A conceptual model
of catchment water balance:
1. Formulation and calibration


Victor M. Ponce
and
A. V. Shetty


Online version 2016

[Original version 1995]



ABSTRACT

A conceptual model of a catchment's annual water balance is developed. The model is based on the sequential separation of annual precipitation into surface runoff and wetting, and wetting into baseflow and vaporization. The separation is based on a proportional relation linking the three variables involved at each step. The generic form of the proportional relation is (X - λZp - Y)/[(1 - λ)Zp = Y/(X - λZp), in which X is an independent variable, Y is a dependent variable, λ is the initial abstraction coefficient, and Zp is the potential value of the difference Z = X - Y. Given a set of model parameters, the method can be used to separate annual precipitation into its three major components: surface runoff, baseflow, and vaporization. Initial application of the method to literature catchment data provided encouraging results. The method can be used for estimates of the annual water yield throughout the climatic spectrum. The model parameters can be estimated from past experience or calibrated using measured data.


1.  INTRODUCTION

A catchment's water yield is a fundamental problem in hydrology, referring to the volume of water available at the catchment outlet over a specified period of time. The yield is expressed for monthly, seasonal, or annual periods. Several approaches are available for the computation of water yield. These vary in complexity from simple empirical formulas to complex models based on continuous simulation. While the empirical formulas have limited applicability (Sutcliffe and Rangeley, 1960; Woodruff and Hewlett, 1970), the continuous simulation models require large amounts of data for their successful operation (Crawford and Linsley, 1966). A practical alternative is represented by conceptual models which use the water balance (or hydrologic budget) equation to separate precipitation into its various components (Hamon, 1963).

In this paper, we develop a conceptual model of annual water balance, suitable for application to a wide range of climatic conditions. The model separates annual precipitation into its three major components: surface runoff, baseflow, and vaporization. It is based on a two-step sequential application of a proportional relation linking two variables X and Y, such that the difference Z = X - Y asymptotically reaches an upper bound as X and Y grow unbounded. The generic form of the proportional relation has two parameters: λ, the initial abstraction coefficient, and Zp, the potential value of Z = X - Y. Initial testing using L'vovich's (1979) catchment data shows that the model is applicable throughout a wide range of climatic settings.


2.  WATER BALANCE EQUATIONS

A water balance equation applicable to an individual storm is:


P = Q + L

(1)

in which P is precipitation, Q is surface runoff (also rainfall excess), and L is losses, or hydrologic abstractions (in millimeters, centimeters, or inches). The losses for an individual storm consist of interception, surface storage, and infiltration.

A water balance equation applicable on an annual basis is:


P = R + E

(2)

in which P is precipitation, R is runoff, including surface and subsurface runoff, and E is evapotranspiration (in millimeters, centimeters, or inches). In this equation, the term 'evapotranspiration' comprises all three types of evaporated moisture: (1) evaporation from vegetated surfacce, i.e. evapotranspiration, (2) evaporation from nonvegetated surfaces (bare ground), and (3) evaporation from water bodies.

Equations 1 and 2 are highly simplified models of a segment of the hydrologic cycle. For example, Eq. 1 does not describe the portion of surface storage that may eventually infiltrate into the ground. Likewise, Eq. 2 does not explicitly describe soil moisture, which is included in both runoff and evaporation. In fact, as shown by L'vovich (1979), a catchment's annual water balance is better described by a set of equations.

Annual precipitation P can be separated into two components (Fig. 1):


P = S + W

(3)

where S is surface runoff, i.e. the fraction of runoff originating on the land surface, and W is catchment welting, or simply 'wetting', the fraction of precipitation not contributing to surface runoff. (Note that Eq. 3 is the annual equivalent of Eq. 1.)

Dimensionless relative wave celerity vs dimensionless wavenumber

Fig. 1  Elements of the water balance.

Likewise, wetting consists of two components:


W = U + V

(4)

where U is baseflow, i.e. the fraction of wetting which exfiltrates as the dry-weather flow of rivers, and V is vaporization, the fraction of wetting returned to the atmosphere as water vapor (Lee, 1970). Deep percolation, i.e. the portion of wetting not contributing to either baseflow or vaporization, is a very small fraction of precipitation [a global average of less than 1.5%, according to L'vovich (1979)], and is neglected here on practical grounds.

Vaporization, which comprises all moisture returned to the atmosphere, has two components:


V = E + T

(5)

where E is nonproductive evaporation, hereafter referred to as 'evaporation', and T is productive evaporation, i.e. that resulting from plant transpiration, hereafter referred to as 'evapotranspiration.'

Evaporation has two components: (1) En, evaporation from nonvegetated areas of the Earth's surface and near-surface, i.e. from bare soil and small surface storage (puddles); (2) Ew, evaporation from sizable water bodies such as lakes, reservoirs, and rivers. Evapotranspiration is the evaporation from vegetated surfaces such as leaves and other parts of plants, a function of the physiological need of plants to pump moisture from the soil to maintain turgor and avail themselves of nutrients.

From Eqs. 3, 4, and 5, runoff consists of two components:


R = S + U

(6)

Likewise, precipitation P consists of two components:


P = R + V

(7)

Equation 7 is analogous to the well-known annual water balance Eq. 2. However, Eq. 7 is preferred here, because the term 'vaporization' clearly identifies the three sources of evaporated moisture (Eq. 5).

The set of Eqs. 3-7 constitute a set of water balance equations. Combining Eqs. 6 and 7 leads to:


P = S + U + V

(8)

that is, annual precipitation is separated into its three major components: surface runoff, baseflow, and vaporization. Significantly, Eq. 8 assumes that the change in soil-moisture storage from year to year is negligible, an assumption which is useful as a first approximation.

Equations 4 and 7 allow the definition of water balance coefficients. The baseflow coefficient is (L'vovich, 1979):


          U            U
Ku = ____ = ________
         W        U + V

(9)

and the runoff coefficient is:


          R           R
Kr = ____ = ________
         P         R + V

(10)

These coefficients vary as a function of prevailing climate, theoretically within the range 0-1. The prevailing climate is characterized by global wind circulation, mean latitude and altitude, aspect, geomorphology, proximity to the ocean, type of vegetation, and land use. In arid and semiarid regions, the water balance coefficients are low, typically less than 0.2, and may approach zero in hyperarid cases. In humid tropical regions, the coefficients are high, often greater than 0.5, with the runoff coefficient Kr approaching unity in some unusual cases of extreme humidity (L'vovich, 1979). However, there are exceptions such as a swamp or marsh (wetland), where Kr is typically a low value because of the increased evapotranspiration potential.


3.  THE CONCEPTUAL MODEL

A significant feature of the water balance equations (Eqs. 3-7) is that they all have the same structure, in which a quantity X is expressed as the sum of two components Y and Z:


X = Y + Z

(11)

L'vovich (1979) has shown that Eq. 3 can be modeled by a proportional relation such that wetting asymptotically reaches an upper bound (WWp) as precipitation and surface runoff increase unbounded (P → ∞; S → ∞). It is noted that a similar relation is the basis of the SCS runoff curve number model, which solves Eq. 1 (US Department of Agriculture Soil Conservation Service (USDA SCS), 1985). L'vovich (1979) has also shown that Eq. 4 can be modeled by the same type of relation, i.e. one where vaporization reaches an upper bound asymptotically (VVp) as wetting and baseflow increase unbounded (W → ∞; U → ∞). In this way, the sequential two-step separation of annual precipitation into its three major components, surface runoff, baseflow, and vaporization, is accomplished.

The generic form of the proportional relation is, according to the SCS model (USDA SCS, 1985),


  X - λ' Zp' - Y              Y
_______________ = ____________
          Zp'                X - λ' Zp'

(12)

in which Y = f(X) and λ' is the initial abstraction coefficient. The initial abstraction is Ia = λ' Zp', in which Zp' is the potential value of Z', i.e. an upper bound to Z' = X - λ' Zp' - Y.

In this paper, the initial abstraction is alternatively defined as Ia = λ Zp' in which Zp is the potential value of Z = X - Y (Fig. 2). This leads to a slightly modified form of the proportional relation:


   X - λ Zp - Y               Y
_______________ = ____________
      (1 - λ)Zp             X - λ Zp

(13)

Dimensionless relative wave celerity vs dimensionless wavenumber

Fig. 2  Definition sketch for initial abstraction coefficient λ and potential Zp.

Equation 13 has a significant advantage over Eq. 12. Unlike λ', which has no theoretical upper bound, λ is limited in the range 0 ≤ λ ≤ 1. The initial abstraction coefficient λ is dimensionless; the units of Zp are those of X and Y (millimeters, centimeters, or inches).

For the special case of zero initial abstraction (λ = 0), Eq. 13 reduces to:


   X - Y          Y
_________ = ____
      Zp           X

(14)

Solving Eq. 13 for Y = f(X) leads to:


             (X - λ Zp)2
Y = __________________
          X + (1 - 2λ) Zp

(15)

subject to X > λ Zp; Y = 0 otherwise.

Using Eq. 15, the surface runoff submodel is:


             (P - λs Wp)2
S = ____________________
          P + (1 - 2λs) Wp

(16)

subject to P > λs Wp, and S = 0 otherwise, where λs is the surface-runoff initial abstraction coefficient. From Eq. 3,


W = P - S

(17)

Likewise, the baseflow submodel is:


             (W - λu Vp)2
U = ____________________
          W + (1 - 2λu) Vp

(18)

subject to W > λs Vp, and U = 0 otherwise, where λu is the baseflow initial abstraction coefficient. From Eq. 4,


V = W - U

(19)

Thus, given annual precipitation and a set of initial abstraction coefficients λs and λu and potentials Wp and Vp, Eqs. 16-19 are used to separate annual precipitation into surface runoff, baseflow, and vaporization. Then, runoff is calculated by Eq. 6, and the baseflow and runoff coefficients Ku and Kr are calculated using Eqs. 9 and 10, respectively.


4.  MODEL CALIBRATION

In the absence of data, the initial abstraction coefficients and potentials are estimated from past experience in similar climatic settings. When data are available, the model parameters can be calibrated. For λ = 0, Zp is solved from Eq. 14:


         X (X - Y)
Zp = ___________
              Y

(20)

For λ > 0, Zp is solved from Eq. 13:


         X + [1/(2λ)] {(1 - 2λ) Y - [(1 - 2λ)2 Y 2 + 4λ (1 - λ) X Y]1/2}
Zp = ________________________________________________________
                                                  λ

(21)

To calibrate the model parameters based on X-Y data, the following recursive procedure is suggested:

  1. Set λ = 0 and Δλ = 0.01.

  2. Use Eq. 20 or Eq. 21 to calculate a Zp for each X - Y pair, i.e. a Zp array.

  3. Calculate the mean, standard deviation, and coefficient of variation of the Zp array.

  4. Stop when the coefficient of variation of the Zp array has reached a minimum. Otherwise, increase λ by Δλ and go back to Step 2.

The calibrated λ is that for which the coefficient of variation of the Zp array is a minimum. The calibrated Zp is the mean of the Zp array corresponding to the calibrated λ.

This procedure was applied to L'vovich's catchment data (1979), and the results of the calibration are shown in Tables 1 and 2. On the basis of this initial application, a tentative classification of parameter ranges is suggested: initial abstraction coefficient λ (dimensionless): low (0 < λ ≤ 0.1), average (0.1 < λ ≤ 0.3), high (0.3 < λ ≤ 0.5), and very high (0.5 < λ ≤ 1); potential Zp (mm): low (0 ≤ Zp ≤ 1000), average (1000 < Zp ≤ 3000), high (3000 < Zp ≤ 5000), and very high (Zp > 5000).

Table 1 shows calibrated values of surface-runoff initial abstraction coefficient λs and wetting potential Wp for nine P - S data sets included by L'vovich (1979) (Fig. 3). Analysis of Table 1 leads to the following conclusions:

  1. Arctic-subarctic plains and subarctic forests in Canada have λs, = 0, and low to average Wp (889-1578 mm). The nearly frozen ground produces an immediate surface runoff response, in relation to the amount of precipitation.

  2. Mountain conifer forests in Africa, and mountain meadows, steppes, and savannas in South America have low λs (0.02-0.06), and average Wp (1789-2164 mm).

  3. Savannas in Africa and South America have average λs (0.18-0.22), and average (bordering on high) Wp (2627-2944 mm).

  4. Evergreen sclerophyll forests and scrub in Africa, and wet evergreen forests in the mountains of South America have average to high λs (0.23-0.31), and average Wp (1326-1517 mm).

  5. Humid evergreen forests in Africa have very high λs (0.54) and average Wp (1970 mm). The very high initial abstraction produces a sluggish surface runoff response.

TABLE 1. Calibration of surface-runoff initial abstraction coefficient λs
and wetting potential Wp.
Curve Description λs Wp
Africa
1 Mountain conifer forests 0.02 2164
2 Evergreen sclerophyl forests and scrub 0.23 1517
3 Savannas 0.18 2944
4 Humid evergreen forests 0.54 1970
Canada
1 Artic-subartic plains (wooden tundra) 0.00 889
2 Subartic forest (taiga) 0.00 1578
South America
1,2 Mountains meadows, steppes, and savannas 0.06 1789
3 Wet evergreen forests in the mountains 0.31 1326
4 Steppes and savannas in the plains 0.22 2627
Precipitation-surface-runoff data to develop this table were taken from L'vovich (1979): Figs. 10 (Africa), 11 (Canada), and 12(b) (South America).


Dimensionless relative wave celerity vs dimensionless wavenumber

Fig. 3  Precipitation-surface runoff relations for selected biogeographical regions in (a) Africa,
(b) Canada, and (c) South America (See Table 1 for explanation of curve numbers).

Table 2 shows calibrated values of baseflow initial abstraction coefficient λu and vaporization potential Vp for 11 W - U data sets included by L'vovich (1979) (Fig. 4). Analysis of this table leads to the following conclusions:

  1. Southeastern wet forests in North America have λu = 0 and very high Vp (6110 mm). This produces an immediate baseflow response, in relation to the amount of wetting. However, this is tempered by the very high value of vaporization potential.

  2. Subarctic forests and temperate steppes in North America have low λu (0.09-0.10), whereas Vp is low for the subarctic forests (796 mm) and high for temperate steppes (4246 mm).

  3. Mixed forests with moderate continental climate in North America and wet evergreen forests in the mountains of South America have low to average λu (0.10-0.13) and average Vp (1294-1856 mm).

  4. Plains and wooded steppes in North America and high mountain meadows in South America have average λu (0.19-0.25), whereas Vp is low for high mountain meadows (977 mm) and average for plains and wooded steppes (2047 mm).

  5. Mountain conifer forests, evergreen sclerophyll forests and scrub, and mountain landscapes in Africa, and savannas in South America have high λu (0.35-0.48), with low to average Vp (903-1797 mm).

TABLE 2. Calibration of baseflow initial abstraction coefficient λu and vaporization potential Vp.
Curve Description λu Vp
Africa
1 Mountain conifer forests 0.35 903
2 Evergreen sclerophyl forests and scrub 0.37 1405
3 Mountain landscapes 0.38 1797
North America
1 Subartic forest (taiga) 0.10 796
2 Mixed forest with moderate continental climate 0.13 1294
3 Plain and wooden steppes 0.19 2047
4 Temperate steppes 0.09 4246
5 Southeastern wet forests 0.00 6110
South America
1 Savannas 0.48 1721
2 Wet evergreen forests in the mountains 0.10 1856
3 High mountain meadows 0.25 977
Wetting-baseflow data to develop this table were taken from L'vovich (1979): Figs. 34 (Africa), 35 (South America), and 36 (North America).


Dimensionless relative wave celerity vs dimensionless wavenumber

Fig. 4  Wetting-baseflow relations for selected biogeographical regions in (a) Africa,
(b) Canada, and (c) South America. (See Table 2 for explanation of curve numbers).

Table 3 shows a comparison of baseflow and runoff coefficients reported by L'vovich (1979) and predicted by the conceptual model developed herein. It is seen that the conceptual model is reasonably predictive for a wide range of climatic conditions. Figure 5 shows calculated baseflow and runoff coefficients for selected biogeographical-climatic regions shown in Table 3. It is seen that the coefficients predicted by the conceptual model are responsive to variations in precipitation input.

Dimensionless relative wave celerity vs dimensionless wavenumber

Dimensionless relative wave celerity vs dimensionless wavenumber

Fig. 5  Baseflow and runoff coefficients as a function of mean annual precipitation:
(a) Cases 1, 2, and 3; (b) Case 4. (See Table 3 for explanation of curve numbers).

The wetting potential Wp is an upper bound to the fraction of annual precipitation that can be retained by a given catchment. A typical catchment is likely to feature several areas, each with a wetting potential Wpi covering a partial area Ai. A wetting potential spatially weighted for the entire catchment is:


         ∑ (Wpi Ai )
Wp = ___________
              ∑ Ai

(22)

The vaporization potential Vp is an upper bound to the fraction of annual wetting that can evaporate from a given catchment. A typical catchment is likely to have three distinct types of surfaces: (1) vegetated surfaces, with an evapotranspiration potential Tpv covering a partial area Av; (2) nonvegetated surfaces, with an evaporation potential Epn covering a partial area An; (3) free-water surfaces, with an evaporation potential Epw covering a partial area Aw. A vaporization potential spatially weighted for the entire catchment is:


          Tpv Av + Epn An + Epw Aw
Vp = ___________________________
                     Av + An + Aw

(23)


5.  SUMMARY

A conceptual model of a catchment's water balance is formulated. The model is based on the sequential separation of annual precipitation into surface runoff and wetting, and wetting into baseflow and vaporization. The separation is based on a proportional relation linking the three variables involved at each step. The generic form of the proportional relation is (X - λZp - Y)/[(1 - λ) Zp] = Y/(X - λZp), in which X is the independent variable, Y is the dependent variable, λ is the initial abstraction coefficient, and Zp is the potential value of the difference Z = X - Y.

Given a set of model parameters, the model can separate annual precipitation into its three major components: surface runoff, baseflow, and vaporization. Furthermore, baseflow and runoff coefficients are characterized as a function of climate. The model can be used for estimates of annual water yield throughout the climatic spectrum. An initial application of the model to L'vovich's (1979) catchment data provided encouragjng results. Additional research is needed to determine initial abstraction coefficients and potentials for a wide range of associated biogeographical regions and climatic settings.


6.  ACKNOWLEDGMENTS

The present study was performed in Spring 1994, while A.V. Shetty was at San Diego State University, on leave from the Hard Rock Regional Centre, National Institute of Hydrology, Belgaum, Karnataka, India. His leave was funded by the United Nations Development Programme.


REFERENCES

Crawford, N. H. and Linsley, R. K., 1966. Digital simulation in hydrology: the Stanford watershed model IV. Stanford Univ. Dep. Civ. Eng. Tech. Rep. 39.

Hamon, W.R., 1963. Computation of direct runoff amounts from storm rainfall. Int. Assoc. Sci. Hydrol. Publ., 63: 52-62.

Lee, R. 1970. Theoretical estimates versus forest venter yield. Water Resour. Res., 6(5): 1327-1334.

L'vovich, M. I, 1979. World water resources and their future. Original in Russian. English translation, American Geophysical Union, Washington, DC.

Sutcliffe, J. V. and Rangeley, W. R., 1960. Variability of annual river flow related to rainfall records. Int. Assoc. Sci. Hydrol. Publ., 76: 182-192.

U.S. Department of Agriculture Soil Conservation Service (USDA SCS), 1985. National Engineering Handbook, Section 4: Hydrology. SCS, Washington, DC.

Woodruff, J. F. and Hewlett, J. D., 1970. Predicting and mapping the average hydrologic response for the Eastern United States. Water Resour. Res., 6(5): 1312-1326.


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