General procedure to initialize the cyclic soil water balance by the Thornthwaite and Mather method

The original Thornthwaite and Mather method, proposed in 1955 to calculate a climatic monthly cyclic soil water balance, is frequently used as an iterative procedure due to its low input requirements and coherent estimates of water balance components. Using long term data sets to establish a characteristic water balance of a location, the initial soil water storage is generally assumed to be at field capacity at the end of the last month of the wet season, unless the climate is (semi-) arid when the soil water storage is lower than the soil water holding capacity. To close the water balance, several iterations might be necessary, which can be troublesome in many situations. For (semi-) arid climates with one dry season, Mendonça derived in 1958 an equation to quantify the soil water storage monthly at the end of the last month of the wet season, which avoids iteration procedures and closes the balance in one calculation. The cyclic daily water balance application is needed to obtain more accurate water balance output estimates. In this note, an equation to express the water storage for the case of the occurrence of more than one dry season per year is presented as a generalization of Mendonça’s equation, also avoiding iteration procedures.


Introduction
Climatologic soil water balance estimation is an important tool for edaphological characterization (Martin et al., 2008).Among the methods to estimate the soil water balance from simple soil and climate data, the method proposed by Thornthwaite and Mather (1955) is one of the most widely used.This procedure allows estimating the actual evapotranspiration, soil water deficit and excess.Therefore, it is especially useful for evaluating the effectiveness of agricultural practices (Dunne and Leopold, 1970;Black, 1996;Silva et al., 2006;Bruno et al., 2007;Sparovek et al., 2007).
Soil water balance as estimated by the Thornthwaite and Mather (1955) method (monthly scale) can be applied for climate classification, hydrological characterization for water management, environmental studies; and agricultural planning to define land use and agricultural practices.To initiate the calculation procedure, soil water storage is assumed to be at field capacity at the end of the last month of the wet season.However, in arid and semi-arid regions this soil water storage will be lower than the soil water holding capacity.
Sci. Agric.(Piracicaba, Braz.), v.67, n.1, p.87-95, January/February 2010 For (semi-) arid climates with one dry (and one wet) season, Mendonça (1958) derived an equation to quantify the soil water storage at the end of the last month of the wet season, without the need of iterations.Many authors have used the original Thornthwaite and Mather's model (Alley, 1984), while others have shown that this procedure can also be applied to smaller time scale (Eaton, 1995;Swanson, 1996), modifying the original model in an effort to improve certain components of the water balance.Steenhuis and Van Der Molen (1986) and Rushton et al. (2006) presented a study of the Thornthwaite's method at a daily scale.
The uses of a monthly scale in water balance models can, for example, lead to as much as a 25% underestimate of groundwater recharge (Rushton and Ward, 1979).The recharge estimation tends to decrease if the time scale is lower.If the accounting period is longer than ten days, Howard and Lloyd (1979) also demonstrated that large errors could occur in the water balance.
A modified Thornthwaite and Mather's model was applied by Swanson (1996) in Wisconsin.The recharge was considerably lower than needed for the successful calibration of a regional groundwater flow model (Krohelski et al., 2000).One of the main explanations for the low estimates was related to the use of a monthly scale in the water balance calculations.Thornthwaite and Mather (1957) gave a brief example of a daily application, because this is needed for more accurate water balance output estimates.They reported that their procedure could theoretically be used at a daily scale.
In this note, an equation that generalizes the approach of Mendonça (1958) is proposed to determine the initial soil water storage without making use of iterations, for locations with more than one dry and wet seasons per year, a situation which becomes especially relevant in arid and semi-arid climates.

Theoretical background
The monthly cyclic water balance (Thornthwaite and Mather, 1955) The basic equation to estimate the actual soil water storage (A, mm) is (Thornthwaite and Mather, 1955) where L is the accumulated potential water loss (mm), defined as the accumulated sum of the difference between pluvial precipitation (P, mm) and potential evapotranspiration (ETo, mm) (equation 3) and A C is the soil water holding capacity (mm): where Z e is effective root depth (mm), θ f and θ w are, respectively, the field capacity and the wilting point soil water contents (cm 3 cm -3 ).
Using i as an index to number the chosen period during the year (for the monthly case, i = 1, 2, ..., 12), we have: If (P i -ETo i ) < 0 (case I -dry season): If (P i -ETo i ) ≥ 0 (case II -wet season): and The actual evapotranspiration ( i ETa , mm), for the period i, can be computed as follows: where ΔA i is the soil water storage change between the periods i and i-1.
The soil water deficit (D, mm) and excess (E, mm) can be calculated as follows: case I (dry season): case II (wet season): Procedures to estimate soil water storage at the end of the wet season

Thornthwaite and Mather (1955)
The soil water storage A i is assumed to be at its maximum value A C (field capacity) at the end of the last month of the wet season.Hence, for this month corresponding to the last month of the wet season, L i-1 = 0 (Figure 1).The iterative calculation procedure begins in month i, the first month of the dry season using equations (3) and (4), as schematically presented in Figure 1.In the case of arid and semi-arid conditions, for which field capacity is not even reached at the end of a wet season, this procedure applied iteratively until conver-Sci.Agric.(Piracicaba, Braz.), v.67, n.1, p.87-95, January/February 2010 gence of the monthly values of soil water storage is reached leads to an inconvenient calculation routine.

Mendonça (1958)
To avoid the need of the iterative computing procedure described above for the case of a dry climate, Mendonça (1958) proposed a procedure (Figure 2) to initialize the cyclic monthly soil water balance for those cases in which one wetter season can be identified.Using the expressions: A system of two equations and two unknown variables x and y (Figure 2) can be written based on equations (13), ( 14) and (15): ( ) where a and b are the numbers of order of the first and the last month of the dry season, respectively.
To estimate L in the last month of the wet season of a cyclic monthly soil water balance, Mendonça (1958) proposed: This equation allows L to be obtained in a straightforward way whereas it would take several cycles to find L by convergence using the original Thornthwaite and Mather procedure.

The proposed procedure
A general procedure is proposed analogous to the Mendonça procedure, but for cases where more than one dry and wet seasons can be identified, irrespective of the time step within the period considered for analysis.This approach becomes relevant especially when using daily or weekly time scales (calculation steps, input data).The general equation to define L at the last period of the first wet season, in case of two (Figure 3) and three (Figure 4) dry and wet seasons, allowing the computation of soil water storage at the first period of the subsequent dry season are, respectively (Appendix B and C): If k = 1, equation ( 25) reduces to equation ( 22) as proposed by Mendonça (1958) for just one dry season (Table 1).

Results and Discussion
The application of the Thornthwaite and Mather (1955) procedure to initialize the climatic cyclic soil water balance (L = 0 mm) (Tables 2 and 4, with one wet season in March; and Tables 3 and 5, with two wet sea-sons in February and November) for Petrolina (State of Brazil), results in L = 399.8460mm and L = 436.2771mm, respectively, after four (q = 4) iterations (in both cases).
Using the Mendonça (1958) procedure (ME) (Table 1), for the case of a single dry period in Petrolina (Table 2 and 4), the accumulated potential water loss (L, mm) at the end of the last month of the wet season is calculated directly using parameters shown in Table 6.Using the proposed procedure (PP) (Table 1, equation 25), the accumulated potential water loss (L 2 , mm) at the onset of the first dry season (February) is calculated directly (Table 7).
The Thornthwaite and Mather (1955) soil water balance method is popular in regions with low data availability.To apply this method in dry climates, an initial     Table 3 -The Thornthwaite and Mather procedure to initialize the climatic cyclic water balance (two wet seasons -February and November).Petrolina-PE, Brazil (year: 1976).
accumulated potential water loss must be estimated which, according to Thornthwaite and Mather (1955) can be done in an iterative procedure.As this iterative method is quite cumbersome, we propose a straightforward calculation procedure to calculate this initial accumulated potential water loss; a similar equation had already been proposed by Mendonça (1958), but unlike his equation, the one we deduced allows to express the water storage for the case of the occurrence of more than one dry seasons per year, a situation which becomes especially relevant when the soil water balance is estimated at a daily scale in arid and semi-arid climates.The procedure is meant to substitute for the originally proposed iterative routine.An example of application shows a perfect agreement between the proposed procedure and the original (iterative) procedure.4 -Climatic cyclic water balance (one dry season -March) using the Thornthwaite and Mather method.Petrolina-PE, Brazil (period: 1975to 2006).20).p: accumulated potential water loss during the wet season (L w , mm) per unit of soil water holding capacity (Ac, mm) (equation 21).

Sci
x: accumulated potential water loss at the last period of the wet season (L 3 , mm) per unit of soil water holding capacity (A c = 125 mm) (equation 22).
Figure 5 -A schematic example to illustrate the procedure of beginning the cyclic soil water balance, for arid and semi-arid areas, with k dry seasons.
The integral solution becomes:

Appendix B
The proposed procedure for two dry and wet seasons Applying equation ( 13) for the two dry and wet seasons case: Combining the equations (B1) and ( B2) and ( B3) and (B4): Rewriting the equations (B5) and (B6) in convenient forms: Then, by Kramer's rule:

Appendix C
The proposed procedure for three dry and wet seasons Applying equation ( 13) for the three dry and wet seasons case: [C1] y is A per unit of A C (y = α) in last month of the dry season, x is L per unit of A C (x = λ) in the last month of the wet season, and L d , L w , n and p are defined as: Figure 1 -A schematic example of the Thornthwaite and Mather's (1955) procedure to initialize the cyclic monthly soil water balance.

Figure 2 -
Figure 2 -A schematic example of theMendonça (1958) procedure to initialize the cyclic monthly soil water balance for arid and semi-arid areas with just one wet season.

Figure 4 -
Figure 4 -A schematic example to illustrate the procedure of beginning the cyclic soil water balance, for arid and semi-arid areas, with three dry seasons.

Figure 3 -
Figure 3 -A schematic example to illustrate the procedure of beginning the cyclic soil water balance, for arid and semi-arid areas, with two dry seasons.

n 1
and n 2 : accumulated potential water loss during the first and second dry seasons (Ld 1 and Ld 2 , mm) per unit of soil water holding capacity (Ac, mm) (equation 20).p 1 and p 2 : accumulated potential water loss during the first and second wet seasons (Lw 1 and Lw 2 , mm) per unit of soil water holding capacity (Ac, mm) (equation 21).x 1 and x 2 : accumulated potential water loss at the last period of the first and second wet seasons (L 2 and L 11 , mm) per unit of soil water holding capacity (A c = 125 mm) (equation 23).

Table 1 -
Thornthwaite and Mather (1955)ather -TM, Mendonça -ME, and the proposed procedure -PP) to estimate initial parameters for a cyclic soil water balance based on theThornthwaite and Mather (1955)method, defining the accumulated potential water loss L at the last period of the wet season.

Table 2 -
The Thornthwaite and Mather procedure to initialize the climatic cyclic soil water balance (one wet season -March).Petrolina-PE, Brazil (period: 1975 to 2006).

Table 5 -
Climatic cyclic water balance (two dry seasons) using the Thornthwaite and Mather method.Petrolina-PE,  Brazil (year: 1976).accumulated potential water loss during the dry season (L d , mm) per unit of soil water holding capacity (Ac, mm) (equation