Calibration of the ångström-Prescott Equation for Solar Radiation Estimation in Minas Gerais

Figueiró, Lucas Santos do Patrocínio; Meira, Lucas Fernandes; Santos, Lucas da Costa

doi:10.1590/0102-778640240045

Abstract

Solar energy emerges as a promising sustainable source of energy, capable of meeting energy demand and mitigating issues related to climate change. To evaluate its availability, we can use empirical models such as the Angstrom-Prescott (AP) equation. In this context, our objective was to calibrate and assess the AP model to estimate global solar radiation (Rs) in Minas Gerais. We utilized solar radiation and insolation data from seven municipalities in the state, extracted from the National Institute of Meteorology (INMET), calibrating the model's coefficients through linear regression. After calculating Rs with the calibrated linear and angular coefficient values, we evaluated the accuracy of the estimative by comparing it with observed data. For this, we used the following statistical indicators: linear, angular, determination, and correlation coefficients; mean absolute error; root mean square error; concordance index; and performance index. The results indicated that the model can be an effective tool for estimating Rs in the region (performance index above 0.80, except in one municipality). The use of the calibrated coefficients presented in this work or the standard method are both effective and show no difference between them. The Angstrom-Prescott model proved to be effective in estimating solar radiation in Minas Gerais, with results highlighting its accuracy and applicability.

Keywords
available energy; empirical coefficients; insolation; linear regression; solar energy

Resumo

A energia solar é uma fonte sustentável consolidada, com potencial para suprir a demanda energética e mitigar problemas relacionados às mudanças climáticas. Para avaliar sua disponibilidade, podemos utilizar modelos empíricos como a equação de Angstrom-Prescott (AP). Diante disso, objetivamos calibrar e avaliar o modelo AP para estimar a radiação solar global (Rs) em Minas Gerais. Utilizamos dados de radiação solar e insolação de sete municípios do estado extraídos do Instituto Nacional de Meteorologia (INMET), calibrando os coeficientes do modelo por meio de regressão linear. Após a o cálculo da Rs com os valores dos coeficientes linear e angular calibrados, avaliamos a precisão da estimativa comparando com os dados observados. Para isso, utilizamos os seguintes indicadores estatísticos: coeficientes linear, angular, de determinação, e de correlação, erro absoluto médio, raiz do erro quadrático médio, índice de concordância e índice de desempenho. Os resultados indicaram que o modelo pode ser uma ferramenta eficaz para estimar a Rs na região (índice de desempenho acima de 0.80, exceto em um município). A utilização dos coeficientes calibrados apresentados neste trabalho ou a utilização do método padrão são eficazes e não apresentam diferença entre si. O modelo de Angstrom-Prescott mostrou-se eficaz na estimativa da radiação solar em Minas Gerais, com resultados que destacam sua precisão e aplicabilidade.

Palavras-chave
Coeficientes empíricos; energia renovável; energia solar; insolação; regressão linear

1. Introduction

Solar energy stands out as one of the most promising sustainable sources, with the potential to meet energy demand and help address issues related to climate change. Its availability and accessibility in nearly every part of the world have made it a reliable energy source (Bowonda et al., 2023). Thus, accurate information is essential to ensure the efficiency, planning, sizing, and proper performance of systems (Mohammadi et al., 2022). Although electricity generation is the primary application of solar radiation data, this information is also valuable in fields such as meteorology, agriculture, water resource management, and forestry (Benti et al., 2022).

However, to obtain an accurate assessment of solar energy availability at the surface, it is necessary to conduct measurements using equipment such as solarimeters, pyranometers, or actinographs. To ensure the accuracy of these measurements, the use of data acquisition systems or recorders, along with the presence of trained professionals, is essential. Both manual and automatic measurements of global radiation incur significant costs and require frequent maintenance of the instruments. Thus, the use of mathematical models to estimate this meteorological parameter has been worldwide used as a viable alternative, especially in situations where instruments are not available. Empirical models are particularly appealing due to their low computational requirements, reduced costs, and the accessibility of input data (Auler and Minuzzi, 2022).

With the aim of estimating the amount of available radiant energy at the surface, daily sunshine (hours n of direct solar beam registered by heliographs) can be a simple and efficient indicator of daily irradiation (Rs). Angström (1924) proposed to approximate the ratio Rs/Rcs (Rcs = clear-sky irradiation) with a linear function of sunshine ratio n/N (N = daily or “theoretical” clear-sky sunshine duration). Prescott (1940) proposed the use of Ra (extraterrestrial daily irradiation) instead of Rcs, be:

(1)

R s = R a * (a + b \frac{n}{N})

This expression enhances model's simplicity. Recognizing contribution of both researchers, Eq. (1) is currently known as the Angström-Prescott (AP) equation (Santos et al., 2022).

The AP model is recommended by the Food and Agriculture Organization of United Nations if pyranometric data are not available (Allen et al., 1998), and is widely used around the world. However, coefficients a and b depend on local climate and require fitting with local measurements of Rs, posing a significant challenge for AP application (Liu et al., 2023). Local measurements in a set of regional sites might allow interpolation within the whole region; in this context, a number of applications can be found in literature. Among the works that include data from the state of Minas Gerais, one can mention the Solarimetric Atlas of Minas Gerais Volume I and II (CEMIG, 2012, 2016), and in Brazil, the Solarimetric Atlas of Brazil (Tiba, 2000), which use pyranometers or satellite estimates. It is worth noting that the latter relies on data from the previous century, and thus, these works can be an important contribution to the construction of time series.

Specifically in Minas Gerais, which according to IBGE (2022) covers an area of 586,513.983 km², empirical models become an appealing alternative given the scarcity of meteorological stations distributed throughout the state. In light of this context, the present study aimed to calibrate and evaluate the ångström-Prescott model for estimating global solar radiation in the state of Minas Gerais, aiming to contribute to climate understanding, especially in filling local time series gaps.

2. Materials and Methods

The territory of Minas Gerais is located between latitudes 14°13’58” and 22°54’00” South, and longitudes 39°51’32” and 51°02’35” West of Greenwich. The diversity of vegetation in the region is influenced by various factors, including climate, topography, and watersheds (Minas Gerais, 2024). The state, with its significant territorial extent, exhibits a wide climatic diversity (BSh, BWh, Aw, Cwa, Cwb), according to Köppen's classification (Martins et al., 2018).

Meteorological data were obtained from the National Institute of Meteorology (INMET) for seven municipalities in the state of Minas Gerais (Fig. 1). These municipalities are equipped with automatic weather stations (source of global solar radiation data-Rs) and conventional stations (sunshine duration data-n). The study period covered the years from 2003 to 2023, as presented in Table 1.

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Table 1
Municipalities, geographic location, and period covered by each automatic weather station located in the state of Minas Gerais.

Figure 1
Map of the location of the municipalities used in the study.

Relationship between daily solar irradiation Rs (in MJ m^-2) and sunshine n (in hours) was fitted following Angström-Prescott model (Eq. (1)). Coefficients a and b were obtained by minimal squares procedure, on a monthly, seasonal, and annual scale.

The daily values of extraterrestrial solar radiation were obtained from equations (2) to (5), and the photoperiod (N) from equation (6) (Pereira et al., 2002).

(2)

R a = \frac{J o}{π} * {(\frac{d}{D})}^{2} * [(\frac{π}{180}) * h n * \sin ϕ * \sin δ + \cos ϕ * \cos δ * \sin h n]

where Ra: extraterrestrial solar radiation (MJ m⁻² d⁻¹); Jo: solar constant (37.6 MJ m⁻² d⁻¹); (d/D)²: correction factor for Jo, given by:

(3)

{(\frac{d}{D})}^{2} = 1 + 0.033 * \cos (\frac{360 * Julian Day}{365})

hn: Earth's rotation angle between local sunrise and noon, given by:

(4)

h n = a r c o s [- \tan ϕ * \tan δ]

ϕ: latitude; δ: solar declination, given by:

(5)

δ = 23.45 * \sin [\frac{360 * Julian Day - 80}{365}]

The photoperiod (N) is given by:

(6)

N = \frac{2 * h n}{15}

Prior to data analysis, assessments and data treatment procedures were conducted according to the guidelines described by Tymvios et al. (2005), Liu et al. (2009) and Moradi (2009), specifically: i) If there were missing data for sunshine or global solar radiation, the corresponding observation day would be excluded; ii) If the clearness index (Rs/Ra) or the sunshine ratio (n/N) exceeded 1, the data for that day would be discarded; iii) If the value of Rs was less than 0.03 times the value of Ra, the data for that day would be excluded. The number of data points obtained after this evaluation is presented in Table 1.

With the observed Rs data and the estimates using the values of a and b obtained from the regression, statistical analysis was conducted to evaluate the effectiveness of the calibrated equation for each municipality. For this purpose, the following statistical parameters were employed: linear coefficient (a) and angular coefficient (b), coefficient of determination (R²) found through regression, correlation coefficient (r), mean absolute error (MAE), root mean square error (RMSE), concordance index (d) from Willmott et al. (1985) and performance index (c) from Camargo and Sentelhas (1997).

Simple linear regression aims to determine the coefficients of a linear equation to predict the behavior of a dependent variable (quantitative). This prediction is based on a single independent variable and assumes a linear relationship between the variables, represented by a straight line. From this relationship, it is possible to estimate values for the dependent variable when the variables influencing its behavior are known (Moreira et al., 2020).

The coefficient of determination (R²) is a metric that evaluates the quality of the fit of the linear regression model. It indicates the proportion of the variation in the dependent variable (Y) explained by the independent variable (X), meaning how much of the variation in Y can be attributed to X. The value of the coefficient of determination ranges from 0 to 1, with values close to 1 suggesting that the model is suitable for describing the phenomenon in question (Oliveira, 2016).

The correlation coefficient (r) is an index that expresses the degree of relationship between two quantitative variables. This dimensionless coefficient can range from -1 to 1. Values close to -1 indicate a strong negative correlation, meaning the variables are inversely proportional; values close to 1 indicate a strong positive correlation, meaning the variables are directly proportional; and values close to 0 indicate no correlation between the variables (Oliveira, 2016).

MAE represents the difference between the observed and simulated data, indicating that the smaller its value, the closer the simulated values are to the observed values. On the other hand, RMSE evaluates the degree of dispersion of the simulated data in relation to the observed data (Oliveira et al., 2021).

The Willmott concordance index (d) reflects the precision of the estimated values in relation to the observed values. It is also a dimensionless index that ranges from 0 to 1. A value of d = 0 indicates total disagreement between the values, while d = 1 indicates total agreement (Willmott et al., 1985).

The performance index (c), developed by Camargo and Sentelhas (1997), aims to combine the correlation coefficient (r), which measures the degree of dispersion of the data relative to the mean, with the Willmott concordance index, which evaluates the deviation between the estimated and observed values. The index c is ranked in Table 2.

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Table 2
Criteria for evaluating performance according to the performance index proposed by Camargo and Sentelhas (1997).

After that, a significance test was performed for the regression on the annual scale for the entire state of Minas Gerais, as well as the process of validating the calibrated average coefficients. For the latter, we proceeded with the calculation of Rs using the calibrated AP method for the municipalities of João Pinheiro, Porteirinha, Uberaba, and Unaí, as described in Table 3.

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Table 3
Municipalities, geographic location, and period covered by each automatic weather station located in the state of Minas Gerais, used for the validation of the calibrated coefficients.

3. Results and Discussion

Table 4 presents the linear (a) and angular (b) coefficients of the ångström-Prescott equation for municipalities in Minas Gerais, relating the daily clearness index (Rs/Ra) to the sunshine ratio (n/N) for the seven municipalities used in the study, along with the coefficient of determination (R²) on a seasonal and annual scale. In Paracatu, the best fit was observed for determining the coefficients a and b, both annually and seasonally, except in winter, achieving a coefficient of determination of 0.87 or higher. In contrast, Uberlândia exhibited an R² of 0.62 on the annual scale, reaching the lowest value in autumn at 0.41, indicating a poor fit of the model, which can be explained by the limited number of available observational records for analysis (898 data points according to Table 1), moreover, the fact that the observational values, obtained from an official source (INMET), are older (2003-2006) may pose a problem related to the instruments and the quality of the records. Therefore, caution is necessary when using the results from the station. Additionally, fire counts and smoke aerosol loads in the Cerrado during this period may have contributed to transmissivity blockage, affecting the Rs and/or n records (Rosário et al., 2022).

The other municipalities (Barbacena, Belo Horizonte, Diamantina, Pirapora, and Salinas) achieved satisfactory performance with determination coefficients equal to or greater than 0.70, except for Pirapora in Autumn and Winter, with values of 0.62 and 0.63, respectively.

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Table 4
Coefficients a and b of the ångström-Prescott equation for municipalities in Minas Gerais.

In Table 4, it is also evident that, in most of the municipalities, atmospheric transmissivity reached its highest values in winter and its lowest values in summer. This occurs because, during summer, there is a greater concentration of humidity in the atmosphere, resulting from cloud formation processes, as it is the rainy season in the state. Conversely, in winter, there is a low concentration of water vapor (the dry season of the year). Furthermore, according to Querino et al. (2011), precipitation not only affects various physical, chemical, and biological processes on the Earth's surface but also helps clean the atmosphere. During precipitation, many of the suspended particles in the air are removed and returned to the surface, resulting in a cleaner atmosphere. This fact explains why, immediately after the rainy season, there is a greater incidence of radiation at the Earth's surface.

In Paracatu, this does not occur; in other words, transmissivity is higher in seasons other than winter. The lowest transmittance is observed during July, August, and September, which, as noted by Rosário et al. (2022), is historically the period with the highest incidence of fires in the Cerrado. These fires contribute to greater radiation blockage due to the increased smoke aerosol load. This justification was also presented by Belúcio et al. (2014) in a study conducted in Macapá/AP.

By comparing the observed data, that is, those recorded by INMET, with the values estimated by the ångström-Prescott method using the values of a and b found for the respective municipalities on an annual scale, the statistical values were obtained, as presented in Table 5.

According to the performance index, the estimation of Rs by the empirical model showed satisfactory behavior, being classified as great or very good, with a value of c greater than or equal to 0.81, except in Uberlândia, which was classified as not good (c = 0.60). The values of the performance index demonstrate that the model exhibits accuracy in its estimates (d ≥ 0.82) and low values of MAE and RMSE, remaining below 2.80 MJ m⁻² d⁻¹, except for Uberlândia.

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Table 5
Statistical parameters obtained from the relationship between observed solar radiation and solar radiation estimated by the ångström-Prescott method using the calibrated coefficients on an annual scale for municipalities in Minas Gerais.

Since the aim of the study was to obtain the parameterized coefficients of the ångström-Prescott equation for the entire state of Minas Gerais, data from all municipalities were aggregated to obtain this information. The average results across temporal scales-monthly, seasonal, and annual-are presented in Table 6.

Analyzing the seasonal period, the best fits were observed in spring and summer, with R² values of 0.81 and 0.79, respectively. On the monthly scale, the values of a varied from 0.25 in November to 0.31 in June, and b ranged from 0.46 in June to 0.52 in November. We observed an increase in a during autumn/winter and in b during winter/spring.

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Table 6
Average coefficients a and b of the ångström-Prescott equation at different temporal scales for Minas Gerais based on the calibration of seven locality.

We identified a linear adjustment trend in the relationship between these variables, the clearness index and the sunshine ratio, resulting in a minimum coefficient of determination of 0.7354 in March, with a maximum value obtained in January (0.8300). These values are higher than the R² obtained by Santos et al. (2022), who observed values of 0.69 and 0.79 for Itumbiara and Rio Verde, respectively, and by Pilau et al. (2007) in the Araras/SP region, with an annual average of 0.7147 and a minimum in February (0.496).

By adopting the annual values of a (0.2772) and b (0.4930), we calculated Rs using the ångström-Prescott model and compared the estimated data with the observed data for each municipality. Subsequently, we conducted a statistical analysis of the results, as presented in Table 7.

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Table 7
Statistical parameters obtained from the relationship between observed solar radiation and solar radiation estimated by the ångström-Prescott method using the annual coefficients, for municipalities in Minas Gerais on an annual scale.

In descending order, according to the performance index, the best fits obtained were: Paracatu, Salinas, Belo Horizonte, and Barbacena (great), followed by Diamantina and Pirapora (very good), and Uberlândia (not good). When analyzing the Pearson correlation coefficient, all municipalities were classified as “very high” (r ≥ 0.73), with four classified as “almost perfect”. There was a slight variation in the values of the performance index when compared to the individual values of a and b for each municipality; however, this did not change the classification.

With the exception of Uberlândia, the values of the concordance index (d) ranged from 0.93 to 0.97, indicating good accuracy in estimating global solar radiation using the annual calibrated AP model. Similar results were found by Medeiros et al. (2017), where the index varied from 0.91 to 0.97 for the state of Rio Grande do Norte.

These results from the annual calibration did not yield an improvement in the R² value compared to the methodology proposed by Glover and McCulloch (1958), for calculating the coefficients of the AP model, where a = 0.29 cos ϕ (latitude), and b = 0.52. This indicates the good applicability of the standard methodology for the coefficient values.

Generalizations cannot be made given the climatic diversity of the state of Minas Gerais, often related to the region's altimetric contrast. Therefore, the validation of the calibration performed for the AP equation enhances the reliability of the obtained results. According to Table 8, the calibration was consistent, with a performance index for the correlation between observed and estimated data equal to or greater than 0.86, classified as excellent for all evaluated municipalities. Additionally, it demonstrated high precision (d = 0.95) and accuracy (r ≥ 0.91).

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Table 8
Statistical parameters of the validation process obtained from the relationship between observed solar radiation and solar radiation estimated by the ångström-Prescott method using the annual coefficients, for municipalities in Minas Gerais on an annual scale.

Figure 2 presents the relationship between the observed (INMET) and estimated (AP model) Rs data, using the calibration coefficients based on the aggregated data (a = 0.2772, and b = 0.4930). There is a good accuracy of the model in estimating Rs, except for Uberlândia. In all municipalities, it can be observed from the regression line and the 1:1 line that, for lower Rs values, the model tends to overestimate the results, while the opposite situation occurs at higher radiation values, where the model begins to underestimate. Nevertheless, the values tend to align with the 1:1 line, indicating accuracy in the estimates, along with a high precision demonstrated by the data points, confirmed by the high values of the coefficient of determination (R²).

Figure 2
Relationship between the observed solar radiation values (MJ m^-2 d^-1) and the solar radiation estimated by the ångström-Prescott method using the coefficients a and b adjusted for the annual scale in the municipalities of Barbacena (A), Belo Horizonte (B), Diamantina (C), Paracatu (D), Pirapora (E), Salinas (F), Uberlândia (G), in the state of Minas Gerais.

The significance test revealed that there is a regression of Rs/Ra in relation to n/N, meaning that n/N is explaining Rs/Ra (p < 0.05), as shown in Table 9.

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Table 9
Analysis of variance of the linear regression.

4. Final Considerations

The calibration of the ångström-Prescott (AP) model showed consistent results in estimating global solar radiation when compared to the data observed by the automatic weather stations of INMET in Minas Gerais.

On the average, AP model exhibits high correlation between daily irradiation provided by pyranometers of INMET automatic network and estimation based on sunshine hours. This is observed on monthly, seasonal and annual epochs (R² always greater than 0.75, except for Uberlândia). Anomalous regression clearness index versus sunshine index observed in Uberlândia case needs further analysis of original data.

When considered the whole data set of seven stations as only one data set, coefficients of AP model still allow estimation of daily irradiation with high correlation level. This fact suggests the use of only one set of AP coefficients could be suitable enough for estimation of local Ra from sunshine hours all over Minas Gerais state. Further critical research for sites with especial climatic characteristics should be performed.

The analysis applies to period 2003-2020. The results of Rs estimation may be useful to complete previous time series applying AP model for 20th century data.

We also concluded that the standard methodology adopted for obtaining the values of coefficients a and b of the equation (a = 0.29 cos ϕ and b = 0.52) fits well for the state.

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Publication Dates

Publication in this collection
27 June 2025
Date of issue
2025

History

Received
09 Oct 2024
rev-request
24 Mar 2025
Reviewed
11 Apr 2025
Accepted
22 May 2025

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Municipalities	Latitude	Longitude	Altitude (m)	Number of data	Interval
Barbacena	21 °S	44 °W	1169	5141	2003-2018
Belo Horizonte	20 °S	44 °W	854	6266	2006-2023
Diamantina	18 °S	44 °W	1359	4490	2007-2020
Paracatu	17 °S	47 °W	705	896	2018-2020
Pirapora	17 °S	45 °W	505	3858	2007-2019
Salinas	16 °S	42 °W	487	5927	2007-2023
Uberlândia	19 °S	48 °W	875	898	2003-2006

Values of “c”	Interpretation
> 0.85	Great
0.76-0.85	Very Good
0.66-0.75	Good
0.61-0.65	Medium
0.51-0.60	Not Good
0.41-0.50	Bad
≤ 0.40	Terrible

Municipalities	Latitude	Longitude	Altitude (m)	Number of data	Interval
João Pinheiro	18 °S	46 °W	877	425	2007-2008
Porteirinha	15 °S	44 °W	454	1568	2007-2012
Uberaba	20 °S	48 °W	778	446	2017-2018
Unaí	17 °S	47 °W	641	682	2007-2009

Municipalities	Period	Calibration AP	a + b	R²
Barbacena	Summer	Rs/Ra = 0.28 + 0.48 (n/N)	0.76	0.81
	Autumn	Rs/Ra = 0.32 + 0.45 (n/N)	0.77	0.74
	Winter	Rs/Ra = 0.30 + 0.48 (n/N)	0.78	0.80
	Spring	Rs/Ra = 0.25 + 0.54 (n/N)	0.79	0.80
	Annual	Rs/Ra = 0.29 + 0.50 (n/N)	0.79	0.79
Belo Horizonte	Summer	Rs/Ra = 0.26 + 0.49 (n/N)	0.75	0.83
	Autumn	Rs/Ra = 0.29 + 0.48 (n/N)	0.77	0.77
	Winter	Rs/Ra = 0.25 + 0.54 (n/N)	0.79	0.79
	Spring	Rs/Ra = 0.25 + 0.51 (n/N)	0.76	0.86
	Annual	Rs/Ra = 0.26 + 0.52 (n/N)	0.78	0.84
Diamantina	Summer	Rs/Ra = 0.29 + 0.44 (n/N)	0.73	0.70
	Autumn	Rs/Ra = 0.29 + 0.49 (n/N)	0.78	0.75
	Winter	Rs/Ra = 0.30 + 0.50 (n/N)	0.80	0.77
	Spring	Rs/Ra = 0.27 + 0.49 (n/N)	0.76	0.79
	Annual	Rs/Ra = 0.28 + 0.49 (n/N)	0.77	0.77
Paracatu	Summer	Rs/Ra = 0.28 + 0.49 (n/N)	0.77	0.91
	Autumn	Rs/Ra = 0.27 + 0.46 (n/N)	0.73	0.88
	Winter	Rs/Ra = 0.31 + 0.42 (n/N)	0.73	0.71
	Spring	Rs/Ra = 0.25 + 0.55 (n/N)	0.80	0.92
	Annual	Rs/Ra = 0.28 + 0.46 (n/N)	0.74	0.87
Pirapora	Summer	Rs/Ra = 0.30 + 0.48 (n/N)	0.78	0.76
	Autumn	Rs/Ra = 0.34 + 0.43 (n/N)	0.77	0.62
	Winter	Rs/Ra = 0.28 + 0.47 (n/N)	0.75	0.63
	Spring	Rs/Ra = 0.27 + 0.50 (n/N)	0.77	0.74
	Annual	Rs/Ra = 0.29 + 0.48 (n/N)	0.77	0.74
Salinas	Summer	Rs/Ra = 0.30 + 0.45 (n/N)	0.75	0.79
	Autumn	Rs/Ra = 0.27 + 0.49 (n/N)	0.76	0.88
	Winter	Rs/Ra = 0.26 + 0.49 (n/N)	0.75	0.86
	Spring	Rs/Ra = 0.26 + 0.51 (n/N)	0.77	0.84
	Annual	Rs/Ra = 0.27 + 0.49 (n/N)	0.76	0.84
Uberlândia	Summer	Rs/Ra = 0.28 + 0.52 (n/N)	0.80	0.64
	Autumn	Rs/Ra = 0.31 + 0.52 (n/N)	0.83	0.41
	Winter	Rs/Ra = 0.30 + 0.56 (n/N)	0.86	0.59
	Spring	Rs/Ra = 0.31 + 0.48 (n/N)	0.79	0.49
	Annual	Rs/Ra = 0.29 + 0.55 (n/N)	0.84	0.62

Municipalities	α	β	R²	r	MAE	RMSE	d	c	ICS
Barbacena	2.54	0.85	0.82	0.90	1.78	2.50	0.95	0.86	GT
Belo Horizonte	2.31	0.88	0.85	0.92	1.53	2.08	0.96	0.88	GT
Diamantina	3.26	0.83	0.78	0.88	2.03	2.79	0.94	0.83	VG
Paracatu	3.51	0.81	0.90	0.95	1.20	1.65	0.97	0.92	GT
Pirapora	4.96	0.76	0.76	0.87	1.76	2.71	0.93	0.81	VG
Salinas	2.72	0.86	0.87	0.93	1.55	2.20	0.96	0.90	GT
Uberlândia	8.52	0.53	0.53	0.73	3.28	4.25	0.82	0.60	NG

Month	a	b	R²
January	0.2726	0.4936	0.8300
February	0.2800	0.4754	0.7867
March	0.2926	0.4588	0.7354
April	0.2917	0.4725	0.7875
May	0.2934	0.4779	0.7662
June	0.3116	0.4578	0.7427
July	0.2979	0.4737	0.7550
August	0.2956	0.4722	0.7501
September	0.2751	0.5023	0.7918
October	0.2584	0.5052	0.7897
November	0.2533	0.5175	0.8198
December	0.2666	0.5139	0.8068
Summer	0.2814	0.4767	0.7881
Autumn	0.2972	0.4720	0.7695
Winter	0.2886	0.4838	0.7680
Spring	0.2600	0.5106	0.8085
Annual	0.2772	0.4930	0.7991

Municipalities	α	β	R²	r	MAE	RMSE	d	c	ICS
Barbacena	2.34	0.84	0.82	0.91	1.79	2.50	0.95	0.86	GT
Belo Horizonte	3.01	0.85	0.84	0.92	1.54	2.12	0.96	0.88	GT
Diamantina	3.23	0.82	0.78	0.88	2.03	2.79	0.94	0.83	VG
Paracatu	3.18	0.86	0.89	0.94	1.27	1.70	0.97	0.91	GT
Pirapora	4.61	0.78	0.76	0.87	1.76	2.71	0.93	0.81	VG
Salinas	2.79	0.87	0.87	0.93	1.54	2.21	0.96	0.90	GT
Uberlândia	8.26	0.53	0.53	0.73	3.37	4.34	0.81	0.60	NG

Municipalities	α	β	R²	r	MAE	RMSE	d	c	ICS
João Pinheiro	2.24	0.95	0.87	0.93	1.52	2.03	0.95	0.88	GT
Porteirinha	2.06	0.92	0.83	0.91	1.45	2.14	0.95	0.87	GT
Uberaba	3.49	0.84	0.82	0.91	1.45	2.20	0.95	0.86	GT
Unaí	2.43	0.85	0.84	0.92	1.53	2.00	0.95	0.87	GT

	DF	SS	MS	F	p-value
Regressão	1	555.82	555.82	109247.88	< 0.05
Resíduo	27474	139.78	0.005
Total	27475	695.59	0.005

Calibration of the ångström-Prescott Equation for Solar Radiation Estimation in Minas Gerais

Calibração da Equação de ångström-Prescott para Estimativa da Radiação Solar em Minas Gerais

Abstract

Resumo

1. Introduction

2. Materials and Methods

3. Results and Discussion

4. Final Considerations

References

Publication Dates

History