The applied algorithms are based on the Mie theory and the geometric optics approach (GOA) along with the Fraunhofer diffraction (for details see (Fomin and Mazin, 1998) [12]). The last one is used when direct computations using the Mie theory become too slow in cases wherein the scattering particle size is much greater than the wavelength. In this case, we apply an effective ‘inverse’ algorithm in order to calculate scattering parameters at any angle (not interval). It makes it easy to combine calculations based on the Mie theory (for “small” particles), which defines scattering parameters at determined angles, with calculations based on GOA (for “large” particles). Thus the angular resolution in our calculations is practically unrestricted, as demonstrated in Figure 2. This figure shows a part of the phase function around rainbows calculated using GOA (solid line) and the Mie theory (dashed line). For this illustration we used a “large” water drop (diameter D = 1 mm, wavelength λ = 0.412 μm).

There is a problem with the GOA approach that needs to be discussed. It is divergence of the light intensities at the angles associated with rainbows, where the ratio of the solid angles of the scattering and initial rays becomes infinite. However, the integration of intensity over the scattering angle gives ray energy that is limited. So inside the neighborhoods of singular points (~0.1°) the average values are used, which are calculated analytically as suggested in (Fomin and Mazin, 1998) [12]. Here it is useful to note that the use of a phase matrix for the radiative transfer calculations generally leads to such integration and this algorithm can be considered as “exact” for the most of practical applications.

For the spherical polydispersions all the scattering parameters of a single particle must be integrated over radius a using the given size distribution function n(a). We apply the Mie algorithms when a < (λ/2π)X and after that the GOA algorithms. The X parameter depends on the accuracy of the calculations and usually lies in the interval (100, 1,000). The special numerical technique for this integration is also described in (Fomin and Mazin, 1998) [12]. Usually, it takes only several minutes to obtain optical properties of a typical cloud or aerosol model in the whole spectral region (from microwave to UV) using an ordinary computer (e.g., notebook).

Generally, such calculations are made in advance for the entire spectral range and the optical properties of clouds and aerosols are stored in the database. However, the remote sensing problems often require performing calculations in a narrow spectral region (e.g., near infrared) and only a few spectral points but for many polydispersions. In this case calculation is performed in seconds and if necessary, this block can be included in the model, which provides its flexibility.

The monochromatic volume absorption coefficient, K(ν), at any j-th wavenumber point ν_j usually can be calculated by formula (1) where f_i(ν) is the absorption profile of the i-th spectral line. As was shown above it takes to consider at least ~10³–10⁴ points ν_j in each spectral channel of the satellite instruments, whereas a number of functions f_i(ν) may reach tens of thousands and more. Moreover the absorption profile f_i (ν) depends on temperature, pressure, gas volume mixing ratio and constants associated with the i-th spectral transition. So the problem of efficient LbL algorithms is of extreme importance. In FLBLM we use the algorithm suggested in (Fomin, 1995) [17], which is one of the most effective for the Summation (1) as has been tested by Kuntz and Hopfner (1998) [18]. In these algorithms we use a set of the uniform wavenumber (frequency) grids with doubling wavenumber steps. It allows the possibility for calculating each profile f_i(ν) independently of others: in detail at the line center and more rarely in the far wings as shown in Figure 3.

Usually it takes less than 100 wavenumber points to approximate each f_i(ν) with the required accuracy (~1%). When all the lines are taken into account, we apply the simple (parabolic) interpolation procedure to obtain the absorption spectrum at the required ~10³–10⁴ equidistant wavenumber points. It should be stressed that the procedure is performed at once to the all spectral lines that takes negligible computer times. This algorithm thus allows for increasing the calculation speed up to ~(10³–10⁴)/100 = 10–100 times and the spectra calculations usually require less computer time than further solving of the radiative transfer equations. Finally it should be mentioned that the HITRAN-2008 spectral database (Rothman et al., 2009) [19] is used for the line profile calculation as well as the MT_CKD-2.5 water vapor and other continuum models from the Line-by-Line Radiative Transfer Model (LBLRTM) (Clough, 2005) [20].

In this version, photon’s trajectories are obtained by MC for scattering but non absorbing medium. The selective gas absorption is then taken into account with the help of the photon’s “weight”: (2) where M* is the trajectory’s point and u_ν(M*) is the photon’s optical path. Due to the above effective LbL technique the consideration of the gaseous absorption usually needs less computer time than the MC calculations so that it is not necessary to apply the K-distribution technique.

The scalar version of the model is a usual forward MC (Marchuk et al., 1980) [21], (Evans and Marshak, 2005) [22] combined with the LbL technique. It has been developed as long ago as in the 1990s (Fomin, et al., 1993) [23]. We will now describe the polarized version, which can calculate the Stokes vector S = (I, Q, U, V) (Chandrasekhar, 1960) [24]. The applied technique is based on the “local estimation” MC and follows the technique that has been described in detail in a paper by Cornet et al. (2010) [16]. There are two differences in the vector approach compared to the scalar one. The first difference is related to the fact that the scattering azimuth angle of a photon now is not a uniformly distributed random value between 0° and 180° degrees but depends on the initial polarization state and the scattering zenith angle. To overcome this difficulty we applied an algorithm suggested by Cornet et al. (2010) [16]. The second difficulty is related to the fact that the scattering plane rotates after each scattering so that it takes to perform a pair of rotations between the meridian and this planes. Here we follow method described in (Sushkevich, 2005) [0].

In FLBLM these monochromatic MC algorithms for the scattering but non absorbing media were combined with the above LbL algorithms to take into account the particulate and gaseous absorption by means of the photon “weights” (see (Marchuk et al., 1980), (Evans and Marshak, 2005)) [21,22]. The presented model is based on the “local estimation” MC because it allows reducing the statistical errors in calculated radiances and consequently this method is more effective for applications in the remote sensing problems.

To perform computations for any spectral regions (e.g., the instrument channels) we use two methods. In the first method, photons are uniformly distributed (usually randomly) over the given spectral region. This method has been used for a long time in the scalar version and was found very effective for calculations in the wide spectral regions due to statistical errors in each wavenumber points, which have a tendency to annihilate each other. Thus the final errors mostly depend on the total number of photons. The second method (Romanov et al., 1991) [26] wherein the same group of the photon’s trajectories (in nonabsorbing media) is used at each wavenumber point is more suitable for the very high-resolution computations: e.g., spectra themselves. Both methods are demonstrated in Figure 4, wherein a spectrum of the normalized Q-parameter at the zenith angle = 50° is shown.

In our model the cloud and aerosol optical properties are constant in the spectral range no wider than 10 cm⁻¹ (calculation is performed in such intervals). For the next interval these optical properties are recalculated. Here we use the fact that the optical properties of aerosol and cloud vary much more smoothly compared to selective gas absorption. It should be stressed that the first method is more rapid in practical calculations and we will discuss it further in Section 4.

For these calculations we used the aerosol model from the recent intercomparison with benchmark results (Kokhanovsky et al., 2010) [9] (see Section 3). The molecular scattering is also taken into account at λ ~ 0.76 μm (to illustrate situation in the A-band O₂), which is comparable with the aerosol scattering at the given angle. Also we used the standard “mid-latitude summer” atmospheric model with the homogeneous aerosol layers between 0.0 and 1.0 km above the underlying surface. The spectral region is a part of the A-band O₂, which is rather popular in the remote sensing.

A measured high-resolution spectrum represents N experimental values that make it possible to evaluate J retrieval values. It should be stressed that J << N due to experimental errors and a fact that the “inverse” problem is ill-conditioned in many practical situations. Also J << N because of the statistical errors in the N measurements as well as several points contain identical information about clouds and aerosols.

From high-resolution spectra of Stokes parameters measured in the molecular absorption bands of the reflected solar radiation, it should be possible to obtain several parameters describing the cloud and aerosol structures. This statement is confirmed by Figure 10, which demonstrates clearly distinguishable differences in a spectrum of the reflected solar radiation caused by changes in the aerosol structure only. The reason of the difference between homogeneous aerosol layer and a layer with extinction coefficient increasing from the top to the bottom is obvious. Indeed, the first layers near the top are more reflective leading to a weaker absorption of the photons by the atmospheric molecules. In the same way, the differences in signals between the channels are linked to the absorption and could give information at different levels.

An important question arises in such kind of satellite experiments: how do errors in the retrieval values depend on the spectral resolution and the Signal to Noise Ratio (SNR)? To make this question clear and indirectly to evaluate applicability of our model, we considered a simple but realistic numerical experiment, which simulates investigation of aerosol structures in the boundary layer. In this experiment it is supposed that the solar and satellite zenith angles = 0.0° and the aerosol optical thickness is small enough to use the “single scattering” approximation. The spectral region and atmospheric model are the same as in Figure 10. The aerosol volume scattering coefficient ρ is defined by expression (4) where z is altitude, the nodes z_j are equidistant points from 0.0 to 1.0 km and x_j are the retrieval values. Thus in our numerical experiment the signal y_i (normalized intensity I, Q or U component) in each spectral point is equal to (5) where ε_i is the Gaussian (normal) distributed errors with variance σ²_i ~ (y_i*SNR)². The B^j_i coefficients are (6) where τ ^j_i is the molecular absorption optical depth at the z_j altitude and the i-th spectral point. It should be stressed that these coefficients are realistic because have been calculated by the LbL method for the given atmospheric model. We have been performed statistical analysis using the theory of maximum likelihood estimation (Rodgers, 2000) [27] that gives a system of J linear equations for J unknown values x_j: (7) where N is a number of the spectral points. For the selected spectral region 5,600–5,800 cm⁻¹ in our numerical experiments we have tested N = 6,400, 800 and 200 (spectral resolution = 1/32, 1/4, and 1 cm⁻¹) and J = 3, 4… Statistical errors of the x_j-values, which are related to experimental errors ε_i, can be evaluated using the Fisher information matrix (Rodgers, 2000) [27] or by means of the series of L repetitions of the experiment. We chose the second method as more universal, wherein we have selected L = 1,000 repetitions. In each l-th repetition y_i^(l) have been obtained by Formula (5), where initial values <x_j> = 0.1 × 2/[J(J + 1)] × (J − j + 1) (it corresponds to the linear aerosol profile with optical thickness = 0.1). Errors ε_i^(l) have been obtained using a program that independently generates the normally distributed quantities (the diagonal covariance matrix of y_i is supposed). Thus in each l-th “retrieval” (l =1, 2, …, L) we have obtained x_j^(l) values for the statistical data processing. The result is shown in Table 1, where relative errors in percent η_j for the x_j values are calculated by expression (8)

These preliminary results confirm that retrieval of the aerosol structure by the discussed method is possible even in the thin boundary layer. However, it needs to take into account spectra consisting from thousands of the spectral points and SNR must be greater than ~300 (as in TANSO-FTS). Also it can be expected that number of the independent parameters (degree of freedom) hardly exceeds J = 3–4 for real instruments.

As a whole errors η_j, as was expected, are inversely proportional to a factor SNR*(N)^0.5. So it may seem that it is the same to use different spectral resolutions and spectral intervals if only the number of points N is the same. In Table 2, we have tested the effect of the spectral resolution on the relative error by keeping the number of spectral points to N = 200 and repeated our experiment in the spectral regions 6,200–6,206.25 cm⁻¹ and 6,200–6,400 cm⁻¹ with spectral resolutions 1/32 cm⁻¹ and 1 cm⁻¹, respectively.

As one can see errors in the high-resolution retrieval are essentially less than in other case and, consequently, the high-resolution measurements are preferable: evidently because they can better use more informative parts of a spectrum. Also it should be mentioned that measurements of the Q and U components of the Stokes vector allow better investigation of aerosols and clouds because they often are the main polarizing factor. Thus the high-resolution polarized measurements seem rather promising.

To investigate applicability of the model to remote sensing as a “forward” model we have performed some numerical experiments with an imagined hybrid of POLDER and TANSO-FTS. The result was a simulation of the measurements by FTS with the spectral resolution equals to 0.5 cm⁻¹ (see Figure 1) in a band 13,000–13,200 cm⁻¹. In this section we present some computer time estimations for calculation of simulated spectra measurements. It took ~10 hours for the calculations performed by the second method (see Section 2.3) using an ordinary computer (CPU 3.20 GHz) with a spectral resolution equal to 1/32 = 0.03125 cm⁻¹ in the spectral region 13,000–13,200 cm⁻¹ (the POLDER’s channel 768.35–757.45 nm) for a triplet of the relative azimuth angles (0, 90 and 180 degrees) and ten of the zenith angles (0, 10, 20 … 90 degrees). The number of photons was 2,000,000 in each 8.0 cm⁻¹ spectral subinterval, where the scattering properties are supposed constant. But the number of photons can be essentially decreased if we apply the more rapid first method. It took only 20,000 instead of 2,000,000 photons and ~10 minutes of the computer time. Figure 11 demonstrates the initial and convoluted spectra obtained by the both methods for the aerosol layer between 0 and 1 km.

Nevertheless the visible statistical errors are less than the spectrum distortion because of the convolution. It should be noted that these errors become practically invisible for ~100,000 photons and it takes ~1 hour of the computer time. Due to the fact that the statistical errors mostly depend on the total number of photons (not spectral interval) the computer time is inversely proportional to the spectral resolution. Thus calculations for POLDER need ~100 times less computer time than for TANSO-FTS (several minutes only).