stoPET v1.0: a stochastic potential evapotranspiration generator for simulation of climate change impacts

. Potential evapotranspiration (PET) represents the evaporative demand in the atmosphere for the removal of water from the land and is an essential variable for understanding and modelling land–atmosphere interactions. Weather generators are often used to generate stochastic rainfall time series; however, no such model exists for the generation of a stochastically plausible PET time series. Here we develop a stochastic PET generator, stoPET, by leveraging a recently published global dataset of hourly PET at 0.1 ◦ resolution (hPET). stoPET is designed to simulate realistic time series of PET that capture the diurnal and seasonal variability in hPET and to support the simulation of various scenarios of climate change. The parsimonious model is based on a sine function ﬁtted to the monthly average diurnal cycle of hPET, producing parameters that are then used to generate any number of synthetic series of randomised hourly PET for a speciﬁc climate scenario at any point of the global land surface between 55 ◦ N and 55 ◦ S. In addition to supporting a stochastic analysis of historical PET, stoPET also incorporates three methods to account for potential future changes in atmospheric evaporative demand to rising global temperature. These include (1) a user-deﬁned percentage increase in annual PET, (2) a step change in PET based on a unit increase in temperature, and (3) the extrapolation of the historical trend in hPET into the future. We evaluated stoPET at a regional scale and at 12 locations spanning arid and humid climatic regions around the globe. stoPET generates PET distributions that are statistically similar to hPET and an independent PET dataset from CRU, thereby capturing their diurnal/seasonal dynamics, indicating that stoPET produces physically plausible diurnal and seasonal PET variability. We provide examples of how stoPET can generate large ensembles of PET for future climate scenario analysis in sectors like agriculture and water resources with minimal computational demand.

. This limits its utility and relevance, particularly for the many data-sparse regions across the globe (Yadeta et al., 2020). The lack of adequate local meteorological data necessitates reliance on empirical methods of PET estimation, which require intensive calibration (Kingston et al., 2009), which can in turn limit the accuracy of the resulting PET product.

5
Temperature is one of the major climate variables influencing PET (Allen et al., 1998). Therefore, with increasing temperature under climate change, there is a need to simulate historical and future PET in a consistent way across the globe, especially in data-sparse regions without good in situ meteorological stations. While global climate models do provide outputs of climatic variables used to estimate PET, we found that none of them directly output PET. Hence, users require a higher computational resource to estimate the PET from these climate model output variables. Another challenge for PET estimation is how to 10 characterise evaporative demand under climate change scenarios, which is an important need for assessing possible future climate change impacts (Xu et al., 2014). Climate models focus on predicting the effects of greenhouse gas emissions on global water and energy transfer, and thus they output climate variables (e.g. temperature, radiation, surface pressure, wind speed, and rainfall) under specific scenarios of climate change at generally coarse spatial and temporal resolutions. These scaling considerations may make climate model output unsuitable for computing PET for application to certain water balance 15 applications, especially where diurnal changes in PET are important for a specific location. Downscaling techniques are commonly used to generate the parameters needed to estimate PET by the PM method at the appropriate resolution from global climate models, but this increases the computational resource requirement (Tukimat et al., 2012) and adds additional uncertainty to the calculations. 20 Given the inherent uncertainty in climatic drivers on the terrestrial water balance and the need to incorporate current and future PET trends in hydrological models, stochastic PET simulation provides a flexible and useful tool to fill this need. While several stochastic weather generators exist and are used to generate physically consistent time series of rainfall (Fatichi et al., 2011;Peleg et al., 2017;Singer et al., 2018;De Luca et al., 2020), no similar model exists for generating stochastic PET time series. This paper addresses this gap and introduces a new stochastic PET generator, stoPET, for simulating hourly time series of PET 25 at 0.1° spatial resolution for the global land surface. stoPET enables the user to characterise the uncertainty in PET for historical and future climate scenarios. It supports the generation of unlimited unique realisations of PET in a computationally efficient way. To support analyses of climate change, stoPET incorporates different methods to account for potential changes in atmospheric evaporative demand in response to rising global temperature, supporting flexibility in simulating various climate scenarios. The importance of including options to simulate multiple future PET time series emanates from the unpredictability 30 of future climate and the need to assess the impacts of climatic changes on the water balance.
Below we provide a comprehensive description of the stoPET model and its potential application for predicting the evolution of water resources in drylands, estimating future crop water demand, assessing flash flood potential, or providing actionable information on expected climatic impacts on the water balance. Section 2 describes the concept and design of the model with 35 a brief note about its implementation. Section 3 describes the model verification at regional and point scale. Section 4 describes the methods used to incorporate temperature changes in the stoPET model. The paper concludes with a discussion of the potential application of stoPET (Sect. 5). A user manual for stoPET is included as a supplement, and all the model scripts and input parameters are freely available on figshare (Sect. 6). https://doi.org /10.5194/gmd-2022-128 Preprint. Discussion started: 11 May 2022 c Author(s) 2022. CC BY 4.0 License. Model concept and design

Concept
The stoPET model generates hourly PET values based on sine function parameters estimated from hPET (Singer et al., 2021), a new hourly PET dataset calculated based on ERA5-Land climatic variables using the Penman-Monteith method (Allen et al., 1998). The resulting PET generated from stoPET retains the diurnal and seasonal variations in PET from the hPET dataset 5 but injects randomness (stochasticity) in the simulated series via a noise factor. The procedure begins using hPET as an input, from which the following steps are taken, each of which is outlined in more detailed in subsequent sections below: 1) Estimate the average diurnal cycle of PET for each month using a sine function 2) Fit a skewed normal distribution to the difference between all hourly values for the diurnal curve and the average diurnal curve, for of each month 10 3) Generate stochastic PET timeseries for a particular month by multiplying that month's average diurnal cycle with a sequence of draws from the corresponding skewed normal distribution

2.2
Model implementation

Sine function parameter estimation
The stoPET model is based on fitting a sine function to the average diurnal cycle calculated from hPET for each month and 15 for each grid cell. The sine function, defined in Eq. (1), provides the four parameters required to represent the characteristic of hourly PET for each month at each grid cell: Where represents the diurnal amplitude (mm h -1 ), is the frequency (h -1 ), is the phase shift (-), is the vertical shift (mm h -1 ). is time (h), and is the new PET value (mm h -1 ) generated from the sine function. 20 The sine fit for each month is based on the average of values from hPET for each hour of a day over the period of record (for this application, 1981-2020). The sine fit is only done based on values for daylight hours (sunrise to sunset), so we assume nighttime PET values are zero.
An example of the sine function representing hPET data for a single grid location (Wajir in Kenya -1.73° N, 40.09° E) for January is shown in Fig. 1. The grey shaded area represents the range of the hourly PET obtained from all days of January 25 within the 40 years record of hPET data, while the black line shows the average of those values. This average diurnal cycle is used to fit the sine function representing a given month of the year. The red line shows the fitted sine function (i.e., in Eq. (1)). The four parameters from Eq. (1) are estimated at each 0.1° grid location for each month and then saved as input for simulating synthetic PET. Figure 2 shows, for illustration, the spatial variability of parameters across the globe for January.
For each month of the year all four parameters, plus the sunrise and sunset hours (which are required to identify daytime and 30 nighttime periods), are provided as an input file with the model script.

Random noise estimation
PET shows variability within each month (Fig. 1), which is represented stochastically in stoPET using a "noise ratio" parameter ( ) (Eq. (2)): (ℎ, , ) = (ℎ, , ) ̅̅̅̅̅̅(ℎ, ) Eq. (2) Where (ℎ, , ) is the PET for every hour (h) and day (d) of each month (m) and ̅̅̅̅̅̅ (ℎ, ) is the average PET of 5 each hour over all days of the month. A skewed normal distribution is then fitted to noise ratios of each month calculated using Eq. (2). The fitted skewed normal distribution parameters (skewness, location, and scale), defined at each grid cell and month, are used as input to stoPET to generate stochastic variability around the sine function by sampling from this skewed distribution. Figure 3 shows the values of the three noise ratio parameters over the spatial domain, estimated for January. By way of a worked example, Fig. 4a shows the monthly distribution of the noise ratio for a single location in Wajir (Kenya), while Fig. 4b shows the randomly generated noise ratio array for January and the parameters representing it. The steps followed to create these noise ratio values were as follows: 15 1) Calculate the average hourly PET for each month from the 40 years hPET data. This gives a characteristic diurnal curve from which we can determine the average hourly PET value for each month (the black line in Fig. 1).
2) Divide each hourly PET for every day in each month (e.g., Jan 1) by its average from step 1. This will give us the noise ratio array (Fig. 4a).
3) Fit a skewed normal distribution to the noise ratio array based on Eq. (2) for each month and save the parameters 20 ( Fig. 4b).

Figure 4: (a) Noise ratio box plot for a single location in Wajir (Kenya). The box plots indicate that the noise ratio is variable over each month with the green triangle showing the mean and the red line in the box plot indicating the median (b) A histogram for
5 the January noise ratio is shown in blue, with the fitted skewed normal distribution shown in red. The corresponding distribution parameters are indicated in the top left of the plot.

2.2.3
Generating stochastic hourly PET stoPET generates new stochastic PET values for a particular month by multiplying the respective sine function ( Fig. 1) by the noise ratio sampled from the corresponding skewed normal distribution (Fig. 4b). For instance, for a particular simulation of 10 January, stoPET will generate 31 random noise ratios, producing 31 diurnal cycles of PET that amplify (or dampen) the mean diurnal PET sine wave for the month. Synthetic PET can then be generated for each month based on the number of years the user chooses.

3
Model verification

Regional representation
stoPET is set up to generate synthetic plausible hourly PET time series within any defined spatial area between 55° N and 55° S. The high latitude areas were not included because some months do not have a clear sunset and sunrise times during summertime creating potential errors in the sine function fitting. We have evaluated the stoPET model for six continental regions (North America, South America, Europe, Africa, Asia, and Australia). Figure 5 shows the average annual PET 20 climatology of Africa for five years of simulated PET from stoPET ( Fig. 5a) against five randomly selected years for hPET, where we have also removed the nighttime PET values (Fig. 5b) since stoPET considers the nighttime PET to be zero. Figure   6 also shows a similar comparison for Europe (stoPET, Fig. 6a and hPET, Fig. 6b). The comparison indicates that stoPET estimate the annual PET value with only an average percentage difference of ~5 % (see Fig. S1 to

Single point representation 5
To verify the performance of stoPET more quantitatively, analysis was carried out on twelve points chosen to be representative of climates across the range of aridity variations observed across the global land surface (Fig. 7). Ten ensembles, each comprising 20 years of synthetic PET data, were generated using stoPET and compared against the hPET dataset over the period 2001-2020, substituting the nighttime PET values of hPET with zeros to ensure consistency with stoPET. Next, the hourly PET values were aggregated to daily average PET values for each month for all the datasets at the twelve locations.
Eq. (4) Where represents the monthly average PET of hPET for each month, is the monthly average PET estimated by stoPET and is the number of months.

5
Based on these tests, we find that PET estimated by stoPET is statistically comparable to hPET historical datasets ( Fig. 8 and   Fig. 9). The pBias values range between -16 % to 13 % indicating that stoPET is not systematically overestimating or underestimating PET values relative to hPET (  (Table 1). stoPET produces PET values that are comparable in terms of capturing the seasonal cycle and variability ( Fig. 8 and Fig. 9).     Fig. 7) as an example; however, the results and plots of the other locations are provided in the supplementary material ( Fig. S13 to Fig. S23). The scatter plot (Fig.  10 10a) indicates that stoPET generates PET values comparable to hPET with a correlation value of 0.83. The box plots (Fig.   10b) show that stoPET produces a comparable mean (green triangle in Fig. 10b), median (red line in Fig. 10b) to hPET and captures the overall variability. Figure 11 shows the density plots of the hPET and stoPET data, and it indicates that the randomly generated stoPET values well represent the PET of the arid location in North America (and other locations, see supplemental figures). Figure 12 shows an hourly time series for 15 days to illustrate how stoPET and hPET appear over 15 several diurnal cycles of the simulation, with good consistency and evidence of the desired stochasticity in the simulated series. https://doi.org/10.5194/gmd-2022-128 Preprint. Discussion started: 11 May 2022 c Author(s) 2022. CC BY 4.0 License. Fig. 7). Fig. 7). The data represent the 5 daytime hourly PET from 2001 to 2020. Future climate is predicted to be warmer due to anthropogenic forcing (Hoegh-Guldberg et al., 2018, IPCC, 2021. This increased atmospheric temperature should lead to higher evaporative demand, which can have substantial impacts on the water balance. stoPET incorporates three methods to account for changes in atmospheric evaporative demand to climate change, supporting flexibility in simulating various climate scenarios. The three methods described below with examples, provide 5 choices for users to explore what fits their study goals. 6) The adjusted hourly PET is then obtained based on the summation of the PET from step 1 and the hourly changes of PET from step 5.

Method 2: Step change in PET based on a user-defined change in atmospheric temperature
Climate change is often characterised in terms of a specified rise in atmospheric air temperature (Randalls, 2010), which may 25 vary for different locations across the globe but is typically communicated as a global mean temperature change (e.g., 1.5 degrees of warming based on the Paris Climate Treaty) (Kriegler et al., 2018). We fully acknowledge that PET (especially based on the PM method of calculation) is not only driven by temperature changes but by changes in solar radiation, wind speed and humidity (Xu et al., 2014). Nevertheless, to isolate the influence of temperature alone, we created within stoPET a method to calculate temperature-based changes in PET, with all other non-temperature related variables remaining unchanged. 30 This is simply implemented, transparent and aligns directly with global climate discussions and policies (IPCC, 2013;Blunden and Arndt, 2020;NOAA, 2021). Method 2 accounts for a user-defined temperature change and its propagation into hourly PET, which works as follows within stoPET: 1) We recalculated hPET globally with uniform homogenous air temperature increment increases of 0.5°C (e.g., 0.5°C, 1.0°C, 1.5°C, 2.0°C, 2.5°C) per hour, with all other non-temperature related variables remaining unchanged. 35 2) hPET, which was calculated based on the current temperature with no adjustment, was subtracted from newly calculated PET values containing the temperature adjustment. This step revealed that the rate of change of the PET increase is uniform on average (Fig. 13); hence we can use the rate of change in PET and the user-defined temperature change as a multiplicative factor to represent the change in annual PET. in temperature (R 2 = 0.998), as an example, every increase by 0.5°C yields ~55 mm of annual PET change for the specified location. stoPET then provides the global annual PET change based on 1°C of warming derived from 20 years of climatology (Fig. 14). These annual PET changes are used as an input and multiplied by the user-defined temperature factor to determine the amount of annual PET change at each grid cell.

5
Method 2 adjusts simulated hourly PET generated by stoPET in similar ways to Method 1 (i.e., steps 3-6 are the same as Method 1), but the first two steps are altered as follows.
1) Generate an hourly stoPET time series for one year and take the annual sum.
2) stoPET multiplies the annual change in PET associated with a 1°C temperature increase (Fig. 14) by a user-defined temperature change (ΔT). 10

Method 3: Progressive change in PET based on the historical trend in hPET.
In some cases, it may be desirable to evaluate the potential impacts if currently observed trends in PET continue into the future.
To support this type of analysis, Method 3 computes the historical trends in hPET for each pixel of the globe and then applies this trend within the stoPET series for every location, leading to progressive change in the simulated PET. stoPET simulates PET via Method 3 as follows, sharing the same steps as Method 1 from step 3 onwards. The first two steps are as follows: 5 1) Generate stoPET for one year and take the annual sum.
2) Estimate the annual PET change using the slope of the linear trend to historical hPET (Eq. (6)). stoPET computes this trend and uses its slope ( ) as an input parameter applied over the number of years of the simulation ( ) to adjust the simulated series from stoPET that would be generated based on a 'no climate change' scenario.

Examples of stoPET-generated PET under climate change by the three methods
As a demonstration of these methods, we simulated PET under climate changes for arid and humid locations used for model evaluation (Fig. 7). Specifically, we present time series of annual PET for a 5 % user-defined percentage increase in PET (Method 1), a user-defined 1.5°C increase in temperature (Method 2), and by imposing the historical trend from hPET into the future (Method 3) ( Fig. 15; Fig. S24 to Fig. S28). These plots demonstrate the built-in flexibility in stoPET for 15 simulating changes to evaporative demand under climate change. For example, they illustrate that under Method 1, there is simply an elevated simulated time series of PET, while the higher values for Method 2 result from propagating a temperature increase through the calculation of PET, and Method 3 shows a clear trend that departs from the historical mean (Fig. 15). https://doi.org /10.5194/gmd-2022-128 Preprint. Discussion started: 11 May 2022 c Author(s) 2022. CC BY 4.0 License. Discussion As the global community works to determine the potential impacts of climate change, it is critical to address how changes to atmospheric evaporative demand will affect the water balance and associated water resource availability. Here, we have presented a novel stochastic PET generator (stoPET), which fills a gap in current capabilities to simulate historical and future evaporative demand across the globe. stoPET is a parsimonious, flexible, and computationally efficient way of generating 5 plausible hourly PET timeseries anywhere on the Earth's land surface. It has the potential for improving climate-related impact studies on the water balance for applications including, but not limited to ecology, ecohydrology, agriculture, and water resources in a wide range of environments across the globe.
The water balance is very sensitive to atmospheric evaporative demand, so the characterization of diurnal and seasonal 10 variability in PET across the globe is a critical component for a wide range of climate impact studies. stoPET is particularly important for the prediction of water resource availability, estimation of future crop water demand, assessment of flash flood risk, and provision of actionable information on expected climatic impacts on the water balance. Given inherent uncertainties in climatic drivers of the water balance (rainfall and PET), simulated assessments of the water balance under potential future climate change would be best framed in a probabilistic way. Stochastic weather generators may provide projections of rainfall 15 and temperature (Chen et al., 2012;King et al., 2015;Steinschneider et al., 2019), but there is currently no standardised capability to simulate plausible time series of PET under a range of future scenarios. It is also not currently possible to retrospectively assess the impact of climate forcing on the historical water balance based on PET. This information gap on PET undermines efforts to drive hydrological, agricultural, and land surface models. We provide a few potential applications of stoPET in this context below. 20 PET significantly influences the partitioning of the long-term water balance into different stores and fluxes that vary over time and space (Bai et al., 2016;Quichimbo et al., 2021). Key water balance components, including groundwater storage, evapotranspiration, runoff and streamflow are challenging to assess without accurately constraining evaporative demand (Bowman et al., 2016;Condon et al., 2020). An obvious example is flood hazard, which is especially sensitive to antecedent 25 moisture conditions within a drainage basin based on the prevailing PET over the period between rainstorms, which affects the subsequent partitioning of rainfall between infiltration and runoff, the downslope flow of both surface and subsurface water, and correspondingly, the magnitude of flood waves in channels. These influences impact the strength of the watershed response to rainfall events and corresponding flood hazard (Zoccatelli et al., 2019) in a range of environments. stoPET derived PET will thus support more realistic analyses of the water balance for the purposes of assessing flood hazard (and potential 30 mitigation measures).
Hydrological and land surface models require PET to close the water and energy balance and to resolve its key components (e.g., parsimonious distributed hydrological model for DRYland Partitioning-DRYP, (Quichimbo et al., 2021); PARallel Flow-ParFlow, (Maxwell and Miller, 2005)). Such models are often assessed in terms of the uncertainty in spatiotemporal rainfall 35 used to drive them, but there is additional uncertainty in PET that is typically unconstrained and especially for scenarios of future climate change (Van Osnabrugge et al., 2019). stoPET can generate multiple realisations of PET, supporting the assessment of uncertainty in atmospheric demand and providing key information on PET to support forecasting and risk assessment associated with water availability and agricultural water demand, especially for a wide range of meteorological conditions (Dimitriadis et al., 2021). The stoPET model fills this gap by providing physically realistic PET time series that 40 vary in space and honour the inherent diurnal and seasonal variability. https://doi.org /10.5194/gmd-2022-128 Preprint. Discussion started: 11 May 2022 c Author(s) 2022. CC BY 4.0 License.
Water availability to plants is one of the limiting constraints for crop production and food security (Funk et al., 2008;Funk and Brown, 2009;Kang et al., 2009;Ayyad and Khalifa, 2021), but also for the health and functioning of the vegetative ecosystem in natural settings (Mayes et al., 2020;Sabathier et al., 2021;Warter et al., 2021). Forecasts of crop water requirement and irrigation demand for major crops like maize, barley, and wheat (Ewaid et al., 2019) are paramount for preparing advisories related to the timing of planting, crop choice, and irrigation scheduling, especially in arid and semi-arid 5 regions, where high atmospheric evaporative demand and erratic rainfall make farming a risky economic activity (Nyakudya and Stroosnijder, 2011). Crop models require estimates of PET to quantify how much water can be lost to the atmosphere over the diurnal cycle and over the entire season of crop growth. In natural settings, PET is necessary to predict both water availability to plants and the timing of plant phenology, including the timing of green-up and senescence cycles, which have broader implications for ecosystem functioning in a range of environments. In this context, stoPET can be used to simulate the 10 PET and thus assess the hourly availability of water in the soil and its variation over the growing season for a wide range of plants. Our new model also supports analyses of future climatic changes and their impact on natural and agricultural plants, as well as irrigation demand for major crops.
Finally, stoPET can potentially be used in concert with rainstorm generators such as the STOchastic Rainstorm Model 15 (STORM) (Singer et al., 2018), wherein rainfall and interarrival times are simulated to obtain inputs to other models. Rainfall and PET may be straightforwardly interlinked such that PET in stoPET is reduced (due to cloud cover and high humidity) on any simulated rainy day in STORM, thus lowering evapotranspiration losses during rainy periods. In this way, STORM and stoPET would provide consistent sequences of raw data required to close the water balance in terms of key climatically derived variables. 20 Other future improvements of the model that we envisage may be to incorporate other variables apart from temperature change that are likely to be non-stationary and affect PET, such as radiation and wind speed. Additionally, the noise factor sampling used to perturb the stochastic PET is currently independent of adjacent grid points, so there is essentially no spatial autocorrelation, which may be undesirable. The impact of this on the realism of the output is not known a priori. Therefore, applying spatial smoothing to the stoPET output across a grid of simulated values might be a potential future improvement of 25 the model.
In summary, stoPET generates stochastic hourly PET across the globe at high spatial resolution and can estimate future PET under a range of potential future climate changes. The model can be used to evaluate different land surface and water balance models, which are used to predict water availability and other metrics related to the impacts of climate on sectors like agriculture and water use.