1. INTRODUCTION

The relevance of the variability of incoming longwave radiation can be justified in several ways, as being important for: (1) Surface Energy Balance (SEB); (2) Numerical Weather Forecasting (NWF); (3) simulations carried out at several different scale models; (4) the global radiation balance; (5) climate studies; (6) agrometeorological studies (predictions for night frosts, mist, air temperature variation and cloudiness in a particular region); (7) other applications ( ^{Iziomon et al., 2003} ; ^{Aguilar et al., 2015} ; ^{Morcrette, 2002} ; ^{Ma et al., 2014} ). Incoming longwave radiation is surely the most difficult component of the radiation balance to measure. One significant reason is that the pygmometer emits radiation at wavelengths and intensities comparable to those that it seeks to measure, which creates a need for correction ( ^{Liou, 2002} ).

The Amazon is the region encompassed by the Amazon River basin, the most extensive in the world, comprising 25,000 Km of navigable rivers, in about 6,900,000 Km^{2}, of which approximately 3.8 million Km^{2} are in Brazil. The Legal Amazon, established in Article 2 of the Law nº 5.173, October 1966, covers the states of Acre, Amapá, Amazonas, Mato Grosso, Pará, Rondônia, Roraima, Tocantins, part of Maranhão and five municipalities of Goiás. It is represents 59% of the Brazilian territory, distributed over 775 municipalities, where 20.3 million people (12.32% of the national population) were resident in 2000, with 68.9% of this contingent being in urban areas ( ^{IBGE, 2015} ).

In this context, it is important to understand the dynamics of the region, as well as the mechanisms that interact with incoming longwave radiation and its relationship to forest functioning ( ^{Satyamurty et al., 2010} ; ^{Almeida et al., 2017} ). Incoming longwave radiation monitoring, in this region, requires spatially well-distributed, high density automatic weather stations (AWS) with an operating pygmometer. Currently, the AWS does not possess such equipment, and consequently, there is no measurement of incoming longwave radiation ( ^{Vilani et al., 2010} ; ^{Aguiar et al., 2011} ). Recently, ^{Aguilar et al. (2015)} showed that the difficulty for data collection is due to the price of the equipment and its calibration. Despite this, some studies have been developed in the region, such as ^{Vilani et al. (2010)} and ^{Aguiar et al. (2011)} . However, these studies only estimate the variable over the time scale, and are not being spatialized using geostatistics.

Conventional weather stations (CWS) and AWS coming from the Instituto Nacional de Meteorologia (INMET) still present many flaws in the data series, which in turn also harm the estimate of incoming longwave radiation in the region by way of *in situ*. Additionally, the Amazon region is extensive and is difficult to access due to dense forest areas. It therefore contains a low number of weather stations, which can cause significant uncertainty when trying to understand the different processes related to dynamic climate mechanisms and their spatial variability.

The United Nations Environment Programme (UNEP), and the World Meteorological Organization (WMO) established the Intergovernmental Panel on Climate Change ( ^{IPCC, 2007a} , ^{b} ) in 1988. The Assessment Report presented different CO_{2} emission scenarios, containing the A2 (high emission, pessimistic) and B2 (low emission, optimistic) scenarios. Currently, climate changes and their projections point to an average increase in air temperature of up to 4.0 °C. Therefore, in the context of changes occurring in the climate, many researchers from different regions around the world have used several numerical models. Particularly, Coupled Model Intercomparison Project Phase 5 (CIMP5) used by ^{Zomer et al. (2014)} in Yunnan Province, China and the regional model HadRM3 used in Europe and India ( ^{PRECIS, 2002} ) and, lastly, in the Western Amazon region ( ^{Justino et al., 2013} ; ^{Delgado et al., 2014} ).

Spatial transitive theoretical models have been applied to the space-time estimation of many variables, especially to model the phenomena and simulations of climate scenarios ( ^{Gondim et al., 2008} ; ^{Correa et al., 2014} ; ^{Gois et al., 2015} ). According to ^{Castro et al. (2010)} , the quality of data interpolation depends on the distribution and knowledge of the points used in the calculation, in addition to the correlation statistical methods for the phenomena being studied.

There are few studies to estimate the incoming longwave radiation based on existing equations in the literature, especially for the Amazon region in the spatial and temporal scales. Additionally, using future scenarios adopted by the IPCC there are still no scientific studies of the region. Given this, the present study aims i) to calculate and spatialize the incoming longwave radiation using Ordinary Kriging method (geostatistics) based on ^{Swinbank’s (1963)} equation; the output of the regional HadRM3 model for air temperature from the IPCC’s A2 scenario, for the Legal Amazon was used, in the annual periods 2070, 2080 and 2090; and ii) interpolation method evaluations of the mathematical transitive theoretical models for incoming longwave radiation.

2. MATERIAL AND METHODS

2.1. Study area

According to Köppen’s classification, the Amazon region shows climate type A, with three subclimates: rainy equatorial (“Af”), the monsoon tropical (“Am”) and the dry and wet tropical (“Aw”), with the monsoon tropical covering most of the region. According to ^{Figueroa & Nobre (1990)} , the average air temperature presents no large-scale variation, which is due to high levels of solar radiation throughout the year. Average air temperature values are between 24 to 26 °C with an annual range from 1 to 2 °C. The average rainfall in the Amazon region is approximately 2,300 mm.year^{-1}. The rainy season is between November and March, and the dry season is between May and September. The months of April and October are months of transition between regimes ( ^{Satyamurty et al., 2010} ; ^{Almeida et al., 2017} ).

The studied case presents a grid of 38 points, which represent the grid points of the HadRM3 model in the Amazon. It has a 50 Km spatial resolution with 19 levels in the atmosphere (from the surface to 30 Km into the Stratosphere) and 4 levels into the soil. Additionally, it has full representation of the physical processes of the atmosphere and the earth's surface, and includes the sulfur cycle (S). This grid, the air temperature data were extracted grid points of HadRM3 model on the Amazon ( Figure 1 ). We used daily air temperature data, which were converted into yearly values. These data were extracted from the model output for the future A2 scenario (2070, 2080 and 2090) in Legal Amazon.

2.2. HadRM3 model

The Hadley Centre Regional Model Version 3 is a configuration of the HadCM3 model designed to provide high-resolution climate simulations of limited areas. It is based on the atmospheric component of HadCM3 with some modifications for the physical model. HadRM3 is typically configured with a horizontal resolution of 0.44 degrees of latitude and 0.44 degrees of longitude (approximately 50 Km) and uses a 5-minute time step to maintain numerical stability. The model was designed to take boundary conditions from coarser global resolution model simulations, in this case HadAM3, and provides a higher spatial resolution of the local topography and more realistic simulations of fine-scale climate features. The HadRM3 model was first used by the Hadley Centre for the UK Climate Impacts Programme 2002. It was subsequently used for a number of other high-resolution regional climate simulation studies, including a number of ensemble projects.

2.3. Calculation of incoming longwave radiation

The interrelationship between the air temperature variables and incoming longwave radiation was determined using Swinbank’s equations (1963), which is physically substantiated. Therefore, the calculation of annual incoming longwave radiation can be obtained by Equation 1 ( ^{Swinbank, 1963} ).

wherein, σ is Stefan-Boltzmann’s constant (5.6697. 10^{-8} W.m ^{-2}.K^{-4}) and T is the average air temperature (K) of the 38 points of the Legal Amazon obtained from the HadRM3 model. The estimate has a probable error of less than 5 W.m^{-2}, according to ^{Swinbank (1963)} .

2.4. Geostatistical analysis

The spatial interpolation method used to analyze and calculate the spatial distribution of incoming longwave radiation for the Legal Amazon was the Ordinary Kriging (OK). Unlike other spatial interpolation methods, Kriging estimates a spatial covariance matrix to determine the weights attributed to different samples, data redundancy, the neighborhood to be considered in the inferential procedure and the error associated with the estimated value ( ^{Goovaerst, 1997} ).

In this study, we adopted the OK, which allows the calculation of averages sites, limiting the average stationary domain to a local neighborhood centered on the point to be estimated. According to ^{Felgueiras (2001)} , quota *z* values are estimated using unobserved spatial coordinates (x _{j}, y_{j}), without the need to know the stationary average (μ), from a linear combination of the values of a local sampling subset. The OK estimator is given by Equation 2:

wherein, is the quota in position (x_{j}, y_{j}) and λ_{i} (x_{j}, y_{j}), the OK weight to the specified quota.

For the geostatistical analysis of interpolated data from incoming longwave radiation, we adopted three mathematical transitive theoretical models (Spherical, Exponential and Gaussian). Further details on transitive theoretical models can be found in ^{Gois et al. (2015)} .

For analysis and selection of the theoretical mathematical model that best fit the experimental semivariogram, we used the Degree of Spatial Dependence (DSD %) and the errors attributed to each model. DSD is calculated by Equation 3:

wherein C_{0} = intercept or nugget effect; C_{0} + C = level and C = contribution.

The methodology proposed by ^{Cambardella et al. (1994)} was adopted in this study wherein a DSD ≤ 25% of the data possesses strong spatial dependence, DSD between 25 and 75% shows moderate spatial dependence, DSD ≥ 75% presents weak spatial dependence, and, finally, a DSD = 100% presents a spatially independent variable. In this case, it has a semivariogram with pure nugget effect. Besides the DSD, the error attributed to each model was adopted as a criterion to select the best model, according to the Equation 4:

wherein, RMSE (W.m^{-2}.year^{-1}) is Root Mean Square Error; O _{i} is equal to the observed values of incoming longwave radiation (W.m^{-2} .year^{-1}); X_{i} is equal to the estimated values of incoming longwave radiation (W.m^{-2}.year^{-1}) and n is equal to number of data.

The coefficient of variation (CV, %) is given by Equation 5:

wherein,

3. RESULTS AND DISCUSSION

3.1. Data analysis

In Figure 2 and Table 1 , the exploratory (boxplot) and descriptive (average, maximum, minimum, standard deviation, variance and coefficient of variation) analyses of incoming longwave radiation data for the IPCC A2 scenario in Legal Amazon is presented. The presence of outliers was observed for all years, with the highest averages and outliers occurring in 2090 in comparison with the other years evaluated. These high values are correlated with high Greenhouse Gas Emissions (GHG) and increasing the air temperature in the pessimistic A2 scenario adopted by the IPCC. Discrepant values found in the study showed that they behave differently in relation to that presented by the majority of values estimated by ^{Swinbank’s (1963)} equation through the HadRM3 model output. Lower average values for incoming longwave radiation were found in 2070 (394 W.m^{-2}.year^{-1}). In the global solar radiation results obtained by ^{Delgado et al. (2014)} in the State of Acre, northern Brazil, a significant increase between simulated years, calculated for both IPCC scenarios was also found, which is explained by the significant changes in the CO_{2} concentration in the A2 scenario.

Year |
Average(Wm ^{-2}year^{-1}) |
Maximum(Wm ^{-2}year^{-1}) |
Minimum(Wm ^{-2}year^{-1}) |
Standard deviation |
Variance |
CV (%) |
---|---|---|---|---|---|---|

2070 | 394.8 | 431.1 | 376.4 | 8.8 | 78.0 | 2.2 |

2080 | 403.9 | 447.3 | 387.0 | 10.5 | 109.3 | 2.6 |

2090 | 413.0 | 461.3 | 394.4 | 11.4 | 129.2 | 2.8 |

In the numerical simulations of the HadRM3 model to estimate incoming longwave radiation for 2080 and 2090, the highest standard deviations were found. This indicates that the estimated values show a large spread, unlike 2070 which presented a low standard deviation, which in turn indicates that the estimated values of the incoming longwave radiation tend to be near the average and present lower dispersion ( Table 1 ), followed by the CV values that were higher for 2080 (2.6%) and 2090 (2.8%), respectively. The high CV values for 2090 demonstrated the existence of a considerable increase in the variable air temperature in the study area, which in turn is the input variable in Swinbank’s equation (1963). This is due to the maximum GHG concentration and abrupt increase in average air temperature global/regional simulated by the HadRM3 model. ^{Vilani et al. (2010)} found similar results based on ^{Swinbank’s (1963)} equation.

3.2. Semivariogram

2070 presented the lowest DSD and RSME values, regardless of the transitive theoretical model used ( Table 2 ). This year, the Gaussian model obtained the best DSD (9.9%) and RSME (9.1 W.m^{-2} .year^{1}) values. For 2080, the transitive theoretical models were the same with respect to RSME (10.5 W.m^{-2}.year^{-1}), however the Gaussian model was the only one which showed a strong spatial dependence (DSD = 24.4%). The other models (spherical and circular) showed moderate spatial dependence, precluding the spatial distribution of the results. For 2090, all transitive theoretical models showed moderate spatial dependence. Although the results found by ^{Santos et al. (2011)} are distinct, strong independence of information makes it impossible to spatially distribute the incoming longwave radiation for 2090 in the Legal Amazon. ^{Wanderley et al. (2013)} and ^{Uliana et al. (2013)} observed similar results, where the Gaussian model obtained the better fit for annual rainfall in the states of Alagoas and Espírito Santo, respectively.

Years |
C_{0} |
C_{0}+C |
a |
DSD (%) |
RMSE (W.m^{-2}.year^{-1}) |
Models |
---|---|---|---|---|---|---|

2070 | 10.3 | 74.5 | 13.8 | 13.8 | 9.2 | Spherical |

9.7 | 74.6 | 12.5 | 13.0 | 9.2 | Exponential | |

7.6 | 77.1 | 12.5 | 9.9 | 9.1 | Gaussian | |

2080 | 30.0 | 88.1 | 14.5 | 34.1 | 10.5 | Spherical |

34.0 | 86.0 | 18.8 | 39.5 | 10.5 | Exponential | |

24.1 | 94.9 | 13.8 | 24.4 | 10.5 | Gaussian | |

2090 | 40.6 | 101.6 | 14.1 | 40.0 | 11.4 | Spherical |

46.7 | 96.7 | 15.7 | 48.3 | 11.4 | Exponential | |

32.5 | 110.6 | 13.1 | 29.4 | 11.3 | Gaussian |

C_{0} = nugget effect; C_{0}+C = level; a = reach; DSD = Degree of Spatial Dependence; RMSE = Root Mean Square Error.

3.3. Transitive theoretical models in 2070

By comparing the transitively interpolated theoretical mathematical models, it was firstly observed that despite sampling the grid points of the HadRM3 model, it did not cover a good grid ( Figure 3 ). The difference of the interpolated maps of incoming longwave radiation by OK for the three models for 2070 was evaluated. Spatial variability of incoming longwave radiation in the studied region was observed, due to low RMSE and the closeness of DSD values for each interpolated model. In this study, the highest estimated values of incoming longwave radiation were found in the southwest-northeast direction (SW-NE) in the order of 400 W.m^{-2}.year ^{-1}, while the lowest values were observed in the West (W), Northwest (NW), South (S) and East (E) regions with approximate values of 390 W.m^{-2}.year^{-1} . According to ^{Herrero & Polo (2012)} , the acquisition of this information is important because the incoming longwave radiation controls temperature rates near the surface.

3.4. Case study of the Gaussian model in 2080

In Figure 4 , the highest estimated values of incoming longwave radiation were found in the central portions and in the SE and Northeastern directions with values higher than 400 W.m^{-2}.year ^{-1}. The lowest incoming longwave radiation values were concentrated in the W, NW and S regions with approximate values of 400 W.m^{-2}.year^{-1}. The observed and spatially simulated results were in accordance with the considerable temperature and CO_{2} increases for this year simulated by the regional HadRM3 model.

4. CONCLUSIONS

The ordinary Kriging method is applicable throughout the Legal Amazon for 2070 and 2080 but not 2090 due to moderate spatial dependence, which precludes its application. The Gaussian model is the most suitable to estimate the incoming longwave radiation for the Amazon region.

The estimated incoming longwave radiation based on the air temperature output obtained from the HadRM3 regional model and applied in Swinbank’s equation showed promising results for the Legal Amazon.

The results of this study will enable and support new studies with the HadRM3 regional model, especially in intercomparative studies involving other regional models and other existing equations in the literature to estimate incoming longwave radiation.