Journal of Hydrology 519 (2014) 2312–2317

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Journal of Hydrology

journal homepage: www.elsevier .com/locate / jhydrol

Estimating long-term changes in actual evapotranspiration and waterstorage using a one-parameter model

http://dx.doi.org/10.1016/j.jhydrol.2014.10.0140022-1694/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Tata-Cornell Agriculture and Nutrition Initiative,Charles H. Dyson School of Applied Economics and Management, Cornell University,Ithaca, New York 14853, United States. Tel.: +1 607 255 7227.

E-mail address: ans62@cornell.edu (A.N. Sharma).

Asha N. Sharma a,b,⇑, M. Todd Walter a

a Department of Biological and Environmental Engineering, Cornell University, Ithaca, New York 14853, United Statesb Tata-Cornell Agriculture and Nutrition Initiative, Charles H. Dyson School of Applied Economics and Management, Cornell University, Ithaca, New York 14853, United States

a r t i c l e i n f o s u m m a r y

Article history:Received 5 March 2014Received in revised form 1 August 2014Accepted 4 October 2014Available online 12 October 2014This manuscript was handled byKonstantine P. Georgakakos, Editor-in-Chief,with the assistance of Matthew McCabe,Associate Editor

Keywords:StorageEvapotranspirationClimate change

Estimations of long-term regional trends in evapotranspiration (E) and water storage are key to ourunderstanding of hydrology in a changing environment. Yet they are difficult to make due to the lackof long-term measurements of these quantities. Here we use a simple one-parameter model in conjunc-tion with Gravity Recovery and Climate Experiment (GRACE) data to estimate long-term E and storagetrends in the Missouri River Basin. We find that E has increased in the river basin over the period1929–2012, consistent with other studies that have suggested increases in E with a warming climate.The increase in E appears to be driven by an increase in precipitation and water storage because potentialE has not changed substantially. The simplicity of the method and its minimal data requirements providea transparent approach to assessing long-term changes in hydrological fluxes and storages, and may beapplicable to regions where meteorological and hydrological data are scarce.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Terrestrial water storage is critical to many ecosystem services(Brauman et al., 2007). Moreover, it is expected that increasedhydrologic variability associated with climate change is likely toincrease human dependence on some components of terrestrialwater storage (Taylor et al., 2013). All the components of terrestrialwater storage (henceforth referred to simply as ‘‘storage’’), namely,groundwater, soil moisture, snow and surface reservoirs, are them-selves susceptible to a variety of factors including changes in cli-mate (e.g. Döll, 2009; Seneviratne et al., 2010; Taylor et al., 2013;Trenberth, 2011), land cover (e.g. Nosetto et al., 2012; Wanget al., 2012), and direct human use (Famiglietti et al., 2011;Rodell et al., 2009). They may also have important feedback effectson climate (e.g. Seneviratne et al., 2010).

Sustainable management of water resources would be aidedconsiderably by an understanding of long-term patterns of storagevariability and its responses to a variety of climate forcings. Yetlong-term records of storage are sparse in most regions. Even inregions with active monitoring programs, data cover only a fewdecades and there is considerable uncertainty about the reliability

of regional extrapolations from point or sub-regional data.The difficulty in generating long records of water storage is inti-mately related to the difficulty of estimating another key compo-nent of the terrestrial water balance: evapotranspiration,resulting in a one equation- two unknown quantities problem.

Over the past decade, the Gravity Recovery and Climate Exper-iment (GRACE) has generated data on regional changes in terres-trial water storage. Data from GRACE have been used tocharacterize changes in water storage in many regions across theworld (e.g. Crowley et al., 2006; Famiglietti et al., 2011; Rodellet al., 2009, 2007). Because GRACE data have existed only for a dec-ade, many applications of GRACE data have tended to focus on thisperiod. However, when combined with models, these data havepotential in extending the understanding of hydrologic patternsbeyond the period of their record. Some studies have used GRACEdata to calibrate elaborate models (e.g. Qiao et al., 2013; Sun et al.,2012; Werth and Güntner, 2010; Werth et al., 2009; Xie et al.,2012). Here we show that GRACE data can be used in combinationwith a one-parameter model to estimate long-term patterns ofevapotranspiration and ‘‘active’’ storage.

The simple model used in this study is based on the assumptionthat evapotranspiration (E) approaches potential evapotranspira-tion (Eo) asymptotically with increasing active storage (Tuttleand Salvucci, 2012). The one parameter in the model determineshow rapidly the E approaches the Eo with increasing storage. Theadvantage of this parsimonious approach is that it is simple and

A.N. Sharma, M.T. Walter / Journal of Hydrology 519 (2014) 2312–2317 2313

gives reasonable estimates of storage variations without the needfor many data other than precipitation, maximum and minimumtemperatures, elevation, and river discharge. The model’s simplic-ity also makes it very transparent, consistent with the ‘‘behavioral’’modeling approach proposed by Schaefli et al. (2011) for investi-gating hydrology-climate change interactions. The objective of thisstudy is to demonstrate the utility of this approach to extendingthe water storage time series in the Missouri River Basin. We usethese modeled estimates of storage and E to evaluate long termtrends (1929–2012) in the hydrologic budget.

Fig. 1. Map of the Missouri River Basin and GHCN stations (points) and USGSdischarge station (triangle, bottom right) used in this study.

2. Model description

We use the standard watershed water budget:

DS ¼ P � Q � E ð1Þ

where E is the evapotranspiration, P is the precipitation, Q is the dis-charge, and DS is the change in storage (all in units of depth pertime). We assume that there are reasonably good P and Q databut DS and E are typically unknown. As described below, wereduced E to a one parameter function.

The ratio of E to Eo can be modeled as a function of storage(Tuttle and Salvucci, 2012):

b ¼ 1� e�aS ð2Þ

where b = E/Eo, a is an empirical parameter, and S is the active stor-age. Although not theoretically proven, this relationship has theintuitive appeal of E asymptotically approaching Eo as storageincreases. For positive a and S, b can only vary between 0 and 1.Equations of this general form, where the ratio of E to Eo, the‘‘wet’’ condition evapotranspiration, depends on some estimate ofstorage in the system, have been in use for a long time (e.g.Thornthwaite and Mather, 1955; Budyko, 1974). E is a function ofmany different factors (Brutsaert, 1982), however, with only oneparameter to calibrate, this equation has the advantage of parsi-mony. Moreover, it can be argued that a, which determines howrapidly b approaches 1, captures most of these variations over a suf-ficiently large temporal and spatial scale. Note however, that awould be expected to vary by location, since different conditions,such as vegetation, soil types, etc. may be expected to controlhow b varies according to S. This concept has been captured forthe annual time scale by some more recent formulations of theBudyko curve (e.g. Fu, 1981; Choudhury, 1999; Zhang et al., 2004;Li et al., 2013), although they do so more indirectly, relying onEo/P as an indicator of storage in the system.

Potential evapotranspiration was calculated on a daily time stepfor each Global Historical Climate Network (GHCN) station. ThePriestley and Taylor (1972) method was used, i.e.

Eo ¼ aeD

Dþ cðRn � GÞ ð3Þ

where ae is the Priestley–Taylor constant, assumed here to be 1.26,D is the slope of the saturated vapor pressure curve at the meantemperature of the day, c is the psychrometric constant, Rn is thenet incoming radiation, and G is the ground heat flux, assumed tobe negligible at this time step (Allen et al., 1988). The psychrometricconstant, c, is dependent on pressure:

c ¼ cpPek¼ 0:665� 10�3p ð4Þ

where c is in kPa �C�1 , cp is the specific heat at constant pressure(1.013 � 10�3 MJ kg�1 �C�1), p is pressure (kPa), e is the ratio ofmolecular weights of water vapor to dry air (0.622) and k is thelatent heat of vaporization (2.45 MJ kg�1). Pressure for each station

is calculated based on elevation using a simplification of the idealgas law:

p ¼ 101:3293� 0:0065z

293

� �5:26

ð5Þ

where p is the pressure (kPa) and z the elevation above sea level (m)(Allen et al., 1988). The p calculated this way for each station corre-sponded well with the mean pressure recorded at each station inthe Global Summary of Day (GSOD) database.

The net radiation was calculated using the method described byWalter et al. (2005) and Archibald and Walter (2013, 2014). TheEcoHydRology R package (Fuka et al., 2013) was used to build thismodel with modifications to c as stated above.

3. Data and methods

Climate variables were obtained from the GHCN (Vose et al.,1992), except where mentioned from the GSOD database (http://www.climate.gov/global-summary-day-gsod, last accessed July31, 2014). Daily discharge data were obtained from the UnitedStates Geological Service (USGS) gauge number 06934500 at Her-mann, MO. GRACE data, used to estimate changes in storage forthe period beginning January 2003, were obtained in the form ofmonthly grids of equivalent water thickness from http://grace.-jpl.nasa.gov/data/gracemonthlymassgridsland/ (last accessed July31, 2014; Landerer and Swenson, 2012; Swenson and Wahr,2006). The GRACE satellites detect changes in Earth’s gravitationalfield which may be caused by a number of factors apart fromchanges in storage. The 1� gridded data used here were pre-pro-cessed to remove the effect of the post-glacial rebound signal aswell as of atmospheric and oceanic effects. The gridded data alsotake into account the ‘‘stripes’’ that are an artifact of the process-ing. The dataset used here is derived from the spherical harmoniccoefficient dataset generated by GRACE using a 200 km Gaussianfilter and a degree cutoff of 60 for the gravity field solutions. Loca-tions of the GHCN and USGS gauges as well as the boundaries ofthe basin are shown in Fig. 1.

Daily Eo and precipitation were first aggregated for each stationto a pentad (five day) time step and then interpolated over theriver basin using the inverse distance weighting method with anexponent of 2 (Brutsaert, 2005). For stations with 1 day or less ofmissing data for the pentad, it was assumed that the Eo or P wouldbe reflected adequately by the mean value of the rest of the days. Ifmore than one day of data were missing, that station was ignoredfor that pentad. There are 1731 GHCN stations in the MissouriRiver Basin (Fig. 1).

We calculated DS from October 1928 to June 2013 usingexhaustive combinations of a (from 0.001 cm�1 to 0.5 cm�1 inincrements of 0.001 cm�1) and initial S (from 0 to 100 cm in incre-ments of 1 cm). The DS at each time step was estimated from Eq.(1). The DS obtained at each time step was added to the storage

2314 A.N. Sharma, M.T. Walter / Journal of Hydrology 519 (2014) 2312–2317

at that time step in order to produce the storage for the subsequenttime step. Any time series in which S was negative at any point wasneglected since it is physically impossible to have negative storage;that some time series would have negative S is unsurprising sincewe do not initially constrain combinations of a and S except byspecifying the ranges mentioned above.

The most likely values of a and S were selected on the basis ofthe residual sum of squares (SSR) between S anomalies and DSfrom GRACE and the modeled values:

SSRS0 ¼Xi¼n

i¼1

ðS0GRACEi� S0modi

Þ2 . . . ð6Þ

SSRDS ¼Xi¼n

i¼1

ðDSGRACEi� Smodi

Þ2 . . . ð7Þ

where S0GRACE denotes storage anomalies from GRACE data (withrespect to the January 2003 to June 2013 time period), S0mod denotesstorage anomalies from the modeled time series, DSGRACE denoteschanges in storage from GRACE data, DSmod denotes changes in stor-age from the modeled time series, and n is the number of GRACEdata points in the time period January 2003 to June 2013.

Since the estimated values are in time steps of pentads, esti-mated S is first interpolated to the GRACE time periods to obtainthe S anomaly (S0mod) and DS series (DSmod) with the same timeperiods as the GRACE time series. The model was fed ten years ofmean weather (climatological) and discharge data in order to allowit to ‘‘spin-up’’ (Rodell et al., 2005).

Long-term (1929–2012) trends were estimated using the non-parametric Theil-Sen method in the wq R package (Jassby andCloern, 2012). All computations were performed in R (R CoreTeam, 2012). More intensive computations, such as spatial interpo-lation of climate data, were performed using the computationalfacilities at the National Center for Atmospheric Research(Computational and Information Systems Laboratory, 2012).

4. Results and discussion

For this basin, streamflow is an order of magnitude smaller thanthe other fluxes in Eq. (1). For a few months in the late-fall to early-spring (ONDJFM), E estimated from Eq. (1) and GRACE data is eithernegative or greater than the Eo. Most were in NDJF when both E andEo are both very small, and therefore this is likely to be an artifactof the error associated with each of the variables in Eq. (1). Despitethe high degree of relative variability in the late fall to early spring,the values of ‘‘observed’’ b tend to be much more constrained in thelate spring-early fall (AMJJAS), when they range between 0.23 and0.85, with �75% of these values between 0.4 and 0.65. However,modeled values of b were much more constrained, rangingbetween 0.22 and 0.67. That the modeled values of b should be lesserratic is easily explained by the form of Eq. (2). For b to be nega-tive, as in the case of E < 0 but Eo > 0, aS has to be negative. Sincewe reject any time series with negative S and allow only positive

Table 1Values of a (cm�1), mean storage S (cm) and Pearson’s product moment correlation (r) ob

a (minimum SSR)a range (SSRs within 5% of minimum)

S (minimum SSR)S range (SSRs within 5% of minimum)r between modeled and observed S0 (minimum SSR)r between modeled and observed S0 (SSRs within 5% of minimum)r between modeled and observed DS (minimum SSR)r between modeled and observed DS anomalies (SSRs within 5% of minimum)

values of a, their product can never be negative. For b to be greaterthan 1, e�aS has to be negative, which is not possible since itasymptotes to zero with increasing values of aS.

The two methods (Eqs. (6) and (7)) give very similar results inmost respects (Table 1 and Figs. 2 and S1), although the mean S pre-dicted by the SSRS0 method has a narrower range. The S predictedhere is similar to the values obtained by Tuttle and Salvucci(2012) (see their Fig. 4). Although the size and location of the water-sheds are different, the similarity of the estimates provides furtherconfidence in these results. While a and S values vary over a factorof two, this does not result in a big difference in the predicted valuesof changes in storage and storage anomalies which are the mainpoints of interest in this study (Fig. 2). We also compared a valuesobtained by maximizing the correlation between S0 and Q (rQ–S0), asTuttle and Salvucci did in their study. The values of a obtained bymaximizing rQ–S0 are similar to the values obtained by minimizingSSRS0 or SSRDS (Fig. S2) although the range is a somewhat wider(0.010–0.035 cm�1). The high degree of correlation with GRACEdata (Figs. 2 and S1) as well as the narrow distribution of the resultsindicate the usefulness of this parsimonious approach. When mod-eled and observed storage anomalies are deseasonalized, the corre-lation decreases but is still quite good (Fig. S3).

Long term (1929–2012) annual time series of P, Q, Eo, andmodel-estimated E and DS are shown in Fig. 4. P, Q and E haveincreased (Table 2 and Fig. 3) but DS does not show a significant(p < 0.05) trend. Increase in E at regional and global scales has beensuggested in many previous studies (e.g. Brutsaert and Parlange,1998; Brutsaert, 2006; Milly and Dunne, 2001; Szilagyi et al.,2001; Walter et al., 2004). Brutsaert and Parlange (1998) suggestedthat global decreases in pan evaporation can be explained by anincrease in actual E because of the Bouchet-Morton ‘‘complemen-tary relationship.’’ In this study, Eo has not changed, while E hasincreased. Thus the apparent potential evaporation term, the quan-tity often assumed to be measured by pan evaporation, has, ineffect, decreased, supporting Brutsaert and Parlange (1998). Whileother studies (e.g. Szilagyi, 2001; Szilagyi et al., 2001) have foundthat Eo could have changed over the conterminous US, possiblydue to changes in the net radiation, we find no significant trendin it. This difference may be explained by the different methodsused to estimate net radiation terms and Eo given limited data.Despite these difficulties, the concept of an Eo is a very usefulone because it allows us to express E as some scaled version ofits potential based on the water availability in the system, whichin this case is represented by S.

The increasing trend in E of 0.055 cm y�2 seen in this study is ofthe same order of magnitude obtained in other studies. Brutsaert(2006) found a global trend in E of 0.044 cm y�2 for 1950–2000.Milly and Dunne (2001) and Walter et al. (2004) found trends inE equivalent to 0.069 (1949–1997) and 0.11 (1950–2000) cm y�2

for the Mississippi river basin, of which the Missouri is a sub-basin.These studies also found somewhat higher trends for precipitation,0.178 cm y�2 (Milly and Dunne, 2001) and 0.176 cm y�2 (Walteret al., 2004) than the trend of 0.084 cm y�2 found in this study,

tained by the two methods.

SSRS0 SSRDS

0.029 0.0280.022–0.038 0.018–0.04023.4 22.6

17.6–29.5 16.8–35.90.79 0.790.76–0.81 0.76–0.820.87 0.870.87–0.88 0.87–0.88

Fig. 2. Modeled and GRACE data found by minimizing SSRS0: (a) S anomalies plotted as a time series; black lines are GRACE data, orange lines are the best (lowest SSR)modeled data, and gray lines are other modeled data within 5% of the lowest SSR; (b) DS plotted as a time series; black lines are GRACE data, green lines are the best (lowestSSR) modeled data, and purple lines (barely visible) are other modeled data within 5% of the lowest SSR; (c) S anomalies, modeled (lowest SSR) v. observed, and (d) DS,modeled (lowest SSR) v. observed. In (c) and (d), lines shown to facilitate comparison have intercept = 0 and slope = 1. Plots for minimum SSRDS are shown in Fig. S1. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 2Trends in fluxes and slopes of cumulative terms. For S, the quantity in the ‘‘fluxes’’column is DS, and in the cumulative terms column, it is the slope of the storageanomaly (S0). For E, DS and S0 , slopes given are the median values for all time serieswhich had SSRs within 5% of the minimum SSR. In most cases, the median values forthe SSRS0 and SSRDS methods were the same.

Trends in fluxes (cm y�2) Slope of cumulative terms (cm y�1)

P 0.084* 50.75*

Q 0.030* 5.48*

Eo �0.017 95.44E 0.055* 45.21*

DS �0.001 0.042*(SSRS0); 0.043* (SSRDS)

* Significant slopes (p < 0.05).

Fig. 3. Potential evapotranspiration (Eo, closed circles), annual precipitation (P,crosses), evapotranspiration (E, open circles), discharge (Q, open diamonds), andchange in storage (DS, triangles) for 1929–2012. Note 1928 and 2013 were left outof trend analysis since data are incomplete for these years. The E and DS valuesplotted are for all time series with SSRS0 within 5% of the minimum SSRS0 . Lines showslopes of each variable (values in Table 2); Eo and DS slopes shown are 0, since theydo not have a significant trend.

A.N. Sharma, M.T. Walter / Journal of Hydrology 519 (2014) 2312–2317 2315

indicating a spatial variation in the trends within the Mississippibasin, i.e., other parts of the Mississippi basin have larger increasesin E than in the Missouri River sub-basin.

Since Eo has not increased significantly (Table 2), the increasein E is driven by the increase in water availability from increasedprecipitation. Although there is no significant trend in the DS,there is a significant positive trend in S0 (Table 2 and Fig. 4). Thismay at first seem paradoxical but may be resolved as follows. Thechange in storage, DS, is in the difference between the flux termsP, Q and E. However the storage itself is a result of the integrationof these flux terms over time. By comparing the slopes of cumu-lative P, Q and E, one gets a value of change in storage(0.064 cm y�1) over this period that is close to the trend in themodeled S (�0.043 cm y�1). This trend is an order of magnitudegreater than the 0.008 cm y�1 (1965–2000) trend in groundwaterstorage that Brutsaert (2008) found for Illinois using long-termwell data and recession slope analysis. While the magnitudes

Fig. 4. Annual change in storage (DS, black circles) and storage anomalies (S0 , graylines) for 1929–2012. Note 1928 and 2013 were left out of trend analysis since dataare incomplete for these years. The DS and S values plotted are for all time serieswith SSRS0 within 5% of the minimum SSRS0 .

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are not comparable because of the different ‘‘quantities’’ mea-sured as well as the spatial differences, there is a concurrenceamong these studies that storage is increasing in this region. Notethat the number of weather stations has varied over the years,with only 427 and 520 stations having at least 90% of the neces-sary data for P and Eo estimates, respectively. Using only the datafrom these stations with long-term records results in trends ofthe same sign but larger magnitudes. The more conservativeresults obtained by using all the stations available at any giventime are reported here.

Often, because of the lack of storage data, studies make theassumption that change in storage is negligible (e.g. Walter et al.,2004). In this particular region, the assumption that E may be esti-mated by the difference in P and Q is valid and suggests a trendvery similar to the modeled estimate. However, this study high-lights a nuance that is often overlooked in long-term hydrologicalstudies. It is seen here that a significant (p-value < 0.05) trend instorage may be accompanied by no significant trend in year to yearchange in storage (Table 2). Usually the change in storage is verysmall compared to other quantities, however, the implications ofthe assumption that change in storage is negligible requires carefulconsideration. Wang and Alimohammadi (2012) found that espe-cially under water-limited conditions, evapotranspiration may beoverestimated if change in storage is neglected. This study pro-vides a simple approach for incorporating changes in storage withlimited data.

The increase in fluxes in this region is consistent with other evi-dence that suggests that the hydrological cycle is intensifying(Huntington, 2006) . The use of this method is limited by the res-olution of the GRACE data, which allows its application only for rel-atively large river basins. However, this limitation may beaddressed by future missions, and even the current resolution pro-vides information on long term regional trends which may be sup-plemented by studies at smaller scales. The use of data from themost recent decade to estimate long-term constant values of aand S raises the question of stationarity. We assume here thatthe parameters a and ae remain constant over the span of eightdecades. In the absence of adequate long-term and spatiallyresolved information on the variation of the Priestley–Taylorparameter ae, the use of a value of 1.26 typically found in the liter-ature is a reasonable first-step. For a, we find that ‘‘reasonable’’ val-ues of this parameter vary within a relatively narrow range, andthe trends in fluxes and storage remain essentially the same withinthis range. Thus, while future work may look into the factors affect-ing a, for the purposes of this study, a constant a is a reasonableassumption. The trends in E found here are in agreement withother studies which use very different approaches, showing thatthe assumptions used here are reasonable. More complicated mod-els typically require the use of multiple parameters which are

based on studies conducted in completely different locations,scales, or time-periods. There is little reason to believe that theassumptions used in this study are less reasonable than the morenumerous assumptions used in multi-parameter models. Further,our understanding of long-term trends in the absence of adequatedata is likely to benefit from the use of approaches that vary bothin complexity as well as in their underlying assumptions.

5. Conclusions

Estimating patterns in storage and evapotranspiration areimportant steps to understanding the hydrological cycle. Thisstudy presents a simple method to estimate changes in these quan-tities for a large river basin. Given the simplicity of the model, andthe errors inherent in the estimation of its inputs (such as P, Eo, Qand DS), it performs remarkably well. We used this model to showlong term increases in E and storage in the Missouri River Basin,consistent with other studies which have shown the intensificationof the hydrological cycle.

Acknowledgements

We would like to thank Drs. Michael Walter, Calum Turvey, twoanonymous reviewers, and the associate editor, Dr. McCabe, fortheir insightful suggestions for this manuscript. We would also liketo thank Dr. Daniel Fuka and Josephine Archibald for their contri-butions to the EcohydRology package in R, which were used here.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jhydrol.2014.10.014.

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