Periodicity of crop coefficient and soil water depletion fraction in a climatological water balance

In comparison to a measured field water balance (FWB), we aimed to evaluate the impact of using different functions to daily estimate the crop coefficient (Kc) and soil water depletion fraction (p) in a climatological water balance (CWB), and verify that the grouping of output variables provides improved results. The FWB was conducted in Telêmaco Borba, Southern Brazil. The data were collected at weekly intervals in 2009, in an area of loblolly pine with 6 years-old. The CWB considers different equations to estimate daily Kc and p values. The output components of the CWB were estimated daily, then weekly and monthly grouped for comparison with the FWB. Linear correlation analysis, index “d” of concordance, index “c” of performance, mean error, mean absolute error and root mean square error were performed in order to compare the water balances, based on the soil-water storage variation (S) and actual evapotranspiration (ETa). The use of a Kc measured weekly improved the CWB, providing high correlation and small errors in relation to a measured water balance, independent of the comparison scale. On the other hand, the use of a Kc that considers climate variables (Kck) had the worst levels of accuracy and precision, and the biggest mistakes in all analyzes and all tested variables. There was no significant improvement with the daily variation of p, both grouping weeks as in months. The proposed equations do not represent any gain in the CWB, in comparison with the use of a constant p value over time. The estimate of the CWB and its subsequent grouping in months for comparison provided greater degree of accuracy and precision for the variables analyzed, but caused the biggest mistakes. Therefore, the calculation of CWB should be performed with the highest periodicity possible, and grouping the CWB output variables should only be performed for comparison.


INTRODUCTION
The field water balance (FWB) is the accounting of inputs and outputs of water at any given volume control over a specific time interval.It can be calculated by means of crop evapotranspiration (ETc) measurement instruments as lysimeters and evapotranspirometers, or by measuring the soil moisture.The monitoring of soil water storage, combined to the understanding about crop needs have been considered important tools to the agroforestry activities planning, improvement of soil water use efficiency by irrigation practices and agroclimatic zoning (Yan et al., 2012;Khazaei and Hosseini, 2015).
The study of water demand in soils under plantation of exotic woods, such as Pinus taeda, assists in the development of alternatives to the rational use of water, which implies in no compromising of the environmental balance and promoting the development of silvicultural activity (Rigatto et al., 2005).
Currently, the FWB is more used in scientific researches and their measures are commonly used to verify mathematical models, which are developed to simulate and perform estimations.Thus, many researchers have sought to develop indirect methods to estimate it from climatic variables, in parts because complete field measurements are time consuming, costly and experimentally difficult depending of the size of the area to be monitored (Zhang et al., 2004;Praveena et al., 2012;Yan et al., 2012).
In this context, the estimated climatological water balance (CWB) is required.However, some input components of CWB do not represents the real conditions of the crop in the field, especially for perennial crops, such as forest species, in relation to the variations in time.Due to the lack of specialized studies and local complexity measurements, the water components are usually estimated empirically and are considered constant over time, such as the crop coefficient (Kc) and soil water depletion fraction (p) (Allen et al., 1998).Using a constant value for these variables can significantly affect the output components, such as crop evapotranspiration (ETc); soil water storage (S); actual evapotranspiration (ETa).Since consistent, the highest frequency of the input data in the CWB generally improve their sensitivity to small variations over time (Khazaei and Hosseini, 2015), making it more reliable for the silvicultural planning.
In comparison to a measured field water balance, we aimed to evaluate the impact of using different equations to daily estimate the crop coefficient (Kc) and soil water depletion fraction (p) in a climatological water balance, and verify that the grouping of output variables provides improved results.

MATERIALS AND METHODS
A field water balance (FWB) was conducted in Telêmaco Borba, state of Paraná, Southern Brazil, 24°13′19"S, 50°32′33"W, 700 m altitude (Figure 1).The region has a climate type transitional wet subtropical to temperate ("Cfa/Cfb") with an average temperature in the coldest month below 16°C including frost events, and an average temperature in the warmest month above 22°C (Alvares et al., 2013).
This experiment served as a witness, being considered the actual values for comparison with the models proposed in this manuscript.The data were collected at weekly intervals in 2009, in an area of 12.5 ha of 6-years old Pinus taeda planting, with standard spacing of 2.0 × 3.0 m (1,667 trees/ha) in a clay Oxisol with undulated relief.For details of the methodology (Souza et al., 2016).
The output components (S -soil-water storage variation and ETa -actual evapotranspiration) of the FWB, were compared with a climatological water balance (CWB).ETa in the FWB was calculated as follows: Where: ETa is actual evapotranspiration (mm week -1 ); S is soilwater storage variation at the root zone (mm week -1 ); P is precipitation (mm week -1 ); D is downward drainage out of the root zone (mm week -1 ); U is upward capillary flow across root zone (mm week -1 ).
Soil water storage (S) was calculated preliminarily, with the ΔS obtained from the difference between previous (Sj) and current water storage (Sj+1): Where: Sj is soil water storage in j-th week year (mm); I is volumetric moisture in i-th soil depth (cm 3 /cm 3 ); zi is soil depth (m); j is weeks over year that samples were taken ; I is sample collection depths (m).
A daily climatological water balance (CWB) was estimated according to Thornthwaite and Mather (1945) methodology.The daily data series of precipitation (P) used in the simulations were the same used in FWB (Souza et al., 2016).Reference evapotranspiration (ETo) was estimated by Penman-Monteith method (Allen et al., 1998).Daily climatological data were provided by an automatic weather station.Soil water storage (S) was estimated from cosine equation (Rijtema and Aboukhaled, 1975).The initial value of S for 2009 was recorded on December 31, 2008, being equal to 52.5 mm, and was considered an average of total available water (TAW) equal to 174 mm, with no variation in the effective root depth (z = 0.80 m).
Different methodologies and functions to estimate the Kc were used to calculate the CWB (Figure 2).A basic value that did not change over time was used (KcA), this value was proposed by Allen et al. (1998) to conifers species.In addition, measured values were used, obtained in the FWB cited, which were grouped weekly and monthly (Kcm and Kcmonth, respectively).Finally, we tested an equation, proposed by Allen et al. (1998), that uses climate variables to estimate the daily Kck:   The output components were calculated daily in the CWB (SCWB and ETaCWB), and then grouped in weeks and months for comparison with the FWB (SFWB and ETaFWB).The evaluation was performed using the coefficient of determination (R²), index "d" Willmott et al. (1985), index "c" of Camargo and Sentelhas (1997): c > 0.85 = great accuracy; c from 0.85 to 0.76 = very good; c from 0.75 to 0.66 = good; c from 0.65 to 0.61 = average; c from 0.60 to 0.51 = tolerable; c from 0.50 to 0.41 = bad; and c ≤ 0.40 = very bad.Mean error (ME) were also used; mean absolute error (MAE) and root mean square error (RMSE).It is important to note that the comparisons were made with the S instead S, because the differences between methodology of FWB and CWB, and the SCWB and ETaCWB results were demonstrated in mm month -1 and mm day -1 , respectively, to facilitate discussion and comparison to Where: ME = mean error; MAE = mean absolut error; RMSE = root mean square error; n = number of observations (dimensionless); Ei = estimated value in the i-th day; Oi = observed value in the i-th day.

RESULTS
As expected, the ETo in 2009 showed typical trend throughout the year, with the lowest and highest values in winter and summer, respectively.Despite, P average was atypical regarding P normal of Telêmaco Borba region (Figure 4).The annual P average was higher than P normal presenting a total value of 1,608.1 and 1,490.0mm, respectively, with poor distribution of precipitation throughout 2009 and significant accumulation from September to December.Historically characterized as a month of low precipitation, July presented mean P average 38% higher than P normal .It is important to note that 2009 may be considered as an atypical year, especially in relation to observed precipitation (Figure 4), in which there has been much lower values (March and April) or higher values (July, September and October ) in relation to P normal .The major occurrence of inaccuracies was related to the atypical series of P average , when the precipitation was much higher or lower than the historical data.
There were larger differences in S CWB compared to S FWB when the mean P average was higher, and ETa CWB in relation to ETa FWB when P average was lower (Figure 5).The poorly adjustment of the S CWB occurred in July and September, when P average overcame P normal by 38 and 52%, respectively.However, there were also minor differences when the situation was the opposite.
Regarding the ETa CWB the greatest errors occurred when the mean P average was below P normal , especially in March and April.The ETa CWB had its highest values in the same periods when the largest precipitation occurred (Figures 4 and 5).This relation is similar to obtained by Silva et al. (2009) with corn in Piracicaba, Southestern Brazil.
There was no significant improvement, in both the S and ETa, with the daily variation of p, both grouping weeks as in months.The use of the proposed equations does not represent any gain in the CWB, in comparison with the use of a constant p value over time (Tables 1  and 2).
The use of Kc m represented the highest degree of accuracy and precision, and minor errors, both to S and ETa,, independent of the comparison scale.On the other hand, Kc k had the worst levels of accuracy and precision, and the biggest mistakes in all analyze and all tested variables.The use of the equation proposed to daily estimate Kc was inadequate and did not contribute to the CWB, on the contrary, because the equation showed worse results than even Kc A , which is constant over time.
The estimate of the CWB and its subsequent grouping in months for comparison provided greater degree of accuracy and precision for the variables analyzed, but caused the biggest mistakes.

DISCUSSION
There is definitely influence of precipitation on S and ETa.Zhang et al. (2004) report that the S directly influences ETa, in the extent that the soil water deficits reduces the ETa.On the other hand, Praveena et al. (2012) found that the excess water leads to increase in ETa.Farré and Faci (2006) found that the factors that most influence ETa are the S and P. The reason is due to a higher evaporation in the surface layers up to 0.40 m deep (Cruz et al., 2005).When ETa CWB was very low (March, April and May) there was small P, and low variations within the months came from deeper layers (0.60 and 0.80 m), which have a higher water retention capacity, contributing to the root water uptake (Souza et al., 2013).
According to Souza et al. (2013) when long periods without precipitation occurs, there is the process of soil water drying, with variation of the humidity, especially within the first 0.20 m deep.In this condition, a large evaporative demand by atmosphere cannot be attended by soil, because the amount of water available on the surface is restricted, and the water conductivity begins to influence evaporation.At this stage, the evaporation rate is controlled by the vapor transfer mechanisms and adsorption on the soil solid matrix.Many authors attested the influence of p in crop productivity (Doorenbos and Kassan, 1979;Tao et al., 2003;Steduto et al., 2009), however, there was no improvement in the adjustment of the component values of the CWB to the FWB, even varying p daily (p DK and p Ai ) in relation to the constant value (p A ) over time.It may be that P average allowed high S throughout the year.As a consequence, the soil was constantly in the wet zone (in other words, when S  TAW (1 -p)), and ETa and crop evapotranspiration (ETc) have been showed almost the same values under this condition.Bruno et al. (2007) using a Kc obtained by lisymeter and then grouped in four phenological phases, to estimate the CWB for coffee in Piracicaba, is crop coefficient recommended by Allen et al. (1998) (dimensionless); u2 is daily average wind speed at 2 m height (m s -1 ); RHmin is minimum daily average relative humidity (%); h is average plant height (m).*Corresponding author.E-mail: brunogurski@ufpr.br.Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License

Figure 1 .
Figure 1.Location of the study area in Southern Brazil.

Figure 3 .
Figure 3. Different soil water depletion fraction used in 2009 for pine, as follows: Constant over time (pA); value recommended by Doorenbos and Kassan (1979) (pDK); and depending on the daily crop evapotranspiration (pAi).

Figure 5 .
Figure 5. Soi water storage (S) and actual evapotranspiration (ETa) in the field water balance for pine, in Telêmaco Borba, in 2009.

Table 2 .
Comparison of actual evapotranspiration (ETa) obtained in a field (FWB) and climatological (CWB) water balances, grouped weekly and monthly, with different crop coefficients (Kc) and soil water depletion fraction (p) for pine in Telêmaco Borba, in 2009.