The study proposes some methods of engineering calculation of outgoing infrared radiation on the basis of the solution of the energy transfer equation for a stationary radiation field in the Earth surface – atmosphere system with allowance for permissible approximations. According to the experimental data, a mathematical model has been developed and verified, which makes it possible to calculate the spectral distributions and integral values of the infrared radiation intensity for a given angular position of the observer outside the atmosphere.

Предложена методика инженерного расчета уходящего инфракрасного излучения на основе решения уравнения переноса энергии для стационарного поля излучения в системе поверхность Земли – атмосфера с учетом допустимых приближений. Разработана и верифицирована по экспериментальным данным математическая модель, позволяющая рассчитывать спектральные распределения и интегральные значения интенсивности инфракрасного излучения для заданного углового положения наблюдателя за пределами атмосферы.

The problem of developing an algorithmic apparatus for engineering calculation of the Earth’s limb infrared (IR) radiation, in particular, radiation of the Earth’s surface – atmosphere system, in order to form background environment when modelling space-based optoelectronic facilities [

When seen from the outer space, the Earth looks like a cold body with the effective temperature of about 255 K, given that the average temperature across the Earth’s surface is 290 K [

Fig. 1. Total averaged 10-year Earth’s energy budget [3]

The absorbing gases in the Earth’s atmosphere are mainly the polar molecules, because it is only the molecules with a constant or induced dipole moment that effectively interact with the electromagnetic field. There also exist spectral regions with low absorption – the so-called atmospheric transparency windows: short-wave window in the wavelength range of 3.5…4.1 µm and longwave window in the range of 8…13 µm (Fig. 2).

Fig. 2. Spectral transmission functions of the entire depth of the atmosphere from its upper boundary to the Earth’s surface, individually for O3, СО2, H2О and the entire atmosphere as a mixture of gases on the whole

The atmosphere components can be divided into constant ones, whose relative abundance is constant up to the altitudes of about 80 km, and variable ones, whose content varies depending on the altitude, temperature, and geographic position. Of the constant components, carbon dioxide CO2 absorbs IR radiation the most intensively [

The assumption applied to engineering calculations that radiation of the Earth’s surface – atmosphere system is supposed to be equal to radiation of a black body is not sufficient for the type of devices under consideration, because it has a substantial spectral ripple [

The objective of this study consists in development of a mathematical model and calculation of the Earth’s surface – atmosphere system radiation in the middle and far IR bands of the electromagnetic spectrum in order to form the initial data in the selected angular direction towards the Earth’s limb.

Radiation intensity variation (radiation intensity is energy contained in a unit wavelength interval within a unit solid angle and passing through a unit area, which is perpendicular to a given direction, per a unit of time) in passing through a medium element is conditioned by two processes: attenuation (decrease in intensity) and emission (increase in intensity) [

Let us take a beam of direction s and analyse a medium element in the form of a cylinder with a unit cross-section, whose axis coincides with the beam direction (Fig. 4). Let the beam be crossing the bases, which are perpendicular to it and lie at distance ds from one another, in points M and M ′ . The radiation intensity in these points is equal to Iλ (M, s) and Iλ (M′, s), respectively.

Fig. 4. Radiation propagation geometry

According to the Bouguer law, in accordance with which linear dependence (in the differential form) of attenuation processes is established depending on the radiation intensity and amount of substance, if the temperature, pressure and content of the latter remain unchanged, any variation in intensity under substance interaction with the radiation field is defined by the sum

where βλ – attenuation coefficient; Jλ – source function; da – amount of substance in the cylinder.

In a general case, the amount of substance da can be determined using different methods. For product βλda to be dimensionless, mass attenuation coefficient βλ is used in the described mathematical model. In so doing, the magnitude of a is calculated as mass per unit area:

a = ρs, where ρ – substance density.

If the infinitesimal increment of the monochromatic optical path be determined as

duλ = −βλ da,

then, substituting duλ in expression (1), we have the equation of radiation transfer in the medium

As mentioned above, the radiation and scattering processes, with account of the introduced approximations, are linear ones, therefore the attenuation coefficient βλ can be represented as the sum of absorption kλ and scattering σλ coefficients:

βλ = kλ + σλ.

For the IR band, we can take into account the absorption and radiation processes only, since the Rayleigh scattering is negligible for long-wave radiation [

Let us assume that the segment of considered medium (atmosphere) is in the state of local thermodynamic equilibrium [

where Bλ (T) – spectral density of radiance within a unit solid angle; – Planck constant; c – light velocity in vacuum; k – Boltzmann constant.

Let us introduce a plane-parallel model of the atmosphere (Fig. 5), for which:

where z – vertical coordinate; θ – zenith angle.

We represent equation (2) as follows:

With account of the introduced approximations, it is further sufficient to consider the upwelling radiation only. Beam directions s form in this case an area of directions (see Fig. 5).

Fig. 5. Plane-parallel atmosphere model

If the variables included in equation (3) be represented as the function of z, an ordinary first-order differential equation can be obtained, whose solution will look as follows:

where C – constant of integration; v, w – integration variables along axis z.

To specify the boundary condition for function at level z = 0, intrinsic heat emission of the Earth’s surface is considered (with no account for the reflected flow of downwelling heat flux), with the temperature T0 :

Here, ελ (θ) – Earth’s surface radiation coefficient.

Let us introduce monochromatic transmission function τλ (θ, z1, z2), which characterises the portion of radiation passed by the atmosphere layer between levels z1 and z2 at angle θ to the vertical plane:

where uλ (z1 , z2 ) – optical thickness at θ = 0.

Thereby,

where ρ(z) – average density of absorbing substance in layer z1 − z2.

Using the transmission function for the total depth of the atmosphere, we have for :

To calculate radiation intensity for a given observer’s position beyond the Earth’s limb, sphericity of the Earth’s atmosphere must be taken into account (Fig. 6):

R – Earth’s radius; h0 – distance from Earth’s surface to the atmosphere layer under consideration, reckoned in the direction towards zenith.

Fig. 6. Diagram of atmosphere splitting into layers with account for its sphericity

Then the transmission function will have Fig. 5. the following view:

where h1, h2 – lower and upper boundary of the layer under consideration.

Further simplification of expression (4) consists in transition from the integral to the algebraic sum of intensities of each layer of the stratified atmosphere [

where N – number of atmosphere layers;

It can be seen from expression (5) that the intensity of IR radiation into the upper hemisphere of directions is the sum of the Earth’s surface radiation intensity attenuated by absorption in the atmosphere, and the intrinsic radiation of each atmosphere layer, also attenuated by absorption in the upper layers.

If the line of sight does not cross with the Earth’s surface, only the right-hand part in formula (5) is used, and layer n = 1 corresponds to altitude h0 (see Fig. 6).

Taken as a whole, the Earth’s surface – atmosphere system is an extremely complex time-dependent thermodynamic system, precise modelling of which for solving this type of engineering problems is impractical. In this respect, the developed model employs thermodynamic parameters of the atmosphere as initial data.

Four atmospheric models are used in the calculations: mid-latitude, Arctic winter and summer, and tropics. The initial data of altitude distributions of partial concentrations of water vapours, ozone, carbon dioxide and other substances, pressure and temperature across the atmosphere were taken from an RFM model [

The values of atmospheric gas absorption spectral coefficients were taken from SPECTRA information system [

As of today, the databases of SPECTRA information system are the most comprehensive and reliable open-data source of information on the parameters of atmospheric gases, recognised by the international scientific community. These parameters were obtained through solving quantum mechanics equations and confirmed by the results of multiple direct measurements using a special-purpose Fourier-transform spectrometer McMath-Pierce (Arizona, U.S.).

There is a number of theoretical models of the Earth’s surface – atmosphere system implemented in the form of special-purpose software [

In scientific literature there are virtually no spectral data in the required wavelength range providing the values of outgoing IR radiation of the Earth’s surface – atmosphere system in the direction towards the Earth’s limb, obtained from direct experimental measurements with the use of special-purpose space vehicles.

Model verification was conducted in two stages. At the first stage, the magnitude of atmosphere’s spectral transmission was calculated at different altitudes in the direction toward the limb (Fig. 7) and the results was compared with values of processed experimental data obtained with the help of MetOp, a special-purpose meteorological satellite (equipped with the IASI instrument). In so doing, appropriate conditions of satellite observations were accepted as the initial data of the atmosphere state [

Fig. 7 (start). Variations in spectral transmission of the atmosphere in the direction towards the limb depending on the viewing altitude: а – MetOp satellite data (IASI instrument) [13]

Fig. 7 (complete). Variations in spectral transmission of the atmosphere in the direction towards the limb depending on the viewing altitude: b – calculation data

At the second stage, for the specified state of the atmosphere, calculation of the spectral density magnitude of the radiation flux from the Earth’s surface – atmosphere system towards the nadir was performed, and the result was compared with the experimental data obtained from the MetOp satellite (IASI instrument) [

Fig. 8. Comparison of radiation spectra: а – as per MetOp satellite data (IASI instrument) [5]; b – as per computational model

Fig. 9. Comparison of radiation spectra: а – as per Nimbus 4 satellite data [20]; b – as per computational model

It can be well seen that radiation of the Earth’s surface – atmosphere system basically repeats the shape of the Planck function curve (dashed lines in Figs. 8, 9) at a certain temperature, but for studying optoelectronic systems it is necessary to account for the atmospheric gas absorption bands.

Analysing the curves, it can be seen that the described model does not fully take into account all the absorption spectral lines present in the experimental data, which restricts the possibilities of its practical application when considering narrow spectral ranges (in the wave number range below ~100 cm–1). As mentioned above, this disadvantage of the model can be eliminated by introducing additional initial data concerning the content of substances in the atmosphere.

The absolute values of radiance spectral density obtained by calculation are slightly different from the experimental data with respect to the results obtained from the Nimbus 4 satellite only (see Fig. 9), which can be explained by the absence of accurate initial temperature data for the atmosphere layers in given geographic region and climatic conditions.

The obtained results show that the accuracy of modelling radiation in the direction towards the Earth’s limb is on the whole sufficient for engineering calculations, even with account of the basic substances only: water vapour, carbon dioxide, and ozone. The absence of multi-component thermodynamic equations in the model, such that consider phase states of substances and various quantum effects, ensures its relative high-speed performance under software implementation.

The developed mathematical model, verified as per the experimental data, makes it possible to calculate both the spectral distribution and the integral value of radiation fluxes for a given spectrum region and specified angular position of the observer beyond the atmosphere limits. The physical limitations put into the mathematical model allow to use it in the wavelength range of 3…30 µm. The calculation accuracies vary depending on the input initial data on the properties of substances and temperature profiles of the atmosphere.

The advantage of this development, as compared with other solutions implemented in resource-intensive and expensive software (see, for example [

The authors declare that there are no conflicts of interest present.