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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

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Data sourcesData for this analysis were extracted from the American Community Survey (ACS) 5-year estimates for 2014–2018 via Integrated Public Use Microdata Series – National Historic Geographic information System (IPUMS-NHGIS)26, and from the Environmental Protection Agency’s (EPA) Enforcement and Compliance History Online (ECHO) Exporter27. Data were extracted at the county level for all 50 states, Washington DC, and Puerto Rico. The ACS is an ongoing survey of the United States which documents a wide variety of social statistics ranging from simple population counts to housing characteristics. Due to the staggered sampling structure of the ACS, it takes 5 years for every county to be sampled. Because of this, researchers must use 5-year intervals to ensure complete data coverage. The data from these 5 years are projected into estimates for all counties in the United States for the 5-year period in question. As of this study, 2014–2018 was the most recently available data.ECHO collates data from EPA-regulated facilities across the United States of America to report compliance, violation, and penalty information for all facilities for the most recent 5-year interval. ECHO data is updated weekly and the data for this paper was extracted on 18 August 2020. This means that the data in our analysis represents the status of each community water system or Clean Water Act permittee, as reported by the EPA, as of 18 August 2020. Only those community water systems or Clean Water Act permittees listed as Active by ECHO were included in this analysis. As ECHO data is at the level of the water system, permittee, or utility, we aggregated data up to the county level.Safe Drinking Water Act data was geolocated using QGIS 3.10 based upon latitude and longitude. This was done because other geographic identifiers for the Safe Drinking Water Act data were often missing. In line with prior work4,5,7,8, and in order to facilitate a cleaner dataset, we only focus on those water systems labeled community water systems for our analysis. Community water systems were geolocated based upon the county in which their latitude and longitude were located, if a community water system had latitude and longitude over water, a nearest neighbor join was used. In total, 1334 out of 49,479 community water systems were dropped because of there being no reported latitude or longitude. Of these, a total of 4.0%, or 54 community waters systems, were reported as in serious violation.Active Clean Water Act permittees were first identified by listed county. This was done because 345,176 out of 350,476 permittees had a county reported. Those without a county reported were located using latitude and longitude in the same manner as community water systems. There were 10 permittees without latitude and longitude or county listed which were excluded from our analysis. Of these, seven were in significant noncompliance and three were not. Due to some Clean Water Act permittees having latitude and longitude placements far away from the United States, those over 100 km from their nearest county were excluded from analysis. Finally, for community water systems and Clean Water Act permittees, some counties (76 for community water systems and 13 for Clean Water Act permittees) had no reported cases. Those counties were treated as zeroes for cartography and as missing for modeling purposes.Similar to prior work in this area4,5,8, we restrict our analysis to the scale of the county for reasons related to data limitations and resulting conceptual validity. Although counties are arguably larger in geographic area than ideal for an environmental injustice analysis, if we were to use a smaller unit for which data is available such as the census tract, the conceptual validity of the analysis would be limited due to the apolitical nature of these units. As outlined above, ECHO data is messy and missing many geographic identifiers. What is provided is generally either the county or latitude and longitude. If only the county is provided, then we are constrained to using the county regardless of conceptual validity. However, even when latitude and longitude are provided—which is the case for many observations—the provided point location says nothing about which households the water system or permittee serves or impacts. Due to this, whatever geographic unit we use carries the assumption that those in the unit could be plausibly impacted by the water system or permittee. Given that counties are often responsible for both regulating drinking water, as well as maintaining and providing water infrastructure29, we were comfortable with this assumption between point location and presumed spatial impact when using the scale of the county. However, we believe this assumption would have been invalid and untestable for smaller apolitical units for which demographic data is available such as census tracts.Beyond the issues presented by ECHO data, the county is also the appropriate scale of analysis for this study due to the estimate-based nature of the ACS. ACS estimates are based on a rolling 5-year sample structure and often have very large margins of error. At the census tract level, these standard errors can be massive, especially in rural areas30,31,32. Due to this variation, and the need to include all rural areas in this analysis, the county, where the margins of error are considerably smaller, is the appropriate unit for this study. All of this said, the county is, in fact, a larger unit than often desired or used in environmental justice studies. Studies focused on exclusively urban areas with clearer pathways of impact can and should use smaller units such as census tracts. It will be imperative for future scholarship focused on water hardship across the rural-urban continuum to gain access to reliable data on sub-county political units, as well as data linking water systems to users, to continue documenting and pushing for water justice.Dependent variablesThe dependent variables for this analysis were assessed in both a continuous and dichotomous format. For descriptive results and mapping, continuous measures were used. For models of water injustice, a dichotomous measure which classified counties as either having low levels of the specific water issue or elevated levels or the specific water issue, was used due to the low relative frequency of water access and quality issues relative to the whole United States population. For all three outcomes, we benchmark an elevated level of the issue as what would be viewed as an unacceptable level under United Nations Sustainable Development Goal 6.1, which states, “by 2030 achieve universal and equitable access to safe and affordable drinking water for all”1. As this goal focuses on ensuring all people have safe water, we deem a county as having an elevated level of the issue if >1% of households, community water systems, or permittees had incomplete plumbing, were in Significant Violation, or Significant Noncompliance, respectively. Although we could have used an even stricter threshold given the SDG’s emphasis on ensuring access for all people, we use 1% as our cut-off due to its nominal value and ease of interpretation.For water access, the continuous measure was the percent of households in a county with incomplete household plumbing as reported by the ACS. The ACS currently asks respondents if they have access to hot and cold water, a sink with a faucet, and a bath or shower. Up until 2016, the question also included a flush toilet33. As we must use the most recent 2014–2018 5-year estimates to establish full coverage of all counties, this means that incomplete plumbing in this item may, or may not include a flush toilet depending on when the specific county was sampled. The dichotomous version of this variable benchmarked elevated levels of incomplete plumbing as whether or not 1% or more of households in a county had incomplete plumbing.Water quality was assessed via both community water systems from the Safe Drinking Water Act, and from permit data via the Clean Water Act. For Safe Drinking Water Act data, the continuous measure was the percent of community water systems within a county classified as a Safe Drinking Water Act Serious Violator at time of data extraction. The EPA assigns point values of either 1, 5, or 10 based upon the severity of violations of the Safe Drinking Water Act. A Serious Violator is one who has “an aggregate score of at least eleven points as a result of some combination of: unresolved more serious violations (such as maximum contaminant level violations related to acute contaminants), multiple violations (health-based, monitoring and reporting, public notification and/or other violations), and/or continuing violations”27. The dichotomous measure benchmarked elevated rates of Safe Drinking Water Act Significant Violation as whether or not >1% of county community water systems were classified as Serious Violators.For Clean Water Act permit data, the continuous measure was the percent of permit holders listed as in Significant Noncompliance at the time of data extraction. Significant Noncompliance in the Clean Water Act refers to those permit holders who may pose a “more severe level of environmental threat” and is based upon both pollution levels and reporting compliance27. The dichotomous measure again set the threshold for elevated levels of poor water quality at whether or not >1% of Clean Water Act permittees in a county were listed as in Significant Noncompliance at time of data extraction.Independent variablesThe independent variables we include in models of water injustice are those frequently shown to be related to environmental injustice in the United States. These include age, income, poverty, race, ethnicity, education, and rurality17,18,19,20,21,22,23,24,25. Age was included as median age. Income was included as median household income. Poverty was the poverty rate of the county as determined by the official poverty measure of the United States34. Race and ethnicity was included as percent non-Latino/a Black, percent non-Latino/a indigenous, and percent Latino/a. Because the focus was on indigeneity, percent American Indian or Alaska Native was collapsed with Native Hawaiian or Other Pacific Islander. We did not include percent non-Latino/a white due to issues of multicollinearity. Finally, rurality was included as a three-category county indicator of metropolitan, non-metropolitan metropolitan-adjacent, and non-metropolitan remote, as determined by the Office of Management and Budget in 201035. The OMB determines a county is metropolitan if it has a core urban area of 50,000 or more people, or is connected to a core metropolitan county by a 25% or greater share of commuting35. A non-metropolitan county is simply any county not classified as metropolitan. Non-metropolitan metropolitan adjacent counties are those which immediately border a metropolitan county, and non-metropolitan remote counties are those that do not.Water injustice modeling approachWater injustice was assessed by estimating linear probability models for the three dichotomous outcome variables with state fixed effects to control for the visible state level heterogeneity and differences in policy, reporting, and enforcement (e.g. the clear state boundary effects in Fig. 3). We employ cluster-robust standard errors at the state level to account for both heteroskedasticity and state similarities. All modeling was performed in Stata 16.0 and mapping was performed in QGIS 3.10. We assessed all full models for multicollinearity via condition index and VIF values and the independent variables had an acceptable condition index of 5.48, well below the conservative cut-off of 15, as well as VIF values of 20). All indications of statistical significance are at the p  More