A forward model for the emission signal and absorption path of each limb measurement of an orbit is set up and inverted for the number densities of the emitting species on a 2D latitude–altitude grid.

The mathematical representation of the forward model is 4πI=∫LOSγn(se)f(∫n(sa)dsa)dse, with I being the wavelength integrated emitted radiance, emissivity γ, density n and an absorption part – along the line of sight (LOS) and along the line from Sun (LFS) to the point of scattering into the LOS (sa stands for both absorption paths) – f. The emitted radiance is the integrated product of the density n and the emissivity γ along the emission path se, which is furthermore attenuated by self-absorption f (see Eq. () for f). Equation () is discretized on the 2D latitude–altitude grid and inverted for the number density n. The changes with respect to the Mg/Mg+ retrieval described by lie in the emissivity γ and self-absorption f calculation, while the rest of the retrieval algorithm remain unchanged, beside marginal changes (e.g., there is no correction for inelastic scattering needed for the Na retrieval, because the ratio of emission radiance to Rayleigh scattered radiance is high enough that this effect is negligibly small). The emissivity γ is calculated as follows: γ︸photonss=P(θ)︸Phase function×∫πF(λ)︸photonsscm2nm×σ(λ)︸cm2dλ×Aji∑kAjk︸rel. Einstein coeff., with πF(λ) being the solar irradiance (note that it is convention to use πF(λ) (see e.g., ) and π belongs to the symbol and is not meant as a factor 3.14) and ∫σdλ=14πϵ0πe2mec2fijλij2︸integ. abs. cross section in nm cm2.

The process causing the emission is resonance fluorescence (A detailed derivation of Eq. () from basic principles as well as a discussion of the phase function P is given in the appendix of .). A solar photon is absorbed by a Na atom, which is excited from the lower state i to the higher state j and is immediately re-emitted, which returns the atom to the lower state i. The two relevant transitions are from the lowermost excited states 32P12 for D1 and 32P32 for D2 to the ground state 32S12. The relative Einstein coefficient, the probability of the resonant transition compared to all other possible transitions from the upper state to lower states, is 1 for both lines, because only the two lowest excited states are involved as upper states and the transition between the P states is highly improbable. The Na specific parameters, i.e., oscillator strength fij and transition wavelength λij, are taken from the NIST (National Institute of Standards and Technology) atomic spectra database (). The scattering angle θ dependent phase function P is a linear combination of the phase function for Rayleigh scattering and an isotropic part: P(θ)=34E1(cos⁡2(θ)+1)+E2. P is normalized to 4π, which is already considered in Eq. (). The factors E1 and E2 depend on the change in angular momentum Δj and are taken from (see Table ). The factors E1 and E2 are different for both D lines. The D1 line has a purely isotropic phase function (E1=0 and E2=1), while the D2 line has a mixture of both components E1=0.5 and E2=0.5. The wavelength-integrated absorption cross section has to be distributed over the correct shape function of the emission line and the resulting absorption cross section profile is multiplied by the wavelength-dependent solar irradiance. This product is then integrated over all wavelengths, yielding the true combination of the second and third factor of the emissivity in Eq. ().

A high-resolution spectrum of the solar irradiance in the vicinity of the Na D1 and D2 lines needs to be used. Figure  shows solar spectra for this wavelength region from SCIAMACHY, and , from which only the latter fully resolves the Na D lines. Note that the approximate formula for the spectrum measured by is only valid in the vicinity of the center of the lines. Following , the solar irradiance in the vicinity of the Na D1 and D2 lines can be calculated for x (x is defined further below): I(x)=I0×e|x|xeA. The ratio I0Ibaseline of the intensity at the line center and the baseline intensity at the edge of the Fraunhofer lines is stated in . To scale this to the SCIAMACHY spectrum, a solar irradiance of 5.44×1014photonscm2nm is used for the edge of the line; for the D2 line, the following values are used: I0=0.0444×5.44×1014photonscm2nmA=2.16xe=σ^kline center=0.228cm-116 973cm-1=13.4×10-6x=k-(kline center-shifts)kline center=k-kline centerkline center+shiftskline center. The formula is given for wavenumbers k and the parameters x and xe are normalized to the wavenumber of the line center. The width parameter of the line in terms of wavenumbers is denoted by σ^ (see ).

Wavenumber shifts between the solar spectrum and the absorption cross section in the mesosphere are considered. Here, a positive shift value leads to a red shift as the line center is moved toward shorter wavenumbers and, therefore, longer wavelengths. Since the width of the solar Fraunhofer line is not much larger than the width of the lines in the mesospheric absorption cross sections, these shifts have a non-negligible influence on the emissivity. In a constant red shift for the solar lines of gravshiftskline center=2.7×10-6 is measured, which is a combination of the gravitational red shift and other smaller shifts, e.g., pressure shifts. Additionally, Doppler shifts from the rotation of Earth and the change of the Earth–Sun distance along the elliptical orbit of Earth are considered, which are similar in magnitude to the constant shift and have a combined maximum amplitude of ± 3.2 ×10-6. For the D1 line the following parameters are used: I0=0.0495×5.44×1014photonscm2nmA=2.14xe=12.8×10-6. Because there are hyperfine splittings for both D lines, the line center is the weighted average of the individual line wavelengths and strengths.

Na has only one stable isotope, i.e., 1123Na, and therefore has no isotope effect. The stable isotope has a nuclear angular momentum of I=32, which leads to a hyperfine splitting of the energy levels. The splitting for the lower 32S12 state is stronger than the splitting of the upper states 32P12 and 32P32. This can be explained phenomenologically by the smaller distance of the valence electron and the nucleus in the S state, which leads to a larger overlap of the nucleus and electron wave functions and therefore a stronger perturbation of the electron state. Due to the stronger splitting of the S state compared to the P states, the D1 and D2 lines each split into two groups of lines in close spectral vicinity (this is called an “s-resolved blend” in ). The Doppler width of the Na lines in the mesosphere is approximately 1.25 pm. The two groups of adjacent lines have a separation of about 2 pm and thus can be separated. However, the lines inside a group are too narrow to be resolved in the mesosphere. The existence of several degenerate lines, however, is important for the correct weighting of the two s-resolved lines. This is, e.g., well explained in .

The solar spectrum as well as the mesospheric absorption cross section for the Na D2 line are shown in Fig. . The Na density is large enough that a non-negligible part of the incoming solar irradiation is either absorbed along the path from the Sun to the point of resonance fluorescence, or along the line of sight, after the emission. This reduction of emissivity is considered in the self-absorption factor f: f=∫σ(λ)πF(λ)×e-σ(λ)gdλ∫σ(λ)πF(λ)dλ, with the integrated true slant column density g=∫nds. Note that the integral in Eq. () contains n as a linear factor, but also has a nonlinear dependence on n because of the self-absorption factor f. The equation is linearized, employing an iterative approach using the retrieved density of the previous step in the nonlinear part to retrieve the linear density n. As noted by , setting f=1, which corresponds to no self-absorption, is a good starting step for the iteration.

For every single limb measurement, Eq. () is discretized on a latitude–altitude grid with m elements to yield the formula: I=c1∑i=1mPixiΔsLOSi⋅f(∑j=1mxjΔsLOSgci,j+∑j=1mxjΔsLFSgci,j). c1 is a constant including all parameters that are path independent in the integral in Eq. (). For every grid element i along the LOS, there is the path element in this grid element ΔsLOSi for the emission path. Furthermore, for every grid element i along the LOS, there are two matrices with path elements. One matrix contains the path elements from the satellite to the grid element with matrix elements ΔsLOSgci,j. The other matrix contains the path elements from the Sun to the grid element with matrix elements ΔsLFSgci,j. Both matrices are needed for the absorption calculations of the attenuation factor f. Pi is the phase function in grid element i, and xi is the density in grid element i that should be retrieved. Figure  is a sketch of the considered paths.

The argument of f is summarized to gi(x)=∑j=1mxjΔsLOSgci,j+∑j=1mxjΔsLFSgci,j.

To invert the forward model Kx=y, the equation system JTJx=JTx needs to be solved, with J being the Jacobian of Kx. If K was independent of x, J simply was equal to K. The elements of the Jacobian J are calculated with the following formula: ddxkI=c1PkΔsLOSkf(gk(x))+c1∑i=1mPiΔsLOSixif′(gi(x))(ΔsLOSgci,k+ΔsLFSgci,k). If f were independent of x, f′=∂f(gi(x))∂gi(x) would be zero and the second addend in Eq. () would also be zero. Then J would be J=K. In practice, omitting the second addend in Eq. () reproduces synthetic model density profiles that are forward modeled by Eq. () and inverted with the retrieval algorithm very well and this simplification J=K(x) is used. However, K still explicitly depends on x, which has to be later considered in the retrieval step.

Figure  is a sketch illustrating the crucial steps for the calculation of the path matrices. Note that this is a minimalistic description and more details are given in . The path segments of the altitude grid are calculated with right-angled triangle geometry. The right-angled triangles considered for the LOS are formed by the path between the center of Earth and tangent point b, the right angle between b and the segment of the line of sight a and the path between the center of Earth and a grid element's upper altitude limit c. As b and c and the right angle are known, a can be calculated for each grid element. The path length within each altitude grid element is calculated by the difference of a for neighboring grid elements. For the LFS, the tangent point of the LFS needs to be found first and the position of the grid element with respect to the tangent point needs to be evaluated. For the calculation of the absorption part, it is crucial to separate the LOS at the tangent point and do the respective calculation of the factor f for each side of the tangent point, before adding both parts of the Jacobian together. In spherical coordinates, positions with the same latitude (north and south) can be calculated by a double cone equation. These double cone equations for the latitude boundaries of a grid element are set equal with the straight line equation r=TP+λeLOS^, where TP is the position of the tangent point, eLOS^ a vector in LOS direction and λ the distance between tangent point and the considered point on the line of sight. λ is determined for each latitude element. Once this is done, the latitude grid boundaries λ and the altitude grid boundaries a are sorted and the path lengths are determined by differences of neighboring boundary parameters (see Fig. , where red crosses are latitude changes and blue circles are altitude changes). Figure  shows the calculated path lengths for one limb measurement. The latitudinal separation of consecutive limb measurements is around 7∘ at the equator. It is smaller near the poles at the cost of a larger variation in local time, which is considered by splitting the orbit in an ascending satellite movement and a descending satellite movement part. For daily averages, the latitudinal separation is roughly halved, as SCIAMACHY has an alternating scanning pattern of limb and nadir scans for consecutive orbits. The largest paths are found in the close vicinity of the tangent point. The emission signal strength for each grid element is roughly the product of density and path length. For the around 4∘ latitudinal separation, the overlap region of consecutive limb measurements at the same altitude is less than 5 km above the tangent altitude. However, note that neighboring measurements in the averaged data come from different longitudes; thus this is not a real overlap of the same volume of air.

The equation system to be solved JTJx=JTx is nonlinear as J explicitly depends on x, because of the self-absorption contributions in f and f′. The equation system is linearized, by using initial values x for the calculation of J and retrieving new x values that are closer to the real values of x. This is iteratively done until convergence of x is achieved. Test retrievals showed that after roughly five iteration steps, convergence is achieved; i.e., the largest step by step changes in the retrieved densities are far less than 1 %. In practice, 20 iteration steps are used.

As JTJx=JTx is typically an ill-posed mathematical problem, solutions oscillate and are not physically correct if no constraints are applied. Therefore, three different smoothing constraints are applied: latitudinal smoothing, which penalizes solutions with differences in densities of grid elements with neighboring latitudes; altitudinal smoothing, which penalizes solutions with differences in densities of grid elements with neighboring altitudes; as well as Tikhonov regularization () with a zero a priori xa which in general favors solutions with the smallest oscillations and overall smallest distance from zero. The final equation that has to be solved is Eq. (): JTSyJ+Sa+λhSHTSH+λϕSϕTSϕx=JTSyy+Saxa︸=0︸=0. The a priori covariance matrix Sa is in fact a scalar (λapriori) multiplied by an identity matrix. SH and Sϕ are the matrices for altitudinal and latitudinal constraints (large sparse matrices with only two diagonals of non-zero values) and λh and λϕ are the scalar weighting factors for both constraints. x is the vector of number densities. On the right-hand side, there is the covariance matrix for the slant column densities (SCDs) (Sy), which is assumed to be diagonal: the SCDs y and the a priori xa. As there should not be any bias on the form of the profile, xa=0 is used.

There is some arbitrariness in the choice of the constraint strength λh, λϕ and λapriori. We choose a ratio of these three of 10:2:1. The constraint strength should be chosen so strong that the solution is affected but not dominated by the smoothing. As will be shown in Sect. , there is a range of 2 orders of magnitude in the choice of the constraint parameters which only results in moderate changes of the final result, which shows that the arbitrariness in the decision of which parameter is finally used does not significantly influence the result.

The approach discussed in this section is based on the expectation to retrieve the same densities from both D lines. Therefore, the albedo factor is assumed to be the factor where both density profiles are nearly the same. In practice, however, this method fails too often; hence another method is used, which is described in Sect. .

The albedo factor is determined as the factor for which both D lines yield the same Na number densities. For typical Na slant column profiles shown in Fig. , the identification of the optimal albedo factor is illustrated in Fig. . It should be noted that the phase function (Eq. ) is changed to Pnew=Pold+(albedo factor-1)albedo factor. This takes into account the increased part of multiply scattered radiation, which we assume is unpolarized and therefore effectively increases E2 in Eq. ().

This approach is only reasonable if the retrieval result is nearly independent of the applied constraint parameters (e.g., vertical smoothing, necessary to reduce oscillation of the result). This is not the case for the Mg/Mg+ retrieval by . For Na the statistical errors are sufficiently small; thus the Na retrieval is much less sensitive to smoothing constraints than that of Mg. However, the dependence on the variation of the constraint parameter is unfortunately not entirely negligible, especially when the densities are large, as is the case in Fig. . Figure  shows retrievals for different constraint parameters. For moderate constraint parameter (1×10-7 to 5×10-10) the retrieved peak density is nearly independent of the choice of the constraint parameter for approximately 2 orders of magnitude. A factor of 5–10 in the constraint parameter has a similar effect to a change of the albedo factor of 0.1 in Fig. . The three highest constraint parameters (5×10-7–5×10-5) show smoothing that is too strong, while the lowest constraint parameters (5×10-9–5×10-10) show oscillations at high altitudes. Note that a stronger smoothing leads to the need of a lower albedo factor to match the D1 and D2 density, so a systematic error in one property is rather reduced than increased if the other one is tuned, which results in some robustness in the method.

Unfortunately, this method failed quite often for the following reasons: the densities were too small and insensitive to the albedo factor, the D2 slant column densities were already larger than the ones for D1, or the differences between the D1 and D2 slant column densities initially were too large; therefore the D2 line could yield larger densities than the D1 even for albedo factors smaller than 1. However, although this algorithm failed the matching of densities retrieved from both Na lines, it is a good indicator for how good the calibration of the data and the radiative transfer model used work.

This approach calculates the albedo factor based on the ratio of the measured radiance, compared to the simulated single scattered radiance in the vicinity of the line. This radiance should only come from Rayleigh scattering of the major atmospheric constituents (N2, O2, Ar etc.), whose concentrations in the atmosphere are well known; therefore the Rayleigh scattering is easy to simulate. However, like other similar instruments, SCIAMACHY has a straylight contamination issue at mesospheric altitudes; indeed, an altitude region must be used for this calculation, where all possible error sources are small. This altitude lies typically below 40 km and is not scanned by the SCIAMACHY limb MLT states. Nevertheless, retrieving the albedo factor from collocated nominal measurements and matching the nominal profile to the MLT measurement profile yields reasonable results.

The Rayleigh scattered background radiance in the vicinity of the Na D lines can be used to calculate the total to single scattering ratio for a limb measurement. A simple approach to obtain the albedo factor is to use the ratio of the measured limb radiance and the limb radiance calculated with a radiative transfer model. The radiative transfer model SCIATRAN is able to calculate the single and total Rayleigh scattered electromagnetic radiation for known measurement geometries and atmospheric parameters. Figure  shows the ratio of the measured limb radiance and the simulated single scattering radiance for different tangent altitudes, as well as the ratio for the simulated total scattered simulated radiance and the single scattered radiance for different ground albedos.

As was reported by the modeled total to single scattering ratio only shows a weak dependency on the tangent altitude. The measured limb radiance, however, has a completely different behavior and shows a nearly exponential increase in the total to single scattering ratio above 50 km altitude. The true Rayleigh scattered limb radiance is roughly proportional to the density at the tangent altitude and exponentially decreasing limb radiances with increasing tangent altitudes are expected. We assume that there is a small straylight component from lower tangent altitudes that reaches the instrument for high tangent altitudes and that this component is only weakly dependent on altitude. Above 50 km this additional straylight component is on the order of magnitude as the actual limb radiance at this tangent altitude and becomes much bigger than the actual Rayleigh scattered component. This nearly constant offset to the radiance along with the exponential decrease of the Rayleigh scattered radiance with altitude explains the nearly exponential rise of the measured to simulated single scattering ratio. In the troposphere and lower stratosphere, clouds significantly influence the radiance, and the simple approach using an albedo factor fails there. Therefore, we assume that in a region above 20 and below 45 km, there is a region where the limb measurement to simulated single scattering ratio is very close to its simulated value with the right ground albedo.

Any unwanted straylight also affects the dark signal measurement at 350 km tangent altitude, which is usually subtracted from the limb radiances at the 30 other tangent altitudes. Instead of simply subtracting the dark signal measurement, another approach is used here: we assume that a part of the total incoming radiation Iinc(h,λ) is proportional to the simulated single scattering radiance Iss(h,λ), which we call the multiplicative part aIss with the multiplicative component a, while a part of the straylight component and the actual dark radiance is an additional component b of the light: Iinc(h,λ)=a(h)Iss(h,λ)+b(h) (with tangent altitude h and wavelength λ). To determine the multiplicative and additive components, the wavelength region between 650 and 660 nm is used. This spectral region is not affected strongly by atmospheric absorption in the mesosphere and upper stratosphere, and includes the H α Fraunhofer line at 656 nm, which is a clear solar signature; hence fitting the multiplicative and additive component is not an ill posed problem (where a smaller a could be compensated by a higher b etc.). The estimated value for the total to single scattering ratio is the minimum of the multiplicative component a above 20 km altitude. Figure  shows the fit and Fig.  the results of the multiplicative and additive component for one example profile.

This estimation, however, can only be done for the nominal SCIAMACHY limb measurements measuring from ground to 90 km altitude. The MLT measurements start at around 53 km altitude; therefore the minimum between 20 and 45 km can not directly be found. However, the latitudinally and longitudinally co-located nominal measurements from the days of the same time period show very similar profile shapes, which can be fitted to MLT data to retrieve the albedo factor. Figure  shows the final fit of the albedo factor for an example measurement.

First the multiplicative components for all nominal and MLT measurements are found. The median for the days in the same time period (± 200 orbits were used here) of nominal limb measurements is formed for all altitudes. The albedo factor A for the median nominal measurements is found. Between 50 and 70 km the logarithms of the nominal and the MLT measurements are fitted as factor B (ln MLT =B ln nominal). Fitting the logarithm puts effectively more weight on the matching of the lower albedo factor values at lower altitudes, which considers that the perturbing effect is smallest there. The albedo factor for the MLT measurements is then given by the product AB. The resulting albedo factors show similarities to the simulated total to single scattering ratios, which are high at scattering angles at around 90∘ and close to 1 for low (around 0∘) and high (around 180∘) scattering angles.

found out that more than 90 % of SCIAMACHY limb measurements are influenced by tropospheric clouds. A typical spread of the albedo due to this can be seen in the spread of the nominal albedo ratios in Fig. . Assuming Fig.  represents the typical spread for this estimation using the factor A only results in an error of about 20 %, which is rather large for a number, whose value is known to be larger 1 but unlikely larger than 2.5. The additional fit of factor B should reduce this error to less than 10 %.