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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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How dangerous is Africa’s explosive Lake Kivu? Read by Benjamin Thompson
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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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Structure of the modeling frameworkThe coevolutionary macroeconomy and river system simulation framework introduced in this study consists of two modeling components: (a) the Egyptian economy and (b) the Nile river system. The modeling framework accounts for the coevolutionary dynamics of river and economic systems using an iterative process. This multi-sector framework is designed for river systems with multiyear storage dams and a mix of hydro and non-hydro electricity generation. The two modeling components are described separately below, followed by a description of their interaction, which characterizes two-way hydro-economic feedbacks. The application of the coevolutionary framework to the Nile is then discussed.Economy-wide modeling componentThe Egyptian economy-wide modeling component is based on the IFPRI (International Food Policy Research Institute) standard open-source CGE model43. The model was modified to include water, energy, and land components and run dynamically (i.e., for a multiyear period). In previous studies, water, energy, and land resources have been included in the productive activities of CGE models in a variety of ways. A recent review of the literature distinguished between CGE models that treat water as an explicit factor of production, those that include water as an implicit factor of production (i.e., embedded in land productivity), and those that treat water as a commodity (i.e., an intermediate input)58. Energy-oriented CGE models typically combine energy with capital in the production structure of goods and services59,60. The inclusion of energy in CGE models is straightforward compared to water because energy is a marketed commodity that can be easily reallocated to different sectors. The reallocation of water supplies across space and time requires storage and network infrastructure and is often constrained by unpredictable supplies (stochastic hydrology). Moreover, raw water supplies are typically unpriced61,62,63,64; thus, the economic value of water is not included in economic data (e.g., social accounting matrices and input–output tables).In this study, we modified IFPRI’s standard CGE model such that economic activities produce commodities using a three-level process (Supplementary Fig. 5). At the top level, composite intermediate inputs and the value-added-energy bundle are combined to produce commodities using a Leontief Function65. The function maintains fixed proportions of inputs (composite intermediate inputs and value-added energy in this case) for each unit of output (commodity). At the second level, energy and value-added are aggregated using a Constant Elasticity of Substitution function (CES)66, such that the optimal input quantities of value-added and energy for each activity are determined based on relative prices subject to substitution elasticity similar to energy-oriented CGE models59. At the third level, substitution is allowed between the electricity commodity and other energy commodities using a CES function. A CES function is also used to combine labor, capital, and land into value-added.The model is customized to allow each household group to allocate its consumption budget to the purchase of commodities based on a nested linear expenditure system (LES)67 and CES (Supplementary Fig. 5). At the top level, a LES function is used to divide the consumption budget between essential and nonessential demands68. The nonessential consumption budget is divided between five commodity categories using fixed shares. Each category includes different commodities that can substitute each other based on CES functions.We modified the IFPRI CGE model to include four types of capital: (a) hydro capital used by a hydropower activity to produce electricity, (b) non-hydro capital used by a non-hydro activity to produce electricity, (c) water capital used by a municipal water activity to produce municipal water, and (d) general capital used by other activities. The use of land and water capital varies endogenously based on their rents. Logistic functions are used to simulate the response of the use of land and water capital to their rents. General and non-hydro capital grow based on past investments. Investment is allocated between these two capital types according to their relative rates of return. Given the increase in electricity demand resulting from economic growth, this specification of investment behavior allows for an incremental expansion of non-hydro electricity generation capacity; hydropower capacity does not grow endogenously with the year-to-year investment allocation. It is assumed that no new hydropower investments are made over the 30-year simulation period. To connect the economy-wide model with the river system model, dynamic exogenous shocks on land, water capital, hydro capital, and non-hydro capital are introduced to the economic model based on the river system modeling component, which simulates water and electricity availability, as explained below.River system modeling componentPywr, a generic open-source Python library for simulating resource system networks42, is used to model the water system, including hydropower generation, in addition to an aggregated representation of non-hydro electricity generation. Pywr allows building resource system elements using input (e.g., catchments), output (e.g., water abstraction), and storage nodes (e.g., reservoirs). Nodes are linked in a network fashion to enable the flow and allocation of resources such as water and electricity. Pywr uses a time-stepping linear programming approach to drive resource allocation using priorities and system operating rules. Any time step resolution can be selected for Pywr simulations (e.g., hourly, daily, weekly, and monthly). Pywr’s multi-scenario simulation allows consideration of uncertainty in resource systems, e.g., hydrologic stochasticity.The simulation results of Pywr can be processed, observed, and/or saved through “recorders.” We extended Pywr “recorders” to enable annual aggregation of the water and electricity metrics required for integration with the economy-wide modeling component. These metrics include annual irrigation and municipal water supply fractions, annual electricity generation from hydropower dams, and annual electricity generation from non-hydro energy generators.Coevolutionary macroeconomy and river system simulationSupplementary Fig. 6 shows a flowchart of the novel coevolutionary macroeconomy and river system generalized hydro-economic69 modeling framework. The figure shows the interaction between the economy-wide modeling component (with an annual time step) and the river system modeling component (with a monthly time step) within each annual time step. Dynamic-recursive multiyear simulations are performed by repeating the procedure in Supplementary Fig. 6 multiple times.The dynamic behavior of CGE models is typically driven by external drivers, such as capital growth (determined as a function of previous investment levels), labor growth, and productivity trends. In the first iteration, the CGE model solves based on its external drivers and determines changes to annual water and electricity demands and non-hydro electricity generation capacity relative to the economy’s initial year. Changes produced by the CGE model in relation to the irrigated area, the water capital, the demand for the electricity commodity, and the non-hydro capital are used as an estimate in the river system model for changes in irrigation water demand, municipal water demand, electricity demand, and non-hydro electricity generation capacity, respectively. The first CGE iteration assumes no irrigation deficit and electricity generation equal to that of the base year. The CGE and the river system models iteratively correct the initial assumptions of water curtailments and electricity generation, as explained in more detail below.CGE models typically have an annual time step, but river system models run at smaller time intervals (e.g., monthly, weekly, daily, hourly). River system models have finer temporal resolutions to enable simulation of the spatial and temporal constraints of river basin resource systems, i.e., to better capture the consequences of stochastic hydrology and infrastructure constraints (e.g., reservoir storage). Although the iterative framework presented in Supplementary Fig. 6 is based on a monthly river system model, models with smaller time steps could also be used. The river system model uses the changes in irrigation water demand, municipal water demand, electricity demand, and non-hydro electricity generation capacity, computed by the economy-wide modeling component, to scale the corresponding water and electricity parameters. The river system model then performs a monthly simulation over a 12-month period based on monthly river flow data and the modified water and electricity demands and non-hydro capacity. The river system model then computes the fractions of annual irrigation and municipal water demands that can be met in addition to the annual hydro and non-hydro electricity generation. Water supply and electricity generation depend on the spatial and temporal availability of natural resources (e.g., river flow), infrastructure capacities (e.g., non-hydro and hydro capacities), and infrastructure operating rules.After the river system modeling component determines water supply fractions and electricity generation, two tests are performed to determine (a) whether the models have converged and (b) when to stop iterating. These tests indicate whether to proceed to the next iteration or accept the current state of the CGE and the river system models as a solution for the annual time step. Passing either of the two tests terminates the iterative convergence process. The CGE and the river system models pass the convergence test when the difference between the current and the previous iteration’s values of an annual economy, water, or energy metric falls below a certain convergence threshold. The value of the threshold depends on the desired level of accuracy and available computational capacity. The stopping test imposes a maximum number of iterations at which the current state of the CGE and the river system models is considered a solution for the annual time step. The stopping test acts as a safeguard to prevent excessively long iteration over one annual time step. The convergence test is performed starting from the second iteration. Thus, at least two iterations are performed within each annual time step to ensure convergence.Failure in the convergence and stopping tests results in proceeding to the next iteration. In the next iteration, annual water supply fractions and electricity generation of the previous iteration are applied to the CGE model to compute new changes to annual water and electricity demands and non-hydro electricity generation capacity relative to the initial year of the economy (i.e., the base year). The irrigation and municipal water supply fractions, computed by the river system modeling component, are introduced to the CGE model as shocks to the land and water capital, respectively. The ratio between current electricity generation and electricity generation in the initial year of the economy is calculated for each of the two electricity generation technology groups (i.e., hydro and non-hydro) and introduced as shocks to the hydro and non-hydro capitals.Implementation of the coevolutionary frameworkThe open-source Python Network Simulation (Pynsim) framework44 was extended and used to integrate the economy-wide and river system modeling components and to manage their iteration, sequencing, and time stepping. Each of the two components was specified as a Pynsim “engine”44. Although the IFPRI CGE model is written in the General Algebraic Modeling System (GAMS)70, it was linked to Pynsim through the GAMS Python Application Programming Interface. Eight Pynsim integration nodes were created for data exchange between the economy-wide and river system modeling components. Four of the integration nodes transfer changes in annual water (irrigation and municipal) and electricity demands and non-hydro electricity generation capacity from the economy-wide to the river system modeling components. The other four integration nodes transfer the annual water (irrigation and municipal) supply fractions and hydro and non-hydro electricity generation from the river system to the economy-wide modeling components.Eastern Nile River system modelSupplementary Fig. 7 shows a schematic of the monthly river system model of the Eastern Nile Basin. The model uses naturalized inflow data for the period 1901–2002, obtained from the Eastern Nile Technical Regional Office54. The Eastern Nile River System model contains all major dams and water consumers in the basin, including the GERD and the HAD. The baseline water withdrawal targets are shown in Supplementary Fig. 8. Supplementary Table 1 reports the main characteristics of the dams included in the Nile River System model. The model was calibrated and validated at eight locations across the basin based on historically observed river flows and reservoir water levels over 1970–2002. This period was chosen based on the availability of observed data. Supplementary Fig. 9 and Supplementary Table 2 show the performance of the Eastern Nile River system at eight locations. In the model, non-hydro electricity generation is used to fill the gap between hydropower generation and electricity demand, subject to generation capacity. This assumption is valid since hydropower in Egypt is a by-product of other activities. Furthermore, the historical evolution of the Egyptian electricity mix shows relatively regular annual hydropower generation with a steady increase in electricity generation from other technologies to fill the supply-demand gap8.Initial filling assumptions of the Washington draft proposalSupplementary Table 3 describes the 5-year plan for the initial filling of the GERD in the Washington draft proposal assuming normal or above-average hydrological conditions. We assumed that after achieving the water retention target of the first year (4.9 bcm), two 375 MW turbines become operational. The rest of the turbines become operational after achieving the second year’s water retention target (18.4 bcm). We assumed that once the filling targets of year-1 or year-2 are achieved, reservoir storage is always maintained above these targets in order to keep the turbines operational. In the Washington draft proposal, water retention is limited to July and August, with a minimum environmental release of 43 Mm3/day. During the initial filling period, from September to June, releases from the GERD equal the inflow to the reservoir. However, if a drought occurs during the 5-year initial filling plan specified in Table S3, the Washington draft proposal has provisions for implementing delays in filling the GERD (our assumptions for these provisions are described in a later section).Long-term operation assumptions of the Washington draft proposalThe Washington draft proposal’s operating rules for the long-term operation of the GERD begins when reservoir storage reaches 49.3 bcm. We assumed that when reservoir storage is at or above 49.3 bcm, water is released through the GERD’s turbines to maintain a constant monthly energy production of 1170 GWh to maximize the 90% power generation reliability71. If reservoir storage drops below 49.3 bcm, the target monthly energy production is reduced to 585 GWh. The purpose of reducing the energy generation target is to enable the GERD storage to recover above 49.3 bcm. Water releases designed to maintain a regular power rate depend on the reservoir water level at the beginning of the time step (the higher the water level, the lower the releases required). A minimum environmental release of 43 Mm3/day is maintained throughout the year when possible. Additional water releases may be made following drought mitigation mechanisms that resemble those of the Washington draft proposal, as described below.Drought mitigation assumptions of the Washington draft proposalThe Washington plan includes three mechanisms to mitigate the adverse effects of droughts, prolonged droughts, and prolonged periods of dry years on the downstream riparians46. The mechanism for mitigating droughts is triggered when the GERD’s annual inflow is forecast to be ≤37 bcm. This first mechanism requires Ethiopia to release a minimum annual water volume, depending on the forecast annual inflow and GERD storage at the beginning of the hydrologic year (see Exhibit A in Egypt’s letter to the United Nations Security Council dated 19 June 202046).The effectiveness of the mechanism for mitigating droughts depends on the accuracy of the forecast of the annual inflow for the upcoming hydrological year. To implement the Washington plan in this study’s river simulation model, we do not forecast annual flows for the next hydrological year. Instead, drought mitigation conditions are checked in March of every hydrologic year, by which time, on average, about 96% of the river’s annual flow is already known because it occurs from June to February. If necessary, water releases during the remaining 3 months of the hydrological year (March–May) are increased to achieve the minimum annual releases specified in the mechanism for mitigating droughts. These increased releases during March–May effectively offset any deviations from water releases specified by the drought mitigation mechanism given the dam inflows and releases in the previous 9 months of the current hydrologic year.The mechanism for mitigating prolonged droughts requires that the average annual release over every 4-year period equal at least 39 bcm (37 bcm during the initial filling). In the implementation of this prolonged drought mitigation mechanism of the Washington draft proposal in our river simulation model, we check in March of every hydrological year to ensure that this annual average release over the previous 4-year period is achieved. Although this mechanism does not depend on reservoir inflow, it is also checked for in March to provide flexibility to GERD operation during the rest of the year.The mechanism for mitigating prolonged periods of dry years is similar to the prolonged drought mitigation mechanism, except the period over which annual releases are averaged is longer (5 years) and the average annual release is higher (40 bcm). We implement this mechanism in our river simulation model in the same way, checking in March of every hydrological year to ensure that the annual average release over the previous 5-year period is achieved. Supplementary Fig. 10 shows the exceedance probability of the annual, 4-year average annual, and 5-year average annual flow of Blue Nile at the location of the GERD over the period 1901–2002. The drought mitigation thresholds of the Washington draft proposal are marked in the figure to show their probability of occurrence in the river flow data.If a deficit from the minimum releases of any of the three mechanisms is identified at the beginning of March, water releases over March–May are increased equally in each month to offset the deficit.Initial filling assumptions of the coordinated operationThe coordinated operating strategy for the initial filling of the GERD is similar to the Washington plan, except for the retention of inflows to meet the targets in Table S3 is not constrained to July and August. The coordinated operation requires that a minimum environmental release of 43 Mm3/day be maintained throughout the year when possible. If physically possible, releases from the GERD are also greater than or equal to (1) Sudan’s monthly water withdrawal targets along the Blue and Main Nile, plus (2) Egypt’s monthly water release target from the HAD if HAD storage is below 50 bcm (156 m a.s.l.). This operating strategy enables Ethiopia to avoid delays in filling the GERD as long as HAD storage is at or above 50 bcm. In simulating coordinated operation, the operations of the Roseires, Sennar, and Merowe dams have been adapted to pass GERD releases intended to benefit Egypt. It was assumed that two of the GERD turbines become operational after achieving the first year’s water retention target, and the rest of the turbines become operational once the second year’s filling target is achieved. After achieving the filling targets of year-1 or year-2, reservoir storage is always maintained above these targets (i.e., 4.9 or 18.4 bcm) to keep the turbines operational.Long-term operation assumptions of the coordinated operationAs with the Washington draft proposal, the long-term operation of the GERD begins as soon as reservoir storage reaches 49.3 bcm. Also the same as the Washington plan, it was assumed that when reservoir storage is at or above 49.3 bcm, water is released through the GERD’s turbines to maintain a constant monthly energy production of 1170 GWh to maximize the 90% power generation reliability71. If reservoir storage drops below 49.3 bcm, the target monthly energy generation is reduced to 585 GWh. A minimum environmental release of 43 Mm3/day is maintained throughout the year when physically possible. The key difference between the Washington draft proposal and coordinated operation is that when physically possible, the coordinated operation ensures that the GERD releases are greater than or equal to Sudan’s water withdrawal targets on the Blue and Main Nile plus Egypt’s target releases from the HAD if HAD storage is below 50 bcm (156 m asl). This provides Ethiopia more flexibility in the operation of the GERD as long as HAD storage is at or above 50 bcm.Drought mitigation assumptions of the coordinated operationThe coordinated operation strategy does not include drought mitigation measures that are based on minimum annual water releases. Instead, a dynamic mechanism is used to help reduce downstream water deficits during periods of water scarcity, as explained in previous sections. Such an approach provides flexibility to Ethiopia in GERD operation and increases the basin-wide and national water, electricity, and economic gains.Economy-wide model of EgyptThe CGE model of Egypt represents a dynamic-recursive, single-country, open-economy, including four agent types: households, enterprises, the government, and the rest of the world. Households are classified into ten groups based on location (urban or rural) and income (five quintiles). The model includes 11 production activities: agriculture, light industry, heavy industry, construction, transport, hydropower, non-hydro, other energy, municipal water supply, public services, and other services. Each of the 11 activities produces a distinct commodity except hydropower and non-hydro, which produce a similar commodity (i.e., electricity). Production activities use six factors of production to produce commodities: labor, land, general capital, water capital, hydro capital, and non-hydro capital. Labor and general capital are assumed to be mobile across production activities, whereas land, water capital, hydro capital, and non-hydro capital are specific to agriculture, municipal water supply, hydropower, and non-hydro, respectively. Labor is updated exogenously to follow the projected changes in the 16–64 age group of the shared socioeconomic pathways (SSPs) “middle of the road” scenario72. Total factor productivity is also updated exogenously to follow economic performance under the “middle of the road” scenario.The CGE model of Egypt assumes fixed price of commodities on the international market following the small open-economy assumption, i.e., that the economy participates in international trade but does not affect world prices73. Government spending is simulated as a fixed share of total absorption (total demand for marketed goods and services). The model follows the saving-investment identity (savings are equal to investment) assuming fixed saving propensities. Foreign savings are assumed fixed, and the exchange rate is flexible.The baseline model was calibrated to a 2019 Social Accounting Matrix (SAM) of Egypt. The 2019 SAM was generated based on a 2011 SAM using an expansion factor equal to the ratio between the Egyptian GDP in the 2 years. We compared the generated SAM with the structure of Egypt’s economy based on the most recent data in the World Bank Database; no significant differences were found in the economy’s structure. Supplementary Fig. 11 shows this comparison.Nile River system–Egypt’s economic integrationThe Eastern Nile River system model and the CGE model of Egypt run dynamically over a 30-year simulation period (2020–2049) and multiple scenarios. For each 30-year simulation, the CGE model executes 30 annual time steps, and the river system model executes 360 monthly time steps (30 years × 12 months). The CGE and river system models are integrated through the water and electricity sectors, as described earlier. The convergence test is performed using the GDP at market prices with an assumed convergence threshold of US$ 5 million. A maximum of 50 iterations is specified for each annual time step. All simulated time steps converged in More
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