## 1. Introduction

The accurate estimation of evapotranspiration (ET) is needed for determining agricultural water demand, reservoir losses, and driving hydrologic simulation models. In typical hydrological and agricultural practice, evapotranspiration is calculated from reference evapotranspiration (ET_{0}), where ET_{0} is the evapotranspiration from a well-watered reference surface. In an effort to provide a common, globally valid standardized method for estimating ET_{0}, the FAO-56 Penman–Monteith (PM) equation (Allen et al. 1998) was adopted by the Food and Agricultural Organization (FAO) of the United Nations. While the physically based PM equation has been shown to accurately estimate ET_{0} (Chiew et al. 1995; Garcia et al. 2004; López-Urrea et al. 2006; Yoder et al. 2005), it requires a large amount of meteorological data that are often not available in many regions.

Forecast output from numerical weather prediction models (NWPMs) and global climate models (GCMs) are potentially useful for ET_{0} forecasting. However, local application usually requires a finer resolution than is currently available from most coarse-scale NWPM and GCM output (Fowler et al. 2007). Downscaling techniques are able to address this problem by using dynamical or statistical methods. Dynamical downscaling focuses on nesting a regional climate model (RCM) in a NWPM or GCM to produce spatially complete fields of climate variables, thus preserving some spatial correlation as well as physically plausible relationships between variables (Maurer and Hidalgo 2008). However, dynamical downscaling suffers from biases introduced by the driving model and from high computational demand (Abatzoglou and Brown 2012; Hwang et al. 2011; Plummer et al. 2006). Statistical downscaling methods develop empirical mathematical relationships between output from NWPM–GCMs and local climate observations (Barsugli et al. 2009). The advantage of statistical downscaling is computational efficiency and the ability to be applied across multiple models to develop ensembles for scenario building (Abatzoglou and Brown 2012). Because of these advantages, extensive research has been conducted on statistical downscaling for a variety of purposes in recent years. For example, Maurer and Hidalgo (2008) downscaled reanalysis precipitation data over the western United States using both constructed analogs (CA) and the bias correction and spatial downscaling (BCSD) method of Wood et al. (2004). Another study compared three statistical downscaling methods: BCSD, CA, and a hybrid of the two [bias correction and constructed analogs (BCCA)] to downscale reanalysis data and used the downscaled data to drive hydrologic models (Maurer et al. 2010). Abatzoglou and Brown (2012) compared two statistical downscaling methods: BCSD and multivariate adapted constructed analog (MACA) to downscale reanalysis data for wildfire applications in the western United States.

Several studies have been conducted on ET_{0} forecasts in recent years. Cai et al. (2007, 2009) developed and applied ET_{0} forecasts using weather forecast messages produced by the China Meteorological Administration. Several studies have focused on the use of artificial neural network (ANN) or other empirical models to simulate or forecast ET_{0} (Chattopadhyay et al. 2009; Dai et al. 2009; Kumar et al. 2011; Landeras et al. 2009; Ozkan et al. 2011; Pal and Deswal 2009; Wang et al. 2011). A recent review of ANN modeling of ET_{0} can be found in Kumar et al. (2011). Comparatively fewer studies have been conducted to dynamically or statistically downscale ET_{0} forecasts using NWPMs or GCMs. Ishak et al. (2010) used the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) to dynamically downscale 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) data in the Brue catchment in southwest England, and found ET_{0} to be overestimated by 27%–46%. However, Ishak et al. (2010) noted that there were clear patterns in downscaled weather variables that could be used to correct the bias in the results. Silva et al. (2010) found that statistically corrected MM5 estimates of ET_{0} improved results compared to raw model output in the Maipo basin in Chile. Direct statistical downscaling methods (i.e., methods that downscale directly from coarse-scale NWPM or GCM output) that specifically address the needs for ET_{0} forecasts appear to be generally lacking to date.

Archives of NWPM reforecasts have recently been made available for diagnosing model bias and improving forecasts, including the reforecast datasets using the National Centers for Environmental Prediction’s (NCEP’s) Global Forecast System (GFS; Hamill et al. 2006) and that from ECMWF (Hamill et al. 2008; Hagedorn et al. 2008). A series of articles have introduced the use of the GFS reforecast dataset, including week-2 forecasts (Hamill et al. 2004; Whitaker et al. 2006), short-range precipitation forecasts (Hamill and Whitaker 2006; Hamill et al. 2006, 2008), forecasts of geopotential heights and temperature (Hagedorn et al. 2008; Hamill and Whitaker 2007; Wilks and Hamill 2007), and streamflow predictions (Clark and Hay 2004; Muluye 2011; Werner et al. 2005). The GFS reforecast dataset includes over 31 years of 1–15-day 15-member forecasts for multiple variables at a T62 (approximately 200 km) resolution (Hamill et al. 2006). Hamill and Whitaker (2006) demonstrated that a forecast analog technique could produce simple, skillful probabilistic forecasts at a high spatial resolution using the GFS reforecast archive. The analog technique has been found to perform, in general, as well or better than other statistical downscaling methods (Timbal and McAvaney 2001; Wilby et al. 2004; Zorita and von Storch 1999). The GFS reforecast dataset includes forecasted variables that include wind speed, temperature, and relative humidity, which are important input data for the PM equation, and thus may be useful for generating probabilistic ET_{0} forecasts. However, the GFS reforecast dataset does not include archives of incoming solar radiation, nor maximum and minimum temperature (forecast variables are archived in 12-h intervals), which are also important input data for the PM equation.

In this work, we employ the GFS reforecast dataset to generate 1–15-day probabilistic daily ET_{0} forecasts and downscale the forecasts using a forecast analog technique over the states of Alabama, Florida, Georgia, North Carolina, and South Carolina in the southeastern United States. Sections 2 and 3 provide the data and methodology used in this work. The description of results and discussion are presented in section 4. Concluding remarks are given in section 5.

## 2. Data

*u*

_{10}), 2-m temperature (

*T*), and 700-mb relative humidity (RH), all of which may be potentially useful to forecast ET

_{0}. For this work, the 15-member forecasts were averaged to the ensemble mean; 12-h 2-m

*T*, 10-m

*u*

_{10}, and 700-mb RH data were averaged into daily values; and daily maximum and minimum temperature (

*T*

_{max},

*T*

_{min}) were estimated by comparing the 12-h

*T*values on each day, where the larger

*T*was

*T*

_{max}and the smaller

*T*was

*T*

_{min}. Wind speed at 2-m height (

*u*

_{2}) was estimated from

*u*

_{10}(Allen et al. 1998):

*z*is the measurement height (10 m). For this work, we used data from 1 October 1979 to 31 September 2009—30 years of reforecasts.

Since the GFS reforecast archive does not include solar radiation output, and includes variables at a relatively coarse temporal resolution (12 h) this work also employed the R2 dataset. The R2 dataset includes variables such as incoming solar radiation (*R _{s}*) and daily maximum and minimum temperature. The R2 dataset is available from 1979 to present at a T62 resolution (Kanamitsu et al. 2002). Daily climatological mean values of the R2 dataset were calculated using a running ±30-day window.

As the long-term continuous observed meteorological data needed for the estimation of ET_{0} are generally not available, the North American Regional Reanalysis (NARR) dataset (Mesinger et al. 2006) was used for forecast verification. The NARR dataset contains all of the variables required for estimation of ET_{0} (as described below) and is available at an approximately 32-km-grid resolution. While the NARR dataset may contain biases (e.g., Markovic et al. 2009; Vivoni et al. 2008; Zhu and Lettenmaier 2007), the NARR data used to calculate ET_{0} were taken as a surrogate for long-term observations in this work.

## 3. Methods

### a. *ET*_{0} calculation methods

_{0}

_{0}(mm). The FAO-PM equation is written as

*R*is the net radiation at the crop surface (MJ m

_{n}^{−2}day

^{−1});

*G*is soil heat flux density (MJ m

^{−2}day

^{−1}), and is considered to be negligible for daily calculations;

*T*is the mean daily air temperature at 2-m height (°C);

*u*

_{2}is wind speed at 2-m height (m s

^{−1});

*e*is saturation vapor pressure (kPa);

_{s}*e*is actual vapor pressure (kPa); Δ is the slope of the saturation vapor pressure curve (kPa °C

_{a}^{−1}); and

*γ*is the psychrometric constant (kPa °C

^{−1}). Further details on the calculation of each of the terms in Eq. (2), as well as the standard methods used to estimate variables that are not available (i.e., solar radiation) in the GFS reforecast archive, can be found in appendix A.

_{0}, the PM method requires a large number of inputs including solar radiation, maximum temperature, minimum temperature, wind speed, and relative humidity or dewpoint temperature. Given the limited number of variables archived in the GFS reforecast dataset, it is worthwhile to compare with the Thornthwaite equation to estimate ET

_{0}(Thornthwaite 1948). The advantage of the Thornthwaite equation is that it only requires mean temperature data as input (Vicente-Serrano et al. 2010):

*L*is the maximum number of sun hours (

*h*):

*ω*is the sunset hour angle (defined in appendix A). Here

_{s}*N*is the number of days in the month;

*I*is a heat index calculated as the sum of 12 monthly values of

*i*:

*T*is monthly mean temperature (°C); and

_{m}*m*is dependent on

*I*:

### b. Forecast analog method

The ET_{0} forecasts were produced using a moving spatial window forecast analog approach, as described by Hamill and Whitaker (2006) and Hamill et al. (2006). The moving window approach uses a limited number of GFS reforecast grid points, which increases the likelihood of finding skillful natural analogs (van den Dool 1994). The moving window forecast analog procedure used the calculated value of forecasted ET_{0} at a given lead using one of the approaches in Table 1 to find a subset of analog forecasts from the historical reforecast archive that were most similar (based on root-mean-square error) within the limited spatial region. Once the dates of the nearest analogs were chosen, the corresponding fine resolution estimates were obtained for these dates from the 32-km-resolution-grid ET_{0} values computed from the NARR. For this work forecast analogs were chosen from a subset of nine grid points (Fig. 1). This subset of nine points was used to determine the finescale analogs within the interior of the domain (Fig. 1). This process was then repeated across the region of interest. Following Hamill and Whitaker (2006), analog forecasts were selected within a ±45-day window around the date of the forecast and the best 75 analogs were chosen to construct the forecast ensemble. A cross-validation procedure was employed where dates from the current year were excluded from the list of potential analogs.

Summary of methods used to find forecast analogs.

Since the GFS reforecast archive does not include solar radiation output, and only includes variables at a 12-h temporal resolution, this work employed the R2 dataset in the selection of forecast analogs by substitution of long-term climatological mean daily values or bias correction of GFS reforecasts (Table 1). The rationale to the substitution of climatological mean values from the R2 dataset is that it effectively provides an appropriate estimate (though not perfect) of the parameter in question. In doing so, it lessens the parameter’s importance in the selection of forecast analogs (since all of the potential analogs within the ±45-day window will have been calculated using the same/similar values) and the analog selection becomes weighted toward other terms in Eq. (2).

*i*at grid

*j*was calculated as

*F*(

*x*) and

*F*

^{−1}(

*x*) denote a CDF of daily data

*x*and its inverse, and subscripts GFS reforecast and R2 indicate the GFS reforecast raw data and R2 data, respectively. This bias-correction process was able to remove both bias of underprediction and overprediction assuming direct correspondence between GFS reforecast and R2 exceedance probabilities.

### c. Evaluation procedure

*S*) is defined as

*P*and

_{f}*P*are the probabilities of the forecast and observation from their respective empirical cumulative distributions. Values of

_{o}*S*range from 2 to −1 and correct forecasts at the extremes score higher than those in the middle of the distribution, which means it gives relatively more penalty when forecasting events around average values, but gives relatively higher scores and less penalty for correct forecasts of extreme events (Potts et al. 1996; Sobash et al. 2011; Zhang and Casey 2000). The value of

*S*can be expressed as a skill score SK with a range 100% to −100%, where a skill score of 100% indicates a perfect forecast and a skill of 0% indicates a skill score equivalent to climatology:

*S*depending on whether the numerator is positive or negative. If the numerator is positive,

*S*is calculated as the best possible forecast, using Eq. (8) with

_{m}*P*=

_{f}*P*. If the numerator is negative,

_{o}*S*is calculated as the worst possible score using Eq. (8), with

_{m}*P*= 0 for

_{f}*P*≥ 0.5 and

_{o}*P*= 1 for

_{f}*P*< 0.5.

_{o}_{f}is the Brier score of the forecast and BS

_{c}is the Brier score of climatology. Climatology was computed from daily values ±30 days of the forecast date. The BS

_{f}and BS

_{c}are calculated as

*n*is the number of forecasts and observations of a dichotomous event;

The resolution and reliability of categorical forecasts were evaluated using the relative operating characteristic (ROC) diagrams (e.g., Sobash et al. 2011; Wilks 2006) and reliability diagrams (e.g., Wilks 2006), respectively. The ROC diagram compares hit rates to false alarm rates at different forecast probability levels and is a measure of how well the probabilistic forecast discriminates between events and nonevents. An ROC curve that lies along the 1:1 line indicates no skill and a curve that is far toward the upper-left corner indicates high skill. The reliability diagram indicates the degree that forecast probabilities match observed frequencies. An overall measure of the reliability of the forecasts can be assessed by the deviation of the reliability curve from the diagonal. For a perfectly reliable forecast system, the reliability curve is aligned along the diagonal. Curves below (above) the diagonal indicate overforecasting (underforecasting). The nearer the curve is to horizontal, the less resolution in the forecast.

## 4. Results and discussion

Table 2 and Table 3 show the overall mean LEPS and BSS skill scores for lead days 1 and 5 for all ET_{0} methods used to find forecast analogs. Overall, PM_RH and PM_RHRs, which used 700-mb RH data, were more skillful than the other methods. For lead day 1, PM_RHRs had the highest skill for overall results (based on the LEPS skill score), lower extremes, lower terciles, and upper terciles, while methods Thorn and PM_Rs had the highest skill for middle terciles and upper extremes, respectively (Table 2). Among the five methods that did not use GFS reforecast RH data, Thorn, which used the Thornthwaite equation with only mean temperature, had the highest skill for middle terciles (Table 2); PM_Rs, which used the PM equation with the combination of climatological mean values of *R _{s}* from the R2 dataset and temperature and wind speed from the GFS reforecast archive (Table 1), showed the highest forecast skill for overall results, lower extremes, lower terciles, upper terciles, and upper extremes. For lead day 5, the overall skill (based on the LEPS skill score) of PM_RH, Thorn, PM_Rs, and PM_RHRs is approximately 8.0, which outperformed PM_GFS, PM_RsT, and PM_BC in terms of the overall forecast (Table 3). The lower extreme and middle tercile forecasts showed no skill in lead day 5 for all the seven methods, while some forecasts were skillful for the lower tercile, upper tercile, and upper extreme categories, with PM_Rs and PM_RHRs showing the highest skill for lower tercile and upper tercile, and PM_Rs and PM_RsT showing the highest skill for the upper extreme forecast.

The overall average LEPS skill score and BSS for lead day 1. LEPS skill score and BSS are, respectively, evaluating the overall skill and categorical skill; the five categories represent <10%, <⅓, ⅓–⅔, >⅔, and >90%. The highest scores are highlighted in bold in each skill category.

### a. Evaluation of reference evapotranspiration methods in time

Figures 2–4 show the skill of ET_{0} forecasts, by method, for lead day 1 and lead day 5 in terms of LEPS skill score (Fig. 2), BSS of forecasted extreme values (Fig. 3), and BSS of forecasted terciles (Fig. 4). Overall, ET_{0} estimated by PM_RH and PM_RHRs were more skillful than the methods that did not use GFS reforecast 700-mb RH data. According to the LEPS skill score (Fig. 2), PM_RH and PM_RHRs lead day 1 forecasts showed similar patterns of skill and were generally greater than other methods in cold months, with PM_BC showing slightly higher skill in warmer months (May–August). The LEPS skill score in lead day 5 shows the skill is higher in cold months than in warm months for all the seven methods. PM_RHRs still performed the best in cold months, while PM_GFS and PM_BC forecasted equally best in warm months. In terms of BSS for lower extreme forecasts (Fig. 3), PM_RHRs was the most skillful for lead day 1 and lead day 5 when the BSS was above zero. For upper extreme forecasts, PM_Rs and PM_RsT generally showed the highest skill during the cold months, while PM_BC was showed the greatest skill in July and August for both lead day 1 and lead day 5. In terms of BSS for tercile forecasts (Fig. 4), PM_RHRs showed the greatest skill for both lead day 1 and lead day 5 for lower tercile forecasts, with PM_BC showing slightly higher skill in June and July; for the middle tercile forecast, the BSS of all the seven methods over the year were within a range of −0.1 to 0.1, and no single method was the best over other methods in terms of BSS in different months; for the upper tercile forecast, PM_RHRs had the highest skill in cold months, while PM_BC and PM_Rs showed the highest in the other months.

Comparison of LEPS skill score for the seven methods as a function of time of the year: (a) LEPS skill score at lead day 1 and (b) LEPS skill score at lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of LEPS skill score for the seven methods as a function of time of the year: (a) LEPS skill score at lead day 1 and (b) LEPS skill score at lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of LEPS skill score for the seven methods as a function of time of the year: (a) LEPS skill score at lead day 1 and (b) LEPS skill score at lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of (top) lower and (upper) BSS for the seven methods extreme forecasts as a function of time of the year: BSS at lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of (top) lower and (upper) BSS for the seven methods extreme forecasts as a function of time of the year: BSS at lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of (top) lower and (upper) BSS for the seven methods extreme forecasts as a function of time of the year: BSS at lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 3, but for tercile forecasts and addition of middle tercile BSS.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 3, but for tercile forecasts and addition of middle tercile BSS.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 3, but for tercile forecasts and addition of middle tercile BSS.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

While the GFS reforecast RH data used here was at 700-mb height [and not at 2-m height as typically required for Eq. (2)], it nevertheless played an important role in improving the forecast skill in the humid southeastern United States. In addition, using external R2 *R _{s}* climatology was found to improve skill in cooler months, while the PM_BC approach produced higher skill in warmer months. In the PM equation, Δ and

*u*

_{2}were calculated using readily available output from the GFS reforecast archive, while the values of

*e*,

_{a}*e*, and

_{s}*R*require, or can be estimated from (in the case of

_{n}*R*),

_{n}*T*

_{max}and

*T*

_{min}. The

*T*

_{max}and

*T*

_{min}were estimated from the available 12-h output from the GFS reforecast archive, and daily RH

_{mean}was also determined from 12-h output and at 700-mb height, which arguably makes these terms the most uncertain in Eq. (2). Comparison of PM_GFS, Thorn, PM_Rs, PM_RsT, and PM_BC suggests that using the Thornthwaite equation with only mean temperature data was able to achieve similar skill to using the PM equation with

*T*

_{mean}and

*u*

_{2}data and approximate (12 h) GFS reforecast

*T*

_{min}and

*T*

_{max}data (PM_GFS), indicating that the PM equation failed to improve forecast skill alone without RH

_{mean}. This result also suggests that all the methods with either the PM or Thornthwaite equations show better skill compared to climatology, albeit less than when GFS reforecast RH data were included. Comparison of PM_RH (which did not use R2

*R*climatology) with PM_RHRs (which used R2

_{s}*R*climatology) suggests that the addition of

_{s}*R*contributed only slightly to improvement in skill and that the majority of skill improvement was due to the inclusion of GFS reforecast RH. Comparison of PM_GFS, PM_Rs, PM_RsT, and PM_BC suggests that bias correction of all the GFS reforecast variables with R2 (PM_BC) can improve the forecast skill above that from replacing all of those variables with R2 climatology (PM_RsT). For the ET

_{s}_{0}calculated by PM_RHRs, it is important to note that this method used R2 climatology of

*R*, so the selection of candidate analogs were likely more weighted based on GFS reforecast

_{s}*T*

_{mean}, RH, and

*u*

_{2}. As PM_GFS, PM_RH, and Thorn suggest,

*u*

_{2}was likely less important than

*T*

_{mean}and RH in the selection of analogs.

In Figs. 5 and 6 the BSS of PM_RH and PM_RHRs are repotted to show the relative skill between the five categories. For both methods, the upper and lower terciles showed the greatest skill for both lead day 1 and lead day 5, the middle tercile the least for lead day 1 and the extreme forecasts the least for lead day 5, and all forecasts in five categories were skillful over all 12 months for lead day 1 except the middle tercile forecast in June, August, and September. For the lead day 5 forecasts of the two methods, only the upper tercile, lower tercile, and lower extreme forecasts were skillful from December to July, while there was no skill in other categories and other months.

Comparison of the categorical forecasts in time for PM_RH in lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of the categorical forecasts in time for PM_RH in lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of the categorical forecasts in time for PM_RH in lead day (a) 1 and (b) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 5, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 5, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 5, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Figures 7 and 8 show the ROC and reliability diagrams for the five categorical forecasts for PM_GFS, PM_RH, and PM_RHRs for both lead day 1 and 5. All of the ROC diagrams in Fig. 7 show that the lower tercile forecast had the greatest resolution for all methods and both lead days, this was followed by the upper tercile, upper extreme, lower extreme, and middle tercile, respectively. The ROC diagrams show that the resolution of the three methods was very close, and similar results can be found for the lead day 1 and 5 forecasts. In Fig. 8, there were few differences in reliability among the three methods and the two lead days. For all methods there was some overforecasting at low probabilities. All methods showed some overforecasting bias, especially for the upper and lower tercile and upper extreme forecasts. High probabilities for the upper extremes were overforecasted for all the three methods. Although the LEPS and BSS showed that PM_RH and PM_RHRs were more skillful than PM_GFS, and the skill in lead day 1 was greater than lead day 5, the ROC and reliability diagrams show the resolution and reliability for the three methods and two lead days were similar. While the BSS indicated no skill in several instances, the corresponding ROC and reliability diagrams showed positive skill. One possible explanation is that the climatological ET_{0} may have significant variability across locations and across seasons, which can produce artificially high skill as explained in Hamill and Juras (2006).

ROC diagrams: (a) PM_GFS lead day 1, (b) PM_GFS lead day 5, (c) PM_RH lead day 1, (d) PM_RH lead day 5, (e) PM_RHRs lead day 1, and (f) PM_RHRs lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

ROC diagrams: (a) PM_GFS lead day 1, (b) PM_GFS lead day 5, (c) PM_RH lead day 1, (d) PM_RH lead day 5, (e) PM_RHRs lead day 1, and (f) PM_RHRs lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

ROC diagrams: (a) PM_GFS lead day 1, (b) PM_GFS lead day 5, (c) PM_RH lead day 1, (d) PM_RH lead day 5, (e) PM_RHRs lead day 1, and (f) PM_RHRs lead day 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 7, but for reliability diagrams.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 7, but for reliability diagrams.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 7, but for reliability diagrams.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Figure 9 shows the mean monthly and annual ET_{0} forecasts for lead days 1 and 5 using PM_RHRs. Also shown are the 10th and 90th percentiles of the forecasts and observations. The figure indicates the mean of forecasted monthly ET_{0} matched quite well with the mean of NARR monthly ET_{0}, with a slight underprediction in May–September and overprediction in November–January for both lead day 1 and 5. Similarly, for the 10th percentile of the monthly ET_{0}, there was slight overprediction in warm months and underpredictions in cool months; on the contrary, 90th percentile forecasts were found to underpredict during warmer months and overpredict in cooler months. Figure 9 also indicates that the annual variation of ET_{0} forecasts generally followed observations for both lead day 1 and 5.

Comparison of the (top) monthly and (bottom) annual ET_{0} over October 1979–September 2009 for NARR ET_{0} and PM_RHRs ensemble mean ET_{0} analogs: lead day (a) 1 and (b) 5. The gray zones represent 10th to 90th percentile of NARR ET_{0}, reflecting spatial variation of the monthly and annual ET_{0} over the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of the (top) monthly and (bottom) annual ET_{0} over October 1979–September 2009 for NARR ET_{0} and PM_RHRs ensemble mean ET_{0} analogs: lead day (a) 1 and (b) 5. The gray zones represent 10th to 90th percentile of NARR ET_{0}, reflecting spatial variation of the monthly and annual ET_{0} over the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Comparison of the (top) monthly and (bottom) annual ET_{0} over October 1979–September 2009 for NARR ET_{0} and PM_RHRs ensemble mean ET_{0} analogs: lead day (a) 1 and (b) 5. The gray zones represent 10th to 90th percentile of NARR ET_{0}, reflecting spatial variation of the monthly and annual ET_{0} over the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

### b. Evaluation of reference evapotranspiration methods in space

Based on the LEPS skill score, PM_RHRs forecasts showed greater overall skill than PM_RH for both lead day 1 and lead day 5 (Fig. 10). The forecast skill for both lead day 1 and lead day 5 were highest in the west and northeast. Forecast skill was lowest over Florida and near the coast and in the more mountainous region at the confluence of North Carolina, South Carolina, and Georgia. This may be due to the inability of coarse-scale analogs to accurately match local-scale phenomena related to topographic effects and the influence of the sea breeze over the Florida peninsula (e.g., Marshall et al. 2004; Misra et al. 2011).

The average LEPS skill score of the (top) PM_RH and (bottom) PM_RHRs across the five states for lead day (left) 1 and (right) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

The average LEPS skill score of the (top) PM_RH and (bottom) PM_RHRs across the five states for lead day (left) 1 and (right) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

The average LEPS skill score of the (top) PM_RH and (bottom) PM_RHRs across the five states for lead day (left) 1 and (right) 5.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

Figure 11 shows the BSS of the categorical forecasts for PM_RH and PM_RHRs. For the upper extreme forecasts, PM_RHRs showed greater skill than PM_RH; the greatest skill found in three states (North Carolina, South Carolina, and Alabama) and northern Georgia, while the least skill were mostly in coastal areas. The lower extreme forecasts were less skillful than upper extreme forecasts but showed a similar pattern in space. The upper and lower tercile forecasts showed a similar spatial pattern to the upper and lower extreme forecasts, with PM_RHRs showing greater skill than PM_RH in most of the area. PM_RHRs showed slightly greater skill than PM_RH for lead day 1 middle tercile forecasts and the forecast skill in space was comparably homogeneous.

The average BSS of the (columns one and three) PM_RH and (columns two and four) PM_RHRs for (top to bottom) the five categorical forecasts across the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

The average BSS of the (columns one and three) PM_RH and (columns two and four) PM_RHRs for (top to bottom) the five categorical forecasts across the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

The average BSS of the (columns one and three) PM_RH and (columns two and four) PM_RHRs for (top to bottom) the five categorical forecasts across the five states.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

### c. Evaluation of reference evapotranspiration methods by forecast lead day

Figures 12 and 13 show the BSS and LEPS skill score as a function of month and lead day for PM_RH and PM_RHRs, respectively. Overall, for PM_RH, the forecasts were more skillful in cooler months when the skill scores were above zero. The LEPS skill score showed that the overall forecasts were skillful for the first nine lead days. For the lower extreme and lower tercile forecasts, the BSS were mostly above zero before lead day 5 and the early half of the year were found to be more skillful at later lead days than later half of the year. For the upper extreme forecasts, the warmer months showed more skill at later lead days than cooler months. For the upper tercile forecasts, lead day 7 was still skillful for the months of January, December, and July; however, the skill was modest. For the middle tercile forecasts, the BSS was greater than 0 from lead day 1 to 3 for most months; after lead day 3 the forecasts showed no skill over the year. PM_RHRs showed similar forecast patterns, except it has more skillful lead days in some months (Fig. 13).

BSS of the PM_RH for five categorical forecasts as a function of time and the lead time of the forecast: (a) lower extreme forecast, (b) lower tercile forecast, (c) middle tercile forecast, (d) upper tercile forecast, (e) upper extreme forecast, and (f) overall forecast.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

BSS of the PM_RH for five categorical forecasts as a function of time and the lead time of the forecast: (a) lower extreme forecast, (b) lower tercile forecast, (c) middle tercile forecast, (d) upper tercile forecast, (e) upper extreme forecast, and (f) overall forecast.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

BSS of the PM_RH for five categorical forecasts as a function of time and the lead time of the forecast: (a) lower extreme forecast, (b) lower tercile forecast, (c) middle tercile forecast, (d) upper tercile forecast, (e) upper extreme forecast, and (f) overall forecast.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 12, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 12, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

As in Fig. 12, but for PM_RHRs.

Citation: Journal of Hydrometeorology 13, 6; 10.1175/JHM-D-12-037.1

## 5. Summary and concluding remarks

A forecast analog technique was successfully used to downscale 1–15-day 200-km-resolution ET_{0} forecasts using the GFS reforecast archive and R2 climatology data using seven methods over the states of Alabama, Georgia, Florida, North Carolina, and South Carolina in the southeastern United States. The 32-km-resolution ET_{0} calculated from the NARR dataset using the PM equation was used to evaluate the forecast analogs. The skill of both terciles and extremes (10th and 90th percentiles) were evaluated. The ET_{0} forecast methods, which included GFS reforecast 700-mb RH data in the PM equation (PM_RH and PM_RHRs), showed greater skill compared to the methods that did not use RH. The inclusion of R2 solar radiation data with GFS reforecast data (PM_Rs, PM_RHRs, and PM_RsT) was found to provide a modest increase of forecast skill. The forecasts using both GFS reforecast 700-mb RH data and R2 solar radiation (PM_RHRs) were found to produce slightly greater skill that forecasts using GFS reforecast 700-mb RH data alone (PM_RH). The bias correction of all the GFS reforecast variables with R2 (PM_BC) was found to improve the forecast skill compared to substitution of several variables with R2 climatology (PM_RsT). While the five categorical forecasts were skillful, the skill of upper and lower tercile forecasts was greater than those of lower and upper extreme forecasts and middle tercile forecasts. Most of the forecasts were skillful in the first 5 lead days.

Forecasting ET_{0} using GFS reforecasts is advantageous in many respects. Previously applied ANN models for evapotranspiration forecasts are black-box models, and ANN models developed for one location cannot be implemented in another without local training (Kumar et al. 2011). In contrast, using ET_{0} from NWPM–GCM output preserves the physical relationships between different variables and preserves the spatial correlation of the output. Compared to the work of Cai et al. (2007), who used daily weather forecast messages to forecast ET_{0} deterministically at eight stations over China, the data availability and the forecast resolution for downscaling NWPM–GCM forecasts is arguably more objective and at a finer resolution. There is no evidence to show that statistical downscaling of forecasts are better than dynamical downscaling forecasts (Abatzoglou and Brown 2012) in terms of forecast skill, but statistical methods have an advantage in requiring significantly less computational resources.

The advantage of analog selection based on calculated values of ET_{0} rather than finding analogs for each variable individually is that the PM equation appropriately weights the input variables according to their importance. The relative importance likely changes at different times of year and is captured by the analog approach used. Physically plausible relationships between variables and correlation between variables are also preserved. The disadvantage of this approach is that analogs are found based on the magnitude of ET_{0} and not the relative contribution of advective or radiative forcing. For example, high ET_{0} analog days may be found where wind and relative humidity played a larger contributing role compared to incoming solar radiation and vice versa.

This work showed that a forecast analog approach using the Penman–Monteith equation with several approximated terms could successfully downscale ET_{0} forecasts. However, the need to approximate several terms in the Penman–Monteith equation was likely a limiting factor in forecast skill and indicates the importance in archiving relevant variables in future, next-generation reforecast datasets. Future work in evaluating ET_{0} forecasts generated from retrospective forecast datasets should include comparison to direct model outputs of operational models in order to clearly demonstrate the value of statistical postprocessing relative to direct model output.

## Acknowledgments

This research was supported by the NOAA’s Climate Program Office SARP-Water program Project NA10OAR4310171. The GFS reforecasts data, R2 data, and NARR data were provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, from their website at http://www.esrl.noaa.gov/psd/. The authors thank the reviewers for helpful comments.

## APPENDIX A

### Parameters’ Calculation for the Penman–Monteith Equation

The terms of the Penman–Monteith equation are calculated or estimated following the recommendations of Allen et al. (1998):

*T*air temperature [°C];

*γ*is calculated using Eqs. (A2) and (A3):

*P*is atmospheric pressure (kPa) and*z*is elevation above sea level (m);

*e*is calculated using Eq. (A4):

_{s}*T*is air temperature (°C);

*e*can be calculated using Eq. (A5) with

_{a}*T*

_{dew}, and thus

*T*

_{dew}is dewpoint temperature (°C);

*e*can also be calculated using Eq. (A6) with RH

_{a}_{mean}:

_{mean}is the mean relative humidity.

*R*

_{ns}(MJ m

^{−2}day

^{−1}) is net shortwave radiation, which can be calculated from Eqs. (A7) and (A8):

*a*is the albedo which is 0.23 for the hypothetical grass reference crop. The

*R*(MJ m

_{s}^{−2}day

^{−1}) is incoming shortwave solar radiation, which can be derived from air temperature differences:

*T*_{max}is maximum air temperature (°C),*T*_{min}is minimum air temperature (°C), andis adjustment coefficient [0.175 (°C ^{−0.5})].

*R*(MJ m

_{a}^{−2}day

^{−}1) is daily extraterrestrial radiation [Eq. (A9)]:

*G*_{sc}is solar constant = 0.0820 MJ m^{−2}min^{−1},*d*is inverse relative distance Earth–Sun [Eq. (A10)],_{r}*w*is sunset hour angle (rad) [Eq. (A11)],_{s}*j*is latitude (rad), and*d*is solar decimation (rad) [Eq. (A12)]:

where

*J*is Julian date.

*R*

_{nl}(MJ m

^{−2}day

^{−1}) is net longwave radiation, which can be calculated from Eq. (A13):

*s*is Stefan–Boltzmann constant (4.903 × 10^{−9}MJ K^{−4}m^{−2}day^{−1}),is maximum absolute temperature during the 24-h period , is minimum absolute temperature during the 24-h period , *e*is actual vapor pressure (kPa),_{a}*R*is solar radiation [see Eq. (A8)],_{s}*R*_{so}is calculated clear-sky radiation (MJ m^{−2}day^{−1}) [Eq. (A14)],

*z*is station elevation above sea level (m), andfor

*R*, see (A9)._{a}

*R*can be calculated from Eq. (A15):

_{n}*u*

_{2}can be calculated from Eq. (A16):

*z*is height of measurement above ground surface (m)when

.

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