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This section describes how we used freely-available data to compile Classic driver input files for the VIC model. First, we created parameter files for VIC-5 Classic, then we converted them to NetCDF format for VIC-5 Image. VIC-5 Classic requires three parameter files: a soil parameter file, a vegetation parameter file, and a vegetation library file. An optional elevation band file can be provided to resolve sub-grid variability in elevation, which is important in regions with complex topography. The parameters are arranged as a relational database: each grid cell has a unique identifier, called a grid cell number, in the soil parameter file, that VIC uses to find the corresponding rows of data in the vegetation parameter and elevation band files. The Image driver uses a different setup, with all parameters stored in a single NetCDF file.Soil parametersThe soil parameter file for VIC-5 Classic is an ASCII text file that includes soil parameters such as hydraulic conductivity and porosity, but also other kinds of static parameters, such as average precipitation and time zone offset from GMT. Each row of the soil parameter file represents one grid cell, and each column represents a different variable. We compiled the soil parameter file using MERIT20 elevation data, soil texture data from the FAO HWSD, pedotransfer tables relating soil texture to other soil properties, and interpolated weather station data (WorldClim21). Any remaining parameters were set to suggested values from the VIC model’s documentation2. The following sections describe the estimation of each variable in the soil parameter file, summarized in Table 1.Table 1 VIC model parameters for the soil parameter file.Full size tableElevation and land maskThe VICGlobal soil parameter file uses the Multi-Error-Removed Improved-Terrain (MERIT20) digital elevation model (DEM) to define the elevations, latitudes, and longitudes of each land grid cell. The MERIT DEM is an error-corrected and extended version of the SRTM DEM, with 3 arc-second resolution and coverage from 60°S to 85°N and 180°W to 180°E. Specifically, MERIT is a combination of the SRTM, AW3D, and Viewfinder Panoramas’ DEMs, corrected for striping, speckle, absolute bias, and tree height bias. We used bilinear interpolation to aggregate MERIT to 1/16° resolution and derive a 1/16° MERIT-based land mask and DEM (Figure S1).Soil texture dataSoil texture (percent sand, silt, and clay) and bulk density were obtained from the FAO HWSD, a gridded soil parameter dataset derived from in-situ measurements of the soil column. We used a 0.05° resolution NetCDF dataset converted from the original HWSD Microsoft Access database by Wieder et al.22. We resampled the HWSD soil data from 0.05° to 1/16° resolution using bilinear interpolation with the MATLAB® function griddedInterpolant. While HWSD has near global coverage, there are missing data in some places around the world, notably Greenland and northern Africa. We filled in these missing data using inpainting, a gap-filling method from the field of image processing. We used the MATLAB® function inpaintnans23, which uses a partial differential equation method to fill in missing data, to fill gaps in the HWSD data over the MERIT land mask. Figure 1 shows the HWSD bulk density data before and after inpainting.Fig. 1Bulk density data from the Harmonized World Soils Database (HWSD). The top panel shows HWSD bulk density data resampled to 1/16° resolution, the middle panel shows bulk density after infilling holes, and the bottom panel shows the difference.Full size imageThe HWSD data are divided into “topsoil” and “subsoil” parameters. The first 30 cm of the soil column are considered topsoil and the lower 70 cm subsoil. VIC is typically run with three soil layers, so we created a three-layer soil parameter file by breaking up the 30 cm HWSD topsoil layer into two soil layers: one of 10 cm and one of 20 cm, so the final soil parameter file has three layers, with thicknesses of 10 cm, 20 cm and 70 cm, from top to bottom of the soil column. Ten centimeters has been a common choice for the uppermost layer soil depth in VIC modeling applications since its use by Liang et al.24. Soil layer depths are typically used as calibration parameters. VICGlobal values should be considered a starting estimate.Calculating soil parameter values based on soil texturesPedotransfer functions (e.g. Cosby et al.25) relate readily available soil properties, such as soil texture, to less easily-observed properties, such as hydraulic conductivity. After resampling the HWSD data from 1/4° to 1/16° resolution, we estimated soil parameters by classifying each grid cell’s USDA soil texture class and assigning physical soil properties based on a lookup table included with the VIC documentation2,26. The lookup table (Table 2) relates the 12 USDA soil texture classes to bulk density, field capacity, wilting point, porosity, saturated hydraulic conductivity, and slope of the soil water retention curve in Campbell’s equation. We classified soil textures using the USDA soil texture triangle, as implemented by the MATLAB® function soil_classification27. Figure 2 shows the derived USDA soil texture map. We used these along with the lookup table to estimate saturated hydraulic conductivity (Ksat), the exponent in Campbell’s equation for hydraulic conductivity (expt), fractional soil moisture at the critical point (wcrfract), where the critical point is about 70% of field capacity, fractional soil moisture at the wilting point (wpwpfract), quartz content, and porosity for each soil layer. The lookup table26 did not include quartz content, so we supplemented it with the soil texture-quartz content lookup table from Peters-Lidard et al.28.Table 2 USDA soil texture class lookup table.Full size tableFig. 2USDA soil texture classifications based on HWSD. Topsoil is soil from 0–30 cm below the surface, and subsoil is soil between 30–100 cm deep.Full size imageWe set the variable infiltration capacity parameter ({b}_{infilt}=0.2), the maximum baseflow fraction threshold ({d}_{s}=0.001), and maximum soil moisture threshold ({w}_{s}=0.9), their suggested values from the VIC documentation. These parameters, along with maximum baseflow velocity (dsmax) and soil depth, are typically calibrated. We set the baseflow curve exponent c = 2, the soil thermal damping depth dp = 4 m, soil density = 2685 kg/m3, surface roughness = 0.001 m, and snow roughness = 0.0005 m, also based on guidance from the VIC documentation. The soil moisture diffusion parameter phis is not used in the current version of VIC, so we set it to the no-data value (−999). The final few soil parameters — dsmax, initial soil moisture (initm), and bubbling pressure (bubble)— were calculated using the following equations, based on guidance from the VIC documentation.$$dsmax=slopeast {bar{K}}_{sat}$$
(1)
$$initm=wc{r}_{fract}ast porosityast {t}_{l}$$
(2)
$$bubble=0.32ast expt+4.3$$
(3)
Equation (1) estimates dsmax for each grid cell as the product of soil-column average Ksat and land surface slope, which was calculated from the elevation data using the MATLAB® function gradientm29g. Equation (2), where tl is the thickness of soil layer l, assumes that initial soil moisture is equal to the fractional soil moisture content at the critical point. Equation (3) calculates bubbling pressure as a function of expt, based on linear regression of bubbling pressure vs. expt30. Figures S2–S9 in the Supplementary Information show maps of each soil parameter. We assumed residual soil moisture, the amount of soil moisture that cannot be removed from the soil by drainage or evapotranspiration, was zero.Elevation bandsVIC uses an elevation band file (also called a snow band file) to account for subgrid heterogeneity in grid cell elevations. The assumption of uniform elevation over an entire grid cell can lead to modeling errors in mountainous regions, where higher topography is associated with cooler temperatures and higher precipitation rates. The elevation band file accounts for subgrid variability in topography by dividing each grid cell into a number of elevation bands, each of which is simulated separately. VIC adjusts temperature, pressure, and precipitation depending on the elevation in each band. We prepared an elevation band file with five elevation bands by comparing the 1/16° DEM used for the soil parameter file with a 30 arc-second DEM. Both DEMs were derived by aggregating MERIT data. For simplicity, we assumed precipitation was evenly distributed among elevation bands within a grid cell. The elevation band file is provided with the caveat that using elevation bands requires more computing power; users may wish to turn elevation bands on or off (via the VIC global parameter file) depending on their needs.Vegetation parametersVIC-5 Classic uses a vegetation parameter file to define the fractional cover of different vegetation types within each grid cell and some of their physical properties. Other vegetation parameters are stored in the “vegetation library” file. (VIC-5 Image simply stores all parameters in a single “parameter” file.) The VIC-5 Classic vegetation parameter file consists of information about fractional cover of each land cover type in each grid cell, and their corresponding root zone depths and root fractions within each root zone. The vegetation parameter file can optionally include time-varying LAI, fractional canopy cover, and albedo data, but it is simpler to specify these in the vegetation library (at the cost of not representing some spatial heterogeneity).We used MODIS land cover data from the 0.05° MODIS MCD12C1 Collection 6 data product31 to assign fractional land cover values to each grid cell by calculating the average land cover for MCD12C1 observations over the 2017 calendar year. We chose 2017 because it was the most recent year with data in all the MODIS-based datasets used for this study, and there is very low interannual variability of land cover32 in MCD12C1 Collection 6. Figure 3 shows majority land cover types from the 2017 MCD12C1 observations.Fig. 3MODIS MCD12C1 majority land cover types (IGBP classifications). .Full size imageLike all global land cover data products, MCD12C1 makes classification errors. Sulla-Menashe et al.32 reported 67% overall IGBP classification accuracy for 2001 land cover. Classification errors are more common in the “mixed” land covers, such as cropland/natural vegetation mosaic, shrublands, grasslands, and savannas. Fortunately for our purposes, the vegetation parameters for commonly-confused land covers tend to be fairly similar themselves, which reduces the impact of misclassification on land surface modelling results. For example, the LAI of open shrubland is not too different from the LAI of closed shrubland.We calculated root fraction as a function of land cover class following the method of Zeng33, who defined the following formula (Eq. 4) for use in parameterizing land surface models:$$Y=1-frac{1}{2}left({e}^{-ad}+{e}^{-bd}right)$$
(4)
where Y = cumulative root fraction, d = depth, and a and b are empirical parameters defined by Zeng33 for each International Geosphere–Biosphere Programme (IGBP) land cover type, based on a rooting depth database compiled from more than 200 field surveys. We used this formula with depths of 0.1 m, 0.7 m, and dr, corresponding to three root zones. The value of dr, the maximum rooting depth for each IGBP land cover type, was taken from Zeng33. This method assumes that the depth and distribution of roots depends only on the land cover type; we assume that land cover type is the primary control on root characteristics. Table 3 shows root fractions and root zone depths for each IGBP land cover type.Table 3 Root zone depths (m) and fraction of roots in each zone for IGBP land cover classes.Full size tableLike previous large-scale VIC vegetation cover datasets, our vegetation parameter file neglects land cover change over time. However, it does have a few other advantages over past vegetation parameter datasets. The land cover classification used in the N2001 and L2013 VIC parameter sets is referred to as “UMD-NLDAS” because it is a modified version of the AVHRR-based University of Maryland (UMD) land cover product34. The UMD-NLDAS classification was modified for the North American Land Data Assimilation project (NLDAS35) to exclude open water, urban, and snow and ice land cover classes (see BV2019). VICGlobal uses 17 IGBP land cover classes, including urban, barren, perennial snow and ice, and inland water bodies, permitting better description of land cover variability than the 11 UMD-NLDAS classification.Vegetation library fileThe vegetation library maps each land cover type to a set of vegetation parameters (Table 4). We adapted the LDAS vegetation library36 for use with the 17 IGBP land cover classes, taking monthly average LAI, fractional canopy cover (fcanopy), and albedo values obtained from recent MODIS data products. We set architectural resistance (r0) and minimum stomatal resistance (rmin) to values from literature (described below). The rest of the parameters, which are described in the N2001 paper, were left to their original LDAS vegetation library values. This section describes how we estimated LAI, fcanopy, albedo, r0, and rmin, and how we transferred the remaining parameters from the 11 UMD-NLDAS land cover classes to the 17 IGBP land cover classes.Table 4 VIC model parameters for the vegetation library file.Full size tableWe used MODIS observations from the year 2017 to calculate monthly average LAI, fcanopy, and albedo for each IGBP land cover type. We calculated LAI and albedo from the MODIS-based Global LAnd Surface Satellite dataset (GLASS37,38,39) and fcanopy from NDVI observations (MCD13C140) The expression used for fcanopy follows BV2019:$$fcanopy={left(frac{NDVI-NDV{I}_{min}}{NDV{I}_{max}-NDV{I}_{min}}right)}^{2}$$
(5)
where NDVImin and NDVImax are the minimum and maximum values of NDVI observed for that month. Monthly LAI, fcanopy, and albedo values were calculated by averaging over all grid cells of the same land cover type, counting only cells that were at least 90% homogenous, to avoid noise from grid cells with multiple land covers. Excepting perennial snow and ice land cover, the vegetation parameters in the VIC vegetation library should describe snow-free vegetation. Therefore, before calculating LAI, fcanopy, and albedo for each land cover class, we used fractional snow cover data from MOD10CM41, a global 0.05 degree monthly snow cover dataset, to exclude grid cells with more than 90% snow cover. Additionally, we set albedo to 0.05 for open water, and we set LAI and fcanopy to 0 for open water and perennial snow and ice.The resistances rmin and r0 play a role in determining how much plant transpiration occurs. Higher resistance means less transpiration. Stomatal resistance is resistance to the release of water through the plant stomata, and architectural resistance is the aerodynamic resistance between the leaves and the canopy top42. Two sets of resistance parameters have been used in past large-scale VIC implementations. N2001 ran VIC over the entire globe using rmin values adapted from Dorman and Sellers’ global database of rmin values43 computed using the Simple Biosphere Model44 (SiB). The Nijssen et al.45 r0 values were taken from Ducoudre et al.’s SECHIBA land surface parameterization42. The other set of rmin and r0 parameters are those used in the LDAS vegetation library and in studies such as Livneh et al.12. This set of rmin values comes from Mao et al.46 and Mao and Cherkauer47. We used the rmin values from SiB44 and the r0 values from SECHIBA42 for VICGlobal as they appeared to be the better documented values.For the other parameters in the vegetation library file (displacement height, roughness length, etc.), we assigned values using the existing LDAS vegetation library. Since there are 17 IGBP land cover classes, and only 11 UMD-NLDAS land cover classes in the LDAS vegetation library, we re-assigned some IGBP land cover classes to take the parameters of UMD-NLDAS land cover classes. We remapped barren land, permanent wetlands, snow and ice, urban land, and water bodies to take the parameters of “grasslands” from the LDAS vegetation parameter file. While the characteristics of the barren, snow and ice, urban, and water land cover types clearly differ from those of grasslands, their low LAI and fcanopy values, corresponding to sparse vegetation, essentially “turns off” the other vegetation parameters in the VIC model, as pointed out by BV2019. The other remappings were more straightforward. Croplands and croplands/natural vegetation mosaics inherited values from “croplands,” savannas became “wooded grasslands,” and woody savannas became “woodlands.” We were thus able to assign vegetation parameter values to the each of the 17 IGBP land cover classes.To calculate global average time series of seasonally-varying vegetation parameters would be of limited interest as the seasonal cycle would average out across the equator. Therefore, we calculated average monthly fcanopy, LAI, and albedo for each vegetation type in each hemisphere, and we developed two separate vegetation library files: one for the northern hemisphere and one for the southern hemisphere. Maps of January and July LAI, fcanopy, and albedo are shown in Fig. 4. For illustrative purposes, the parameter values in this figure have been averaged over the 17 IGBP land cover classes using area-based weighting. Figures S14–S19 show maps of the remaining vegetation parameters. Figures S20–S22 show the cycle of LAI, fractional canopy cover, and albedo for each vegetation type, averaged separately over each hemisphere.Fig. 4Maps of leaf-area index, albedo, and fractional canopy cover values. Parameter values have been averaged over the 17 IGBP land cover classes using area-based weighting.Full size image More

Description of scenarios used in this studyTo assess future urban water scarcity, we used the scenario framework from the Scenario Model Intercomparison Project (ScenarioMIP), part of the International Coupled Model Intercomparison Project Phase 6 (CMIP6)38. The scenarios have been developed to better link the Shared Socioeconomic Pathways (SSPs) and Representative Concentration Pathways (RCPs) to support comprehensive research in different fields to better understand global climatic and socioeconomic interactions38,39. We selected the four ScenarioMIP Tier 1 scenarios (i.e., SSP1&RCP2.6, SSP2&RCP4.5, SSP3&RCP7.0, and SSP5&RCP8.5) to evaluate future urban water scarcity. SSP1&RCP2.6 represents the sustainable development pathway of low radiative forcing level, low climate change mitigation challenges, and low social vulnerability. SSP2&RCP4.5 represents the business-as-usual pathway of moderate radiative forcing and social vulnerability. SSP3&RCP7.0 represents a higher level of radiative forcing and high social vulnerability. SSP5&RCP8.5 represents a rapid development pathway and very high radiative forcing38.Estimation of urban water scarcityTo estimate urban water scarcity, we quantified the total urban population living in water-scarce areas2,3,7,19. Specifically, we first corrected the spatial distribution of the global urban population, then identified water-scarce areas around the world, and finally quantified the urban population in water-scarce areas at different scales (Supplementary Fig. 1).Correcting the spatial distribution of global urban populationThe existing global urban population data from the History Database of the Global Environment (HYDE) provided consistent information on historical and future population, but it has a coarse spatial resolution of 10 km (Supplementary Table 1)40,41. In addition, it was estimated using total population, urbanization levels, and urban population density, and does not align well with the actual distribution of urban land42. Hence, we allocated the HYDE global urban population data to high-resolution urban land data. We first obtained global urban land in 2016 from He et al.42. Since the scenarios used in existing urban land forecasts are now dated43,44, we simulated the spatial distribution of global urban land in 2050 under each SSP at a grid-cell resolution of 1km2 using the zoned Land Use Scenario Dynamics-urban (LUSD-urban) model45,46,47 (Supplementary Methods 1). The simulated urban expansion area in this study was significantly correlated with that in existing datasets (Supplementary Table 6). We then converted the global urban land raster layers for 2016 and 2050 into vector format to characterize the spatial extent of each city. The total population within each city was then summed and the remaining HYDE urban population cells located outside urban areas were allocated to the nearest city. Assuming that the population density within an urban area was homogeneous, we calculated the total population per square kilometer for all urban areas and converted this back to raster format at a spatial resolution of 1 km2. The new urban population data had much lower error than the original HYDE data (Supplementary Table 7).Identification of global water-scarce areasAnnual and monthly WSI values were calculated at the catchment level in 2014 and 2050 as the ratio of water withdrawals (TWW) to availability (AWR)33. Due to limited data availability, we combined water-scarce areas in 2014 and the urban population in 2016 to estimate current urban water scarcity. WSI for catchment i for time t as:$${{{{{mathrm{WS{I}}}}}}}_{t,i}=frac{{{{{mathrm{TW{W}}}}}}_{t,i}}{{{{{mathrm{AW{R}}}}}}_{t,i}}$$
(1)
For each catchment defined by Masutomi et al.48, the total water withdrawal (TWWt,i) equalled the sum of water withdrawals (WWt,n,i) for each sector n (irrigation, livestock, industrial, or domestic), while the water availability equalled the sum of available water resources for catchment i (Rt,i), inflows/outflows of water resources due to interbasin water transfer ((varDelta {{{{mathrm{W{R}}}}}}_{t,i})), and water resources from each upstream catchment j (WRt,i,j):$${{{{{mathrm{TW{W}}}}}}}_{t,i}={{sum }_{n}{{{{mathrm{WW}}}}}}_{t,n,i}$$
(2)
$${{{{{mathrm{AW{R}}}}}}}_{t,i}={R}_{t,i}+varDelta {{{{mathrm{W{R}}}}}}_{t,i}+mathop{sum}limits_{j}{{{{mathrm{W{R}}}}}}_{t,i,j}$$
(3)
The changes of water resources due to interbasin water transfer were calculated based on City Water Map produced by McDonald et al.3. The number of water resources from upstream catchment j was calculated based on its water availability (AWRt,i,j) and water consumption for each sector n (WCt,n,i,j)49:$${{{{{mathrm{W{R}}}}}}}_{t,i,j}=,max (0,{{{{mathrm{AW{R}}}}}}_{t,i,j}-{{sum }_{n}{{{{mathrm{WC}}}}}}_{t,n,i,j})$$
(4)
For areas without upstream catchments, the number of available water resources was equal to the runoff. Following Mekonnen and Hoekstra36, and Hofste et al.33, we did not consider environmental flow requirements in calculating water availability.Annual and monthly WSI for 2014 were calculated directly based on water withdrawal, water consumption, and runoff data from AQUEDUCT3.0 (Supplementary Table 1). The data from AQUEDUCT3.0 were selected because they are publicly available and the PCRaster Global Water Balance (PCRGLOBWB 2) model used in the AQUADUCT 3.0 can better represent groundwater flow and available water resources in comparison with other global hydrologic models (e.g., the Water Global Assessment and Prognosis (WaterGAP) model)33. The annual and monthly WSI for 2050 were calculated by combining the global water withdrawal data from 2000 to 2050 provided by the National Institute of Environmental Research of Japan (NIER)34 and global runoff data from 2005 to 2050 from CMIP6 (Supplementary Table 1). Water withdrawal ({{{{{mathrm{W{W}}}}}}}_{s,m,n,i}^{2050}) in 2050 for each sector n (irrigation, industrial, or domestic), catchment i, and month m under scenario s was calculated based on water withdrawal in 2014 (({{{{{mathrm{W{W}}}}}}}_{m,n,i}^{2014})):$${{{{{mathrm{W{W}}}}}}}_{s,m,n,i}^{2050}={{{{mathrm{W{W}}}}}}_{m,n,i}^{2014}cdot [1+{{{{mathrm{WW{R}}}}}}_{s,m,n,i}cdot (2050-2014)]$$
(5)
adjusted by the mean annual change in water withdrawal from 2000 to 2050 (WWRs, m, n, i), calculated using the global water withdrawal for 2000 (({{{{{mathrm{W{W}}}}}}}_{{{{{mathrm{NIER}}}}},m,n,i}^{2000})) and 2050 (({{{{{mathrm{W{W}}}}}}}_{{{{{mathrm{NIER}}}}},s,m,n,i}^{2050})) provided by the NIER34:$${{{{{mathrm{WW{R}}}}}}}_{s,m,n,i}=frac{({{{{mathrm{W{W}}}}}}_{{{{{mathrm{NIER}}}}},s,m,n,i}^{2050}/{{{{mathrm{W{W}}}}}}_{{{{{mathrm{NIER}}}}},m,n,i}^{2000})-1}{2050-2000}$$
(6)
Based on the assumption of a constant ratio of water consumption to water withdrawal in each catchment, water consumption in 2050 (({{{{{mathrm{W{C}}}}}}}_{s,m,n,i}^{2050})) was calculated as:$${{{{{mathrm{W{C}}}}}}}_{s,m,n,i}^{2050}={{{{mathrm{W{W}}}}}}_{s,m,n,i}^{2050}cdot frac{{{{{mathrm{W{C}}}}}}_{m,n,i}^{2014}}{{{{{mathrm{W{W}}}}}}_{m,n,i}^{2014}}$$
(7)
where ({{{{{mathrm{W{C}}}}}}}_{m,n,i}^{2014}) denotes water consumption in 2014. Due to a lack of data, we specified that water withdrawal for livestock remained constant between 2014 and 2050, and used water withdrawal simulation under SSP3&RCP6.0 provided by the National Institute of Environmental Research in Japan to approximate SSP3&RCP7.0.To estimate water availability, we calculated available water resources (({R}_{s,m,i}^{2041-2050})) for each catchment i and month m under scenario s for the period of 2041–2050 as:$${R}_{s,m,i}^{2041-2050}={R}_{m,i}^{{{{{mathrm{ols}}}}},2005-2014}cdot frac{{bar{R}}_{s,m,i}^{2041-2050}}{{bar{R}}_{m,i}^{2005-2014}}$$
(8)
based on the amount of available water resources with 10-year ordinary least square regression from 2005 to 2014 (({R}_{m,i}^{{{{{mathrm{ols}}}}},,2005-2014})) from AQUEDUCT3.0 (Supplementary Table 1). ({overline{R}}_{m,i}^{2005-2014}) and ({overline{R}}_{s,m,i}^{2041-2050}) denote the multi-year average of runoff (i.e., surface and subsurface) from 2005 to 2014, and from 2041 to 2050, respectively, calculated using the average values of simulation results from 10 global climate models (GCMs) (Supplementary Table 2).We then identified water-scarce catchments based on the WSI. Two thresholds of 0.4 and 1.0 have been used to identify water-scarce areas from WSI (Supplementary Table 4). While the 0.4 threshold indicates high water stress49, the threshold of 1.0 has a clearer physical meaning, i.e., that water demand is equal to the available water supply and environmental flow requirements are not met36,37. We adopted the value of 1.0 as a threshold representing extreme water stress to identify water-scarce areas. The catchments with annual WSI >1.0 were identified as perennial water-scarce catchments; the catchments with annual WSI equal to or 1.0 were identified as seasonal water-scarce catchments.Estimation of global urban water scarcityBased on the corrected global urban population data and the identified water-scarce areas, we evaluated urban water scarcity at the global and national scales via a spatial overlay analysis. The urban population exposed to water scarcity in a region (e.g., the whole world or a single country) is equal to the sum of the urban population in perennial water-scarce areas and that in seasonal water-scarce areas. Limited by data availability, we used water-scarce areas in 2014 and the urban population in 2016 to estimate current urban water scarcity. Projected water-scarce areas and urban population in 2050 under four scenarios were then used to estimate future urban water scarcity. In addition, we obtained the location information of large cities (with population >1 million in 2016) from the United Nations’ World Urbanization Prospects1 (Supplementary Table 1) and identified those in perennial and seasonal water-scarce areas.Uncertainty analysisTo evaluate the uncertainty across the 10 GCMs used in this study (Supplementary Table 2), we identified water-scarce areas and estimated urban water scarcity using the simulated runoff from each GCM under four scenarios. To perform the uncertainty analysis, the runoff in 2050 for each GCM was calculated using the following equation:$${R}_{s,g,m,i}^{2050}={R}_{m,i}^{2014}cdot frac{{R}_{s,g,m,i}^{2041-2050}}{{R}_{g,m,i}^{2005-2014}}$$
(9)
where ({R}_{s,g,m,i}^{2050}) denotes the runoff of catchment i in month m in 2050 for GCM g under scenario s. ({R}_{g,m,i}^{2005-2014}) and ({R}_{s,g,m,i}^{2041-2050}) denote the multi-year average runoff from 2005 to 2014, and from 2041 to 2050, respectively, calculated using the simulation results from GCM g. Using the runoff for each GCM, the WSI in 2050 for each catchment was recalculated, water-scarce areas were identified, and the urban population exposed to water scarcity was estimated.Contribution analysisBased on the approach used by McDonald et al.2 and Munia et al.50, we quantified the contribution of socioeconomic factors (i.e., water demand and urban population) and climatic factors (i.e., water availability) to the changes in global urban water scarcity from 2016 to 2050. To assess the contribution of socioeconomic factors (({{{{{mathrm{Co{n}}}}}}}_{s,{{{{mathrm{SE}}}}}})), we calculated global urban water scarcity in 2050 while varying demand and population and holding catchment runoff constant (({{{{{mathrm{UW{S}}}}}}}_{s,{{{{mathrm{SE}}}}}}^{2050})). Conversely, to assess the contribution of climate change ((Co{n}_{s,CC})), we calculated scarcity while varying runoff and holding urban population and water demand constant (({{{{{mathrm{UW{S}}}}}}}_{s,{{{{mathrm{CC}}}}}}^{2050})). Socioeconomic and climatic contributions were then calculated as:$${{{{{mathrm{Co{n}}}}}}}_{s,SE}=frac{{{{{mathrm{UW{S}}}}}}_{s,{{{{mathrm{SE}}}}}}^{2050}-{{{{mathrm{UW{S}}}}}}^{2016}}{{{{{mathrm{UW{S}}}}}}_{s}^{2050}-{{{{mathrm{UW{S}}}}}}^{2016}}times 100 %$$
(10)
$${{{{{mathrm{Co{n}}}}}}}_{s,CC}=frac{{{{{mathrm{UW{S}}}}}}_{s,{{{{mathrm{CC}}}}}}^{2050}-{{{{mathrm{UW{S}}}}}}^{2016}}{{{{{mathrm{UW{S}}}}}}_{s}^{2050}-{{{{mathrm{UW{S}}}}}}^{2016}}times 100 %$$
(11)
Feasibility analysis of potential solutions to urban water scarcityPotential solutions to urban water scarcity involve two aspects: increasing water availability and reducing water demand2. Approaches to increasing water availability include groundwater exploitation, seawater desalination, reservoir construction, and inter-basin water transfer; while approaches to reduce water demand include water-use efficiency measures (e.g., new cultivars for improving agricultural water productivity, sprinkler or drip irrigation for improving water-use efficiency, water-recycling facilities for improving domestic and industrial water-use intensity), limiting population growth, and virtual water trade2,3,18,32. To find the best ways to address urban water scarcity, we assessed the feasibility of these potential solutions for each large city (Supplementary Fig. 2).First, we divided these solutions into seven groups according to scenario settings and the scale of implementation of each solution (Supplementary Fig. 2). Among the solutions assessed, water-use efficiency improvement, limiting population growth, and climate change mitigation were included in the simulation of water demand and water availability under the ScenarioMIP SSPs&RCPs simulations34. Here, we considered the measures within SSP1&RCP2.6 which included the lowest growth in population, irrigated area, crop intensity, and greenhouse gas emissions; and the largest improvements in irrigation, industrial, and municipal water-use efficiency34.We then evaluated the feasibility of the seven groups of solutions according to the characteristics of water-scarce cities (Supplementary Fig. 2). Of the 526 large cities (with population >1 million in 2016 according to the United Nations’ World Urbanization Prospects), we identified those facing perennial or seasonal water scarcity under at least one scenario by 2050. We then selected the cities that no longer faced water scarcity under SSP1&RCP2.6 where the internal scenario assumptions around water-use efficiency, population growth, and climate change were sufficient to mitigate water scarcity. Following McDonald et al.2,3 and Wada et al.18, we assumed that desalination can be a potential solution for coastal cities (distance from coastline More

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Overview of the SWE Retrieval methodThe SWE processing system relies on Bayesian assimilation which combines ground-based data with satellite-borne observations2. The method applies two vertically polarized satellite-based brightness temperature observations at 19 and 37 GHz and a scene brightness temperature model (the HUT snow emission model4). First, snow microstructure described by an ‘effective snow grain size’ is estimated for grid cells with a coincident weather station SD observation. Effective snow grain size is used in the HUT model as a scalable model input parameter to optimize agreement with the satellite measurements. These values of grain size are used to interpolate a background map of the effective grain size, including an estimate of the effective grain size error. This spatially continuous map of grain size is then used as an input for a second HUT model inversion to provide an estimate of SWE. In the inversion process, the effective grain size in each grid cell is weighed with its respective error estimate and a constant value of snow density is applied. The spatially continuous SWE map obtained from the second run of the HUT snow model described above is fused with a background SD field (converted to SWE using 0.24 g cm−3) to obtain a final estimate of SWE using a Bayesian non-linear iterative assimilation approach (which weights the information sources with their estimated variances). The background SD field is generated from the same weather station SD observations used to estimate the effective snow grain size using kriging interpolation methods.The microwave scattering response to SWE saturates under deep snow conditions ( >150 mm) and model inversion of SD/SWE over areas of wet snow is not feasible because the microwave signal is absorbed rather than scattered. For these reasons, the method decreases the weight of satellite data for deep dry snowpacks and wet snow by assessing the modeled sensitivity of brightness temperature to SWE within the data assimilation procedure2,3.Before SWE retrieval, dry snow is identified from brightness temperature data7. For the autumn snow accumulation season (August to December), the dry snow detection is used to construct a cumulative snow presence mask to track the advance of snow extent (SWE estimates are restricted to the domain indicated by the cumulative snow presence mask). During spring the overall mapped snow extent is determined from the cumulative mask, which (as the melt season proceeds) is reduced using a satellite passive microwave derived estimate for the end of snow melt season for each grid cell8.The snow part of the applied scene brightness temperature model is based on the semi-empirical HUT snow emission model which describes the brightness temperature from a multi-layer snowpack covering frozen ground in the frequency range of 11 to 94 GHz4,5. Input parameters to the model include snowpack depth, density, effective grain size, snow volumetric moisture and temperature. Separate modules account for ground emission and the effect of vegetation and atmosphere. Comparisons of HUT model simulations to airborne and tower-based observations, reported elsewhere (e.g.9,10), demonstrate the ability of the model to simulate different snow conditions and land cover regimes. Intercomparisons with other emission models show comparable performance when driven by in situ data11,12 or physical model outputs13, although the HUT model has the tendency to underestimate brightness temperatures for deep snowpacks12.Basic underlying assumptionsPassive microwave sensitivity to SWE is based on the attenuating effect of snow cover on the naturally emitted brightness temperature from the ground surface. The ground brightness temperature is scattered and absorbed by the overlying snow medium, typically resulting in a decreasing brightness temperature with increasing (dry) snow mass. The scattering intensity increases as the wavelength approaches the size of the scattering particles. Considering that individual snow particles tend to range from 0.5 to 4 mm in the long axis direction, high microwave frequencies (short wavelengths) will be scattered more than low frequencies (long wavelengths). The intensity of absorption can be related to the dielectric properties of snow, with snow density largely defining the permittivity for dry snow. Absorption at microwave frequencies increases dramatically with the inclusion of free water (moisture) in snow, resulting in distinct differences of microwave signatures from dry and wet snowpacks.Initial investigations pointed out the sensitivity of microwave emission from snowpacks to the total snow water equivalent14. This led to the development of various retrieval approaches of SWE from the earliest passive microwave instruments in space (e.g.15,16). From the available set of observed frequencies, most SWE algorithms employ the ~37 GHz and ~19 GHz channels in combination. These two frequencies are available continuously since 1979. The scattering from snow at 19 GHz is smaller when compared to 37 GHz, while the emissivity of frozen soil and snow is estimated to be largely similar at both frequencies. The brightness temperature difference of the two channels can be related to snow depth (or SWE), with the additional benefit that the effect of variations in physical temperature on the measured brightness temperature are reduced (relative to the analysis of single frequencies). Similarly, observing a channel difference reduces or even cancels out systematic errors of the observation, provided that the errors in the two observations are similar (e.g. due to using common calibration targets on a space-borne sensor). Typically, the vertically polarized channel at 19 and 37 GHz is preferred due to the inherent decreased sensitivity to snow layering (e.g.17).A basic assumption in the data assimilation procedure that combines spaceborne passive microwave observations and synoptic weather station data to estimate snow depth is that the background snow depth field, interpolated from weather station data, provides meaningful information on the spatial patterns of snow depth. A limitation of the methodology is that this assumption does not hold for complex terrain (mountains). Further, the methodology is not suitable for snow cover on top of ice sheets, sea ice or glaciers.Input dataThe main input data are synoptic snow depth (SD) observations and spaceborne passive microwave brightness temperatures from the Scanning Multichannel Microwave Radiometer (SMMR), Special Sensor Microwave/Imager (SSM/I) and Special Sensor Microwave Imager/Sounder (SSMIS) data from Nimbus-7 and DMSP F-series satellites. The most important frequencies for SWE retrieval and snow detection are 19 GHz (reference measurement with very little scattering from the snow volume) and 37 GHz (sensitive to volume scattering by dry snow), which are available in all instruments. The satellite datasets are described in detail in Data Records section.Ground-based SD data were acquired from the Finnish Meteorological Institute (FMI) weather station observation database, augmented from several archive sources, including the European Centre for Medium-Range Weather Forecasts (ECMWF), The United States National Climatic Data Centre (NCDC), The All-Russia Research Institute of Hydrometeorological Information-World Data Centre (RIHMI-WDC) and The Meteorological Service of Canada (MSC) archives, as described in the Data Records section.In the assimilation of SD values with space-borne estimates, a density value of 0.24 g cm−3 is assumed in estimating SWE. In the assimilation procedure the spatial small-scale variability of SD is considered by assigning a variance of 150 cm2 to the weather station observations over forested areas, and a variance of 400 cm2 for open areas. These variance estimates describe how well a single-point SD observation describes the snow depth over a larger area surrounding the measurement site, and were determined from available FMI, Finnish Environment Institute (SYKE) and Environment and Climate Change Canada (ECCC) snow transect measurements, as well as experimental field campaign data from across Finland and Canada.Daily SD background fields were generated from observations at synoptic weather station locations acquired from multiple archives for the years 1979–2018. For each measurement, the exact location, date of measurement, and SD are required. The long-term weather station data is pre-processed before utilization in the SWE retrieval to remove outliers and improve the overall consistency of the data, as described in the Methods section.Land use and, most importantly, forest cover fraction are derived from ESA GlobCover 2009 300 m data18. Stem volume is required as an input parameter to the emission model to compensate for forest cover effects4,19; average stem volumes are estimated by the ESA BIOMASAR20 data records as described in the Methods section.The following auxiliary datasets are used to mask out water and complex terrain (mountain) pixels:

ESA CCI Land Cover from 2000: water fraction is aggregated to the 25 km grid cell spacing of the SWE product, pixels with a water fraction >50% are masked as water.

ETOPO521: if the standard deviation of the elevation within a 25 km grid cell is above 200 m it is masked as complex terrain.

The Forward model applied in SWE retrievalCalculation of brightness temperature for a satellite sceneFor a satellite scene consisting of a mixture of non-forested terrain, forests, and snow-covered lake ice, the bottom-of-atmosphere brightness temperature TB,BOA is calculated so that:$${T}_{B,BOA}=left(1-FF-LFright){T}_{B,snow}+FFcdot {T}_{B,forest}+LFcdot {T}_{B,lake}$$
(1)
where FF is the forest fraction and LF the lake fraction of a given grid cell. ({T}_{B,snow}), ({T}_{B,forest}), and ({T}_{B,lake}) are the brightness temperatures emitted from non-forested terrain (ground/snow), forested terrain, and lake ice, respectively. Land cover fractions FF and LF are determined from ESA GlobCover data resampled to the 25 km EASE grid. A statistical approach is used to calculate top-of-atmosphere brightness temperatures from TB,BOA, statistics are based on studies covering the Northern Hemisphere4,22,23.Brightness temperature from snow-covered groundThe brightness temperature ({T}_{B,snow}) for snow-covered, non-forested terrain is calculated using the HUT snow emission model4. The model is a radiative transfer-based, semi-empirical model which calculates the emission from a single homogenous snowpack. The current approach utilizes multi-layer modification which allows the simulation of brightness temperature from a stacked system of snow or ice layers5.The absorption coefficient in the HUT model is determined from the complex dielectric constant of dry snow, applying the Polder-van Santen mixing model for the imaginary part24. The calculation of the dielectric constant for dry snow as well as effects of possible liquid water and salinity inclusions, are described through empirical formulae25. Emission from the snow layer is considered as both up- and down-welling emission. These are, in turn, reflected from interfaces between layers (air-snow, snow-ground). The transmission and multiple reflections between layer interfaces are calculated using the incoherent power transfer approach.Applying the delta-Eddington approximation to the radiative transfer equation, the HUT model assumes that most of the scattered radiation in a snowpack is concentrated in the forward direction (of propagation) due to multiple scattering within the snow media, based on26, which assumes that losses due to scattering are approximately equal to generation of incoherent intensity by scattering. However the omission of the backward scattering component as well as omission of trapped radiation will lead to underestimation of brightness temperature for deep snowpacks12. In the HUT model, the rough bare soil reflectivity model27 is applied to simulate the upwelling brightness temperature of the soil medium.Brightness temperature from forest vegetationThe brightness temperature over forested portions of the grid cell ({T}_{B,forest}) is derived from ({T}_{B,snow}) using a simple approximation so that:$${T}_{B,forest}={t}_{veg}cdot {T}_{B,snow}+left(1-{t}_{veg}right)cdot {T}_{veg}+left(1-{t}_{veg}right)cdot left(1-{e}_{snow}right)cdot {t}_{veg}cdot {T}_{veg}$$
(2)
where ({t}_{veg}) is the one-way transmissivity of the forest vegetation layer, ({T}_{veg}) the physical temperature of the vegetation (considered to be equal to air, snow and ground temperatures, ({T}_{veg}={T}_{air}={T}_{snow}={T}_{gnd}=-,{5}^{^circ }{rm{C}})) and ({e}_{snow}) the emissivity of the snow covered ground system. The choice of −5 °C is based on experimental data28 and follows the previous publications2,3,4. Moreover the impact of physical temperature is minimal on the simulated brightness temperature difference of two frequencies applied in the retrieval (typically More

The development of the proposed decision-support system requires the undertaking of interdisciplinary research brought about by a diverse team. It is in this context that researchers from the chemical engineering department and the geomatics sciences department at Laval University, in Quebec, Canada, have developed a nutrient stakeholder platform (NutriPlatform-QC), i.e. a regrouping of actors from research institutions, industry, governmental authorities, municipalities, and agricultural organizations, among others, that are active in the field of organic waste management. Since 2017, regular meetings have been organized with the members of the platform in order to frame the objectives and methodology of such interdisciplinary research, as well as to adapt the scope of the research to the stakeholder needs.As such, the authors initiated the design and development of a decision-support software tool that allows setting up optimal organic waste value chains for the province of Quebec, with Primodal Inc. and Chamard Environmental Strategies as industrial partners. The system, named optim-O (www.optim-o.com), applies a holistic modelling approach that focuses on minimization of costs and greenhouse gas emissions throughout the entire value chain. The scope (Fig. 1) includes the generation and collection of organic waste across the province (including urban, suburban, peri-urban and rural areas), the treatment of the waste through biomethanation, composting, and/or nutrient recovery, and the distribution of the end-products such as biogas, digestate, compost, and recovered mineral fertilizers. All of these items are geolocated in order to account for transport distances and potential traffic nuisance. Regulatory and market restrictions for product distribution are also taken into account.Fig. 1: Scope and four use cases of the optim-O decision-support system.Scope and use cases.Full size imageThe software tool integrates three key components: (1) a multidimensional spatiotemporal database system (including georeferenced and non-georeferenced data), (2) a model-based decision module (for simulation and optimization) and (3) a user-friendly interface (to facilitate knowledge transfer and interpretation). Table 1 provides an overview of the data included in the system. Generally speaking, georeferenced data includes data that is location-specific, such as population, commerce, services and industry (position and size), road networks, hydrographic networks, existing infrastructure (wastewater treatment plants, biogas and composting facilities), agricultural parcels (location, size, crop, nutrient saturation index) and associated regulatory and market constraints (fertilizer application limitations). Non-georeferenced data includes costs and other factors used for economic assessments, greenhouse gas emission factors, technical process-related factors (used for the mathematical process models), and social factors (odour emissions, population density, the latter also being part of the georeferenced data). Default values are provided for the non-georeferenced data, but the user can modify these if case-specific data would be available. A prototype of the developed tool is currently being validated using two major biomethanation plants in Quebec. The tool also has the flexibility for extension with other resource recovery processes.Table 1 Georeferenced and non-georeferenced data included in the optim-O decision-support software tool.Full size tableFigure 1 presents the four use cases for which the tool can be used. It concerns decision problems related to (1) the collection of organic waste, (2) the treatment process operation, (3) the end-product distribution and (4) the integration of the three previous use cases as one global optimization problem. In each case, the tool can be used to either simulate and evaluate various scenarios defined by the user, or to solve the optimization problem taking into account optimization criteria defined by the user, as described in the examples below.In the first use case, i.e. the collection of organic waste, the tool allows for the estimation of organic waste generation based on data from households, services, businesses and industries, with associated organic matter generation rates for each, either based on the number of members in a household, employees or clients, as well as the type of service, business or industry. As presented in Table 1, all of this information is geolocated, allowing users to locate sources of organic waste across a territory, as presented in Fig. 2. From here, using the treatment plant location and road networks, various waste collection routes can be simulated. The user can also select specific modelling objectives, for example: maximising organic waste collection, evaluating the potential to collect a certain waste type, assessing long-distance travel (for example, through transfer stations), as well as associated optimization objectives, for example, reducing GHG emissions, reducing costs or reducing both at the same time.Fig. 2: Geolocated organic waste generation throughout the southern area of the province of Quebec, as estimated by the decision-support system.Geolocated organic waste generation Southern Quebec.Full size imageIn the second use case, i.e. treatment process operation, the system can evaluate processing outcomes through a mathematical model library developed for this tool. It includes models for anaerobic digestion, composting and processes to recover nutrients as mineral fertilizers from digestate, and allows easy extension with other process models in the future. The models are numerically simple, requiring basic data inputs (e.g., key physico-chemical waste characteristics), and are coded directly in the database. By selecting this approach, a balance was sought between model complexity and simulation times, with the aim to minimize computational efforts, while maximizing usability. Using the models, one can aim at evaluating the impact of varying substrates on the process performance, seeking to optimize certain parameters (e.g., minimizing GHG emissions, maximizing product quality or minimizing process duration/size). Moreover, different treatment process combinations can be evaluated and compared, for example the implementation of anaerobic digestion as sole technology vs the implementation of anaerobic digestion with nitrogen recovery from the liquid fraction of digestate and composting of the solid fraction of digestate.In the third use case, users can simulate and optimize locations for end-product distribution. In this case, an estimation of quantity and quality factors for the end-products (biogas, digestate, compost, recovered mineral fertilizers), either provided as model outputs or entered by the user, are considered as data inputs. From here, agricultural lands can be evaluated regarding their receptivity for the product. This receptivity is based on the quality of the product, size of the plot, the phosphorus saturation status of the soil, the nitrogen pollution status of the surrounding water bodies and the nitrogen requirements for crop production, which all determine how much product can be accepted on the land under study. Distribution networks can then be set up and optimized using spatial analysis, identifying the nearest receptive lands.Finally, a fourth use case concerns the integrated assessment of the above three use cases. Indeed, the outputs of one module can serve as the inputs to another module. As such, the outputs of the waste collection module can be used as inputs to the treatment process module, providing a certain quantity and quality of substrate(s). The process models can then be run to determine the optimal treatment process chain, as well as quantity and quality factors for the end-products. The latter can then be used to search for an optimal agricultural site for end-product distribution. This process can be undertaken iteratively by the system seeking to meet desired criteria and/or propose a few scenarios of interest to decision makers. The fourth use case can also be applied to select the optimal position of a new treatment plant, taking into account the organic waste availability and the access to agricultural land for end-product distribution. Moreover, such integrated approach can allow users to understand the impact of changing waste collection strategies on existing treatment process chains, or to evaluate how a change in process conditions can affect end-product distribution. More

Study sites and data collectionThe data used for this study were obtained from a previous multi-site study on post-distribution FRC decay collected from refugee settlements in South Sudan, Jordan, and Rwanda19. This dataset was selected as process-based models have been used to produce FRC targets for these sites, which provide a useful comparison to the risk-based targets generated in this study. Details of the data collected at these sites, as well as important site characteristics are included in Table 3. Two datasets were collected from Jordan: one from the summer of 2014 and one 9 months later from the late winter of 2015. The original study treated these as two separate datasets due to differences in environmental conditions between the two datasets (10 °C difference in average temperature) and amount of time between the two datasets19. To ensure a consistent comparison with the original study, we have also treated the 2014 and 2015 data from Jordan as two distinct datasets.Table 3 Summary of Key Site Characteristics19,59,60,61.Full size tableThe dataset for each site includes FRC as well as other water quality parameters, which are routinely collected in humanitarian water systems operation including total residual chlorine, EC, water temperature, turbidity, and pH. Data were collected using paired sampling whereby the same unit of water was sampled at the following points along the post-distribution water supply chain:

From the tap at the point-of distribution

In the container immediately after collection

In the container immediately after transport to the dwelling

After a follow-up period of storage in the household

This study only used the measurements at the point-of-distribution and point-of-consumption to reflect data collection practices that are more feasible for humanitarian operations. In preparing the dataset, observations were removed if the point-of-distribution water quality did not meet humanitarian drinking water quality guidelines. Supplementary Table 2 in the Supplementary Information includes the full list of data cleaning steps that were used to prepare the data for use in the ANN models.EthicsThe initial field work in South Sudan received exemption from full ethics review by the Medical Director of Médecins sans Frontières (MSF) (Operational Centre Amsterdam) as data collected was routine for the on-going water supply intervention at the study site. For subsequent field studies in Jordan and Rwanda, ethics approval was obtained from the Committee for Protection of Human Subjects (CPHS) of the Institutional Review Board at the University of California, Berkeley (CPHS Protocol Number: 2014-05-6326). Informed consent was provided throughout all data collection.Input variable selectionTwo input variable combinations were considered for predicting the output variable, the point-of-consumption FRC concentration. The variables considered are all variables that are routinely monitored in humanitarian water system operations. The first input variable combination (IV1) included FRC at the water point-of-distribution and the elapsed time between the measurement at the point-of-distribution and the point-of-consumption. This input variable combination represents the minimum number of variables that would be regularly collected under current humanitarian drinking water quality guidelines31. Additionally, these are the only two variables included in the process-based model developed in a past study for these sites19, so this input variable combination allows for a direct comparison of the ANN ensemble models with the process-based models. The second input variable combination (IV2) included the variables from IV1 as well as additional water quality variables measured from the point-of-distribution (directly after water had left the water distribution point): EC, water temperature, pH, and turbidity. These additional variables are recommended for collection in some humanitarian drinking water quality guidelines29,30,31, and as such, may also be available in humanitarian response settings. This larger input variable set allowed us to investigate the usefulness of additional water quality variables for forecasting point-of-consumption FRC concentrations.Base-learner structure and architectureThe ensemble base learners (the individual ANNs in the ensemble models) were built as multi-layer perceptrons (MLPs) with a single hidden layer using the Keras 2.3.0 package48 in Python v3.749. This structure was selected because it has been shown to outperform other data-driven models and ANN architectures for predicting FRC in piped distribution systems20,21. The weights and biases of the base learners were optimized to minimize mean squared error (MSE) using the Nadam algorithm with a learning rate of 0.1. An early stopping procedure with a patience of 10 epochs was used to prevent overfitting.The hidden layer size of the base learners was determined through an exploratory analysis by consecutively doubling the hidden layer size until performance decreased or ceased to improve substantially from one iteration to the next. Based on this analysis, we selected a hidden layer size of four hidden neurons at all sites for the models using the IV1 variable combination for all sites. For the models using the IV2 input variable combination, we selected a hidden layer size of 16 hidden nodes for South Sudan and Jordan (2015), and a hidden layer size of eight hidden nodes for Jordan (2014) and Rwanda. The full results of the exploratory analysis into hidden layer size are included in Supplementary Figs 13–20 in the Supplementary Information.Data divisionThe full dataset for each site and variable combination was divided into calibration and testing subsets, with the calibration subset further subdivided into training and validation data. The testing subset was obtained by randomly sampling 25% of the overall dataset. The same testing subset was used for all base learners so that each base-learner’s testing predictions could be combined into an ensemble forecast. The training and validation data were obtained by randomly resampling from the calibration subset, with a different combination of training and validation data for each base learner to promote ensemble diversity. The ratio of data from the calibration set used for training and validation, respectively, was selected to avoid both overfitting and underfitting through an exploratory analysis using a grid search process. In all but two cases, we selected a validation set that was twice the size of the training set, for an overall training-validation-testing split of 25–50–25%. The two exceptions to this were for the Jordan (2014) model when using the IV1 input variable combination where we found that a training-validation-testing split of 50–25–25 produced better performance, and for the Jordan (2015) model when using the IV1 input variable combination where a training-validation-testing split of 30–45–25 performed substantially better. The full results of the exploratory analysis for data division are included in Supplementary Figs 21–28 in the Supplementary Information. Descriptive statistics for the calibration and testing datasets are included in Supplementary Tables 3 and 4 of the Supplementary Information, and histograms of the input and output variables are provided in Supplementary Figs 5–12 in the Supplementary Information to provide context of the range and patterns in the data used to train the ANN base learners.Ensemble model formationThe ensemble models in this study were used to generate probabilistic forecasts of post-distribution FRC by combining the predictions of each base learner into a probability density function (pdf). Thus, for each observation of FRC at the point-of-consumption, the ensemble model outputs a pdf representing the predicted probability of point-of-consumption FRC concentrations. This pdf can then be used to identify ensemble confidence intervals (CIs) for the expected point-of-consumption FRC concentration. To ensure a good representation of the full output space in the final pdfs, two approaches were taken to ensure ensemble diversity. First, as discussed above, the data used to train the base-learner ANNs was randomly sampled from the calibration set, so each ANN was trained on a different subset of the data. Second, the initial weights and biases were randomized for each base learner in a random-start process. Both of these are implicit approaches to ensuring ensemble diversity as they do not directly create diversity and instead the diversity arises through the randomization of the training data and the weights and biases50. The benefit of implicit approaches is that the differences between the base learners are derived from randomness in the data50.The ensemble size (number of base learners included in the ensemble) was also determined through an exploratory analysis using a grid search procedure This exploratory analysis showed that in general, performance increased with larger ensemble sizes, but improvements in performance plateaued at ensemble sizes ranging from 50 members to 250 members. Based on this, a standard ensemble size of 250 members was selected for all sites and variable combinations. The full results of the exploratory analysis for ensemble size are included in Supplementary Figs 29–36 in the Supplementary Information.Ensemble post-processingWe used ensemble post-processing to attempt to improve the forecasts generated by the raw ensembles. We used the kernel dressing method to post-process ensemble predictions51. This method follows a two-step process: first a kernel function is fit centred on the base-learner prediction for each observation, then each member’s kernel is summed together to produce the post-processed pdf, which is a non-parametric mixture distribution function. We used a Gaussian kernel function in keeping with past studies27,28,38,51, though the selection of the specific kernel function is not critical28. The kernel bandwidth was defined using the best member error method where the bandwidth for all kernels is the variance of the absolute error of the prediction that is closest to each observation in the calibration dataset51.Ensemble verification and performance evaluationWe used ensemble verification metrics to evaluate the performance of the raw and post-processed ensembles for each site and variable combination. Ensemble verification metrics differ from traditional measures of performance (e.g. Nash Sutcliffe Efficiency, MSE, etc.) as they assess the performance of the probabilistic forecasts of an ensemble whereas traditional measures typically evaluate the average performance of an ensemble model or the predictions of a deterministic model52. Throughout the following section, (O) refers to the full set of observed FRC concentrations at the point-of-consumption and (o_i) refers to the (i^{{mathrm{th}}}) observation, where there are (I) total observations. (F) refers to the full set of probabilistic forecasts for point-of-consumption FRC, where (F_i) is the probabilistic forecast corresponding to observation (o_i) and (f_i^m) is the prediction by the (m^{{mathrm{th}}}) base learner in the ensemble on the (i^{{mathrm{th}}}) observation. For the following metrics, it is assumed that the predictions of each base learner in the ensemble are sorted from low to high for each observation such that (f_i^m le f_i^{m + 1}) from (m = 0) to (m = M).Percent capturePercent capture measures the percentage of observations which are captured within the ensemble forecast and provides a useful indication of how well the model can reproduce the full range of observed values, and, as such, can indicate if a model is underdispersed. For a raw ensemble forecast, the (i^{{mathrm{th}}}) observation is captured if (f_i^0 le o_i le _i^M). For a post-processed forecast, the (i^{{mathrm{th}}}) observation is captured if the probability of (o_i) in the mixture distribution is greater than 0. While not commonly used for ensemble verification, a similar metric has been used for evaluating other probabilistic or possibilistic models, especially neurofuzzy networks, referred to either as the percent capture or the percent of coverage53,54,55,56. The percent capture was calculated both for the overall set of observations, as well as for observations with point-of-consumption FRC below 0.2 mg/L. The latter is a useful indicator of how well the model can predict if water will have sufficient FRC at the point-of-consumption, which is an important indicator of the degree of confidence we have in the risk-based targets generated using these ensemble models.CI reliability diagramReliability diagrams are visual indicators of ensemble reliability, where reliability refers to the similarity between the observed and forecasted probability distributions with the ideal model having all observations plotted along the 1:1 line showing that the observed probabilities are equal to the forecasted probabilities. These diagrams plot the observed relative frequency of events against the forecast probability of that event, though the reliability diagram has been adapted in past studies as the CI reliability diagram which compares the frequency of observed values within the corresponding CI of the ensemble. For raw ensembles, the CIs are derived from the sorted forecasts of the base learners (for example, the ensemble 90% CI would include all of the forecasts between (f^{0.05M}) and (f^{0.95M})) and for post-processed ensembles, the CIs are calculated directly from the probability distribution. In this study, we extended the CI reliability diagram further by plotting the percent capture of each CI within the ensemble against the CI level. For each ensemble model we plotted the CI reliability for the 10–100% CI levels at 10% intervals as well as at the 95 and 99% CI. We used this to develop a numerical score for the CI reliability diagram, which is calculated as the squared distance between the percentage of observations captured within each CI and the ideal percent capture in that CI. This was calculated for each CI threshold, k, from 10 to 100% in 10% increments as shown in Eq. 1.$$CI;{mathrm{Reliability}};{mathrm{Score}} = mathop {sum }limits_{k = 0.1}^1 left( {k – {mathrm{Percent}};{mathrm{Capture}};{mathrm{in}};CI_k} right)^2$$
(1)
The CI reliability score measures the horizontal distance between the percent capture and the 1:1 line for each CI. The ideal value for this score would be 0, indicating all points fall on the 1:1 line. The worst possible score will depend on the number of CI’s included in the calculation of the score; for this study the worst score is 3.9, which would only occur if no observations were captured in any CI of the ensembles. The CI reliability score was calculated for both the overall dataset and for forecast-observation pairs where the observed household FRC concentration was below 0.2 mg/L.Continuous Ranked Probability ScoreThe Continuous Ranked Probability Score (CRPS) is a common metric for evaluating probabilistic forecasts that evaluates the difference between the predicted and observed probabilities of continuous variables and is equivalent to the mean absolute error of a deterministic forecast57,58. The CRPS measures not only model reliability but also sharpness, which is an indicator of how closely the ensemble predictions are clustered around the observed values. Thus, the CRPS can be a useful measure of overdispersion and can provide an indication if improvements in reliability are being obtained at the expense of excess overdispersion. The CRPS is measured as the area between the forecast cumulative distribution function (cdf) and the observed cdf for each forecast-observation pairing58. Since each observation is a discrete value, the observation cdf is represented with the Heaviside function (H{ x ge x_a}), which is a stepwise function with a value of 0 for all point-of-consumption FRC concentrations below the observed concentration and 1 for all point-of-consumption FRC concentrations above the observed concentration. The equation for calculating the CRPS of a single forecast-observation pair is given in Eq. 2. Note that Eq. 2 shows the calculation of CRPS for a single forecast-observation pair. To evaluate the ensemble models, the average CRPS, (overline {{mathrm{CRPS}}}), is calculated by taking the mean CRPS overall forecast-observation pairs.$${mathrm{CRPS}} = {int nolimits_{-infty }^infty} left( {F_ileft( x right) – Hleft{ {x ge o_i} right}} right)^2dx$$
(2)
For the post-processed probability distributions, we calculated CRPS directly from Eq. 2 using numerical integration. For the raw ensemble, we treated the forecast cdf as a stepwise continuous function with (N = M + 1) bins where each bin is bounded at two ensemble forecasts and the value in each bin is the cumulative probability58. (overline {{mathrm{CRPS}}}) is calculated using (overline {g_n}), the average width of bin (n) (average difference in FRC concentration between forecast values (m) and (m + 1)) and (overline {o_n}) the likelihood of the observed value being in bin (n)58. Using these values, the (overline {{mathrm{CRPS}}}) for an ensemble can be calculated as58:$$overline {{mathrm{CRPS}}} = mathop {sum }limits_{n = 1}^N overline {g_n} [(1 – overline {o_n} )p_n^2 + overline {o_n} left( {1 – p_n} right)^2]$$
(3)
Where (p_n) is the probability associated with each bin, (p_n = frac{n}{N})58.Generation of risk-based targetsTo generate the risk-based FRC targets, the trained ensembles of ANNs were used to forecast the point-of-consumption FRC for a series of point-of-distribution FRC concentrations from 0.2 to 2 mg/L in 0.05 mg/L increments. For each point-of-distribution FRC concentration, the predicted risk of insufficient FRC was calculated from the forecast pdf as the cumulative probability of FRC at the point-of-consumption being below 0.2 mg/L. Using this predicted risk, the target FRC concentration for the point-of-distribution was then selected as the lowest FRC concentration at the water point-of-distribution that provides the desired level of protection. For this study we selected the FRC concentration that resulted in negligible risk of FRC being below the 0.2 mg/L threshold (i.e. the lowest FRC concentration where the predicted risk is 0), though operationally any level of protection could be used and the risk of insufficient FRC at the point-of-consumption should be balanced against risks associated with high FRC concentrations, such as DBP formation and taste and odour concerns.For comparison with the previously published results, we used a storage duration of 10 h when generating the FRC targets for South Sudan, and 24 h for all other sites19. Since the IV2 model also requires values for EC, water temperature, pH, and turbidity, two scenarios were considered. First, an “average” scenario was used where the median observed value for all other water quality parameters were selected. The second scenario considered was a “worst-case” scenario, where we simulated a scenario where water quality conditions were unfavourable for maintaining chlorine residual. A partial correlation analysis, which assesses the correlation between an input variable and the output variable while controlling for the impacts of other input variables, was used to determine the least favourable conditions for each input variable. The partial correlation analysis is performed by first developing multiple linear regression predictions of both the output variable (point-of-consumption FRC) and the input variable of interest using the remaining input variables as the predictors to the linear regression models and then taking the Pearson correlation coefficient of the residuals between the two regression models. Partial correlation was used to assess the directionality of the effect of the additional water quality variables included in IV2 to assess whether high or low values of these inputs would create a worst-case scenario. Once the directionality of the impact of the different variables had been established, the 95th or 5th percentile observed value of that variable was used at each site to simulate the worst-case scenario. More