We use 108 years of meteorological observations to quantify the sensitivity of potential evaporation when calibrated using a non-stationary climate for various natural vegetation classes using the two-step approach. We investigate how empirical Eref methods and empirical K values affect the validity of the estimated potential evaporation for different vegetation classes, by applying empirical coefficients outside their calibration period. We vary the calibration period in both length (2–30 years) and reference period (in 1906–2013).

First (Sect. 2.3), we simulate reference evaporation according to the process-based Penman–Monteith equation (Eref_PM), which is considered the international standard method for estimating reference evaporation (Allen et al., 1998). In addition, we apply four empirical equations that contain constants derived for a calibration period (Fig. 2; Sect. 2.3). From these simulations, we identify deviations between each empirical Eref method and the Eref_PM (Fig. 2; Sect. 2.3).

Secondly (Sect. 2.4), we generate time series of the main components of potential evaporation, i.e., synthetic series of Et, Es and Ei, for five different vegetation classes, using the SWAP soil–vegetation–atmosphere transfer (SVAT) scheme (Kroes et al., 2009; Van Dam et al., 2008) (Fig. 2; Sect. 2.4). SWAP allows users to simulate potential evaporation for different vegetation classes directly (i.e., a one-step approach) by parameterizing the Penman–Monteith equation for each vegetation class implicitly rather than by using crop factors. These synthetic series are considered “observations” throughout the paper for all comparisons with estimates from the two-step approach.

Finally (Sect. 2.5), we derive monthly crop factors for each vegetation type (5×) and for each Eref method (5×) based on the synthetic data of Et, Es and Ei for a calibration period (e.g., 1906–1935) to simulate crop factor estimation using field measurements (Fig. 2; Sect. 2.5). We use different (3×) definitions of crop factors: for transpiration (Kt), for transpiration plus soil evaporation (Kts) and for total evaporation (Ktot). Next, we apply the two-step approach, using Eref and crop factors from the calibration period, to calculate daily “predicted” evaporation components (3×) for each vegetation class (5×) and each Eref method (5×) for the entire period (1906–2013) (Fig. 2; Sect. 2.6). Doing so, the empirical Eref methods and crop factors are applied outside their calibration range. From these simulations, we quantify the deviations introduced by the use of Eref and K, by comparing the evaporation components obtained with the two-step approach to the synthetic “observations” (Fig. 2; Sect. 2.6). Each of these steps, which are executed for all calibration periods during the period 1906–2013 (2697×), are described in greater detail in subsequent sections.

Although SWAP may be expected to provide adequate evaporation values, its absolute accuracy is not discussed in this paper, because we focus on the sensitivity of the two-step approach using synthetic (hypothetical) data only. Therefore, the actual accuracy of SWAP is irrelevant for this paper. To ensure that our analysis is not biased by a specific choice of SWAP parameter settings, we considered different vegetation classes ranging from grasses to shrubs and forests. Doing so, we include different parameter sets that affect the variability and relative proportions of the calculated E components. For a detailed discussion of the SWAP model and its accuracy, please refer to Kroes et al. (2009) and Van Dam et al. (2008). By comparing potential evaporation components obtained from the two-step approach with the synthetic “observations” as simulated using the physical SWAP model, we are able to quantify the deviations introduced by using different Eref methods in combination with crop factors, as no other source of uncertainty is involved.

We use meteorological data from De Bilt, the Netherlands, covering the period 1906–2013, which were provided by the Royal Netherlands Meteorological Institute (KNMI). De Bilt (longitude = 5.177∘ E, latitude = 52.101∘ N, altitude = 2 m) is the main meteorological site of the KNMI, located in the center of the Netherlands. Daily records are available for minimum and maximum temperature, sunshine hours, wind speed, and precipitation from 1906 onwards, and for global radiation from 1957. The observations are continuous, except for April 1945, where values from April 1944 are used instead. All required input variables are calculated for the period 1906–2013 following Allen et al. (1998). Observed global radiation was used to derive the Angstrom coefficients needed to calculate daily global radiation (Allen et al., 1998) from 1906 onwards. For consistency, we only use these simulated values for further analysis; simulations agree very well with observations (1957–2013, Radj2 = 0.96). Wind speed, measured at different heights, was scaled to the reference height of 2 m following Allen et al. (1998) and corrected for systematic differences between measurement periods. Figure 3 shows the annual values and the 30-year moving averages of the variables used to calculate evaporation from De Bilt.

Although the results are only valid for the site and period for which they were developed, the times series of radiation for De Bilt station resembles the global trends of global dimming/brightening. Values of global radiation (Rs) from De Bilt show a similar trend to the observations for Stockholm, as presented in Wild (2009). The data (Fig. 3) show an increase in temperature consistent with previous studies (Solomon et al., 2007) and a pattern of sunshine duration consistent with dimming and brightening for northwestern Europe identified by Sanchez-Lorenzo et al. (2008).

Long time series of meteorological observations will, to some extent, not be homogeneous, for example due to changes in measurement devices over time. However, this does not affect the calculations herein, as the aim is to investigate the sensitivity of the two-step potential evaporation methodology to non-stationary climate, rather than to produce an exact reconstruction of the last century's climate conditions. In this way, changes in measurement accuracy with time simply represent another non-stationary trend in this data set.

Several methods are available for calculating reference evaporation, differing in complexity and empiricism (Sperna Weiland et al., 2012; Bormann, 2011; Federer et al., 1996). Here, we analyze five of these methods, given in Table 1: the physically based Penman–Monteith equation (PM), the radiation-based methods of Makkink (Mak) and Priestley–Taylor (PT), and the temperature-based methods of Hargreaves (Har) and Blaney–Criddle (BC).

The FAO-56 method (Allen et al., 1998), using PM parameterized for reference grass, is recommended as the international standard for calculation of Eref. Given the physical basis of PM, it can be used globally, without the need to estimate or calibrate its parameters (Droogers and Allen, 2002). In contrast, the methods of Mak, PT, Har, and BC contain empirical coefficients, derived for specific meteorological conditions and sites. Following Farmer et al. (2011), we consider Eref_PM as the best approximation of Eref. In order to reduce any systematic differences between Eref values, we estimate the empirical factors C1, C0, α′, β, a, b, c, d of the other four Eref methods (Table 1) by least squares regression against the simulated daily Eref_PM, for a specific calibration period. Subsequently, daily values of Eref are calculated for each method during the full period, i.e., 1906–2013, and deviations between the empirical Eref methods and Eref_PM are calculated. The sensitivity of Eref to the choice of calibration period is evaluated for each of the methods using Eref_PM as a basis.

Synthetic time series of the three evaporation components are derived to systematically unravel the use of empirical crop factors. The synthetic time series are based on the SWAP physical model (Van Dam et al., 2008; Kroes et al., 2009) from which Et, Es and Ei can be simulated separately. From these simulations, we derive monthly K values for each Eref method (5×) and vegetation class (5×) (Fig. 2; Sect. 2.5), which are subsequently used to derive the corresponding potential evaporation components (5×5×3) using the two-step approach (Fig. 2; Sect. 2.6).

Standard values for the vegetation classes and their schematization are taken from the National Hydrologic Instrument (NHI, http://www.nhi.nu/nhi uk.html; De Lange et al., 2014) of the Netherlands. The vegetation schematization is constant throughout the period 1906–2013; i.e., dynamic vegetation is not simulated. We consider five natural vegetation classes: grassland (height = 0.5 m and no full soil cover, i.e., not to be confused with the reference grass), heather, deciduous forest, pine forest and spruce forest. Parameters are chosen following NHI (2008) and are provided in the Supplement. It should be noted that we do not discuss the exact validity of the parameter values used, as we are only concerned with evaporation sensitivity to non-stationary climate within the range of typical vegetation.

SWAP simulates the potential evaporation components of a crop or vegetation class based on the aerodynamic resistance, height, leaf area index (LAI), and albedo. SWAP uses the Penman–Monteith equation, parameterized for each vegetation class to simulate Et (potential transpiration) and Es (potential soil evaporation). In the case of intercepted precipitation, the values of Et and Es are reduced (Van Dam et al., 2008). Interception, which partly evaporates (Ei) and partly drips to the ground, is estimated following Von Hoyningen-Hüne (1983) and Braden (1985) for short vegetation and Gash et al. (1995) for forests. For an extended description of SWAP and the procedures for calculating Et, Es and Ei, we refer the reader to Kroes et al. (2009) and Van Dam et al. (2008). Given the international recognition of the SWAP model and successful testing, we assume that the model is able to produce representative synthetic estimates of each evaporation component.

As Kt and Kts are defined for a vegetated surface with a dry canopy (i.e., without interception) and Ktot includes interception (see Sect. 1), two different SWAP runs are performed for each vegetation class, without and with interception. Throughout the paper, Et and Es are valid for conditions with a dry canopy, whereas Etot includes interception and its limiting effect on transpiration and soil evaporation.

We derive Kt, Kts and Ktot for each vegetation class (5×) and Eref method (5×) based on the synthetic Et, Es and Etot time series, and the equations given in Table 2. Similar to the calibration of Eref methods, K values are derived for a specific calibration period (e.g., 1906–1935). K values for each vegetation class and Eref method are derived as monthly averages over the calibration period.

Potential evaporation components, Et, Et plus Es (hereafter Et & Es) and Etot, for each vegetation class and method, are calculated from the daily Eref values by multiplying it by the corresponding K values, Kt, Kts and Ktot, respectively, for each vegetation class. Using these three definitions of crop factors separately allows quantification of the error that is made by correcting for each evaporation component.

Eref estimates that are calibrated for a specific period, combined with K values determined for the same period, are used to calculate daily values of Et, Et & Es, and Etot for the full period, i.e., 1906–2013. This procedure corresponds to what is commonly done using the two-step approach, where the empirical parameters of an Eref method are fixed for the region in question, along with the corresponding K values. Here, we determine the deviation that is potentially introduced when this approach is applied outside its calibration range (period and region/site) in a changing environment, by comparing Et (Eref, Kt), Et & Es (Eref, Kts), and Etot (Eref, Ktot) obtained by the two-step approach with the synthetic “observed” Et, Et & Es, and Etot series.

Figure 4 shows the 30-year backwards-looking moving average Eref according to PM, Mak, PT, Har and BC, with the four latter models calibrated to fit the simulated Eref_PM for the first 30-year period, i.e., the calibration period 1906–1935. The minor differences seen between all 30-year mean Eref values during the calibration period (Fig. 4a, year 1935) indicate that each method was calibrated successfully. Using the calibrated equations, Eref's are calculated for the period 1906–2013, i.e., also outside the calibration period. All empirical models are evaluated with respect to the physically based Eref_PM, which was also used when calibrating the empirical coefficients. The radiation-based methods, Mak and PT, deviate only slightly from PM on average, with no consistent bias (Fig. 4d), which can be explained by the relatively strong correlations between the trend in 30-year average Eref_PM (Fig. 4a) and Rs and Rn (Fig. 4f and g); Pearson's r equals 0.85 and 0.70 for Eref_PM vs. Rs and Eref_PM vs. Rn, respectively. The temperature-based methods, Har and BC, deviate systematically from PM, which can be explained by the relatively weak correlation between the trend in 30-year average Eref_PM (Fig. 4a) and T‾ (Fig. 3b); Pearson's r equals 0.17 for Eref_PM vs. T‾. This shows that, as the energy used for evaporation mainly comes from direct solar radiation and, to a lesser extent, from air temperature, temperature-based models fail if radiation and temperature trends are weakly correlated (see Fig. 3; Pearson's r for 30-year average Rs vs. T‾ equals 0.22). Additionally, Har and BC deviate in different directions (Fig. 4b and d): Har consistently underestimates Eref, whereas BC consistently overestimates Eref. This opposite trend is related to the decreasing trend in Tmax - Tmin (used in Har), while T‾ increases (Fig. 3). All four empirical models are unable to reproduce the extreme high evaporation values predicted by PM, especially Har and BC (Fig. 4c). The deviations from Eref_PM are considerably larger for individual years (Fig. 4d) than for the 30-year moving average (Fig. 4b).

In practice, 30-year observed time series of evaporation are rarely available for calibration. Therefore, Fig. 5 shows the effect of calibration period length on estimates of Eref for the current climate (1984–2013). This effect is expressed as the maximum absolute deviation of the 30-year average with respect to Eref_PM. Figure 5 was compiled by first calibrating the empirical Eref coefficients for all possible calibration periods (in 1906–2013) with a given length (2–30 years) and then simulating Eref for the period 1984–2013 using the calibrated coefficients. The largest deviations occur for shorter calibration periods, as expected. Specific years may cause large deviations when the obtained empirical coefficients are applied outside the calibration period. Deviation decreases notably with increasing calibration periods, suggesting that using more calibration data should result in more stable and accurate Eref estimates. As the calibration period length decreases, deviations in the 30-year average Eref for 1984–2013 increase exponentially.

It should be noted that only deviations in 30-year averages are shown for varying calibration period lengths; deviations in the underlying yearly values are larger, as indicated by Fig. 4d. Additionally, the amplitude of the deviations shown in Fig. 4b and d would increase when calibrated using periods shorter than 30 years (Fig. 5).

Figure 6 gives monthly average synthetic evaporation components Et, Es and Etot, which were used to derive monthly crop factors (three methods: Table 2) for five vegetation classes and five Eref methods (Table 1), i.e., 3×5×5 crop factors for each calibration period. In contrast to the reference grass surface, the grassland of Fig. 6 does not fully cover the soil, which results in higher Es and lower Et. Figure 7 shows simulated Et (the two-step approach) for the period 1906–2013, using empirical coefficients for each Eref method and matching Kt values, all calibrated on the period 1906–1935. The general patterns in Et correspond to those of Eref (Fig. 4b), meaning that the deviations introduced by the two-step approach are mainly determined by the empirical coefficients in the Eref methods.

The deviation introduced for Et derived from Eref_PM and Kt_PM is relatively minor compared to what is found for the empirical Eref methods, especially for short vegetation. Apparently, Eref_PM follows the trend in Et (also obtained using the Penman–Monteith equation, but parameterized for each vegetation class, Sect. 2.4), and the ratio of Et and Eref_PM, used to estimate Kt_PM, changes little with time for short vegetation. More significant effects of Kt_PM are seen for taller vegetation, as climate-induced temporal changes in Eref_PM show a height-dependent non-linear relation to changes in Et (Allen et al., 1998). Therefore, the deviation introduced when using Eref_PM is larger for forests than for the short vegetation classes (Fig. 7). Similar to what is seen in Fig. 4, the deviations for individual years can be considerably larger than the climatic averages.

Figure 8 shows the sensitivity of crop factors, K, with respect to the calibration period length for heather and spruce forest. The variation in K decreases with increasing calibration length for all methods, but, except for Eref_Mak, the variability of K values for the empirical Eref methods is larger than for Eref_PM. These differences are especially notable for forests and illustrate that a poor relationship between the Eref method and the synthetic potential evaporation component (Sect. 2.4) is compensated for by K values that thus show a larger variation over time. Remarkable is the low variability in Ktot values for heather (Fig. 8c), which indicates that the variability seen for Kt (Fig. 8a) is reduced by interception. However, for spruce forest, for which interception is much more dominant, interception increases the variability in Ktot.

From Fig. 8a and d, it can be concluded that the deviations shown in Fig. 7 will increase when shorter calibration periods are used, irrespective of the applied Eref method. Figure 9 shows the effect of period (years and length) on the maximum absolute deviation made by the two-step approach for each Eref method and for Kt, Kts and Ktot. Figure 9 confirms that deviations in climatic average evaporation components obtained by applying the two-step approach will generally increase when shorter calibration periods are used. Additionally, Fig. 9 illustrates that deviations are (i) larger for tall vegetation than for short vegetation and (ii) larger for Ktot than for Kt and Kts for vegetation classes with high interception, as is the case for spruce forest. The large deviations for Etot for spruce forest confirm the remark by De Bruin and Lablans (1998) that, for wet forest evaporation, the crop factor approach will not be sufficient. Nevertheless, when derived for a sufficiently long time series, the deviations level out and there is no detectable bias.

In contrast to Fig. 9, which only shows the maximum absolute deviations for the 30-year average potential evaporation components for the years 1984–2013 as a function of the calibration period length, Fig. 10 includes the results of all underlying deviations for heather and spruce using Eref_PT. Figure 10 demonstrates that climate variability induces systematic overestimation or underestimation of the calculated potential evaporation components, depending on the calibration period used. The sign of the error strongly varies with the calibration period, and the inclusion of a single anomalous year can change the sign of the error.

Figure 10 further shows that anomalous years or multi-annual climate patterns tend to propagate considerable errors outside the calibration period to the current climate (1984–2013). The patterns of deviations from the synthetic “observations” show similarities to the global dimming and brightening periods (see Sect. 1): the first warming period (about 1925–1945) causes a systematic overestimation up to calibration lengths of 30 years, although specific calibration years may result in an underestimation for shorter calibration lengths. The succeeding period of cooling leads to a systematic underestimation, while the second warming period (starting around 1975) results in a more variable pattern. The latter may be linked to the finding of Wang and Dickinson (2013) that, in contrast to the years until the 1970s, there is no significant relationship between variations in temperature and global radiation in following years.

The patterns are comparable for Et, Et & Es, and Etot, based on Kt, Kts, and Ktot, respectively, for short vegetation classes. However, for tall vegetation classes with high interception capacity, e.g., spruce (Fig. 10f), using Ktot results in a noisier pattern due to specific years of high precipitation. Additionally, including interception may shift the sign of the error.

In this paper, we systematically unraveled and quantified how empirical coefficients in the two-step approach affect estimates of potential evaporation. We used the past century's time series of observed climate containing non-stationary signals of multi-decadal atmospheric oscillations, global warming, and global dimming/brightening (Suo et al., 2013; Stanhill, 2007; Wild, 2009; Wild et al., 2005) to evaluate the sensitivity of the two-step approach to both the length of the reference calibration period and the reference years. To this end, we calibrated the empirical coefficients of the two-step approach based on different periods and showed that using the thus obtained empirical coefficients outside their calibration range may lead to systematic differences between Eref methods, and to systematic errors in estimated potential E components. The signs of the errors for calculated climatic average evaporation components differ, depending on the Eref method used, and on the specific period (length and years) of calibration. Hooghart and Lablans (1988) stated that, for the two-step approach, the correctness of empirical coefficients for the estimation of Eref is of minor importance, as these are compensated for by K. However, here we have shown that while this may be true within the calibration period, this statement does not hold when extrapolating. As potential evaporation is a key input in hydrological models, input errors will propagate to estimates of related processes, such as the soil moisture budget, droughts, recharge and groundwater processes.

These results are important because the two-step approach, including extrapolating empirical coefficients, is frequently applied in hydrological modeling studies, as mentioned in the introduction. Ehret et al. (2014) state that “in hydrological modeling, it is often conveniently assumed that the variables presenting climate vary in time while the general model structure and model parameters representing catchment characteristics remain time-invariant.” There is a clear parallel of this statement with the approach presented herein where meteorological conditions vary in time, while climate-dependent (empirical) parameters are often fixed values.

In practice, long time series of observed evaporation are rare and not evenly distributed spatially. As such, for many applications, hydrologists must rely on incomplete calibration data, use analogous stations with similar characteristics, or simply default to published values for crop factors and Eref model parameters. Such published values for empirical factors of the different Eref methods (Table 1) are C1 = 0.65, C0 = 0 (De Bruin, 1987), α′ = 1.3, β = 0 (De Bruin and Lablans, 1998), a = 0.0023, b = 17.8 (Droogers and Allen, 2002), and c = 0, d = 1 (Sperna Weiland et al., 2012). Besides absolute values of the calibrated empirical factors, our analysis provides insight into the sensitivity of the results, i.e., the parameter values, to the calibration period. Calibrated model parameters for 30-year calibration periods are (standard deviations between parentheses) C1 = 0.64 (0.01), C0 = 0.37 (0.03), α′ = 1.06 (0.01), β = 0.57 (0.02), a = 0.0022 (0.002), b = 20.7 (3.0), and c = -2.31 (0.06), d = 1.73 (0.03). For a more realistic calibration period of, e.g., 3 years, the standard deviations increase by a factor of 2–10, depending on the Eref method used.

For the Netherlands, published crop factors (Ktot for Eref_Mak with coefficients C1 = 0.65 and C0 = 0) are 1.0, 0.8, 1.1, 1.2 and 1.3 for grass, heather, deciduous forest, pine forest and spruce forest, respectively (Spieksma et al., 1996). Values from our study for, e.g., the mean 30-year Ktot values, were 0.8, 1.03, 1.02, 1.07 and 1.25, respectively. Problems arise in the applicability of the published and frequently re-used crop factors, as the climatic conditions used for fitting are rarely documented. The analysis herein provides insight into the uncertainty ranges that could be expected using published empirical coefficients (Fig. 8) and their potential impact on simulated potential evaporation components.

This study has shown that potential evaporation estimates are most accurate and stable with a long calibration period. However, even when using a long observed record, estimates may include errors due to the assumption of constant empirical coefficients in a non-stationary climate; i.e., the calibration period is not representative of current conditions. Evaporation estimates outside the calibration period are even more susceptible to non-stationarity when the calibration period is relatively short, as with areas where observed evaporation data are sparse. Finally, estimating evaporation based on published typical values without calibration is most susceptible to errors, as these parameters are typically global averages but also contain the non-stationary reference period issues identified in this paper. To remove bias by systematic input errors, as in, e.g., evaporation, it is common practice to tune models by calibration (Ehret et al., 2014, and references therein). Although model calibration may compensate for biased input data, resulting in more accurate results and comparable model efficiencies, such calibration limits the general applicability of models when the bias is not constant over time (Andréassian et al., 2004). Figures 4 and 7 show that such non-constant bias occurs for both Eref and potential E estimates, thus limiting their application outside the calibration range.

Although extrapolations to future periods will always include uncertainty, it is important to quantify and limit this uncertainty. This analysis provides such a quantification, identifying the sensitivity of evaporation estimates to extrapolation and representing information on ways to reduce potential errors; e.g., Fig. 7 quantifies the error that is made in extrapolations. A similar modeling approach as presented here could be used to identify climate-induced changes in potential evaporation components. Moreover, such information can be used to reduce potential errors, if the errors can be explained from, e.g., differences in climatic conditions between the periods of calibration and application. We did so for the errors identified for potential evaporation in Fig. 7, and found that for all Eref methods except for Har, and for all vegetation classes, the error correlates well with differences in relative humidity (R2 > 0.78) (Fig. 3). This makes the errors predictable, and provides opportunities to correct for them.

Although we advocate using process-based evaporation simulations where possible, it should be emphasized that the two-step approach still can be a valuable concept, especially in regions with limited data availability. However, some considerations may strengthen the robustness of the two-step approach. First, our results show that applying radiation-based methods is preferred over temperature-based methods. Second, ideally, independent of the type of empirical method used, the coefficients should be recalibrated against measurements. Third, as such recalibration will practically often not be feasible, we advocate identifying changes in climatic conditions for the period of application and the calibration period, and quantifying, using a sensitivity analysis, how they may impact potential evaporation estimates. This provides uncertainty ranges that advance the interpretation of modeling exercises.

Poor transferability of parameter estimates made during calibration can have potentially large impacts for studies under non-stationary conditions (Coron et al., 2014), e.g., for climate change impact studies (Bormann, 2011; Karlsson et al., 2014). To improve the temporal robustness of hydrological modeling, Coron et al. (2014) propose, while adding it to the framework of the new “Panta Rhei” IAHS scientific decade (Montanari et al., 2013), to especially advance our abilities to estimate temporal variations in evaporation fluxes. This study contributes to this larger objective.

For climate change impact studies, applications of empirical models are particularly problematic, as empirical methods closely approximate observations of natural processes, but do not capture the underlying physics. When extrapolating to new climate regimes, these assumptions are not guaranteed to remain valid (Kay and Davies, 2008; Bormann, 2011; Arnell, 1999). Similar to our findings, simulating historic non-stationary climatic conditions, Kay and Davies (2008) demonstrate that Eref_PM and temperature-based Eref methods give different projected evaporation estimates when applied to future climate model data. Additionally, Haddeland et al. (2011) show, using the WATCH climate forcing data (Weedon et al., 2011), that global hydrological models that differ in their choice of evaporation schemes show significantly different evaporation estimates. These large discrepancies in an important part of the water cycle may have a large effect on the modeled hydrological impacts of climate change and increase the uncertainty of impact estimates (Bormann, 2011; Kay and Davies, 2008; Haddeland et al., 2011).

To show the implications of using different empirical Eref methods in hydrological applications under recent climate change, without the need for numerous extensive model runs, we calculated the Standardized Precipitation and Evaporation Index (SPEI) for the period 1906–2013, with the empirical coefficients calibrated for the 30-year period 1906–1935.

The SPEI (Beguería et al., 2014; Vicente-Serrano et al., 2010) is a commonly used meteorological drought index, which is a variant of the WMO-recommended Standardized Precipitation Index (SPI) (Guttman, 1999; Hayes et al., 2011; McKee et al., 1993). Unlike the SPI, which calculates precipitation accumulated over a period and then normalizes the accumulated value based on typical seasonal conditions, the SPEI instead normalizes the accumulated difference of the climatic water balance, defined as the difference between precipitation and Eref. This produces a time series of normalized values, such that an SPEI of 0 refers to typical conditions, an SPEI of -1 refers to a condition where Σ (P - Eref) is 1 standard deviation drier than typical, and vice versa for +1. For this example, the SPEI6 was calculated, normalizing the climatic water balance summed over the preceding 6 months, following the fitting procedures outlined in Stagge et al. (2015) and Gudmundsson and Stagge (2014).

Figure 11 shows the results of this analysis, with the assumed accurate SPEI6, based on Eref_PM, shown at the top, and the difference between this and SPEI6 for all other empirical reference evaporation models shown below. As with the results of Eref simulations, the Mak and PT models are closest to the observed signal (differences in the range from -0.2 to 0.2), while the Har and BC models produce greater variability (ΔSPEI6 = -0.5 to 0.5). Differences in this magnitude can make a large difference when interpreting drought risk. For example, the year 1947 produced a severe drought at the De Bilt site (SPEI6 = -2.2); however, all other methods underestimate Eref, producing SPEI6 values between -1.5 and -1.9. This in turn changes the interpretation of this drought from an event expected to occur once every 72 years to an event expected to occur once every 15–35 years. This is a significant difference in risk level that can be attributed to differences among the evaporation methods and a potentially non-representative calibration period. SPEI sensitivity to the Eref method is analyzed in greater detail in Stagge et al. (2014).