective resolution.
The average precipitation value over the domain (represented by the dots in Figure 8a) is also related to the effective resolution of a reanalysis.However, the energy spectra highlight the differences between the reanalyses more clearly and allow us to determine the range of spatial scales where significant differences emerge.For example, ERA-20CM exhibits a larger average precipitation value over the domain in certain regions such as South Asia (SAS) and a smaller value in Southeast Asia (SEA).However, when considering the energy spectra, ERA-20CM is consistently similar to ERA-20C.
The reanalysis ensemble members show identical energy spectra, suggesting that the common physical model used in the ensemble contributes to the similarity in the spectra.The climatological energy spectrum can be considered a model signature.
Comparing the different reanalyses, ERA-20C often exhibits similar energy spectra of energy percentages to ERA-20CM, but ERA-20CM occasionally has greater absolute energy (e.g., in AFR, CAS, MEA, SAS).The assimilation of observations does not significantly alter the energy spectra of energy percentages, but it may reduce the absolute energy in certain regions.CERA-20C spectra are very similar to those of ERA-20C and ERA-20CM, but CERA-20C tends to have slightly more energy on smaller scales (e.g., in East Asia, EAS).
ERA5 has the highest energy across all scales and allocates larger percentages of energy to smaller spatial To compare the reanalyses, we use the stable bounded efficiency (SBE) (Casati et al., 2023) computed for all reanalyses by using ERA5 as the common reference.The SBE is an overall performance measure that combines the energies of the two reanalyses and their mean squared error.It is symmetric, meaning it is not influenced by the order of the reanalyses being compared.A perfect match results in an SBE of 1, while an SBE of 0 indicates uncorrelated spatial patterns between the two reanalyses.
The expected behaviour of the SBE when comparing two reanalysis precipitation fields decomposed over spatial scales is that it is small for smaller scales and grows as the spatial scale increases.However, due to the nature of reanalyses and their data assimilation cycles, the SBE may be positive even for small scales.When considering the macroscale or the larger part of the mesoscale, the SBE should be closer to 1.
Figure 9 shows the SBE over Africa, obtained by considering all daily precipitation fields from 1959 to 2010.Similar figures for other regions are included in Supporting Information.The thick lines in Figure 9 represent the spatial scales where the SBE grows as the scale increases.Each thick line starts as a dashed line for smaller scales.
In our analysis, we have a rough idea of the minimum effective resolution of a reanalysis, which is given by the grid spacing specified by the reanalysis producers (see section 2).We conservatively assume that energies on spatial scales smaller than the grid spacing are more indicative of the regridding procedure used in the pre-processing of precipitation rather than the actual performance of the reanalysis.Therefore, we show scales smaller than the grid spacing as dashed lines.
The SBE behaviour is expected to deviate from the typical trend at smaller scales due to various factors such as attempting to reconstruct scales that are too small for the reanalysis or systematic spatial mismatches between features at those scales.This discrepancy could arise from differences in geographical information (e.g., topography or land-sea masks) used between ERA5 and other reanalyses, resulting in systematic differences at specific scales and locations (e.g., mountainous regions).The interpretation of performance differences at these scales is complex.Therefore, we use the SBE as an indicator for the scales on which to compare the reanalyses, excluding the problematic scales.It is worth noting that the scales not considered also tend to have low energy values for the reanalyses.

| Do the energy spectra change over time?
Figures 10 and 11 show the comparison of energy spectra over Africa for two different 30-year periods (1961-1990 and 1981-2010) for ERA5 and CERA-20C, respectively.These figures help explain the interpretation of the summary presented in Figure 12 for Africa.Similar summaries for other regions are available in Supporting Information.
F I G U R E 1 2 Summary of the comparison of energy spectra over Africa for the different reanalyses in the period 1959-2010.Each row corresponds to the reanalysis indicated on the y-axis.The ensemble members are shown separately.The colours stand for trends in absolute energy, while the symbols stand for trends in energy percentages.Colour legend: red stands for significant positive trend larger than 2% of the mean energy; pink for smaller significant positive trend; grey for no significant trend; blue for significant negative trend larger than 2% of the mean energy; light blue for smaller significant negative trend.Symbol legend: up-pointing triangle stands for significant positive trend; circle for no trend; down-pointing triangle for significant negative trend.Small black dots stands for time series where we do not consider the trends meaningful.[Colour figure can be viewed at wileyonlinelibrary.com] In this paragraph, we focus on Figures 10 and 11 to explain the interpretation of the summary in Figure 12 for Africa.At each spatial scale, we estimate the linear trend of the energy time series for daily precipitation over the period 1959-2010.We use the least-squares method to obtain the best-fitting linear trends for both absolute and percentage energies.We then perform two-sided Mann-Kendall trend tests to assess the statistical significance of the trends.The rejection level chosen is 1%.
In Figure 12, each symbol represents the linear trend in energy for a specific reanalysis at a specific scale.Only trends from reanalyses that are comparable to ERA5 (based on the SBE analysis) are shown.For spatial scales where the SBE is represented as a dashed line in Figure 9 (or in Supporting Information), small black dots are displayed in Figure 12.For ERA5, small black dots are shown for scales smaller than 31 km, which is the nominal grid spacing of ERA5.Note that this often occurs for scales where the relative energy is less than 5% of the total energy.
The colours in Figure 12  The last column represents the energies of the approximation, while all other columns represent the energies of the detail fields.It is important to note that the temporal trend of the approximation may deviate from the trends observed in the annual mean depicted in Figure 6.This disparity arises because the annual mean in the figure is calculated exclusively from grid points where the daily precipitation equals or exceeds 1 mm.
For example, a sequence of up-pointing triangles for smaller spatial scales followed by a sequence of downpointing triangles for larger scales indicates a shift in the model signature over time, with smaller scales becoming more important in defining the spatial structure of daily precipitation.If all symbols are red, there is a general increase in energy occurring simultaneously with the shift in the energy spectra.This implies that, on average, the total mass of precipitation over the region is increasing, and precipitation patterns are more likely to include smaller features that are more intense than before.
In the first row of Figure 12 (ERA5), we compare it with Figure 10.The prevalence of warm colours in the summary corresponds to the increase in absolute energy shown in Figure 10a.The sequence of up-pointing triangles for smaller scales followed by down-pointing triangles for larger scales suggests the shift towards smaller scales observed in Figure 10b.The last symbol in the first row represents a down-pointing blue triangle, indicating that the time series of the averaged daily precipitation over the African domain (i.e., the approximation) is decreasing over time, and the relative energy of the approximation is becoming smaller compared to the energies of the details.Note that the approximation is not shown in Figure 10.
As a second example, we compare the first row of the CERA-20C block in Figure 12 with Figure 11.In this case, note that compared to ERA5, the calculations of trends become meaningful for larger spatial scales (more small black dots), which aligns with the larger effective resolution of CERA-20C.The prevalence of warm colours in the summary corresponds to the increase in absolute energy shown in Figure 11a.The two spectra of relative energies are very similar for the two 30-year periods, as seen in Figure 11b.Thus, the symbols in the row of Figure 12 are all circles.
We can compare Figure 13 with Figure 10.If we consider the non-overlapping normal periods (1961-1990 and 1991-2020), which are only available for ERA5 products, we observe a more pronounced shift in the energy spectra in Figure 13.This indicates an acceleration in the variations of the energy spectra over time.A similar increase in the rate of variation of certain climate indicators over the period 2010-2020, compared to previous periods, can be seen in Figures 6 and 7.

| CONCLUSIONS
The indications of trends in daily precipitation can vary among different reanalyses, particularly regarding changes over time in the spatial structure of precipitation events.Interpretation of variations in the climatological energy spectrum becomes challenging when comparing spectra among different reanalyses, as each possesses a unique signature.Nonetheless, despite these differences, certain regions exhibit consistent trends across all reanalyses, particularly when considering relative anomalies and accounting for differences in effective resolutions.In such cases, the agreement among multiple reanalyses strengthens the confidence in the represented climate signal.
In summary, our analysis reveals the following main conclusions regarding the temporal trends in daily precipitation: 1. Africa (AFR) and Middle East/North Africa (MENA): All reanalyses, except ERA20-CM, exhibit an upward trend in both mean and extreme precipitation values for grid points where daily precipitation equals or exceeds 1 mm.This is accompanied by a decrease in the overall mean precipitation and its spatial coverage (i.e., area fraction with 1 mm as threshold), although there is an increase in the spatial coverage when using a 20 mm threshold.Moreover, the energy spectra demonstrate an increase in precipitation energy across most spatial scales, with ERA5 reanalyses indicating a shift towards smaller mesoscale scales.2. Europe (EUR), North Europe (NEU), and the Mediterranean (MED): All reanalyses indicate a gradual rise in both mean (more evident when we consider 5-year running averages) and extreme precipitation values, along with an increase in the proportion of precipitation events exceeding 20 mm.The variations appear to be more pronounced in NEU compared to MED.In terms of the energy spectra, there is a general trend of increased energy across various spatial scales, without notable shifts in spatial structure.3. North America (NAM), Central America (CAM) and South America (SAM): most of the reanalyses show an increase in precipitation, with ERA20-CM generally exhibiting a less pronounced trend.The area fraction of precipitation exceeding 20 mm increases in all regions, while the precipitation area fraction decreases in South and Central America.The precipitation energy increases for some spatial scales, particularly in South and Central America.ERA5 reanalyses exhibit a shift towards smaller scales in the energy spectra over South and Central America.4. In both Australia (AUS) and Southeast Asia (SEA), most reanalyses indicate an increasing trend in precipitation values over time.Notably, ERA5 and ERA20-CM show good agreement in this regard.Conversely, CERA-20C does not exhibit any significant trend and instead fluctuates around zero.The rate of increase is particularly pronounced in SEA compared to AUS.Regarding area fractions, the variations are less definitive, although there is a tendency towards an increased area fraction with precipitation exceeding 20 mm in Southeast Asia.The energy spectra consistently demonstrate an upward trend in energy across various spatial scales, with ERA5 reanalyses revealing a shift towards smaller mesoscale scales.5. Central Asia (CAS), East Asia (EAS) and South Asia (SAS): Central Asia shows no clear trends in precipitation or area fractions, and the energy spectra remain relatively stable.In our definition of CAS, we encompass several subregions characterized by diverse precipitation patterns (refer to IPCC AR6 ch.8).The absence of a distinct trend may arise from the intricate interplay of these subregional trends.Consequently, further research is needed to interpret the climate signals within this regions.East and South Asia display an increase in the 99th percentile of precipitation, but trends in mean and 75th percentile are less clear.The precipitation area fractions decrease over time, while the area fraction exceeding 20 mm increases.The energy spectra show an increase in energy for most spatial scales, with ERA5 reanalyses indicating a shift towards smaller scales.
ERA5 reanalyses, being the most recent and higherresolution datasets, often exhibit the most significant trends in climate indicators and variations in the spatial structure of precipitation.This could be attributed to the benefits of higher spatial resolution in resolving atmospheric dynamics across a wider spectrum of spatial scales.However, further studies and high-resolution datasets are required to fully understand the causes and implications of these trends, especially the observed acceleration of variations in ERA5 reanalyses from 2010 onwards.

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fig. 8.7  in IPCC AR6 ch.8 displays linear trends in annual mean precipitationfrom 1901 to 1984 and 1985 to 2014.According to IPCC AR6 ch.8, there is likely an overall increase in annual mean precipitation over midlatitude land areas in the Northern Hemisphere, with medium confidence noted after 1951.Moreover, there is evidence of a more pronounced increase since the 1980s.Concurrently, global warming has intensified droughts in several regions.An increased contrast between wet and dry areas in the Tropics and subtropics have been observed.This indicates increased rainfall in wet regions and slight decreases in dry regions from 1988 to 2019.For a review of wetting and drying trends trends under climate change, readers are directed toZaitchik et al. (2023).A comprehensive review of regional precipitation trends can be found in IPCC AR6 ch.8, section 8.3.1.3.It is worth noting that within each of the 13 selected regions, multiple subregional precipitation trends may exist, making our analysis a summary of a more intricate subregional situation.For an exploration of the causes and mechanisms behind large-scale responses of the water cycle to current climate change, we recommend referring toAllan et al. (2020).They highlight that "Rapid adjustments to forcings, cooling effects from scattering aerosols, and observational uncertainty can explain the current challenges in detecting observed global precipitation responses.However, these responses are expected to become more detectable and accelerate as global warming increases and aerosol forcing diminishes".

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I G U R E 2 Daily precipitation totals for July 14, 2021, in the Central and Northern Europe (NEU) region.The left panel displays the original ERA5 data over NEU, represented in the geographical coordinate system.In the middle panel, the ERA5 data has been regridded using a nearest neighbour resampling technique onto the intermediate transformation grid, utilizing the Lambert Azimuthal Equal-Area (LAEA) Projected Coordinate System (units are km).Lastly, the right panel illustrates the ERA5 data resampled from the intermediate transformation grid to the target dyadic grid, still within the LAEA Projected Coordinate System (km).[Colour figure can be viewed at wileyonlinelibrary.com]resolution.The detail fields capture edges and features such as horizontal, vertical and diagonal patterns in the precipitation field.Each detail coefficient is associated with a 2 by 2 matrix of wavelet basis functions that correspond to translated and dilated versions of the wavelet function.

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I G U R E 3 This diagram illustrates the regridding procedure using the Lambert Azimuthal Equal-Area (LAEA) coordinate reference system.The panels on the top row display three extracts from the upper edge of the region depicted in Figure 2b: panel (a) represents the top left corner, panel (b) is positioned in the middle of the top edge, and panel (c) shows the top right corner.Similarly, the middle row panels (d, e, f) present three extracts from the central band of NEU in Figure 2b, showcasing the left, middle and right sections of the central band.Finally, the bottom row panels (g, h, i) exhibit three extracts from the lower edge of NEU in Figure 2b, representing the left, middle, and right sections of the lower edge.In each panel, the red squares indicate the original ERA5 grid points, which are irregularly spaced in the LAEA reference system.The blue dots represent the grid points of the intermediate transformation grid, which are equally spaced, albeit with different spacings in the easting and northing directions.The blue circles mark the grid points of the target grid, which is dyadic and equally spaced in both directions.Additionally, a grey reference grid is displayed in the panels, consisting of squares with dimensions of 10 km by 10 km.[Colour figure can be viewed at wileyonlinelibrary.com]The transformation can be stopped at any desired spatial scale, denoted as m, where 0≤m<n.The m-grid has a grid size of 2 m by 2 m and a grid spacing of δ Á 2 n − m ð Þ .The approximation field on the m-grid is obtained by downscaling the approximation field from the m −1 ð Þ-grid using nearest neighbour interpolation and summing it with the three detail fields.In our study, we have used n= 10, resulting in dyadic grids with 1024 by 1024 grid points.The approximation F I G U R E 4 ERA5 daily precipitation over Europe (EUR) on February 15, 2020.The left panel displays the precipitation map (mm), while the right panel shows the energy spectrum.The transition between white and beige background colours marks the progression from one atmospheric scale to the next, namely from the smallest to the largest scale (see section 3): meso-γ, meso-β, meso-α, and macroscales.[Colour figure can be viewed at wileyonlinelibrary.com]F I G U R E 5 ERA5 daily precipitation over Europe (EUR) on July 14, 2021.The layout of the figure is similar to Figure 4, displaying the daily precipitation data for the specified date.[Colour figure can be viewed at wileyonlinelibrary.com]F I G U R E 6 Legend on next page.

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I G U R E 6 Time series of statistics depicting the daily precipitation amount over Africa (AFR).The time series displays 5-year running averages of the daily statistics, calculated solely from grid points where the daily precipitation exceeded or equalled 1 mm.The left column shows the original statistics spanning from 1901 to 2022, while the right column exhibits the relative anomalies compared to the mean values of the period 1961-1990 for the most recent timeframe of 1959-to 2022.Panels (a) and (b) portray the annual mean of the daily 99th percentile.Panels (c) and (d) depict the annual mean of the daily 75th percentile.Lastly, panels (e) and (f) illustrate the annual mean amount.[Colour figure can be viewed at wileyonlinelibrary.com]F I G U R E 7 Legend on next page.
Time series of statistics representing the daily area fractions of precipitation over Africa (AFR).The time series exhibits 5-year running averages of the daily statistics.The left column displays the original statistics spanning from 1901 to 2022, while the right column shows the relative anomalies compared to the mean values of the period 1961-1990 for the most recent timeframe from 1959 to 2022.Panels (a) and (b) depict the fraction of the domain where precipitation equalled or exceeded 20 mm.Panels (c) and (d) portray the fraction of the domain where precipitation equalled or exceeded 10 mm.Lastly, panels (e) and (f) illustrate the fraction of the domain where precipitation equalled or exceeded 1 mm.[Colour figure can be viewed at wileyonlinelibrary.com] scales.ERA5-EnDA closely follows the characteristics of ERA5, although its coarser resolution results in lower energy values on smaller scales.

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I G U R E 8 Energy spectrum depicting the energy as a function of spatial scale for various reanalyses over Africa, covering the time period from 1959 to 2010.In panel (a), the energies of the detail coefficients are represented by lines, while the energies of the approximation at the larger scale are denoted by dots.Panel (b) illustrates the energy percentages of the detail coefficients, indicated by lines.The shaded colours in the background serve to demarcate the atmospheric scales.[Colour figure can be viewed at wileyonlinelibrary.com]F I G U R E 9 Stable bounded efficiency (SBE) for the different reanalyses over Africa based on the time period 1959-2010.The thick lines show the SBE where it is deemed as significant.The dashed lines show the SBE at the other spatial scales.The shaded colours in the background mark the atmospheric scales.[Colour figure can be viewed at wileyonlinelibrary.com]

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I G U R E 1 0 Comparison of ERA5 energy spectra over Africa in the two 30-year periods 1961-1990 (blue) and 1981-2010 (red).The lines show the energy spectra (i.e., mean energy), the areas with shading lines is delimited by the 1st and the 99th percentiles of the energy distributions.Panel (a) shows the absolute energies.Panel (b) shows the energy percentages.The shaded colours in the background mark the atmospheric scales.[Colour figure can be viewed at wileyonlinelibrary.com]F I G U R E 1 1 Comparison of CERA-20C energy spectra over Africa.The layout is similar to Figure 10.[Colour figure can be viewed at wileyonlinelibrary.com] represent trends in absolute energy.Darker colours indicate more significant trends.Warm colours indicate positive trends, while cold colours indicate negative trends.Grey represents a flat time series.The symbols represent trends in relative energy.Up-pointing triangles indicate a significant positive trend, indicating an increasing importance of energy at that scale over time.Down-pointing triangles indicate a significant negative trend, indicating a decreasing importance of energy at that scale.Circles indicate no significant trend.

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I G U R E 1 3 Comparison of ERA5 energy spectra over Africa in the two 30-year periods 1961-1990 (blue) and 1991-2020 (red).The layout is similar to Figure 10.[Colour figure can be viewed at wileyonlinelibrary.com]