Diurnal Evolution and Estimates of Hourly Diffuse Radiation Based on Horizontal Global Radiation, in Cerrado-Amazon Transition, Brazil

: In the Cerrado-Amazonian ecotone in the State of Mato Grosso, intensely altered by anthropic action, the knowledge and processes of energy conversion and energy balance are still incipient, making the monitoring and modeling of diffuse radiation essential for several applications. The objective of this study was to evaluate the seasonality of the diurnal evolution and estimate the hourly diffuse radiation ( H hd (cid:17) and incident radiation in the horizontal plane between June 2011 and October 2016. The instantaneous measurements (5 min) of diffuse radiation underwent geometric, astronomical, and anisotropic corrections, with subsequent hourly integrations. The seasonality of diffuse radiation and its radiometric fractions was evaluated. The estimates were made considering total and seasonal data groupings (water stations in the region) and in different cloudiness classes (atmospheric transmissivity index— K hT ). The diurnal behavior of diffuse radiation ( H hd ) was similar to that of global radiation and at the top of the atmosphere, with maximum values at solar noon. The correlations between K hd and K hT showed third-order polynomial behavior, with maximum observed values of K hd ranging from 0.8 to 0.9, for K hT less than 0.2. Estimation equations based on radiometric fractions underestimated the values of diffuse radiation, with a better performance presented by models adjusted in annual data groupings. Among the parameterized models for estimating diffuse radiation obtained in the literature, those calibrated regionally in this study, together with those developed for tropical regions, presented better statistical performances.


Introduction
Brazil is in full technological development in the areas of renewable energy (photothermal and photovoltaic conversion, biomass and biodiesel), agriculture (increased efficiency based on the physical and physiological properties of crops and animals), and civil construction (construction materials, micrometeorological aspects), among others. This Brazilian scenario reflects what happens worldwide, with continuous increases in energy supply and demand, especially from renewable sources, due to limited fossil fuel resources and the problems associated with greenhouse gases [1]. In this context, solar energy plays an important role in the global and national sustainable infrastructure [2], being considered a clean energy source that does not use fuels with fossil origins [3].
The need to supply energy on an ever-increasing scale, and at the same time economically and sustainably, together with the profile of the high national agrosilvopastoral potential, make up a growing demand for the knowledge of seasonal variations in solar radiation levels, considering the spectral and atmospheric attenuation components in the Brazilian territory. The temporal variation of the amount of solar radiation incident at any location on the Earth's surface depends on astronomical, geographic, and atmospheric factors. The main variations in the seasonal levels of each component of solar radiation originate from the interaction with the atmosphere since some atmospheric constituents are relatively constant in concentration (permanent gases), and others are highly variable in time and space (such as CO 2 , methane, aerosols, and water vapor). This variability allows the current composition and concentration of gases in the atmosphere to depend on the geographic position, altitude, and time of year, influencing solar radiation's absorption, reflection, and transmission. In turn, among the local characteristics, variations in altitude, inclination (declivity), orientation (azimuth), and shading can affect the energy levels available on the surface, mainly with changes in the geometry of incidence of direct solar fluxes [4,5].
The solar radiation that reaches the earth's surface, called global radiation, can be divided into two components, direct and diffuse, which are transmitted directly through the atmosphere (without attenuation) and result from the scattering action in the atmosphere, respectively [6,7]. Measurements of global radiation fluxes are normally obtained in the horizontal plane and are available in instantaneous, hourly, daily, monthly, and annual partitions.
Diffusion is a fundamental physics process associated with electromagnetic waves and their interaction with matter, in which particles in the path of an electromagnetic wave radiate their energy in all directions, occurring for all wavelengths within the electromagnetic spectrum. The size of the particles interferes directly in the dispersion process, given by the proportion between the circumference of the particle and the incident wavelength (λ), that is, x = 2πa/λ (where 'a' is the radius of the particle); in this case, if x < 1 (when the particle diameter is less than 10% of the size of the incident wavelength), the dispersion is called selective or isotropic (Rayleigh scattering). For particles whose sizes are similar to or larger than the wavelength, i.e., x ≥ 1, the diffusion is known as non-selective or anisotropic (Mie scattering) [4,5].
In Brazil, the knowledge of diffuse radiation levels is restricted to locations close to research institutions and universities since the country does not have a solarimetric monitoring network. Recently, the main Brazilian studies focused specifically on the characterization and estimation of diffuse radiation were developed in the Southeast regions [8][9][10][11][12][13][14][15], Northeast [16][17][18], and South [19][20][21]. Through a partnership between the Federal University of Mato Grosso and the Laboratory of Solar Radiometry of the Faculty of Agricultural Sciences of UNESP, between 2011 and 2016, global and diffuse radiation monitoring was carried out at the Federal University of Mato Grosso, University Campus of Sinop, standing out as the only site with these observations in the northern region of Mato Grosso [22,23]. This article presents the quantification of energy levels of diffuse radiation in this locality, in hourly temporal partitions, and in different seasonal periods, in addition to contributing to the understanding of the behavior of local atmospheric phenomena, serves as an environmental indicator since it reflects the air load of aerosols in suspension because of anthropic activities such as forest fires.
In several parts of the world, as well as in Brazil, the global radiation is measured in horizontal planes in many locations, however, the beam and diffuse radiation data can be obtained in restricted periods and are very scarce. The diffuse solar radiation levels generally follow the seasonal radiation behavior at the top of the atmosphere and vary throughout the year according to local atmospheric conditions (precipitation, cloudiness, aerosols, among other factors) [24]. The greatest difficulties in obtaining diffuse radiation are linked to the variation in cloudiness, for conditions ranging from clear to cloudy skies; under clear sky conditions they can be calculated theoretically using various climatic and geographic (local) parameters. However, there is not good method for computing the diffuse radiation under a cloudy sky. Regarding this topic, many researchers have recently carried out several studies with empirical formulas based on different parameters for different regions of the world [3,[25][26][27], new estimation methodologies such as machine learning models and hybrids models [28][29][30][31][32], and remote sensing applications [33,34].
This work aimed to analyze the seasonality and propose statistical models for estimating the hourly diffuse radiation incident on the horizontal in the Cerrado-Amazon transition, Mato Grosso state, Brazil. For this purpose, the annual and seasonal diurnal evolution of global and diffuse radiation and their radiometric fractions were characterized; in addition, statistical models for estimating hourly diffuse radiation were calibrated based on the atmospheric transmissivity coefficient, and comparisons were made with parameterized models that allow the estimation of hourly diffuse radiation.

Characterization of the Study Region
Global and diffuse radiation data, as well as other meteorological variables used in this study, were obtained from an Automatic Meteorological Station (AMS) in Sinop, Mato Grosso, located at latitude 11.864 • S and longitude 55.485 • W (altitude 371 m) ( Figure 1). The evaluated database provides measures between June 2011 and October 2016, which, despite being short and not updated, is representative of the atmospheric behavior of the region, as it is a 5-year time series.
Atmosphere 2023, 14, x FOR PEER REVIEW 3 of 20 geographic (local) parameters. However, there is not good method for computing the diffuse radiation under a cloudy sky. Regarding this topic, many researchers have recently carried out several studies with empirical formulas based on different parameters for different regions of the world [3,[25][26][27], new estimation methodologies such as machine learning models and hybrids models [28][29][30][31][32], and remote sensing applications [33,34]. This work aimed to analyze the seasonality and propose statistical models for estimating the hourly diffuse radiation incident on the horizontal in the Cerrado-Amazon transition, Mato Grosso state, Brazil. For this purpose, the annual and seasonal diurnal evolution of global and diffuse radiation and their radiometric fractions were characterized; in addition, statistical models for estimating hourly diffuse radiation were calibrated based on the atmospheric transmissivity coefficient, and comparisons were made with parameterized models that allow the estimation of hourly diffuse radiation.

Characterization of the Study Region
Global and diffuse radiation data, as well as other meteorological variables used in this study, were obtained from an Automatic Meteorological Station (AMS) in Sinop, Mato Grosso, located at latitude 11.864° S and longitude 55.485° W (altitude 371 m) (Figure 1). The evaluated database provides measures between June 2011 and October 2016, which, despite being short and not updated, is representative of the atmospheric behavior of the region, as it is a 5-year time series. The municipality of Sinop is located in the Mid-North Region of the State of Mato Grosso, spanning 3942.23 km 2 with approximately 130,000 inhabitants. With large population growth in the last seven years (around 20%), the use and occupation of the soil has been intensely altered, with the conversion of vegetated areas into urban and agricultural areas and an increase in the demand for energy and other natural resources [35]. The municipality of Sinop is located in the Mid-North Region of the State of Mato Grosso, spanning 3942.23 km 2 with approximately 130,000 inhabitants. With large population growth in the last seven years (around 20%), the use and occupation of the soil has been intensely altered, with the conversion of vegetated areas into urban and agricultural areas and an increase in the demand for energy and other natural resources [35].
According to the Köppen climate classification, the climate of the region is hot and humid Aw tropical, characterized by the presence of two well-defined seasons: rainy For the observational characterization of the seasonality of global and diffuse radiation, it was decided to group the months according to the rainfall regime of the region, avoiding the seasonal evaluation by the seasons. This type of seasonal analysis is favorable in regions with frequent atmospheric changes resulting from rainfall patterns, as recommended by [37] for the analysis of solar radiation estimates in the Amazon. Souza et al. [36] emphasized that the State of Mato Grosso is representative of great environmental complexity, conditioned, among other factors, to water availability. By observing the behavior of rainfall (Figure 2), the following groupings were adopted: (i): rainy season (December to February); (ii) dry season (June to August); (iii) rainy/dry transition (March to May); (iv) dry/rainy transition (September to October).

Instrumentation and Data Analysis
Instantaneous global and diffuse radiation data (5-min values) were monitored by Kipp and Zonen CM3 pyranometers positioned at a height of 1.0 m on a metallic platform ( Figure 3). The sensors had a response sensitivity of ±10-35 µV/Wm 2 , a response time of 18 s, a temperature response of ±1.0% for the range from −40 to 80 °C, and deviations for the cosine effect of ±2% (0 < z < 80°). Data were recorded by a CR1000 datalogger For the observational characterization of the seasonality of global and diffuse radiation, it was decided to group the months according to the rainfall regime of the region, avoiding the seasonal evaluation by the seasons. This type of seasonal analysis is favorable in regions with frequent atmospheric changes resulting from rainfall patterns, as recommended by [37] for the analysis of solar radiation estimates in the Amazon. Souza et al. [36] emphasized that the State of Mato Grosso is representative of great environmental complexity, conditioned, among other factors, to water availability. By observing the behavior of rainfall (Figure 2), the following groupings were adopted: (i): rainy season (December to February); (ii) dry season (June to August); (iii) rainy/dry transition (March to May); (iv) dry/rainy transition (September to October).

Instrumentation and Data Analysis
Instantaneous global and diffuse radiation data (5-min values) were monitored by Kipp and Zonen CM3 pyranometers positioned at a height of 1.0 m on a metallic platform ( Figure 3). The sensors had a response sensitivity of ±10-35 µV/Wm 2 , a response time of 18 s, a temperature response of ±1.0% for the range from −40 to 80 • C, and deviations for the cosine effect of ±2% (0 < z < 80 • ). Data were recorded by a CR1000 datalogger (Campbell Scientific, Logan, UT, USA), operating at 1 Hz of frequency. For the measurement of diffuse radiation, the pyranometer was positioned under the MEO shading ring [38], constantly remaining below the shadow projected by the ring. In contrast, for global (Campbell Scientific, Logan, UT, USA), operating at 1 Hz of frequency. For the measurement of diffuse radiation, the pyranometer was positioned under the MEO shading ring [38], constantly remaining below the shadow projected by the ring. In contrast, for global radiation the measurement, the sensor remained in full sun. The shading ring used was 0.1 m wide and 0.4 m in radius.
In addition to the pyranometers and MEO shading ring, the following sensors were used: psychrometer with thermometric shelter Vaisala CS 215 installed at 2 m height, Vaisala TE 525 rain gauge at 1.5 m height, heliograph at 1.5 m height, for the monitoring meteorological elements of air temperature and air relative humidity, rainfall, and insolation, respectively. The solar radiation data were submitted for analysis to observe inconsistencies generated by the collection and storage system. By integrating instantaneous partitions, global ( ) and diffuse ( ) hourly irradiation were obtained. The hourly extraterrestrial radiation ( ) was obtained according to Iqbal [4]. After the hourly integrations, the data were submitted to geometric and astronomical corrections proposed by Dal Pai [39] and Oliveira et al. [40] by applying the isotropic/geometric correction factor-FC (Equations (1) and (2)). For the location, FC ranges from 0.99 to 1.00. Anisotropic correction factors proposed by Dal Pai et al. [12,41] are dependent on atmospheric transmissivity ( ), following the sky cover classification described by Escobedo et al. [42] (Table 1).
where: Fp represents the portion of diffuse radiation intercepted by the shading ring; bring width (0.1 m); R-ring radius (0.4 m); δ-solar declination; w-hour angle; ϴZ-zenith angle. In addition to the pyranometers and MEO shading ring, the following sensors were used: psychrometer with thermometric shelter Vaisala CS 215 installed at 2 m height, Vaisala TE 525 rain gauge at 1.5 m height, heliograph at 1.5 m height, for the monitoring meteorological elements of air temperature and air relative humidity, rainfall, and insolation, respectively.
The solar radiation data were submitted for analysis to observe inconsistencies generated by the collection and storage system. By integrating instantaneous partitions, global (H h G ) and diffuse (H h d ) hourly irradiation were obtained. The hourly extraterrestrial radiation (H h 0 ) was obtained according to Iqbal [4]. After the hourly integrations, the H h d data were submitted to geometric and astronomical corrections proposed by Dal Pai [39] and Oliveira et al. [40] by applying the isotropic/geometric correction factor-FC (Equations (1) and (2)). For the location, FC ranges from 0.99 to 1.00. Anisotropic correction factors proposed by Dal Pai et al. [12,41] are dependent on atmospheric transmissivity (H h T ), following the sky cover classification described by Escobedo et al. [42] (Table 1).
where: Fp represents the portion of diffuse radiation intercepted by the shading ring; b-ring width (0.1 m); R-ring radius (0.4 m); δ-solar declination; w-hour angle; θ Z -zenith angle. After processing the data (n = 23,704 h of observation of solar radiation), a sample "n" of 22,506 h was obtained, that is, there was a loss of approximately 5% of the data.

Radiometric Fractions of Diffuse Radiation
The hourly atmospheric transmissivity coefficients (K h T ) and the radiometric fractions K h d and K h d were obtained (Equations (3), (4), and (5), respectively).
Subsequently, statistical equations were generated for the correlations between the hourly global radiation diffuse fraction (H h G ) and the atmospheric transmissivity coefficient (K h T ) for the annual and seasonal data groupings (four hydrological seasons). In this case, the data series was divided into two parts (in a 2:1 ratio) for calibration/generation and statistical performance analysis of the estimates (validation). In this case, an organization was adopted so that the months were contemplated proportionally in the two databases, resulting in 42 and 23 months for generation and validation, respectively, uniformly distributed over the years.
The correlations between K h d and K h T allowed adjustments of third-order polynomials, as recommended by Abreu et al. [43]. Polynomial regression equations were also generated for the K h T intervals 0 ≤ K h T < 0.55 (which includes cloudy and partially cloudy sky coverage) and K h T ≥ 0.55 (partly open and open sky) to improve the performance of the estimates. The K h d × K h T correlations were generated in total (annual) and seasonal grouping of data.

Estimates of Diffuse Radiation by Parameterized Models
The correlations proposed in this study were compared with 17 parameterized models for estimating hourly diffuse radiation based on the K h d × K h T correlation available in the literature for the most different regions of the globe ( Table 2). Some of the models are partitioned, with estimation equations for different K h T intervals, totaling 42 equations. The table presents the location for which the equations were developed, and the corresponding K h T intervals. Spencer's [44] equations were adjusted for local latitude.

Statistical Performance Evaluations of Estimation Models
In evaluating the performance of the estimation equations and models, the statistical indicators MBE (Mean Bias Error), RMSE (Root Mean Square Error), and Willmott's "d" and the coefficient of determination (R 2 ) (Equations (6)-(9), respectively), were indicated by Abreu et al. [43] as the most used. In evaluating the performance of the estimation equations and models, the statistical indicators MBE (Mean Bias Error), RMSE (Root Mean Square Error), and Willmott's "d" and the coefficient of determination (R 2 ) (Equations (6) to (9)) were indicated by Abreu et al. [43] as the most used.
The MBE and RMSE values represent, respectively, the mean deviation and the actual value of the error produced by the model. Negative MBE values indicate underestimations of the tested model, and vice versa. The smaller the absolute value of MBE, the better the performance of the tested model. The same applies to the RMSE. The concordance index "d" reflects the precision of the estimated values of the observed ones. It takes values between 0 to 1, and the closer to 1, the more perfect the agreement. The determination coefficient measures how well the model describes the observed data-the higher the value, the more adequate the proposed model is [56][57][58].  at 12 h are 4.57 ± 0.03, 2.25 ± 0.12, and 0.53 ± 0.21 MJ m −2 h −1 , respectively. In the hours of rising and setting sun, the average values observed for 0 ℎ are 0.35 ± 0.07 and 0.41 ± 0.06 MJ m −2 h −1 , and for ℎ they are 0.04 ± 0.03 and 0.08 ± 0.02 MJ m −2 h −1 ; being the global radiation incident at these times practically composed only by the diffuse portion ( ℎ = 0.03 ± 0.04 and 0.06 ± 0.00 at 6 a.m. and 6 p.m., respectively).  [13] in a study conducted in the city of Rio de Janeiro-RJ (−22.86°; −43.23°), justified by the fact that the reduction in precipitation in the dry period is related to the decrease in cloud cover, which raises the levels of ℎ .

Radiations and Fractions Radiometrics
Due to the high cloudiness during the rainy season, the lowest averages of ℎ and the highest averages of ℎ (at solar noon) were also observed in this period, 2.02 ± 0.15 and 0.66 ± 0.17 MJ m −2 h −1 , respectively. The minimum averages of ℎ at solar noon occur in the dry period 0.35 ± 0.04 MJ m −2 h −1 . The lowest standard deviation values for global and diffuse radiation are observed during the dry season due to the greater stability of atmospheric conditions (Table 3).   (Table 3). Figure 5 shows the annual diurnal evolution of the radiometric fractions K h T , K h d , and K h d during the four hydrological seasons. The maximum value of atmospheric transmissivity of global radiation occurs at solar noon, with incident radiation levels on the Earth's surface corresponding to approximately 50% of H h 0 . The minima were observed at sunrise for a better analysis of atmospheric transmissivity throughout the year.
K h d has the opposite behavior to K h T , with higher values at the beginning and end of the day. At these times, global radiation is predominantly composed of diffuse radiation, corresponding to about 70% of energy levels (67.66% and 75.22%, respectively). The hourly average values of K h d range from 0.15 to 0.78, with a higher hourly average in the rainy season (0.53 ± 0.13) due to the higher concentration of water vapor in the atmosphere. The lower values of standard deviations for K h d in the dry season are justified by the greater stability in atmospheric conditions. Abbreviation: SD = standard deviation. Figure 5 shows the annual diurnal evolution of the radiometric fractions ℎ , ℎ , and ℎ ′ during the four hydrological seasons. The maximum value of atmospheric transmissivity of global radiation occurs at solar noon, with incident radiation levels on the Earth's surface corresponding to approximately 50% of 0 ℎ . The minima were observed at sunrise for a better analysis of atmospheric transmissivity throughout the year. ℎ has the opposite behavior to ℎ , with higher values at the beginning and end of the day. At these times, global radiation is predominantly composed of diffuse radiation, corresponding to about 70% of energy levels (67.66% and 75.22%, respectively). The hourly average values of ℎ range from 0.15 to 0.78, with a higher hourly average in the rainy season (0.53 ± 0.13) due to the higher concentration of water vapor in the atmosphere. The lower values of standard deviations for ℎ in the dry season are justified by the greater stability in atmospheric conditions.
The atmospheric transmissivity coefficient ℎ ranges from 0.07 to 0.64, with a higher average value in the dry season (0.53 ± 0.14) due to low cloud cover, which allows for greater passage of direct radiation. The lowest mean value, consequently, is observed in the rainy season, 0.33 ± 0.11, while in the transition seasons ℎ ranges from 0.36 to 0.42.
Through the analysis of the average values of ℎ , and according to the sky cover classification established by Escobedo et al. [42], it can be stated that the behavior of the sky in Sinop, in the dry season, varies from cloudy to partially cloudy at the beginning of The atmospheric transmissivity coefficient K h T ranges from 0.07 to 0.64, with a higher average value in the dry season (0.53 ± 0.14) due to low cloud cover, which allows for greater passage of direct radiation. The lowest mean value, consequently, is observed in the rainy season, 0.33 ± 0.11, while in the transition seasons K h T ranges from 0.36 to 0.42. Through the analysis of the average values of K h T , and according to the sky cover classification established by Escobedo et al. [42], it can be stated that the behavior of the sky in Sinop, in the dry season, varies from cloudy to partially cloudy at the beginning of the day, and during the hours it is partially open. In the rainy season, the sky remains cloudy or partially cloudy throughout the day.
In all hydrological stations, the sky is cloudy in the early morning, with a reduction in the atmospheric transmissivity coefficient at the end of the day, except in the dry season, when K h T values remain high. Marques Filho et al. [13] stated that the high values of K h T at the end of the day are due to surface reflections due to the low values of the solar elevation angle at this time of year. Table 4 describes the hourly average values of the components and fractions of solar radiation at 12 noon in the different hydrological stations. Zamadei et al. [59] observed the diurnal evolution of K h T in the municipality of Juína-MT, 360 km from Sinop, MT, between 10/2007 and 01/2013, and found that the highest values of K h T occurred when the Sun had an angle of elevation greater than 45 • in relation to the surface, being higher in winter (dry period). As observed in this study, the greatest deviations occurred in the afternoon, indicating an increase in the water vapor content in the atmosphere due to the evapotranspiration process that occurred in the region throughout the day. According to these authors, the highest frequencies of clear skies occurred in the months of May, June, and July (dry period), while in the period from November to March there was a cloudy sky condition (rainy period), behavior similar to that observed for Sinop, MT.
In Table 5 it is possible to observe the behavior of the sky throughout the year through the frequency of K h T within each hydrological station. In the dry season, the condition of partially open sky prevails (31.22%), while in the rest of the year, there is a greater frequency of times with cloudy sky conditions (above 42%).

Estimates Based on the Atmospheric Transmissivity Coefficient
The K h d × K h T correlations are displayed in Figure 6. As observed and described in the literature, with the increase in the clarity index (K h T ), the diffuse fraction tends to decrease, since there is a decrease in the isotropic effects [60]. Figure 6 shows the behavior of the correlation in the different hydro stations. It is observed that in the dry season the point cloud is more concentrated for K h T values greater than 0.4; in this season, sky conditions II, III, and IV predominate ( Table 6). In the rainy season and rainy/dry transition there is greater data dispersion, indicating that for the same value of there is greater data dispersion, indicating that for the same value of ℎ there is great variability in the values of the diffuse fraction. Borges et al. [37] observed a similar behavior in a study carried out in the State of Rondônia (−10.75°; −62.35°), attributing this fact to the greater atmospheric variations (cloudiness) that occur during the seasons.  The lowest values of the diffuse fraction when ℎ tends to 0 can be attributed to the lower horizontal brightness in this region when compared to places at high altitudes and with rugged relief. According to Perez and Seals [61], the horizon zone is infinitesimally thin at 0° elevation.  The lowest values of the diffuse fraction when K h T tends to 0 can be attributed to the lower horizontal brightness in this region when compared to places at high altitudes and with rugged relief. According to Perez and Seals [61], the horizon zone is infinitesimally thin at 0 • elevation.
The equations generated for the entire K h T interval (0 to 0.82) performed better than the sectioned ones, with R 2 above 0.77. In this same interval, the dry and rainy seasons presented better adjustments, which indicates that the atmospheric conditions in the transition periods are more unstable, making estimates difficult. This instability can be explained by the high load of aerosols from biomass burning, an anthropic activity with greater incidence in the months of April and September, corresponding to the periods in question.
The values of the statistical indicators MBE, RMSE and Willmott's d for the estimation equations generated can be observed in Table 7. The seasonal equations, in the intervals of 0.0 ≤ K h T ≤ 0.82 and 0.0 ≤ K h T < 0.55, tended to underestimate the diffuse radiation values during the year. A similar behavior was observed by Borges et al. [37] and Oliveira et al. [52] in the correlations established in their studies. In the range of K h T ≥ 0.55, there was an inversion of this behavior, with a tendency to overestimate. Table 7. Statistical performance indicators of the hourly diffuse solar radiation estimation equations, generated through the K h d × K h T correlation in different data groups in Sinop, MT, Brazil. Regarding the annual equations, when applied to water stations, there was also a tendency to underestimate the values of H h d , especially for K h T ≥ 0.55. The RMSE values ranged from 112.8 to 206.3 kJ m −2 h −1 , with higher scattering rates observed in the rainy season due to the high variation in atmospheric conditions.

Seasonal Annual
The seasonal equations showed better statistical performances when compared to the annual ones in the same periods, except for the rainy season in the intervals 0.0 ≤ K h T ≤ 0.82 and 0.0 ≤ K h T < 0.55, in which the annual equation developed for each partition showed better statistical indicators than the equation for each specific period.
Regarding the annual equations applied to the entire data set, there is a tendency for underestimations in the intervals 0.0 ≤ K h T ≤ 0.82 and K h T ≥ 0.55, and overestimations of the values when 0.0 ≤ K h T < 0.55. Scattering over all K h T intervals is about 160 kJ m −2 h −1 , and the fit index values are best for intervals 0.0 ≤ K h T ≤ 0.82 and 0.0 ≤ K h T < 0.55. Table 8 presents the statistical performance indicators of the 42 evaluated equations (17 models) and the three equations generated in this study (annual grouping in Table 7). In order to assess the performance of the tested estimation models, the method of position values (Pv) of the statistical indicators was used, in which weights from 1 to "n" are assigned to each statistical indicator in each model, with "n" corresponding to the number of evaluated equations. In the end, the best model (equation) is the one considered with the lowest accumulated Pv value, obtained by summing the Pv of each equation in each indicative statistic [62]. The models (equations) were classified by the accumulated value of accumulated Pv considering five groups: (1) Pv1-models with total data grouping (K h T < 0.82); (2)     For the analysis of models that present sectioned regression equations for different intervals of the atmospheric transmissivity coefficient, those that determine fixed values for K h d were disregarded. Subsequently, the statistical performance was analyzed regarding the estimate of diffuse radiation (H h d ). For cloudy sky conditions, six equations were analyzed (4, 15, 18, 21, 28, and 31), and the two models generated by Maduekwe and Garba [51] showed the best statistical performances (Table 8-Pv3 group). These equations, despite having low determination coefficients (0.08 and 0.10), were significant at 5% probability, and when compared with measured values they showed good results for the agreement index "d" (0.9735 and 0.9592) and low scattering indexes (55.73 and 78.22 kJ m −2 h −1 , respectively). Oliveira et al. [63] stated that the coefficient of determination should not be used individually in the analysis of statistical performance, but can help in decision making when comparing different regression models.

Estimates by Parameterized Models
For intermediate sky cover conditions (partly cloudy and partly open) fourteen equations were evaluated, and the best estimates were obtained with the equations generated in this study (  [47] and Jacovides et al. [48] presented better performances when compared to the others (Pv.5).
Of the ten models that showed the best performance for estimating the diffuse fraction in the different K h T ranges, eight were developed or adjusted for tropical regions (latitudes between 6.58 • and −23.56 • ). Of these, there were five for Brazil, which confirms the influence of this climatic factor on the diffuse radiation incident on the Earth's surface.

Discussion
When comparing the annual averages of H h d in the morning and afternoon periods, a higher value was observed during the afternoon (0.27 ± 0.21 and 0.33 ± 0.21 MJ m −2 h −1 ) for the morning and afternoon periods, respectively. Soil heating, and consequently evapotranspiration, are highest in the afternoon, which causes greater attenuation of solar radiation due to the high concentration of water vapor present in the atmosphere.
Diffuse radiation behaves similarly to global radiation in hydrological stations, with the exception of the dry season, in which case there is no symmetry between the morning and afternoon periods (Table 3). This behavior may be due to multireflection processes associated with the elevation of the zenith angle, especially in the first hours of sun exposure, with a reduction in the dispersion of suspended particles in the atmosphere.
The increase in the variability of the diffuse fraction of global radiation can also be attributed to a phenomenon called "cloud gap effect" [64][65][66]. According to the authors, at a given solar elevation angle, a decrease in the atmospheric transmissivity coefficient generally indicates an increase in cloud thickness. However, there is an exception when clouds are not continuously distributed across the sky. Ground surfaces illuminated by the Sun, located at the end of paths of solar beams passing through gaps formed between clouds, may receive greater irradiance than under a clear sky, due to the scattering and reflection of the radiation beam from the side of adjacent clouds. This effect can increase the irradiation incident on the ground by up to 20%. In short, you can increase the K h T without indicating that the sky is open.
If we compare the amplitude of the K h d × K h T curve generated in this study ( Figure 6) with that of other regions available in the literature, it is possible to notice that the highest number of points for K h T < 0.20 is concentrated between 0.80-0.90 K h d , while in other works this value is usually above 0.90. Some authors generate sectioned regression equations, partitioning this K h T interval with fixed values for K h d [48,49], including in Brazil [8,52,55]. It should be noted that smaller time slices respond more sensitively and quickly to atmospheric changes, generating greater variability and detailing of the punctual distribution of solar radiation, which makes estimation difficult. These effects are minimized when values are integrated into daily and monthly partitions [39,60].
Among the five models (equations 1, 2, 3, 24, and 43) that consider the entire K h T interval, the best estimates were obtained by the locally calibrated equation (generated in this study). In this case, the equation proposed by Marques Filho et al. [13] presented the second best accumulated Pv, however, this model was calibrated for the city of Rio de Janeiro (RJ, Brazil), which has atmospheric characteristics distinct from the Cerrado-Amazon transition region, and, therefore, presented overestimates of radiation time difference of up to 323.39 kJ m −2 h −1 . It is interesting to mention that these same authors also evaluated some models evaluated in the present study [40,[46][47][48] to estimate the local diffuse radiation. However, better statistical performances were observed when using the locally developed correlation model.
Singh [67] compared the efficiency of diffuse radiation estimation models considering the entire range of the atmospheric transmissivity coefficient and intervals of cloudiness classes, and concluded that, in general, the models present similar statistical performances for regions with well-defined water seasonality (rainy and dry seasons). These observations by Sing [67] corroborate those found in the present study, through the analysis of the statistical performance of the 45 equations (group Pv2), since the five aforementioned (which consider the entire K h T value) were classified as 14th, 30th, 34th, 38th, and 40th, respectively. In this case, the equations that estimate K h d in intermediate sky cover performed better than the others of the same model.
The diffuse radiation database obtained in the region of Sinop, MT (Brazil), can be considered short (5 years), however, it allows applications directed mainly towards the calibration of models of estimates of diffuse radiation. The studies focused on the analysis of seasonality and estimates on daily integration have already been presented by Zamadei et al. [22,23]. The network of stations of the National Institute of Meteorology (INMET) routinely monitors hourly global horizontal radiation, and, with good estimates of diffuse radiation, it is possible, by difference, to obtain information on direct radiation in horizontal planes.
The modeling of estimation equations with the insertion of a larger number of meteorological variables is recommended for future works to verify the performance in relation to the models generated in this study, and to verify the evaluation of those existing in the literature. The correlation between the diffuse fraction and the insolation ratio can also be worked on in order to propose models with an easily obtainable variable.
The models proposed in this study can contribute to the development of solar energy utilization in places where diffuse radiation measurements are not available. Thus, in today's conditions where global warming is a threat and greenhouse gas emissions are causing warming, the use of clean energy sources such as solar energy can reduce numerous environmental, social, and economic impacts. This study can contribute to providing necessary data on cleaner and more ecologically correct production technologies. More studies can be performed on models based on satellite data for the estimation of diffuse radiation in different climatic regions where the diffuse solar component does not exist.
It should also be noted that there is an urgent need to implement a network of solarimetric stations in the Cerrado and Amazon regions, considered important Brazilian biomes. There are no scientific reports of any radiometry monitoring in the aforementioned regions. This type of monitoring is essential to guide environmental and engineering studies in the most varied areas of knowledge. These stations must, above all, routinely monitor at intervals of 5 or 10 min, in horizontal planes, the spectral components of solar radiation (ultraviolet, visible, and infrared) and diffuse, direct radiation, and reflected radiation.

Conclusions
The hourly diffuse radiation in the region of Sinop, MT, presents similar behavior to the incident radiation at the top of the atmosphere and globally, with maximum values at the lowest zenith angles. Seasonality indicates higher hourly diffuse radiation levels K h d in the rainy season in the region.
The radiometric fractions also show characteristic behavior during the different water seasons. The highest values of the atmospheric transmissivity coefficient K h T were observed in the dry season due to low cloud cover.
The K h d × K h T correlation established showed a peculiar behavior, with the amplitude of the curve characteristic of regions located at low latitudes (closer to the Equator). As in other studies, when K h T tends to 0.0, K h d tends to 1.0; however, for K h T values lower than 0.20, the maximum observed K h d values ranged from 0.80 to 0.90. Polynomial equations were generated to estimate the diffuse radiation considering three intervals of K h T , and these equations presented better statistical performances when compared with parameterized equations from the literature.
Among the 17 models (42 equations) for estimating the parameterized diffuse radiation evaluated, it is recommended to use the polynomials developed in this study or the one elaborated by Marques Filho et al. [13] for estimates of the K h d fraction, in the range of 0.0 ≤ K h T ≤ 1.0, for regions climatically similar to the Cerrado-Amazon transition of Mato Grosso. Funding: This study was financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES), Finance Code-001. The authors wish to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for their support with a productivity grant (Process 308784/2019-7).

Informed Consent Statement: Not applicable.
Data Availability Statement: Study data can be obtained upon request to the corresponding author or the second author, via e-mail. The data are not available on the website as the research project is still under development.