Atmospheric Chemistry and Physics Discussions Interactive comment on “ First space-borne measurements of the altitude distribution of mesospheric magnesium species ”

First space-borne measurements of the altitude distribution of mesospheric magnesium species M. Scharringhausen, A. C. Aikin, J. P. Burrows, and M. Sinnhuber Institute of Environmental Physics, University of Bremen, Bremen, Germany Institute for Astrophysics and Computational Sciences, The Catholic University of America, Washington, D.C., USA Received: 9 January 2007 – Accepted: 26 March 2007 – Published: 2 April 2007 Correspondence to: M. Scharringhausen (scharr@iup.physik.uni-bremen.de)


Introduction
The major source for metal species in the upper atmosphere is assumed to be influx of meteoric particles and cosmic dust. These particles enter the atmosphere at high velocities (12-72 km/s) and evaporate in the middle and upper atmosphere due to frictional heating with the ambient air.

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The most stable neutral magnesium species is the dihydroxide Mg(OH) 2 . It is formed by reaction of an Magnesium oxide with water (Eq. 13). The only destruction reaction to matter in the upper atmosphere is reaction with atomic hydrogen (which leads to recovery of neutral Mg (Eqs. 14,8). Low number densities of atomic hydrogen on the one hand and increasing number densities of water vapour in lower altitudes on the 5 other hand make Mg(OH) 2 a very efficient reservoir species for Mg.
We use SCIAMACHY limb measurements in an altitude region of 70-92 km to simultaneously analyze Mg and Mg + profiles. Thus, the partitioning as well as the spatial 10 variabilities of Mg and Mg + can be investigated.

The SCIAMACHY instrument
The European ENVISAT satellite launched in March 2002 carries three instruments designed to perform observations of the atmosphere. Beside the Fourier transform spectrometer MIPAS and the stellar occultation device GOMOS there is the SCIAMACHY in-15 strument (Scanning Imaging Absorption Spectrometer for Atmospheric Chartography). SCIAMACHY covers the wavelength range from 220 to 2380 nm with a spectral resolution of 0.2 to 1.5 nm and provides nadir, limb and occultation viewing modes (Noel et al., 1999), see Fig. 3. The limb viewing mode covers altitude levels from 0 to 92 km. This altitude range covers a large part of the mesosphere. Limb scans are used to within a layer of thickness ∆h. This value can be chosen by the user (see Sect. 3.4).
Though radiation emitted due to resonance fluorescence is unpolarized, the polarization of Rayleigh scattering depends on the scattering angle. As SCIAMACHY consists of grid spectrometer devices, the radiation intensity detected depends on the polarization of the light entering the instrument. This polarization sensitivity of the instrument 10 is accounted for.
The radiative transfer model presented here will be denoted by MARS (Mesospheric Atmospheric Radiative Transfer Simulator). It has been compared to the radiative transfer model SCIARAYS (see Kaiser, 2001). Results of limb calculations agree within 2.5% in an altitude range of 60-95 km, a wavelength range of 250-300 nm, and a range 15 of the solar angles of 0-88 for zenith and 0-180 for azimuth.

Rayleigh scattering and ozone absorption cross sections
We use the Cabannes form (Chandrasekhar, 1960) of the Rayleigh scattering cross section for gaseous species. The following formula is slightly simplified according to the fact that the refractive index of the gaseous species under consideration is close to unity:

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where the wavelength λ is given in nanometers, N L =2.69×10 19 cm −3 denotes the Loschmidt number, and F K is the King correction factor: Here, ρ is the depolarization factor of air. The commonly used value ρ=0.0295 (used e.g. in Rozanov, 2001) is used in this radiative transfer model. The refractive index of standard air n s is calculated using the Edlén formula (Edlen, 1966): The Rayleigh phase scattering function is given by where θ denotes the scattering angle (Eichmann, 1995). The probability of scattering into a solid angle d Ω under the scattering angle θ is given by P (θ)d Ω/4π. Ozone absorption cross sections are taken from laboratory measurements .

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The case of resonance fluorescence to the ground state is considered only. That is, the retrieval species are assumed to be excited by sunlight of wavelength λ i j from the ground state i to an energetically upper state j . De-excitation then leads to isotropic and unpolarized radiation of the same wavelength. Due to low number densities of all species considered here de-excitation by quenching is neglected.
The g-factor links number density and emitted radiance, e.g.

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Quantities indexed with i j correspond to the transition from the upper state j to the lower state i . The g-factor is calculated as the product of the actinic flux, the absorption cross section σ i j and the relative Einstein coefficient of spontaneous emission (Anderson and Barth, 1971). Here, σ i j depends on the classical electron radius as well as on the transition wavelength λ i j and the oscillator strength f i j . The relative Einstein 5 coefficient presents the probability of relaxation to the lower state i . Note that there may be a number of lower states reachable from state j . This is accounted for by normalising A i j by the sum of all respective absolute Einstein coefficients.
10 All values necessary for numerical calculations are obtained from the NIST database (NIST, 2005).

Irradiance and radiance computations
The extinction of radiation traversing the atmosphere follows the Lambert-Beer-Law. The absorption coefficient k(x, λ) at a point x within the atmosphere depends on the scattering/absorption cross sections at wavelength λ as well as on the number densities of the scatterers/absorbers. Throughout this paper, the total number of species under consideration will be denoted by nspecies. All cross sections are assumed to be independent of the point of evaluation x. In particular, the ozone cross sections are assumed to be independent of temperature. This leads to the following expression for Using the Lambert-Beer's law leads to the optical thickness τ sun , τ sat (indexed with sun, respectively. sat to account for different light paths from the sun to the scattering point respectively. from the scattering point to the satellite). To calculate the solar irradiance F (Q, λ) present at a point Q within the atmosphere, it is necessary to integrate the absorption coefficient from Q to the TOA along the light path connecting the sun with 5 Q: A parameterization of the light path can be obtained by using the solar zenith and azimuth angle Φ, Ψ.

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The conversion of irradiance F to radiance I is done by applying the Rayleigh phase function P and the emissivity, respectively. the scattering cross section. As scattering and emission by resonance fluorescence are mathematically not distinguishable, the total emissivity can be written in a similar way as the absorption coefficient: Here, the quantity ε i denotes the scattering cross section respectively the g-factor of species i . The vector ε ∈ R nspecies merges these values. The scattering angle necessary for scattering into the LOS is then just the angle between the sun direction 4606 EGU at Q and the LOS. It depends on the local solar zenith angle as well as on the local solar azimuth angle. Let the scattering angle be denoted by θ Q . Note The light scattered in the LOS and entering the instrument is subject to extinction once again. The extinction between Q and the instrument can be calculated analogously: Here, +TOA denotes the top-of-atmosphere towards the satellite.
To obtain the total radiance at the instrument, the radiances at all points Q along the LOS have to be integrated and weighted with the respective optical thickness: Here, Q(s) denotes a parametrization of the LOS, and -TOA corresponds to the TOA away from the satellite. The second equation is a more convenient form of the first. These integrals are evaluated by numerical quadrature using an adaptive grid (see Sect. 3.4). That is, the atmosphere is divided into altitude layers and all quantities are evaluated at discrete locations along the respective light paths. The trapezoid rule is 10 used throughout the retrieval, as it is simple, fast and accurate at the same time.
Let n sun , n sat , n LOS be the number of points the light paths are divided into. The c sun,i , c sat,i , c LOS,i denote air mass factors (AMF) along the light paths, the discrete correspondences of the differential operators d s:

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The following equations for the absorption coefficients and the total radiance hold: In limb mode, the SCIAMACHY instrument performs 31 to 35 limb scans from the surface to ≈92 km tangent altitude in steps of approx. 3.3 km. A last measurement is taken at 150-200 km, which is not included into the forward model. The instrument has a vertical field-of-view (FOV) of 2.6 km. Thus, the instrument convolves all radiation 10 within the FOV. This is accounted for by calculating the radiance on a fine altitude grid (stepsize 1 km) and then convolving the values with the SCIA FOV slit function of full-width-half-maximum (FWHM) W =2.6 km. For this purpose two slit functions have been tested. Rectangular: Gaussian, order 10:

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The standardization constant for the Gaussian is found using the following method: The integral ∞ −∞ S(h)dh is evaluated for a number of values for 1≤F ≤10. The relationship between F and the value of the integral is virtually linear, and the integral vanishes for F =0. The regression coefficient is found to be between 1.068 and .9941, and the relative error is well below .5% for all values of F . This method has been tested for 5 Gaussians of order 2 . . . 20.
Differences between the results using different slit functions are found to be small and thus a Gaussian of order 10 is used throughout all calculations.

The retrieval
3.2.1 Basic principles 10 Taking into account nal t altitude levels and nl am wavelengths, the observed limb radiances can be joined together in a measurement vector y. The same holds for the number density profiles of all species (remember that nspecies denotes the total number of species), these are merged in an atmospheric state vector x: The forward computations are treated in an analogous way using a discrete forward model operator F that depends on the atmospheric state x: However, due to discretization and other effects such as uncertainties in measuring the observational parameters (SZA, altitude, etc) one has to allow for a forward error δ: The aim of the retrieval is to invert F to find x. However, this problem is ill-posed, e.g. due to noise in the measurement it cannot be assumed that y (or y−δ) lies within the 4609 Introduction EGU range of F . That is, in general, Eq. (37) has no exact solution. Moreover, the altitude resolution is limited as one limb view traverses the entire atmosphere. All altitudes above (and, due to the finite width of the field-of-view, to a certain extend even below) the tangent altitude contribute to the observed radiance. The latter is true even for a ideal measurement with no noise. Thus, the best one can do is to minimize where S y denotes the measurement covariance matrix. This way, knowledge about the measurement uncertainties can be introduced as weighting factors in the retrieval. However, in general there is an infinite number of atmospheric state vectors x that minimize (38). Additionally, small perturbations of y due to noise may lead to large variations of the retrieval result. This may lead to large oscillations.

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Thus, some kind of regularization has to be applied to restrict space of possible solutions. The most common regularization scheme is the Tikhonov regularization using a priori information x a : Here, S a presents the a priori covariance matrix. This matrix contains a priori uncertainties of the atmospheric parameters. This approach has been adapted and reformulated by C. Rodgers (see Rodgers, 1976) to derive the Optimal Estimation method.

Weighting functions
The derivations of the limb radiances with respect to the atmospheric state parameters x j constitute the weighting function matrix K :

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The derivative of the radiance at a certain tangent altitude is calculated using the above formulas: These integrals are evaluated using quadrature algorithms as well.

Minimization method
The functional (39) is minimized using a Levenberg-Marquardt-style algorithm. Set The gradient and the Hessian of G are computed as follows: A newton step consists of the solution of a linear system of equations: 4612 EGU Unfortunately, the newton algorithm converges only locally and only in case the Hessian is positive definite. An algorithm with global but slow convergence is the method of gradient descent. This method uses the unit matrix I instead of the Hessian. Thus, the update step is To overcome the disadvantages of both methods, the retrieval implemented here uses the Levenberg-Marquardt method, i.e. doing step like Eq. (48) using a dynamic convex combination of the unit matrix and the Hessian instead of the Hessian alone. If the Hessian is positive definite, Eq. (48) is performed. If the Hessian fails to be positive definite, the algorithm performs The parameter θ is chosen minimal such that B is positive definite. Different stepsizes in θ are possible. The iterations may be stopped at a point x * ∈ x ∈ R nspecies·nalt if one of the following criteria is fulfilled: -The norm of ∇G(x) falls below a certain (relative) threshold (e.g. one percent of 5 the initial norm).
-The residual ||F (x)−y|| does not improve significantly in a certain set of successive iterations.
-The value ||x−x a || does not change significantly in a certain set of successive 10 iterations. The method of Optimal Estimation does not provide an exact result, but rather a probability density function (PDF) of the true state of the atmosphere. This PDF is assumed to be of Gaussian shape, and the retrieval solution x R actually constitutes the mean value. The covariance matrix can be written as follows: The diagonal values of this matrix are the variances σ 2 i of the state vector elements x * i , and hence the standard deviations σ i can be used as an estimation of the retrieval error. Throughout this paper, these values are used as estimations of the profile errors.
Let x t be the true state of the atmosphere. A useful information is contained in the averaging kernel matrix This derivative reflects the influence of the true state on the retrieved one. An ideal 5 measurement and retrieval would result in an unity matrix A. As a real instrument like a limb sounder has a limited spatial resolution, the retrieved number density at an altitude h(k) may be influenced by number density at lower and higher altitudes. This is quantified by the off-diagonal elements of the k-th row of A. The vertical resolution can thus be estimated as the full-width-half-maximum (FWHM) of the averaging kernel 10 function (which is discretely represented by the k-th row of A) for this altitude. Moreover, the sum of the k-th row of A can be used as an estimation of the measurement response. When it is close to 1, the retrieval result is completely determined by the measurement and not by the a priori.
The measurements of Mg and Mg + presented here exhibit very poor averaging ker-15 nels (see Figs. 24,26). However, this is just due to the fact that the a priori itself contains much information from the measurement due to the preconditioning (see Sect. 3.4.6). 4614 However, as a inherent feature of limb geometry, information about higher tangent altitudes is contained in every tangent altitude covered by SCIAMACHY. To gain information about the thermospheric content, the topmost limb scan is treated as a quasinadir measurement of the thermosphere. Self-absorption is very weak (however, it is accounted for in the retrieval) and absorption by scattering out of the LOS as well as 10 absorption by O 3 is negligible at these high altitudes. Thus the signal of a certain emission species at 92 km depends almost linearly on the column density of this species (see Fig. 10).
The one-dimensional Newton method is used to estimate the thermospheric content. Let y(ρ) be the forward computation vector corresponding to a thermospheric 15 content ρ (given e.g. in cm −2 ). The measurement at 92 km tangent altitude may be denoted y m (92). The minimum ρ T of the function F (ρ)=||y(ρ)−y m (92)|| 2 then gives a good estimation of the thermospheric content.

Numerical issues
Though the general approach of discretization and numerical quadrature will lead to 20 results in principle, the retrieval has to be designed to deal with a fundamental computational issue. That is, all calculations have to be done fast and accurate at the same time. To fulfill this need the following optimization steps have been applied: The forward modelled radiance at high altitudes depends on the choice of the TOAhigher values of TOA will result in higher radiances. Although this is true for arbitrarily high TOA, the change in radiances tends to zero as the TOA increases. It has been found that radiances at all wavelengths within the range considered here increase less 5 than 1% when changing the TOA from 110 to 120 km. Thus, a TOA of 120 km is chosen throughout the whole radiative transfer code.

Adaptive quadrature grids
The accuracy of the numerical quadrature depends crucily on the number and position of the quadrature points. 10 First, the LOS is divided in more parts if the tangent altitude decreases. The length of the LOS within the atmosphere varies from more than 1000 km at low tangent alitudes of 55-60 km to about 200 km at the topmost tangent height 92 km. This is accounted for. The lower the tangent altitude the more often the RTE is evaluated.
Second, the traverse length of a sun-bound ray increases if the SZA increases. At 15 30 SZA the way a ray has to travel is about 70% less than at 60 SZA. Changing from 60 to 80, the path length increases by a factor of more than 2. This is accounted for. The higher the SZA, the more quadrature points are chosen along a ray.

Separation of retrieval for different species
Rayleigh backscatter is a broadband effect. Line emissions, however, only feature ob-20 servable signals at very few isolated wavelengths. Moreover, the backscatter radiances at wavelengths different from the emission/excitation wavelengths are not correlated with the emissions itself. A two-step retrieval has been developed to exploit this fact. In a first step, the air and O 3 densities are determined from backscatter radiances near the emission wavelength. It is assumed that instrumental errors such as calibra-EGU tion behave similar for nearby wavelengths. Thus one may assume that a good background fit obtained from the first retrieval step will stay good when the wavelengths under consideration are changed a little bit for the second retrieval. The second retrieval then treats air and O 3 density as fixed while the number densities of the emission species are determined. These two steps provide a more reliable separation of 5 the background radiance from the emission than using a fixed profile of air and O 3 for the retrieval of the emission species.

Relative deviations instead of absolute values
The radiative impact of the various species involved in the retrieval is highly variable. For example, the number density of mesospheric ozone is roughly six orders of magnitude smaller than the air density within the same altitude region. Though the absorption cross section of O 3 is larger than the Rayleigh scattering cross section of air, the absorption coefficients differ by up to four orders of magnitude, depending on the wavelength region.
To make computations more homogeneous and comparable, the estimated retrieval parameters (i.e. the atmospheric state vector) are not the number densities of the atmospheric species itself but rather their deviations from a respective given a priori profile. That is, where the actual retrieval result is given by

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The usual first order Taylor series expansion for the forward model operator F thus can be rewritten in the following manner: where K is given by According to this notation, K i j denotes the fractional variation of the radiances with respect to to the fractional variation of the atmospheric parameters.

Improvement of S/N
The emission signals are very small and heavily contaminated by noise. To get a more distinct signal, the measured spectrum is divided through by a solar spectrum (see 5 Fig. 12). To prevent artefacts that are due to the different wavelength scales of the limb and the sun spectrum, the wavelength grid of the latter is adjusted to the grid of the first using a shift-and-squeeze algorithm.

Preconditioning by onion-peeling
Though the Levenberg-Marquardt is at least in theory a globally converging algorithm, it 10 is desirable to provide an initial state vector x that is as good as possible. Starting with no information and using an arbitrarily chosen initial state vector x (e.g. a zero profile) 4618 To obtain a preliminary estimate of the number density at each tangent height, an onion-peeling method is used as a first step. It uses similar, multiply times applied method as the thermospheric estimation (Sect. 3). After determination of the thermospheric content, the following steps are performed for each tangent altitude H(k) from the top tangent height on downward. Note that for each species these steps have to 10 be performed likewise.

Consider the altitude interval
centered around tangent altitude H(k). Keep all number densities at higher altitudes fixed.
2. Assume number densities to be equal at all altitudes within A k (assign the name ρ, say, to this value. See Fig. 13a) and minimize the functional Here, y m (k) denotes the measurement at tangent altitude k and y(ρ) denotes the modelled radiances that are obtained by convolution over A k , assuming all num-15 ber densities to have equal value ρ. The regularization factor γ may be interpreted as a one-dimensional covariance matrix S a in a classical optimal estimation retrieval. It prevents the profile from suffering from large oscillations. The a priori value ρ 0 may take any value mirroring a priori knowledge. Here, ρ 0 =0 is chosen.

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Note that in case the radiative transfer is fairly linear, the above functional is well approximated by a parabola, any optimization method such as Newton's method or Regula falsi (the latter is chosen here due to computational time reasons) will do well.
The number density profiles with respect to the fine altitude grid are obviously piece-5 wise constant. Interpolation to the retrieval grid then delivers profiles that are used as new initial profiles for the conventional optimal estimation retrieval (Fig. 13b).

Sensitivity analysis
A number of retrieval runs has been performed to analyse the dependency and stability of the retrieved profiles with respect to the input parameters (absorption/scattering 10 cross sections) as well as the measured data (irradiances, radiances, wavelength grid, SZA, SAA, tangent altitude grid). As a typical setup of SCIAMACHY measurements, the following parameters have been chosen to produce a synthetic measurement:

Measurement noise 20%
The retrieval is run with the following setup: EGU be wrong by up to 1.5 km (v. Savigny et al., 2006). For completeness, a downshift is considered as well as an upshift by 1.5 km. The tangent height grid of the synthetic measurement is decreased resp. increased by 1.5 km and the retrieved profile is compared to the true profile. Surprisingly at first sight, the offset between the retrieved and the true profile has the same sign but is approximately half the 5 offset of the respective tangent height grids. This may be due to the limited height resolution of SCIAMACHY, which is found to be approximately 5 km (twice the value of the tangent height step), see Sect. 4.1. Mesospheric air density is retrieved from Rayleigh backscattered radiance. As Rayleigh scattering is highly wavelength dependent, the retrieval is supposed to obtain information from a very large wavelength range. However, computational capacity and time limits restrict the coverable range. To overcome this disadvantages, 15 a number of microwindows consisting of a small number of detector pixels is selected. The complete set of these windows covers a large part of SCIAMACHY UV channel 1 (see Fig. 19). The wavelength microwindows are chosen in a way not to contain any atmospheric emission features that would perturb the Rayleigh information.

Mesospheric air density
At lower wavelengths (≤260 nm) the O 3 absorption in the Hartley-Huggins bands 20 has a large impact on the radiative transfer. The ozone number density is contained as an additional retrieval species. However, very little information is contained in the mesospheric measurement. Thus no scientific benefit is gained from these results. An MSIS profile (Hedin, 1991) is used as a priori for air density and O 3 .

Results
The retrieved profile of air density (Fig. 20) is in reasonable agreement with the model profile. The strong increase at altitudes above 85 km is due to straylight in SCIA-MACHY's upper tangent heights (van Soest, 2006). However, the spectral data of SCIAMACHY have not been corrected for the pointing error yet. This offset accounts 5 for values of up to 2 km (v. Savigny et al., 2005). That is, retrieved profiles might be shifted upwards by this value. Note that the offset between the model profile and the retrieved one is approximately 1.5 km. Note, however, there is no correlation yet between the choice of the a priori and the geolocation of the tangent point.
The averaging kernels as shown in Fig. 21 exhibit a FWHM of ≈5 km. This can be 10 used as an estimate of the instrument's altitude resolution. The information content is near unity, indicating that the results are determined primarily by the measurement and not by the a priori. See Sect. 3.2.4 for a stringent derivation of averaging kernels. Using support by mesospheric and thermospheric atmosphere models, possible applications for the retrieval of mesospheric air density contain retrievals of temperature 15 as well as pressure. However, the retrieval results of air and O 3 are used primarily to have a good approximation of the background radiance, which is essential for the retrieval of emission species. Thus, the results in terms of air density are not discussed in further detail here.

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The measured radiances are divided through by a solar spectrum of the same day measured by SCIAMACHY. A retrieval using the absolute radiances is possible in principle but leads to very poor results compared to those using relative radiances.
Before running the actual Optimal Estimation retrieval, a preconditiong is performed to provide good a priori knowledge. Due to this, the influence of the measurement on 5 the final retrieval result as calculated from the averaging kernels (see Sect. 3.2.4) is very weak. Note, however, that the a priori itself contains much information from the measurement.

Results
The Mg profile shows a pronounced peak around 85 km, see Fig. 23. These values are 10 significant in terms of the retrieval/measurement error as well as in terms of the information content that can be read from the averaging kernels. This result is consistent with model calculations (Fritzenwallner and Kopp., 1998;Plane et al., 2003). These calculations suggest the maximum abundance of neutral Mg being located around 86-89 km and the peak concentration of Mg + located around 95-100 km (see Figs. 9a, The profile of the ionized species Mg + is virtually zero below 85 km, though the limb signal is well pronounced (see Fig. 25). This indicates that the major abundance of ionized Magnesium is in the thermosphere at altitudes above the top tangent altitude of SCIAMACHY. The thermospheric column densities agree well with LIDAR observations 20 of the total column done over Wallops Island (36.9 N, 1.7×10 1 0 cm −2 ) and Sardinia (36.2 N, 2.1×10 9 cm −2 ) (comp. Fritzenwallner and Kopp., 1998).
As can be read from the averaging kernels shown in Figs. 24 and 26, the vertical resolution of the retrieval is approximately 5 km. The information content for Mg is well around 1 for altitudes above 75 km. Mg + , however, performs worse. The measurement A forward model for radiative transfer calculations in the mesosphere has been developed. It has been coupled to an augmented Optimal Estimation Retrieval. Numerical improvements and stabilizations (such as preconditioning) have been applied A joint retrieval of Mg, Mg + and mesospheric air density is now available and has been tested. The retrieval has been optimized to the corresponding species by adaptive choice of spectral microwindows. The altitude resolution (5 km) at mesospheric altitudes is found to be worse than what can be expected from the actual limb tangent altitude step and FOV width (both approx. 3 km). The pointing error of SCIAMACHY is not corrected for yet.
Retrieval results of Mg, Mg + and air density show reasonable agreement with corre-15 sponding models. Column densities of the metallic species are in well agreement with previous rocket measurements.   (Fritzenwallner and Kopp., 1998). Maximum MgI and MgII abundances are predicted between 90 and 100 km. (c): Comparison of the MgII layer predicted by model calculations, taken from (McNeil et al., 1998). The four curves correspond to different rate constants r (given in cm −6 s −1 ) for the reaction Mg + +2N 2 →MgN + 2 +N 2 . Curves are denoted by − ln(r). Scattered: Ion mass spectrometry measurements published in Grebowsky et al. (1998). These measurements suggest highest abundances of MgII to occur at altitudes above 90 and below 105 km. Model calculations predict a layer with maximum abundance at altitudes between 87 and 105 km, depending on r.  Fig. 26. Corresponding averaging kernels for Mg + (Fig. 25). The dashed line shows the sum of all kernels. As can be read from the averaging kernels, the information content of the measurement drops rapidly to zero for altitudes smaller than 82 km. The FWHM is approximately 5 km.