Introduction

Evapotranspiration (ET) is an important element in the interaction between soil, vegetation and the atmosphere, is a central factor in the quantitative evaluation of water balance and surface energy, since many processes and parameters of the environment are influenced by this phenomenon such as soil moisture content, vegetation productivity, nutrient absorption, and water balance, among others, particularly in arid and semi-arid areas where water availability becomes increasingly critical (^{Liu et al., 2013}).

Most methods used to measure or estimate evapotranspiration are space-limited; however, from a hydrological and water resource management point of view, large-scale estimates are required, which can be obtained using techniques such as scintilometry and remote sensing, ET validation estimated from remote sensing techniques has become a central research topic in different parts of the world, mainly due to the complexity of the terrestrial surface, caused by the heterogeneity of the vegetation cover and the variability of the surface topography (^{Kleissl et al., 2009}; ^{Gao et al., 2011}; ^{Samain et al., 2012}); and for this, the scintilometry technique is considered the only one capable of performing measurements of sensible heat fluxes (H) in dimensions or areas comparable to the size of a pixel or several pixels of a satellite image (^{Kleissl et al. , 2008}).

The objective of this paper is to evaluate the triangular space method (Ts-Fr) using satellite data from the MODIS sensor, and the energy balance equation; and the use of a scintillometer (BLS) to measure sensible heat flux (H) and a vortex covariance system (EC) to measure the latent heat flux (LE) on a homogeneous surface of Bell pepper in the Culiacán Valley.

Materials and methods

This work was carried out in the Culiacán Valley in the central area of the state of Sinaloa, México, in an area of 90 ha planted with Bell pepper (Capsicum annumm, L), the central geographical coordinates of the lot are 24.59569 north latitude and 107.51875 west longitude (Figure 1). The considered period was 81 days, which includes from February 2^nd to April 23^th, 2014, a detailed description of the study site can be consulted in ^{López et al. (2015)}.

Figure 1 Geographical location of the studied area.  

Estimation of latent heat flux (LE) using the triangular space method with MODIS data

For the triangular space method with satellite information, the surface temperature data (Ts) of the product MOD11A2 of the MODIS sensor installed on the TERRA satellite were obtained with a spatial resolution of 1 km and a temporal resolution of 8 days, while the data of normalized difference vegetation index (NDVI) were obtained from the product MOD13Q1 of the same sensor and with spatial resolution of 250 m and temporal resolution of 16 days. The MODIS sensor satellite data was obtained from ^{ORNL DACC (2008a} y ^2008b) on the following platform: http://daacmodis.ornl.gov/cgi-bin/modis/glbviz-1-glb/modis-subset-order-global-col5.pl. It is important to mention that digital image processing was not applied to obtain the NDVI and surface temperature values, since the satellite data was already processed from the mentioned web platforms.

The equation (equation 3) proposed by ^{Jiang and Islam (1999)} and adapted from Priesley-Taylor (1977) was used to calculate the latent heat flux (LE) value.

LE=∅ΔΔ+γRn-G (3)

Where: LE= latent heat flux (W m^-2); Rn= net solar radiation (W m^-2); G= heat flux in the soil (W m^-2); Δ= slope of the water vapor saturation pressure curve (kPa ^oC^-1); γ= psychometric constant (kPa ^oC^-1); the value of φ is calculated with temperature information of the earthʼs surface and values obtained from the determined geometric space by equation 4.

∅=Tsmax,i-TsiTsmax,i-Tsmin∅max,i-∅min,i+∅min,i (4)

Where: Ts_i= surface temperature (°K) and is obtained from the product MOD11A2 of the MODIS sensor; Ts_min= minimum value of the surface temperature; φmax= worth 1.26.

In order to generate the triangular space, the surface temperature (Ts) and an index (equation 5) known as the vegetation cover fraction (Fr) were used, which is calculated as a function of the vegetation index (NDVI), which seems to be more representative of the relative proportionality between the soil and the vegetation within the pixel (^{Tang et al., 2010}).

Fr=NDVI-NDVIminNDVImax-NDVImin2 (5)

In order to define the limits (edges) of the triangular space (Ts-Fr), first the total range of values of Fr was divided into intervals and the maximum surface temperature value corresponding to each interval was obtained, then a linear regression was applied between the maximum values of Ts and each interval of Fr, obtaining a linear equation (equation 6).

Tsmax,i=a+bFr (6)

Where T_Smax= a and T_Smin= b + T_Smax

Equation 6 defined the “dry” and “wet” edges of the triangular space, which is formed with the surface temperature (Ts) and the vegetation cover fraction (Figure 2), so that the ends values match Ts_{max, i}= Ts_max, Fr= 0 and Ts_{min, i}= Ts_min, Fr= 1.

Figure 2 Schematic diagram of the triangular space concept.  

To adjust the spatial resolution of the Ts images (MOD11A2) a 9*9 pixel mesh was generated, with a spatial resolution of 250*250 m covering the entire study area. Since the spatial resolution of Ts data is 1000*1000 m, each original pixel was divided into 4 pixels of 250*250 m, and to assign them the value of Ts a linear interpolation was performed in all directions considering the values of the original contiguous pixels, in this way it was possible to match each pixel of the Ts mesh with the pixels of the NDVI mesh.

^{Girolimetto et al. (2011)} worked by dividing the pixels of high-resolution Ts (10*10 km) into pixels with lower resolution (2*2 km) and comparing the results obtained with MODIS images of 1*1 km resolution, obtaining reasonably accurate results of the evaporable fraction (EF), the authors agree that the geometric space generated (NDVI-Ts) summarizes the energy balance of the region under study, and that the surface temperature (Ts) is the most dynamic variable and therefore the most limiting one. The final mesh contained 81 pixels and each pixel corresponded to a value of Ts and NDVI.

The satellite data for 11 Julian days were analyzed, where the Ts and NDVI data coincided were: 33, 41, 49, 57, 65, 73, 81, 89, 97, 105 and 113. To maintain the robustness of the results, it was taken care that the triangular space was generated with at least 80% of the set data.

Once the triangular space was defined, the pixels outside the triangular space were removed. And only with the pixels within the triangular space, the evaporable fraction value (EF) was calculated, (equation 7) for each of them, and finally the average value of all was considered. It was assumed that the evaporable fraction (EF) remains constant during the day, so that the ET is obtained by the FE product by the energy available during the considered period of the day (^{Jiang et al., 2009}).

FE=∅ΔΔ+γ (7)

In order to determine the EF of the 81 days included within the study period, a linear interpolation was performed with the existing EF data, thus every day had their Fe value and available energy (Rn-G) measured, with which was used to calculate the ET, the influence of the cloudiness on the study area would basically be reflected in the net solar radiation that was used to calculate ET. Subsequently, the behavior of ET was analyzed statistically with respect to the results obtained with the other two methods.

Results and discussion

The ET measured with the vortex covariance (EC) system, as well as the estimated using scintilometry (BLS) and remote sensing triangular space method using data (MODIS) is shown in Table 1. The total ET estimated for the evaluated period was 275.2, 255.4 and 262.0 mm for BLS, MODIS and CE; as can be observed, the mean ET for MODIS and CE is the same as the minimum ET, while for BLS the mean ET is slightly higher and the minimum ET slightly lower than those determined with the other two methods, the maximum ET is very variable in the three models used, which is due to the nature of the variables that are measured in each method (CE has a punctual character, BLS uses the average of the variation of a variable in a length, whereas MODIS uses the average of the variation of two variables in an area determined by the pixels resolution of the satellite image).

Table 1 Evapotranspiration (ET) reference values estimated and measured during the study period. 

It is important to note that the maximum ET value estimated with BLS was 4.5 mm d^-1 and occurred on March 31, coinciding with the maximum ET date estimated by MODIS, whose value was 3.9 mm d^-1, ET value measured with EC for that same day was 4.3 mm. While the maximum ET measured with EC occurred on March 23 (5.3 mm d^-1), while the values for MODIS and BLS for that same day were 3.1 and 3.4 mm d^-1, respectively. In a review of the meteorological information looking for the possible cause of the EC measurement, no cause was found to justify ET elevation during that day, considering the value of sensible heat flux (H) measured with EC and applying equation of energy balance (LE= Rn - G - H), the estimated ET value decreases considerably, suggesting that there is an error in latent heat measurement (LE) with the vortex covariance system due to lack of balance closure with this technique.

The negative values of ER in Table 2 indicate a MODIS underestimation to estimate ET with respect to the other two methods, while comparing the estimated results with BLS and those measured with CE resulted in an overestimation of the data obtained with BLS. At this point, it is important to note that the BLS and MODIS measurement scale has a regional amplitude, while in EC it is on a space-determined scale.

Table 2 Relative error (ER) and standard error (ES) among the used methods. 

^{Gordillo et al. (2014)} when comparing ET results at daily level with 12 satellite images with the METRIC method and vortex covariance (EC) indicate an ET overestimation determined with METRIC with respect to EC, where they obtained ER= 7.273% and ES= 0.208 mm d^-1, while in the present investigation, ER= -7.203% and ES= 0.233 mm d^-1were obtained, considering that in this research ET is considered daily and for a greater number of days.

On the other hand, ^{Liu et al. (2013)} when comparing ET results obtained in annual periods and for several years, using the scintilometer method and the product MOD16 ET, found a different behavior, with ER values between -14.52 and 25.16%, while R varied from 0.76 to 0.97.

While ^{Tang et al. (2010)}, when comparing the sensible heat flux (H) estimated with scintilometry and covariance techniques of vortices, show that the scintillometer underestimated the value of H measured with EC, implying that when applying the energy balance equation , the latent heat flux (LE) value was higher and, therefore, the scintillometer overestimated the ET calculation with respect to the value measured with EC, similar to what happened in this paper; also, applying the triangular space method with MODIS data to estimate the latent heat flux (H) indicates an overestimation of H and underestimation of LE with respect to the scintilometer, therefore a lower ET estimated with the triangular space method with MODIS data, which ir a similar behavior to the one on this research.

The estimation of ET with any of the methods depends, among other factors, on the variables and main assumptions that the method, the meteorological and soil conditions of the study site consider; the type, phenology and general heterogeneity of the crop or vegetation, as well as the spatial and temporal scale considered for the calculation or measurement (^{Tang et al., 2010}; ^{Muiu et al., 2011}; ^{Samain et al., 2012}; ^{Liu et al., 2013}).

Figure 3 shows the daily behavior of evapotranspiration with the three analyzed methods, where three values of “peak” ET measured with the CE system are clearly observed, the observer values were 4.4, 5.3 and 4.5 mm for days 171, 195 and 211 after transplantation (DDT).

Figure 3 Daily behavior of ET during the study period, estimated and observed with the techniques used (MODIS, BLS, CE).  

The largest discrepancies between ET estimated with MODIS and BLS were observed during 20 days of the total of the considered period, which represented 24.6% of the evaluated days. During these days, a higher average relative humidity was present in the environment, probably generating an overestimation of sensible heat flux (H) and consequently an ET reduction estimated with BLS, ^{Tang et al. (2010)}, when comparing the results of H obtained with MODIS and scintilometer, posed a possible overestimation of sensible heat flux due to the presence of heavy rains in the study area, which probably increased the environmental relative humidity, reducing ET.

Figures 4, 5 and 6 show the results of the statistical analysis of the estimated and observed data with the three methods, the root mean square error (RMSE), bias (BIAS), Willmott’s concordance coefficient (d) and the Pearson coefficient of determination (R). A value of RMSE and BIAS, equal to zero indicate a perfect fit of the estimated data with respect to the observed data, are considered acceptable when RMSE does not exceed 20% of the data set mean. The coefficient of concordance (d) ranges from 0 to 1, the closer to 1 the d value is found, the more accurate the estimation model will be. Finally, the Pearson (R) correlation coefficient has a variation between -1 and 1, in this investigation it is sought that the estimation model replicate the measured data, therefore the closer to 1 the R value, there will be a greater correlation between data, although the use of this parameter to evaluate forecast or estimation models shows certain limitations and is thus little used (^{Willmott, 1981}).

Figure 4 Statistical relationship Values between ET estimated with MODIS and BLS.  

Figure 5 Statistical relationship values between ET estimated with MODIS and observed with EC.  

Figure 6 Statistical relationship values between ET observed with EC and estimated with BLS.  

Conclusions

In this paper were compared the ET values obtained with three methods based on the energy balance applied to the land surface, two methods of a regional character and one of a space-limited nature.

With this research it is possible to conclude that the triangular space method using MODIS data, is an adequate tool to estimate the evapotranspiration, it was obtained as a residual value of the energy balance applied in the terrestrial surface; in order to estimate the ET values on a daily basis, data from NDVI and Surface temperature (Ts) obtained from MODIS were used, comparing ET values obtained with this technique, with those obtained with scintilometer (BLS) and vortex covariance (EC), showed a relative error (ER) of -7.2% and -2.5%, and a standard error of 0.233 mm d^-1 and 0.32 mm d^-1, respectively. The best statistical relation was between ET estimated with MODIS and BLS, where Willmott՚s coefficient of agreement (d) was 0.8, RMSE was 0.44 mm d^-1, BIAS -0.245 mm d^-1 and R of 0.75.

The values obtained from the statistical parameters indicate that there is no significant difference when comparing ET estimated with the three methods, probably due to the size of the surface and the conditions of the crop that did not allow sufficient heterogeneity to show differences between the methods of regional and limited nature. Research on larger surfaces and diverse vegetation is necessary to increase heterogeneity.

It is concluded that the triangular space method with MODIS data is a good alternative to estimate ET in homogeneous crop surfaces, because it was posible to obtain a good fit with the data estimated with scintilometer and vortex covariance methods.