Resources

Kidney progression projectThe Kidney Progression Project was initiated in 2017 in the Wilgamuwa Divisional Secretariat, a highly endemic CKDu area of 40,000 people in the lowland dry zone area of the Central Province (Supplementary Fig. 1). All protocols were reviewed and approved by review boards at the University of Connecticut in the US and National Hospital in Kandy, Sri Lanka. The detailed methodological approach including a description of behavioral and clinical and expanded environmental variables is described in Vlahos et al. (2018)13. Briefly, in 2016, the Ministry of Health conducted a screening of urine and blood in Wilgamuwa for residents 11 years and older to identify those with CKDu. Using the resulting serum creatinine values obtained during this screening effort, the KiPP team calculated CKD-EPI eGFR23, which resulted in a total of 330 people at Stage 3 and 4 of CKDu (eGFR in the range of 20-60 ml/min/1.73 m2), who did not have identifiable cause for CKD with evidence of chronic interstitial nephritis in renal biopsies or small echogenic kidney. Of these, 304 agreed to participate but ultimately 293 answered the baseline questionnaire and came for at least one serum creatinine measurement and were included in this analysis.Baseline survey componentsAll participants were administered a baseline survey that focused on environmental exposure, behavioral and occupational factors, and clinical values as described in the KiPP protocol13. We probed water sources in detail. Water sources in the study area and the dry zone in general include household wells dug by hand that are 10 meters deep or shallower, tube wells dug to a depth of 20–30 m with drilling equipment, and lesser-used sources including surface water (tanks, channels and river water), rainwater collection, natural spring water, publicly supplied pipe water, and public water delivered to individual houses by truck (bowsers) and stored in large roof containers. The rise in CKDu cases led the government to invest heavily in reverse osmosis (RO) units and nanofiltration membrane technology for many dry zone villages14. These were installed at the end of 2017 and early 2018 to provide rationed, free drinking water.Baseline water samples and analysisThe wells of each participant household were sampled once for target agrochemicals as described in Shipley et al.24. In all, 272 household wells were sampled with 31 households sharing wells.Agrochemical analysesAgrochemical analyses follow methods of Shipley et al., (2022)24 and EPA (2018)25. Briefly, 1 L well water samples were collected at each participant’s home and pre-filtered through a 0.45 µm nominal GFF to remove particulates. The sample was then extracted using 3 mL Chromabond C-18 SPE cartridges and a Supelco Visiprep SPE vacuum manifold. Three deuterated surrogate standards (chrysense d12, acenaphthene d10, and 1,4-dichlorobenzene d4) were loaded onto the cartridge before elution with 5 ml of acetonitrile and nitrogen reduction to 1 ml. Recoveries ranged from 70 to 101%.An initial non-targeted analysis was run on samples in scan mode which identified over 100 compounds, including pyrolytic compounds that are likely the result of field burning practices in preparation for the new season. We supplemented these analyses with data from a local list of agrochemicals for the year 2017–2018 supplied by the Sri Lankan Ministry of Environment. Based on these data, targeted analyses were performed for 30 agrochemicals using selective ion mode.Inorganic analysesPhosphate in samples was measured with an Ion Chromatograph (Thermo Dionex ICS-1100). For repeated analyses of selected samples, an analytical precision better than ±5% of relative standard deviations was achieved. Total hardness was determined by EDTA titration method (APHA 2012)26.Follow up: From December 2017 to the beginning of 2020, study participants had quarterly follow-up visits assessing behavioral changes including water consumption and serum creatinine testing. Serum creatinine was tested using an IDMS-calibrated enzymatic assay and converted to estimated glomerular filtration using the CKD-EPI equation.GIS Analysis: Using GPS coordinates recorded by the field team for the domestic wells of each participant, individual eGFR at baseline and eGFR slopes over the study period were plotted over the ArcMap World Topographic map. For the baseline eGFR map, values were separated into five categories using Jenks Natural breaks provided by the ArcGIS software. The uppermost category was manually set to 65 mL/min/1.73 m2 and points with null or More

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Water access data processingData on drinking water coverage by region was acquired from the WHO/UNICEF JMP. The JMP acts as official custodian of global data on water supply, sanitation and hygiene2 and assimilates data from administrative data, national census and surveys for individual countries, and maintains a database that can be accessed online through their website. We accessed data tables for national and subnational drinking water service levels from https://washdata.org.JMP datasets are not geographically linked to official boundary files. We joined the tables to GIS boundaries obtained from the following open-source collections: GADM (https://gadm.org), the Spatial Data Repository of the Demographic and Health Surveys Program of USAID (DHS) and the Global Data Lab of Radboud University (GDL)2,50,51,52,53. Subnational regions reported by the JMP are unstructured, representing various regional administrative levels (province, state, district and others).The JMP national and subnational data were joined to GIS boundaries using a custom geoprocessing tool built in Python and ArcGIS 10. The tool joins the available JMP subnational-level survey data to the closest name match of regional boundary names from a merged stack of GADM (admin1, admin2 and admin3), DHS and GDL boundaries worldwide. The JMP national-level survey data is then joined to GADM national (admin0) boundaries for countries which have no subnational data available. Finally, the two boundary-joined datasets (national and subnational) are merged, processed and exported as a seamless global fabric of water-stressed-population data at the highest respective spatial resolutions available (Fig. 1a).JMP does not report the breakdown between the SMDW and basic service level within subnational regions, and instead reports a combined category called ‘at least basic’ (ALB). To estimate the SMDW values in subnational regions, a simple cross-multiplication was performed using the splits at the national level:$${{rm{SMDW}}}_{{rm{subnational}}}=frac{{{rm{SMDW}}}_{{rm{national}}}}{{{rm{ALB}}}_{{rm{national}}}}{times {rm{ALB}}}_{{rm{subnational}}},$$where ALBnational, ALBsubnational and SMDWnational are known values.Validation of the cross-estimation of share of SMDW from ALB for subnational regions was conducted on a reference dataset of nationally representative household surveys that collected data on all criteria for SMDW54, shown in Extended Data Fig. 2. We report regression results of R2 = 0.87 and a standard error of 3.67, indicating a bias which over-reports SMDW share and a probable underestimate of people living without SMDW in our study. This discrepancy comes from JMP calculations of SMDW that rely on the minimum value of multiple drinking water service criteria (free from contamination, available when needed and accessible on premise) rather than considering whether individual households meet all criteria for SMDW55.The fraction of population without SMDW was multiplied by residential population values in the WorldPop top-down unconstrained global mosaic population count of 2017 at 1 km spatial resolution56 (https://www.worldpop.org). WorldPop was accessed online as a TIF image and imported to Google Earth Engine. The year 2017 was chosen to more closely match water access data from JMP. The percentages reported by JMP are probably not uniform within most regions57, introducing an unknown error to Fig. 1b, but represent the best estimate available to us given the limitations of these regionally reported data.Climate input and conversion approximationsGHI and reference planeWe used GHI (in W m−2) as solar energy input data. GHI has good availability in climate datasets and introduces the fewest number of assumptions. Since GHI describes the irradiance in a locally horizontal reference plane, this approximation is only exact for devices having a horizontally oriented solar harvesting area. Annually averaged comparisons between horizontal and optimal fixed-tilt panels show negligible differences in direct plus diffuse radiation in tropical latitudes, and ratios below 25% in locations within 50° north and south latitudes58. Those seeking precise absolute predictions for tilted devices or higher latitudes are encouraged to adapt the provided code to their specific assumptions.Conversion from SY to AWH outputAs discussed in the main text, solar-driven AWH devices typically have one of two predominant energy inputs: thermal (converted directly from incident sunlight on the device) or electrical (from PV). Here, the energy units used to calculate yield in l kWh−1 are incident solar energy directly from GHI. The various assumptions are made in relation to the reported values based on their source. The thermal limits33, target curves, and experimental results reported by TRP15 and MOFs were assumed to have direct (100%) conversion from sunlight to heat. For the ZMW device, the table provided by the manufacturer accounts for system losses, so the table values were directly converted in our model35. For ref. 34 and the cooler–condenser limits from ref. 32, which both assume work input instead of heat, we applied a typical PV conversion efficiency of 20% to convert from sunlight kWh (GHI) to kWhPV (electrical work) input to the device59.Sufficiently short sorbent cycling timesAWH-Geo assumes continuous or quasi-continuous AWH. AWH-Geo considers each 1-h timestep independently and is thus stateless. Aside from edge cases, this is a safe assumption for mass efficient SC-AWH devices, which typically have time constants shorter than 1 h, both for sorbent cycling and for most of the thermal time constants. For devices with longer time constants, batch devices or processes with slow (de)sorption kinetics, this assumption may introduce increased error, and may require further adaptation of the provided code.Climate time-series calculationAWH-Geo is a resource-assessment tool for AWH. It consists of a geospatial processing pipeline for mapping water production (in litres per unit time) around the world of any solar-driven continuous AWH device that can be characterized by an output table of the form output = f(RH, T, GHI).Output tables show AWH output values in l h−1 or l h−1 m−2 across permutations of the 3 main climate variables in the following ranges: RH between 0 and 100 % in intervals of 10%, GHI between 0 and 1,300 W m−2 in intervals of 100 W m−2, and T between 0 and 45 °C in intervals of 2.5 °C (2,145 total output values). The tables are converted into a 3D array image in Google Earth Engine and processed across the climate time-series image collection for the period of interest. Finally, these AWH output values are composited (reduced) to a single time-averaged statistic of interest as an image.Climate time-series data was acquired from the ERA5-Land climate reanalysis from the European Centre for Medium-Range Weather Forecasts (ECMWF)60, accessed from the Google Earth Engine data catalogue. ERA5-Land surface variables were used in 1-h intervals and 0.1°× 0.1° (nominal 9 km). The 10-year analysis period (2010–2019, inclusive) was used for this work, and represents a period long enough to provide a reasonable correction for medium-term interannual climatic variability.Climate variables GHI and T were matched to ERA5-Land parameters ‘Surface solar radiation downwards’ (converted from cumulative to mean hourly) and ‘2 metre temperature’ (converted from K to °C), respectively. RH was calculated from the ambient and dew point temperature parameters in a relationship derived from the August–Roche–Magnus approximation61 rearranged as:$${rm{RH}}=100 % times frac{{{rm{e}}}^{left(frac{a{T}_{{rm{d}}}}{b+{T}_{{rm{d}}}}right)}}{{{rm{e}}}^{left(frac{{aT}}{b+T}right)}}$$where a is 17.625 (constant), b is 243.04 (constant), T is the ERA5-Land parameter ‘2 metre temperature’ converted from K to °C, and Td is the ERA5-Land parameter ‘2 metre dewpoint temperature’ converted from K to °C.Spot validation of the climate parameters and the mapped output was performed manually in Google Earth Engine across several timesteps in 2016 in Ames, Iowa (using the Iowa Environmental Mesonet AMES-8-WSW station62) and showed insignificant error (99.99 to 95.80% for thermal absorbers, depending on the level of angular selectivity63.Rearranged, Kim’s model yields$$frac{{dot{V}}_{{rm{water}},{rm{out}}}}{A}le {E}_{{rm{GHI}}}times left(1-frac{{T}_{{rm{ambient}}}}{{T}_{{rm{hot}}}}right)times {left[frac{1}{{omega }_{{rm{air}},{rm{in}}}-{omega }_{{rm{air}},{rm{out}}}}({e}_{{rm{air}},{rm{out}}}-{e}_{{rm{air}},{rm{in}}})+{e}_{{rm{water}},{rm{out}}}right]}^{-1}times frac{1}{{rho }_{{rm{water}}}}$$where, in addition, ({dot{V}}_{{rm{water}},{rm{out}}}) is the production rate of liquid water by volume, ({A}) is the area harvesting sunlight (see approximation section below), ({E}_{{rm{GHI}}}) is GHI in Wsun m−2, and ({rho }_{{rm{water}}}) is the density of water.This is now a function of the three key climate variables: GHI (in the first term), ambient temperature (in the second and hidden in the third term) and RH (entering the third term). This was converted to an output table and processed through the AWH-Geo pipeline and presented in Fig. 3a. While this can be run for any choice of parameter ({T}_{{rm{hot}}}), we present figures here for ({T}_{{rm{hot}}}) = 100 °C, a temperature still achievable in low-cost (non-vacuum) practical devices without tracking or sunlight concentration. Higher driving temperatures increase the upper bound for water output. For the limits analysis, values of RH above 90% are clamped to prevent unrealistically high theoretical outputs as Kim’s equation goes to infinity at 100% RH. A further assumption is made that new ambient air is efficiently refreshed.Figure 3b maps the maximum yield for active cooler–condensers without recuperation of sensible heat—all given work input and an optimum coefficient of performance of the cooling unit at a condenser temperature that maximizes specific yield as modelled by Peeters32, which we digitized from their fig. 11. Peeters chose to set yield to zero whenever frost formation would be expected on the condenser. Since Peeters assumes work input, we convert from solar energy (GHI) to kWhPV as discussed above.Figure 3c maps Zhao’s experimental results from a TRP using a logistic regression curve fit to their reported SYs of 0.21, 3.71 and 9.28 l kWh−1 at 30, 60 and 90% RH, respectively15. The terms of the curve fit are reported in the next section.Custom yellow to blue map colours are based on www.ColorBrewer.org, by C. A. Brewer, Penn State64.Specific yield and target curvesTwo simple characteristic equations, linear and logistic, were used to fit a limited set of SY and RH pairs from laboratory experiments or reported values and plotted through AWH-Geo using calculated output tables. Hypothetical curves of similar form whose terms were adjusted iteratively in AWH-Geo to goal-seek a target output (5 l d−1) and user base, and are reported here (for 1-m2 devices). In the following equations, RH in % is taken as a fraction (for example 55% is equivalent to 0.55).The linear target curve is a simple linear function which crosses the y-axis at zero:$${rm{SY}}({rm{RH}})=atimes {rm{RH}}$$where a is set to 1.60, 1.86 and 2.60 L/kWh to reach targets of 0.5, 1.0, and 2.0 billion people without SMDW, respectively, and RH is input RH (fractional).The logistic target curve is a logistic function:$${rm{SY}}({rm{RH}})=frac{L}{1+{{rm{e}}}^{-k({rm{RH}}-{{rm{RH}}}_{0})}}$$where L is set to 1.80, 2.40 and 4.80 L kWh−1 to reach targets of 0.5, 1.0 and 2.0 billion people without SMDW, respectively, k is the growth rate set to 10.0, and ({rm{RH}}) and ({{rm{RH}}}_{0}) are input RH (fractional), and 0.60, respectively.The SY values reported by Zhao for TRPs (which they term ‘SMAG’) were fit to a logistic function of the same form with the following parameters: L set to 9.81 L kWh−1, k set to 11.25 and RH0 set to 0.645.The resulting fitted SY profile is expanded into an output table. As with all reports providing SY values instead of full output tables, this forces an assumption of linearity in heat rate (approximately equal to GHI), which may introduce error at lower GHI levels. Zhao reports SY of the TRP material is consistent across temperature below 40 °C—the material’s lower critical solution temperature—above which its performance drops precipitously. Accordingly, we set the SY to 0 l kWh−1 for temperatures ≥40 °C in the output table.Bagheri reported performance of three existing AWH devices across several climate conditions using an ‘energy consumption rate’ in kWh/L, which can be considered to be the SEC, and the simple reciprocal of SY. Instead of fitting a logistic curve to the reciprocals, we fit an exponential function to the average SEC of the three devices in conditions above 20 °C of the equation:$${rm{SEC}}({rm{RH}})=9.03{{rm{e}}}^{-2.99{rm{RH}}}$$where SEC is specific energy consumption in kWhPV l−1 and RH is fractional.This was applied to RH and taken as reciprocal in an output table and run through AWH-Geo. Since Bagheri reports the equivalent of kWhPV, we scale to adapt to GHI input with a photovoltaic conversion efficiency as discussed above.For performance of the ZMW device (the company’s ~3 m2 SOURCE Hydropanel), we used values from the panel production contour plot in the technical specification sheet available from the manufacturer’s website35. The decision for inclusion was made owing to the importance as an early example of a SC-AWH product with commercial intent. Values in l per panel per day were taken at each 10% RH step at 5 kWh m−2, assumed to represent kWh m−2 d−1, and divided by 15 kWh (~3 m2 × 5 kWh m−2) to convert to SY in l kWh−1. From the resulting SY curve, an output table was generated and processed with AWH-Geo.Coincidence analysis and population sumsThe coincidence analysis was run through AWH-Geo across 70 threshold pairs given the full permutation set of RH from 10 to 100% and GHI from 400 to 700 W m−2 threshold intervals, using binary image time series. The resulting mean multiplied by 24 represents average hours per day thresholds are met simultaneously, giving ophd. Below is a functional representation of this time-series calculation:$${langle ({{rm{RH}}}_{t,{rm{px}}} > {{rm{RH}}}_{{rm{threshold}}}){{rm{& & }}}_{{rm{simultaneous}}}({{rm{GHI}}}_{t,{rm{px}}} > {{rm{GHI}}}_{{rm{threshold}}})rangle }_{{rm{time; average}}}$$where ({{rm{RH}}}_{t,{rm{px}}}) is the RH in the map pixel ({rm{px}}) at time (t), ({{rm{RH}}}_{{rm{threshold}}}) is the threshold of RH above which the device is assumed to operate, ({{rm{GHI}}}_{t,{rm{px}}}) is the GHI in the map pixel ({rm{px}}) at time (t), and ({{rm{GHI}}}_{{rm{threshold}}}) is the threshold of GHI above which the device is assumed to operate.The population calculation was then conducted on these images in Google Earth Engine.Zonal statistics were performed on the mean ophd images as integers (0–24) using a grouped image reduction (at 1,000-m scale) summing the population integer counts on the population without SMDW distribution image created previously (derived from WorldPop). This reduction was performed at 1,000 m. Validation was performed in Google Earth Engine on single countries within single ophd zones and showed insignificant error ( More

General description of a geoscientific model and parameter calibrationA model for both non-dynamical and dynamical systems can be generically written for site i as$${left{{{{{{{bf{y}}}}}}}_{t}^{i}right}}_{tin T}=fleft({left{{{{{{{bf{x}}}}}}}_{t}^{i}right}}_{tin T},{{{{{{boldsymbol{varphi }}}}}}}^{{{{{{boldsymbol{i}}}}}}},{{{{{{boldsymbol{theta }}}}}}}^{{{{{{boldsymbol{i}}}}}}}right)$$
(1)
where output physical predictions (({{{{{{bf{y}}}}}}}_{t}^{i}={left[{y}_{1,t}^{i},{y}_{2,t}^{i},cdots right]}^{T}), with the first subscript denoting variable type) vary with time (t) and location (i), and are functions of time- and location-specific inputs (({{{{{{bf{x}}}}}}}_{t}^{i}={left[{x}_{1,t}^{i},{x}_{2,t}^{i},cdots right]}^{T})), location-specific observable attributes (({{{{{{boldsymbol{varphi }}}}}}}^{i}={left[{varphi }_{1}^{i}{{{{{boldsymbol{,}}}}}}{varphi }_{2}^{i}{{{{{boldsymbol{,}}}}}}{{{{{boldsymbol{cdots }}}}}}right]}^{T})), and location-specific unobserved parameters that need to be separately determined (({{{{{{boldsymbol{theta }}}}}}}^{i}={left[{theta }_{1}^{i}{{{{{boldsymbol{,}}}}}}{theta }_{2}^{i}{{{{{boldsymbol{,}}}}}}{{{{{boldsymbol{cdots }}}}}}right]}^{T})). θ may be unobservable, or it may be too expensive or difficult to observe at the needed accuracy, resolution, or coverage. This formulation also applies to dynamical systems if ({{{{{{bf{x}}}}}}}_{{{{{{boldsymbol{t}}}}}}}^{{{{{{boldsymbol{i}}}}}}}) includes previous system states ({{{{{{bf{y}}}}}}}_{t-1}^{i}) (i.e. ({{{{{{bf{y}}}}}}}_{t-1}^{i}{{{{{boldsymbol{subset }}}}}}{{{{{{bf{x}}}}}}}_{t}^{i})), and the rest of the inputs are independent (e.g. meteorological) forcing data. In a non-dynamical system, ({{{{{{bf{x}}}}}}}_{t}^{i}) is independent of ({{{{{{bf{y}}}}}}}_{t-1}^{i}).Given some observations$${{{{{{bf{z}}}}}}}_{t}^{i}=hleft({{{{{{bf{y}}}}}}}_{{t}}^{{i}}right)+{{{{{{boldsymbol{varepsilon }}}}}}}_{t}^{i}$$
(2)
where h(·) relates model outputs to observations and ({{{{{{boldsymbol{varepsilon }}}}}}}_{{{boldsymbol{t}}}}^{{{boldsymbol{i}}}}={big[{varepsilon }_{1,t}^{i},{varepsilon }_{2,t}^{i},cdots big]}^{T}) is the error between the observations (Big({{{{{{boldsymbol{z}}}}}}}_{{{{{{boldsymbol{t}}}}}}}^{{{{{{boldsymbol{i}}}}}}} = {big[{z}_{1,t}^{i},{z}_{2,t}^{i},cdots big]}^{T}Big)) and the model predictions (({{{{{{bf{y}}}}}}}_{{{{{{boldsymbol{t}}}}}}}^{{{{{{boldsymbol{i}}}}}}})), we adjust the model parameters so that the predictions best match the observations. This is traditionally done individually for each location (here generically referring to a gridcell, basin, site, river reach, agricultural plot, etc., depending on the model):$${hat{theta }}^{i}={{arg }},{{{min }}}_{{{{{{{boldsymbol{theta }}}}}}}^{i}}mathop{sum }limits_{tin T}{Vert {{{{{{boldsymbol{varepsilon }}}}}}}_{t}^{i}Vert }^{2}={{arg }},{{{min }}}_{{{{{{{boldsymbol{theta }}}}}}}^{i}}mathop{sum}limits_{tin T}{Vert h(f({{{x}_{t}^{i}}}_{tin T},{varphi }^{i},{theta }^{i}))-{z}_{t}^{i}Vert }^{2}$$
(3)
where i ∈ I and where (I=left{{1,2},ldots ,{N}_{I}right}). Note that the superscript i suggests that this optimization is done for each site independently.The process-based hydrologic model and its surrogateThe Variable Infiltration Capacity (VIC) hydrologic model has been widely used for simulating the water and energy exchanges between the land surface and atmosphere, along with related applications in climate, water resources (e.g., flood, drought, hydropower), agriculture, and others. The model simulates evapotranspiration, runoff, soil moisture, and baseflow based on conceptualized bucket formulations. Inputs to the model include daily meteorological forcings, non-meteorological data, and the parameters to be determined. Meteorological forcing data include time series of precipitation, air temperature, wind speed, atmospheric pressure, vapor pressure, and longwave and shortwave radiation. More details about VIC can be found in Liang et al.39.LSTM was trained to reproduce the behavior of VIC as closely as possible while also allowing for gradient tracking. In theory, if a hydrologic model can be written into a machine learning platform (as in our HBV case), this step is not needed, but training a surrogate model is more convenient when the model is complex. To ensure the surrogate model had high fidelity in the parameter space where the search algorithms want to explore, we iterated the training procedure multiple times. We first trained an LSTM surrogate for VIC using the forcings, attributes, and parameters from NLDAS-2 as inputs, and the VIC-simulated surface soil moisture (variable name: SOILM_lev1) and evapotranspiration (ET, variable name: EVP) as the targets of emulation. Then, as the search algorithms (SCE-UA or dPL) went near an optimum, we took the calibrated parameter sets, made perturbations of them by adding random noise to these parameters, and retrained the network with added data. The perturbation was done to better represent the parameter space close to optimal solutions. We repeated this procedure four times so that the NSEs of the parameters, obtained from the CPU-based VIC model, converged. At 1/82 sampling density (sampling one gridcell from each 8 × 8 patch), this results in fewer overall forward runs than a 1/8-degree NLDAS-2 simulation. Also note that this effort is needed similarly for both dPL and SCE-UA. If we did not use the surrogate model, SCE-UA would also have needed to employ the O(102) more expensive CPU-based VIC model. We evaluated the accuracy of the surrogate model, and the median correlations between VIC and the surrogate simulation were 0.91 and 0.92 for soil moisture and ET, respectively (Supplementary Fig. S2). When we connected the trained surrogate model to the parameter estimation network, the weights of the surrogate model were frozen and prevented from updating by backpropagation, but the gradient information could pass through. This was implemented in the PyTorch deep learning framework36.The long short-term memory networkThe long short-term memory network (LSTM) was originally developed in the artificial intelligence field for learning sequential data, but has recently become a popular choice for hydrologic time series data26. As compared to a vanilla recurrent neural network with only one state, LSTM has two states (cell state, hidden state) and three gates (input gate, forget gate, and output gate). The cell state enables long-term memory, and the gates are trained to determine which information to carry across time steps and which information to forget. These units were collectively designed to address the notorious DL issue of the vanishing gradient, where the accumulated gradients would decrease exponentially along time steps and eventually be too small to allow effective learning48. Given inputs I, our LSTM can be written as the following:$${{{{{mathrm{Input}}}}}}; {{{{{mathrm{transformation:}}}}}}quad {x}^{t}={{ReLU}}({W}_{I}{I}^{t}+{b}_{I})$$
(4)
$${{{{{mathrm{Input}}}}}}; {{{{{mathrm{node:}}}}}}quad {g}^{t}={tanh }({{{{{mathscr{D}}}}}}({W}_{{gx}}{x}^{t})+{{{{{mathscr{D}}}}}}({W}_{{gh}}{h}^{t-1})+{b}_{g})$$
(5)
$${{{{{mathrm{Input}}}}}}; {{{{{mathrm{gate:}}}}}}quad {i}^{t}=sigma ({{{{{mathscr{D}}}}}}({W}_{{ix}}{x}^{t})+{{{{{mathscr{D}}}}}}({W}_{{ih}}{h}^{t-1})+{b}_{i})$$
(6)
$${{{{{mathrm{Forget}}}}}}; {{{{{mathrm{gate:}}}}}}quad {f}^{t}=sigma ({{{{{mathscr{D}}}}}}({W}_{{fx}}{x}^{t})+{{{{{mathscr{D}}}}}}({W}_{{fh}}{h}^{t-1})+{b}_{f})$$
(7)
$${{{{{mathrm{Output}}}}}}; {{{{{mathrm{gate:}}}}}}quad {o}^{t}=sigma ({{{{{mathscr{D}}}}}}({W}_{{ox}}{x}^{t})+{{{{{mathscr{D}}}}}}({W}_{{oh}}{h}^{t-1})+{b}_{o})$$
(8)
$${{{{{mathrm{Cell}}}}}}; {{{{{mathrm{state:}}}}}}quad {s}^{t}={g}^{t}odot {i}^{t}+{s}^{t-1}odot {f}^{t}$$
(9)
$${{{{{mathrm{Hidden}}}}}}; {{{{{mathrm{state:}}}}}}quad {h}^{t}={tanh }({s}^{t})odot {o}^{t}$$
(10)
$${{{{{mathrm{Output:}}}}}}quad {y}^{t}={W}_{{hy}}{h}^{t}+{b}_{y}$$
(11)
where W and b are the network weights and bias parameters, respectively, and ({{{{{mathscr{D}}}}}}) is the dropout operator, which randomly sets some of the connections to zero. The LSTM network and our whole workflow31 were implemented in PyTorch36, an open source machine learning framework.Here we do not use LSTM to predict the target variable. Rather, LSTM is used to (optionally) map from time series information to the parameters in our gz network as described below.The parameter estimation networkWe present two versions of the dPL framework. The first version allows us to train a parameter estimation network over selected training locations Itrain where some ancillary information A (potentially including but not limited to attributes in φi used in the model) is available, for training period Ttrain (illustrated in Fig. 1b):$${hat{{{{{{boldsymbol{theta }}}}}}}}^{i}={g}_{A}left({{{{{{bf{A}}}}}}}^{i}right){{{{{{rm{for}}}}}}; {{{{{rm{all}}}}}}; iin I}_{{{{{{{mathrm{train}}}}}}}}$$
(12a)
$${hat{g}}_{A}(cdot )={{{{{rm{arg }}}}}},{{{{{{rm{min }}}}}}}_{{g}_{A}(cdot )}mathop{sum}limits_{tin T,iin {I}_{{{{{{rm{train}}}}}}}}{Vert h(f({x}_{t}^{i},{varphi }^{i},{g}_{A}({{{{{{bf{A}}}}}}}^{i})))-{z}_{t}^{i}Vert }^{2}$$
(12b)
Essentially, we train a network (gA) mapping from raw data (A) to parameters (({{{{{boldsymbol{theta }}}}}})) such that the PBM output (f) using ({{{{{boldsymbol{theta }}}}}}) best matches the observed target (({{{{{bf{z}}}}}})). We are not training to predict the observations – rather, we train gA on how to best help the PBM to achieve its goal. The difference between Eq. 12 and Eq. 3 highlights that the loss function combines the sum of squared differences for all sites at once.The second version is applicable where some observations ({left{{{{{{{bf{z}}}}}}}_{t}^{i}right}}_{tin T}) are also available as inputs at the test locations:$${hat{{{{{{boldsymbol{theta }}}}}}}}^{i}={g}_{z}left({left{{{{{{{bf{x}}}}}}}_{t}^{i}right}}_{tin T},{{{{{{bf{A}}}}}}}^{{prime} ,i},{left{{{{{{{bf{z}}}}}}}_{t}^{i}right}}_{tin T}right){{{{{rm{for}}}}}}{,,}{{{{{rm{all}}}}}}{,,}{iin I}_{{{{{{{mathrm{train}}}}}}}}$$
(13a)
$$widehat{{g}_{z}}(cdot )={{{{{rm{arg }}}}}},{{{{{{rm{min }}}}}}}_{{g}_{z}(cdot )}mathop{sum }limits_{tin {T}_{{{{{{rm{train}}}}}}},iin {I}_{{{{{{rm{train}}}}}}}}{Vert h(f({x}_{t}^{i},{varphi }^{i},{g}_{z}({{{{{{{{bf{x}}}}}}}_{t}^{i}}}_{tin T},{{{{{{bf{A}}}}}}}^{{prime} ,i},{{{z}_{t}^{i}}}_{tin T})))-{z}_{t}^{i}Vert }^{2}$$
(13b)
Essentially, we train a network (({g}_{z})) that maps from attributes (A′), historical forcings (x), and historical observations (({left{{{{{{{bf{z}}}}}}}_{t}^{i}right}}_{tin T})) to a suitable parameter set (({{{{{boldsymbol{theta }}}}}})) with which the PBM output best matches the observed target (({{{{{bf{z}}}}}})) across all sites in the domain. Ancillary attributes A′ may be the same as or different from A used in gA, and in the extreme case may be empty. Succinctly, they can be written as two mappings, gA: A → θ and gZ: (A′,x,z) → θ. gZ can accept time series data as inputs and here we choose LSTM as the network structure for this unit. There is no circular logic or information leak because the historical observations (({left{{{{{{{bf{z}}}}}}}_{t}^{i}right}}_{tin T})) are for a different period (T) than the main training period (Ttrain). In practice, this distinction may not be so crucial as the PBM acts as an information barrier such that only values suitable as parameters (({{{{{boldsymbol{theta }}}}}})) can produce a reasonable loss. As LSTM can output a time series, the parameters were extracted only at the last time step. For gA, only static attributes were employed, and so the network structure amounts to a multilayer perceptron network. After some investigation of training and test metrics, we set the hidden size of g to be the same as for the surrogate model.The whole network is trained using gradient descent, which is a first-order optimization scheme. Some second-order schemes like Levenberg–Marquardt often have large computational demand and are thus rarely used in modern DL49. To allow gradient accumulation and efficient gradient-based optimization and to further reduce the computational cost, we can either implement the PBM directly into a differentiable form, as described in the global PUB case below, or first train a DL-based surrogate model (f^{prime} left(bullet right)simeq fleft(bullet right)) and use it in the loss function instead of f(·),$$g(cdot )={{{{{rm{arg }}}}}},{{{{{{rm{min }}}}}}}_{g(cdot )}mathop{sum}limits_{tin {T}_{{{{{{rm{train}}}}}}},iin {I}_{{{{{{rm{train}}}}}}}}{Vert h(f{{mbox{‘}}}({x}_{t}^{i},{varphi }^{i},g(cdot )))-{z}_{t}^{i}Vert }^{2}$$
(14)
where (g(bullet )) generically refers to either gA or gZ with their corresponding inputs. gA can be applied wherever we can have the ancillary inputs A, while gZ can be applied over areas where forcings and observed responses (x, z) are also available, without additional training:$${hat{{{{{{boldsymbol{theta }}}}}}}}^{i}={g}_{z}({{{{{{{{bf{X}}}}}}}_{t}^{i}}}_{tin T}{{{{{boldsymbol{,}}}}}}{{{{{{boldsymbol{varphi }}}}}}}^{i},{{{{{{{{bf{Z}}}}}}}_{t}^{i}}}_{tin T}){,,}{{{{{rm{or}}}}}},,{hat{{{{{{boldsymbol{theta }}}}}}}}^{i}={g}_{A}({{{{{{boldsymbol{varphi }}}}}}}^{i}); {{{{{rm{for}}}}}},{{{{{rm{any}}}}}}, i,{{{{{rm{and}}}}}},{{{{{rm{any}}}}}},{{{{{rm{reasonable}}}}}},T$$
(15)
We tested both gA and gZ, which work with and without forcing-observation (x-z) pairs among the inputs, respectively. Since SMAP observations have an irregular revisit schedule of 2–3 days and neural networks cannot accept NaN inputs, we have to fill in the gaps, but simple interpolations do not consider the effects of rainfall. Here we used the near-real-time forecast method that we developed earlier30. Essentially, this forecast method uses forcings and integrates recently available observations to forecast the observed variable for the future time steps, achieving very high forecast accuracy (ubRMSE  More

NATURE PODCAST
03 October 2021

Audio long-read: How dangerous is Africa’s explosive Lake Kivu?

A lake in central Africa could one day release a huge amount of greenhouse gases, threatening the lives of millions.

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Nicola Jones is a science journalist based in Pemberton, Canada.

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Lake Kivu, nestled between the Democratic Republic of the Congo and Rwanda, is a geological anomaly that holds 300 cubic kilometres of dissolved carbon dioxide and 60 cubic kilometres of methane.The lake has the potential to explosively release these gases, which could fill the surrounding valley, potentially killing millions of people.Researchers are trying to establish the likelihood of such an event happening, and the best way to safely siphon the gases from the lake.This is an audio version of our feature: How dangerous is Africa’s explosive Lake Kivu?Never miss an episode: Subscribe to the Nature Podcast on Apple Podcasts, Google Podcasts, Spotify or your favourite podcast app. Head here for the Nature Podcast RSS feed

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