Model OLD₀: A Physical Parameterization for Clear-Sky Downward Longwave Radiation

Modelo OLD₀: Uma Parametrização Física para Fluxo Descendente de Onda Longa em Condições de Céu Claro

Abstract

Downward longwave irradiance DLF is one of the main components of the surface radiation balance (SRB), but its direct measurement is currently limited. Clouds modulate its behavior, and clear-sky DLF₀ is predominant in composition of the final DLF value. It is shown that in mid-latitude and tropical atmospheres, DLF₀ can be represented as the sum of fluxes from three distinct spectral regions: R1 (λ < 7.5 µm), R2 (7.5 to 14 µm), and R3 (λ > 14 µm). R1 and R3 are closely described by blackbody radiation at screen temperature (T_scr), while R2 exhibits a mean emissivity that primarily depends on total precipitable water (w). It is presented a simple yet consistent physically-based model (hereafter denoted by OLD₀), suitable for estimation of DLF₀ at ground level. Validation of OLD₀ with ground-based data of a worldwide set of 21 stations shows fair accuracy with bias MBE lower than 6 W.m^-2 and spread (standard deviation STD) lower than 12 W.m^-2 for typical values DLF₀ ~ 300-400 W.m^-2, compatible with surface pyrgeometer measures. The proposed algorithm outperforms existing methods, achieving a mean bias error (in module |MBE|) of approximately 2.8 W.m^-2. In contrast, other widely used algorithms typically exhibit |MBEs| ranging from 8.1 to 15.9 W.m^-2.

Keywords
downward longwave radiation; clear sky radiation; atmospheric longwave model; model OLD₀

Resumo

A irradiância descendente de onda longa DLF é uma das principais componentes do balanço de radiação à superfície (SRB, Surface Radiation Balance); porém, sua medição não é frequente. A cobertura de nuvens modula o seu comportamento e o fluxo de céu claro DLF₀ tem peso predominante no valor final de DLF. O presente trabalho mostra que, nas atmosferas tropical e de latitudes médias, DLF₀ pode ser representado como as contribuições em três intervalos espectrais específicos: R1 (λ < 7.5 µm), R2 (7.5 a 14 µm) e R3 (λ > 14 µm). R1 e R3 são descritos em boa aproximação pelo espectro de corpo negro à temperatura de abrigo (T_scr), enquanto R2 possui emissividade média que depende principalmente da água precipitável (w). Apresenta-se um modelo simples e fisicamente consistente (OLD₀ no que segue), apropriado para estimar DLF₀ à superfície. A validação de OLD₀ com dados de superfície (21 estações em escala mundial) mostra acurácia com viés MBE inferior a 6 W.m^-2 e dispersão (desvio padrão STD) menor que 12 W.m^-2 para valores típicos DLF₀ ~ 300-400 W.m^-2 compatíveis com a precisão dos pirgeômetros). O algoritmo proposto mostra desempenho superior ao dos usuais, exibindo uma média dos desvios absolutos MAE ~ 2.8 W.m^-2. Em contraste, outros algoritmos frequentemente usados exibem MAE entre 8.1 e 15.9 W.m^-2.

Palavras-chave
radiação de onda longa descendente; radiação de céu claro; modelo de onda longa atmosférica; modelo OLD₀

1. Introduction

Surface radiation balance (hereafter SRB) is the main component in energy exchange between ground and atmosphere, having direct impact on evapotranspiration of vegetated surfaces, on natural environment and ultimately on regional climate. Downward long-wave component DLF is much less documented than solarimetric fluxes, being measured mainly in specialized networks like worldwide Baseline Surface Radiation Network (Zhang et al., 2015), Atmospheric Radiation Measurement (ARM) Climate Research Facility of the United States Department of Energy and SURFRAD (Augustine et al., 2000), both in USA, or SONDA in Brazil. Satellite-based estimations have become the clue for high-resolution global-scale mapping. Such estimations have been undertaken by programs like NASA-GEWEX SRB project, based on massive satellite data sets along the last 25 years; also, estimations over Meteosat monitoring area are performed since 2005 (Schulz et al., 2009; Trigo et al., 2010).

DLF can be thought in terms of clear-sky flux DLF₀ corrected by cloud influence. A number of published algorithms assess DLF₀ using ground-based meteorological data; corrections for cloud influence use to be based on empirical expressions that depend on cloud cover registered by observer, or on shortwave clearness indexes (Niemelä et al., 2001; Flerchinger et al., 2009). For estimations of DLF by Numerical Circulation Models as well as by satellite-based monitoring, mean values over a limited region and short-time interval are given by (Schmetz et al., 1986; Gupta, 1989):

(1) $$DLF=\left(1-Ac\right)DL{F}_0+AcDLFc=DL{F}_0+\frac{\partial DLF}{\partial A}Ac$$

Here, Ac represents the cloud cover fraction, while DLF₀ and DLF_c denote the downward longwave flux for clear and overcast skies, respectively. The partial derivative depends on difference between DLF_c and DLF₀ and allows evaluation of cloudiness effects. Schmetz (1969) reported that for clear-sky conditions, the layers above 500 m contribute to DLF₀ in 20% (midlatitude winter) and 16% (tropical atmosphere). Considering cloud base at 2 km, cloud contribution in overcast conditions represents one third of DLF in mid-latitude winter and only 8% in tropical environment. While clouds can significantly influence local dynamic and thermodynamic processes, downward longwave flux from the atmosphere, primarily driven by DLF₀, remains the dominant component in Eq. (1). This paper will therefore concentrate on estimating DLF₀. The overcast flux (DLF_c) will be addressed in a further study. DLF_c uses to be assessed as a blackbody total irradiance at near-surface air temperature T_a, corrected by an emissivity ɛ_a dependent on atmospheric conditions. Various currently used algorithms claim to estimate atmospheric emissivity ɛ_a as a function of near ground temperature T_a and/or local humidity represented by near-ground water vapor pressure e_a. Their foundations have been increasingly complex in time. Earlier approach by Brunt (1932) proposed ɛ_a to be a function of square root of vapor pressure (√e_a); he found reasonable empirical linear fit with measures at various sites as Benson (England), Upsala (Sweden), Algeria and France. Swinbank (1963) showed that ɛ_a deduced from Benson data would be proportional to T_a² rather than dependent on humidity, assumption confirmed for Aspendale and Kerang (Australia), and equatorial Indian Ocean. Similar fitting behavior was reported for northern Germany (Schmetz et al., 1986). Both models were essentially empirical although justified by some physical reasoning. Brutsaert (1975) integrated Schwartzschild's equation (Liou, 1980) using a profile similar to US standard atmosphere (exponential decrease of pressure, temperature and vapor density) and wideband emissivity of a H₂O + CO₂ slab (fitted as potential function of vertical optical path). This procedure leads to an emissivity proportional to a power of near-surface vapor density, ɛ_a ∝ (e_a/T_a)^1/7. Prata (1996) pointed out that CO₂ effect would not be properly described in drier atmospheres and proposed an ad-hoc algorithm for waveband emissivity of the same H₂O + CO₂ slab, leading to a more complex expression of ɛ_a (a function of precipitable water w parameterized through variables e_a, T_a). In order to validate results, DLF₀ was estimated by Prata's formula and by LOWTRAN7 processing of a set of measured (radiosonde) and standard TIGR atmospheric profiles (Chedin et al., 1985) all over the world. Last but not least, Dilley and O'Brien (1998) considered Schwarzschild equation and its solution for monochromatic diffuse irradiance. This one is an exponential integral function E(τ) (Liou, 1980) with τ being a “grey optical depth” assumed as linear combination of T_a and √w. Proper coefficients are found by fitting DLFo estimation to results of spectral integration of Schwarzschild equation using actual τ(λ) provided by LOWTRAN7 and TIGR profiles. Performance of all models is in general satisfactory, but it is recommended their improvement by adaptation of coefficients to local DLF₀ measurements (e.g. Flerchinger et al., 2009; Kruk et al., 2010) algorithms in order to perform fitting between multispectral signal and surface measured fluxes (see Lee and Ellingson, 2002; Tang and Li, 2008; Wang and Liang, 2009; Wu et al., 2012; Masiri et al., 2017). On the other hand, Gupta et al. (1992) developed an algorithm that uses satellite-based atmospheric retrieval and a spectral radiative code; integrated DLF₀ is fitted as a function of precipitable water and temperature at three atmospheric levels. It has been tested with a high number of profiles and seems applicable in any situation, being applied to worldwide DLF estimation by NASA-GEWEX SRB project (Zhang et al., 2015).

It is desirable to have a DLF₀ physical (universal) model which can be easily applied to ground-based data as well as to satellite information. Complex or not, usual models need to define proper coefficients by empirical fit of model to local observed values (measured or computed) considering integral values for DLF₀. In this paper we show that a closer analysis of DLF₀ spectral behavior allows substantial simplifications. In section 2 we deduce a model numerically simple, physically robust and explicitly based on only two atmospheric parameters: near-ground temperature T_a and precipitable water w. The limiting condition is a planetary boundary layer with water column of at least 1 g.cm^-2 (10 kg.m^-2). Section 3 describes the data sets and criterions used for validation. Section 4 shows model performance for a worldwide set of 21 sites and comparison with five usual estimators. Proposed DLF₀ estimator shows good performance within temperate, semiarid and tropical climates, and (as far as possible) does not need fitting to local atmospheric profile.

2. Clear-Sky Model and Parameterization for DLF₀

Atmosphere as a whole is not a body in thermodynamic and radiative equilibrium. For practical purposes, current algorithms estimate DLF₀ in terms of surface meteorological variables through a Stefan-Bolzmann-like expression

(2) $$DL{F}_0={\varepsilon }_a\sigma T_a^4$$

Here T_a is near ground air temperature (compatible with screen temperature T_scr conventionally obtained at a meteorological station, so that T_a = T_scr), σT_a⁴ the corresponding blackbody emittance in infrared spectral interval (in practice, 4-100 µm) and ɛ_a is an effective atmospheric emissivity which depends on temperature and atmospheric profiles within height interval z:(0, ∞). The main source of DLF₀ is emission from water vapor (H₂O), carbon dioxide (CO₂) and ozone (O₃). Its main spectral characteristics were early depicted by Simpson in 1920 decade (Johnson, 1954, chapter 4) who showed that both H₂O and CO₂ strongly absorb (hence also emit) infrared thermal radiation throughout IR spectrum, except for an interval known as “atmospheric window”. Within this window, additional (weak) radiative flux is due to the 9.6 μm O₃ band and to H₂O continuum. Kondratyev (1969), Idso (1981), Schmetz (1989), Niemelä et al. (2001) and Cucumo et al. (2006), among others, have also described this spectral characteristic. A parameterization of atmospheric emissivity ɛ_a refers to a weighted value of ɛ(λ) over the infrared spectrum. It is currently obtained by computing total DLF₀ for a standard atmosphere or by parameterizing its fit to results of a bunch of profiles (TIGR collection, for instance). Subsequently, it has been usual to “correct” parameters by comparison with local measurements of DLF₀. These bulk results do not clearly show the relationship between ɛ(λ), the infrared spectrum and the atmospheric profile.

In order to better understand DLF₀ spectral structure, we used the radiative code SBDART (“Santa Barbara DISORT Atmospheric Radiative Transfer”, Ricchiazzi et al., 1998). It is based on LOWTRAN 7 radiation database and has 20 cm^-1 spectral resolution; in its standard mode it uses 1 km layers for atmospheric profiles but better vertical resolution is allowed. SBDART standard profiles are close (but not identical) to the ones published by McClatchey et al. (1972). They include several minor gases; CO₂ concentration is 360 ppm.

Firstly, a diagnostic test was performed considering potential contribution of single gases in a tropical atmosphere. Figure 1 shows the results for six gaseous components (CO₂, O₃, CH₄, N₂O, O₂, H₂O) as well as the overall spectrum of DLF₀. There is a strong downward flux due to CO₂ in the 14 μm band as well as a weaker flux in 4.8 and 5.2 μm bands, and a moderately intense pair of bands at 10.4 and 9.4 µm. Ozone (O₃) exhibits irradiance in the bands of 4.75, 9.6 (most intense) and 14.3 μm. Minor gases like CH₄ and N₂O, as well as O₂, show somewhat intense bands between 6 and 8 μm and also at 17 μm. Concerning the overall effect, the most relevant aspect is water vapor contribution to DLF₀. Indeed, H₂O spectral flux itself closely describes the spectrum of downward long-wave radiation for λ < 7.5 μm and λ > 14 μm. The intermediate region λ: (7.5-14 μm) shows predominance of the water vapor continuum, as well as the presence of O₃ emission from the upper atmosphere. Thence, we recognize three well defined spectral regions (hereafter named R1, R2 and R3). Closer inspection of Fig. 1 makes evident that: 1) the global effect of CO₂, O₂, N₂O and CH₄ bands overlapping H₂O band in region R1 (λ < 7.5 μm) yields a spectrum close to that of H₂O alone, with some influence of the weak CO₂ 4.8 μm band; 2) R3 region (λ > 14 μm) closely follows H₂O spectrum except for weak influence of CO₂ 14 μm band; 3) R2 [7.5-14 μm] is mainly described by H₂O plus O₃ properties only.

Figure 1
Spectral contribution of various gases to ground level DLF₀ (labeled “DLW”). Atmospheric profile: Tropical atmosphere used in SBDART. “Others” refers to CH₄, N₂O, O₂ footprints.

A second fact is highlighted by spectral DLF₀ in different atmospheres. Figure 2 illustrates SBDART application for tropical, mid-latitude summer and subarctic summer profiles. Brightness temperature TB(λ) in lieu of spectral irradiance E_λ is shown. The variable TB corresponds to a blackbody emitting the same irradiance E_λ at wavelength λ. We find the same spectral intervals R1, R2, R3 for all atmospheres; in particular, it is seen that regions 1 and 3 closely follow a blackbody spectrum with temperature of the lowest layer. However, this apparent behavior could be induced by the unrealistic physics of this layer (described as 1 km thick, homogeneous and isothermal).

Figure 2
Spectrum of brightness temperature at ground level for three model atmospheres used in SBDART: Tropical atmosphere, Mid-latitude Summer and Subarctic Summer.

In order to clarify lowest layers behavior, SBDART was applied to several detailed atmospheric profiles. Figure 3 illustrates results for a real sounding case (pressure, temperature, humidity at Santa Maria site, in Southern Brazil). Layers had vertical resolution of 50-70 m within the first kilometer and 500 m for upper altitudes. Curves represent spectral DLF₀ at ground level (z = 0) in terms of brightness temperature TB(λ), for atmospheric profile on 16 November 2012, 1200 UTC. Temperature profile was conserved, but atmosphere was “dried” (setting null humidity) for layers above z = 180, 360, 550, 960 and 2700 m (obtaining five TB spectra). Legend indicates the water column (in g.cm^-2) resulting from each drying process. The total precipitable water is 2.03 g.cm^-2; calculations for z > 2700 m has shown that radiative contributions of upper layers do not increase the near-surface R2 spectrum. Concentration of other gases follows SDBART profile for a tropical atmosphere. It is seen that spectral TB (thus E_λ) increases with increasing water content and converges to a constant value TB₀ for regions R1 and R3. Within R1, TB₀ is virtually reached by a layer of about 360 m thickness (except for a persistent footprint of CO₂ at 4.75 μm). R3 needs a 960 m thick layer in order to reach TB₀ within the interval 16-25 μm.

Figure 3
DLF spectra at ground level obtained by considering successive H₂O layers within the atmospheric profile at Santa Maria, Rio Grande do Sul, sounding at 16 November 2012 by CHUVA project. Legend describes integrated water columns and total precipitable water, in g.cm^-2. Geometrical depths are 180, 360, 550, 960 and 2700 m. Vertical profile of other atmospheric gases remained unchanged. SI units: 1 g.cm^-2 = 10 kg.m^-2.

Within intervals 5-7 μm, 14-16 μm and beyond λ = 26 μm, the strong absorption allows for constant TB₀ even for thicknesses lower than 150 m; thus, a water column of 1 g.m^-2 is enough to reach a TB spectrum close to constant TB₀. It is also seen that irradiance within R2 is strongly dependent on precipitable water. The remarkable point is that TB₀ value is very close to thermodynamic temperature of the first layer (in this case, T_a = 293 K), which in turn should be close to screen temperature T_scr. Santa Maria is located in Southern Brazil (with warm temperature and fully humid climate according to Köppen classification); profiles of Amazonian region (with higher water vapor concentrations) should reinforce these characteristics.

It is concluded that DLF₀ can be partitioned in three well defined spectral intervals: R1: (λ ≤ 7.5 μm); R2:(7.5 < λ ≤ 14 μm); R3:(λ > 14 μm). At least for mid-latitude and tropical atmospheres with precipitable water column w ≥ 1 g.cm^-2, regions R1 and R3 are fitted by a blackbody spectrum with temperature T_a representative of surface boundary layer (expectedly, screen value T_scr). Irradiance in region R2 is produced by H₂O continuum and O₃ absorption/emission (with slight, if any, contribution of CO₂ bands) and seems mainly relied to variations of precipitable water. These conclusions suggest a model assessing DLF₀ irradiance by simply adding contributions of regions R1, R2, R3. Regions R1 to R3 correspond to fractions f₁, f₂, f₃ (f₁ + f₂ + f₃ = 1) of blackbody radiation at temperature T_a, with spectral region R2 having an emissivity ɛ₂ < 1, such that

(3) $$DL{F}_0=f_1\sigma T_a^4+E_2+f_3\sigma T_a^4=\left(f_1+f_2{\varepsilon }_2+f_3\right)\sigma T_a^4,$$

(4) $$DL{F}_0=\left[1-f_2\left(1-{\varepsilon }_2\right)\right]\sigma T_a^4 = {\varepsilon }_a\sigma T_a^4.$$

Equations (3) and (4) provide the basic model for estimating atmospheric emissivity ɛ_a. Hereafter the model will be labeled OLD₀ (Onda Longa Descendente = Downward Longwave, in Portuguese). It depends on fraction f₂ (which is dependent on T_a) and on mean emissivity in region R2; this last is related to vertical profile of water vapor and temperature, but essentially should be a function of precipitable water w and at a less extent of stratospheric O₃ contribution (better transmitted in dryer atmospheres). As a matter of fact, when w > 1 g.cm^-2 the emission by CO₂ and other gases do not actually contribute to DLF₀ within R1 and R3 spectral regions. In such atmospheres, the key for DLF₀ assessment by satellite sensors would depend essentially on w assessment and on proper retrieval of near-ground temperature.

2.1. Model parameterization

Fraction f₂(T) refers to a blackbody spectrum and is easily estimated by integration of Planck's function πB(λ,T) within interval R2: (7.5-14 μm), divided by σT⁴. Earth-atmosphere temperatures lie mostly in interval T: (183-353 K), within which f₂ is fairly well fitted by a second-degree polynomial (determination coefficient r² = 0.9981):

(5) $$f_2=-0.72+0.65\left(\frac{T}{100}\right)-0.092{\left(\frac{T}{100}\right)}^2$$

Cucumo et al. (2006) found similar result assuming an atmospheric window δλ: (8-14 μm). Concerning emissivity in R2, four Brazilian stations located in different climates were considered: Porto Alegre (POA, Köppen class Cfa), Brasilia (BRB, class Aw), Manaus (MNS, class Af) and Petrolina (PTR, class Bsh). Their location and climate characteristics are described in Tables 1 and 2. Spectral fluxes in region R2 were assessed applying SBDART to 243 atmospheric profiles during year 2006 (RAOB stations in Table 1) and divided by irradiance f₂ σ T_a⁴, assuming the lower-level temperature value T_RAOB as proxy of T_a. Lower layers of atmospheric profiles have poor resolution (about 500 m) but allow reliable w estimation; in addition, DLF₀ spectrum within R2 is smoothly w-dependent so that downward R2-fluxes may be described with simple algorithms. Figure 4 illustrates results for atmospheric conditions corresponding to a wide variety of situations (winter and summer in four different climates, covering water columns from 0.5 to 5.5 g.cm^-2). A polynomial fit provides the expression (w in g.cm^-2)

(6) $${\varepsilon }_2=0.211+0.138w-0.0075w^2$$

with standard deviation lower than 0.02 for difference δɛ between calculated and fitted values. The close concentration of points along function described by Eq. (6) suggests a rather weak dependence on temperature and water vapor profile. Hereafter, it will be assumed close coherence of ɛ₂ with water column only.

Table 1
Locations in Brazil and abroad providing surface and atmospheric data for this paper. RAOB stations do not provide DLF. G: solar irradiance. T_scr (station), T_rad (radiosonde): near ground temperature. Providers in Data Source: see Acknowledgements section.

Site	lat/lon	altitude	Data Source	period	Köppen
Model Parameterization (R2 emissivity) - data: Trad, profiles, w
Santa Maria (BR)	29.70S; 53.70W	95	Radiosonde	2006	Cfa
Porto Alegre (BR)	30.05S; 51.44W	47	Radiosonde	2006	Cfa
Brasília (BR)	15.60S; 47.71W	1023	Radiosonde	2006	Aw
Manaus (BR)	2.00S; 60.00W	93	Radiosonde	2006	Af
Petrolina (BR)	9.06S; 40.31W	387	Radiosonde	2006	BSh
Model Validation - data: 1) Tscr, G, DLF; 2) Trad, profiles, w
01 MAN - Manacapuru (BR)	3.28S; 60.61W	60	CHUVA1,2	2014/2015	Af
02 SPQ - Santa Rita do P. Q. (BR)	21.64S; 47.79W	752	EMBRAPA1	2005	Aw
03 RIO - Rio de Janeiro (BR)	22.90S; 43.20W	10	UFRJ1,2	2012	Am
04 CTB - Curitiba (BR)	25.49S; 49.33W	891	SONDA1,2	2015	Cfb
05 FLN - Florianópolis (BR)	27.60S; 48.51W	31	SONDA1,2	2014	Cfa
06 BRB - Brasília (BR)	15.60S; 47.71W	1023	SONDA1,2	2014	Aw
07 PTR - Petrolina (BR)	9.06S; 40.31W	387	SONDA1,2	2015	BSh
08 PAY - Payerne (Switzerland)	46.81N; 6.94E	491	BSRN1,2	2016	Cfb
09 TAT - Tateno (Japan)	36.05N; 140.12E	25	BSRN1,2	2017	Cfa
10 DAA - De Aar (South Africa)	30.66S; 23.99E	1287	BSRN1,2	2004	BSk
11 MOM - Momote (Papua N. Guinea)	2.06S; 147.42E	6	BSRN1,2	2011	Af
12 SBO - Sede Boqer (Israel)	30.85N; 34.77E	500	BSRN1,2	2011	BSh
13 CNR - Cener (Spain)	42.81N; 1.60W	471	BSRN1,2	2014	Cfb
14 CAR - Carpentras (France)	44.08N; 5.05E	100	BSRN1,2	2017	Cfb
15 PAL - Palaiseau (France)	48.71N; 2.20E	156	BSRN1,2	2016	Cfb
16 NIA - Niamey (Niger)	13.63N; 2.30E	223	ARM1,2	2006	Bsh
17 GER - Black Forest (Germany)	48.27N; 8.18E	511	ARM1,2	2007	Cfb
18 PEN - Penn State (USA)	40.72N; 77.93W	376	SURFRAD1,2	2018	Dfb
19 FOR - Fort Peck (USA)	48.30N; 105.1W	634	SURFRAD1,2	2018	BSk
20 DES - Desert Rock (USA)	36.62N; 116.01W	1007	SURFRAD1,2	2018	BWk
21 BON - Bondville (USA)	40.05N; 88.37W	230	SURFRAD1,2	2018	Dfa
22 GWC - Goodwin Creek (USA)	34.25N; 89.87W	98	SURFRAD1,2	2012	Cfa

Table 2
Basic key for definition of main Köppen climates. Adapted from Kottek et al. (2006). More details in: Alvares et al. (2014). Tmin = coldest month temperature. Rainfall: Rann = annual, Rmin = in driest month, Rth = variable threshold, Rsmin = min in summer, Rsmax = max in summer, Rwmin = min in winter, Rwmax = max in winter.

Code	Climate	Criterion
A	Tropical climate	Tmin ≥ +18 °C
Af	Rainforest, fully humid	Rmin ≥ 60 mm
Am	Monsoon	Rann ≥ 25*(100-Rmin)
As	Savannah, dry summer	Rmin < 60 mm in summer
Aw	Savannah, dry winter	Rmin < 60 mm in winter
B	Arid climate	R ann < 10 Rth
BS	Steppe	Rann > 5 Rth
BW	Desert	Rann ≤ 5 Rth
C	Warm temperature climate	-3 °C < T min < +18 °C
Cs	With dry summer	Rsmin < Rwmin, Rwmax > 3*Rsmin (Rsmin < 40)
Cw	With dry winter	Rwmin < Rsmin, Rsmax > 10*Rwmin
Cf	Fully humid	Neither Cs nor Cw
D	Snow climate	T min ≤ -3 °C
Ds	Dry summer	Same as Cs
Dw	Dry winter	Same as Cw
Df	Fully humid	Neither Ds nor Dw

Figure 4
Mean emissivity ɛ₂ as a function of precipitable water. SBDART was applied to 243 profiles over four sites in 2006 (RAOB - Radiosonde Observation stations in Table 1).

3. Data Sets and Procedures for Model Validation

Five sounding stations contributed to model development (see Figs. 3 and 4). Surface data and sounding profiles of 22 sites (Brazil and abroad) were used for validation.

Table 1 summarizes information about data sources, including Köppen climatic classification (Kottek et al., 2006). Geographical distribution is shown in Fig. 5, illustrating the global scale of stations data. Table 2 shortly describes Köppen criterion for the first two characters, related to temperature and rainfall; the third character in climatic label includes additional details described in Alvares et al. (2014). Brazilian stations cover Tropical, Arid and Warm temperate climates, and so do the foreign stations (exception: US stations Penn State and Bondville are classified as Snow, colder climate). It is interesting to note that geographical distribution of stations within Brazil implies dimensions as large as Lisbon-Moscow distance in Europe, displaying multiple climates except Snow and Desert.

Figure 5
Geographical distribution of 25 stations considered in this paper.

The radiosonde data (RAOB) are collected at Altitude Meteorological Stations maintained by the Department of Air Space Control (EMA - DECEA). Only the 12Z time slot was used in this work.

Daily cycle of solar radiation helped to choose clear-sky situations, as described below in this section. Brazilian SONDA radiometric network has information on DLF and solar radiation observed at the surface from a PIR model pygeometer (Eppley) and a CM11 model pyranometer (Kipp & Zonen), respectively. The quality control applied to the data of these stations is the same as for BSRN (Baseline Solar Radiation Network).

Manacapuru station is located at about 100 km from Manaus-AM. DLF and solar radiation data were collected as part of the GoAmazon 2014/5 campaign (Green Ocean Amazon, Martin et al., 2016) and are available on the ARM website. ARM obtains information on solar and thermal radiation through a platform called SKYRAD, measuring the downward components of radiation every minute. This platform measures solar radiation with a PSP (Precision Spectral Pyranometer) with Eppley Model V1 ventilator, and DLF with a PIR (Precision Infrared Radiometer) with Kipp & Zonen Model 2-AP automatic solar tracker and Eppley Model V1 ventilator (Andreas et al., 2018).

Santa Rita do Passa Quatro station was located in a sugarcane plantation area, supported by EMBRAPA (Brazilian Agricultural Research Corporation). Global solar radiation data was obtained by a CM6B pyranometer (Kipp & Zonen) and long-wave radiation was measured by a CG1 model pygeometer (Kipp & Zonen). Data are half-hour means.

SURFRAD stations (Surface Radiation Budget Observation network, Augustine et al., 2000) are also part of the BSRN project. The network uses an Eppley PSP pyranometer for solar radiation and a PIR (Eppley) model pyrgeometer for DLF.

Variables T_scr, DLF and G had 1-min frequency (excepting Santa Rita do Passa Quatro). Concerning precipitable water, CHUVA, SURFRAD and ARM sites provided also sounding profiles; for other locations, sounding data were obtained at the University of Wyoming site. For Santa Rita do Passa Quatro (with no closer radiosonde launching), precipitable water was estimated following Prata (1996) algorithm. CHUVA, SURFRAD and ARM exhibit soundings, T_scr e DLF together at the same experimental site. In case of multiple daily soundings, 12 UTC profile was chosen.

Concerning validation of the model by comparison with “ground truth”, it is worthwhile to assess its dependence on atmospheric variables. Eqs. (5) and (6) provide the parameters f₂ and ɛ₂ for estimating DLF₀ with the model OLD₀ (Eq. (4)). Assuming that fluctuations δT and δw are not too high during a time interval and measures are placed not too far between them, Eq. (6) shows that relative fluctuations of DLF₀ (at constant T or w) for OLD₀ algorithm are

(7) $$\left(\delta DL{F}_0/DL{F}_0\right){\left.\hspace{0.17em}\right|}_w=\left(\partial {\varepsilon }_a/\partial f_2.d{f}_2/dT\right).\delta T/{\varepsilon }_a+4\delta T/T=\beta \delta T,$$

(8) $$\left(\delta DL{F}_0/DL{F}_0\right){\left.\hspace{0.17em}\right|}_T=\left(\partial {\varepsilon }_a/\partial w\right).\delta w/{\varepsilon }_a=\gamma \delta w.$$

Deviations may be associated to uncertainty in definition of a “near surface temperature” T_a and “local column of water vapor” w obtained from radiosonde or satellite retrieval. For typical values w = 2.5 g.cm^-2 and T = 298 K we have DLF₀ (OLD₀) = 340 W.m^-2. From Eqs. (7) and (8) we find β = -0.00054 + 0.0134, showing that the second term of sum in Eq. (7) is largely predominant. Thus, deviations of T_a estimates with average <δT> = +1 K imply in DLF₀ overestimation of about 1-2%. On the other hand, contribution of precipitable water exhibits γ = +0.055: thence, δw ~ +0.5 g.cm^-2 induces overestimation of about 3%. This is a delicate matter, since daily changes of water column might be large. Nevertheless, pyrgeometer may show about 5% error in DLF measurements (Wang and Dickinson, 2013), be 10-15 W.m^-2. These facts suggest that errors inherent to OLD₀ model may be of the same order than uncertainty of surface instruments themselves. The actual issue in DLF₀ data seems to be a reliable diagnosis of clear-sky situation (in order to eliminate cloud influence). We used simultaneous solar irradiance G for helping in this matter (thus, OLD₀ validation was constrained to daytime intervals).

OLD₀ algorithm was validated against surface data of 22 stations described in Table 1 and Fig. 4. Daytime data included at least one atmospheric profile. Precipitable water w for 12 UTC profile was assumed for the whole day. It was assumed T_a (model) = T_scr (station screen measure). In order to perform better comparisons between OLD₀ and DLF ground data, a suitable day should present (during a long-time interval) situations as far as possible compatible with clear-sky situations. Solar irradiance G was used as indicator of such situations. All 22 stations furnished one-minute G(t) data, and a clear-sky model G₀(t) (Ceballos, 2000) was considered as reference. The fluctuating difference dG(t) = G - G₀ was smoothed with an 11-min moving average DG(t). The moving standard deviation SG(t) over eleven data was also calculated. The choice of days and validation intervals followed a “Clear Longwave Criterion” (hereafter CLW₀). It required (1) G(t) > 50 W.m^-2 (instants not too close to start or end of daily period); (2) deviation |dG| < 50 W.m^-2 (in order to prevent larger deviations from model G₀ associated to aerosol or clouds), together with (3) SG(t) < 4 W.m^-2 (indicating small fluctuations in short time intervals). At list in principle, condition (3) prevents cumulus presence. Finally, (4) conditions 1 to 3 should be observed during more than 240 min (4 h) in that day (this time interval helps in the choice of a predominantly clear-sky day). CLW₀ limits are empirical, subject to reformulation according to available data and/or atmospheric modelling which include, for instance, cirrus effects.

Figure 6 illustrates results for one day in Brasilia, with more than 240 min required by CLW₀ criterion. The left figure includes the ratio GG = G/G₀ (red/black points). This fraction must tend to values close to unity for clear-sky conditions. We see that CLW₀ situations (black points) happen during a rather long interval, with smooth evolution of shortwave flux G. On the right, black and blue boxes show CLW₀ limits for differences G = G - G₀ (orange for original values and black for 11-sized moving average) and the moving average of standard deviation, S(t) (11-sized sample, in blue). Green box defines a bound |DLF - OLD₀| < 30 W.m^-2. During the “suitable interval”, deviation of measured DLF from DLF₀ (OLD₀) amounts only a few W.m^-2. In the following, the CLW₀ criterion holds for defining days with “clear-sky working conditions” for data collection for OLD₀ validation.

Figure 6
(left) Daily evolution of relevant parameters at Brasilia, 01 July 2009: temperature (10*t °C, magenta), solar irradiances G (measured, black) and G₀ (clear-sky estimated, orange), longwave fluxes DLF (measured) and estimated DLF₀ (OLD₀) in green. Ratio 1000*G/G₀ (red + black) is illustrative as clearness index; superimposed black points indicate CLW₀ situations. (right) Differences between measurements and models: OLD₀ - DLF (green), δG = G₀ - G (orange); moving average of δG +50 (black); moving standard deviation of δG + 100 (blue). CLW₀ criterion tested during 10 h (within 12 UTC ± 5). Green box bounds differences |DLF-OLD₀| < 30.

Given the set of deviations δE between model and measurements in a set of N clear-sky events, model performance was evaluated by four parameters: mean deviation (bias) MBE, standard deviation STD, mean squared error RMSE and mean absolute deviation MAE, calculated as follows

(9) $$\delta E_i={\left(DL{F}_0-DLF\right)}_i,i=\mathrm{1,2},\dots ,N$$

(10) $$MBE=\left(1/N\right)\sum _i\delta E_i=<\delta E>$$

(11) $$STD={\left[{\left(N-1\right)}^{-1} \sum _i{\left(\delta E_i-MBE\right)}^2\right]}^{1/2}$$

(12) $$RMSE={\left[N^{-1} \sum _i\delta E_i^2\right]}^{1/2}=<\delta E^2>$$

(13) $$MAE=\left(1/N\right)\sum _i\left|\delta E_i\right|=<\left|\delta E\right|>$$

Note that for a moderately high number N it is RMSE² ≈ MBE² + STD².

4. Performance of Model OLD₀: Results and Discussion

Figure 7 illustrates some results of OLD₀ for Brazilian locations using instant T_scr and 12 GMT precipitable water values (provided by radiosonde). Hundreds and even thousands of instants were selected during one year following CLW₀ criterion (see Table 3). They are close to 1:1 relationship with measured fluxes. Better estimations are obtained for instants closer to sounding routine (12 UTC), presumably due to instantly better w estimates. Points shifted towards higher measured values may be due to cloudy conditions not detected by CLW₀ criterion (for instance persistent cirrus, or even fair weather cumulus fields) which enhance DLF fluxes. From Eqs. (7-8), we could expect errors of about 4-5% associated to δT = 1 °C and δw = 0.05 g.cm^-2, that is about 15 W.m^-2; observed deviations δE = DLF₀(OLD₀) - DLF₀(measure) are strongly bounded by about ±20 W.m^-2. For the metropolitan region of São Paulo, Barbaro et al. (2010) have reported a difference of about +20 W.m^-2 between all-days mean <DLF> and mean <DLF₀>; it can be expected that overcast days present appreciably higher differences.

Figure 7
Application of OLD₀ to four Brazilian sites, based on observed T_scr and w (radiosonde 12 UTC). All detected clear-sky situations in the year (CLW₀ criterion). Red dots correspond to time close to radiosonde schedule (11 to 13 UTC).

Figure 8 illustrates OLD₀ behavior in five countries, all with temperate climate; they include all cases detected by CLW₀ during one year. Figures 7 and 8 suggest a satisfactory performance of OLD₀ model within and outside Brazil. Table 3 resumes performance of OLD₀ model and five current algorithms for 22 worldwide distributed stations. Prata (1996), Brutsaert (1975), Swinbank (1963) and Brunt (1932) are widely mentioned in meteorological applications. Model A of Dilley and O'Brien (1998) was considered.

Figure 8
Comparison of OLD₀ (modeled) against DLF₀ (measured) in stations of Germany, France, USA, Israel, Japan.

Table 3
Performance of OLD₀ and other five algorithms, showing mean deviation (bias MBE) and its spreading (standard deviation STD). Stations details in Table 1. Np: number of clear-sky instants (CLW₀ criterion). Adopted w: 12 UTC sounding.

			OLD 0		Dilley and O'Brien, 1998		Prata, 1996		Brutsaert, 1975		Swinbank, 1966		Brunt, 1932
site	year	Np	MBE	STD	MBE	STD	MBE	STD	MBE	STD	MBE	STD	MBE	STD
02 MAN	2015	34524	2.9	8.1	-1.7	9.6	9.2	10.2	11.7	10.2	2.8	17.8	6.1	10.7
03 BRB	2014	59401	2.6	6.3	-4.7	13.4	13.0	9.4	10.9	9.8	25.6	17.6	-8.4	10
04 PTR	2015	41354	2.0	7.3	-2.9	12.1	9.3	9.0	9.6	8.7	17.1	20.2	-1.6	9.2
05 SPQ	2005	2102	12.9	15.0	6.2	15.3	19.6	15.0	19.0	14.7	23.2	24.7	1.3	14.5
08 RIO	2012	26767	-3.8	12.2	-10.6	18.4	5.0	14.5	6.4	14.9	7.0	8.5	-10.0	15.9
09 CTB	2015	24610	0.4	7.7	3.5	13.0	16.8	11.8	16.0	11.5	20.4	19.2	-2.7	11.4
10 FLN	2014	33395	0.1	7.1	7.0	9.6	19.6	9.5	20.2	9.6	16.5	14.3	3.9	10.3
11 GWC	2018	63338	-1.2	9.0	-16.3	12.8	0.1	9.1	-2.7	9.4	7.1	18.3	-19.3	8.9
12 PAY	2016	8910	1.9	7.1	1.1	9.8	10.6	10.0	7.7	10.8	10.0	16.4	-10.4	10.4
13 TAT	2017	401	10.2	9.6	-7.2	12.2	9.8	9.5	-4.8	12.1	17.0	16.1	-17.3	9.8
14 DAA	2004	11265	-3.7	10.1	-19.7	17.9	-0.3	12.9	-10.1	14.8	12.6	28.3	-25.9	13.1
15 MOM	2011	4585	-1.2	5.7	-9.0	7.3	6.5	7.9	9.9	7.7	-3.0	11.4	-0.1	7.1
16 SBO	2011	6675	0.6	8.8	-9.5	16.2	9.2	11.2	6.6	12.9	21.7	21.8	-11.8	13
17 CNR	2014	50951	5.5	8.8	-0.3	10.7	12.5	10.4	9.4	10.6	16.7	19.9	-9.0	10.3
18 CAR	2017	8893	-0.5	12.1	-11	12.2	0.3	11.8	-6.1	10.9	2.8	16	-22.7	10.9
19 PAL	2016	25806	2.5	7.0	-2.2	9.8	9.4	9.1	6.8	9.6	12.3	16.7	-11.6	9.6
20 NIA	2006	19189	-2.6	9.3	-55.4	41.9	3.3	10.9	-11.4	21.0	57.1	43.5	-24.7	19.9
21 GER	2007	16345	-1.3	6.8	-7.5	10.5	4.4	8.6	1.2	9.0	8.1	17.5	-17.3	8.7
22 PEN	2018	24331	0.3	10.3	-8.3	11.9	6.3	9.9	0.7	12.3	10.0	16.0	-14.8	10.6
23 FOR	2018	49737	0.9	8.9	-13	14.4	3.2	9.8	-4.0	12.4	13.5	23.5	-20.5	10.2
24 DES	2018	98120	-4.7	12.3	-54.7	29.8	-6.5	14.6	-24.4	13.9	41.7	22.3	-38.3	14.7
25 BON	2018	45425	0.3	9.7	-8.7	12.2	4.1	9.9	-1.0	12.5	4.4	16.9	-15.3	10.9
Mean		29824	1.1	9.1	-10.2	14.6	7.5	10.7	3.3	11.8	15.7	19.4	-12.3	11.4
Mean*			1.1	9.1	-5.7	12.5	7.5	10.7	4.6	11.7	13.7	18.3	-11.1	11.2
<|MBE|*>			2.8		11.8		8.1		9.1		15.9		13.3

Table 3 shows observed MEB and STD for various models over 22 sites, their mean values <MEB>, <STD> and also the mean MEB*, STD* obtained by excluding cases outside interval <MEB> ±2 <STD> (denoted by italics in the table). It is seen that performance of Dilley & O'Brien model becomes significantly better; no further analysis will be performed here about outsider cases. There is a fair fit of OLD₀ estimates to measured DLF₀ over a worldwide region (South and North America, Europe, two locations in Asia and three in Africa and Oceania). Bias lies below 6 W.m^-2 (except in two sites) and standard deviation about is 12 W.m^-2 or less; for typical DLF₀≈ 350 W m^-2, they amount 1.7% and less than 3.4% respectively. OLD₀ validation yields bias MBE and spread STD compatible with error expected for (T_a, w) and uncertainty of measured DLF. It is worthwhile to note that, as far as accuracy of validation depends on proper choice of “clear-sky status” situation, CLW₀ criterion appears as a valuable tool for this purpose.

It is seen that all six models have typical STD ≈ 10 W.m^-2 (except Swinbank, which is based on T_scr only). OLD₀ algorithm exhibits better performance on the mean. Dilley & O'Brien, Prata and Brutsaert have similar behavior, with frequent DLF underestimation for the first one and frequent overestimation for the others (expectedly similar). On the mean, Swinbank tends to overestimate and Brunt to underestimate DLF₀ with |MBE| > 10 W.m^-2. It is interesting to note that Brunt shows better performance over Brazil.

The results suggest that the algorithm for window emissivity ɛ₂ (Eq. (8)) is closely representative of an extended set of atmospheres, actually being a function on w (independently of atmospheric profile) when w > 1 g.cm^-2. Such emissivity affects blackbody irradiance in the R2 region, which spectral width depends on temperature T_a. Of course, Eq. (8) is an approximation subject to further improvement, especially for low values of water column.

5. Conclusions

Detailed analysis of downward longwave irradiance for several atmospheric profiles shows that DLF₀ exhibits three distinct spectral behaviors controlled by H₂O absorption/emission. They are: a) R1: (λ < 7.5 μm) shows blackbody behavior with near-surface temperature T_a; b) within an atmospheric window R2:(7.5 < λ < 14 μm), mean emissivity ɛ₂ related to the same blackbody is a function of precipitable water w only; c) R3:(λ > 14 μm) behaves as R1 if w >1 g.cm^-2. This spectral behavior allows to build the simple but physically robust model OLD₀ described by Eqs. (4), (5) and (8). It is a function of near-surface temperature T_a and total precipitable water w, independent of atmospheric profile. The weight of emissivity ɛ₂ is controlled by the R2 spectral width, which depends on blackbody spectrum at temperature T_a.

Concerning proper validation of OLD₀ model against DLF measurements, a “CLW₀ criterion” applied to one-minute shortwave values G(t) allows detection of clear-sky-like events. Comparison with one-year data of a worldwide set of 22 stations (7 in Brazil, 5 in United States, 5 in Europe, 3 in Asia and Oceania, 2 in Africa) showed deviations δE = DLF₀(OLD₀) - DLF₀ (measure) mainly bounded by ±20 W.m^-2 (6% of a typical value DLF₀ = 340 W.m^-2), compatible with radiometer error itself. The averages of 22 mean values MBE and STD were <MBE> = 1.1 W.m^-2 and <STD> = 9.1 W.m^-2 without any adaptation of OLD₀ to local climate. Such low value <MBE> could hide a statistical compensation of high discrepancies with DLF₀ measures; however, the average <|MBE|> amounts 2.8 W.m^-2 only. These results suggest that OLD₀ model is accurate in a large variety of climates provided that w >1 g.cm^-2. DLF₀ estimation using OLD₀ is impacted in 1-2% by errors δT ≈ +1 K and 3% if δw ~ +0.5 g.cm^-2. As far as accuracy also depends on proper choice of “clear-sky status” situation, CLW₀ criterion appears as a valuable tool for diurnal estimations.

In general, behavior of OLD₀ model shows higher stability (lower bias and at least similar STD) when compared with some currently used algorithms. This can be due to explicit use of the variable “precipitable water” in lieu of proxy variables as near surface humidity or vapor pressure. Failing meteorological network data, variables (T_a, w) can be provided by Numerical Circulation Models (Figueroa et al., 2016) or retrieved by proper sensors as MODIS in Terra-Aqua satellites (Menzel et al., 2002) and (expectedly) ABI in GOES-16. Being a simple as well as low-cost algorithm, OLD₀ may be an accurate proxy of DLF₀ in numerical circulation models and SRB estimation.

6. Data Access

Brazilian data: DECEA (Departamento de Controle do Espaço Aéreo, RAOB stations at Manaus, Brasília, Porto Alegre and Santa Maria). CHUVA/GOAMAZON projects (Manacapuru station. SONDA (Brasilia, Petrolina, Curitiba and Florianópolis sites). CT-Hidro Project (Santa Rita do Passa Quatro, site held by EMBRAPA-Meio Ambiente and University of São Paulo). UFRJ (Federal University of Rio de Janeiro, experimental site at Department of Meteorology, LabMiM - Laboratório de Micrometeorologia e Modelagem). Brazilian and worldwide networks: BSRN, ARM, SURFRAD.

Acknowledgements

This work is a product of collaborative activity with Graduate Course of Meteorology at CPTEC-INPE; co-authors were partial or totally granted by fellowships of CNPq (Brazilian National Council for Scientific and Technological Development) and CAPES (Coordination of Superior Level Staff Improvement) during their graduate studies. The authors are also thankful to Dr. Osvaldo Cabral (EMBRAPA - Meio Ambiente) as well as to Dr. Edson Pereira Marques Filho of LabMiM (Laboratório de Micrometeorologia e Modelagem, UFRJ), for kindly providing data of Santa Rita do Passa Quatro and Rio de Janeiro, respectively.

References

• ALVARES, C.A.; STAPE, J.L.; SENTELHAS, P.C.; GONçALVES, J.L.M.; SPAROVEK, G. Köppen's climate classification map for Brazil. Meteorol. Z., v. 22, n. 6, p. 711-728, 2014. doi
  » https://doi.org/10.1127/0941-2948/2013/0507
• ANDREAS, A.; HABTE, A.; REDA, I.; DOORAGHI, M.; KUTCHENTEITER, M.; et al. Solar Infrared Radiation (SKYRAD), Ground Radiation (GNDRAD), and Broadband Radiometer Station (BRS) Instrument Handbook. United States: ARM Climate Research Facility, 2018.
• AUGUSTINE, A.J.; DELUISI, J.J.; LONG, C.N. Surfrad - A national surface radiation budget network for atmospheric research. Bull. Am. Meteorol. Soc., v. 81, n. 10, p. 2341-2357, 2000. doi
  » https://doi.org/10.1175/1520-0477(2000)081%3C2341:SANSRB%3E2.3.CO;2
• BARBARO, E.; OLIVEIRA, A.P.; SOARES, J.; CODATO, G.; FERREIRA, M.J. Observational characterization of the downward atmospheric longwave radiation at the surface in the city of São Paulo. J. of Appl. Meteor. and Climatology, v. 49, n. 12, p. 2574-2590, 2010. doi
  » https://doi.org/10.1175/2010JAMC2304.1
• BRUNT, D. Notes on radiation in the atmosphere. Q. J. R. Meteorol. Soc., v. 58, n. 247, p. 389-420, 1932. doi
  » https://doi.org/10.1002/qj.49705824704
• BRUTSAERT, W. On a derivable formula for long-wave radiation from clear skies. Water Resour. Res., v. 11, n. 5, p. 742-744, 1975. doi
  » https://doi.org/10.1029/WR011i005p00742
• CEBALLOS, J.C. Estimativa de radiação solar à superfície com céu claro: um modelo simplificado. Rev. Bras. Meteorol., v. 15, n. 3, p. 113-122, 2000. doi
  » https://doi.org/10.1590/0102-7786333014
• CHEDIN, A.; SCOTT, N.A.; WAHICHE, C.; MOULINIER, P. The Improved Initialisation Inversion method: A high resolution physical method for temperature retrievals from the TIROS-N series. J. Clim. Appl. Meteorol., v. 24, n. 2, p. 124-143, 1985. doi
  » https://doi.org/10.1175/1520-0450(1985)024%3C0128:TIIIMA%3E2.0.CO;2
• CUCUMO, M.; DE ROSA, A.; MARINELLI, V. Experimental testing of correlations to calculate the atmospheric “transparency window” emissivity coefficient. J. Sol. Energy, v. 80, n. 8, p. 1031-1038, 2006. doi
  » https://doi.org/10.1016/j.solener.2005.06.012
• DILLEY, A.C.; O'BRIEN, D.M. Estimating downward clear sky long-wave irradiance at the surface from screen temperature and precipitable water. Q. J. R. Meteorol. Soc., v. 124, n. 549, p. 1391-1401, 1998. doi
  » https://doi.org/10.1002/qj.49712454903
• FIGUERIA, S.N.; BONATTI, J.P.; KUBOTA, P.Y.; GRELL, G.A.; MORRISON, H.; et al. The Brazilian Global Atmospheric Model (BAM): performance for tropical rainfall forecasting and sensitivity to convective scheme and horizontal resolution. Wea. Forecasting, v. 31, n. 5, p. 1547-1572, 2016. doi
  » https://doi.org/10.1175/WAF-D-16-0062.1
• FLERCHINGER, G.N.; XAIO, W.; MARKS, D.; SAUER, T.J.; YU, Q. Comparison of algorithms for incoming atmospheric long-wave radiation. Water Resour. Res., v. 45, n. 3, p. 1-13, 2009. doi
  » https://doi.org/10.1029/2008WR007394
• GUPTA, S.K. A parameterization for longwave surface radiation from sun-synchronous satellite data. J. Climate, v. 2, n. 4, p. 305-320, 1989. doi
  » https://doi.org/10.1175/1520-0442(1989)002%3C0305:APFLSR%3E2.0.CO;2
• GUPTA, S.K.; DARNELL, W.L.; WILBER, A.C. A parameterization for longwave surface radiation from satellite data: Recent improvements. J. Appl. Meteorol., v. 31, n. 12, p. 1361-1367, 1992. doi
  » https://doi.org/10.1175/1520-0450(1992)031%3C1361:APFLSR%3E2.0.CO;2
• IDSO, S.B. A set of equations for full spectrum and 8-14 µm and 10.5-12.5 µm thermal radiation from cloudless skies. Water Resour. Res., v. 17, n. 2, p. 295-304, 1981. doi
  » https://doi.org/10.1029/WR017i002p00295
• JOHNSON, J.C. Physical Meteorology. Cambridge: Massachusetts Institute of Technology, 1954.
• KONDRATYEV, K.Y.A. Radiation in the Atmosphere. New York: Academic Press, 1969.
• KOTTEK, M.; GRIESER, J.; BECK, C.; RUDOLF, B.; RUBEL, F. World map of the Köppen-Geiger climate classification updated. Meteorol. Z, v. 15, p. 259-263, 2006. doi
  » https://doi.org/10.1127/0941-2948/2006/0130
• KRUK, N.S.; VENDRAME, I.F.; ROCHA, H.R.; CHOU, S.C.; CABRAL, O. Downward longwave radiation estimates for clear and all-sky conditions in the Sertãozinho region of São Paulo, Brazil. Theor. Appl. Climatol., v. 99, p. 115-123, 2010. doi
  » https://doi.org/10.1007/s00704-009-0128-7
• LEE, H-T.; ELLINGSON, R.G. Development of a nonlinear statistical method for estimating the downward longwave radiation at the surface from satellite observations. J. Atmos. Ocean. Technol., v. 19, n. 10, p. 1500-1515, 2002. doi
  » https://doi.org/10.1175/1520-0426(2002)019%3C1500:DOANSM%3E2.0.CO;2
• LIOU, K.N. An Introduction to Atmospheric Radiation. Cambridge: Academic Press, 1980.
• MARTIN, S.T.; ARTAXO, P.; MACHADO, L.A.T.; MANZI, A.O.; SOUZA, R.A.F.; et al. Introduction: Observations and modeling of the green ocean Amazon (GoAmazon2014/5). Atmos. Chem. Phys., v. 16, n. 8, p. 4785-4797, 2016. doi
  » https://doi.org/10.5194/acp-16-4785-2016
• MCCLATCHEY, R.A.; FENN, R.W.; SELBY, J.E.A.; VOLZ, F.E.; GARING, J.S. Optical Properties of the Atmosphere. In: Environmental Research Papers. Washington, D.C.: United States Air Force, 1972.
• MASIRI, I.; JANJAI, S.; NUNEZ, M.; ANUSASANANAN. A technique for mapping downward longwave radiation using satellite and ground-based data in the tropics. Renew. Energy, v. 103, p. 171-179, 2017. doi
  » https://doi.org/10.1016/j.renene.2016.11.018
• MENZEL, W.P.; SANACONDA P.S.W.; LI, J.; GUMLEY, L.E. MODIS Atmospheric Profile Retrieval Algorithm Theoretical Basis Document. Madison: University of Wisconsin-Madison, 2002.
• NIEMELä, S.; RäISäNEN, P.; SAVIJäRVI, H. Comparison of surface radiative flux parameterizations. Part I: Longwave radiation. Atmos. Res., v. 58, n. 1, p. 1-18, 2001. doi
  » https://doi.org/10.1016/S0169-8095(01)00084-9
• PRATA, A.J. A new long-wave formula for estimating downward clear-sky radiation at the surface. Q. J. R. Meteorol. Soc., v. 122, n. 533, p. 1127-1151, 1996. doi
  » https://doi.org/10.1002/qj.49712253306
• RICCHIAZZI, P.; YANG, S.; GAUTIER, C.; SOWLE, D. SBDART: A Research and teaching software tool for plane-parallel radiative transfer in the Earth´s atmosphere. Bull. Am. Meteorol. Soc., v. 79, n. 10, p. 2101-2114, 1998. doi
  » https://doi.org/10.1175/1520-0477(1998)079%3C2101:SARATS%3E2.0.CO;2
• SCHMETZ, J. Towards a surface radiation climatology: Retrieval of downward irradiances from satellites. Atmos. Res., v. 23, n. 3-4, p. 287-321,1989. doi
  » https://doi.org/10.1016/0169-8095(89)90023-9
• SCHMETZ, P.; SCHMETZ, J.; RASCHKE, E. Estimation of daytime downward longwave radiation at the surface from satellite and grid point data. Theor. Appl. Climatol., v. 37, p. 136-149, 1986. doi
  » https://doi.org/10.1007/BF00867847
• SCHULZ, J.; ALBERT, P.; BEHR, H.-D.; CAPRION, D.; DENEKE, H.; et al. Operational climate monitoring from space: the EUMETSAT Satellite Application Facility on Climate Monitoring (CM-SAF), Atmos. Chem. Phys., v. 9, n. 5, p. 1687-1709, 2009. doi
  » https://doi.org/10.5194/acp-9-1687-2009
• SWINBANK, W.C. Long-wave radiation from clear skies. Q.J.R. Meteorol. Soc., v. 89, n. 381, p. 339-348, 1963. doi
  » https://doi.org/10.1002/qj.49708938105
• TANG, B.; LI.; Z-L. Estimation of instantaneous net surface longwave radiation from MODIS cloud-free data. Remote Sens. Environ., v. 112, n. 9, p. 3482-3492, 2008. doi
  » https://doi.org/10.1016/j.rse.2008.04.004
• TRIGO, I.F.; BARROSO, C.; VITERBO, P.; FREITAS, S.C.; MONTEIRO, I.T. Estimation of downward long-wave radiation at the surface combining remotely sense data and NWP data. J. Geophys. Res., v. 115, n. D24118, 2010. doi
  » https://doi.org/10.1029/2010JD013888
• WANG, K.; DICKINSON, R.E. Global atmospheric downward longwave radiation at the surface from ground-based observations, satellite retrievals, and reanalyses. Rev. Geophys., v. 51, n. 2, p. 150-185, 2013. doi
  » https://doi.org/10.1002/rog.20009
• WANG, W.; LIANG, S. Estimation of high-spatial resolution clear-sky longwave downward and net radiation over land surfaces from MODIS data. Remote Sens. Environ., v. 113, n. 4, p. 745-754, 2009. doi
  » https://doi.org/10.1016/j.rse.2008.12.004
• WU, H.; ZHANG, X.; LIANG, S.; YANG, H.; ZHOU, G. Estimation of clear-sky land surface longwave radiation from MODIS data products by merging multiple models. J. Geophys. Res., v. 117, n. D22, p. 1-15, 2012. doi
  » https://doi.org/10.1029/2012JD017567
• ZHANG, T.; STACKHOUSE JR. P.W.; GUPTA, S.K.; COX, S.J.; MIKOVITZ, J.C. The validation of the GEWEX SRB surface longwave flux data products using BSRN measurements. J. Quant. Spectrosc. Radiat. Transfer, v. 150, p. 134-147, 2015. doi
  » https://doi.org/10.1016/j.jqsrt.2014.07.013

Internet Resources

• ARM, https://www.arm.gov/
  » https://www.arm.gov/
• SONDA, http://sonda.ccst.inpe.br/index.html
  » http://sonda.ccst.inpe.br/index.html
• NASA-GEWEX SRB project, http://gewex-srb.larc.nasa.gov/
  » http://gewex-srb.larc.nasa.gov/
• Eppley Model V1 ventilator, https://www.arm.gov/publications/tech_reports/handbooks/sirs_handbook.pdf
  » https://www.arm.gov/publications/tech_reports/handbooks/sirs_handbook.pdf
• BSRN, https://bsrn.awi.de/
  » https://bsrn.awi.de/
• SURFRAD, https://www.esrl.noaa.gov/gmd/grad/surfrad/
  » https://www.esrl.noaa.gov/gmd/grad/surfrad/
• University of Wyoming site, http://weather.uwyo.edu/upperair/sounding.html
  » http://weather.uwyo.edu/upperair/sounding.html

Publication Dates

• Publication in this collection
  27 Jan 2025
• Date of issue
  2025

History

• Received
  02 July 2024
• Accepted
  14 Nov 2024