Total water storage in the Arkansas-Red River basin

Authors


Abstract

[1] This paper presents a diagnostic approach to study the spatial and temporal variability of total water storage (TWS) properties of hydrologic basins. The approach utilizes a simple water balance equation based on the physical law of mass conservation. Thirty-six years of monthly historical records of precipitation and streamflow discharge data were collected from 27 basins in the Arkansas-Red River basin. Five different assumptions were made to estimate evapotranspiration. The study found that the assumption that considers both initial total water storage and precipitation amount in the current period provides the most consistent long-term seasonal evapotranspiration, when compared to seasonal evapotranspiration derived from the atmospheric water budget in the Arkansas-Red River basin. The TWS variability range has an apparent east-west gradient and is highly correlated to climatological factors such as the ratio of annual precipitation and potential evapotranspiration and vegetation information. Wetter basins tend to have a larger TWS variability range than dry ones. The intra-annual TWS variability range is small compared to year-to-year variability. This phenomenon is more noticeable for dry basins, implying that longer data sets are needed to study the water balance of dry regions. Another finding is that the TWS variability range is shown to be scale dependent, with intermediate-scale-area basins exhibiting larger variability than large-scale-area basin.

1. Introduction

[2] Total water storage (TWS) in the land surface refers to water stored in the soil zone, groundwater aquifers, streams and surface reservoirs. Land surface water storage controls the partitioning of precipitation into evapotranspiration and runoff, the partitioning of net radiation energy into latent and sensible heat fluxes and the occurrence of base flow in the streams. Water storage properties are important at both short and long timescales [Milly, 1993, 1994], and have important effects on weather and climate [Yeh et al., 1984; Ookouchi et al., 1984; Bonan and Stillwater-Soller, 1998; Koster and Suarez, 2001].

[3] This paper is concerned with the study of TWS properties over intermediate-scale (100–1000's km2) and large-scale (10,000–100,000's km2) hydrologic basins. Particularly, it investigates how TWS and its variability ranges vary in space and in time and how they relate to basin soils, vegetation and climate characteristics. Special attention is paid to the TWS variability range (hereinafter referred as TWS range) of a basin and its relationship to the basin TWS capacity (TWSC), a parameter that appears in most hydrologic models in one form or another. Hydrologic model parameters, including TWSC, play pivotal roles in regulating water and heat fluxes of the land surface [Liston et al., 1994; Ducharne and Laval, 2000]. How to estimate those parameters has been identified as a key research issue in hydrologic modeling by the Science Plan of the Global Energy and Water Cycle Experiment's (GEWEX) Continental-scale International Project (GCIP) [World Meteorological Organization (WMO), 1992]. This continues to be an important issue for the GEWEX Americas Prediction Project (GAPP) [Office of Global Programs, 2000].

[4] Investigation of basin-scale TWS is subject to uncertainty because they cannot be measured directly because of the extreme heterogeneous nature of its distribution patterns and its existence in both the root zone and in ground water aquifers. Some limited measurements of soil moisture at a few locations and at different vertical depths are available which could yield information about the local TWS [Robock et al., 2000]. However, questions have been raised on how well these point measurements represent an areal quantity [Loague, 1992]. National Oceanic and Atmospheric Administration (NOAA) operates an aerial gamma remote sensing system that measures soil moisture and snow water equivalent along a flight line [Peck et al., 1992]. Recently National Aeronautics and Space Administration (NASA) initiated a series of remote sensing experiments to obtain areal soil moisture measurements [Schmugge and Jackson, 1996]. However, those satellite and aircraft measurements of soil moisture apply only to the top few centimeters of the soil surface. The recently launched Gravity Recovery and Climate Experiment (GRACE) mission by NASA will be able to detect TWS changes over large-scale areas [Rodell and Famiglietti, 2001].

[5] There were numerous past research efforts directed at understanding TWS and its role in the hydrologic cycle. Milly [1994] presented an analytical approach to study the relationships between climate, interseasonal TWS variation and annual water balance. He found that realistic relationships between these quantities need a reasonable representation of runoff generating mechanisms that account for interseasonal variability of precipitation, spatial variability of TWSC, and infiltration capacity. Koster and Suarez [2001] used an autocorrelation equation to study climatic predictability based on soil water storage memory. They found that soil water storage memory is controlled by four factors: seasonality of atmospheric forcing, functional dependence of evaporation on soil water storage, functional dependence of runoff on soil water storage, and correlation between soil water storage and subsequent atmospheric forcing.

[6] Roads et al. [1994] followed an atmospheric water budget approach to estimate the large-scale hydrologic climatology of the United States using moisture flux divergence derived from the National Meteorological Center (NMC)'s (now known as National Center for Environmental Predictions (NCEP)) operational global 2.5 degree analysis and using gridded analyses of observed precipitation and streamflow data, produced an estimate of TWS range of just less than 20 cm for the period 1984 to 1989 for the entire United States and almost identical results for the Mississippi River basin. Ropelewski and Yarosh [1998] also used an atmospheric water budget approach with twice daily radiosonde measurements together with precipitation and streamflow measurements to estimate water storage variability during the period 1973–1992 for the central United States. During this period, the range of variability of the running 12-month water storage was about 40 cm.

[7] The most common approach in the past to analyze TWS has been to conduct water balance simulations. Hydrologic models are used to simulate water storage changes in response to observed precipitation and energy (e.g., potential evapotranspiration) forcing. Early prominent work on water balance modeling includes that of Thornthwaite and Mather [1955], who used the surface water balance as a basis to study climatology of different parts of the world. Other efforts have concentrated on constructing more elaborate water balance models to predict runoff as well as evapotranspiration (see, e.g., the Stanford Watershed Model by Crawford and Linsley [1966] and the Sacramento Soil Moisture Accounting Model by Burnash et al. [1973]). Eagleson [1978] proposed a macroscale water balance model to study soil, vegetation and climate interaction as an integrated system. Other researchers [Schaake and Liu, 1989; Schaake, 1990; Cayan and Georgakakos, 1995] developed similar, but less complicated models for the same purpose. Mintz and Walker [1993] used a simple bucket water balance model to produce global fields of hydrologic variables including soil moisture. An estimate of the plant-available water-holding capacity equal to 15 cm was used by Mintz and Walker [1993]. Milly [1994] explains that this estimate is equivalent to assuming that the water holding capacity of the soil is about 15% of its volume and that the maximum root depth is about one meter. Although the range of water holding capacity of soils is reasonably well known, there is uncertainty in the actual water holding capacity of the soils in a given basin. Mintz and Walker [1993] concluded in their study that better water budgets can be achieved if soil water storage capacity accounts for spatial variability. Sankarasubramanian and Vogel [2002] used a water balance model to develop an annual climatology of the United States. Their approach utilized a climatic index based on the ratio of annual precipitation and potential evapotranspiration to formulate the relationships between actual evapotranspiration, runoff and soil water storage.

[8] In the aforementioned studies, observed precipitation and potential evapotranspiration were directly used in the water balance simulation, but streamflow, actual evapotranspiration and soil water storage were estimated on the basis of various assumptions about runoff and evapotranspiration processes. Observed streamflow was not used as an input, rather was used only to validate model performance and to tune model parameters. Water balance simulation is therefore dependent on model representations. Accordingly, Koster and Milly [1997] showed that different water storage levels were simulated by several different models applied to the same data sets and that these differences depended on model assumptions regarding both evapotranspiration and runoff.

[9] Hydrologic models of land surface processes also require reasonable estimates of model parameters such as TWSC. Because all hydrologic models are limited approximations of reality and because all of the model components are interdependent, there is a tendency in tuning model parameters to compensate for limitations of the model. Accordingly, it is not clear if TWSC implied by model parameters is a good measure of the actual TWSC or if it is either an artifact of the model or the method used to estimate model parameters.

[10] It is desirable to analyze basin-scale TWS and TWSC by limiting uncertainty due to the factors mentioned above. One way to accomplish this is to represent the water balance using historical precipitation and streamflow observations with minimum model parameterizations. The advantages of this approach include the following: (1) The resulting water balance equations involve fewer approximations and are less dependent on spatial scale, and (2) the climatic data needed to perform such studies are readily available for many parts of the world.

[11] In this study, observed monthly streamflow data, along with observed monthly precipitation and climatic potential evapotranspiration is used in the analysis so that the partitioning between runoff and evapotranspiration is not model dependent. However, the month-by-month partitioning of the sum of storage change and evaporation is dependent on the rule used to estimate the actual monthly evaporation. Accordingly, five different evaporation schemes are considered in this study to investigate the sensitivity of the conclusions to this assumption. Guetter and Georgakakos [1996] used a diagnostic approach similar to that used in the present analysis to study the surface water budget in the central and southeastern United States for the period 1960–1988. Their focus was on characterizing water storage variability across spatial scales. They concluded that in the central United States, larger regions possess a greater range of temporal variability than smaller embedded regions and that extreme water storage anomalies possess scales of spatial coherence from 105 to 103 km2, with stronger anomalies characterized by the smaller scales. They found the strongest spatial coherence of water storage anomalies in the wetter southeastern United States.

[12] This study takes on a different angle from previous studies to examine the space/time variability of water storage properties. This paper is organized as follows. First the study region and data are described. Then a diagnostic water balance analysis approach is presented. The space and time variability of the TWS is analyzed. The relationship between the TWS range and soils, vegetation and climate is examined. Finally a summary and conclusions are given.

2. Study Region and Data Sites Used

[13] The Arkansas-Red River basin (shown in Figure 1) is located in the southwestern part of Mississippi River basin. This area is termed large-scale area-southwest (LSA-SW) by GCIP [WMO, 1992]. With a total area of 538,382 km2, it encompasses part or all of eight Southwestern states. Within the LSA-SW are a number of smaller river basins termed intermediate-scale areas (ISAs) by GCIP [WMO, 1992]. Thirty-six years (from 1950 to 1986) of monthly mean areal precipitation and streamflow discharge data were assembled for 27 unregulated ISA basins. Long-term annual potential evapotranspiration for the 27 basins were estimated from maps of free water evaporation from the NOAA evaporation atlas [Farnsworth et al., 1982]. Also available for these basins are the corresponding land surface characteristics including soil texture and the derived soil hydraulic properties, land cover, vegetation, and climatology. The geographical locations of the 27 individual basins are shown in Figure 1.

Figure 1.

Arkansas-Red River basin (LSA-SW) and location of study basins.

[14] The climatology of the LSA-SW is mostly humid in the east and arid or semi-arid in the west. Vegetation composition is made of deciduous forest in the east, grassland in the west, with a big portion covered by seasonal crops. Large east-west gradients of climatic and hydrological variables, diverse geological and morphological factors, coupled with the availability of long-term climatic data from this region are the primary reasons for selecting Arkansas-Red River basins for this study. Figure 2 shows the maps of climatic annual precipitation, runoff and potential evapotranspiration for LSA-SW. The precipitation map (Figure 2a) is based on the 1961–1990 precipitation climatology developed by Daly et al. [1994]. Precipitation varies from east to west, with the southeastern corner recording annual totals of over 1300 mm and the western parts receiving about 300 mm. The runoff map for LSA-SW (Figure 2b) is obtained by analyzing the historical streamflow data of all unregulated headwater basins. A high runoff value of about 570 mm is seen in the Northeast and a low value of less than 10 mm in the west. The potential evapotranspiration map (Figure 2c) is extracted from the NOAA evaporation atlas [Farnsworth et al., 1982]. It shows that potential evapotranspiration ranges from around 950 mm to 1700 mm.

Figure 2.

Spatial variability of annual averages of (a) precipitation, (b) runoff, and (c) potential evapotranspiration.

[15] One interesting way to look at the climatology of a region is to examine the relationships between the ratio of actual to potential evapotranspiration and the ratio of precipitation to potential evapotranspiration. A number of empirical relationships were described in the literature [Dooge, 1997; Sankarasubramanian and Vogel, 2002]. One such empirical relationship is the Turc-Pike equation:

equation image

where E is the actual climatic annual evapotranspiration, EP is the climatic annual potential evapotranspiration, and P is the climatic annual precipitation. Depending on the values of P/EP ratio, a region can be classified into desert, steppe, forest and tundra, respectively. Figure 3a displays the theoretical Turc-Pike curve along with the observed data from the 27 ISA study basins. Note that the basins fall evenly into three categories: desert, steppe and forest. Figure 3b shows an alternative Turc-Pike curve where the vertical axis is the ratio of runoff to precipitation, Q/P, also known as the runoff coefficient. Figures 4a and 4b display the spatial distributions of P/EP and Q/P. The east-west gradients showing a dry west and a humid east are obvious in those maps.

Figure 3.

Theoretical and observed relationships (a) between the ratio of actual to potential evapotranspiration and the ratio of precipitation to potential evapotranspiration and (b) between the ratio of runoff to precipitation and the ration of precipitation to potential evapotranspiration.

Figure 4.

Spatial distributions of (a) the ratio of precipitation to potential evapotranspiration and (b) the ratio of runoff to precipitation (or runoff coefficient).

3. Water Balance Analysis

[16] A diagnostic water balance analysis framework is presented below. In this framework, the water storage term includes water available for evapotranspiration as well as for base flow production. The absolute level of water storage is dependent on assumptions of how evapotranspiration occurs.

[17] Let the time step for water balance calculation be a month, and assume that the transfer of groundwater across basin boundaries is negligible. Then, the water balance equation can be expressed as

equation image

where t denotes month, Pt, Et, and Qt are, respectively, average precipitation, evapotranspiration, and total streamflow from the basin during the tth month and where, St+1 and St are total water stored within the basin at the beginning and the end of tth month.

[18] To calculate water balance, values of Pt, Qt and Et in equation (2) must be measured or estimated. For many basins, two of the variables, Pt and Qt, are routinely monitored and thus historical observations are readily available. Since there are no direct observations of Et, assumptions have to be made to estimate Et. Assuming that it is proportional to the potential evapotranspiration, Ep,t, Et can be computed as

equation image

where f(St, Pt, Φ) is the evapotranspiration function dependent on variables St, Pt, and parameters Φ =1, φ2, …}.

[19] Schaake and Liu [1989] and Schaake [1990] developed a procedure for estimating climatic potential evapotranspiration Ep,t. It was assumed that the long-term average potential evapotranspiration Ep,t varies periodically during the year, but not from year to year. For the United States, this seems reasonable because the coefficient of variation of annual pan evaporation is less than 10% everywhere in the continental United States. Also, the dominant component of the seasonal variation of Ep,t is the annual cycle which explains more than 99% of the variance of monthly Ep,t. Accordingly, the value of Ep,t can be approximated by

equation image

where Ep,avg is the annual average, ε is the amplitude, ω is the frequency of the annual cycle (equal to 2π/12), and θ is the phase angle. On average, ε is about 0.70, indicating that the computed Ep,t ranges from 30% of Ep,avg during the winter to 170% during the summer [Schaake and Liu, 1989; Schaake, 1990]. Substituting Ep,t in equation (3) with the right-hand term in equation (4) and then placing them into equation (2), the following expression is obtained:

equation image

Assuming that function f(St, Pt, Φ) is given, parameters Ep,avg, ε and θ are prescribed, the initial water storage, S0, is known, Pt and Qt are obtained from historical records, the values of St can then be determined recursively for all corresponding time periods. It should be noted that the values of Ep,avg for the conterminous United States can be determined from the NOAA annual free-water evaporation map [Farnsworth et al., 1982]. On the basis of the same NOAA annual free-water evaporation map and the May–October free-water evaporation, J. C. Schaake and Q. Duan (unpublished notes, 1997) have also created maps of ε as well as θ.

[20] If observational data for Pt and Qt are reliable, most of the uncertainty in equation (5) can be attributed to evapotranspiration and water storage terms. In this paper, five different evapotranspiration functions, f(St, Pt, Φ), are investigated. They are listed in Table 1 and illustrated in Figures 5a–5f. A detailed description of the five functions is given in Appendix A.

Figure 5.

(a–f) Schematic illustrations of five evapotranspiration functions.

Table 1. Definitions of Five Evapotranspiration Functions
FunctionDefinition
1
equation image
2
equation image
3
equation image
4
equation image
5
equation image

[21] Note that in all evapotranspiration functions, a water storage deficit term, Dt (= DmaxSt), is used, rather than the water storage term, St. Negative values of Dt may occur and represent a TWS beyond the water holding capacity of the soil zone. If we substitute St in equation (5) with Dt, equation (5) becomes

equation image

4. Determination of Parameters in Evapotranspiration Functions

[22] Depending on which function is selected, one or more other parameters must be specified. The values of the exponents, α0, α1, and α2, used in the function determine how the rate of evapotranspiration changes according to the relative saturation level. For a vegetated surface, α1 is set to 0.5, indicating that the transpiration rate decreases gradually at first when roots are able to extract water from the soil and more rapidly as soil dries out. For bare soil, α2 is set to 2.0, implying that the rate of evaporates decreases rapidly at first, then more gradually, as water storage changes from being wet to being dry. For a mixed surface, α0 is 1.0 to account for both bare soil and vegetation.

[23] Dmax is a critical parameter in all evapotranspiration functions except function 3 and similar parameters are commonly employed in many land surface hydrology models. This parameter determines the maximum moisture deficit that can be produced in the root zone during drought periods. The main effect of Dmax is to influence the timing of how long it takes for infiltrated water to evaporate. This influences the evolution of the simulated total water storage and therefore may have an influence on the maximum range of total water storage. For most of the analysis presented in this paper, Dmax, was assigned the value of the saturated available water capacity (SAWC) (i.e., the space between soil porosity and wilting point) from soil surface to bedrock, as determined from the State Soil Geographic (STATSGO) Database [Miller and White, 1998]. Alternative methods to define Dmax were tested and their effects on the TWS range are presented in section 5.6.

5. Temporal Analysis of TWS Range

5.1. Water Balance Simulation

[24] The water balance as expressed in equation (6) was applied to 27 basins from the LSA-SW using 36 years of historical precipitation and streamflow data. The initial water storage deficit was set equal to the last deficit value from a full-record spin-up run. The parameters used in calculating potential evapotranspiration, ε and θ in equation (6), were determined on the basis of the maps created by J. C. Schaake and Q. Duan (unpublished notes, 1997). Other parameters in the five evapotranspiration functions, except Dmax, are specified in Table 2.

Table 2. Parameter Values for the Five Evapotranspiration Functions
FunctionEvapotranspiration Parameters and Their Specified Values
1α0 = 1.0
2α0 = 1.0, β = 0.25
3φ - determined by ratio of climatic potential and actual evapotranspiration
4α1 = 0.5, α2 = 2.0, ξt determined from digital maps of vegetation fraction
5φ = 0.5, α1 = 0.5, α2 = 2.0, ξt determined from digital maps of vegetation fraction

[25] Figures 6a–6c show simulated water storage deficit time series over a 10-year period for three typical basins, representing respectively wet, moderate and dry climate. Function 5 was employed in these examples. Note that the deficit values are allowed to be negative (see Figure 6c), implying the basin's being fully saturated and evapotranspiration taking place at potential rate during these time periods.

Figure 6.

(a–c) Water storage deficit time series for three representative basins.

[26] Figures 7a–7c display the long-term monthly average of the water balance elements for the same three basins. Note that there are great differences in the amount of seasonal distribution of runoff produced in the three basins, and that evaporation in the dry basin is closely tied to precipitation but not in the wet basin. Figure 8 displays the corresponding long-term average water storage deficits for these basins. The seasonal fluctuation of water storage for the wet basin is the largest. In the dry basin, there is little average change in water storage. The mean level of the deficit is partly an artifact of the assumed rules for evapotranspiration and partly due to the interannual variability of the hydrology. Note that the amplitudes of the mean seasonal TWS change increase from dry to wet.

Figure 7.

(a–c) Long-term average of hydrologic variables for three representative basins.

Figure 8.

Long-term average of water storage deficit for three representative basins.

5.2. Definition of TWS Range RT

[27] Denoting Xt as the monthly change in TWS, the following expression is obtained:

equation image

Inserting Xt into equation (6), we get

equation image

During a period T beginning at time t, the maximum and minimum values of Dt are

equation image
equation image

The variability range of water storage RT can be expressed as follows:

equation image

5.3. RT as a Function of the Analysis Period T

[28] The TWS range RT, as defined in equation 10, is analyzed for all 27 basins. Figures 9a–9c show estimates of RT as a function of analysis period T for three representative basins. The solid lines in Figure 9 are the sample average values of RT. The averages are taken over all T-year values in the available data period. The dotted lines represent the average values plus/minus the sample standard deviation, while the dashed lines represent the sample maximum and minimum. The figure shows that RT increases monotonously as T increases from one to 36 years. It is also clear that the wet basin has the largest variability range while the dry one has the smallest range. In each case there appears to be an upper limit to the total storage capacity, but this is approached asymptotically.

Figure 9.

(a–d) TWS range for three representative basins and the LSA-SW.

[29] A spatial analysis of the data for these basins was done to produce area-averaged estimates of RT for the entire LSA-SW. The spatial analysis was done in the following steps. A 10-min grid is used for the spatial analysis. First, at each grid point a local trend surface was fitted to the RT values at nearby basins as a linear function of the P/PE ratios at the same sites. Then, a local deviation from the trend surface at each grid point was estimated using an inverse distance weighted combination of the surrounding deviations. The RT for LSA-SW is the average of all grids for a given T. Figure 9d presents the resulting RT versus T curve for the entire LSA-SW. Also shown is the average ISA basin RT versus T curve, which lies above the composite LSA-SW function. The value of RT is much higher for ISA basins than that for the LSA-SW (340 versus 250 mm). Since the size of the ISA basins varies between 530 and 6800 km2, compared to 538,382 km2 for LSA-SW, the results may indicate that RT is scale dependent. This scale dependency may be expected to apply for scales smaller than the ISA scale as well. However, this is difficult because sufficiently accurate measurements of precipitation do not exist for most very small stream basins. The results shown in Figure 9d differ from the results presented by Guetter and Georgakakos [1996], who concluded that larger regions possess a greater range of temporal soil water variability than smaller embedded regions. This difference can be attributed to the fact that our observation is based on the comparison of the TWS range of all 27 ISA basins to that of the LSA-SW, while Guetter and Georgakakos [1996] drew their conclusion on the basis of single sample TWS anomalies of the embedded regions. The single sample anomalies of the embedded regions may not be typical for their respective spatial scales.

[30] A further observation from the figures is that the value of RT rises faster initially for the wet basin than for the dry one, implying that it initially takes longer to observe the full range of storage variability for dry basins. This figure might suggest that at least 20–30 years of data are required to get an approximation of the asymptotic RA for the ISA basins. For an LSA basin, a shorter period may be adequate for the same purpose.

5.4. Sensitivity of RT to Evapotranspiration Assumptions

[31] This section examines the sensitivity of the results to different evaporation assumptions. Figure 10 shows typical RT versus T curves, which are derived from different evapotranspiration functions. Three of the functions (e.g., 1, 2 and 4) produce similar results. Function 5 has the lowest RT, while function 3 has the highest. These results imply that RT is not sensitive to evaporation assumptions if evaporation is assumed to be dependent only on initial soil moisture of the current month, but not on current precipitation. This assumption is apparently erroneous because part of the precipitation does evaporate in the same month as it occurred. Function 5 takes this into account in calculating evaporation. Consequently, it requires less storage. Function 3 has the greatest RT because, according to this function, water evaporates and transpires at a constant fraction of potential evapotranspiration throughout the entire analysis period. This constant fraction produces the correct total volume of evaporation over the long run but evaporation by this function is insensitive to the temporal variation of precipitation or water storage. Accordingly, more evapotranspiration takes place in dry years and less in wet years than actually occurs. Since storage is the only water balance component that can absorb the error in calculating evapotranspiration, the amplitude of storage variability is unreasonably enlarged as a result.

Figure 10.

Comparison of TWS range for five different evapotranspiration functions.

[32] Different values for φ in function 5 can also result in different RT. Figure 11 shows average RTT relationship of all ISA basins for φ = 0.0, 0.25, 0.5, 0.75 and 1.0, respectively. The values for RT become smaller as φ increases from 0 to 0.5. When φ reaches the value of 0.5 or more, the RT values become indistinguishable. However, when φ = 1.0 (i.e., the Thornthwaite-Mather assumption), the RTT curve departs from other cases and RT becomes very large as T increases, indicating water storage taking unrealistic values. The results here clearly suggest that the Thornthwaite-Mather assumption is inconsistent with the reality. This divergence of RT occurs because none of the precipitation in a month can be runoff when φ = 1.0.

Figure 11.

Comparison of TWS range for different values of φ in function 5.

[33] Function 5 incorporates more physical notions in its formulation than the other 4 functions. Results presented later shows that function with φ = 0.5 gives very reasonable seasonal evapotranspiration. The analysis done thereafter uses function 5 unless otherwise specified.

5.5. Sensitivity of RT to Precipitation Errors

[34] Estimates of basin average precipitation are subject to estimation error because they are based on a weighted combination of point measurements. However, because a monthly time step is used in this analysis, the magnitude of this error is not expected to be very large. The exact magnitude of the error depends on the spatial decorrelation structure of the precipitation and on the number of gages available for each basin. A detailed analysis of the actual errors was not done in this study, but an eigenvalue analysis of the monthly data for the 27 basins was made. If the data for an individual basin contains only noises, all of the eigenvalues would have comparable values. The eigenvalue analysis showed that 99% of the total variability of the monthly basin precipitation in each month could be explained by only five eigenfunctions. This implies that it is unlikely a large part of the data sets are noises. Nevertheless, to understand how precipitation errors might affect this analysis, a normally distributed random multiplier with a mean of 1 and standard deviation of 0.1 was used to create parallel data sets to represent the actual precipitation that might have occurred. Precipitation errors must be “stored” until they evaporate. They cannot appear in the streamflow because measured, not computed, streamflow is used. Introducing this random error had only a negligible rise (<3%) in the values of RT.

5.6. Sensitivity of RT to Dmax Values

[35] As pointed out previously, parameter Dmax influences the timing of evapotranspiration, and therefore may influence RT. A sensitivity study was done to examine the impact of different Dmax values on RT. Four sets of Dmax are tested: (1) Dmax,1, which is the saturated available water capacity (SAWC) adjusted by soil depth; (2) Dmax,2, which is SAWC but assumed a uniform soil depth of 2 m; (3) Dmax,3, which is 2.5 m available water capacity (AWC) [Miller and White, 1998]; and (4) Dmax,4, assumed to be constant at 500 mm. The difference in RT using the four sets of Dmax is very small, less than 5%.

[36] Figure 12 plots the scattergram of Dmax versus R36. For Dmax,3, R36 values lie above the 45° line for most of the basins. This suggests that the AWC in the top 2.5 m is too small to store all the water needed for evapotranspiration. This figure also shows that even other Dmax values are used, R36 values for some basins still exceed the Dmax values. This implies that water in the deeper groundwater storage may have participated in the water cycle. Figure 12 also suggests that even though Dmax plays a significant role in partitioning water between evaporation and runoff, its influence on total water storage range is very limited. A further implication is that Dmax as defined by available space in the soil to store water should not be used to estimate total water storage capacity (TWSC), a parameter commonly used in many hydrologic models. The asymptotic value of RT is probably a more appropriate indicator of TWSC.

Figure 12.

Scattergram of Dmax versus R36.

5.7. Comparison of RT Against the Theoretical Curve

[37] TWS range analysis has much similarity with the reservoir storage analysis. The TWSC value needed in a model to produce a sequence of monthly storage changes is similar to the classical problem of the required capacity of a water supply reservoir. Salas-La Cruz and Boes [1974] discuss the stochastic properties of the capacity of reservoir storage required to supply a constant demand, μ · Q, where μ is a fraction of the mean streamflow, Q, into the reservoir. In the land surface storage problem, μ assumes the value of 1. The temporal statistical properties of RT have been studied extensively in relation to the planning and design of water resources storage reservoirs [Anis and Lloyd, 1953; Salas-La Cruz, 1974; Salas-La Cruz and Boes, 1974]. One question is, what storage capacity is needed to meet a water demand equal to the mean streamflow. This leads to the analysis of the statistical properties of the range, RT, of variation in storage levels in an infinitely large reservoir. Assuming that monthly storage changes, Xt, is a normally distributed independent random time series, then the expected value of RT, E(RT), can be estimated from the following expression [Anis and Lloyd, 1953]:

equation image

where σx is the standard deviation of Xt. Asymptotically for large T, equation (11) can be approximated by

equation image

which agrees with Feller's [1951] result. Equation (11) does not account for seasonality or for the effects of serial correlation. Hurst [1951] found that persistence in the inflow time series caused RT to increase in proportion to TH, where Hurst coefficient, H > 0.5. Equations (11) and (12) apply to systems with infinite storage capacity. If capacity is limited, then RT is less than given by equation (12) for large values of T [Fiering, 1967].

[38] The RT versus T curves obtained for the individual basins were compared to the theoretical curve as defined by equation (12) for an infinite reservoir. To facilitate the comparison, monthly storage changes were rescaled to have a unit variance for each month. The values of RT were then computed for each of the 27 basins. In order to reduce the sampling variability of RT for large T, average values of RT across the basins were computed. The results are compared in Figure 13. The sample values agree well with the theory for small values of T. However, sample values of RT are clearly less than the theoretical value for large T. The main reason for the expected values of RT, E(RT), found in this study to deviate from equation (12) is the effect of limited surface water storage capacity.

Figure 13.

Comparison of theoretical TWS range [from Anis and Lloyd, 1953] and standardized average TWS range of ISA-scale basins.

6. Spatial Analysis of RT

[39] The RT values for T = 1, 5, 10, and 36 years were gridded into spatial plots (see Figures 14a–14d). These figures show that RT increases from the west to the east. The spatial differences are greater for smaller T. As in Figures 4a and 4b, the RT plots reveal an apparent east-west gradient. The following subsections examine if RT is related to basin soil, vegetation and climate.

Figure 14.

(a–d) Spatial plots of TWS range for different time lengths.

6.1. Relationship Between RT and Basin Characteristics

[40] An analysis was done to see if differences in RT among the 27 basins can be explained by basin characteristics. R10 is used for the analysis, instead of R36, because R36 is more subject to the uncertainty in data than R10. Figure 15 is a scattergram of R10 versus P/PE. It is clear that R10 increases with P/PE, echoing previous findings that wetter basins have larger RT than drier ones. The correlation between R10 and P/PE is relatively high, with R2 = 0.806. Figure 16 is a plot of R10 versus a measure of normalized deviation of vegetation index (NDVI5) developed by Koren and Kogan [1995]. NDVI5 is a 5-week average of NDVI during the peak growing season. A high NDVI5 value suggests a high state of vegetative activity. R10 is shown to increase with NDVI5, suggesting that vegetated regions have larger values for R10 than desert regions.

Figure 15.

Scattergram of the 10-year TWS range R10 and the P/EP ratio for 27 ISA basins.

Figure 16.

Scattergram of the 10-year TWS range R10 and NDVI5 for 27 ISA basins.

[41] Figure 17 is a scattergram of R10/R1 versus R1. A high value for R10/R1 indicates that the year-to-year TWS variability is very high compared to intra year variability, R1. It can be noted that R10/R1 decreases as R1 increases. This suggests that the dry basins, which have smaller R1, have higher year-to-year TWS variability than the wet basins and therefore need longer data periods to observe full range of water storage variability.

Figure 17.

Scattergram of the R10/R1 ratio versus R1 for 27 ISA basins.

6.2. Use of Soil and Vegetation Information to Estimate RT

[42] Soil and vegetation type information has often been used to estimate parameters of hydrologic models a priori. A statistical analysis was therefore done to see if this kind of information would help to explain the variability of RT in space. The 1 km soil texture data developed by Pennsylvania State University [Miller and White, 1998] and the 1 km vegetation type data developed by International Geosphere-Biosphere Program (IGBP) [1992] were used for this analysis. There are 12 soil texture classes and 17 vegetation classes. Tables 3 and 4 give the definitions of the soil and vegetation type. Figures 18a and 18b are the histograms of the soil and vegetation type for the entire LSA-SW. As shown in the figures, only some of the soil and vegetation type are present in LSA-SW, with silt loam as the most dominant soil type and open shrubland as the most dominant vegetation type. From Figures 14a–14d, we can determine the corresponding RT values for each 1 km soil and vegetation pixel. The vertical bars in Figures 19a and 19b show the distribution of R10 for each soil and vegetation type. The five horizontal lines on each bar represent from bottom to top, respectively, the 5, 25, 50, 75 and 95 percentile of R10. If we calculate the soil and vegetation type distribution for each basin and then treat the 50 percentile R10 values (i.e., the median) as the representative values for each soil and vegetation type, we can derive estimated R10 values for each basin. The estimated R10 values are plotted against the “observed” R10 values in Figures 20a and 20b (The “observed” values are the values obtained from water balance study using historical data). Figure 20a indicates that there is no correlation between R10 estimated from soil type information and the “observed” R10. On the other hand, Figure 20b shows that R10 estimated from vegetation type information is correlated with the “observed” R10, with a R2 value of 0.65. This is very consistent with the finding in Figure 16. These results reveal that vegetation type information is more indicative of TWS variability than soil type information.

Figure 18.

(a and b) Distribution of soil and vegetation type in LSA-SW.

Figure 19.

(a and b) Distribution of R10 for different soil and vegetation type.

Figure 20.

(a and b) Comparison of the “observed” R10 and the estimated R10 based soil and vegetation type for 27 ISA basins.

Table 3. Definition of Soil Texture Type
Soil Texture TypeSoil Texture Description
1sand
2loamy sand
3sandy loam
4silt loam
5silt
6loam
7sandy clay loam
8silty clay loam
9clay loam
10sandy clay
11silty clay
12clay
Table 4. Definition of IGBP Vegetation Type
IGBP Vegetation TypeVegetation Description
1evergreen needleleaf forest
2evergreen broadleaf forest
3deciduous needleleaf forest
4deciduous broadleaf forest
5mixed forest
6closed shrublands
7open shrublands
8woody savannah
9savannahs
10grasslands
11permanent wetlands
12croplands
13urban and built up
14cropland/natural vegetation mosaic
15snow and ice
16barren or sparsely vegetated
17water bodies

7. Evaluating Estimated Seasonal Evapotranspiration Using Atmospheric Divergence Data

[43] The results presented in section 5.4 show that TWS range can be sensitive to the evapotranspiration functions used. We believe that function 5 provides the most reasonable results because it incorporates realistic physical notions in its formulation. Because no observed evapotranspiration data are available, it is difficult to verify this claim. One indirect way to do this is to use atmospheric divergence data, which were provided to us by E. Wood of Princeton University [Lohmann et al., 1998].

[44] Long-term monthly average evapotranspiration over the entire LSA-SW was computed for all five evapotranspiration functions using a 10-year data period from 1979 to 1988. The values, along with the evapotranspiration derived from atmospheric divergence over the corresponding period, Eatm, are listed in Table 5. Also included in the table are the annual totals and the values of R2, which measure the correlation between estimated evapotranspiration values and Eatm. Note that the Eatm values are subject to a certain degree of uncertainty and the annual total for Eatm is at least 10% higher than estimated by water balance simulations. Table 5 shows that the annual evapotranspiration amount of function 3 is quite different from those of other functions, confirming the observation from section 5.4. The table also shows that the seasonal evapotranspiration pattern of function 5 matches that of Eatm most closely, lending more support that function 5 is the most reasonable of the five evapotranspiration schemes.

Table 5. Comparisons of Evapotranspirationa
 EatmF1F2F3F4F5
  • a

    F1–F5, functions 15.

January17.3316.6017.3615.0819.4418.31
February26.4725.1626.3921.4929.7327.35
March53.1447.2149.4339.8454.0351.86
April69.4268.1070.9857.4775.6864.51
May101.3391.1894.8976.9699.3794.97
June103.6290.2290.7985.0895.2390.25
July84.8095.8194.9989.1990.3675.88
August73.5181.2379.1880.3470.2065.79
September73.6649.9445.0861.9338.5751.72
October43.4730.7226.5444.4424.2745.97
November41.4719.5517.9326.0517.4129.34
December22.6615.2615.4916.1616.2019.57
Year710.88630.98629.05614.03630.49635.52
R20.890.870.900.840.95

8. Summary and Conclusions

[45] This paper examined basin TWS properties and their relationships to soils, vegetation and climate by using a diagnostic approach that relies on observed precipitation and streamflow data for water balance analysis. This approach involves no assumptions on how runoff occurs and can be applied easily anywhere in the world.

[46] It is found that TWS range is sensitive to assumptions on how water evapotranspirates from the land surface. However, the pattern that TWS range increases with the study period is similar, no matter what assumption is used for evapotranspiration. Comparison with evapotranspiration derived from atmospheric divergence data shows that the evapotranspiration function that considers both initial soil moisture and the amount of precipitation in the period provides the most consistent seasonal evapotranspiration results. This study also showed that TWS range is not sensitive to precipitation errors or to the values of model parameter Dmax, which regulates the time of the evapotranspiration.

[47] Climate plays a significant role in how the TWS range varies in space and time. An east-west gradient is apparent in TWS variability. Wetter regions exhibit larger variability than dry regions. Dry regions have very small intra annual variability, compared to year-to-year variability, suggesting that longer data sets are especially important to model dry region water balance. TWS range shows strong correlation to climatological factors such as P/PE ratio and vegetation index such as NDVI5. Vegetation type information is a much better indicator of TWS range than soil type information. TWS range was found to be spatial scale dependent, with ISA scale displaying larger ranges than larger LSA scale.

[48] Much emphasis of this paper has been placed on TWS range. This is because it can be regarded as a surrogate to TWSC, a common parameter that appears in most hydrologic model in one form or another. Intuitively, TWS range can be regarded as the lower bound for TWSC. Understanding of TWS range in space and time and its relation to climate, soils, and vegetation should help modelers to develop regionalization equations for hydrologic model parameters such as TWSC. Further research to investigate the relationship between TWS range and TWSC is planned by the authors. A follow-up study to examine the space-time variability of TWS properties over the entire continental United States by applying the same analysis to basins selected from different parts of the country will also be conducted in the future.

Appendix A:: The Five Evapotranspiration Functions Used in the Analysis

[49] Five evapotranspiration functions were used in the analysis presented in this paper to test the sensitivity of the conclusions from this paper to different evapotranspiration rules. A detailed description of these is given below.

[50] In function 1, the estimated evapotranspiration Et is assumed to be linearly proportional to the relative water storage saturation level. In function 2, Et is assumed to be proportional to the water storage saturation level until the relative saturation level reaches a certain level, β, where evapotranspiration would take place at potential rate. Variations of the first two functions can be founded in many existing land surface hydrologic models. In function 3, Et is assumed to be equal to a fixed fraction, φ, of potential evapotranspiration. It is the only function that is independent of the water storage saturation level. To satisfy the long-term water balance, φ is set to a value such that the estimated long-term evapotranspiration is equal to the long-term observed precipitation minus the long-term observed runoff. Function 3 was used as a boundary condition on evapotranspiration in early atmospheric models [Mitchell et al., 1998]. In function 4 an attempt was made to differentiate evaporation from bare soil and transpiration from vegetated surface. Time-dependent parameter ξt is the vegetation greenness fraction from the basin. For this study, ξt is assumed to vary seasonally but not from year to year. The values for ξt are derived from a 10 × 10 km2 gridded map of vegetation fraction developed by Gutman and Ignatov [1998]. In the first four functions described, the evapotranspiration is assumed to be independent of the precipitation input as occurred during the current period. For water balance conducted at a monthly time step, this assumption is erroneous. Thornthwaite and Mather [1955] assumed that all precipitation occurred during a month is available for evapotranspiration until the potential evapotranspiration demand is met. The rationale is that the surface is wet after it rains. Therefore evapotranspiration does not depend on the water storage at the beginning of the month. In function 5, parameter φ is introduced, which is a fraction of precipitation input in current month available to satisfy evapotranspiration demand. If this is less than potential evapotranspiration Ep,t, the residual evapotranspiration demand is met in the same manner as in function 4. If φ = 0, function 5 is the same as function 4. When φ = 1.0, evapotranspiration is computed under the same assumption used by Thornthwaite and Mather [1955].

Acknowledgments

[51] The authors gratefully thank the anonymous reviewers for their helpful comments and suggestions. This work was part of the NOAA GCIP/GAPP Core Project supported by NOAA's Office of Global Programs.

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