1. Introduction

It is generally recognized that stratospheric aerosol particles consist of a submicron sulfate haze concentrated between the tropopause and an altitude of roughly 30 km. These particles are known to play an influential role in climate and chemistry. In addition, they can interfere with the remote sensing of trace gases (Bhartia et al. 1993). Stratospheric sulfates are also known to possess large spatial and temporal variability, especially as one of the main sources is through the conversion of SO₂, injected into the stratosphere via volcanic eruptions. Thus, a thorough knowledge of their widespread distribution as well as their physical and optical properties is essential to a better quantitative understanding of heterogeneous chemistry processes, climate change, and their sources, sinks, and transport. It is also essential for the accurate calculation of stratospheric radiation and photolysis rates. Finally, specific to the ER-2 flights, it is important to have an accurate description of stratospheric aerosol for forward modeling necessary when retrieving trace gases such as NO₂ or BrO.

From 1993 to the present, a spectroradiometer has been flown on the NASA ER-2 high altitude research aircraft. This instrument, part of the Composition and Photodissociative Flux Measurement (CPFM), measures the horizontal and vertical polarization components of limb radiance along a 10-point scan from approximately 5°–8° above to 5°–8° below the local horizon in the wavelength range 300–770 nm (McElroy 1995; McElroy et al. 1995). The aircraft typically cruises at an altitude near 20 km. Also measured are the nadir radiance and the horizontal flux, which have been used to estimate surface albedo (McLinden et al. 1997). The absolute limb radiance uncertainties are estimated to be about 8% in the visible and 10% in the near-UV. Radiance and polarization have proven to be very useful in determining aerosol mass characteristics (Brogneiz et al. 1997; Herman et al. 1986). Aerosol retrievals are performed using measurements from the Photochemistry of Ozone Loss in the Arctic Region in Summer (POLARIS) campaign.

The remainder of this study is organized as follows. First, a description of the vector radiative transfer model and the retrieval algorithm are given. Following this are results from two POLARIS flights, which include number density profiles and size distributions. Finally, discussion and conclusions are presented.

2. Radiative transfer model including polarization

The model used in this study solves the vector radiative transfer equation in a plane-parallel inhomogeneous atmosphere (McLinden 1998). All polarization information is contained in the Stokes vector, I, a four-element column matrix (often written as a row matrix for convenience), {I, Q, U, V}. The first element, I, represents the total radiance. The second, Q, is the radiance linearly polarized in the directions parallel and perpendicular to a reference plane. The CPFM spectroradiometer measures the parallel and perpendicular components of limb radiance, denoted I_‖ and I_⊥ respectively. The relationship between these and the Stokes vector elements are

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so that the fraction of light linearly polarized is given by

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The vector radiative transfer equation is solved using the successive orders of scattering solution technique. In this method the Stokes vector elements are calculated for photons scattered once, twice, three times, etc., with the total of each Stokes vector element being the sum of overall scattering orders (e.g., Hansen and Travis 1974). In addition, the azimuthal dependence of the Stokes vector elements and phase matrix elements are expanded out in a Fourier series.

The atmosphere is the U.S. Standard Atmosphere at 70°N for spring and extends from 0 to 100 km. To improve modeling of the limb, spherical corrections have been implemented. Integration of the source vector is done using spherical shells and by taking it as varying linearly with optical depth through each shell. In addition, the direct solar beam is attenuated in a spherical atmosphere. The surface is assumed to be both Lambertian and depolarizing. All angular integration is performed using Gaussian quadrature. Rayleigh scattering cross sections are calculated from an empirical formula (Nicolet 1984) and the depolarization factor (due to the anisotropy of N₂ and O₂) is taken as 0.0279 (Young 1980). Aerosol phase matrix elements and cross sections are calculated numerically using a Mie scattering code. The model has been thoroughly tested against tabulated results of other polarized radiative transfer models (Garcia and Siewert 1989; Evans and Stephens 1991; Stammes et al. 1989). All comparisons were made for homogeneous Mie or Rayleigh plane-parallel atmospheres using a wide range of optical thicknesses and solar angles. In all cases differences were found to be <0.1% (McLinden 1998).

3. Aerosol retrieval algorithm

The quantities to be retrieved are the aerosol number density vertical profile and the aerosol size distribution, assumed to be independent of altitude. It is further assumed that the refractive index for a 0.75 H₂SO₄ + 0.25 H₂O aerosol composition is appropriate. Refractive indices for this composition have been measured at 300 K (Palmer and Williams 1975) and they are adjusted to stratospheric temperatures using the Lorentz–Lorenz relation (e.g., Steele and Hamill 1981). The size distribution is described in terms of an effective radius, r_eff, and an effective variance, υ_eff. These are useful quantities as they take into account the fact that larger particles tend to be more efficient scatterers. The mathematical definitions for r_eff and υ_eff are given in the appendix.

The ER-2 flights used were from 26 April 1997 and 6 May 1997, both out of Fairbanks, Alaska (65°N, 148°W). The two flight tracks are shown in Fig. 1. Results are presented from 2100 UTC (83°N, 148°W; SZA = 70°) 26 April 1997 and 2215 UTC (75°N, 108°W; SZA = 61°) 6 May 1997. All limb scans used were during clear-sky conditions. The surface is believed to be predominantly snow and ice-covered snow, which is modeled as a Lambertian reflector. The albedo was obtained using the nadir and horizontal flux fields, also measured by the CPFM, using the general method described in McLinden et al. (1997). Only limb scans in which the previous and subsequent scans had similar radiance and polarization values were selected. This is important as it takes about 15 min to complete a scan, during which time the aircraft has traveled about 150 km.

As the CPFM spectroradiometer measures radiance from 300 to 770 nm, there is a great deal of flexibility as to what wavelength(s) to use. As an initial constraint, all wavelengths at which significant gaseous absorption occurs were ruled out. This eliminates 300–330 nm and 480–680 nm due to absorption by ozone. Absorption by NO₂ occurs from 400 to 460 nm. There are also O₂ absorption bands centered near 760 nm. Therefore, three wavelengths are selected: 340, 475, and 750 nm. In general, longer wavelengths should be better suited because with a λ⁻⁴ dependence on Rayleigh scattering the fraction of the signal due to scattering by aerosols will be greater.

The retrieval algorithm can be roughly divided into two parts: 1) determining the extinction coefficient profiles using the limb radiances and 2) determining the size distribution using the limb polarization and the previously determined extinction coefficient profile.

a. Extinction coefficient profiles

The limb radiances are sensitive mainly to the aerosol optical thickness along the line of sight and less so to the details of the size distribution as long as α ≥ 1, where α = 2πr_eff/λ is the aerosol size parameter and λ is wavelength. This is because the phase function (or the P₁₁ element of the phase matrix) changes slowly with particle size. The exception is when the Rayleigh limit is approached, or roughly when α < 1. For r_eff = 0.2 μm, a typical size for stratospheric sulfates, α = 1.7 at 750 nm and α = 3.7 at 340 nm. Thus, radiances at shorter wavelengths are quite insensitive to changes in aerosol size, assuming the optical thickness is constant. Changes in r_eff by ±50% resulted in a variation in radiance of <1% for a CPFM simulated limb scan. At 750 nm, varying r_eff by +50% resulted in a maximum change in limb radiance of <1% but varying it by −50% resulted in a maximum change of about 5%. Further calculations revealed that the phase function remains stable at 750 nm for r_eff ≥ 0.12 μm. This radius is also representative of the lower limit of stratospheric sulfates (Kent et al. 1995), indicating that 750 nm should be useful in determining extinction profiles, irrespective of the size distribution.

On this basis, a constant lognormal size distribution (defined below) having equivalent size parameters of r_eff = 0.2 μm and υ_eff = 0.28 is used in determining the extinction coefficient profile. In addition, only the limb radiances will be used, as polarization is sensitive to the details of the size distribution. For convenience, the extinction coefficient profile is assumed to be well represented by

kλ, zaλbλzcλz²dλz³(3)

following the model of McCormick et al. (1996), where z is the altitude above the tropopause (km) and k is the aerosol extinction coefficient (km⁻¹). The extinction coefficient can be interpreted as the aerosol optical depth per unit (physical) length and is the product of number density and cross section. The advantage of using this approach is that the lower-midstratosphere extinction coefficient profile can be expressed using only four parameters. This is especially useful if the results are to be used in models. A drawback of this approach is that small-scale structure cannot be captured. As a result of the geometry, only limited aerosol profile information can be extracted as there is an information overlap between uplooking scan steps. It was shown in a previous study that limb radiances were largely insensitive to the amount of aerosol below the lowest point along the line of sight and hence the tropospheric aerosol profile is less important (McLinden et al. 1998). The tropospheric aerosol extinction coefficient adopted is from upper-tropospheric Stratospheric Aerosol and Gas Experiment (SAGE) II measurements for background aerosol conditions (Kent et al. 1995).

The coefficients a, b, c, and d are determined by minimizing the difference of the measured and modeled limb radiances. The rms difference, ϵ_o, is the quantity to be minimized,

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where Iⁱ_m represents the measured radiance at the ith step in the limb scan, Iⁱ(a, b, c, d) is the calculated radiance using (3), and M is the number of steps in the scan. Limb scan steps that probed below the tropopause were excluded. An estimate of the uncertainty can be obtained for each parameter by varying it until ϵ_o is 10% larger than the minimum, following the approach of Sato et al. (1996). The value of 10% was chosen as it represents the absolute uncertainty in the limb radiance. Model radiances were observed to be relatively insensitive to the value of d at all wavelengths. This suggests that much of the aerosol information is for altitudes below that at which the d(λ)z³ term from (3) becomes important.

This procedure was carried out at 340, 475, and 750 nm. The limb radiances are shown as a function of elevation angle in Fig. 2 for the scan made at 2100 UTC 26 April 1997. Elevation angle is defined as the angle above (positive) or below (negative) the local horizon. For these scans, the height of the tropopause is estimated to be about 8 km. There is excellent agreement at all three wavelengths. The extinction coefficient profiles and the uncertainties are shown in Fig. 3 at the three wavelengths, while the extinction coefficient parameters are provided in Table 1. The extinction coefficient uncertainties are calculated based on the uncertainties in a, b, c, and d as follows:

kkabzcz²dz³(5)

Note that the uncertainties at 340 nm are quite large as the radiances are relatively insensitive to the aerosol due to increased multiple scattering and molecular scattering. At 20 km, aerosols account for roughly 20% of the total scattering. In contrast, the 750-nm radiance is quite sensitive to aerosol amount and vertical distribution as even slight changes in the aerosol profile induce substantial increases in ϵ_o.

b. Size distribution

It is convenient to employ an analytic expression to represent the size distribution so that the entire distribution can be described by a few parameters. Also, it has been shown that for all but recent posteruption periods, the majority of the aerosols are in a single mode. In this case, all major characteristics of a distribution can be obtained from two parameters, so the lognormal size distribution (Hansen and Travis 1974) is employed:

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where r_g and σ_g are size parameters and n(r) is the size distribution defined such that n(r)dr represents the number of particles with radii between r and r + dr. It is normalized such that ∫^∞₀ n(r) dr = 1. It is assumed that the size distribution is height independent, or nearly so, so that one distribution is representative of the aerosol at all heights in the lower stratosphere. The parameters r_g and σ_g are related to the effective radius and variance via

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Using the previously determined extinction coefficient profile, the size parameters can be determined in an analogous fashion using the limb polarization. Polarization is much more sensitive to the details of aerosol size distribution and hence is ideal for its retrieval. The rms difference is minimized to determine the size parameters r_g and σ_g,

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where LPⁱ_m is the measured polarization at the ith step in the scan and LPⁱ(r_g, σ_g) is the model polarization. Uncertainty estimates in r_g and σ_g are determined by varying each until ϵ_o is 10% larger than the minimum. Hansen and Hovenier (1974) and Sato et al. (1996) used a similar approach to determine the size distribution and refractive index of sulfates in the Venusian atmosphere. The polarization scans at 340, 475, and 750 nm for 2100 UTC 26 April 1997 are shown in Fig. 4. In general, polarization is seen to decrease with increasing wavelength due to a decreasing contribution from Rayleigh scattering. The best agreement between model and measurement occurs at 750 nm and the worst agreement at 340 nm. Even at 750 nm there appears to be some fine structure that the model did not capture. This is likely due to the fact that the actual size distribution is height dependent and this retrieval process can only determine a single “effective” size distribution. The increasingly worse comparison at shorter wavelengths is due in part to error propagation in the extinction profile. Although the model agrees quite well with the 340-nm measured radiances, the estimated error is in the retrieved extinction coefficient profile is observed to be quite large. Again, this is due to the decreased relative contribution of aerosol scattering to total scattering.

The resultant size parameters from the 750-nm polarization are given in Table 2 and the size distributions are shown in Fig. 5 for each of the scans studied. Values for r_eff were found to be 0.20 ± 0.03 μm and 0.17 ± 0.03 μm for the scans from 26 April 1997 and 6 May 1997, respectively. Both are larger than the minimum 0.12 μm required for the extinction coefficient retrieval process to be insensitive to size distribution. Using 475-nm or 340-nm polarization to obtain size information resulted in estimated uncertainties in excess of ±0.08 μm. This, combined with the larger errors from the extinction coefficient profiles at these wavelengths, makes their use less than ideal.

c. Number density profile

The number density vertical profile N(z) can be recovered from extinction coefficient profile using

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where σ_M is the Mie scattering cross section and can be calculated exactly for a given size distribution and refractive index. Using the 750-nm extinction and size distribution, the two number density profiles have been calculated and are shown in Fig. 6. There is maximum at 15 km of 10 cm⁻³ at 2100 UTC 26 April 1997 and a maximum at 12 km of 20 cm⁻³ at 2215 UTC 6 May 1997. Above this there was a steady decrease.

As an internal consistency check, the 20-km extinction coefficient was calculated using

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and compared with the 20-km extinction coefficients at 340 and 475 nm as determined separately using the procedure outlined in section 3a. The agreement was found to be reasonable and within the estimated uncertainties indicating that size parameters are reasonable. At 30 km, the agreement was not quite as good, which is likely due to the fact that the actual aerosol size distribution is not truly independent of height.

d. Measurement intercomparison

1) In situ particles counters

Also flown on the ER-2 were two in situ aerosol counters. The first is the ER-2 Condensation Nucleus Counter II, which counts particles with radii between 0.004 and 1 μm (Wilson et al. 1983). The second is the Focused Cavity Aerosol Spectrometer (FCAS) II, which detects particles with radii between 0.004 and 1 μm (Jonsson et al. 1995). The FCAS II also measures aerosol surface area per unit volume and aerosol volume per unit volume. For comparison with retrieved aerosol densities at ER-2 height, the surface area and volume densities for the lognormal size distribution can be computed:

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where SA(z) is the lognormal surface area density and V(z) is the lognormal volume density. The number, surface area, and volume density comparisons are given in Table 3. The CPFM values were evaluated using the number density value at the ER-2 altitude. For the particle counters, the measured range over the duration of the limb scan is listed. Overall, the agreement is quite good as all CPFM-determined values, within their uncertainties, fall into the in situ measured ranges over a limb scan. The good agreement between surface area and volume densities indicates that the CPFM-derived effective radius and variance are realistic.

2) SAGE II

General comparisons are made against 1.02-μm extinction coefficient profiles SAGE II measurements from the 1989–90 period. It is reasonable to use this period for comparisons as it represents a period of stratospheric minimum after the eruptions of El Chichón and other smaller volcanos (Thomason et al. 1997a). Similarly, the effects of the 1991 Mount Pinatubo eruption have decayed enough to reasonably suggest that the present period is also one of stratospheric minimum.

The 1.02-μm extinction coefficient profiles are calculated using (10) and compared with least squares fits of 1989–90 SAGE II data to (3) (McCormick et al. 1996). This comparison is performed simply to confirm that the extinction coefficient order of magnitude and general shape of the profile is similar. From Fig. 7, it is clear that this is so. SAGE II data from 1989–90 also estimates the effective radius at 70°N and 20 km to be in the range 0.10–0.15 μm (Kent et al. 1995), which is generally consistent with the values of 0.2 ± 0.03 and 0.17 ± 0.03 μm from this study. Also, the CPFM surface areas listed in Table 3 are comparable to the late 1994–early 1995 SAGE II surface areas at 20 km and 70°N (Thomason et al. 1997b).

4. Conclusions

Stratospheric aerosol number density profiles, extinction coefficient profiles, and size distributions have been estimated using ER-2 CPFM polarized limb radiance measurements and a vector radiative transfer model. This was done using the radiances to determine the extinction coefficient profiles and the polarization to determine the size distribution. Results from a limb scan made at 2100 UTC 26 April 1997 indicate a maximum in number density at 15 km of 10 cm⁻³ with an effective radius of 0.20 μm and an effective variance of 0.21. Results from a limb scan made at 2215 UTC 6 May 1997 indicate a maximum in number density at 12 km of 20 cm⁻³ with an effective radius of 0.17 μm and an effective variance of 0.24. The increasing uncertainties of the extinction coefficient with increasing altitude combined with the relative insensitivity of the model radiances to the value of d suggest that these profiles may not necessarily be valid above approximately 30 km. The retrieved number, surface area, and volume densities at 20 km agree well with two in situ particle counters.

Improvements to this basic method should be possible using multiple wavelengths. Future studies will also be directed toward estimating how small errors in the CPFM viewing direction might impact aerosol retrievals. In addition, the feasibility of retrieving tropospheric aerosol information from the nadir radiance and polarization will be examined.