We used a global dataset of monthly precipitation oxygen and hydrogen isotope measurements from 650 and 610 precipitation monitoring stations, respectively. These previously compiled (Jasechko et al., 2016) data were collected from the Canadian Network for Isotopes in Precipitation (Birks and Edwards, 2009; Birks and Gibson, 2013), the US Network for Isotopes in Precipitation (Delavau et al., 2015; Welker, 2000, 2012), and the Global Network for Isotopes in Precipitation (Aggarwal et al., 2011; Halder et al., 2015). Following Jasechko et al. (2016), we characterize seasonal cycles only at monitoring stations that report precipitation isotope compositions for at least eight unique months. Monthly precipitation amounts (or snow-water equivalencies) are also available from 623 of the 650 stations that measured oxygen isotope ratios, and from 603 of the 610 stations that measured hydrogen isotope ratios. All hydrogen and oxygen isotope ratios of precipitation are denoted as δ2H and δ18O, defined by 1δ2H=2H/1Hsample-2H/1HV-SMOW2H/1HV-SMOW×1000‰, and 2δ18O=18O/16Osample-18O/16OV-SMOW18O/16OV-SMOW×1000‰, where V-SMOW refers to the Vienna Standard Mean Ocean Water standard.

We compiled gridded climatological and geographical data for global modeling and for inferring site characteristics of the precipitation monitoring stations (Fig. 1). We downloaded climate maps of monthly precipitation sums and monthly means of daily low, high, and mean temperature, all at 5 arcmin (i.e., 0.083∘) resolution (WorldClim; Fick and Hijmans, 2017). Station climate data were inferred from these gridded products for all but three stations that were on small islands or stationary weather vessels, for which local meteorological data were acquired. The range of mean monthly temperatures was computed at each pixel (and each monitoring station) as the difference between the highest and lowest monthly mean values, using the WorldClim data. Annual mean daily temperature range was calculated as the mean differences between daily minimum and maximum temperatures. The WorldClim data were also used to calculate time of peak precipitation and temperature, and seasonal amplitude of precipitation and temperature, metrics which can together capture global patterns in hydroclimate (Berghuijs and Woods, 2016). We also used a 30 s gridded elevation map (GTOPO30; US Geological Survey, 1996) that was aggregated to 5 min for consistency with the other grids. Monitoring station elevation data were not inferred from the grids, but instead downloaded directly from the isotope network databases. Distance from oceans and seas was calculated in ArcGIS 10.4.1 (ESRI, Redlands, USA) using published coastline data (Wessel and Smith, 1996) for the center of each 5 min pixel and for each monitoring station.

We fitted sine curves (described by the parameters amplitude, phase, and offset) to each monitoring station's monthly measured δ18O and δ2H time series using a nonlinear fitting routine (“fitnlm” in MATLAB R2016B, Mathworks, Natick, Massachusetts, USA). The sine curve is defined with a fixed period of 1 year, 3precipitationδ18Oorδ2H(t)=amplitude×sin⁡(2πt-phase)+offset, where t is the fractional year.All fitted amplitudes and phases were adjusted so that fitted amplitude values are positive, and phase values are between π and -π. Phase was calculated in radians, but we report all values in days from the summer solstice. Allen et al. (2018) previously confirmed that this nonlinear fitting routine yields parameter values and component standard errors that are equivalent to those obtained by fitting sine curves as an additive model of sine and cosine functions with their uncertainties calculated by Gaussian error propagation. It should be noted that standard errors depend on the length of records, and while some stations have datasets that are as long as 57 years, shorter durations are more common (Fig. 2a). We fitted the sine curves by two alternative approaches: (a) using iteratively reweighted least squares with a bisquare weighting function (robustly fitted), and (b) using standard least squares with the influence of each monthly isotope measurement weighted by the amount of precipitation during that month (amount-weighted). These metrics have different limitations. The amount-weighted cycles are less influenced by erratic values that can occur in low-precipitation months but also do not capture the variations during drier seasons as effectively. For example, if there was an anomalously dry month in a short data record and that dry month also had an atypical isotope value (e.g., because it was composed of a single small event), that value could result in a robust-fit exaggerating the true seasonal isotope cycle. If estimates based on that sinusoid were later weighted with typical precipitation amounts, this could introduce errors. Weighted fits could introduce errors if drier season precipitation is important to the study system, but the dry season precipitation has minimal influence on the fits and thus those values are misrepresented. Weighted fits might also mischaracterize the seasonal dynamics of a typical year in regions that are impacted by extreme precipitation in some years (e.g., hurricanes or monsoons) if that extreme precipitation has distinct isotope values and yields volumes that are substantial fractions of annual precipitation (e.g., Price et al., 2008). We focus on the robustly fitted parameters describing the seasonal cycles, but for comparison, the amount-weighted fits are also reported in Supplement 2. We recommend that future users of these data carefully consider the different limitations when selecting between these two approaches.

To characterize spatial variations in precipitation isotope seasonality, we establish relationships between the fitted sine parameters (amplitude, phase, and offset) and site characteristics of the precipitation isotope monitoring stations using multiple linear regression. To characterize the monitoring stations, we used elevation, absolute latitude, distance from the nearest ocean, mean annual temperature, range of mean monthly temperatures, seasonal amplitude of precipitation amount, and mean annual precipitation amount (Fig. 1). We chose these metrics as spatial predictors because global datasets of these metrics are publicly available and they capture aspects of climate and circulation patterns that are known to affect precipitation isotopic composition (Aggarwal et al., 2016; Birks and Edwards, 2009; Rozanski et al., 1993). To determine which predictors should be included in regression models, we used a stepwise model selection approach in which different combinations of predictors were used to maximize R2 values while requiring that all coefficient p values be statistically significant (p<0.05). This step limits model overfitting by excluding redundant or nonsignificant predictors. We found that using these criteria more aggressively removed variables compared to the more standard Akaike information criterion (AIC). To assess collinearity among these variables, we calculated the variance inflation factors (VIFs) associated with a hypothetical model that includes all six variables; we found those factors to range from 1.4 to 7.8, and while no fitted models were actually included all six terms, the variance inflation factors among the six predictors are still all below the often-used threshold of 10 (Marquaridt, 1970). After identifying the appropriate model terms, models were fitted using the “fitlm” function with robust fitting options that reduce the influence of outliers (MATLAB R2016B). In preliminary analyses, we also tested other metrics – precipitation phase, temperature phase, and mean daily temperature range – but determined that they were not consistently important (i.e., when included in the initial model selection, they were mostly excluded). Thus we excluded these other metrics from subsequent analyses to avoid overcomplicating the models; however, they often showed interesting relationships with the sine parameters, so they are provided in Fig. S1 in the Supplement.

For models of phase, we only used data from monitoring stations where there is a distinct seasonal cycle, because phase terms are meaningless and fitted values are unstable where there are no sinusoidal seasonal cycles; these phase values will also be excluded from the supporting information data files to avoid confusion. We characterize distinct seasonal cycles as ones where the phase is well constrained, with standard errors of the fitted phase terms lower than 15 d (and thus 95 % confidence intervals of approximately ±1 month). Roughly 74 % of the sites (n=479) met this criterion. We also tested other criteria for filtering out stations with meaningless phase terms, such as R2>0.3 (n=425) or R2>0.5 (n=232), and those yielded similar regression models for phase. We modeled phase in middle and high latitudes (30 to 90∘; n=349 after removing data without distinct seasonal cycles) separately from phase in tropical and subtropical latitudes (0 to 30∘; n=130 after removing data without distinct seasonal cycles). We took this approach because initial inspections of these data and past examinations of similar data (Bowen and Revenaugh, 2003; Feng et al., 2009; Halder et al., 2015) suggested that phase is relatively consistent within each of these zones, with sharp transitions at approximately 30∘ N and S (roughly corresponding with Hadley Cell boundaries; Birner et al., 2014).

These fitted spatial regression equations for amplitude, phase, and offset were used to map global precipitation isotope seasonality using the gridded site-characteristic data. We did not extend these maps to extrapolate Antarctic isotope seasonality because there are few monitoring stations there. We also mapped the residuals, estimated by subtracting the regression model estimates of amplitude, phase, and offset from the same variables determined from the fitted sine curves at the precipitation monitoring stations. We interpolated those residuals using inverse-distance weighting of the residual values from the three stations that are most proximal to each grid-cell center. For phase, we used nearest-neighbor interpolation, rather than inverse-distance weighting, because averages across unlike phases are poorly representative. We then applied a Gaussian filter to smooth each of the residual adjustment layers, with the standard deviation equal to 3∘, because we assume there are measurement uncertainties and thus the layer should not be fitted exactly to the points; we smoothed the phase residuals separately in absolute latitudes >30∘ versus absolute latitudes <30∘. For final predictive maps, we added the smoothed residual maps to the regression-based maps; wherever negative amplitudes resulted, those values were forced to zero. Errors were evaluated by running this routine again, but while randomly excluding 65 sites (10 %) for subsequent use as independent quality-control checks. Sine parameters for those 65 stations were predicted using models calibrated with the other ∼585 sites; this Monte Carlo procedure was iterated 15 times for both δ18O and δ2H.

We provide these predictive maps of the gridded amplitude, phase, and offset values of δ18O and δ2H. We also provide gridded amplitude, phase, and offset values for precipitation amount, which can be used to scale precipitation isotopic inputs, in applications where amount is important. These maps are provided (Supplement 3).

To explore sub-global variations in performance of the spatial multiple regression models, we also performed regional regression analyses in which we fitted multiple regressions to data from subsections of the globe. Regressions of amplitude, phase, and offset were calculated for 40∘×40∘ windows using the same site characteristics that were used in the global models: absolute latitude, elevation above sea level, distance from coastline, range of mean monthly temperatures, mean annual temperature, and annual precipitation amount. These regional regressions were calculated at all vertices of a 10∘ grid (marking the center of each 40∘ window). We used the same combination of stepwise regression model selection and robust regression fitting as in the global analysis. Only windows that contained more than 25 precipitation isotope monitoring stations were analyzed. We report gridded R2 and root mean square error (RMSE) values to indicate where these relationships are strongest. We also provide fitted sine parameters and site characteristics in the supporting information to facilitate users' development of other regression models for regionally specific applications (Supplement 2).

Globally, 94 % of the precipitation δ18O monitoring stations (n=650) have statistically significant seasonal isotope cycles (p<0.05; t test of the δ18O amplitudes), although those cycles do not always explain the majority of the variance in monthly isotope values (i.e., only 36 % of the stations had R2 greater than 0.5; Fig. 2). Amplitudes range from 0 to 11 ‰ δ18O (Fig. 3), with a median value of 2.3 ‰ δ18O; here, amplitude quantifies the strength of seasonal cycles as deviations from average annual values, so an amplitude of 2.3 ‰ δ18O corresponds to a range of 4.6 ‰ between typical values in the “higher δ18O season” and the “lower δ18O season”. Seasonal isotope variations are larger in colder, higher-latitude, higher-elevation, or more continental regions (Fig. 3), although no individual site characteristic explains the majority of variation in amplitude (Fig. 3; Table 1). The few coastal stations that have strong seasonal cycles are almost exclusively located in high absolute-latitude regions (Fig. 4a). Many of the monitoring sites within tropical latitudes also have substantial seasonal cycles; for example, 27 % of sites in the tropics show amplitudes greater than 3 ‰ δ18O, and they are not all high-elevation sites (Fig. 3b).

Although most stations show a seasonal precipitation δ18O cycle, the ability of sine curves to capture monthly δ18O values varies (Fig. 2). The median percent of variance explained by sine curves is 42 %; the median RMSE of individual monthly deviations from fitted sine curves is 2.2 ‰ δ18O. Stronger fits occur where (a) there is a strong seasonal cycle, (b) the seasonal cycle is the dominant pattern of variation, and (c) sine curves are the appropriate shape to characterize precipitation isotope variations. Accordingly, the spatial pattern in R2 (Fig. 2c) is broadly similar to the pattern in amplitude (r=0.74). However, RMSE also increases with amplitude (r=0.58), demonstrating that greater seasonal variability is also generally associated with greater month-to-month deviations from the seasonal sinusoidal cycle.

The phase term is well constrained (i.e., SE of phase <15 d) at most but not all sites (n=479), and its geographic distribution is surprisingly binary (Fig. 4b). From 30∘ S to 30∘ N (i.e., roughly corresponding with the Hadley cells), peak isotope values occurred 104±43 d before the summer solstice (mean ± SD). By contrast, in the middle- and high-latitude regions, peak isotope values occurred 18.6±24 d after the summer solstice. A few exceptions are found in absolute latitudes near 30∘, which may be attributable to the effects of the Asian monsoon cycle (Cai et al., 2018) or the migration of Hadley cell boundaries, which do not consistently occur at 30∘ (Chen et al., 2014). Peak precipitation isotope values occur within a month of peak temperature at 89 % of the monitoring stations that are in absolute latitudes above 30∘ and have well-constrained seasonal isotopic phases (Fig. S2); however, that pattern was not ubiquitous. On average, phase of δ2H significantly lags δ18O in absolute latitudes over 30∘ (p<0.01), albeit with a median difference of only 2 d (and median absolute difference of 4 d); these observations suggest that precipitation line-conditioned (LC) excess, defined as δ2H-a×δ18O-b (where a is the slope and b is the intercept of the local meteoric water line (LMWL); (Landwehr and Coplen, 2006), may frequently have a seasonal cycle, as previously described in Switzerland (Allen et al., 2018) and suggested in global deuterium-excess variations (Pfahl and Sodemann, 2014).

Offset values, describing the central tendency of the seasonal cycle, span a range of 33 ‰ in δ18O. These values are highest (least negative) in tropical and subtropical regions, and lowest in polar regions (Fig. 4c). Most prominent is the strong temperature trend (0.47 ‰ δ18O per degree Celsius, R2=0.77; Fig. 3; Table 1), consistent with patterns that have been previously described (Dansgaard, 1964; Rozanski et al., 1993). It should be noted that offsets and amplitudes are associated differently with continentality (Fig. 4a, c); while many of the regions with highly negative offsets also have large amplitudes, this is untrue of coastal regions in middle and high latitudes where highly negative offsets and small amplitudes co-occur. For example, in Reykjavik, Iceland, the δ18O offset is -8.0 ‰ and the amplitude is 0.9 ‰; a similar offset is found in continental Iowa, USA (-8.2 ‰), but the amplitude is 4.5 times larger (4.0 ‰).

The spatial patterns in amplitude, phase, and offset can be described as functions of site characteristics. Of the predictors examined, all have significant correlations (at p<0.05) with amplitude, phase, and offset (Table 1; see also Fig. 3). Spearman rank correlations, which are less influenced by extreme values, are also statistically significant for all but one of these relationships (Table 1). However, no variables explain the majority of variation in amplitude, and only temperature explained the majority of variation in offsets (Table 1).

We developed multiple linear regression models of site characteristics and sine parameters, and used them to generate maps of δ18O sinusoidal cycles (Fig. 4). The multiple regression models explain 64 % of the variation in amplitude (RMSE = 1.1 ‰) and 83 % of the variation in offset (RMSE = 2.0 ‰). The multiple regression models for phase have low R2 values (0.19 and 0.21 for absolute latitudes above and below 30∘, respectively) because there is little variation in phase within each latitude band; thus, phase RMSE values are small (12 and 28 d; Table 2). The coefficients of the multiple regression equations describing mapped precipitation δ18O sinusoidal cycles are presented in Table 2 and analogous coefficient tables describing global regression models of δ2H, amount-weighted δ18O, and amount-weighted δ2H cycles are presented in Table S1 in the Supplement.

Residuals from the interpolated sine parameter layers often show clusters of similar values (Fig. 5), implying that sources of geographic variation are not fully captured by the predictors that we have used. Consequently, regionally calibrated models (calculated over moving 40∘×40∘ windows) often yield better fits (Fig. 6). Even in regions where multiple regression models do not effectively explain the variations in precipitation isotope sine parameters (e.g., Central America, south-central Asia), they will necessarily be fitted to the mean regional values, so the regional multiple regression model errors (RMSEs) will usually be smaller than those of the global regression model.

To produce final predictive maps, we adjusted for the geospatially clustered residuals by adding the smoothed residual maps (Fig. 5) to the regression-based maps (Fig. 4). These predictive sinusoidal maps of δ18O seasonality (Fig. 7) and δ2H seasonality (Fig. S3) are made available in the Supplement. They capture 88 %, 97 %, and 96 % of the global variations in amplitude, phase, and offset, respectively. To calculate the prediction errors, we ran this routine again but randomly excluded 10 % of the sites from the calibration so that the sine parameters at those sites were predicted independently; the median amplitude and offset errors were 0.49 ‰ and 0.73 ‰ δ18O (and 4.0 ‰ and 7.4 ‰ δ2H), and median phase errors were 8 and 20 d (for absolute latitudes above and below 30∘, respectively).