Why is the cybernetic hydrologic balance
better suited for yield hydrology
than the conventional approach?


Victor M. Ponce


15 January 2018



Abstract. A comparison between the conventional approach to the hydrologic balance and L'vovich's catchment wetting approach, referred herein as the cybernetic approach, reveals fundamental conceptual differences. The conventional approach is seen to be mostly suited to event hydrology, particularly for applications of flood hydrology and related urban hydrology. On the other hand, the cybernetic approach is suited to yield hydrology, i.e., for determinations of the availability of water resources on an annual basis.


1.  CONVENTIONAL HYDROLOGIC BALANCE

The conventional hydrologic balance, commonly referred to as the hydrologic budget of a catchment, is covered in all standard hydrology textbooks; see, for instance, Ponce (2014).

A hydrologic budget is an accounting of the various transport phases of the hydrologic cycle within a catchment. The following is a complete hydrologic budget, including both surface water and groundwater:

            
ΔS = P - ( E + T + G + Q )
            
(1)

in which ΔS = change in storage, P = precipitation, E = evaporation, T = evapotranspiration, G = groundwater outflow, and Q = surface runoff. Equation 1 states that within a given time span, the change in water volume remaining in storage in a catchment is the difference between the input (precipitation) and the sum of all outputs (evaporation, evapotranspiration, groundwater outflow, and surface runoff).

In hydrologic practice, the terms of Eq. 1 are expressed in units of water depth, i.e., a water volume uniformly distributed over the catchment area. Under equilibrium conditions, ΔS = 0, and Eq. 1 reduces to (Fig. 1):

            
P = E + T + G + Q
            
(2)

a hydrologic budget that considers both surface water and groundwater

Fig. 1  A complete hydrologic budget.

A partial hydrologic budget, which considers only surface water, is:

            
ΔS = P - ( E + T + I + Q )
            
(3)

in which I = infiltration. Within a given time span, under equilibrium conditions (ΔS = 0), Eq. 3 reduces to (Fig. 2):

            
P = E + T + I + Q
            
(4)

a hydrologic budget that considers only surface water

Fig. 2  A partial hydrologic budget, which considers only surface water.

Equation 4 is imperfect, because it assumes that infiltration amounts are permanently lost to the surface-water budget, which is not quite correct. Within a given time span, say one year, infiltrated amounts may actually return to the control volume (and be eventually released) as evaporation from lakes and ponds, evapotranspiration from vegetation, or as the baseflow component of surface runoff.

Notwithstanding its apparent limitations in yield hydrology, Eq. 4 works well in event hydrology (refer to Ponce and Palaniappan, 2014: Facets of Hydrology). In Eq. 4, the hydrologic losses L are defined as the sum of evaporation, evapotranspiration, and infiltration: L = E + T + I This results in:

            
P = L + Q
            
(5)

which leads to the fundamental equation of flood hydrology:

            
Q = P - L
            
(6)


2.  CYBERNETIC HYDROLOGIC BALANCE

On a global annual basis, all precipitation either (1) it returns to the atmosphere as vaporization V (through evaporation and evapotranspiration), or (2) it runs off the land as runoff R (i.e., streamflow). The conventional water balance separates precipitation into: (1) losses, and (2) runoff (Eq. 5). The losses are evaporation, evapotranspirarion, and infiltration. Yet the infiltration portion of the losses may eventually show up as evaporation, evapotranspiration, or even the baseflow component of runoff; hence, the double counting.

L'vovich (1979) solved this problem by introducing the concept of catchment wetting. He separated annual precipitation into two components (Fig. 3):

            
P  =  S  +  W
            
(7)

in which S = surface runoff, i.e., the fraction of runoff originating on the land surface, and W = catchment wetting, or simply, wetting, the fraction of precipitation not contributing to surface runoff.

L'vovich's water balance.

Fig. 3  L'vovich's water balance.

In turn, wetting is separated into two components:

            
W  =  U  +  V
            
(8)

in which U = baseflow, i.e., the fraction of wetting which exfiltrates as the dry-weather flow of streams and rivers, and V = vaporization, i.e., the fraction of wetting returned to the atmosphere as water vapor. Ostensibly, L'vovich's hydrologic budget neglects deep percolation, the portion of wetting which bypasses the surface waters altogether, estimated at less than 2% of precipitation on a global basis (World Water Balance, 1978).

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

            
V  =  E  +  T
            
(9)

in which E = nonproductive evaporation, hereafter referred to as "evaporation," and T = productive evaporation, i.e., that resulting from plant transpiration, hereafter referred to as "evapotranspiration."

In turn, evaporation has two components:

            
E  =  Eg  +  Ew
            
(10)

in which Eg = evaporation from the ground, bare soil, and small surface storage (puddles), and Ew = evaporation from water bodies such as lakes, reservoirs, streams, and rivers.

Evapotranspiration T 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.

Runoff (i.e., total runoff) is the sum of surface runoff and baseflow:

            
R  =  S  +  U
            
(11)

Combining Eqs. 7, 8, and 11:

            
P  =  R  +  V
            
(12)

Equations 7 to 12 constitute a set of water balance equations. Combining Eqs. 11 and 12 leads to:

            
P  =  S  +  U  +  V
            
(13)

Equation 13 separates annual precipitation into its three major components: (1) surface runoff, (2) baseflow, and (3) vaporization. Significantly, Eq. 13 assumes that the change in soil-moisture storage from year to year is negligible, an assumption which is useful as a first approximation.

Under L'vovich's hydrologic budget, two water balance coefficients may be defined: (1) runoff coefficient, and (2) baseflow coefficient. The runoff coefficient is:


            R                R
Kr  =  _____  =  __________
            P           R  +  V

(14)

The baseflow coefficient is:


             U                U
Ku  =  _____  =  __________
            W           U  +  V

(15)

Figure 4 shows runoff and baseflow coefficients calculated by Ponce and Shetty (1995), based on data reported by L'vovich (1979). It is seen that in all cases, runoff and baseflow coefficients increase with annual precipitation.

[Click on either image to display]
Runoff coefficients.
(a)
    
Baseflow coefficients.
(b)
Fig. 4 (a) Runoff coefficients, and (b) baseflow coefficients, for the following data:
1. Africa: evergreen sclerophyll forests and scrub.
2. Africa: mountain conifer forests.
3. North America (Canada); subarctic forests (taiga).
4. South America: wet evergreen forests in the mountains.
5. Asia (india): semideciduous forests in the mountains (Western Ghats).

3.  ANALYSIS

It is seen that the conventional hydrologic balance, Eqs. 1 to 6, is deductive and demonstrably based on a premise of cause-effect. The fundamental approach is subtractive, characterized by the statement "runoff is equal to precipitacion minus losses" (Eq. 6). It works reasonably well for applications in event hydrology (such as the rational method of urban hydrology) but, due to the double counting, it accuracy decreases for the longer-term yield hydrology.

On the other hand, the cybernetic hydrologic balance, Eqs. 7 to 13, is inductive, not based on cause-effect but rather on the system's wholeness. The fundamental approach is additive, characterized by the statement "precipitation is equal to runoff plus vaporization" (Eq. 12). This approach is suited to applications in yield hydrology, i.e., for determinations of annual baseflow and runoff coefficients.


4.  SUMMARY

A comparison between the conventional approach to the hydrologic balance and L'vovich's catchment wetting approach, referred herein as the cybernetic approach, reveals fundamental conceptual differences. The conventional approach is seen to be mostly suited to event hydrology, particularly for applications of flood hydrology and related urban hydrology. On the other hand, the cybernetic approach is suited to yield hydrology, i.e., for determinations of the availability of water resources on an annual basis. An online calculator for the cybernetic water balance rounds up the experience (Online water balance).


References

L'vovich, M. I. 1979. World water resources and their future. Translation from Russian by Raymond L. Nace, American Geophysical Union.

Ponce, V. M., and A. V. Shetty. 1995. A conceptual model of catchment balance: 2. Application to runoff and baseflow modeling. Journal of Hydrology, 173, 41-50.

Ponce, V. M. 2014. Engineering hydrology: Principles and practices. Online text; Second edition.

World water balance and water resources of the Earth. (1978). USSR Committee for the International Hydrologic Decade, UNESCO, Paris.


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