Dave and Mateer (1967) first proposed a technique to estimate total ozone column from nadir backscatter UV measurements taken in the Huggins ozone absorption band (310–340 nm), assuming no SO2 is present. Their algorithm was inspired by the pioneering Dobson spectrophotometer, which measures attenuation of solar irradiance by UV wavelength pairs from which total ozone is derived using the Beer–Lambert law. However, unlike the direct sun technique, radiative transfer calculations show that the top-of-the-atmosphere (TOA) BUV radiances (I) do not follow the Beer–Lambert law. In general, log(I) varies nonlinearly with ozone column amount (Ω), and this relationship is sensitive to the shape of the ozone profile (defined as the ozone density profile normalized to total ozone). To account for this effect, Dave and Mateer (1967) proposed constructing a set of lookup tables (LUTs) based on standard ozone profiles with different total ozone amounts using ozonesonde and Dobson Umkehr data. Since the shapes of the profiles also vary with latitude, they proposed using three sets of profiles for low latitudes, midlatitudes and high latitudes. These profiles are then used to estimate I, which varies with wavelength (λ), observational geometry, surface pressure and surface reflectivity (R). Following the Dobson convention, log(I) is converted to N value which is defined in Eq. (1) as 1N=-100log⁡10IF. F is the extraterrestrial solar irradiance. By linearly interpolating N between total ozone nodes, one forms the N-Ω curves that are a single valued function of Ω representative of a given latitude band and observational geometry. This approach allows Ω to be estimated by matching the measured N value to the interpolated N values.

Over the years several modifications have been introduced to this basic concept. Mateer et al. (1971) proposed a Lambert-equivalent reflectivity (LER) concept to estimate the combined contribution of the surface, clouds and aerosols to BUV radiance. In this concept, the scene at the bottom of the atmosphere is assumed to be a Lambertian reflector, whose reflectivity (Rs) is derived from the measurements at 380 nm where the ozone and SO2 absorption is negligible. The effective pressure of this reflecting surface is assumed to vary with Rs, from a surface pressure at Rs < 0.2 to a cloud pressure 0.4 atm at Rs > 0.6, linearly interpolated at intermediate Rs. The algorithm assumed that Rs, thus derived, did not vary with wavelength. Although in the earlier versions of this algorithm wavelength pairs (313/331, 318/340) were used to derive Ω, Rs was later derived at 331 nm to minimize errors due to the spectral dependence of Rs. This made pairing unnecessary (McPeters et al., 1996).

By explicitly modeling the effect of aerosols using a radiative transfer code, Dave (1978) showed that Rs did not vary significantly with wavelength for non-absorbing aerosols, hence they produced no ozone error. However, for aerosols that might have strong absorption in the UV, that study predicted that Rs would decrease at shorter wavelengths, producing an overestimation of ozone. Since aerosol properties in the UV were not known at that time, no correction for aerosol absorption was applied until the mid-1990s when the effect predicted by Dave (1978) was detected in the Nimbus-7 TOMS data launched in October 1978.

Since the TOMS instrument had three reflectivity channels (331, 340, 380 nm), it was possible to compare the reflectivities derived from them. This comparison showed that Rs increased significantly with wavelength for moderately thick clouds, causing a significant underestimation of Ω (up to 3 %). A modified LER (MLER) concept assuming two Lambertian surfaces, one at the surface and the other at the cloud top was applied to minimize this error (Ahmad et al., 2004).

The most recent version of the TOMS ozone algorithm reverts back to the LER model, but it assumes that clouds are at the surface, which reduces the Rs wavelength dependence (Ahmad et al., 2004). This simple LER (SLER) model is used in our SO2 algorithm. However, since there are many other reasons for such a dependence, including ocean color, non-Lambertian surfaces, such as ocean glint and fogbow and, most importantly, the absorbing aerosol effect predicted by Dave (1978), Rs is assumed to vary linearly with λ; its slope is derived using 340 and 380 nm radiances. This simple omnibus approach works well for most cases, except when the UV absorbing aerosols (smoke, dust and volcanic ash) are very thick. Such data are flagged in the TOMS ozone algorithm. The new MS_SO2 algorithm is an extension of this algorithm into two dimensions (Sect. 3).

Krueger (1983) was the first to suggest that TOMS could be used to retrieve sulfur dioxide from explosive volcanic events. He correctly interpreted the large positive ozone anomaly observed following the explosive eruption of El Chichón in 1982 as being due to the SO2 released into the atmosphere during the event. To estimate the SO2 inside the plume region, he separated the SO2 and O3 signals by computing a residual reflectance, estimated as the difference between the interpolated unperturbed background reflectances outside the plume and the reflectance anomaly inside the plume. This early technique for retrieving SO2 from TOMS ozone estimates became known as the residual method. The residual method, however, failed when the background could not be clearly separated from the ozone anomaly. Krueger subsequently developed the first BUV algorithm that separated the O3 and SO2 radiance contributions, based on an earlier methodology developed by Kerr et al. (1980) to retrieve the SO2 column from the ground with a Brewer spectrophotometer. This method assumed that the BUV radiation was attenuated by the two absorbing species (O3, SO2), leading to an equation describing BUV radiance, I, for a given wavelength, λ, corresponding to the TOMS field of view (FoV): 2Iλ=aFλexp⁡-bλ+Sg(τO3+τSO2). In Eq. (2), F is the incoming solar flux, Sg is the geometrical optical path (air mass factor, AMF), and τO3 and τSO2 are the vertical optical thicknesses for O3 and SO2, while the coefficients a and b depend on the satellite viewing geometry, cloud or surface reflectance, and volcanic ash and sulfate aerosols (Krueger et al., 1995; Krotkov et al., 1997). Equation (2) can be expressed in matrix form, which is then inverted to obtain estimates for the SO2 and O3 vertical column densities and the dimensionless parameters a and b. This algorithm is generally referred to as the Krueger–Kerr algorithm (Krueger et al., 1995). Krotkov et al. (1997) developed radiative transfer path correction, which explicitly accounted for the Rs, ozone and SO2 vertical profiles, replacing the geometrical AMF in Eq. (2). The modified algorithm with empirical background correction has been used offline on a case-by-case basis for the past 2 decades to retrieve SO2 mass tonnage from medium to large explosive eruptions using TOMS BUV measurements (Krueger et al., 2000; Carn et al., 2003).

Our step 1 inversion starts with an initial state vector x0, consisting of first guesses for Σ0, Ω0 and dRs0/dλ shown in Table 2. The final state vector, x, is determined iteratively by inverting the Jacobian matrix K at each iteration step: 4dN=Kdx, where dx represents the relative changes in the state vector from the previous iteration and dN represents the residual vector equal to Nm-Nc, computed as the difference between the measured N values, Nm, and the calculated N values, Nc, at the four TOMS channels at 317, 331, 340 and 380 nm. K represents a 4×4 Jacobian matrix computed from the LUTs. These matrix elements are defined as follows: 5Ki,j=∂Nc,i∂xj,i,j=1,4, where Nc,i is the forward model calculated N value at wavelength i.

The reflectivity Rs is computed analytically using the measured BUV radiance at 380 nm (see Supplement, Eq. S4). Note that since the O3 and SO2 cross sections are negligible at 380 nm, the Rs and ∂N380/∂Rs do not change with the iterations (i.e., dRs=0).

Equation (4) is solved iteratively by zeroing the residuals, dN=Nm-Nc, and recomputing the Nc and the Jacobians at each iteration step for the four used channels. The state vector is then adjusted after each iteration, xk=xk-1+dxk, k=1,2,… until it converges on a solution as described below: 6adNk=Nm-Nc,k-1=Kk-1dxk,6bdxk=xk-xk-1=Kk-1-1dNk. Since O3 and SO2 exhibit small absorption at 340 nm, a nonzero R spectral slope (i.e., dR/dλ≠0) accounts for the radiative effects of aerosols and surface reflectance (e.g., sun glint).

As indicated in Table 2, the algorithm initially assumes zero R-λ dependence (i.e., dR/dλ=0); however, absorbing aerosols (smoke, dust and volcanic ash) cause dR/dλ≠0.

The algorithm uses retrieved spectral slope dR/dλ in Eq. (7) below to update the calculated LERs after each iteration: 7Rj=Rs+∂R∂λ×λj-λRj=1,4, where λj=312, 317, 331, 340 nm and λR=380 nm. When SO2 or aerosol loading is high nonlinear R-λ dependence can cause systematic errors in the retrieval state vector. For this reason, we do not use the shortest 312 nm channel in the retrievals (Eqs. 5–6), but the final residual dN312 is used as a diagnostic of the nonlinearity. A step 2 empirical procedure, described in the next section, was developed to correct for the retrieval bias resulting from these errors.

The MS_SO2 forward model accounts for O3 and SO2 absorption and linear spectral changes in Rs due to the presences of aerosols. The algorithm, however, does not explicitly characterize the absorption and scattering effects of volcanic ash (absorbing) and sulfate (non-absorbing) aerosols. The retrieval errors in Σ and Ω caused by volcanic ash during the first few days after an explosive eruption can be significant in the case of major volcanic eruptions like Pinatubo and El Chichón (Krueger et al., 1995; Krotkov et al., 1997). A step 2 procedure was developed primarily to handle explosive eruptions (Volcanic Explosivity Index, VEI > 3 ), in which large Ω anomalies are identified to occur in conjunction with high ash concentrations. In step 2, a corrected total ozone Ωcor inside the SO2 cloud is interpolated using the retrieved Ω outside the plume along the orbit for each cross-track position. Even if ozone-destroying chemicals are present, such effects can still be considered negligible over the relatively short time periods that SO2 concentrations are high enough to affect TOMS observations.

In deciding whether to apply step 2, the algorithm considers the retrieved Σ, Ω and aerosol index (AI) in Step 1. The AI is estimated from the dR/dλ and the calculated Jacobian dN/dR at 340 nm: 8AI=∂N340∂R∂R∂λλ340-λ380=-40⋅∂N340∂R∂R∂λ. Positive AI (dR/dλ > 0) identifies spatial regions affected by absorbing aerosols (dust, smoke and ash). The step 2 selection criteria first select FoVs where either SO2 > 15 DU (inside the plume) or AI > 6. The additional AI criterion allows for the selection of FoVs around the edges of the cloud, where the SO2 can be less than 15 DU due to high aerosol concentrations. In this case, it is assumed that the step 1 SO2 may have been underestimated due to the ozone error caused by high aerosol concentrations (in these cases, the SO2 retrieved in step 2 may still not exceed 15 DU and therefore would be excluded from the plume in subsequent mass calculations). We describe the methodology for interpolating Ωcor in Eqs. (S5)–(S7). A second retrieval of SO2 and dR/dλ is then performed by inversion using the measured 317 and 340 nm radiances while treating the ozone Ωcor as a constant. This constraint on the ozone bounds the SO2 Jacobians computed from the forward model LUTs. The operational MS_SO2 product files include a step 2 algorithm flag (not applied =0, applied =1).

To illustrate the effects of the step 2 procedure, we consider the 1982 explosive eruption of El Chichón, which emitted ∼7 Tg SO2 (Krueger et al., 2008) the second largest observed in the satellite era (Fig. 1). Figure 3 shows the retrieved AI map during TOMS overpass of the volcano on 4 April 1982, while it was still erupting. High AI values exceeding a value of 10 correspond to biased high step 1 ozone values (Fig. 4a) and underestimated Σ values (Fig. 4c). Figure 4b shows the step 2 corrected Ωcor, making it consistent with the Ω field outside of the volcanic cloud. Figure 4d shows the step 2Σ, which is much higher than step 1. As can be seen in this particular example, the step 2 correction can significantly increase the SO2 mass. In this case, the SO2 mass increased from 2475 (step 1) to 3637 kt (step 2). Peak Σ values increased from 396 to 549 DU in the aerosol affected region. The biases, dΩ and dΣ, for this case are shown in Fig. S1 in the Supplement. Step 2 was developed primarily to handle extreme eruptions (VEI > 3), such as El Chichón and Pinatubo, where large Ω anomalies sometimes occur in conjunction with high ash concentrations. In practice, step 2 corrections tend to be small (or nonexistent) for most of the eruptions detected observed during the observation period covered by TOMS.

The corrected step 2 Ω values inside the volcanic cloud shown in Fig. 4b appear to be fairly consistent with the regional unperturbed ozone field, but it should be noted that a few remaining high Ω values in the boundary of the plume still exist, which were not selected for step 2 (Fig. 4b). These pixels were not corrected because the threshold criteria were not met, thus Σ may be underestimated. However, their contribution to the total SO2 cloud mass is insignificant.

Step 2 follows a methodology similar to the original residual method developed by Krueger (1983), which separated the O3 and SO2 contributions by subtracting the measured BUV reflectance in the unperturbed region from the BUV radiance anomaly associated with the SO2 cloud. The MS_SO2 algorithm corrects the overestimated step 1 ozone inside the plume by correcting the positive ozone bias. Our step 2 procedure is typically only applied when the ash and/or SO2 loading causes the reflectivity dependence to become nonlinear, as the forward model does not explicitly account for volcanic aerosol absorption. This scenario typically lasts for about 1–3 d following a major explosive eruption, during which total retrieved SO2 mass is likely to be underestimated and in some cases could even increase with time due to ash and ice fallout and plume dispersion. For such extreme cases we recommend estimating SO2 to sulfate conversion e-folding lifetime using weeks of measurements of the total SO2 cloud daily mass and extrapolating it back in time to estimate total SO2 mass emitted on eruption day. This “day one” time extrapolated SO2 mass is typically larger than retrieved on days immediately following the eruption (Krotkov et al., 2010).

We assume that the background sulfur dioxide is below TOMS detection limit in regions of the atmosphere far away from SO2 sources (e.g., volcanic, anthropogenic). Random errors associated with the retrieval process, however, are normally distributed around zero. We expect that the true volcanic SO2, Σtrue and the mean of the distribution, 〈Σ〉clean, to equal zero such that 9Σtrue=〈Σ〉clean=0. We examined a sample of 90 TOMS orbits in clean regions of the central Pacific Ocean and found a positive bias of about 3 DU (i.e., 〈Σ〉clean∼3 DU, Fig. 5). A soft calibration procedure was developed for correcting this bias by applying a small constant N340 value adjustment to the measured 340 nm BUV radiances. The details of this procedure are described in Sect. S3.3 in the Supplement. Figure 5 shows probability density functions (PDFs) of the step 1 SO2 before (dashed) and after (solid) applying the correction for 11 November 1981. The mean bias is reduced to < 1 DU after applying the correction.

The random errors in the MS_SO2 retrieval were estimated from the standard deviation in the SO2 from a large data sample that included 90 central Pacific orbits, spanning a 10-year period between 1980 and 1990. Data were restricted to Σ values between -20 and 20 DU (Fig. 6a). Standard deviations were then computed as a function of the TOMS swath position, as shown in Fig. 6b. Figure 6b can be used to characterize the SO2 detection limits for TOMS. In this section, we compare the TOMS error distribution with the UV Ozone Mapping Profile Suite Nadir Mapper (OMPS-NM), a hyperspectral UV instrument onboard the Suomi National Polar-orbiting Partnership (NPP) and NOAA 20 satellites. For this comparison, we selected 1 month of NPP/OMPS spectral data (central Pacific) and applied the MS_SO2 algorithm using the same four wavelength bands of TOMS (Table 2), which were first convolved with the TOMS bandpass function.

Figure 6b shows that TOMS retrieval noise depends on the swath position, varying from ∼6 DU at nadir to ∼4 DU at higher viewing angles, while OMPS is 2–3 times smaller (∼2 DU) and is relatively independent of the cross-track position. Using the MS_SO2 algorithm, we subsequently estimate the SO2 detection limit for TOMS and OMPS-NM to be about 15 DU and 6 DU (∼99 % confidence level), respectively. We note that when applying the Principal Component Algorithm (PCA) (Li et al., 2013) to all the 100–200 wavelengths available from OMPS-NM hyperspectral measurements, the noise is reduced by an order of magnitude to ∼0.2–0.5 DU, allowing detection of large anthropogenic point sources (emissions more than ∼80 kt yr-1) (Zhang et al., 2017).

In this section, we evaluate systematic errors of the MS_SO2 retrievals of volcanic SO2. The two most significant errors are caused by volcanic aerosols (ash and sulfate) and incorrect assumptions regarding the SO2 profile, namely the plume height. The radiance tables used by the algorithm account for ozone and SO2 absorption but do not account for the absorption and scattering by aerosols. The ash errors can be significant during the first couple days after the initial eruption phase (Rose et al., 2003; Guo et al., 2004). The precomputed radiance tables used by MS_SO2 assume an SO2 column amount and an a priori CMA and standard deviation (Sect. 3). An incorrect CMA assumption can cause significant SO2 errors that vary with viewing geometry, ozone and SO2 column amounts. We characterize these error sources by applying the MS_SO2 algorithm to synthetic radiances.

Mount Pinatubo is a large stratovolcano located at 15∘08′ N, 120∘21′ E in western Luzon, Philippines, that erupted explosively on 15 June 1991, following weeks of precursory activity. TOMS SO2 imagery on 15 June shows a narrow, elongated SO2 ash plume extending to the west from the location of the volcano. On the following day TOMS measured a massive SO2 plume to the west of the volcano (Bluth et al., 1992). TOMS continued tracking the daily evolution of the Pinatubo volcanic cloud as it encircled the Earth over a period of about 22 d. Previous estimates of the Pinatubo SO2 height (CMA) range between 18 and 25 km (Self et al., 1996; Guo et al., 2004).

Figure 9 shows TOMS daily SO2 maps produced with the MS_SO2 and the PCA algorithms for the 6 d period from 16 to 21 June. Corresponding ash index (AI) imagery from MS_SO2 is shown in Fig. 10. SO2 and AI imagery for 16 June show a large SO2 ash cloud propagating to the west. AI values range from 4 to above 12 across the plume. The AI values decreased over the following days due to wind advection and wet deposition (Guo et al., 2004). As the SO2 cloud area continues to expand, total SO2 mass remains high, while SO2 peak values decrease, which is expected from cloud dispersion. The MS_SO2 and the PCA imagery show excellent qualitative agreement in resolving the plume area and internal SO2 plume structure, as inferred from the SO2 gradients across the peak regions of the cloud. Note that for 16–19 June, part of the observed cloud is missing due to a known mechanical problem with the TOMS instrument. These missing regions can be clearly identified in the imagery.

Figure 11 shows a scatterplot comparing the MS_SO2 and PCA retrievals for the 6 d time series, which included over 7000 matching FoVs. These results show the retrievals are in close quantitative agreement, with a correlation of 0.993 and a slope of 1.00. Since the two algorithms apply fundamentally different approaches to retrieving SO2, this level of agreement is impressive considered over such a broad range of values.

We further compared quantitative estimates of SO2 cloud mass, peak SO2 and plume area. For this comparison, we also considered results from the Krueger–Kerr algorithm (KK), based on the published results of Guo et al. (2004). Table 3 displays daily estimates of the SO2 cloud mass and peak SO2 amounts for the MS_SO2, PCA and KK algorithms for the 6 d period. Guo et al. (2004) applied a modified version of the KK algorithm that assumes a radiative transfer air mass factor (AMF), which accounts for the a priori ozone and SO2 absorption profiles (Krotkov et al., 1997). The early SO2 mass estimates by Bluth et al. (1992) derived from Pinatubo eruption assumed a geometrical AMF. Also note that Guo et al. (2004) interpolated across the missing data regions of the plume on 16, 18 and 19 June using a punctual kriging statistical analysis. Here, we did not correct for the missing data. The three algorithms are in good overall agreement for the period from 17 to 21 June, with the differences within 10 % compared to MS_SO2. The most significant differences between the three algorithms are observed on 16 June under conditions of heavy ash loading. KK mass tonnage estimates exceeded MS_SO2 by over 24 % even though MS_SO2 and the PCA differ by just 2 %. Some of the difference between KK and the other two algorithms can be attributed to the fact that the Guo et al. (2004) estimates include contribution from the missing data region at the northern boundary of the plume (compare SO2 and aerosol imagery), but this contribution does not nearly account for the total difference in Table 3.

The differences can be explained by considering how each algorithm is affected by aerosols. MS_SO2 accounts for ash by retrieving the spectral dependence at 340 nm, which is then adjusted iteratively to correct the reflectivity at the two absorbing channels. As explained in Sect. 3.2, absorbing aerosols in the column can cause possible ozone anomalies, which decrease Σ. The KK algorithm (Krueger et al., 1995) accounts for ash implicitly by retrieving two linear spectral parameters that adjust calculated Nc to match measured Nm. Like MS_SO2, the KK radiative path LUTs are based on TOMRAD calculations that do not explicitly account for ash (Krotkov et al., 1997). Krueger et al. (1995) estimated that ash aerosols can cause errors in the retrieval up to +30%, depending on the ash size distribution. The PCA algorithm, in contrast, accounts for ash in the separation and ordering of the principal components. The differences between MS_SO2 and KK on 16 and 17 June can be partly ascribed to the effects of aerosols on the retrievals.

By 18 June, the ash and SO2 clouds have mostly separated, though, aerosol indices over 4 are still observed in some regions of the plume. Pinatubo did not erupt again after the major eruption on 15 June, yet the three algorithms show retrieved SO2 mass increases on 17 and 20 June (the PCA and KK retrievals also indicate a small increase on 18 June). Guo et al. (2004) attribute these increases to the sequestering of volcanic SO2 by ice–ash mixtures in the plume. They propose the sequestered SO2 was released at a later time through sublimation of ice in the lower stratosphere. The oxidation of hydrogen sulfide offers another mechanism to account for the observed mass increases in the days following the eruption. The combined results of the three algorithms support the conclusions of Guo et al. (2004) that the observed mass increases in the temporal evolution of the plume are real.

Overall, the PCA retrieved 3 % more total mass tonnage than MS_SO2. These differences are attributed to differences in how the MS_SO2 algorithm handles aerosols and differences in the area of the plume due to differences in the retrieval near the sensitivity threshold (∼15 DU). Ash, sulfates and high SO2 amounts impact the ozone retrieval, for as was seen in Sect. 3.2, systematic errors in SO2 are anticorrelated with errors in O3 (see Fig. S1). For the case of the KK algorithm, the total ozone retrieved inside the SO2 plume can be unrealistically low and even negative in an extreme event like Mount Pinatubo shown in Figs. S4 and S5. Figure S4 compares the KK ozone retrieval with MS_SO2 step 2 ozone retrieval and Fig. S5 compares scatterplots of SO2 and total ozone for 17 and 18 June.

Table 4 provides estimates of the plume area for the MS_SO2 and PCA. The area of the plume is most sensitive to the minimum detection threshold around the edges of the SO2 cloud. MS_SO2 and the PCA algorithms were directly compared by computing the areal sum of all the pixels where Σ > 15 DU (Fig. 9). For the 6 d study period, the plume increased in size from about a little over 2×106 km2 to ∼9×106 km2 The PCA trends observed a larger cloud area for five of the 6 d, with most of the observed differences within 7 %. On 16 June, shortly after the major eruption of 15 June, the estimated area for the PCA is about 15 % greater than for MS_SO2. The fresh plumes are opaque, which result in underestimating of SO2 mass by all BUV algorithms due to the mixing of aerosols (Krotkov et al., 1997). The PCA appears slightly more sensitive to SO2 near the edges of the cloud, where aerosol loading is high (AI > 1.5). It should be noted that the soft calibration applied to the 340 nm channel, described in Sects. 3.3 and S3.3, may also contribute to the lowering the sensitivity around the edges of the plume. This correction effectively lowered the background by about 3 DU.