An algorithm for the retrieval of global stratospheric aerosol profiles is presented using the optical spectrograph and infrared imaging system limb scatter measurements as an example data set. The retrieval utilizes a one-dimensional version of the MART nonlinear relaxation inversion suitable for limb scatter. A height profile of the particle size distribution must be assumed in order to retrieve the aerosol number density. An altitude normalized wavelength ratio measurement vector is employed to minimize effects of upwelling radiation from ground albedo and uncertainties in the neutral density profile. Using a method of numerical perturbation, a formal error analysis is performed that shows that the dominant error term is the measurement noise. Comparison with stratospheric aerosol and gas experiment (SAGE) II and SAGE III coincident measurements of extinction shows agreement with the limb scatter retrievals to within 15% throughout the lower stratosphere for an appropriate choice of particle size distribution. The relatively high sampling resolution of the limb scatter technique makes this data set of interest for studying the dynamics of the stratosphere, stratosphere/troposphere exchange, and for extending the long history of the aerosol data record from the SAGE series of occultation measurements.
 For more than a decade, the upper atmosphere has been unperturbed by any significant volcanic event. Since Mt. Pinatubo in 1991, aerosol in the upper troposphere and the stratosphere has been in a state of relaxation toward background levels. In fact, recent years afford an opportunity to study the background stratospheric aerosol layer at an equilibrium level that has not been observed in the era of space-borne remote sensing of the atmosphere.
 The first measurements of the stratospheric aerosol layer, made by Junge et al.  with balloon instrumentation occurred during a volcanically quiescent period [Stothers, 1996] but the results of the study were limited in nature and so could not provide a complete picture of the background layer. While the important radiative effects of the stratospheric aerosols [McCormick et al., 1995] are greatly diminished in times of low aerosol loading, recent analysis shows that the direct cooling effect of aerosol scattering may have a more dramatic effect than previously thought [Coakley, 2005]. Also, the influence of aerosol, even at background levels, on trace gas chemical reactions is significant. For example, the partitioning of NOx/NOy, which is important in the ozone cycle [Crutzen, 1970], is sensitive to low levels of aerosol surface area.
Hamill et al.  have described, in detail, the life cycle of the background stratospheric layer. The source gases enter the stratosphere in rising tropical air masses, nucleate near the cold point tropopause, grow by condensation, coagulation, and mixing in the tropical stratospheric reservoir, and transport slowly to high latitudes where the main removal mechanism is through tropopause folding events and descent in the polar vortex. As such, satellite measurements of aerosol have been used in previous work to study the dynamics and circulation of the stratosphere [Grant et al., 1996; Trepte and Hitchman, 1992] and stratosphere-troposphere exchange [Menzies and Tratt, 1995].
 It is widely known from in-situ measurements that background stratospheric aerosols are spherical droplets of hydrated sulfuric acid, approximately 75% H2SO4 and 25% H2O. Generally, a lognormal distribution of particle sizes fits the data quite well. The background aerosol is characterized by the smallest particles with a typical mode radius between 0.1 and 0.3 μm and a mode width varying from 1.1 to 1.8 [Bingen et al., 2004; Deshler et al., 2003]. Multimodal distributions characterize the volcanically perturbed state with larger mode radius values ranging beyond 0.5 μm.
 In the “Assessment of Stratospheric Aerosol Properties (ASAP)” document recently released by The WMO/ICSU/IOC World Climate Research Programme [Thomason, 2006], a key recommendation is a commitment to the continued space-based monitoring of stratospheric aerosols into the “foreseeable future” due to their importance in climate and chemistry. Previous decades of space-based measurements of aerosol extinction have been well covered by a long series of occultation instruments. However, with the recent loss of stratospheric aerosol and gas experiment (SAGE) III and polar ozone and aerosol measurement (POAM) III and a lack of upcoming occultation experiments, the future of stratospheric aerosol monitoring is uncertain.
 The optical spectrograph and infrared imaging system (OSIRIS) instrument [Llewellyn et al., 2004], launched on the Odin satellite in 2001 [Murtagh et al., 2002], is one of a number of recently launched instruments that measure tangent height profiles of the ultraviolet (UV)/visible/near infrared limb scattered sunlight. Others include scanning imaging absorption spectrometer for atmospheric chartography [Bovensmann et al., 1999] and SAGE III [Rault, 2005]. The Ozone Profiler and Mapping Suite (OMPS) on the upcoming National Polar-Orbiting Operational Environmental Satellite System Preparatory Project is designed to provide limb scatter measurements for operational trace gas retrievals [Dittman et al., 2002]. The limb scatter technique combines high vertical resolution (1–4 km) and good spatial coverage of the sunlit hemisphere. A number of techniques have been developed for the retrieval of trace gases from limb scattered spectra including spectral analysis techniques [Flittner et al., 2000; von Savigny et al., 2003; Rault, 2005] and the application of differential optical absorbtion spectroscopy [Haley et al., 2003; Rault, 2005]. It is of interest to use the limb scatter technique for the retrieval of stratospheric aerosol parameters because of the relatively high sampling resolution and the need to continue monitoring of aerosol in the absence of an operational occultation experiment. A reliable retrieval of aerosol from limb scatter is also necessary because the retrievals of trace gases such as O3 and NO2 are sensitive to any error in the assumed aerosol profile [von Savigny et al., 2005]. Loughman et al.  show that even for background aerosol loading, uncertainty in the aerosol profile is the second most significant source of error in the limb scatter retrieval of ozone. This paper presents a technique for the retrieval of stratospheric aerosol from the OSIRIS limb scatter measurement set.
 The Canadian optical spectrograph and infrared imaging system (OSIRIS) is one of a new generation of satellite instruments designed to measure the along track atmospheric limb radiance of scattered sunlight [Llewellyn et al., 2004]. The instrument is onboard the Swedish satellite Odin [Murtagh et al., 2002], which was launched on 20 February 2001, and continues full operation to date. The Odin orbit is sun-synchronous and near-terminator with an ascending node local time of 1800 h and a period of 96 min. The orbit inclination of 98° provides near-global coverage as the corresponding sampled latitude range is nominally from 82°S to 82°N. The local time remains very near to 1800 h on the ascending track of the satellite and close to 0600 h on the descending track, and sweeps quickly through noon at high northern latitudes and through midnight at high southern latitudes. As such, the winter hemisphere at Odin local times is not illuminated by the Sun. For two time periods in each year, the Odin orbit track is closely aligned with the solar terminator providing a solar zenith angle at the tangent point of the line-of-sight very close to 90° at all latitudes. The variation of solar zenith angle over a year is between approximately 60° and 120°. The solar scattering angle ranges from approximately 55° to 125°.
 The two subsystems of OSIRIS, suggested by its name, are an optical spectrograph (OS) and an infrared imager (IRI). The IRI is composed of three vertical near-infrared linear array channels that capture one dimensional images of the limb radiance at 1.26, 1.27, and 1.53 μm at a tangent altitude resolution of approximately 1 km. The tomographic inversion of the 1.26 and 1.27 μm oxygen infrared atmospheric (OIRA) band volume emission rate is the heritage of the algorithm used here for the one-dimensional species inversion. The optical spectrograph, which is the focus of this work, is essentially a grating and a CCD detector, and measures spectra of the limb radiance from 280–800 nm with a spectral resolution of approximately 1 nm. Vertical profiles of the limb radiance are obtained by taking OS exposures while performing a repetitive vertical scan of the single line-of-sight through selected tangent altitude ranges from approximately 10 to 100 km. The time required for a single altitude scan is about 1.5 min so allowing for nearly 60 scans per orbit. As the satellite speed is close to 8 km/s, the satellite moves approximately 700 km during a single scan. For a more complete description of the instrument and details of the observations see the work of Llewellyn et al.  and Haley et al. .
3. Retrieval Technique
3.1. Forward Model
 The SASKTRAN spherical radiative transfer model was developed specifically for the efficient retrieval of aerosol and trace gases from limb scatter measurements. The diffuse radiation field that arises from atmospheric multiple scattering and reflection from the ground surface is calculated by successive orders of scattering in a fully spherical geometry. Stratospheric sulfate aerosols are incorporated into the model as scattering and absorbing particles, with cross sections and scattering phase function calculated with a standard Mie code [Wiscombe, 1980] assuming an index of refraction for sulfuric acid tabulated as a function of wavelength [Palmer and Williams, 1975].
 For this work, and for the OSIRIS operational processing, homogeneous spherical shells with a vertical resolution of 1 km are used. A fully spherical geometry is used for all orders of multiple scattering (no pseudospherical assumption is required); however, the multiple scattering contribution to the source term is calculated only at the tangent point and then used at all locations for integration along the line of sight. For the Odin/OSIRIS dawn/dusk orbit sampling geometry, this assumption usually leads to only very small errors, i.e., less than 1%, in the total radiance (D.A. Degenstein, manuscript in preparation, 2006).
3.2. Aerosol Signature in Limb Scatter
 For particles that are spherical droplets of sulfuric acid, the scattering cross section at visible and near infrared wavelengths is several orders of magnitude larger than the absorption cross section. Therefore in a basic sense, where the total atmospheric optical depth is small, stratospheric aerosols enhance the total limb radiance beyond the Rayleigh background level by a small fraction. In regions of high total atmospheric optical depth, addition of aerosols contribute to the extinction of radiation from the scattering point along the line of sight more than they contribute to additional scattering into the line of sight. This results in a reduction of the total limb radiance by a small fraction. Therefore unlike the limb sensitivity to an absorbing gas species, which is always negative, the addition of aerosol in the atmosphere can increase or decrease the limb signal, depending on the altitude and wavelength range.
Figure 1 is a plot of the sensitivity, or kernel matrix, of the limb radiance at selected OSIRIS wavelengths to the aerosol number density profile for an assumed lognormal particle size distribution and under normal OSIRIS viewing conditions (solar zenith angle of 72°, scattering angle of 88°) calculated using the SASKTRAN forward model. Each curve in Figure 1 represents the sensitivity of the radiance at a single tangent altitude calculated using a typical aerosol number density profile that is successively perturbed by +1% at each altitude from 1 to 40 km. The curves tend to peak at the altitude corresponding to the tangent altitude. This is because the majority of the information in a limb spectra comes from the tangent point of the line of sight and is discussed in more detail in section 3.4.
 In general, the sensitivity to aerosol increases at longer wavelengths. At the shortest measured wavelengths (less than 300 nm), the limb radiance is essentially insensitive to stratospheric aerosols. The sensitivity at the other wavelengths, greater than 300 nm, shows an enhancement of the limb radiance due to an increase in aerosol density for upper altitudes, and an extinction of the signal at lower altitudes where the optical thickness is large. Note that the altitude where the sensitivity changes from positive to negative is dependent on wavelength and is lower at the longest wavelengths. This point of insensitivity also moves in altitude depending on the shape and magnitude of the aerosol profile. The increased sensitivity of the limb signal to aerosol at longer wavelengths is partly due to the fact that the Mie scattering cross section of the aerosol particles does not decrease as rapidly with wavelength as the Rayleigh scattering cross section of the neutral density.
 In these calculations, we assume a height profile of a single mode, lognormal size distribution and a number density profile that are simple parameterizations consistent with SAGE II retrievals as calculated by Bingen et al.  for postvolcanic conditions (September 1993). Mode radius varies from 0.4 to 0.3 μm from 10 to 40 km and mode width varies from 1.2 to 1.1 over the same range. We have chosen this size distribution as it is significantly influenced by the larger volcanic aerosols and represents a worst case scenario for the modeling and sensitivity studies of this inversion technique. It is shown in section 5 that a size distribution of smaller particles that better represents the background aerosol state of more recent years is required in order to retrieve OSIRIS extinction that agrees well with SAGE II/III measurements. However, the validity of the technique presented here, which is a retrieval of the number density for an assumed distribution, still applies.
 In the forward model, two parameters are required to characterize the effect of the aerosol particles. One of these is the extinction,
defined as a product of the aerosol number density, na, and the scattering cross section, σ(λ), as calculated by Mie theory based on the particle size distribution. As this is a product, the effect of particle size distribution and number density are intertwined in such a way that, roughly speaking, a small number of large particles can produce the same extinction as a large number of small particles. In a retrieval sense, the extinction is not as strongly affected as the number density by incomplete knowledge of the particle size distribution; however, particle size still has an important effect through the second required parameter of the radiative transfer calculation, the scattering phase function, p(Θ). Conveniently, for stratospheric aerosol particle sizes and visible wavelengths [McLinden et al., 1999] show that the phase function changes slowly with mode radius such that the extinction is not a strong function of particle size.
Figure 2 is a plot of the modeled OSIRIS spectrum at 25-km tangent altitude for a clean, Rayleigh/O3 atmosphere and for the same conditions with a typical stratospheric aerosol load. It is clear that the addition of aerosol enhances the signal at long wavelengths; short wavelengths are relatively insensitive to aerosol loading.
3.3. The Retrieval Vector
 The general formulation of the inverse problem is
where each element of the measurement vector, y, is associated with a measured altitude, F represents the physics of the measurement, i.e., the forward model, and is a function of the desired atmospheric parameters, x, often referred to as the state vector. The forward model is also a function of many other variables, all denoted by b. In this case, the state vector is the unknown aerosol profile.
 A spectral analysis technique, often used in limb scatter retrievals, attempts to construct each element of the measurement vector, y, using an algebraic combination of radiance measurements at different wavelengths and altitudes designed to maximize sensitivity to the state vector parameter and minimize the effects of other unknowns.
 For the retrieval of trace gases, it is also standard practice to normalize the measured and modeled limb radiance profiles to an exposure at a reference tangent altitude [Flittner et al., 2000; Haley et al., 2003]. This technique is similar to the solar reference technique used in limb occultation and removes the need for an absolute calibration. It has been shown previously that this altitude normalization provides a degree of insensitivity to unknown ground albedo [von Savigny et al., 2003] as the contribution from the upwelling radiation to the reference altitude exposure is in a similar ratio for all other tangent altitudes.
 It is tempting to choose a set of individual wavelengths in the OSIRIS spectrum away from trace gas absorption, and, using a normalization technique with an upper tangent altitude, solve equation (2) for the aerosol extinction profile that best makes the modeled radiance profile match the measurements at each wavelength. In this case, one must assume a reasonable scattering phase function for the aerosol (Mie or otherwise). This is certainly a possible method; however, it relies on the assumption that the background Rayleigh atmosphere can be modeled perfectly. Any difference between the modeled and measured radiances that is due to an imperfect knowledge of the neutral density, i.e., a gravity wave, or errors in the assumed profile, is attributed to and fit by the aerosol profile. Figure 3 is a plot of the sensitivity of the limb radiance to a change in the neutral density for the same wavelengths and geometry used to show the aerosol sensitivity in Figure 1. These are also calculated with the SASKTRAN model and each curve represents the sensitivity of the measurement at a single tangent altitude to successive small (+1%) perturbations in a reference neutral density profile at each altitude (all neutral density profiles in this work are derived from ECMWF analysis at OSIRIS scan locations; see http://www.ecmwf.int). At all wavelengths and tangent altitudes, the sensitivity to the neutral density is approximately an order of magnitude larger than the sensitivity to the aerosol profile. For a retrieval of the aerosol extinction using only information from a single wavelength, a small error in the assumed neutral density would translate to a very large systematic error in the retrieved aerosol extinction.
 Therefore we propose a spectral analysis construct of the measurement vector. The ratio of a long wavelength, λl, to a shorter wavelength, λs, characterizes the Mie scattering wavelength dependence of the aerosol particles shown in Figure 2 and provides some insensitivity to uncertainties in the neutral density. Each measured spectrum I(h,λ) at tangent altitude, h, is first normalized to a measurement at a reference altitude, hr.
Then, an element of the measurement vector, yj, is constructed for each measured tangent altitude as the logarithm of the wavelength ratio of the altitude normalized profile.
The logarithm of the ratio is used as it creates a better-behaved measurement vector due to the exponential nature of the radiance profile.
 The choice of wavelengths is of course quite critical. It is important to choose λl and λs with the largest possible separation in order to maximize the spectral characterization of the Mie scattering. However, λs must remain sufficiently long in order to provide sensitivity at lower stratospheric tangent altitudes as the limb optical depth quickly becomes large for blue and UV wavelengths, approximately λ < 450 nm. The OSIRIS spectral order sorter, an unavoidable characteristic of the optical design, contaminates wavelengths from 475 to 535 nm [Llewellyn et al., 2004] and the Chappuis ozone absorption band has a significant cross section between 500 to 680 nm. Therefore we have chosen λs = 470 nm, essentially the longest wavelength on the short wavelength side of the order sorter. For the long wavelength we have chosen λl = 750 nm to maximize separation from λs and avoid the O2 A-band absorption (762 nm) and Woods anomalies from the polarization response of the grating above 780 nm [McLinden et al., 2002]. A reference tangent altitude of hr = 40 km was chosen as this is an upper bound for the retrieval because the aerosol number density is negligibly small above approximately 35 km. This represents a type of calibration point where the signal can be reliably modeled, as it is not greatly affected by the aerosol load.
 The sensitivity of this measurement vector to both the aerosol density and the neutral density calculated with SASKTRAN through successive perturbation of each species at specific altitudes is shown in Figure 4. The relative sensitivity to the neutral density compared to the aerosol density is indeed more favorable than for the single wavelength cases. Above 20-km altitude, the vector is more sensitive to aerosol density than it is to neutral density. For lower altitudes nearer to the tropopause, the sensitivity of the neutral density still grows beyond that of the aerosol, but down to approximately 15 km they remain on the same order of magnitude. It is worthwhile to note that this analysis only considers localized changes, or errors, in the neutral density profile. If the assumed neutral density profile was systematically too low or too high at all altitudes, it would have almost no effect because of the altitude normalization.
 A two-dimensional tomographic retrieval technique implemented by Degenstein et al.  is used to retrieve mesospheric volume emission rates of the oxygen infrared atmospheric band measured by the infrared imaging (IRI) subsection of the OSIRIS instrument. The multiplicative algebraic reconstruction technique (MART) was first developed as a statistical method to deblur Fabry-Perot images [Lloyd and Llewellyn, 1989] and has many similarities to the maximum likelihood expectation maximization technique used in positron emission tomography [Shepp and Vardi, 1982] as well as the Chahine [Chahine, 1970, 1972 and Twomey-Chahine [Twomey et al., 1977] retrieval techniques often used in atmospheric remote sensing. This work uses an adaptation of the tomographic technique to solve the one-dimensional inverse problem for the aerosol extinction profile [equation (2)].
 As the MART technique is very similar in nature to the Chahine relaxation, a brief overview of the Chahine inversion in the context of limb scatter is given here to place the MART in context. In the limb geometry, the sensitivity of observation yj to a change in state xi is strongly peaked in altitude as shown in Figures 1 and 4. The maximum sensitivity occurs where the measurement is tangent through the element that is changed. The Chahine nonlinear relaxation technique relies on this condition where the state vector elements, i.e., points on the altitude profile of the species of the interest, correspond in a one-to-one fashion with those of the observation vector, i.e., the measured tangent altitudes. Note that this requires the length of the measurement vector and the state parameter vector to be equal. This is often done by some type of simple interpolation of the smooth atmospheric state profile. Therefore for limb geometry, the altitude of each of the state vector elements must be the tangent altitude of the corresponding measurement vector element, requiring j = i. The Chahine iterative equation,
is simply the modification of each state element, xi, by the ratio of the corresponding measurement vector element, yj=i, to the forward model calculation, F. Here the superscript n denotes the iteration number and F, the SASKTRAN model vector, is calculated using the current estimate, x(n), of the state vector and other assumed external parameters, b, such as the neutral density and the albedo. In the standard implementation, the forward model is only reevaluated after all elements of the state parameter have been adjusted based on the current iteration. In this formulation, the measurement vector is constructed so that if the measurement value is greater than the evaluation of the current state of the forward model, increasing the value of the state parameter produces the desired effect of bringing the model closer to the observations. The iteration process is repeated until the modeled vector matches the measurement within a tolerance that reflects the measurement error. This method is quite advantageous in its simplicity; however, the stability of the solution suffers from the noise associated with the measurements. Given an infinite number of iterations, this type of relaxation method will fit the retrieved solution to the error in the measurements.
 In this work, we use a similar form of nonlinear relaxation that is an adaptation of the two-dimensional tomographic MART inversion implemented by Degenstein et al. . For a given shell element of the state vector, this technique incorporates information not only from the measurement that samples at the tangent altitude, j, that corresponds to the shell element, i, but also from measurements at tangent altitudes below the element through intersection on the near and far side of the tangent point. The iteration equation is
The terms Wi,j make up a weighting filter matrix that determines the importance of the jth element of the measurement vector to the value of the ith element of the state parameter. In an atmospheric sense, Wi,j defines the contribution of measurements at different tangent altitudes, j, for the retrieval at shell altitude i. It is important to note that W is not the Jacobian of linear term derivatives, or kernel, sometimes used in an inversion procedure. The normalization of the entire weighting filter function for any state parameter element, i, is unity.
 Like the Chahine relaxation, this technique relies on the construction of the measurement vector such that a positive change in yj results in an increase in the state parameter. The spectral ratio measurement vector discussed in the previous section provides this functionality. There is an altitude, at each wavelength, where the optical depth becomes large and the aerosol density kernel is zero as shown in Figure 1. Below this point, the kernel matrix elements are negative. This lack of sensitivity and change in sign are extremely undesirable for any inversion scheme and cannot be used with the MART formulation. However, the spectral ratio measurement vector proposed here allows the short wavelength information to be incorporated in a way such that the kernel is positive for all tangent altitudes, i.e., the requirement for the MART inversion. This is demonstrated in Figure 4 where the sensitivity is positive for all tangent altitudes above 10 km.
 The effectiveness of this method depends on judicious selection of the weighting filter, W. It may be difficult to determine the optimal set of weights; however, it is possible to make a weighting filter that functions quite well in a practical sense. It should be noted that the simplest choice of weighting filter, Wij = δij, reduces this method to the Chahine relaxation. However, this adapted MART technique serves to over-determine the problem in a way that provides stability and some degree of insensitivity to random noise in the measurements. The incorporation of multiple measurements to each retrieval altitude also provides a faster convergence to the solution.
Degenstein et al.  set the weighting filter terms proportional to the path length matrix based on intersections with the grid cell in the two-dimsensional atmosphere used for the tomographic inversion. We implement a simplified, but similar variant of W. For the one dimensional problem that we solve here, the path length through a spherical shell at altitude i is maximum for the observation j that samples with a tangent altitude nearest to i. The path length for observation j falls off quickly for shells at altitudes above i. Therefore all measurements with a tangent altitude lower than the altitude of interest, i, have a successively smaller path length of intersection through shell i. All measurements with tangent altitude above measurement j do not intersect shell i. The weighting filter, W, allows these measurements which are lower in tangent altitude than measurement j, to contribute to the retrieval at shell i through the path length relationship. Since a large separation in tangent altitude of incorporated measurements is not desirable, we allow a maximum of three measurements to contribute to the retrieval of a given element of the state parameter. Measurements near the bottom of the scan are handled as a special case. W is therefore a lower triangular square matrix where each row i corresponds to a retrieval altitude. Since the OSIRIS measurements are not uniformly spaced in tangent altitude and are not aligned with shell boundaries, W is different for each scan. For simplicity, we use the same weighting filter for each scan, loosely based on the path length relationship. A typical example, for a scan of 6 exposures, where i = 1 (the first row) is the lowest tangent altitude, the weighting filter is
With a sufficient number of iterations, any similar choice of weighting filter is equally adequate.
where α is a column vector of the factors used to update the state parameter upon each interation,
and m is a column vector of the ratio of the measurement vector to the forward model evaluated at the current value of the state parameter.
Note that both α and m have the same number of elements. For OSIRIS sampling geometry, this is equal to the number of exposures in a limb scan.
 To test the retrieval, a simulation of the limb measurement using a known aerosol number density profile, labeled “control” in Figure 5, was calculated with SASKTRAN. Then, beginning with an initial guess profile, ten iterations of the MART inversion were performed using the measurement vector defined by equations (3) and (4). Using the SASKTRAN model, this requires computation time of less than 1 min on a standard desktop PC (2.8 GHz Pentium D). The measurement vectors corresponding to both the initial guess profile and the retrieved profile are shown in the left panel of the figure, along with the vector directly calculated from the simulated measurements. The right panel of the figure shows the number density profile for the same three cases. The inversion is successful at adjusting the modeled measurement vector to better match the control case. This is also the case for the number density profile. The retrieval matches the control profile to within 3% except at the lowest altitudes where the difference is closer to 15%. The larger discrepancy at lower altitude is due, in part, to the limb geometry, which causes an “onion-peel” like nature of the retrieval. For a small number of iterations, the values at the highest altitudes converge the earliest. As iterations proceed, the lower altitudes then converge because the measurements that sample these altitudes with the largest sensitivity also have a smaller, but significant, sensitivity to the upper altitudes due to the path length through the upper shells. For a larger number of iterations, i.e., ≥30, all altitudes above approximately 12 km have converged to within a small percent difference. The difference that arises at lower altitudes is addressed further in section 4.1. As the purpose of this work is not to investigate the utility of the MART inversion, but to study the retrieval of aerosol from limb scatter measurements, we leave the study of the MART inversion to another paper, and suffice to say that for this purpose, the technique is sufficient.
3.5. Extinction Product
 The retrieved solution is the number density that makes the forward model best match the measurements for the assumed aerosol particle size distribution profile. So the solution is really only truly meaningful in the light of this assumption. It was noted previously in section 3.2 that the extinction [equation (1)] provides some measure of insensitivity to errors in the assumed particle sizes.
 In order to test the variability of the retrieved extinction with respect to errors in the assumed particle size, three simulations were performed. The SASKTRAN forward model was used to construct a set of simulated observations that each used the same aerosol number density profile (labeled “control” in Figure 5) and a different height profile of the size distribution. Figure 6a shows the aerosol cross section at 750 nm associated with the climatological profile of the size distribution used for the simulated retrieval shown in Figure 5. The cross sections that result from adjusting rg by ±10% are also shown. It is important to note that there is a different scattering phase function associated with each cross section. The aerosol number density was retrieved for the three sets of simulated measurements, each time assuming the same climatological size distribution profile. Effectively, the retrieval was performed for perfect knowledge of the particle distribution and for the cases where the particles in the atmosphere are both smaller and larger than those assumed in the retrieval. These results are shown in Figure 6b.
 Where both the simulated measurements and the retrieval used the same cross sections, the solution is identical (to within a small percentage) to the control profile. When rg was adjusted by −10% in the simulated measurements, the retrieved number density is significantly less, by almost a factor of 2, than the control profile at all altitudes. This is essentially a compensation of the lower optical depth due to the smaller cross section. Because the retrieval still assumes the climatological cross sections, the retrieved density must be lower. The opposite is true for the case where rg is adjusted by +10% and the retrieved density is larger than the control profile.
 It is apparent that the retrieved aerosol number density is quite sensitive to errors in the assumed size distribution. However, as expected the extinction, shown in Figure 7 for each of the three cases, does provide some insensitivity to the size distribution. The known, or reference, extinction is calculated as a product of the control density and the three different cross section profiles. The retrieved extinction is the product of the retrieved density and the cross section for the climatological size distribution that is the same in each case. In the first case shown in Figure 7a, where the forward model and the retrieval use the same cross sections, the difference in extinction in less than 3% at all altitudes. This difference results directly from the inversion and is due to the interpolation of the profile onto the SASKTRAN altitude grid at each iteration and the convergence of the solution (30 iterations were used to improve algorithm performance at lower altitudes). Figures 7b and 7c show a comparison of the known and retrieved extinction for simulated smaller and larger mode radius, respectively. For both of these cases, the extinction profiles agree with the reference to within approximately 10%.
 This larger discrepancy is due to the scattering phase function. Particles used in the forward model to simulate the measurements for cases (b) and (c) have a different phase function than the particles used for the retrieval. By adjusting the number density appropriately, the retrieval is able to compensate for the difference in optical depth caused by the different cross section. However, the difference in phase function causes an unavoidable systematic error in the retrieved extinction.
 For particles, the phase function is strongly peaked in the forward and back directions and is reasonably flat for scattering near to the plane that is perpendicular to the incident light. For the OSIRIS geometry, the angle between the line of sight and the solar vector is always near 90° so that the single scatter contribution to the limb radiance depends only this relatively flat region of the aerosol phase function. However, the diffuse radiation field, which arises from atmospheric multiple scattering and upwelling radiation, requires scattering at all angles. This can explain the fact that for all three cases shown in Figure 7, the agreement is very good, i.e., <3%, at altitudes near 16–20 km. The limb radiance at λs = 470 nm has a very large contribution from the diffuse field due to the large optical depth of the Rayleigh atmosphere, whereas for low values of ground albedo the λl = 750 nm radiance depends largely on the single scatter component. The aerosol kernel matrix elements for wavelengths around 470 nm drop off quickly at altitudes below about 20 km (see Figure 1). Therefore in this region the measurement vector, which depends on the ratio λl/λs, is largely unaffected by the diffuse field and does not have a large sensitivity to the phase function. Thus, the extinction agrees very well regardless of errors in phase function. Below 16 km the discrepancy in extinction increases again because even at 750 nm the optical depth is becoming large and the contribution from the diffuse field is significant.
 Much of the previous work in the retrieval of species from limb scatter measurements uses a statistical matrix inversion. The Rodgers formal error analysis [Rodgers, 1976, 2000] applies directly to these techniques and identifies three distinct error categories that exist in the retrieval process: smoothing error from the finite resolution of the inversion, measurement error, and forward model error, including both approximations in the model and uncertainties in model parameters. In this formulation, the difference between the retrieved profile, , and the true profile, x, is
where xa is the apriori profile and is used as a linearization point to describe the dependence of the forward model on changes in the state parameter. The matrix A is known as the averaging kernel and is defined as the sensitivity of the retrieval to the true state,
Therefore the first term in equation (12) is related to the smoothing error where the solution is considered exact with uncertainty due to the resolution of the retrieval. The second and third terms in equation (12) arise from a second linearization, this time of the inversion itself, and are therefore proportional to the matrix D, known as the contribution function. This matrix describes the sensitivity of the inversion to the measurement vector,
The second term includes errors that come directly from the measurement process, Δy, such as detector random noise. The third term describes error that arises from the forward model, ΔF, from approximations in the physics of the model, and from uncertainty in model parameters such as neutral density, albedo, etc.
 For a relaxation-type inversion, such as the MART used in this work, an algebraic derivation of these matrices in not applicable. However, following the work of [Puliafito et al., 1995], we attempt to determine the matrices A and D numerically through perturbation of the true profile and the measurements using a set of simulated measurements constructed with the forward model such that the true state is known. A standard formal error analysis can then be carried out; however, it should be noted that it is only strictly valid for the profile used in this numerical analysis.
4.1. Smoothing Error
 In the absence of noise, the well-posed relaxation-type inversion always converges to the exact solution in the limit of infinite iterations [Puliafito et al., 1995]. This means that the averaging kernel is very close to the identity matrix, i.e., a change in the true profile is perfectly reflected in the retrieved solution.
To test the magnitude of the smoothing error for the aerosol inversion proposed here, columns of the averaging kernel matrix, A, were calculated by successively performing the simulated retrieval shown in Figure 5, each time perturbing a single altitude of the true state, x, by 1% and computing the differences in the retrieved profile, [see equation (13)]. Note that A is dimensionless. Figure 8 shows a plot of the columns of A for altitudes between 10 and 40 km at 2-km vertical resolution. For altitudes above 20 km, the columns are very close to delta functions centered at the altitude of the perturbation. This means that inversion captured the perturbation of the true state in the retrieval very accurately. Below 20 km, the effect is blurred, or smoothed, over a range of altitudes near the perturbation resulting in a retrieval error that increases with lower altitudes. The reason for this lack of information at the lower altitudes is the very large limb optical depth of the atmosphere. Even in a perfect simulation with no measurement noise or forward model error the inversion cannot perfectly converge on the true state at large optical depths. This is simply a physical limitation of the measurement and interpretation of the retrieved state must be carried out with caution at altitudes below 20 km.
 The right panel of Figure 8 is a plot of the results of a simulated retrieval with an initial guess profile (in this context taken as the apriori profile) and a curve representing the smoothing error due to the inversion. It is calculated as the magnitude of the first term in equation (12) and is equal to the difference between the retrieved state and true state and shows that for a typical case a smoothing error of 30% at lower altitudes is possible.
4.2. Measurement Error
 The error in the retrieval that results directly from error in the OSIRIS measurements is determined through the contribution function matrix, D. Like the averaging kernel, columns of D are calculated numerically by perturbing elements of a simulated measurement vector profile [see equation (14)] and calculating the resulting difference in the retrieved profile. These columns are plotted in Figure 9. Note that D is in number density units as the measurement vector is dimensionless [see equation (4)].
 The main sources of measurement error are detector noise and uncertainty in the altitude registration of each exposure. From analysis of equation (4), random error in the OSIRIS radiances due to detector noise, δI, causes an error in an element of the measurement vector,
The covariance matrix describing the error on the measurement vector is the diagonal matrix, Sɛ, where diagonal elements correspond to the height profile of δyj2. From equation (12), the covariance matrix for errors in the retrieval due to measurement errors is calculated through the contribution function as
Figure 10 shows, in the left panel, a plot of the measurement vector constructed from a typical OSIRIS scan with error bars calculated using equation (16). The right panel of the figure shows the retrieved aerosol number density profile for these measurements and the error bar that results from the measurement error as the square root of the diagonal elements of SM. Near the peak of the number density profile the measurement error due to detector noise is less than 10% and at the uppermost altitudes near 35 km it increases to above 100%.
 A similar analysis can be carried out to determine the error in the retrieval that arises due to uncertainty in altitude registration of the limb radiance measurement. Due to the astronomy requirements of the Odin satellite, the OSIRIS instrument has very good pointing accuracy from on-board star trackers. The error in altitude registration is believed to be better than 200 m at the tangent point. Assuming this uncertainty, a similar analysis can be carried out by recalculating the measurement vector for simulated measurements shifted by 200 m in tangent altitude. This again results in an error in the retrieved density profile of approximately 10% for lower altitudes and larger errors of 30–40% at upper altitudes.
4.3. Forward Model Error
 Error from the forward model can be separated into two distinct types. The first type includes all of the systematics that arise from the approximate nature of the model. These are extremely difficult to gauge as an accurate characterization requires comparison with an exact model that is, of course, impossible to attain. Often this type of error is used to describe the systematics that arise when it is possible to use the forward model in an approximate sense. While we certainly do not claim that the SASKTRAN model is perfect, we do utilize it to the fullest extent for the aerosol retrieval. Fully spherical geometry is always used. The multiple orders of scattering and the resolution of all spatial integrals in the calculation of the ith order diffuse profile are done to a sufficiently high resolution that effectively no change in the retrieved aerosol profile is observed. No attempt is made to quantify the effect of assuming horizontal homogeneity or the variation of the diffuse profile along the tangent point.
 The second type of error in this category, often called forward model parameter error, occurs due to uncertainty in the external inputs specified in the radiative transfer. These errors are most certainly systematic in nature. In this approach, the effect of each parameter is investigated independently through numerical perturbation of the simulated retrieval by an amount in the parameter that represents a realistic uncertainty. Total forward model parameter error is the quadrature sum of each term. Surface albedo and neutral density are the most important inputs to the radiative transfer calculation in terms of their uncertainty affecting the aerosol retrieval. Again, a note about particle sizes must be made. The assumed particle size distribution is, in fact, the most important forward model parameter that affects the aerosol retrieval. However, it is such a crucial aspect of the product that we do not consider it an error. It is simply an assumption. The retrieved density is only meaningful with respect to the assumed size distribution and the two must always be considered together. The accuracy of the product as an optical depth or extinction is shown by the analysis accompanying Figure 7.
 The reflectivity of the earth, or the albedo, represents a significant unknown in the radiative transfer modeling of the limb signal. The signature of the radiation reflected from the ground in the limb radiance is much stronger at red wavelengths than at shorter UV/blue wavelengths because shorter wavelengths suffer greater extinction along the path length to the ground and back up to the scattering point due to the Rayleigh optical depth. The reddening of the spectrum is similar to the effect of an increased aerosol load and results in potential confusion between the signal attributed to aerosol and the signal caused by an error in the assumed albedo. Even though the altitude normalization of the measurement vector tends to cancel out the majority of the effect of the albedo, it is important that the best estimate of the albedo be used in the forward model as an error in the assumed albedo does effect the solution.
Figure 11 is a plot of the same simulated retrieval used to investigate the effect of particle sizes but for different values of albedo. This time the forward model of the measurements is calculated using an albedo of 0.4. The retrievals are performed using assumed albedos of 0.3, 0.4, and 0.5. When the correct albedo is used, i.e., 0.4, the solution is accurate to within the same retrieval error discussed in sections 3.5 and 4.1. With an assumed albedo of 0.3, the retrieval attributes some reddening of the spectrum that is actually due to the upwelling radiation to the aerosol and retrieves an aerosol number density that is systematically about 10% too high at all altitudes. The error is slightly larger at the uppermost and lowermost altitudes. The opposite is true when the assumed albedo is 0.5. That is, some reddening of the spectrum actually caused by the aerosol profile is attributed to the modeled upwelling radiation that is too large. The retrieved solution is too low, by approximately 10%, with a slightly larger error at the altitude extremes.
 Because of these similar effects on the limb radiance, given a set of measurements in the region of the aerosol layer, it is difficult to estimate the albedo. However, at altitudes above the aerosol layer a much more reliable estimate of the albedo can be obtained. For OSIRIS scans, the reference altitude chosen for the measurement vector, hr = 40 km, is also a suitable altitude for determining the albedo. For this work, an estimate of the albedo is obtained by fitting the absolute value of the modeled limb radiance at 750 nm and 40 km altitude to the measured value by adjusting the albedo in the forward model. This wavelength was chosen because of the greater sensitivity to upwelling radiation at longer wavelengths and because the contribution from atmospheric multiple scattering, which could represent some error in the forward model, is small. The reflection from the earth is assumed to be Lambertian with no variation in the horizontal direction and the same value of albedo is used at all wavelengths. Even though these sweeping assumptions must be made, it is better to attempt to estimate the albedo from the measurements directly than to use a climatology of earth reflectivity because of the frequency of clouds that significantly modify the amount of upwelling radiation from that predicted by clear sky conditions and earth albedo.
 The main problem with this technique is that it relies on the absolute calibration of the instrument. Even so, the calibration is believed to be good to within less than 10% for measurements on the long wavelength side of the order sorter. This is based on extensive in-flight radiative transfer testing and the validation of other retrievals that rely on the absolute calibration [Lloyd, personal communication, 2006]. This translates into an uncertainty in the estimated albedo of approximately 20%. In order to relate this uncertainty in the albedo to an uncertainty in the retrieved profile, the Rodgers error analysis requires the sensitivity of the forward modeled measurement vector to the albedo parameter, β,
Note that this is a diagonal matrix as the albedo is a single parameter that affects the entire profile of the measurement vector. It is determined numerically by computing the difference between the forward model simulation of the measurement vector due to a small change in albedo. For an albedo covariance matrix, Sβ, by equation (12), the forward model parameter error covariance is
Cloud cover is handled in this work as a modification of the ground albedo. By using a tangent altitude near 40 km, the retrieved albedo is effective for the entire scene below and compensates for cloud cover modification to the upwelling radiation. A systematic error arises in the calculation of the multiple scattering component in the forward model because the light path is modified from the cloud free scenario assumed in the model; however, this error is small compared to the uncertainty from the absolute calibration.
 The same error analysis is applied to the sensitivity to the neutral density where we assume an uncertainty of 1% at all altitudes. Figure 12 is a summary plot of the error terms discussed here as a percentage of the aerosol number density. The total error is the quadrature sum of all terms. Especially at upper altitudes, the dominant uncertainty arises from detector noise. This error is very large in terms of percentage, but it is quite constant with altitude in terms of number density. The large percentage error reflects the very small aerosol load at high altitude. For this case, the total error is less than 20% between 12 and 23 km and increases rapidly above 25-km altitude.
5. OSIRIS Measurements
 This described technique has been applied to the OSIRIS limb scatter measurements. Figure 13 is a plot of the limb radiance at 470 and 750 nm for a typical midlatitude OSIRIS scan (#06432019, 73 degrees solar zenith angle, 104 degrees scattering angle). Also, shown on the plot is the limb radiance calculated using SASKTRAN with an initial guess aerosol number density and the particle size distribution used in the simulations. Radiances are shown on an absolute scale and no altitude normalization has been performed. The albedo used in the model for this scan is 0.57 and was determined using the technique described above. The largest difference between the measurements and the model is at 750 nm at altitudes below 15 km. Slight discrepancies are noticeable at both wavelengths for all altitudes up to approximately 25 km where the measurements and the model are in close agreement.
 The difference between the measurement and the model is much more apparent in the measurement vector [see equation (4)] plotted in the left panel of Figure 14 for the OSIRIS measurements and for the forward model with an initial guess aerosol profile. Because the measurement vector is constructed in such a way that the kernel matrix elements are always positive, the difference between the measured and modeled vector can be simplistically interpreted as an overestimate of the initial guess aerosol load at altitudes above 13 km and an underestimate below. This is not strictly true due to the coupling between altitudes due to multiple scattering.
 The measurement vector calculated with SASKTRAN after 30 iterations of the MART inversion of the aerosol density is shown in the second panel of this figure. The forward model vector is now in very good agreement with the measurements. Convergence to within 2% in the measurement vector, i.e., ∣αi − 1∣ < 0.02, at all altitudes is obtained except at the lowest altitudes where a larger difference is still present due to the retrieval error discussed in section 4.1. The retrieved aerosol number density for this OSIRIS scan is shown in Figure 15 as an example of a typical result.
 Finally, the modeled calculation of the limb radiance at 470 and 750 nm was repeated using the retrieved aerosol density. The results are shown in Figure 16. Even though the fitting of the measurement vector does not require the agreement of the limb radiance with the measurements, at both wavelengths, close agreement is obtained at altitudes above 15 km. Below this altitude, some discrepancy remains; however, especially at 750 nm, the modeled profile is much closer to the measurements than it was before the inversion. Because the modeled radiance profiles using the retrieved profile closely match the observations, systematic error due to aerosol in the further retrieval of trace gases such as ozone is significantly reduced by this solution.
 While a complete comparison and validation study is left for further work, a brief comparison of a single profile measured by OSIRIS, SAGE II and SAGE III is shown in Figure 17. The coincidence of the measurement, detailed in Table 1, was chosen based on time and latitude. All three measurements occur within 8 hours and less than 1° of latitude. Longitudinal separation is large; however, the relatively constant zonal nature of the stratosphere at midlatitudes should allow for a reasonable comparison of the three results. As both SAGE II and SAGE III measure 1020 nm extinction, the Mie cross section for the assumed particle size distribution for 1020 nm was used to convert the retrieved OSIRIS aerosol number density to extinction per kilometer for this comparison. It is important to note that the cross section at this wavelength is not used in the retrieval. It is only required for the conversion of OSIRIS measurements for comparison with the SAGE extinction measurements.
Table 1. Measurement Locations and Time on 5 January 2004, for Extinction Profile Comparison Between OSIRIS, SAGE II, and SAGE III
 In the top panels of the figure, the OSIRIS retrieval assumes Bingen et al.'s  size distribution from 1993 SAGE II data inversions previously used for the simulations in this work. In the lower panels, the retrieval is performed using background layer size distribution parameters consistent with in situ measurements by Deshler et al.  in 2001 (mode radius of 0.08 μm, mode width of 1.6 at all altitudes). It is clear from the figure that the particle size distribution from 1993, which is skewed to larger size by volcanic aerosols, does not retrieve an extinction that is consistent with that directly measured by the occultation instruments. However, the smaller particle sizes that represent the background layer used in the lower panels results in an extinction that agrees reasonably well with SAGE II and SAGE III. The difference between the OSIRIS extinction and the SAGE II extinction between 15 and 30 km altitude is very similar to the difference between SAGE III and SAGE II. At these altitudes, all three instruments agree to within approximately 15%. Below 15 km, OSIRIS extinction is systematically high. This is likely due in part to the growing smoothing error at low altitudes because of the large optical depth.
 A note should be made here about the nature of this limb scatter inversion. An occultation instrument measures the extinction directly. In the limb scatter case, however, the measurement is of a type of triple product of the aerosol number density, the cross section, and the scattering phase function and is coupled in altitude through the significant contribution from multiple scattering. The limb scatter inversion presented here retrieves a number density that is most consistent with the measurement vector given the cross section and phase function from an assumed size distribution. Although the number density depends completely on this assumption, it is still a geophysically meaningful result in that it is an indication of the global altitude distribution of the aerosol load. The conversion to extinction, while it does provide some measure of insensitivity, in an absolute sense, to the uncertainty in the assumed particle size, it is not the intrinsic measurement, and systematic error arises from the phase function. However, it can be seen from this brief comparison with SAGE II/III, with some knowledge of the particle size, the limb scatter inversion can retrieve the correct extinction.
 To demonstrate the utility of the limb scatter data set, a global map of retrieved aerosol extinction at 750 nm for the one week time period from 27 February to 3 March 2006, at a fixed altitude of 20 km is shown in Figure 18. During this time period, the Odin orbit plane is very close to the solar terminator affording coverage over the full latitude range in a single orbit. Missing data is indicated by white in the color scale. Although we leave the interpretation of this distribution to further work, it is interesting to note that regions of enhanced aerosol load are detected near the Amazon, central Africa, and Indonesia, all of which are often associated with source terms of aerosol through deep convection. A clean northern polar vortex, which arises from adiabatic descent of air within the vortex, is also visible in the figure.
 The limb scatter data set, such as that measured by OSIRIS, provides good opportunity for the study and monitoring of stratospheric aerosol. This paper presents a technique for the retrieval of aerosol number density for an assumed size distribution at altitudes from 15 to 30 km. Forward model radiances using the retrieved aerosol agree very well with OSIRIS measured limb radiances which will result in an improvement in further retrieval of trace gases from the spectra. The absolute value of retrieved aerosol number density is meaningful only with respect to the assumed particle sizes; however, the geophysical utility of the data remains in the relative nature of the global distribution. Conversion of the results to extinction provides some insensitivity to the uncertainty in the size. Thus using an in situ measured size distribution that is representative of the background layer for 2002, the OSIRIS 1020 nm extinction compares well with SAGE II and SAGE III throughout the stratosphere for a coincident case investigated here. Future work to gain insight into the size distribution from the measurements may improve the result.