### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[1] Given increasing demands on finite water supplies, accurate estimates of evapotranspiration (LE) from arid shrublands of the Southwestern United States are needed to develop or refine basin water budgets. In this work, a novel approach to estimating the equilibrium (or wet environment) surface temperature (*T*_{e}) and LE from regionally extensive phreatophyte shrublands is tested using complementary theory and micrometeorological data collected from five eddy correlation stations located in eastern Nevada. A symmetric complementary relationship between the potential LE (LE_{p}) and actual LE is extremely attractive because it is based on general feedback mechanisms where detailed knowledge of the complex processes and interactions between soil, vegetation, and the near-surface boundary layer can be avoided. Analysis of computed LE_{p} and eddy correlation–derived LE indicates that there is unequivocal evidence of a complementary relationship between LE_{p} and LE, where the measured and normalized complementary relationship is symmetric when *T*_{e} is utilized to compute the wet environment LE (LE_{w}). Application of a modified Brutsaert and Stricker advection-aridity (AA) model, where *T*_{e} is utilized to compute LE_{w} as opposed to the measured air temperature, indicates an improvement in prediction accuracy over the standard Brutsaert and Stricker AA model. Monthly and annual predictions of LE using the modified AA model are within the uncertainty of the measurement accuracy, making the application of this approach potentially useful for estimating regional LE in arid shrubland environments. Our observational evidence supports the idea of a symmetric complementary relationship yielding an approach with standard parameters, making it simple to apply with satisfactory accuracy. To our knowledge, this work presents the first application and evaluation of the complementary relationship in phreatophyte shrublands while utilizing the *T*_{e} with comparisons to actual LE via flux measurements.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[2] Given increasing demands on finite water supplies in arid environments, the need for accurate estimates of sustainable groundwater resources is greater than ever. Many Great Basin and greater Southwestern United States drainage areas are considered hydrologically closed, where the entire groundwater recharge volume is consumed by evaporation and evapotranspiration along mountain front and valley floor areas. Because phreatophyte shrubs utilize shallow groundwater for transpiration, phreatophyte evapotranspiration is larger than direct precipitation. For example, in eastern Nevada it has been found that evapotranspiration from phreatophyte shrubs can range from 106% to 162% of the measured direct precipitation [*Moreo et al.*, 2007; *Welch et al.*, 2007].

[3] The amount of groundwater recharge that occurs in a given hydrographic basin is difficult to estimate accurately and is therefore commonly quantified by estimating the groundwater discharge for individual basins or entire flow systems if groundwater flows from one basin to another. Quantifying evaporation and evapotranspiration in the Great Basin has long been a major focus for developing and refining basin water budgets [*Maxey and Eakin*, 1949; *Robinson*, 1958; *Eakin*, 1966]. As such, many phreatophyte shrub evapotranspiration rates have recently been reassessed in the Great Basin region using micrometeorological, energy balance, and remote sensing techniques [*Malek et al.*, 1990; *Nichols*, 1994; *Tyler et al.*, 1997; *Nichols*, 2000; *Steinwand et al.*, 2006; *Moreo et al.*, 2007; *Allander et al.*, 2009]. More basic approaches have also been employed to estimate evapotranspiration (LE) from shrublands, such as multiplying the potential LE (LE_{p}) by the ratio of LE to LE_{p}, where micrometeorological, energy balance, and soil water balance methods are used to quantify this fraction [*Granger and Gray*, 1989; *Steinwand et al.*, 2001]. The LE in arid shrub environments is highly correlated to the amount of direct precipitation. Therefore, fractions of LE/LE_{p} covary in time and space with precipitation, making the application of LE/LE_{p} fractions for different time periods or areas of interest difficult and likely inaccurate without accounting for relative precipitation and soil moisture differences. Applying one- or two-source physically based models to estimate LE that consider energy transport from the soil and canopy [*Shuttleworth and Wallace*, 1985; *Kustas*, 1990; *Shuttleworth and Gurney*, 1990] is equally difficult to apply in arid shrub environments with confidence because of uncertainties in parameters relating complex aerodynamic, canopy, and soil resistances to sensible and latent heat fluxes [*Nichols*, 1992; *Stannard*, 1993].

[4] An approach based on general feedback mechanisms is attractive because it allows us to avoid extremely detailed knowledge of the complex processes and interactions between soil, vegetation, and the near-surface boundary layer. For this reason, methods that employ the complementary relationship (CR) of evapotranspiration have become popular, as they rely on feedbacks between LE and LE_{p}. The CR is related to water availability and near-surface atmospheric feedbacks with the land surface. Simply stated, when there is ample water available, LE increases and approaches the LE_{p}. When water is limited and the available energy is fairly constant in space, energy that would have been used for evapotranspiration is now used in the production of sensible heat flux and the vapor pressure deficit increases because of the lack of LE, thus elevating LE_{p}. *Bouchet* [1963] first hypothesized that there are complementary feedbacks between LE and LE_{p}, and related these fluxes to the available energy-limited wet environment LE, termed equilibrium evapotranspiration (LE_{w}). The equilibrium, or wet environment evapotranspiration rate, LE_{w}, is the LE_{p} of a wet surface having an area large enough to influence the atmospheric variables at a regional scale so that LE_{w} ≤ LE_{p}. The complementary relationship can be expressed as

where *b* is a proportionality constant of unity if the relationship is symmetric. A symmetric CR implies that a unit increase in LE will result in a unit decrease in LE_{p}, and when the surface is saturated, LE = LE_{w} = LE_{p} (Figure 1). *Morton* [1969] and *Brutsaert and Stricker* [1979] further developed the idea and proposed a quantitative approach for estimating LE_{p}, LE_{w}, and LE on the basis of a symmetric CR and combination approach for estimating LE_{p}.

[5] The CR has been the subject of much debate regarding (1) whether the CR has physical basis and is actually complementary [*LeDrew*, 1979; *Lhomme and Guilioni*, 2006; *Szilagyi and Jozsa*, 2008; *Pettijohn and Salvucci*, 2009], (2) the cause of decreasing worldwide panevaporation during a period when air temperatures are increasing [*Brutsaert and Parlange*, 1998; *Roderick and Farquhar*, 2002; *Hobbins et al.*, 2004; *Ramírez et al.*, 2005; *Brutsaert*, 2006], and (3) recent findings that the CR is asymmetric for certain conditions [*Pettijohn and Salvucci*, 2006; *Kahler and Brutsaert*, 2006; *Szilagyi*, 2007; *Pettijohn and Salvucci*, 2009]. Despite some skepticism on its heuristic nature, the CR has been extensively applied to estimate regional-scale LE and has been tested against energy and large-scale water balance estimates of LE [*Morton*, 1983; *Brutsaert and Stricker*, 1979; *Hobbins et al.*, 2001; *Ozdogan and Salvucci*, 2004; *Kahler and Brutsaert*, 2006; *Yang et al.*, 2006; *Szilagyi and Jozsa*, 2008]. Results from these studies have all supported a realistic physical basis of the CR. Recent research has focused on various assumptions of model formulations, such as considering the stomatal conductance in the formulation of LE_{p} [*Pettijohn and Salvucci*, 2006], considering the wet environment temperature when estimating LE_{w} [*Szilagyi et al.*, 2009], and considering two-dimensional analytical and numerical modeling of the dry-wet interface [*Pettijohn and Salvucci*, 2009; *Szilagyi and Jozsa*, 2009a, 2009b].

### 3. Study Sites and Meteorological Data

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[7] Study sites are located in eastern Nevada, within the Great Salt Lake and Colorado regional flow systems (Figure 2). The climate of the study sites is arid to semiarid, where the mean annual precipitation ranges from 150 to 250 mm with approximately 40% of the precipitation occurring in the winter months. The monthly average extreme temperatures range from 30°C in July to −10°C in December. The vegetation surrounding the study sites consists of spatially extensive and fairly homogeneous phreatophyte shrub species dominated by greasewood (*Sarcobatus vermiculatus*) with smaller amounts of rabbitbrush (*Chrysothamnus nauseous*), salt grass (*Distichlis spicata*), and sagebrush (*Artemisia tridentata*) (Figure 3), where the depth to groundwater ranges from 2 to 10 m below land surface [*Moreo et al.*, 2007]. Micrometeorological stations at the study sites were operated and maintained by the U.S. Geological Survey as part of the Basin and Range Carbonate-Rock Aquifer System Study [*Moreo et al.*, 2007; *Welch et al.*, 2007]. *Moreo et al.* [2007] computed LE at each site using the eddy correlation approach, where the average energy balance closure error for all sites averaged 10%. Daily average meteorological measurements of net radiation, ground heat flux, air temperature, vapor pressure, and wind speed are used in this study to estimate LE_{p} and LE_{w}, and the eddy correlation–derived LE is compared to CR-predicted LE. Energy balance closure corrections to LE and or sensible heat (*H*) were not performed because of relatively good closure in the original data (∼10%) and uncertainties related to the measured available energy and closure correction procedures [*Twine et al.*, 2000; *Foken*, 2008]. For specifics regarding data processing and micrometeorological instrumentation at the study sites, refer to *Moreo et al.* [2007].

### 4. Advection-Aridity Approach

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[8] The advection-aridity (AA) model proposed by *Brutsaert and Stricker* [1979] is based on a symmetric CR where *b* is unity and (1) becomes

The potential evapotranspiration, LE_{p}, expressed in terms of water depth equivalent of mm d^{−1}, is estimated by applying the combination approach by *Penman* [1948] to compute the potential evapotranspiration,

where Δ is the slope of the saturation vapor pressure curve at air temperature, *γ* is the psychrometric constant, and *Q*_{n} is the available energy at the surface expressed in terms of water depth equivalent of mm d^{−1}. *E*_{a} (mm d^{−1}) represents the drying power of the air and is expressed here using Penman's original Rome wind function for a wet vegetated or free water surface [*Brutsaert*, 1982] as

where *e*_{s} and *e*_{a} are the saturation and actual vapor pressures (hPa) and *U* is the measured wind speed (m s^{−1}) at a 2 m reference level. The Priestley-Taylor equation [*Priestley and Taylor*, 1972] is used to estimate the wet environment evapotranspiration at a length-scale greater than about 1 km as

where *α* is the well-known Priestley-Taylor coefficient. Commonly, *α* is used as a calibration coefficient; however, here *α* is fixed to the Priestley-Taylor original value of 1.26 to reduce the degrees of freedom.

#### 4.1. The Modified Advection-Aridity Approach

[9] *Szilagyi and Jozsa* [2008] argue that Δ in (5) should be evaluated at the wet environment air temperature as opposed to the available (drying environment) air temperature since LE_{w} is intended to represent the wet environment LE. The wet environment air temperature is generally unknown under water-limited conditions but can be approximated by the wet environment surface temperature, *T*_{e}, because in wet environments the temperature gradient of the air is relatively small. *T*_{e} can be estimated iteratively by employing the Bowen ratio, *B*_{o} for a hypothetical small wet surface surrounded by water-limiting conditions so that ambient air temperature can be used:

where *H* is the sensible heat; *T*_{s} and *T*_{a} are wet surface and measured air temperature, respectively; and *e*_{s}(*T*_{e}) is the saturated vapor pressure taken at the wet environment surface temperature. By applying (3) with the measured *Q*_{n}, *T*_{a}, and *e*_{a} to estimate LE_{p}, all terms are known except for *T*_{e} and *e*_{s}(*T*_{e}) and therefore can be solved iteratively. For *T*_{e} to be less than *T*_{a}, *H* is required to be negative, implying advection of energy over the hypothetical wet area. Equation (6) assumes that (1) the measured *Q*_{n} is spatially and temporally constant for each time step (daily in this case), which is valid given the large homogenous fetch at the sites; (2) the extent of the wet surface is small (making the Penman equation applicable with use of ambient weather data); and therefore, (3) measured air temperature and humidity over the surface are just minimally affected by the wet surface and can be estimated by the measured values under water-limited conditions. The key in the application of (6) is the realization that under a constant *Q*_{n}, required for the CR, the surface temperature of a small wet area would stay constant as the environment dries around it as shown by *Pettijohn and Salvucci* [2009] and *Szilagyi and Jozsa* [2009a, 2009b]. The modified AA model proposed by *Szilagyi and Jozsa* [2008] is identical to the original [*Brutsaert and Stricker*, 1979] except for using the iteratively solved *T*_{e} in computing the wet environment *LE* in (5), i.e.,

In arid environments, differences in computed Δ and LE_{w} using *T*_{e} versus *T*_{a} can be significant. While *Szilagyi et al.* [2009] successfully tested the modified AA model with water balance closure data from watersheds across the conterminous United States, (7) has not been validated with measured LE.

#### 4.2. The Normalized Complementary Relationship

[10] Normalization procedures are attractive because they allow results or formulae to be expressed in a dimensionless form normalized by minimum or maximum values, where the minimum or maximum can change depending on location or environment. *Kahler and Brutsaert* [2006] normalized the CR by scaling LE and LE_{p} by LE_{w}, where *E*_{+} = LE/LE_{w} and *E*_{p+} = LE_{p}/LE_{w}. They formulate (1) as functions of the dimensionless variable (termed evaporative moisture index) *E*_{MI} = LE/LE_{p}, where

Figure 4 illustrates the normalized CR where the scaled LE and LE_{p} are functions of *E*_{MI} (i.e., (8) and (9)). As the environment experiences wet surface conditions, *E*_{MI} increases to unity, where LE and LE_{p} approach LE_{w}. Conversely, as *E*_{MI} approaches zero, the environment experiences drying conditions where LE and LE_{p} diverge from LE_{w}. As shown in Figure 4, *b* is a proportionality constant controlling the shape of the CR. In their application of the normalized CR to pan data and Bowen ratio flux measurements, *Kahler and Brutsaert* [2006] recommended a *b* value of 5. The value of *b* has been described for an evaporation pan as a measure of the energy transfer between the pan and the surrounding environment [*Brutsaert*, 2006]. When applied to pan data, the CR is asymmetric (*b* > 1) because of the fact that the pan is exposed to more energy than the surrounding environment via radiation, conduction, and advection, and has increased mass transfer because of its small size [*Kahler and Brutsaert*, 2006; *Brutsaert*, 1982].

[11] In this work, it is shown that by evaluating (5) at *T*_{e} and estimating LE_{p} via the Penman equation, *b* becomes unity, yielding a symmetric CR. Whether (5) is to be evaluated at *T*_{a} or *T*_{e} becomes an issue only in arid environments where the *T*_{a} − *T*_{e} difference can be large [*Szilagyi et al.*, 2009]. Figure 5 illustrates average monthly *T*_{a} and computed *T*_{e} for each site, where it is evident that *T*_{e} differs significantly from *T*_{a} as *T*_{a} becomes large. Making the CR symmetric through the estimation of *T*_{e} is advantageous because it eliminates the need to calibrate *α* and/or *b*. Note that the application of pan data in the CR for estimation of LE_{p} also requires additional meteorological data (air temperature and radiation) for estimating LE_{w}, and it requires a calibrated pan coefficient *C*_{p}, and/or *b* because of the pan's extreme sensitivity (*b* = 4 ∼ 10) to changes in energy between the pan and the surrounding environment.

### 6. Searching for a Complementary Relationship

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[13] Potential complementary feedbacks that occur between the atmosphere and the surrounding environment are explored by analyzing the LE_{p} and measured LE as a function of the *E*_{MI} for all five sites. Daily 7 day average LE_{p} and LE versus *E*_{MI} spanning from approximately 15 August 2005 to 31 August 2007 for all five sites are plotted together, making a total of 3718 days (Figure 6a). Figure 6a illustrates that there is indeed complementary behavior between LE_{p} and LE; however, data points are quite scattered. As stated by *Kahler and Brutsaert* [2006], a truly universal relationship requires that the formulation be dimensionless, which in this case is accomplished by scaling the LE_{p} and LE by the LE_{w}. Scaled LE_{p} and LE as a function of the *E*_{MI} is shown in Figure 6b, where it is clearly evident that a complementary relationship between LE_{p} and LE exists. Still, there is a large degree of scatter and asymmetric behavior in *E*_{p+}. This scatter occurs during winter months when the LE_{w} is very small (0.05–0.1 mm d^{−1}) and LE_{p} is relatively large (0.5–1 mm d^{−1}), which inflates the *LE*_{p}/*LE*_{w} ratio (i.e., *E*_{p+}), resulting in asymmetry in the CR. As discussed in section 5, weather fronts can decouple the dynamic equilibrium between the land surface and the atmosphere. Most weather fronts occur over the study area during winter periods, while spring and summer months are relatively calm and are often associated with fairly consistent high-pressure weather patterns. Because extremely small values of LE_{w} occur in the denominator of *E*_{p+} during winter periods, data from winter periods (December–February) were eliminated. Figure 6c illustrates a more coherent CR between LE and LE_{p} after removing data during winter periods. Roughly 90% of the total LE from the study sites occur during March–November, which for all practical purposes warrants the exclusion of winter periods for quantitative analysis of the shape of the CR and the calibration of *b*. Eliminating winter periods and normalizing LE and LE_{p} by LE_{w} results in unequivocal evidence that a CR exists in these environments (Figure 6d).

#### 6.1. Asymmetric or Symmetric CR

[14] A symmetric CR is desired, not only to provide a consistent and unified theory that explains feedback processes between the land surface and the atmosphere, but also to provide consistency and simplicity for predicting LE. *Kahler and Brutsaert* [2006] show that the increase in available energy that an evaporation pan experiences causes the CR to become asymmetric. *Szilagyi* [2007] suggested that whenever advection of energy is present around the device (such as a class A pan) that estimates LE_{p}, the time rate of change between LE and LE_{p} is not a constant but is a function of the surface temperature. *Pettijohn and Salvucci* [2006] found that nonconvergence and asymmetry of the CR exists when canopy conductance is not considered in estimating LE_{p}. They recommend that the Penman-Monteith equation be employed with specified maximum conductance terms and stability correction to reduce over estimation of LE_{p}, and hence under estimation of LE.

[15] Rather than explore how variations in estimating LE_{p} affect the shape of the CR and prediction accuracy, as was done previously, we chose the *Penman* [1948] formulation of LE_{p} to explore the CR with estimated values of LE_{w} using the wet environment surface temperature. The shape of the CR is evaluated for spring, summer, and fall months for all five sites combined by optimizing the proportionality factor *b* in (8) and (9) simultaneously. In this analysis, the sum of square errors is minimized between 7 day average predicted and measured *E*_{+} and *E*_{p+}. *T*_{e} is iteratively solved by employing (6), the Bowen ratio for a small wet surface using daily average measured air temperature and vapor pressure. Results indicate that *b* = 0.810 using LE_{p} and LE_{w} and *b* = 1.008 using LE_{p} and LE_{w}(*T*_{e}) for all sites combined (Figure 7). Residuals of predicted *E*_{+} and *E*_{p+} about the theoretical normalized CR curves were found to be correlated to net radiation, *R*_{n}, but not *U*, *T*_{a}, or *e*_{a}. Residuals became positive for higher *R*_{n} values and negative with larger variance for lower *R*_{n} values, indicating that errors were largest during spring and fall periods. Unlike the findings by *Pettijohn and Salvucci* [2006], when the standard forms of the Penman (with the Rome wind function) and Priestley-Taylor (*α* = 1.26) equations are employed to estimate LE_{p} and LE_{w}, a unit decrease in the LE_{p} results in more than a unit increase in LE. Of significant interest is the fact that *b* = 1.008, indicating a symmetric CR when the estimated wet environment surface temperature is utilized for computing LE_{w}. Because (5) is “expecting” a wet environment temperature, it seems that the use of *T*_{e} in (5) is more appropriate for applications of the CR in arid environments because of its reliance on the wet environment LE.

### 7. Application and Results

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[16] Application and evaluation of the CR approach to estimate LE have been explored by several investigators at various temporal scales of annual [*Morton*, 1983; *Hobbins et al.*, 2001, 2004], monthly [*Morton*, 1983; *Xu and Singh*, 2005; *Szilagyi and Jozsa*, 2008; *Szilagyi et al.*, 2009], daily [*Brutsaert and Stricker*, 1979; *Crago and Brutsaert*, 1992; *Kahler and Brutsaert*, 2006; *Granger and Gray*, 1990; *Pettijohn and Salvucci*, 2006; *Crago and Crowley*, 2005; *Qualls and Gultekin*, 1997; *Granger and Gray*, 1989; *Szilagyi*, 2007], and 20 min periods [*Parlange and Katul*, 1992]. In this research, application of 7 day moving average LE_{p,} LE_{w}, and LE_{w}(*T*_{e}) are applied to (2) and (7) (i.e., AA and modified AA models, respectively), and the predicted LE is compared to the measured LE for all five sites. A time series comparison for site SPV-2 is shown in Figure 8, where both approaches predict rapid temporal changes in LE reasonably well considering that predictions are simply relying on symmetric feedbacks between LE and LE_{p}. However, there are periods where LE is overpredicted, mainly during spring months. In winter months the predicted LE is often negative as a result of the LE_{p} exceeding the LE_{w} by a factor of 2. These results are generally consistent among all sites.

[17] Underpredictions occur during winter months primarily because of dry, windy conditions as winter weather fronts pass through the study area. These conditions elevate the Penman equation while suppressing the Priestley-Taylor equation as a result of low available energy. For these reasons negative LE predictions were assumed to be zero. Figure 9 illustrates 1:1 plots of measured and predicted monthly LE using (2) and (7), where *R*^{2} is 0.77 and 0.71 for the AA and AA(*T*_{e}) models, respectively. Although the LE computed with the AA(*T*_{e}) model has more scatter because of uncertainties in the iteratively computed wet environment temperature, the predicted LE is closer to the 1:1 line when compared to the AA model–predicted LE. The average monthly percent bias for all sites improves from 18% to 1% when LE is computed with the AA(*T*_{e}). The average monthly root-mean-square error for all sites is 11 and 13 mm for the AA(*T*_{e}) and AA models, respectively. Figure 10 illustrates the annual total LE for approximate water years (1 October to 30 September) for years 2006 and 2007, where it is evident that the inclusion of the wet environment temperature in the AA(*T*_{e}) model improves the annual total LE as compared to the AA model. Considering that the measurement error is approximately 10%, most of the predicted annual LE using the AA(*T*_{e}) model is generally within the measurement error.

### 8. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[18] Regional estimates of LE using the CR in arid environments require that *E*_{a} be sensitive to fairly rapid changes in surface moisture conditions. A time series of LE_{p}, LE_{w}, *E*_{a}, eddy correlation–derived LE, and available volumetric soil moisture measured at 15 cm depth at SPV-2 is illustrated in Figure 11, where it is evident that the LE_{p}, *E*_{a}, and LE are slightly positively correlated to changes in measured soil moisture in winter months because of non-water-limited conditions. As the LE_{p} and LE_{w} increased during spring months (days 90–150) because of increased *Q*_{n} and *E*_{a}, LE was fairly constant and insensitive to the large decrease in soil moisture. The fairly constant rate of LE during this period was likely due to the utilization of shallow groundwater, where the depth to water at SPV-2 was at an annual minimum of 2.1 m below land surface. During early summer (days 150–200) the soil moisture was continuing to decrease, while the LE was slightly increasing because of the general increase in *Q*_{n} and *E*_{a} and the utilization of shallow groundwater by phreatophyte taproots. During midsummer (days 200–225), soil moisture significantly increased because of summer monsoon rains, LE reached the annual maximum, and *E*_{a} and hence LE_{p} was significantly reduced because of the increase in LE, while LE_{w} remained fairly constant. Late summer and early fall months (days 225–300) experienced a general decrease in LE, which follows the general decrease in LE_{w}, *E*_{a}, and hence LE_{p}. During the later portion of this period there is a marked increase in soil moisture and slight increase in LE. These rapid land surface and lower boundary layer feedbacks are remarkably captured by the AA model in the prediction of LE as illustrated in Figure 8. Annual precipitation for these sites ranged from 150 to 250 mm yr^{−1} during the study period, where the annual measured and predicted LE ranged from 170 to 400 mm yr^{−1} (Figure 10), indicating consumption of shallow groundwater. The AA model is able to predict the utilization of shallow groundwater through the reduction in the measured vapor pressure deficit and LE_{p}. Both seasonal and annual results of this work serve as prime examples of the fundamental concept, fairly robust prediction accuracy, and merit of applying the CR in areas where LE and LE_{p} are correlated not only with precipitation [*Yang et al.*, 2006] but other sources such as shallow groundwater.

[19] Temporal variations in soil moisture have been used to scale LE_{w} or LE_{p} as a means to estimate LE from crops [*Davies and Allen*, 1973], forests [*Flint and Childs*, 1991; *Black*, 1979], and desert vegetation [*Garcia et al.*, 2009]. Rapid increases of LE commonly occur in arid and semiarid environments during spring and summer months because of precipitation events, but these events are not always captured in measured soil moisture at shallow depths. *Garcia et al.* [2009] developed a function relating measured soil moisture to inversely calibrated *α* values and used these soil moisture–dependent *α* values to scale LE_{w} for prediction of bare soil LE in southern Nevada. *Garcia et al.* [2009] reported *α* values ranging from 0.25 to 1.4 for nearly constant volumetric soil moisture of 0.05 and *α* values ranging from 0.75 to 1.75 for nearly constant volumetric soil moisture of 0.20. *Black* [1979] also showed this large variability during dry periods, where LE/LE_{w} ranged from 0.2 to 0.9 for volumetric soil moistures of 0.12 to 0.15 because of precipitation events. Measured soil moisture has been used to develop various forms of the CR as well [*Crago and Brutsaert*, 1992; *Kahler and Brutsaert*, 2006; *Pettijohn and Salvucci*, 2006; *Yan and Shugart*, 2010]; however, similar to findings of *Kahler and Brutsaert* [2006], the use of soil moisture measurements at 15 cm depth to supplement *E*_{MI} in this research did not yield usable results for reasons related to high winter time soil moisture storage and low LE_{w} with relatively high corresponding LE_{p} from passing weather fronts, and rapidly varying soil moisture conditions in the summer time. These findings suggest that measured soil moisture at one depth does not properly characterize the surface soil moisture status, nor does it characterize the rapidly varying evaporative and transpirative conditions that exist in water-limited environments.

[20] The use of *T*_{e} in (5) to estimate the LE_{w} is of particular importance in arid environments because *T*_{a} − *T*_{e} can be quite large, as shown in this work. *E*_{a} was originally calibrated with ambient measurements using the available *T*_{a} and humidity near the experiment site [*Penman*, 1948]. Had these measurements been taken directly over a large free water surface, the parameters of the wind function in (4) would be different. In contrast, (5) was calculated and *α* was optimized under actual regionally wet conditions [*Priestley and Taylor*, 1972]. This leads to the reason why in arid environments (3) does not require *T*_{e} and why (5) is more accurately estimated with *T*_{e}. Our findings demonstrate this argument by introducing *T*_{e} in (5) for estimating LE_{w}, where the CR becomes symmetric from a slightly asymmetric CR using *T*_{a} in (5) where the value *b* = 0.876. This finding is consistent with the idea that advection of energy at the study sites is negligible, otherwise the value of *b* would exceed unity, as *Kahler and Brutsaert* [2006] point out. This leads to a logical explanation of why *b* would have a value less than unity: the estimated LE_{w} was artificially inflated by using the measured *T*_{a}.

### 9. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Objectives
- 3. Study Sites and Meteorological Data
- 4. Advection-Aridity Approach
- 5. Data Preparation
- 6. Searching for a Complementary Relationship
- 7. Application and Results
- 8. Discussion
- 9. Conclusions
- Acknowledgments
- References

[21] Quantifying monthly and annual rates of LE in arid shrub environments is important for updating and developing groundwater budgets in the Southwestern United States. In this study, we demonstrate clear evidence of a CR between LE and LE_{p} in arid shrublands by utilizing eddy correlation data and commonly measured weather variables. We show that the CR is fairly robust for predicting rapid changes in LE, as well as total monthly and annual LE rates; however, winter predictions are underestimated. Furthermore, we show that by employing the wet environment temperature, *T*_{e}, for estimating LE_{w}, the optimized CR becomes symmetric, reduces the bias, and improves the accuracy of the total monthly and annual predicted LE when compared to eddy correlation–derived LE. The fact that CR is symmetric in arid shrub environments, where *b* equals unity and *α* is the original quantity of 1.26, leaves no calibration parameters to estimate LE and requires only commonly measured weather data and measured or predicted *Q*_{n}.

[22] Application of the CR can be used to study complex feedbacks between the land surface and near-surface boundary layer and to complement other studies such as remote sensing, vegetation phenology, and regional-scale hydrologic and atmospheric modeling. This paper summarizes the first application of the CR to estimate LE from phreatophyte shrubs and, it is hoped, will spur wider application to better understand and predict hydroclimatology in arid environments.