Gas absorptions include those from line absorbers and non-line absorbers. In the wavelength region considered here (270 to 2400 nm), line absorbers are H2O, CO2, CO, CH4, and O2. Non-line absorbers are O3, SO2, O2-O2, BrO, HCHO, NO2, and CHOCHO. The collision complexes, such as O2-O2 and N2-O2, absorb light at UV, Vis, NIR, and SWIR (shortwave infrared) wavelengths. The amount of absorption by the collision complexes generally scales with the square of the pressure. In DISAMAR absorption by N2-O2 (in the spectral region of the O2 A-band) and the absorption by O2-O2 are simulated.

The standard database for line-absorbing molecules is the HITRAN 2008 database . Line parameters are read from the HITRAN database for a particular gas and a Voigt profile is used to calculate the absorption cross section. For H2O, CO2, CO, and CH4, line mixing is ignored. For O2, line mixing is taken into account using the model described in for the O2 A-band .

The gas absorptions from line absorbers have to be calculated line-by-line at a high-resolution wavelength grid, and then convolved with the ISRF. Gaussian division points are used to define the wavelength grid, which improves the accuracy of the integration during the convolution of the ISRF.

There are two steps to construct the high-resolution wavelength grid: (1) determining wavelength intervals and (2) dividing each interval with proper Gaussian division points. The full width at half maximum (FWHM), the number of Gaussian division points for one FWHM (N0), and the minimum number of Gaussian division points are specified in the configuration file.

We start from the shortest wavelength with an interval of one FWHM of the ISRF. If there is a strong absorption line in the interval, the boundary of the interval is set to the position of the strong line, making this interval smaller than one FWHM. The next interval starts from the position of the strong line with one FWHM interval again. This process is repeated until the end of the wavelength range.

Figure  shows two 1 nm wide parts of the O2 A-band spectrum, from 757.8 to 758.8 and from 765.2 to 766.2 nm, simulated using DISAMAR. The used spectral resolution (wavelength difference between two grid points) is also shown in Fig. . In the spectral part from 757.8 to 758.8 nm, which is devoid of O2 absorption lines, the wavelength intervals are 0.4 nm, and every interval is divided into 40 Gaussian points (see Fig. a, b). In the spectral part from 765.2 to 766.2 nm, which has many O2 absorption lines, the intervals are smaller due to the absorption lines (see Fig. c, d).

Vertical profiles of absorbing gases are specified using the volume mixing ratio (VMR, in parts per million by volume, ppmv) at pressure grid levels. The volume mixing ratio has the advantage that it remains constant if the temperature changes. The number density of gas molecules is calculated internally using hydrostatic equilibrium. The pressure profile and temperature profile grid can be different for simulation and retrieval: for example, trace gas profile retrievals may be using 50 grid points for the simulation to have high forward model accuracy but only 12 or 18 grid points for the retrieval to represent the limited information content of the measurements.

Two different types of cloud and aerosol layers are distinguished in DISAMAR: an opaque Lambertian reflector and a layer of scattering particles. For the Lambertian type, the cloud and aerosol reflector is located at the top of a pressure interval. For the scattering particles layer, a Henyey–Greenstein phase function or a Mie phase function (or phase matrix in case of polarization) can be used for aerosols and clouds in DISAMAR. If the Henyey–Greenstein phase function is used, the wavelength dependence of the optical thickness is modelled using the Ångstrøm exponent and the absorption is controlled by the single-scattering albedo of the aerosol and cloud particles. A Mie phase function is provided in the form of expansion coefficients in an external file. It is assumed that in general a part of the pixel is cloud-free and that a part is covered by cloud. This part is controlled by the cloud fraction, which can have values in the interval [0, 1].

In DISAMAR the atmosphere is vertically divided into pressure intervals. The surface pressure (e.g. 1000 hPa) defines the lower boundary of the first interval. The intervals are further specified by the pressures of the top of the intervals. For example, if the pressure levels are specified at 700, 600, and 0.1 hPa, the atmosphere has three intervals: from the surface to 700 hPa, from 700 to 600 hPa, and from 600 to 0.1 hPa. The top of the atmosphere (TOA) is at 0.1 hPa. This division is suited to model a cloud layer with boundaries at 700 and 600 hPa. Internally, the pressure levels are translated into altitude levels assuming hydrostatic equilibrium. The formulae are provided in Appendix . For this calculation, the temperatures at the pressure levels should be provided in the configuration file.

In order to perform radiative transfer calculations, each interval is divided into a number of homogeneous layers using Gaussian division points for the altitude. For example, the three intervals from the surface to TOA may be divided into layers using 12, 10, and 28 Gaussian division points, respectively. The standard atmospheres can be used as input for the pressure, temperature, and gas mixing ratio profiles.

Figure  illustrates the pressure intervals and the layers in the intervals used in DISAMAR. It is important to use a sufficient number of Gaussian division points for each pressure interval. Each layer is actually further divided into sub-layers using Gaussian division points, with typically four sub-layers per layer. In this way, an accurate integration over altitude can be performed in order to get the layer quantities. Averaging of the optical properties and radiation quantities at the top and bottom of a layer is not sufficiently accurate if the optical properties differ strongly between the top and bottom of the layer. The VMR profiles of gases are interpolated at the radiative transfer model grid. For optically thick clouds, we suggest using 15–50 Gaussian division points for the cloud layer, particularly if the cloud parameters for that layer are to be fitted. For optically thin clouds, 10 Gaussian division points should be sufficient for the cloud layer. We use Gaussian division points because it improves the accuracy in the integration over the altitude.

The surface below the atmosphere is a Lambertian, i.e. isotropically reflecting, surface. In DISAMAR one can choose between a wavelength-independent surface albedo or a surface albedo that has a polynomial wavelength dependence for each spectral band.

Sun-induced fluorescence from vegetation due to photosynthesis can be observed in the near-infrared spectrum . In order to be able to fit fluorescence in DISAMAR, a polynomial describing surface emission can be used in combination with a wavelength-dependent surface albedo.

The doubling-and-adding method (also called adding method) is described in many books and papers on radiative transfer in planetary atmospheres e.g.. Here we use the notation that follows and . We present the new formulae needed to calculate the derivatives; for the detailed formulae used in the doubling–adding method, we refer to and .

DISAMAR uses the doubling method for the calculation of the scattering properties of the individual atmospheric layers. The adding method is used to add different atmospheric layers, and thus it is used to construct the entire atmosphere. As discussed by and , the doubling–adding method can be used to calculate the internal radiation field at the interface between layers in an efficient manner using principles of invariance. This approach can be extended to obtain the derivatives of the internal radiation field, in a linearization step, with respect to a parameter of the atmosphere–surface system. Because the adding formulae are needed in the derivation of the linearization, we provide them here as well.

Radiative transfer matrices are denoted in bold font, and they operate on incident radiation from right to left. A beam of polarized radiation is represented by the Stokes four-vector I=[I,Q,U,V]T. Incident sunlight is the Stokes four-vector E0=[E0,0,0,0]T, where E0 is the extraterrestrial solar irradiance. The elements of the 4×4 scattering matrix F are expanded into generalized spherical functions, where the expansion coefficients are calculated using the Mie computer program . A generalized form of the addition theorem is used to calculate the Fourier coefficients of the 4×4 phase matrix, denoted by Z. Z is found by rotating F to the local meridian plane of incident and scattered light, respectively. Figure  illustrates the definition of the directions used in DISAMAR.

The doubling method is started with an optically thin layer, with an optical thickness of, e.g. 1×10-5. For a thin layer, the Fourier coefficients (indicated by m) of the reflection matrix Rm and the transmission matrix Tm of the layer illuminated at the top are given by 1Rthinm(μ,μ′)=dzksca(z)4μμ′Zm(-μ,μ′),2Tthinm(μ,μ′)=dzksca(z)4μμ′Zm(μ,μ′). If the thin layer is illuminated at the bottom, the reflection and transmission matrix Fourier coefficients are given by 3Rthin*m(μ,μ′)=dzksca(z)4μμ′Zm(μ,-μ′),4Tthin*m(μ,μ′)=dzksca(z)4μμ′Zm(-μ,-μ′). Here ksca is the volume scattering coefficient of the layer, z is the altitude, and μ and μ′ are the cosines of the polar angles for scattered and incident light, respectively. The “*” indicates quantities with illumination at the bottom. Zm(±μ,±μ′) are the Fourier coefficients of the phase matrix of the layer. We assume that the scattering matrix does not depend on altitude within a homogeneous layer The unit of ksca is per kilometre if z is in kilometres.

Attenuation of an incident beam of light due to extinction (sum of scattering and absorption) by the thin layer is given by the direct transmission matrix Ethin: 5Ethin(μ,μ′)=1-kext(z)μdzδ(μ-μ′)1, where the volume extinction coefficient is kext(z)=ksca(z)+kabs(z), with kabs(z) as the volume absorption coefficient. δ(μ-μ′) is the Dirac delta function, and 1 is the unity matrix. Equation () follows from the Lambert–Beer attenuation law by keeping only the linear term of a Taylor series expansion.

We now consider adding two layers on top of each other. Let Ek, Tk, and Rk be the matrices representing the Fourier coefficients of the direct transmission, diffuse transmission, and reflection for illumination at the top of layer k, respectively. Here we omitted the Fourier index m for brevity. Similarly, let Ek*, Tk*, and Rk* be these matrices for illumination at the bottom of layer k. The matrices for layer k+1 are denoted similarly. Ek is a diagonal matrix. For a plane-parallel atmosphere, its elements are given by 6E4(i-1)+j,4(i-1)+j=exp⁡(-τ/μi), where i is the index of the Gaussian μ point, j=1,2,3,4 is the Stokes parameter index, and τ is the layer optical thickness.

The adding scheme for illumination at the top of the two layers is presented in Eqs. ()–(). The adding scheme for illumination at the bottom of the two layers is presented in Appendix . The adding scheme is used for every Fourier coefficient; at the end of the process, the reflection and transmission matrices are obtained by summing up the Fourier coefficients.

The scheme of adding layer k+1 on top of layer k to create a combined layer is illustrated in Fig. . The adding method formulae are as follows. 7Ek,k+1=EkEk+18Qk,k+1=Rk+1*Rk+Rk+1*RkRk+1*Rk+…9Dk,k+1=Tk+1+Qk,k+1Ek+1+Qk,k+1Tk+110Uk,k+1=RkEk+1+RkDk,k+111Rk,k+1=Rk+1+Ek+1Uk,k+1+Tk+1*Uk,k+112Tk,k+1=TkEk+1+TkDk,k+1+EkDk,k+1 Here Ek,k+1 is the attenuation of incident light by the combined layer. Qk,k+1 is the sum of the repeated reflections between the layers for a unit amount of light incident on the interface between the layers. Dk,k+1 is the diffuse downward light at the interface between the layers, Uk,k+1 is the upward light at the interface between the layers, Rk,k+1 is the reflectance of the combined layer, and Tk,k+1 is the transmission of the combined layer.

The matrices R, T, E, Q, D, and U are so-called super-matrices. The elements of super-matrices contain Gaussian points for the polar angle with weights that are suited for numerical integration by multiplication of two super-matrices. For the definition of super-matrices, we refer to . The total number of polar angles, Nμ, is the number of Gaussian division points, Ng, plus one solar zenith angle and one viewing zenith angle, pertaining to the selected sun-view geometry. The dimension of the super-matrix is (Nμ, Nμ) if polarization is ignored and (4Nμ, 4Nμ) if polarization is included using all four Stokes parameters. In the following a super-matrix is meant when we speak of a matrix.

The formula for repeated reflections at the interface, Eq. (), can also be written as 131+Qk,k+1=(1-Rk+1*Rk)-1, where 1 is the unity matrix. Inserting Eq. () into Eq. () yields 14(1-Rk+1*Rk)(Ek+1+Dk,k+1)=Ek+1+Tk,k+1. Equations () and () are needed in the derivation of the derivatives, as will be discussed next.

Retrieval algorithms that are designed to derive quantitative information about properties of the atmosphere from measured spectra require not only an atmospheric model but also derivatives of the reflectance with respect to the optical properties of the atmosphere, sometimes called Jacobians or weighting functions (see, e.g. ), to simulate reflectance spectra. Obviously, these derivatives can be calculated using numerical differentiation based on perturbation. However, numerical differentiation is a slow process, in particular when altitude profiles have to be determined and the atmosphere consists of 20–100 layers. Here we derive semi-analytical expressions for the derivatives of the reflectance with respect to the optical properties of the atmospheric layers. The semi-analytical derivatives are calculated at the interfaces of the layers, i.e. at levels. The derivatives for the layers themselves can then be calculated from the derivatives at the interfaces using interpolation and integration over the layers. Note that often the derivatives are calculated pertaining to optical properties of atmospheric layers, which differs from our approach.

The derivatives can be found by calculating the change in reflectance at top-of-atmosphere using the changes in atmospheric properties at each height, dR/dz. In the adding scheme, such a change can be invoked by adding a thin layer at altitude z. As shown by Eqs. ()–(), the reflection and transmission of a thin layer can be written analytically as a function of the optical properties of the layer: ksca, kabs, and Z. Therefore, the computation of semi-analytical derivatives should involve a thin layer. If one first calculates the reflection of two layers on top of each other, Rtop+bot, and next calculates the reflection of the two layers with a thin layer in between, Rtop+thin+bot, then dR/dz can be calculated from the following differential: (Rtop+thin+bot-Rtop+bot)/dz.

We divide the atmosphere into two parts, a top part and a bottom part. Let Rtop, Rtop*, Ttop, Ttop* be the Fourier coefficients of the reflection and transmission matrices of the top part; the quantities without an asterisk denote the properties for illumination at the top, and the quantities with an asterisk denote illumination at the bottom. Similarly, matrices with the subscript “bot” denote the properties of the lower part of the atmosphere.

Using the adding formulae, Eqs. ()–() and keeping only constant terms and terms linear in dz, one can derive the derivative dR/dz in a concise form as follows: 15dRdz=[Etop+Ttop*](1+Q*)W(z)(1+Q)[Etop+Ttop], where Q and Q* are given by 16Q=Rtop*Rbot+Rtop*RbotRtop*Rbot+…,17Q*=RbotRtop*+RbotRtop*RbotRtop*+…, and W(z) is given by 18W(z)=ksca(z)Z-+′(z)+ksca(z)Z--′(z)-kext(z)[1/μ]Rbot+Rbotksca(z)Z++′(z)-kext(z)[1/μ]+Rbotksca(z)Z+-′(z)Rbot. Here 1/μ is the super-matrix with the element 1/μ on the diagonal, representing the slant path for extinction. Z′±± are abbreviated notations of Z′(±μ,±μ′), the super-matrices of the Fourier coefficients of the phase matrix. The full derivation of Eq. () is given in Appendix .

Inserting Eq. () into Eq. () yields the separate terms of the derivative: 19dRdz=ksca(z)[Etop+Ttop*](1+Q*)Z-+′(z)(1+Q)[Etop+Ttop]+ksca(z)[Etop+Ttop*](1+Q*)Z--′(z)Rbot(1+Q)[Etop+Ttop]+ksca(z)[Etop+Ttop*](1+Q*)RbotZ++′(z)(1+Q)[Etop+Ttop]+ksca(z)[Etop+Ttop*](1+Q*)RbotZ+-′(z)(1+Q)[Etop+Ttop]-kext(z)[Etop+Ttop*](1+Q*)[1/μ]Rbot(1+Q)[Etop+Ttop]-kext(z)[Etop+Ttop*](1+Q*)Rbot[1/μ](1+Q)[Etop+Ttop]. Since we want to calculate the derivative dR/dz with respect to the scattering and absorption properties of the atmosphere at altitude z, Eq. () has to be expressed in the internal fields D and U at altitude z. Therefore, in Eqs. ()–() the internal fields are expressed using (1+Q) or (1+Q*) and R, T, and E. The relations are as follows. 20Etop+D=(1+Q)(Etop+Ttop)21U=Rbot(1+Q)(Etop+Ttop)22Etop+Δ3D̃=(Etop+T̃top)(1+Q̃)23Δ3ŨΔ3=(Etop+Ttop*)(1+Q*)Rbot Here ̃ indicates the transpose of a super-matrix, and Δ3=diag(1,1,-1,1) is a diagonal matrix that occurs due to polarization. By replacing the terms related to (1+Q) and (1+Q*) in Eq. () with the relations in Eqs. ()–(), we obtain the derivative in terms of the internal radiation fields: 24dRdz=ksca(z)[Etop+Δ3D̃Δ3]Z-+′(z)[Etop+D]+ksca(z)[Etop+Δ3D̃Δ3]Z--′(z)U+ksca(z)Δ3ŨΔ3Z++′(z)[Etop+D]+ksca(z)Δ3ŨΔ3Z+-′(z)U-kext(z)[Etop+Δ3D̃Δ3][1/μ]U-kext(z)Δ3ŨΔ3[1/μ][Etop+D].

First, replace kext with ksca+kabs in Eq. () and separate the terms containing kabs and ksca. Following this, calculate the partial derivative of dR/dz with respect to kabs and ksca using Eq. (). This gives the partial derivative of the reflectance with respect to the volume absorption coefficient (Eq. ) and the partial derivative of the reflectance with respect to the volume scattering coefficient (Eq. ). These equations are as follows: 25∂2R∂kabs∂z=-[Etop+Δ3D̃Δ3][1/μ]U-Δ3ŨΔ3[1/μ][Etop+D],26∂2R∂ksca∂z=[Etop+Δ3D̃Δ3]Z-+′(z)[Etop+D]+[Etop+Δ3D̃Δ3]Z--′(z)U+Δ3ŨΔ3Z++′(z)[Etop+D]+Δ3ŨΔ3Z+-′(z)U-[Etop+Δ3D̃Δ3][1/μ]U-Δ3ŨΔ3[1/μ][Etop+D]. The above relation, Eq. (), is a generic expression for the derivative with respect to the volume scattering coefficient. However, since it contains the phase matrix, it makes a difference whether the change in the volume scattering coefficient is due to a change in the number of scattering molecules in the thin layer or due to a change in the number of aerosol particles. If the number of scattering molecules is changing, the phase matrix of the molecules has to be used in Eq. (), whereas the phase matrix of the aerosol particles has to be used if the number of aerosol particles is changing. In general one has to distinguish between molecules, aerosols, and cloud particles.

If polarization is ignored, the partial derivative of the reflectance with respect to some specific quantity x(z) (e.g. ozone number density or aerosol number density) can be expressed as follows: 27∂2R∂x∂z=∂2R∂ksca∂z∂ksca∂x+∂2R∂kabs∂z∂kabs∂x+∑l=0∞∂2R∂α1l∂z∂α1l∂x, where R is the (1, 1) element of R and α1l is the phase matrix expansion coefficient for unpolarized light. In DISAMAR the third term on the right-hand side is ignored, so it is assumed that the phase matrix of the particles does not change as function of z. The derivatives for specific parameters can be calculated using these basic derivatives . Here we give two examples of the use of the semi-analytical derivatives presented in the previous section.

The first example is the altitude-resolved air mass factor (also called block air mass factor) at wavelength λ, m(z,λ). The altitude-resolved air mass factor is used in DOAS retrievals and represents the relative reduction in the reflectance when a unit amount of absorption is added to the atmosphere in a thin layer located between z and z+dz. So it is the partial derivative of the reflectance due to the absorber at z divided by the reflectance. Writing the reflectance of the atmosphere without the extra absorption as R(λ,0), the altitude-resolved air mass factor is defined as follows: 28m(z,λ)=-1R(λ,0)∂2R(λ,0)∂kabs(z,λ)∂z. The partial derivative on the right-hand side can now be calculated from Eq. ().

The second example is the derivative of the reflectance to the altitude of a Lambertian cloud below an absorbing and scattering gas layer: 29∂R∂z=-∂2R∂kgas,sca∂z(z)kgas,sca(z)-∂2R∂kgas,abs∂z(z)kgas,abs(z). Here the cloud is approximated as a Lambertian surface, without any transmission. When the altitude of the Lambertian cloud increases, the amount of gas above the cloud is reduced, and thus we get a minus sign in Eq. (). The amount of scattering gas that is removed is proportional to the volume scattering coefficient (kgas,sca), whereas the amount of absorbing gas that is removed is proportional to the volume absorption coefficient (kgas,abs). The partial derivatives on the right-hand side of Eq. () can now be calculated from Eqs. () and ().

As noted before, the derivatives can also be calculated numerically. Thus, dR(λ,x)/dxk, with xk element k of the state vector x, is written as the following numerical differential: 30∂R∂xk=R(λ,x+Δx)-R(λ,x)Δxk. Here Δx=0, except for the element k, where Δx=Δxk. By comparing the analytical derivatives in DISAMAR with the numerical derivatives, we can validate the former.

In the adding method as implemented in DISAMAR, the reflection matrix is calculated for a set of solar directions equal to the number of Gaussian division points used for integration over the polar angle plus at least two additional directions, namely the actual viewing direction and the actual solar position. Thus, for satellite retrievals the adding method provides more results than required.

The layer-based orders of scattering method (LABOS) has been developed to obtain the reflection matrix for the actual solar position or, when derivatives are required, for the actual solar position and a solar position corresponding to the viewing direction. The reflection and transmission properties of the individual homogeneous layers are still calculated with the doubling method; only the adding of different layers and the subsequent calculation of the internal field is replaced by the successive orders of scattering method (see Eqs. –). Here an order of scattering represents scattering by an atmospheric layer. This differs from the classical method of successive orders of scattering where the scattering element is a volume element of the atmosphere instead of a layer . In the adding method, one deals with matrix–matrix multiplications, whereas in LABOS one deals with matrix–vector multiplications. However, in LABOS the calculations have to be repeated for the different orders of scattering. Hence, for an optically thin atmosphere with only a few orders of scattering, LABOS requires less calculation time, while adding is more efficient for optically thick atmospheres, e.g. those containing clouds.

In LABOS we deal with the upward and downward internal radiation fields, i.e. U and D. In order to determine the derivatives using reciprocity, we need to consider two directions for the incident light, one corresponding to the actual solar position and one corresponding to the viewing direction. Therefore, U and D are (4N, 2) matrices, with the first column corresponding to incident light for the viewing direction, and the second column corresponding to the solar direction, where the number of polar angles Nμ=Ng+2. If the incident light is polarized, we need (4Nμ, 8) matrices. In order to calculate the total internal fields U and D (see Eqs. –), we first need to calculate local internal fields Ulocal and Dlocal (see Eqs. –).

Let Rk and Tk be the super-matrices for the reflection and transmission of layer k (k=0,1,…K). If k=0, the reflection matrix is that of the surface, and the transmission matrix vanishes. Further, let μ and μ0 be the cosines of the viewing and solar zenith angle, respectively. In order to evaluate different orders of scattering, we first consider the so-called local radiation field, which is due to the reflection by the layer just below the level considered and the transmission by the layer just above this level. Figure  illustrates the local upward and downward radiation fields at the interface k for the first-order scattering (n=1).

The local upward first-order radiation at level k is given by 31Uklocal,1=Rkexp⁡-τkμ0, where τk is the vertical optical depth of level k, measured from the top of the atmosphere. The two columns of matrix Uklocal,1 correspond to the two columns of super-matrix Rk having indices 4Ng+1 and 4(Ng+1)+1, respectively, if the dimension of the Stokes vector is 4. The first-order local downward radiation field at level k is 32Dklocal,1=Tk+1exp⁡-τk+1μ0. Here the two columns of Dklocal,1 correspond to the two columns of super-matrix Tk+1 with indices 4Ng+1 and 4Ng+5. The superscript 1 denotes the first-order scattering for the layers.

From the local radiation fields, we can now derive the total radiation fields. The first-order-scattered upward radiation at level k, Uk1, is calculated as the sum of the contributions of the local upward radiation from the ground up to level k. The downward radiation at level k, Dk1, is calculated as the sum of the contributions of the local downward radiation from level k up to level K (TOA): 33Uk1=∑l=0kexp⁡-τl-τkμUllocal,1,34Dk1=∑l=kKexp⁡-τk-τlμDllocal,1, where the attenuation accounts for the direct transmission of light from level l to level k. We note that Eqs. () and () can be rewritten in terms of the following recurrence relations, which are more efficient to evaluate. 35U01=U0local,136Uk+11=Uk+1local,1+exp⁡-τk-τk+1μUk137Dk1=038Dk-11=Dk-1local,1+exp⁡-τk-1-τkμDk1 The same recurrence relations hold for the higher orders of scattering because the direct transmission of radiation to other levels is the same for each order of scattering.

The following equations are used to determine the local radiation fields for the higher orders of scattering, where n indicates the order of scattering: 39Uklocal,n+1=RkDkn+Tk*Uk-1n,40Dklocal,n+1=Tk+1Dk+1n+Rk+1*Ukn, where Rk+1* is the reflection of layer k+1 for illumination at its bottom. Since the layers are homogeneous, we can use symmetry relations to calculate Rk+1* and Tk* from Rk+1 and Tk, respectively . After the local radiation fields are calculated using Eqs. () and (), the recurrence relations are applied for the transfer to other levels to provide the radiation fields Ukn+1 and Dkn+1 for scattering order n+1.

One can continue calculating the orders of scattering and summing them up until the contribution of higher orders can be ignored. At the end of the calculation, the internal radiation fields at all levels are known. The Fourier coefficients of the upward radiation at the top of the atmosphere provide the reflection, and the internal radiation fields are used to calculate the derivatives. In DISAMAR users can choose to use either the doubling–adding method or the doubling–LABOS method.

The upward radiation at the top of the atmosphere is calculated by assuming that the atmosphere is consisting of homogeneous layers, both in the doubling–adding method and in the doubling–LABOS method. The calculation will converge to the correct value if many homogeneous layers are used to represent the true atmospheric scattering and absorption profile. In DISAMAR the reflection can also be calculated from numerical integration over the source function J to mitigate errors due to layering that is too coarse. The reason is that by using the source function the assumption of homogeneous layers is not used for single scattering but only for multiple scattering. For each azimuthal Fourier term the reflection is then given by 41R(μ,μ0)=exp⁡-τ(0)μU(z=0,μ,μ0)+∫0TOAexp⁡-τ(z)μJ(z,μ,μ0)dz, where the surface is at z=0. Since the source function for the surface is a Dirac delta function in z, its contribution to the reflection occurs as a separate term in Eq. (). The source function for the atmosphere is 42J(z,μ,μ0)=12μ∫01ksca(z)Z(-μ,-μ′)U(z,μ′,μ0)dμ′+12μ∫01ksca(z)Z(-μ,μ′)D(z,μ′,μ0)dμ′+ksca4μμ0Z(-μ,μ0)exp⁡-τ(z)μ0. U(z,μ′,μ0) and D(z,μ′,μ0) are the internal fields at altitude z that are known after doubling–adding or doubling–LABOS calculations. Hence, J can be easily calculated at those altitudes where the internal radiation fields are known.