Methods for calculating phreatic evaporation on bare grounds on rainy and dry days

In order to depict the impact of rainfall on phreatic evaporation, this study analyzes phreatic evaporation and the phreatic evaporation coef ﬁ cient between surface evaporation and soil depth in Shajiang black soil and Fluyo-aquic soil. We have improved the existing commonly used mathematical framework, established two rainless day phreatic evaporation calculation models, and then calculated the calculation model of the phreatic evaporation reduction on rainy days. Finally, rainy day evaporation calculation models on two soils were proposed. The results show that the evaporation coef ﬁ cient is affected by both depth and the evaporation ability of the surface water. The evaporation reduction of Shajiang black soil increased with depth and the increasing trend gradually slowed down until it approached zero. The evaporation reduction of the Fluyo-aquic soil phreatic decreased ﬁ rst and then increased with depth, reaching a minimum at 0.4 m. The reduction of phreatic evaporation in both soils decreased with the increase in rainfall level and decreased with the increase in rainfall duration showing ‘ inverted S-type ’ . In summary, the phreatic evaporation composite calculation models on rainy days and rainless days have good ﬁ tting and prediction results, which can improve the accuracy of phreatic evaporation calculations.


INTRODUCTION
In summary, a lot of research has been done to improve evaporation models and analysis of phreatic water, but there are few studies on the effects of rainfall on phreatic evaporation and quantitative methods that distinguish between rainy and dry days. Based on phreatic evaporation data and meteorological observations from the Wudaogou Experimental Station from 1993 to 2015, this paper analyzes the influence of evaporation and depth of Shajiang black and Fluyo-aquic soil on phreatic evaporation and its coefficient in a bare ground scenario. Based on this law, the empirical formula has been improved to establish a model for phreatic evaporation on rain-free days in two soils and to estimate the effects of rainwater on phreatic evaporation. Comparing the differences between estimations, measurements, and model predictions, this study was able to quantifyin different soils (black and yellow) and under different conditions (rainy and dry)the effect of rainwater and groundwater depth on evapotranspiration. The calculation model of the rainfall duration. Based on the integrated rain-free day phreatic evaporation calculation model and the rainy day phreatic evaporation reduction calculation model, the rainy day phreatic evaporation calculation model of Shajiang black soil and yellow Fluyo-aquic soil was established and verified to provide technical support for exploring the impact of rainfall on phreatic evaporation. gather data on surface temperature, average temperature, water surface evaporation, incident radiation (sunshine), relative humidity, wind speed, saturation difference, absolute humidity, water vapor pressure difference, and rainfall.

Overview of the experimental area
The groundwater level is shallow with fluctuations between 1.5 and 3.5 m.  Table 1.

Mathematical modeling
The model construction in this paper is completed according to the basic steps of mathematical modeling.
Model preparation: Understand the actual background of diving evaporation, clarify the practical significance of the calculation model of diving evaporation on rainy and non-rainy days, and collect various necessary information.
Model establishment: Graphs are used to describe the relationship between variables, and MATLAB is used to quantify their specific mathematical relationships to establish corresponding mathematical structures.
Model solution: Use the obtained data to calculate (estimate) all the parameters of the model.  Phreatic evaporation curve fitting In scientific and engineering problems, several discrete data can usually be obtained by methods such as sampling or experiments. Based on these data, a continuous function (curve) or a more dense discrete equation is obtained that matches the known data. This process is called curve fitting.
In this paper, the following steps are used to establish a fitting curve for phreatic evaporation.
Step rainfall and rainfall duration on rainy days data.
Step 2: Observe and analyze the distribution of the data points, the specific relationship between the independent variable and the dependent variable and determine the function of the fitted curve.
Step 3: According to the principle of generalized least squares, use known data sets combined with computer software to estimate the parameter values in the curve fitting function.
Step 4: Bring the parameter estimates to the fitting function and establish the calculation model of the final rainy day and no rainy day phreatic evaporation.

Least-square estimation
When the model is determined based on the relationship between the variables, the method of determining the parameters in the model according to the principle of minimum squared deviation is called least-squares estimation.
Given a set of data (X, Y) and know the fitting function φ(X), which contains parameters a k (k ¼ 1, . . . , s) where n is the number of observation variables and m is the number of observation samples. To estimate the parameter values in the model, the sum of the square of deviations between Y and φ(X) should be minimized, According to the necessary conditions of the extremum of the multivariate: Solve equations to get parameter estimates and the parameter estimates are substituted into the model to establish the specific functional formula of the model.

Model evaluation index
This paper uses mean absolute error (MAE), square root error (RMSE), and coefficient of determination (R 2 ). These statistical indicators evaluate the accuracy of the model; the formulas for each statistical indicator are as follows: whereŷ i is the calculated value of the phreatic evaporation, y i is the measured value of the phreatic evaporation, and y is the mean of the measured value of the phreatic evaporation.
m is the number of samples, i ¼ 1, Á Á Á , m is the sample number. The more the R 2 metric independent variable interprets the dependent variable, the closer the value is to 1. The higher the degree of interpretation of the dependent variable is, the higher the accuracy of the model. The value of MAE, RMSE is more, the smaller the accuracy of the small model.

Relationship between phreatic evaporation and surface evaporation
The evaporation of phreatic water is affected by the atmospheric evaporation capacity, including meteorological elements, such as surface temperature, average temperature, water surface evaporation, incident radiation (sunshine), relative humidity, wind speed, and rainfall. Relevant research (Yu et al. ) shows that evaporation can be used instead of atmospheric evaporation ability to study the evaporation of phreatic water. In order to avoid the impact of crops on phreatic evaporation, bare land conditions were tested to establish the relationship between phreatic evaporation and surface evaporation. Figure 2 shows the relationship between the average phreatic evaporation and the average surface evaporation of Shajiang black soil and yellow tidal soil under different groundwater depth (H) conditions from 1993 to 2015.
It can be seen from Figure 2 that for both soils under the same conditions (i.e. groundwater depth), the evaporation of the phreatic water and the evaporation of the surface water have a power function relationship, functional relationship is significant, that is, the evaporation of the phreatic water increases with surface water evaporation. When increased to a certain extent, the evaporation of the diving is, generally, constrained to a certain limit. The reason for this phenomenon is that the evaporation of the phreatic water is affected by both the atmospheric evaporation capacity and the soil water transport capacity. When the atmospheric evaporation capacity is less than the soil water capacity, phreatic evaporation is mainly limited by meteorological elements and the surface water evaporation increases. When the atmospheric evaporation capacity is greater than the soil water transport capacity, phreatic evaporation is mainly limited by the soil water transport capacity, slowly increasing and trending around the maximum limit (Yu et al. ). Therefore, when the surface evaporation level is high, the correlation between phreatic evaporation and surface evaporation weakens, which is manifested by the large degree of dispersion of observation point near the fitted curve, and the fitting is worse.
It can also be seen from Figure 2 that the influence of surface water evaporation on yellow Fluyo-aquic soil is greater than that of Shajiang black soil, which is determined by the texture of the two soil types. Texture is the main factor affecting soil water conductivity and soil water movement parameters. The surface evaporation coefficient decreased with the increase of soil physical clay content; the clay content of Shajiang black soil accounted for 13.1%, powder (49.4%), sand (37.5%). In yellow Fluyoaquic soil, clay content accounted for 2.0%, powder (11.5%), and sand (37.5%). The sandy black soil is more viscous than the yellow Fluyo-aquic soil which is less porous and not conducive to water migrationi.e. it has poor water permeability (Wang et al. 2014).   Relationship between evaporation coefficient, surface evaporation, and groundwater depth

Relationship between evaporation and groundwater depth
The phreatic evaporation coefficient is the ratio of phreatic evaporation to surface evaporation in the same period.
The depth of the groundwater determines water transport distance when the phreatic water evaporates. As depth increases, the water transport distance increases, the influ-

Quantitative model of phreatic evaporation under rainless conditions
The existing empirical formulas for phreatic evaporation are as follows: The Averyanov's equation (Parabolic Formula) (Aweliyongrufe ): The Zhang Chaoxing's equation (Jin & Zhang ): The Shen Li-chang's equation (Shen ): The Tsinghua equation (Lei et al. ): According to the relationship between the phreatic evaporation coefficient and water surface evaporation discussed above, it can be seen that phreatic evaporation coefficient is not only related to buried depth but also affected by various underlying surfaces. Accordingly, this paper improves the Ye-Shuiting's exponential formula and establishes a new calculation model for phreatic evaporation under rainless conditions, as shown below: where Eg denotes phreatic evaporation intensity (mm/d);

Calculation model of daily phreatic evaporation under rainy conditions
The phreatic water evaporation is mainly influenced by the atmospheric evaporation intensity, the buried depth of phreatic groundwater and soil texture. Water surface evaporation is the concentrated embodiment of atmospheric evaporation which is affected by some meteorological conditions including solar radiation, temperature, humidity, wind speed, air pressure, etc.
In rainy days, when there is no rainfall, the solar radiation and air temperature decrease and the humidity increases, which results in the decrease in water surface evaporation (i.e. atmospheric evaporation intensity). While during the rainfall period, not only the atmospheric evaporation intensity is reduced, but also the capillary action is From Figure 5, it can be seen that the reduction of phreatic evaporation in lime concretion black soil increases with depth and the trend slows down until it approaches zero.
We can also find that the reduction of phreatic evaporation is basically zero when the depth is 0.8 m. Figure 6 shows the reduction of phreatic evaporation in lime concretion black soil declines when rainfall increases.
When the phreatic water table is shallow, the reduction of phreatic evaporation decreases linearly with increasing rainfall levels. The shallower the buried depth is, the more obvious the decreasing trend is. When the buried depth reaches a certain deeper depth, the reduction of phreatic evaporation is not affected by rainfall level any more. The explanations for these phenomena above are as follows; it is because when the buried depth of the phreatic water table is shallow, rainfall can directly inhibit phreatic evaporation. With increasing depths, water in the phreatic zone still migrates to the upper soil pores in unsaturated zones     by capillary action before the rainfall infiltrates into that layer. There is little difference between the phreatic evaporation under rainfall condition or not, that is, rainfall has less inhibition on phreatic evaporation.
As is shown in Figure 7, the decrease in phreatic evaporation in lime concretion black soil decreases progressively with the increasing rainfall duration in an inverted S-shaped manner. This is because the longer the rainfall lasts, the greater the amount of rainfall infiltrates into the vadose zone and the greater the inhibition effect on phreatic evaporation is. According to the above analysis, the quantitative models (as shown below) establish the amount by which phreatic evaporation reduction varies with the buried depth of phreatic groundwater, rainfall, and rainfall duration in lime concretion black soil and fluvo-aquic soil respectively.
For lime concretion black soil: For fluvo-aquic soil: f(H, P, t) ¼ 1 where f(H, P, t) represents the decrease in phreatic evaporation caused by precipitation, H is the depth of water table or depth of phreatic surface; P represents the rainfall, t represents the rainfall duration a, b, c, d, m, l, n, λ are empirical constants.

Model solution
According to the daily data on phreatic evaporation under rainless conditions and water surface evaporation in lime concretion black soil and Fluyo-aquic soil from 1993 to 2015, the specific expression of phreatic evaporation under rainless conditions for two typical soils each is as follows.

Model fitting index
The fitting results of quantitative model of phreatic evaporation under rainless and rainy conditions are shown in Table 2.
According to the results, the methods discussed above Model of phreatic evaporation under rainless condition in Fluyo-aquic soil: Governing equations of phreatic evaporation under rainless condition in Fluyo-aquic soil:

CONCLUSIONS
This paper analyzed the law and mechanisms of phreatic evaporation and the trends involving the depth of the phreatic water table and surface water evaporation. In order to reveal the laws, mechanisms, and equations governing phreatic evaporation, we probed questions on how the depth of the phreatic surface and surface water evaporation influenced the magnitude of phreatic evaporation and its coefficient.
At the same buried depth, phreatic evaporation increased with an increase in surface water evaporation.
When the surface water evaporation increased to a certain extent, phreatic evaporation tended to a certain limit value accordingly. Under the same surface water evaporation conditions, phreatic evaporation decreased gradually with increasing depth. The greater the water surface evaporation was, the more intense the trend of phreatic evaporation decreases with depth. The coefficient of phreatic evaporation had no relationship with depth. The phreatic evaporation coefficient varied with the depth of the phreatic water table and was influenced by different surface water evaporation levels.  The existing empirical formula of phreatic evaporation was improved and the exponential calculation models of phreatic evaporation under rainless condition for lime concretion black soil and Fluyo-aquic soil were established.
Based on the model, we could calculate the estimated value of phreatic evaporation under rainfall conditions and then calculate the difference between the estimated value and the measured value. After that, the reduction of phreatic evaporation under rainfall conditions relative to that in rainless days was obtained.
The relationship between the decrease in phreatic evaporation and the buried depth of phreatic groundwater, rainfall, and rainfall duration was identified. Based on the measured data collected by the lysimeter, models of phreatic evaporation under rainfall and rainless conditions were proposed in this paper.
These improved functions can be used to calculate phreatic evaporation with a higher degree of accuracy under the conditions accounted for in this paper. It is of great significance to further study and refine the models of phreatic evaporationespecially, as it relates to agriculture in farmlands and crop fields in both rainy and rainless periods.