# Modeling of Earth's limb infrared radiation

### Abstract

#### For citation:

Shumov A.V., Sultangulova A.I. Modeling of Earth's limb infrared radiation. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2017;(4):53-62.
https://doi.org/10.38013/2542-0542-2017-4-53-62

## Introduction

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 [1] remains relevant due to intense implementation of semi-realistic simulation technologies in ground-test rigs operating in real time.

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 [2]. This is associated primarily with intense molecular absorption and re-emission of IR radiation in the atmosphere, as well as with scattering on aerosol particles, clouds and precipitation. In this way, in the IR band (unlike the optical band), intrinsic heat emission of the atmosphere produces a substantial effect on the outgoing electromagnetic radiation of the Earth’s surface – atmosphere system (Fig. 1).

**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 O_{3}, СО_{2}, H_{2}О 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 CO_{2} absorbs IR radiation the most intensively [4]. The main variable component of the atmosphere that absorbs IR radiation is water vapour, whose amount rapidly decreases with altitude. Another variable component of the atmosphere, ozone, is present in very small amounts at the sea level. The ozone amount increases with altitude (reaching the maximum at an altitude of about 26–26 km) and decreases again at higher altitudes.

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 [5], which is associated with a large amount of spectral absorption lines for water, carbon dioxide, ozone, etc. In general, we also should consider differences in the radiation intensity depending on geographic observation points, illumination conditions, season, climactic and weather conditions. Fig. 3 shows the night time data as of April, 2012 [6].

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.

## Description of mathematical model

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) [7].

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 I_{λ} (M, s) and I_{λ} (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 β_{λ} – attenuation coefficient;

J_{λ} – 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 β_{λ}da to be dimensionless, mass attenuation coefficient β_{λ} 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

du_{λ} = −β_{λ} *da*,

then, substituting du_{λ} 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 β_{λ} can be represented as the sum of absorption k_{λ} and scattering σ_{λ} coefficients:

β_{λ} = k_{λ} + σ_{λ}.

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 [7]. In a general case, it is reasonable to take into account scattering on aerosols (tiny particles of a solid or liquid substance residing in suspended state in a gaseous medium – the atmosphere). However, taking into account these processes will add to the complexity of the mathematical model and noticeably tighten the requirements to the accuracy of the initial data on the atmosphere state, due to which this effect is not considered in the method of engineering calculation discussed herein.

Let us assume that the segment of considered medium (atmosphere) is in the state of local thermodynamic equilibrium [7]. The radiation intensity of such segment depends on the wavelength and temperature only. Then the source function J_{λ} (M , r) will be specified by the Planck function:

where B_{λ} (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 T_{0} :

Here, ε_{λ} (θ) – Earth’s surface radiation coefficient.

Let us introduce monochromatic transmission function τ_{λ} (θ, z_{1}, z_{2}), which characterises the portion of radiation passed by the atmosphere layer between levels z_{1} and z_{2} at angle θ to the vertical plane:

where u_{λ} (z_{1} , z_{2} ) – optical thickness at θ = 0.

Thereby,

where ρ(z) – average density of absorbing substance in layer z_{1} − z_{2}.

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;

h_{0} – 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 h_{1}, h_{2} – 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 [9] and transformation of the expression under integral sign [10]. Then we can write the final form of the expression for intensity in a narrow spectral interval:

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 h_{0} (see Fig. 6).

## Initial data for modelling

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 [11]. The data for carbon dioxide were updated, because its concentration in the atmosphere had grown as compared with the data of 2001. As of April 2017, the carbon dioxide concentration was 410 ppmV.

The values of atmospheric gas absorption spectral coefficients were taken from SPECTRA information system [12] for specified values of temperature, pressure, type of line broadening, and spectral range with the specified wave-number resolution.

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.).

## Verification of mathematical model and calculation results

There is a number of theoretical models of the Earth’s surface – atmosphere system implemented in the form of special-purpose software [13]. However, some of the models, such as *MODTRAN (Moderate Resolution Atmospheric Transmission)* [14], used by the U.S. Air Force, or *SciTran (Scientific Transparency, U.S. and Germany)* [15], impose strict requirements on the computation capacity, while being actually unavailable for the companies of the defence industry complex of Russia. Other models use too rough approximations regarding spectral parameters of absorption or a geometric model of the atmosphere, such as *CRTM (Community Radiative Transfer Model, U.S.)* [16], *CRM (Column Radiation Model, U.S.)* [17], *COART (Coupled Ocean-Atmospheric Radiative Transfer code, U.S.)* [18]. For these reasons it does not seem possible to consider these theoretical models in practice as data sources for verification of the software and algorithm complex under development.

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 [19]. The curve differences in the areas marked with red and blue ovals (see Fig. 7) are basically associated with the presence of spectral lines of additional substances in the transmission spectra determined experimentally, which can be easily eliminated in the developed mathematical model by entering additional initial data concerning the content of substances in the atmosphere.

**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) [2] and Nimbus 4 satellite [20] (Figs. 8, 9).

**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.

## Conclusion

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 [14], [21]), consists in a possibility to use the mathematical model in engineering calculations and software applications for semi-realistic simulators in real time.

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### About the Authors

**A. V. Shumov**Russian Federation

Shumov Andrey Valerievich – Candidate of Engineering Sciences, Head of the Department. Science research interests: plasma physics, optics, radiophysics, control systems.

Moscow

**A. I. Sultangulova**Russian Federation

Sultangulova Al'bina Il'darovna – Designer. Science research interests: optics, physics of radiation generation and propagation, solid-state lasers.

Moscow

### Review

#### For citation:

Shumov A.V., Sultangulova A.I. Modeling of Earth's limb infrared radiation. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2017;(4):53-62.
https://doi.org/10.38013/2542-0542-2017-4-53-62