### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. AIRS Instrument
- 3. Data Analysis
- 4. Retrieval Scheme
- 5. Examples and Comparison
- 6. Error Budget
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] The concentrations of volcanic sulphur dioxide (SO_{2}) in the upper troposphere and lower stratosphere are inferred using new infrared measurements made by the Atmospheric Infrared Sounder (AIRS). Column abundance of SO_{2} is derived using the strong SO_{2} absorption feature near 1362 cm^{−1} (*ν*_{3}-band). The retrieval takes into account interference from water vapor across the band. Examples from several recent volcanic eruptions are given to illustrate the technique and the retrievals are compared to contemporaneous SO_{2} ultraviolet measurements from the Ozone Monitoring Instrument (OMI), Global Ozone Monitoring Experiment (GOME) and Total Ozone Mapping Spectrometer (TOMS).

### 4. Retrieval Scheme

- Top of page
- Abstract
- 1. Introduction
- 2. AIRS Instrument
- 3. Data Analysis
- 4. Retrieval Scheme
- 5. Examples and Comparison
- 6. Error Budget
- 7. Conclusions
- Acknowledgments
- References
- Supporting Information

[7] The retrieval scheme is a two-step process. In the first step pixels that contain SO_{2} are identified. In the second step a least squares procedure is used to find the amount of SO_{2} in the pixel, based on off-line radiative transfer calculations. The upwelling radiance received by AIRS is assumed to consist of emission from the surface and from the atmosphere,

[8] *I*_{ν,s} is the radiance emitted from the surface, *B*_{ν} is the Planck function, *ν* is wave number (cm^{−1}), *T*(*z*) is the temperature as a function of height *z*, *τ* is the transmittance and *q*_{i}(*z*), *i* = 1…*n* are constituent profiles of SO_{2}, H_{2}O, CO_{2}, O_{3} etc. The aim of the analysis is to retrieve the column abundance, *u*_{1} of SO_{2}, which is related to the constituent profile by,

[9] A portion of the AIRS spectrum between 1295 and 1405 cm^{−1} is used for the retrieval. Although SO_{2} has absorption features at 2500 cm^{−1} and 1160 cm^{−1} these are not used in the retrieval. The 2500 cm^{−1} feature is often discernible in the data, while only the long wavelength side of the 1160 cm^{−1} feature can be identified in AIRS because there are no channels between 1170 and 1180 cm^{−1}. By using the 1295–1405 cm^{−1} interval, the surface emission term in (1) can be neglected, because the atmosphere is effectively black in this interval. This is demonstrated in Figure 1a where the transmittance at the top of the atmosphere as a function of wave number is shown for a US standard atmosphere, with background SO_{2} (and other gases) using the MODTRAN-3 radiative transfer model. The transmittance profile of the SO_{2}*ν*_{3}-band is also shown. Except for two small regions between 1320–1335 cm^{−1} and 1342–1358 cm^{−1}, the atmosphere appears to be opaque, and this is due principally to water vapor absorption. As the water vapor resides mostly in the lowest layers, closest to the surface, higher in the atmosphere the transmittance is higher. At some point in the atmosphere, which depends on the water vapor and SO_{2} amounts, the atmosphere is sufficiently transparent and the SO_{2} amount sufficiently large that the signal from the SO_{2} dominates over that from the water vapor, and if the signal is greater than the instrumental noise, AIRS is able to detect and quantify the SO_{2} absorption.

[10] The total column background atmospheric SO_{2} in the absence of volcanic activity is typically less than 1 Dobson unit (DU) (1 DU = 2.6849 × 10^{16} molecules cm^{−2}) (<0.2 DU in the boundary layer). We assume that the SO_{2} lies in a layer at *z* = *z*_{1} to *z* = *z*_{2} (above the boundary layer) so that equation (1) can be written,

[11] The AIRS retrieval relies on being able to correctly identify pixels within the image granule that are affected by SO_{2}. To do this we assume that the transmission of radiation within this restricted band, for each pixel, follows the Beer-Bougier-Lambert law:

where *I*′_{ν} is the radiance at wave number *ν* leaving the SO_{2} layer measured at the satellite (term 2 in equation (3)), *I*_{ν,0} is the radiance entering the SO_{2} layer from below and is equivalent to term 1 in (3), and *k* is the absorption coefficient. We assume that the radiance contributions from the atmosphere above the SO_{2} layer (term 3 in (3)) are the same with or without the SO_{2} layer. The radiance contributions reflected off the SO_{2} layer are assumed to be negligible. Figure 1b shows the ratio of the path radiance to the surface emission at the top of the atmosphere for a background atmosphere, as a function of height, together with the same ratio for an atmosphere containing ∼50 DU of SO_{2} placed in a single layer at ∼5 km. This is the ratio of term 2 on the right-hand side of (1) to *I*_{ν}, with the lower limit of the integral in term 2 replaced with *z*. Above the SO_{2} layer the radiances are almost identical and small (less than 10% of the total emission). It can be seen that the difference between the radiance profiles with and without the SO_{2} layer are very small, and we are justified in neglecting this contribution in the light of more serious sources of error. We treat the radiation field as isotropic and assume that the atmosphere is locally horizontally homogeneous below and above the SO_{2} layer. From (4) we can determine the absorbance spectrum,

[12] For a single spectral Lorentz line or band and a single absorbing gas, the absorption can be written,

where *α* is the line half-width and *ν*_{0} is the location of the line center. For a homogeneous path, the integral can be solved to yield [*Goody*, 1964],

where *S* is the line strength, and *L*_{0} and *L*_{1} are modified Bessel functions. The weak, and strong absorption limits give a linear and square-root dependence respectively, of the absorption (*A*) on the absorber amount (*u*). The absorption due to a single spectral line or spectral band is often referred to as the equivalent width measured in units of cm^{−1}. In practice the absorbance is due to many lines, the path is inhomogeneous and there may be lines due to multiple absorbing gases, some with overlapping lines. Because the path is inhomogeneous there is a temperature and pressure dependence of the line parameters with height. A series of MODTRAN-3 simulations was carried out to examine how well this absorption model holds for the 7.3 *μ*m SO_{2}*ν*_{3}-band. Figure 2 shows a plot of the absorption (cm^{−1}) versus absorber amount (milli atm-cm or DU) for an SO_{2} layer inserted at two heights in the atmosphere, for amounts up to 120 DU. Up to about 40 DU the relationship is linear, and follows the weak-line absorption limit. Beyond about 50 DU, the absorption is nonlinear and follows the square-root dependence of the strong-line limit. At high absorber amounts, the absorption tends to a constant and presumably the band is saturated resulting in a low sensitivity. Despite the presence of other interfering gases, the simulations suggest that the Lorentz band model is quite accurate. Calculation of the absorbance spectrum requires identification of a background or reference pixel (*p*_{r}, *l*_{r}), from which the reference radiance *I*_{ν,0} is determined. This pixel is found by comparing the absorbance spectrum of each pixel with a synthetic spectrum calculated using library line strengths (the HITRAN 96 database is used; *Rothman et al.* [2003], MODTRAN-3 and a standard atmosphere perturbed by 100 DU of SO_{2}. Spectra were calculated using varying amounts of SO_{2} (from 10 DU up to about 120 DU) and all have similar shapes and are highly correlated. For different atmospheres with different amounts of interfering gases, there are only small changes in the spectral features as determined by these simulations. The method of determining the optimal reference pixel relies solely on the degree of correlation between the absorbance spectrum computed from,

and that computed from,

where *I*_{s} is the synthetic radiance spectrum (1295–1405 cm^{−1}) with 100 DU of SO_{2} and *I*_{0} is the synthetic spectrum with background SO_{2}. *I*_{pt}*,l*_{t} and *I*_{pr}*,l*_{r} are measured AIRS radiance spectra (functions of ν; the reference to ν has been dropped for notational convenience) for the target pixel [*p*_{t}, *l*_{t}] and the reference pixel, [*p*_{r}, *l*_{r}], respectively, and *p* and *l* represent pixel and line number. The R^{2} correlation is calculated from,

[13] The ordinates of the spectrum occur at discrete values of wave number, *ν*_{i} that are determined by the AIRS instrument characteristic, and *n* is the number of channels used (*n* ≈ 140). and are the normalized measured and synthetic absorbance spectrums, defined as,

[14] The reference pixel is deemed to be that pixel which produces the highest R^{2} correlation over all other reference pixels in the image. Generally, this pixel is geographically close to the target pixel and thus the assumption that the atmospheres of the target and reference pixel be similar is likely to be met for each SO_{2} pixel. The correlation is lowest when other gases interfere with the spectral matching, or when the SO_{2} column amount is low or when clouds and collocated water vapor exist within the SO_{2} layer or above it. The relation between the R^{2} correlation and the column SO_{2} need not necessarily be linear or even positive. High SO_{2} amounts may occur in the presence of enhanced water vapor loadings and the spectral matching may, in this case, produce a low correlation. Note that it is the spectral shape that determines the correlation, not the absolute absorber amount. For thin SO_{2} layers in very dry atmospheres over cold surfaces many of the underlying assumptions of the retrieval scheme break down and the R^{2} correlations can be quite low regardless of the amount of SO_{2} present.

[15] The procedure is time consuming, but is objective and optimal in some sense. Once the reference pixels have been found (there will be potentially up to one less reference pixels as there are target pixels), the absorbance spectra for all pixels with R^{2} correlations above a specified value are calculated and a correlation image (containing the R^{2} correlations between the synthetic and observed spectra for all pixels) determined. The correlation image and the reference pixel arrays are retained during the analysis procedure. Figure 3 shows a comparison between the observed absorbance spectrum (dotted black line) and the synthetic spectrum (continuous red line) for six pixels with different degrees of calculated correlation for the 10 May 2003 eruption of Anatahan (see later). Also shown in Figure 3 are the retrieved SO_{2} column amounts derived during the second stage of the retrieval process (see later). The correlations vary from R^{2} = 0.46 (Figure 3a) to R^{2} = 0.96 (Figure 3f); in this case the pixel with the highest R^{2} correlation also contains the highest SO_{2} column amount. The location (pixel number and geographic coordinates) are also given on Figure 3 and it can be seen that in all cases but one, the reference pixel is on the same line number as the target pixel and in all cases the reference pixel is clear of any retrieved SO_{2} (see Figures 5a and 5b in section 5.1). There are some possible scenarios where the spectral matching might produce erroneous results. These might be when the reference pixel contains some SO_{2}, leading to an underestimate of the target SO_{2} or when the reference pixel is geographically distant from the target pixel, and hence the atmospheric environment might be different. From experience with processing large amounts of AIRS data it seems these cases are rare. When SO_{2} is present in a reference pixel it is expected that the R^{2} correlation will be lower than for another SO_{2}-free pixel. However, it is possible to have an SO_{2} cloud so large that it covers the entire AIRS granule (in that case one might choose to use a second contiguous granule) and we caution that there are other scenarios one could imagine that might affect the correlations.

[16] The column SO_{2} is now determined for pixels exceeding a specified R^{2} from the absorbance spectrum using a linear least squares method. In this method a set of precomputed spectra are determined at discrete levels in the atmosphere at 2 km intervals starting at 6 km and ending at 20 km. These are linearly combined to produce a least-squares “best fit” between the measured and computed spectra. The computed spectra include the effects of a constant predefined water vapor distribution and the least-squares is improved by providing an estimate of the height of the cloud layer, which is assumed to contain only SO_{2} and enhanced water vapor (a constant amount for all retrievals). There is a strong dependence between absorbance and the height of the SO_{2} layer in the atmosphere, which is unknown. The height can be specified from a trajectory model run, or from some other independent source.

[17] The amount of SO_{2} retrieved using the 7.3 *μ*m band is dependent on the location of the SO_{2} layer in the atmosphere. There is a temperature dependence of the line strengths for this band. There is also a strong vertical dependence because of the interfering effects of water vapor. We define the cutoff height for SO_{2} retrievals as the height in the atmosphere where the signal-to-noise ratio reaches unity. The NEΔT for AIRS channels between 3.7 and 13.5 *μ*m is ∼0.2 K at 250 K, which gives NEΔI ≈ 0.1 mW/(m^{2} sr cm^{−1}) for channels in the region 1295–1405 cm^{−1}. The signal strength can be calculated from simulations,

where *I*_{ν,z′} is the spectral radiance from an atmosphere with a prescribed amount of SO_{2} placed at height *z*′ in the atmosphere, and *I*_{ν,0} is the spectral radiance from an unperturbed, background atmosphere. The signal to noise ratio (SNR) is,

We define the cutoff height *z*_{c} = *z*′ when SNR = 1.0. Figure 4 shows the variation of SNR with height (*z*′) for a tropical atmosphere with 50 DU placed at different levels in the atmosphere from the surface up to about 5 km. In this case the SNR = 1 at *z*′ ≈3 km. It is apparent that for this case, below ∼2.5 km, the noise is twice the signal strength. For drier and wetter atmospheres, *z*_{c} decreases or increases, respectively; likewise for emplacements of larger masses of SO_{2} the signal strength would be larger and the cutoff height would decrease. Under most atmospheric conditions absorption by water vapor across this band is significant below ∼3 km and we maintain that the AIRS instrument is effectively “blind” to SO_{2} emissions in the boundary layer, and often below ∼3 km (but note that this cutoff height depends on the SNR). This constraint suggests that the 7.3 *μ*m AIRS channels behave like a filter to reveal mostly upper troposphere/lower stratosphere (UTLS) SO_{2}, which is more likely to be climatically significant.

[18] Another approach to assess the information content of the AIRS channels in the 1295–1405 cm^{−1} interval is to use weighting functions. The weighting function [e.g., *Rodgers*, 2000] is

which appears in the integrals of (3). Weighting functions for channels near 7.3 *μ*m typically peak between 400 and 600 hPa depending on the water vapor, temperature and SO_{2} vertical profiles. This suggests that this waveband has greatest sensitivity to SO_{2} in the mid to upper troposphere (4–8 km). Given knowledge of these three profiles it is straightforward to calculate the *W*s for AIRS channels and determine which atmospheric layers are contributing to the measured radiances. The value in this approach is that it permits an objective means for selecting specific (high information content) AIRS channels for retrieving SO_{2}. Here we employ an ad hoc “two-step” method of first identifying SO_{2} pixels through spectral matching and then retrieving the column abundance through a least-squares procedure and off-line radiative transfer calculations, utilizing all AIRS channels between ∼1320 cm^{−1} to ∼1395 cm^{−1}. We sacrifice any potential vertical SO_{2} profile information in pursuing this approach.

[19] Equation (5) in matrix notation may be written.

where *x* = {*x*_{o}… *x*_{N}} is a vector of layer absorber amounts to be determined, *K* is an *MxN* matrix consisting of absorption cross sections at *M* wave numbers and *N* atmospheric layers, and *y* is a vector of measured absorbances at *M* wave numbers. The absorption cross sections at 8 levels, starting at 6 km and ending at 20 km in 2 km steps, are predetermined from a detailed radiative transfer program [*Griffith*, 1996]. The least-squares solution to (10) is [*Rodgers*, 2000],

[20] Equation (11) is underdetermined because the basis functions *K* are approximately linear functions of each other. The dominant processes affecting the shapes of the spectral lines within the band are pressure and Doppler (thermal) broadening. Temperature decreases through the UTLS (upper troposphere–lower stratosphere) in an almost linear manner, giving absorption cross sections in different layers which are very nearly linear functions of each other. To stabilize **K**, we found it necessary to include the effects of a second gas: water vapor. Thus the basis functions consist of absorption cross sections for a standard profile of water vapor with a single layer of enhanced SO_{2} and H_{2}O at a prescribed level. The retrieval produces a layer abundance of SO_{2}, which we treat as a column amount. The retrieval scheme produces layer amounts in eight layers, which are integrated to obtain a column amount. The information content in the layers is insufficient to expect an accurate vertical profile. Improvements to this scheme would include a better specification of the water vapor profile (potentially this could be determined from AIRS standard retrieval products) and simultaneous retrieval of SO_{2} and H_{2}O. This would also help to identify enhancements in “in plume” water vapor associated with the volcanic cloud. In principle it is also possible to determine some vertical profile information on SO_{2} by judicious choice of “microwindows” within the *ν*_{3}-band.

[21] The SO_{2} is retrieved on a pixel by pixel basis using (11) for only those pixels that exceed a specified value of R^{2}. The units of SO_{2} amount are molecules cm^{−2}, which we convert to milli atm cm or DU. The use of DU for SO_{2} column abundance is done for consistency with the OMI, TOMS and GOME retrievals. The total mass loading (in Tg) is evaluated from:

where *i* is pixel number, *u* absorber amount (in DU), *β* is the area of a pixel (in km^{2}) evaluated assuming elliptical pixels on a spherical Earth, and θ is the AIRS scan angle subtended at the Earth's surface.

[22] Each SO_{2} retrieval is accompanied by an R^{2} correlation map. R^{2} ≈0.3 was found to delineate the boundary for SO_{2} retrievals of 6 DU, which is considered to be the lower limit of SO_{2} detection from AIRS. In this work we compute total SO_{2} mass loadings for pixels with R^{2} = 0.1 and R^{2} = 0.7, which provides a range of certainty and an error bound for the retrieved products. Finally, the height of the SO_{2} cloud is required as input to the retrieval. This information is derived using wind trajectories determined using the HYSPLIT model http://www.arl.noaa.gov/ready/hysplit4.html). In the cases shown, there is no ambiguity in setting the height of the cloud from the trajectory forecast, but we acknowledge that this may not always be the case and it would be preferable to determine the height uniquely by some independent means.