## 1. Introduction

Potential evapotranspiration (PET) is defined as the maximum ability to evaporate under the assumption of a well-watered surface (Thornthwaite 1948; Brutsaert 1982; Shuttleworth 1993). Accurate and timely estimates of PET are essential for agricultural and water resource planning as well as for understanding the impacts of climate variability on terrestrial systems. Our efforts are also motivated by the need for reliable estimates of PET as an input for hydrologic modeling systems in both gauged and ungauged basins. Daily PET inputs can be obtained from pan evaporation measurements or from estimates based on meteorological data that are gathered routinely at many locations around the country. Because of the scarcity and inconsistency of ground-based measurement data, many studies typically use a long-term monthly mean value developed from past climatological records as an alternative to daily estimates (Farnsworth and Thompson 1982). However, evapotranspiration varies significantly from day to day and from year to year. This is especially true in regions that are heavily influenced by monsoonal activity or El Niño–La Niña cycles (Oort and Yienger 1996; Mestas-Nuñez and Enfield 2001; Wang 2002). Therefore, the use of long-term monthly PET estimates that do not account for temporal (or spatial) variation in local conditions may lead to additional uncertainty in hydrologic studies.

In this paper, we explore an alternative methodology to estimate PET by taking advantage of the new generation of Earth Observation Satellites (EOS). The recently launched Moderate Resolution Imaging Spectroradiometer (MODIS) sensor on board the EOS *Terra* and *Aqua* satellites has a higher number of bands (36 spectral bands) and better radiometric resolution than previous sensors [i.e., the Advanced Very High Resolution Radiometer (AVHRR) has five spectral bands]. The *Terra*-MODIS platform provides an opportunity to monitor a number of environmental variables influencing evapotranspiration that were previously unavailable. Our work deals primarily with observations from *Terra* (available from April 2000 to present) in order to obtain a relatively long time series (∼4 yr) for comparison to PET estimates using ground-based data.

Previous studies have utilized remotely sensed data to estimate evapotranspiration, primarily focusing on the use of satellite-based surface temperature and vegetation characteristics [i.e., normalized difference vegetation index (NDVI), leaf area index (LAI), and enhanced vegetation index (EVI)]. Most of the developed algorithms could be considered combination methods, requiring a range of ground-based variables in addition to the remotely sensed inputs. Further, they are designed to apply primarily to clear days (Jackson et al. 1977; Nemani and Running 1989; Goward and Hope 1989; Moran et al. 1994; Yang et al. 1997; Bastiaanssen et al. 1998a, b; Li and Lyons 1999). Other procedures require model (typically simple energy balance) simulations that require estimation of site parameters and relevant input data (Carlson et al. 1995; Margulis et al. 2005). Jiang and Islam (2001) proposed an evapotranspiration (ET) model that primarily uses remotely sensed data and the Priestley–Taylor equation. However, this method uses ground-based net radiation as an input.

While many existing satellite-based methods offer the capability for objectively measuring evapotranspiration from the earth’s surface, they are traditionally limited to clear-sky conditions and for an overpass (instantaneous) time. Thus, previous studies typically utilize a select handful of days (i.e., golden days) under instantaneous clear-sky conditions (Stewart et al. 1999; Jiang and Islam 2001; Nishida et al. 2003; Norman et al. 2003, Venturini et al. 2004). Our developed method can be applied not only to clear-sky days, but also to cloudy days, for development of a continuous daily time series of PET. We base our procedure on the Priestly–Taylor equation, utilizing a net radiation scheme proposed by Bisht et al. (2005) during cloudless days to derive an instantaneous PET value. A sinusoidal model is subsequently implemented to produce a daily PET value from our instantaneous estimate. We then incorporate a simple algorithm that uses a “theoretical clear-sky” net radiation and potential evapotranspiration (linearly interpolated values during clear days), along with several primary cloud factors (i.e., cloud fraction and cloud optical thickness) to estimate net radiation and potential evapotranspiration under cloudy conditions.

We evaluate the MODIS-based PET product over various hydroclimatic regimes by comparing 1-km^{2} (pixel size) estimates to PET estimates derived from point-scale flux-tower observations. We acknowledge that using point-scale data may limit representation of biome heterogeneity and that comparison to MODIS-scale observations inherently adds spatial uncertainty. However, similar to other authors (Bastiaanssen et al. 1998b; Nishida et al. 2003), we undertake initial evaluation of our algorithm against point-scale, ground-based measurements, eliminating the uncertainty that could be associated with model estimates (due to parameter estimation, model structure errors, etc.). The remaining paper is organized as follows: section 2 presents the proposed PET methodology, section 3 compares the performance of the PET product with PET derived from ground-based observations of PET at four study sites, and section 4 presents a discussion and conclusions of our work.

## 2. Methodology

### a. Study and validation sites

To initially validate the MODIS-derived variables and the proposed methodology, a reasonable number of sampling sites, with varying climatology and vegetation, were selected. Three sites within the Ameriflux network and one site from the Oklahoma Mesonet network were chosen based on their biome diversity, the availability of required variables, and a data period (2001–04) that corresponds with the MODIS sensor data (Table 1). The four sites cover a range of ecosystems (from desert grassland to cropland) and also contain a relatively long time series (∼4 yr) to assure our analysis covers a range of hydroclimatic conditions.

The three sites within the Ameriflux network are located in Arizona, Illinois, and Mississippi. These sites provide a sustained set of detailed observations of surface energy fluxes including latent and sensible heat fluxes and related meteorological data at 30-min intervals. While the Arizona and Mississippi sites have been operating from mid-2002, the Illinois site provides data since August 1996. We also use the Oklahoma Mesonet (Elliott et al. 1994) station at Westville, Oklahoma. Relevant data for each of the study sites are provided in Table 1. To extract the ground-based data specific to the *Terra* satellite overpass time, the Ameriflux data (30-min intervals) was linearly interpolated. Linear interpolation was also used to fill small gaps (less than 5 h) when missing data exist. Daily mean values are calculated by averaging. However, daily means were not estimated for days with data gaps greater than 5 h. Additional details of the study sites are available at the Ameriflux Web site (http://public.ornl.gov/ameriflux/index.html) and the Mesonet homepage (http://www.mesonet.org).

### b. Priestley–Taylor formulation

^{−1}),

*γ*is the psychrometric constant (Pa K

^{−1}), and

*α*is the “Priestly–Taylor parameter” that accounts for the complex effects of evapotranspiration. A value of 1.26 is generally accepted in humid regions (Eichinger et al. 1996; Wang et al. 2004). The Priestley–Taylor approach neglects the influence of vapor deficit on the reference-evapotranspiration, primarily relying on radiation and temperature as proxies of PET processes. The equation also does not require wind speed or dewpoint temperature. Although the Priestley–Taylor formulation is fairly simplistic and has associated uncertainty given this simplicity, it has been used extensively and shown to provide reasonable guidance on PET values for agricultural and hydrologic studies (Flint and Childs 1991; Stannard 1993; Sumner 1996). Priestley–Taylor estimates are also found to be highly correlated to Penman-based PET estimates (Pereira and Nova 1992). Use of this equation also satisfies our constraint that each formula variable can be derived from remotely sensed information. We utilize the following methods to derive the necessary Priestley–Taylor variables.

#### 1) Estimating instantaneous PET under clear-sky condition

##### (i) Net radiation

*α*is the surface albedo, Rs is the incoming shortwave radiation (or solar irradiance), Rl

^{↓}is the downward longwave radiation, and Rl

^{↑}is the upward longwave radiation.

*R*) and diffuse irradiance (

_{B}*R*): The direct and diffuse solar irradiance are then computed using the following equations: where

_{D}*E*

_{0}is the solar constant (= 1367 W m

^{−2}), (

*R*/

*R*

_{0})

^{2}is the solar constant correction given the current sun–earth distance, and

*θ*is the solar zenith angle (degrees). For transmittance (

_{z}*T*), five specific attenuation processes are considered: ozone (

*T*

_{O3}), water vapor (

*T*), trace gas absorption (

_{w}*T*), Rayleigh scattering (

_{g}*T*) and aerosol extinction (

_{r}*T*

_{aer}). Here

*γ*is the fraction of diffuse radiation reflected back to the surface [

*γ*= 0.5 for pure Rayleigh scattering from a nonreflecting surface (Paulescu and Schlett 2003)]: where ℓ [Dobson unit (DU), where 1 DU = 2.69 × 10

^{16}molecules(O

_{3}) cm

^{−2}] is the total ozone amount in a vertical column,

*β*is the Angstrom’s turbidity factor (Angstrom 1961),

*w*(g cm

^{−2}) is the water vapor column content,

*P*(hPa) is the local air pressure, and

*P*

_{0}= 1013 hPa. Further details of this scheme can be found in Paulescu and Schlett (2003). The standard air mass (

*m*) can be computed using the Kasten and Young (1989) equation: where

*z*(m) is the station elevation.

_{s}*is surface emissivity,*

_{s}*σ*= 5.67 × 10

^{−8}W m

^{−2}K

^{−4}is the Stefan–Boltzmann constant, and

*T*is the surface temperature.

_{s}*) is determined by water vapor pressure*

_{a}*e*

_{0}(hPa) and air temperature

*T*(K) at the screen level: where ɛ

_{a}_{a}= 1 − (1 +

*ζ*) exp[−(1.2 + 3

*ζ*)

^{0.5}] and

*ζ*= 46.5

*e*

_{0}/

*T*

_{a}.

##### (ii) Soil heat flux (G)

*G*) is estimated from Rn and NDVI using a scheme proposed by Moran et al. (1989): where NDVI is the MODIS-derived normalized difference vegetation index using bands 1 (620–670 nm) and 2 (841–876 nm).

##### (iii) Derivative of saturated vapor pressure *(Δ)*

Air temperature is also required to compute Δ/(Δ + *γ*). Since its effect in the Priestley–Taylor equation is implicit through Δ, Δ/(Δ + *γ*) shows little sensitivity to air temperature with a reported 1%–2% change given a 1°C change in air temperature (Jiang and Islam 2001; Garatuza-Payan et al. 2001). This is another attractive feature of the Priestley–Taylor approach; the insensitivity of *T _{a}* allows the assumption that potential evapotranspiration is directly proportional to net radiation. As a result, we can also estimate potential evapotranspiration using a clear-sky ratio between potential evapotranspiration and net radiation during cloudy conditions when air temperature cannot be retrieved. Once all three quantities (Rn,

*G*, and Δ) are estimated, instantaneous potential evapotranspiration can be readily obtained using Eq. (1) under clear-sky conditions.

#### 2) Conversion of instantaneous PET to daily PET

_{i}are daily and instantaneous net radiation, respectively, DL is the day length (i.e., the difference between sunrise and sunset time), and

*t*is the number of hours after sunrise.

_{i}are daily and instantaneous potential evapotranspiration, respectively.

#### 3) Estimating daily PET for all-sky conditions

We use the following assumptions to develop a “cloudy day” PET estimate: 1) a simple interpolation between adjacent clear days can be used to derive theoretical clear-day shortwave and net radiation values on cloudy days, and 2) the observed cloud fraction and optical thickness are two primary factors modulating the surface evapotranspiration rate.

*N*is the total cloud cover fraction,

*τ*is cloud optical thickness, and

*a*and

*b*are empirical coefficients. In this study, Rs

^{↓}

_{clear}is theoretical clear-day shortwave radiation,

*N*is day-mean cloud fraction, and

*τ*is the cloud optical thickness from MOD08.

*α*is the surface albedo, and

*c*and

*d*are regression coefficients. Once the shortwave radiation is determined through Eq. (13), daily net radiation for all-sky conditions can be determined along with the surface albedo from MODIS.

### c. MODIS products

*Terra*-MODIS datasets. The MODIS geolocation dataset, MOD03, provides the solar zenith angle to compute shortwave radiation. We use surface temperature (

*T*) and the average of bands 31 and 32 to represent land surface emissivity at a spatial resolution of 1 km from MOD11. The MODIS Atmospheric profile product (MOD07) provides several parameters, including air and dewpoint temperature profiles and total ozone, which are implemented in the current method. The spatial resolution of the MOD07 product is 5 km × 5 km, at 20 vertical pressure levels. Air temperature and dewpoint temperature at the two lowest levels (i.e., 1000 and 950 hPa) along with the geopotential heights at each pressure level are linearly interpolated to calculate actual air and dewpoint temperature for the study site elevation under the assumption that the geometric height (i.e., elevation) and the geopotential height are numerically interchangeable in the lower atmosphere. The interpolated air temperature and dewpoint temperature are then taken as surrogates for the temperatures at screen-level height. Our interpolation is formulated as where

_{s}*T*,

_{z}*T*

_{1000hPa}, and

*T*

_{950hPa}are temperatures at a given elevation, 1000 and 950 hPa, respectively;

*Z*

_{1000hPa}and

*Z*

_{950hPa}are the geopotential heights at the two pressure levels; and

*z*is the study site elevation.

_{s}Aerosol optical depth and the Angstrom exponent are obtained from MOD04 to calculate Angstrom turbidity (Angstrom 1961), while total precipitable water vapor content is acquired from MOD05. However, aerosol properties from MOD04 tend to have large gaps in the images due to the complex retrieval from the at-sensor radiance. If these values are not reported for the point measurement site, the closest available pixel value is chosen.

For the albedo calculation, we use the MOD43B3 V004 data product, which consists of black and white albedo for seven spectral bands and three broad bands. We combine the black-sky and white-sky albedos as a function of optical depth obtained from MOD04 to calculate “actual” or blue-sky albedo. Detailed procedures for these formulations can be referenced in Bisht et al. (2005). The Geography Department at Boston University also provides tools that allow users to compute the actual albedo (http://geography.bu.edu/brdf/userguide/tools.html). NDVI (16-day composites) was acquired from the MOD13 land product. Figure 1 illustrates our methodology and the remote sensing products used to derive the final PET estimate.

## 3. Results

### a. MODIS-derived variables

For each study site we select the available clear-sky days from the study period (2001 to 2004). A clear day is defined as the cloud fractional cover from MODIS (MOD06-Instantaneous cloud cover and MOD08-Daymean cloud cover) being less than 0.2 (1.0 represents 100% cloud cover) (Jiang and Islam 2001; Venturini et al. 2004). The total number of selected clear days for each study site is shown in Table 1. However, the number of clear days used in the comparison with ground-based data may differ slightly due to occasional gaps in ground measurements.

Instantaneous PET at the satellite overpass time is highly dependent on the estimation of several key variables, namely, surface temperature, air temperature, dewpoint temperature, and surface radiation. Hence, we first evaluate these variables by comparison with site-specific ground-based measurements (Table 2 and Fig. 2). Overall, both surface and air temperatures from MODIS agree fairly well with ground measurements. However, dewpoint temperature has a slightly larger scatter at all sites (Fig. 2c). Similar results have been reported by Kim et al. (2004) where vertical moisture data from MODIS was underestimated and resulted in a relatively low correlation (*R*^{2} < 0.85), while air temperature showed a fairly good agreement when compared to radiosonde data.

Both air temperature and dewpoint temperature are key variables in the downward longwave radiation scheme (Prata 1996). We found that the Prata scheme generates a systematic positive bias when we compare computed incoming longwave radiation using the Prata scheme with the ground-based measurements of air and dewpoint temperature and the measured longwave radiation (not shown here). The resulting bias is 22 W m^{−2} (6.8% of instantaneous average), 34 W m^{−2} (9.7%), and 48 W m^{−2} (14.5%) at the Bondville, Goodwin, and Audubon sites, respectively. In addition, at the Audubon site, the bias is slightly increased as the magnitude of longwave radiation decreases (with a regression slope of 1.27). Thus, the Prata scheme does not seem to perform as well at the semiarid site. In addition, the correlation error (*R*^{2} ≈ 0.67) associated with dewpoint temperature translates to the modeled longwave radiation having the lowest correlation (*R*^{2} ≈ 0.88) at the Audubon site (Arizona). Since the Mesonet network does not provide upward or downward longwave radiation measurements, comparison could not be undertaken at the Westville site.

Figure 2d shows the comparison of ground observations and estimated downward shortwave radiation using the PS (Paulescu and Schlett) model. The comparisons result in a bias of 27.9 W m^{−2}, RMSE of 76.4 W m^{−2}, and *R*^{2} of 0.89. Several similar parameterization schemes were evaluated in our work, including Zillman’s (1972) scheme used in the Bisht et al. (2005) study; however, the PS model was found to have the best performance. The PS model also takes into account many of the factors influencing solar radiation transmissivity and emissivity such as aerosol, water vapor, and ozone. In addition, the RMSE of 76.4 W m^{−2} (9.5% of the mean shortwave radiation) is very comparable to earlier studies using the Geostationary Operational Environmental Satellite (GOES) solar radiation, which typically results in a 10% error (RMSE is 10% of daily mean) at the daily time step, and 15%–20% error at the hourly time step (Schmetz 1989; Stewart et al. 1999; Garatuza-Payan et al. 2001).

The estimation of outgoing longwave radiation is directly influenced by the accuracy of two primary inputs: surface temperature and emissivity. Thus, as shown in Fig. 2e, fairly high accuracy in the surface temperature estimation results in good agreement between the estimated and observed longwave radiation, with an average RMSE of 19.5 W m^{−2} and an *R*^{2} of 0.97 for the three sites where ground-based data were available.

Comparison between the observed net radiation and the modeled net radiation in Fig. 3a shows a slight scatter due to the cumulative effect of bias in both the shortwave and longwave radiation schemes. The overall RMSE of net radiation derived from MODIS ranges from 62 to 104 W m^{−2}. The largest bias is observed at the Westville site, where measured net radiation is on average 93.8 W m^{−2} lower than the estimated net radiation during the MODIS overpass time. The observed bias at the Westville site is mostly related to the overestimation in the shortwave radiation. However, given that our results are based on a much larger number of cloud-free days (from 161 to 343), we compare favorably to results reported in Bisht et al. (2005), where a RMSE of 74 W m^{−2} and *R*^{2} of 0.89 was achieved with only 15 completely cloud-free days.

### b. Clear-sky instantaneous PET

The remotely sensed PET estimate is compared with a ground-based PET (using the Priestley–Taylor equation but with the ground-measured data). Results for the comparison are shown in Fig. 3b. As PET is nearly proportional to net radiation in the Priestley–Taylor equation, the scatter pattern is similar to that seen in the net radiation comparison. However, the PET comparison is slightly improved over the net radiation in terms of correlation. While the regression slope is not different from the net radiation results, the regression intercept is significantly reduced over all the study sites. This is likely due to the consistent overestimation of ground heat flux using Moran’s scheme (not shown here). We note that the Moran formulation is originally based on AVHRR-derived NDVI. The MODIS and AVHRR sensors have widely differing bandpasses with narrower MODIS red and near-infrared (NIR) bands (620–670; 841–876 nm) relative to those of the AVHRR (570–700; 710–980 nm), which can result in fairly large discrepancies between MODIS and AVHRR NDVI values (Huete et al. 2002). As a result, we assume that a major cause of overestimation in the ground-heat-flux estimation may be attributed to the sensor-dependent NDVI value. We are currently investigating improved ground-heat-flux estimates using MODIS-based formulations.

Even though the magnitude of the ground heat flux is significantly smaller than other surface energy fluxes at most locations, many empirical studies have shown that ground heat flux is neither constant nor negligible on a diurnal time scale (Brutsaert 1982; Humes et al. 1994; Kustas et al. 1993; Kustas and Goodrich 1994). The magnitude of ground heat flux at the Audubon site (semiarid region) is significantly larger than at the other study sites and the impact of a systematic bias is more significant. However, in general, results from the instantaneous PET comparison indicate that there is reasonable agreement between the PET computed by observed measurements and that derived through our MODIS-based method.

### c. Clear-sky daily PET

One of the key assumptions in our sinusoidal model is that Rn begins to rise (become positive) at sunrise and declines at sunset. The sunrise and sunset times obtained from the U.S Navy (http://aa.usno.navy.mil/data/docs/RS_OneYear.php) do not precisely match the observed net radiation flux at our study sites. To examine the sunrise and sunset timing impact, we substituted the observed sunrise and sunset time for net radiation flux into the sinusoidal model to estimate the daily mean net radiation. We found that the impacts of these errors are almost negligible or reduced in the daily averaging of net radiation, as also observed by Bisht et al. (2005).

Comparison between the sinusoidal model generated daily net radiation and measured daily net radiation is shown in Fig. 4a. Statistics are also presented in Table 3. The daytime mean measured net radiation is obtained by averaging cumulative totals having positive measurements. At all sites, the daily mean Rn estimates contain lower RMSE errors when compared to the instantaneous Rn estimates. As stated previously, the magnitude of the daily averaged estimates is smaller than the instantaneous and, subsequently, the relative error is smaller. Although the magnitude of the RMSE errors has been decreased, the correlation coefficient (*R*^{2}) is significantly degraded, especially at the Westville site. This is attributed to days that deviate from the norm due to significant temporal variation in clouds (i.e., these days are not truly clear days).

The daily averaged MODIS-based PET is compared to the daily PET calculated using ground-based data. Results are illustrated in Fig. 4b and statistics are provided in Table 3. Compared to the instantaneous PET results, the RMSE as a percentage of the mean is similar to the instantaneous (<5%) for all study locations except Westville. These results demonstrate that the proposed sinusoidal model performs well for estimating daily net radiation during clear days and our assumption that the ratio of PET to net radiation is constant during the daytime is reasonable for estimating a daily PET.

### d. All-sky daily PET

To determine PET under cloudy conditions, we estimate shortwave radiation using the relationship provided in Eq. (13). We calibrated *a* and *b* coefficients at each site using a least squares method to fit the daily ground-based incoming shortwave radiation. Since *Terra*-MODIS has a partial or total loss of data for some days during our study period, we exclude the days that have missing cloud data (i.e., cloud fraction and cloud optical thickness) for MOD08 or missing ground-based solar radiation. The estimated coefficients and the statistics for the comparisons are listed in Table 4. The best combinations obtained by least squares method for each study site yielded a fairly narrow range of coefficient values (1.01–1.04 for *a*, and 0.32–0.41 for *b*). In a sensitivity analysis of coefficients, it was found that *a* is significantly more sensitive than *b*. The RMSE changes about 20% with ±40% of change in *a*, whereas *b* results in only a 5% change in RMSE with the same variation. In spite of the simplicity of the cloud correction method, the modeled daily solar radiation for all-sky days provides fairly good estimates with less than 70 W m^{−2} RMSE (∼20% of the mean solar radiation). The method also results in a strong correlation (*R*^{2} ∼ 0.9) when compared to ground-based measurements radiation measurements.

To determine the general relationship between net shortwave radiation and net radiation in Eq. (14), we use the daytime average values of measured net radiation and solar radiation at each study site and MODIS-derived albedo and determine the slope and intercept parameters by the same least squares method. This relationship was very clear for most of sites with an *R*^{2} value of around 0.95. However, the relationship for the Audubon site is somewhat weaker (*R*^{2} ≈ 0.86). We hypothesize that this simple single regression model may be inadequate at the semiarid site, since the longwave exchange may be more complex than in other regions. Idso (1968) and Idso and Cooley (1971) have attempted to account for the effect of net longwave exchange by adding another term in Eq. (14). Kustas et al. (1994a) show that improved estimates can be obtained by including soil moisture measurements as a modulating factor on longwave radiation in semiarid regions. Alados et al. (2003) indicate that the use of seasonal models [seasonal fitting of Eq. (14)] can also provide appreciable improvement.

The obtained relationships with all measured data for our study sites are shown in Table 5. This table also includes results obtained with the published relationship (Kaminsky and Dubayah 1997; Kustas et al. 1994b). Kaminsky and Dubayah (1997) and Kustas et al. (1994b) have developed similar relationships using data recorded in central Canada containing various land cover types (from prairie grassland to boreal forest) and semiarid regions in Arizona, respectively.

The results obtained with the coefficients derived by Kaminsky and Dubayah (1997) at three of the study sites (Bondville, Goodwin, and Westville) show that the single regression relationship is sufficient to estimate net radiation within an RMSE of 50 W m^{−2} and can be applied to these regions with varying land cover types (from cropland to grassland), similar to the results found in Kaminsky and Dubayah (1997). At the Audubon site, our calibrated coefficients are compared with those from Kustas et al. (1994b) established in a similar semiarid environment. Although the slope is slightly larger than that determined by Kustas et al. (1994b), the intercept nearly duplicates the value obtained by Kustas et al. (1994b).

We applied the obtained linear relationship between net radiation and net shortwave radiation (see Table 5) to estimate all-sky daily net radiation as a function of daily shortwave radiation retrieved through the all-sky cloud correction scheme and surface albedo. The overall bias is fairly small for all of the study sites, ranging from 1% to 6% of the mean net radiation. Our results (and previous sensitivity analysis) show that development of regional coefficients based on climate and surface characteristics can be undertaken (summarized in lookup tables) to allow for the application of the method over larger scales without calibration at each grid or watershed location.

As a final step, daily PET for cloudy days is estimated using the ratio in Eq. (15). Similar to the net radiation results, the modeled daily PET using the MODIS data compares favorably with the calculated PET value from the ground-based measurements (Fig. 5c), showing an RMSE of 1.81 mm day^{−1} (∼30% of the mean) and an *R*^{2} of 0.87 on average (see Table 4). Particularly, three of the sites (Bondville, Goodwin Creek, and Westville) show a higher correlation (*R*^{2} = 0.89) and small bias of 0.34 mm day^{−1} (6% of the mean) at the daily time step. However, performance at the Audubon site (semiarid biome) is less satisfactory (*R*^{2} = 0.86 and bias = −2.05 mm day^{−1} at the daily time step). Since this negative bias is transferred from the instantaneous PET estimation, we attribute it to the overestimation of ground heat flux that was observed during development of the instantaneous estimates. Figure 6 illustrates the daily time series of PET at the four sites during 2004.

We also examine how the proposed method captures the general pattern of monthly PET. The monthly mean is calculated by taking the average of the daily values for each month. Figure 7 shows that the satellite-derived monthly mean PET compares well with the ground-based PET. For reference, the monthly Penman-based PET estimates (using the same ground measurements) is also shown. The monthly comparison (Priestley–Taylor equation versus Penman equation) for sites (Bondville, Goodwin, and Westville) shows lower uncertainty (19% RMSE for the mean PET) than the daily time step. Generally speaking, the MODIS-derived monthly PET is in excellent agreement with the ground-based PET using both the Priestley–Taylor and Penman equations for all study sites except the Audubon site. The biggest discrepancies occur at the Audubon site when higher temperatures are observed (April–July). This error appears to be propagated from the uncertainty of the ground-heat-flux estimates during clear days (see Fig. 3a). Further work is ongoing to improve the robustness of the proposed method over semiarid regions (focusing on a revised MODIS-based ground-heat-flux algorithm). However, the results presented here suggest that the proposed method can successfully estimate PET at both the daily and monthly time scale, with slightly better performance at sites with more humid climatology.

## 4. Discussion and conclusions

Recently, various attempts have been made to develop spatial estimates of PET or ET at regional to global scales by taking advantage of the wealth of remotely sensed products now available to users. However, most of the developed techniques have been limited to clear-sky conditions at a specific satellite overpass time or require a large number of ground-based measurements/parameters. Development of a *daily time series* of near-real-time PET/ET estimates using primarily satellite remote sensing has not been previously undertaken. Our proposed method uses ground observations only for the calibration of radiation coefficients (which we propose to regionalize and provide to users). Thus, the method has the potential to be useful for remote regions of the globe where in situ data are not readily available. Our algorithm is also flexible enough to incorporate ancillary data if relevant information is available.

We have evaluated our method at the point scale for four sites covering a range of hydroclimatic conditions and biomes with in situ measurement data over a 4-yr period. Results are very promising and appear to provide comparable or better estimates of PET than currently available PET methodologies that use ground-based observations (Narasimhan et al. 2003). RMSE is on the order of 30% of the daily mean at the daily time step, which is quite reasonable when compared to previous relevant study (Jiang and Islam 2001). Further improvement is seen when data are aggregated to the monthly scale, with RMSE less than 23% of the monthly mean and a correlation greater than 0.92 for all study sites.

The uncertainty in our MODIS-derived PET estimates may be attributed to several sources: 1) Although the Priestley–Taylor equation is widely used and has the advantage that extra data inputs (such as those required for Penman-type methods) are not necessary, the equation is often considered less sophisticated (and contains higher uncertainty) than Penman-type equations. This is primarily attributed to its simplistic representation of aerodynamic effects through a constant parameter (*α*). However, the performance of the Priestley–Taylor equation (using *α* = 1.26) was quite comparable to the Penman-based estimates except at the semiarid site (not shown here). Furthermore, we hypothesize that the performance of the Priestley–Taylor equation may be improved by adjustment of the *α* parameter for a given location and for the time scale of interest (Brutsaert and Stricker 1979; De Bruin and Keijman 1979; Morton 1983). 2) The MODIS variables (i.e., surface temperature, air and dewpoint temperature, albedo, etc.) are associated with a series of errors dependent upon the viewing angle, spatial and temporal resolution, atmospheric correction, etc. As our method is solely dependent on these variables, the uncertainty in the measurement of these variables directly propagates to the uncertainty of our methodology. We have validated (and addressed uncertainty) for all variables for which we have ground-based estimates. 3) Incoming longwave radiation estimates were systematically overestimated using Prata’s (1996) scheme, especially at the semiarid site. 4) Comparison of shortwave radiation estimates using the PS model (Paulescu and Schlett 2003) and ground measurements suggest an RMSE of 55–70 W m^{−2} for all study sites. Although the magnitude of this error is within the range for comparable parameterization schemes, if the clear-sky shortwave radiation is generally biased, the cloud corrections may also be biased. 5) The overall influence of clouds on surface energy processes are complex (especially multilayer clouds). Consideration of only the cloud fraction and cloud optical thickness in our cloud parameterization is a fairly simplistic approach for representing the true effect of clouds on the daily variation in net radiation. Furthermore, the spatial resolution of the cloud products from MOD08 (1° × 1°) may add uncertainty, especially when compared to point observations. However, given the simple nature of the cloud estimation scheme proposed here, the method still produces a reasonable estimate of PET.

Given the limitation of the once-a-day nature of MODIS satellite observations and the parsimonious methodology we have developed, the bias and RMSE reported here are reasonable and within the current uncertainty reported in other studies (Jiang and Islam 2001; Narasimhan et al. 2003). The proposed methodology demonstrates a large degree of agreement between the estimated PET from MODIS and the ground-based PET using the Priestley–Taylor and Penman–Monteith equations. Particularly, the monthly time series of PET generated in this study provides an alternative to current monthly estimation methods (typically pan evaporation) for numerous regions where ground-based data are unavailable. Since reliable and timely estimates of PET are necessary for water resource planning and hydrologic modeling, as well as for investigating the impacts of climate variability on terrestrial systems, our proposed method has wide application. One should caution, however, that the reliability of the satellite-derived estimates for arid and semiarid land surfaces appears to have a larger degree of uncertainty. Thus, additional work is being undertaken to further evaluate and improve our methodology over semiarid regions. Future work will also involve regional-scale derivation of the MODIS-based PET and evaluation against operational products [National Centers for Environmental Prediction (NCEP) Land Data Assimilation System (LDAS)] and/or regional land surface model output.

## Acknowledgments

Financial support for this work was provided by a grant from the NASA EOS and NOAA GAPP programs (NNG04GP71G). This support is greatly appreciated. The authors also would like to thank the reviewers for their comments and suggestions, which greatly improved this manuscript.

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Location and characteristics of the four study sites. Annual mean precipitation, evaporation, and temperature are obtained from observations collected at each study site during the available period.

Correlation coefficient (*R*^{2}), bias (model–observation), and RMSE statistics for comparison between instantaneous ground-based observations and MODIS-derived variables at the satellite overpassing times during clear-sky days. Mean of the ground-based observations is also provided.

Correlation coefficient (*R*^{2}), bias (model–observation), and RMSE statistics for comparison of daily mean between ground-based observations and MODIS-derived variables for clear-sky days at the four study sites. Mean of the ground-based observations is also provided.

Regression constants developed for the all-sky cloud correction scheme, along with correlation coefficient (*R*^{2}), bias (model–observation), and RMSE statistics for the four study sites. Mean of the ground-based observations is also provided.

Regression results using Eq. (12) for estimating net radiation (Rn). Correlation coefficient (*R*^{2}), bias (model–observation), and RMSE statistics are calculated between the estimated vs measured net radiation; *N* is the number of data used for fitting.