Our TROPOMI BD2 retrieval algorithm relies on the Look-Up Table COvariance-Based Retrieval Algorithm (LUT-COBRA), previously developed by for BD3. Here, we review the theoretical foundations of this approach. Comprehensive discussions on the joint retrieval of SO₂ plume heights and column densities can be found in .

Let y denote the total slant optical depth (SOD) at the top of the atmosphere (TOA), defined as the negative logarithmic ratio between the measured wavelength-calibrated radiance I and irradiance I0, that is y=-log⁡II0. A general formulation is 1y=ySO2+ybckg+ϵ, where ySO2 describes the SO₂ slant optical depth, ybckg the background contribution (e.g., from O₃ absorption), and ϵ the measurement error. The total SOD can be approximated by the following Taylor expansion, 2y-ySO2,i≈ΔVCD∂ySO2,i∂VCD+ΔLH∂ySO2,i∂LH+ybckg+ϵ, with ySO2,i=ySO2(VCDi,LHi) the SOD at the linearization point i, [ΔVCD, ΔLH] the increments in column density and plume height, and ∂ySO2,i∂VCD,∂ySO2,i∂LH the associated partial derivatives (Jacobians).

Equation () can be solved with LUT-COBRA, an iterative method that models the SO₂ spectrum through a look-up table containing precomputed SO₂ SODs and Jacobians for a wide range of scenarios. Basically, the algorithm starts by extracting these data for conditions close to the measurements, and taking into account a priori values for the LH and VCD. Each iteration then refines the estimates of SO₂ height and column density, which are subsequently used to update the optical depths and Jacobians. This process is repeated until convergence to the solution x^i=[VCDi,LHi], given by 3x^i+1=x^i+kiTS-1ki-1kiTS-1y-ySO2,i-y‾, where ki is a two-column matrix of SO₂ Jacobians, and all the contributions other than SO₂ are treated through a generalized error covariance matrix S and its associated mean spectrum y‾. Indeed, LUT-COBRA does not directly fit the background, but instead, it considers an ensemble of N measured spectra yl′ (l∈[1,N]) that are unaffected by SO₂. These spectra are characterized by a mean optical depth y‾, so that, 4S=1N-1∑l=1Nyl′-y‾yl′-y‾T, is used to statistically represent the combined error term ybckg+ϵ of Eq. (). Moreover, the retrieval errors, calculated as the square root of the diagonal elements of the solution covariance matrix, i.e., 5x^err,i=kiTS-1ki-1, can be used to describe the quality of the solutions found by the algorithm, as demonstrated below.

To evaluate the performance of LUT-COBRA across different wavelength ranges (i.e., spectral bands), a set of synthetic spectra was generated using the Linearized Discrete Ordinate Radiative Transfer (LIDORT) scalar model . Although LIDORT neglects polarization effects, these typically alter TOA radiances by less than 5 % , mainly in a spectrally smoothed way, suggesting only a minor influence on the retrievals. Simulations were performed under typical TROPOMI measurement conditions between 300 and 330 nm, with a spectral sampling of 0.065 nm for BD2 and 0.2 nm for BD3. The observation conditions were arbitrarily fixed to a solar zenith angle (SZA) of 10°, a viewing zenith angle (VZA) of 0°, a relative azimuth angle (RAA) of 0°, a surface height of 0 km, and a surface albedo of 5 %. It should be noted that clouds, aerosols, and rotational Raman scattering were not included in this study. While these factors may introduce uncertainties of a few percent in the radiative transfer , and up to 5 km bias on the SO₂ LH for highly absorbing aerosols collocated with SO₂, our aim was to maintain a simple and controlled synthetic framework. Furthermore, we used the US Standard (US STD) atmospheric profiles of ozone, temperature, pressure, and air density . For sulfur dioxide, a Gaussian function with a standard deviation of 0.5 km was applied to various combinations of SO₂ heights and columns. Each profile was actually centered on a specific layer height, ranging from 1 to 25 km in 1 km increments, and from 25 to 45 km in 5 km increments, and was scaled to the following VCDs: 1, 2, 5, 10, 15, 20, 25, 30, 40, 50, 75, 100, 125, 175, 250, and 300 DU. For high SO₂ columns, a coarser sampling of VCD nodes was used to limit the total number of simulations. However, sensitivity tests indicate that interpolation errors in the SODs and Jacobians are negligible under these conditions, with differences of less than 0.5 % in the radiative transfer, even at shorter UV wavelengths. These profiles were interpolated on an altitude grid from 0 to 50 km, with steps of 0.5 km up to 28 km, then 1 km. The temperature-dependent absorption cross sections of ozone by and sulfur dioxide by were used as input to the simulations, after convolution with a Gaussian function characterized by a standard deviation of 0.47 nm to approximate the TROPOMI Instrument Spectral Response Function (ISRF) in the UV. It is important to stress that the above description stands for our “baseline” set of synthetic spectra. In addition, to investigate the role of influencing quantities, this reference set is complemented by many other synthetic spectra corresponding to perturbed conditions in terms of atmospheric profiles (e.g., temperature, pressure and air density, O₃ and SO₂), absorption cross-section datasets, albedos, and surface heights. This will be further addressed in Sect.  and Appendix .

With this configuration in mind, we processed the simulated spectra for each condition. The outputs comprise the SO₂-free radiance, IO3, which only includes O₃ absorption, and the combined radiance, ISO2+O3, which includes absorption from both SO₂ and O₃. These spectra were then used to compute the SO₂ SOD as, 6ySO2=-ln⁡ISO2+O3IO3. From the different combinations of plume heights and column densities, the LH and VCD Jacobians in Eq. () were derived by finite difference. As an example, the dependence of the SO₂ SOD on the SO₂ altitude and column density is presented on Fig.  for various SO₂ profiles. Overall, the results are in line with the findings of and . In particular, the sensitivity to the plume height increases with higher SO₂ abundances, as the signal becomes more pronounced. The LH Jacobians are also more intense for lower-altitude plumes. This can be attributed to the strong vertical gradient of the air mass factors in the lower atmosphere (see Fig.  in Appendix ), where the measurement sensitivity increases rapidly with the altitude. As a result, small changes in plume height induce significant variations in the SO₂ signal, leading to larger LH Jacobians in that region. In contrast, the VCD Jacobians tend to be larger for higher-altitude plumes, where the AMFs, and thus the sensitivity to the SO₂ column, are higher. However, their amplitude decreases with an increasing SO₂ column, due to the non-linear nature of SO₂ absorption, which leads to a saturation of the signal. Overall, for both Jacobians, the results indicate an improved performance below 310 nm, consistent with the discussions of Sect. . This is especially evident for high plume heights (above 15 km), as noted by .

In a next step, the calculated SO₂ SOD and Jacobians were incorporated into a small look-up table. Before the retrievals, noise levels, representative of BD2 or BD3, could be added to the synthetic spectra to be fitted, as 7y*=y+ynSNR, where yn is a spectrum of random real numbers sampled from a univariate Gaussian distribution with mean 0 and variance 1. In case such noise was introduced, we considered a diagonal covariance matrix, with Sii=1SNRi2 for the ith wavelength, consistent with the data shown in Fig. b. In the absence of noise, however, we used a unit covariance matrix for the retrievals. Note that for the synthetic retrievals, we assumed the background contribution to be perfectly known, which means that SO₂ spectra, instead of total spectra, were fitted (see Eq. ). In practice, the algorithm is considered to have converged when the variation in plume height is less than 0.25 km and the change in VCD is below 5 % between successive iterations. To ensure realistic solutions and computational efficiency, the number of iterations is limited to 10. Results of these synthetic tests are presented below.

In this section, we first evaluate the performance of LUT-COBRA for both UV bands of TROPOMI, accounting for their respective spectral characteristics in terms of sampling, resolution, and SNR (see Sect. ). Then, we assess the systematic errors on the retrievals by perturbing input data of the radiative transfer. These synthetic analyses were carried out to understand how the algorithm responds to forward model and measurement uncertainties. Spectra in BD2 and BD3 were computed from different input SO₂ VCDs and LHs, detailed in Table .

The a priori inputs were arbitrarily selected, but our tests have revealed that this choice has a negligible impact on the retrieval quality. Figure  summarizes the resulting biases for the LH and VCD, in both bands, averaged over 100 noisy spectra. It can be seen that the LH and VCD biases are globally reduced in BD2, especially for lower SO₂ heights and column densities. They are typically within the convergence criteria (0.25 km for the LH and 5 % for the VCD) of the algorithm. We also observe that the standard deviations of the retrievals are, as expected, greatly reduced compared to BD3. Figure  thus demonstrates a better accuracy and indicates that, in principle, there is no fundamental limitation in retrieving the SO₂ plume height and column density from the TROPOMI BD2.

A key question is whether BD2 can enhance the retrieval precision compared to BD3. To investigate this, we estimated the SO₂ LH retrieval error, as defined in Eq. (), for both bands. The results, given in Fig. , correspond to a prescribed SO₂ column of 2 DU, with plume heights going from the lower troposphere to the upper stratosphere. For BD2, retrieval errors are shown for different fitting ranges, varying the lower wavelength while keeping the upper limit fixed at 320 nm. In contrast, BD3 is evaluated using a single spectral range (310–326 nm), similar to . From Fig. , it can be concluded that the use of BD2 results in a remarkable improvement in precision compared to BD3. When the same lower wavelength limit (310 nm) is applied, BD2 already shows slightly lower LH errors in the UTLS relative to BD3, confirming the positive impact of the finer spectral sampling. Figure  further demonstrates that extending the BD2 spectral range to shorter wavelengths significantly enhances the retrieval performance, particularly for high-altitude plumes (i.e., above 25 km). For a lower limit of 305 nm, the retrieval error is reduced by approximately a factor of 2. This is consistent with the behaviour observed for the air mass factors (see Fig. ). At high altitudes, AMFs are greater for wavelengths larger than 310 nm, but they vary less, whereas the information on the plume height primarily comes from this weak altitude dependence and, indirectly, from the temperature dependence of the SO₂ absorption cross sections. However, for shorter wavelengths (or higher SO₂ VCDs), the height dependence of the sensitivity functions is increasing. The algorithm is therefore able to extract information more easily from the spectrum, thereby reducing retrieval errors. On the other hand, including shorter wavelengths in the fit may introduce significant systematic errors in the retrieval. In the remainder of this section, we evaluate the potential sources of uncertainty (e.g., on ozone) and estimate the associated errors.

In this context, the sensitivity of LUT-COBRA in BD2 was tested by inverting noise-free spectra simulated with a single deviation from the baseline conditions of our look-up table (see Sect. ). These perturbations were applied to atmospheric parameters, including temperature, air density (pressure), O₃ and SO₂ profiles, as well as to the SO₂ absorption cross sections, albedo, and surface height. In each case, variations of different magnitudes were applied, as indicated in Table .

For the temperature, we used the climatology of , selecting profiles representative of different seasons in the northern hemisphere. Note that we also analyzed a case in the tropics (not reported in Table ). These distributions, originally provided for different ozone column densities, were interpolated to a fixed VCD of 345.7 DU (matching the standard value) so that only the shape of the profile differs from our baseline conditions. The effect of air density and pressure, in turn, was evaluated by adjusting the air concentration in each layer of the US Standard atmosphere , as defined in our LUT. Regarding the ozone contribution, we examined two sources of uncertainty: the shape of its profile and the corresponding VCD. Given the strong O₃ absorption in BD2, a broader range of scenarios was explored. For the vertical distribution, we considered four profiles , spanning latitudes from 30° S to 90° N (tropics and southern hemisphere profiles are not listed in Table ) and all seasons. As with the temperature analysis, these profiles were scaled to the standard O₃ VCD to test the effect of a change in profile shape independently. Afterwards, to assess the impact of the VCD, we tested values ranging from 310 to 380 DU (only two cases are provided in Table ), while keeping the same standard profile shape. For SO₂, we investigated the impact of different vertical distributions by fitting spectra corresponding to rectangular, Lorentzian, and Gaussian profiles of varying standard deviations (σG), for the LHs and VCDs given in Table . The influence of the SO₂ absorption cross sections was also studied, by comparing the SEOM-IAS dataset with that from . Finally, we looked at uncertainties related to observation conditions, in particular the albedo and surface height.

All the above sensitivity analyses were performed within the 305–320 nm spectral range, selected as an optimal balance, minimizing LH retrieval errors (see Fig. ) and reducing systematic uncertainties related to the ozone absorption. Figure  summarizes the mean systematic errors on the retrieved LHs and VCDs, assuming extreme and more realistic (see Table ) perturbations in each case, for a relatively low column density (5 DU) and UTLS altitude (13.5 km).

Overall, the total systematic errors are estimated by combining the individual effects of each source of uncertainty in quadrature, yielding 1.91 km for the LH and 2.58 % for the VCD. Since some contributions are partially correlated (e.g., temperature and air density profiles, or the O₃ profile and its VCD), these values should be interpreted as indicative and approximate estimates of the total error. The temperature, pressure and air density, as well as the sulfur dioxide profiles appear to have a negligible impact on the retrieval accuracy. Similarly, the albedo and surface height have little effect on the quality of the results, especially since these two observation conditions are generally well constrained in practice. In contrast, the choice of SO₂ absorption cross-section dataset has a more important influence on the retrieved solutions, particularly for the plume height. A comparison of the two datasets (see Fig. ) shows that their temperature dependence is similar, but the dataset from assumes a stronger SO₂ absorption than that of , especially below 310 nm. As a result, SO₂ air mass factors, as defined by Eq. (), decrease at the wavelengths with the highest absorption. Since the SO₂ SOD depends on the product of the AMFs and ACS, the fitted signal is “smoothed”, ultimately leading to systematically higher SO₂ LH. Importantly, we observe from Fig. , that the ozone profile, particularly its vertical shape, plays a key role on the retrievals, as expected given its strong absorption at shorter UV wavelengths. Nevertheless, the obtained LH and VCD biases remain relatively modest. Note that we conducted further tests using alternative baseline conditions (i.e., with various look-up tables) for the temperature, O₃ profile, SO₂ ACS, and albedo. These complementary analyses are detailed in Table  and lead to consistent conclusions. The results are therefore promising, but they highlight the need to mitigate the influence of ozone. Our strategy to better account for its strong contribution is described in Sect. .

Following the synthetic analyses, we developed a more comprehensive look-up table and refined LUT-COBRA for application to actual TROPOMI BD2 observations. As explained in Sect. , a LUT is a set of pre-calculated SO₂ spectra for various conditions, which allows efficient retrievals by avoiding the computational cost of online radiative transfer simulations. In this updated LUT, we considered a broader range of scenarios, with particular emphasis on the role of ozone, which emerged as a key factor affecting the retrieval quality (see Sect. ). As a first step, we performed a statistical analysis of atmospheric profiles, including temperature, air density, and ozone. Our objective was to build a compact set of representative O₃ profiles parametrized based on the total ozone column only. This was motivated by the need to limit the LUT size while adequately representing the ozone profile variability in the retrievals. The idea of using the O₃ VCD to inform about the shape of the O₃ profile is not new ( and references therein). It has been extensively used for satellite total ozone retrievals (e.g., by ) and, in our application, has the potential to reduce the uncertainties around the O₃ profile (see Fig. ), given that the total ozone column is retrieved from TROPOMI with an excellent accuracy . To this aim, we used the climatology of that provides vertical distributions of temperature and ozone density for several O₃ VCDs, latitudes, and seasons. From these data, we derived the following mean atmospheric profiles, 8P¯i=∑j,kαi,j,kPi,j,k∑j,kαi,j,k, with Pi,j,k the vertical distribution (of temperature, air density or ozone) for a VCD i, latitude j and season k, and αi,j,k a corresponding weighting factor which is calculated from a frequency distribution based on the OMI Level-3 ozone product . As an example, Fig. a shows the mean ozone density profile for 385 DU, resulting from different contributions, with their respective weighting factors. This highlights the regions and seasons where such a high concentration of ozone is most likely to be found. Similarly, Fig. b presents the calculated mean ozone distributions for various column densities. They thus correspond to the most probable profiles for such O₃ VCDs, and it can be seen that the O₃ peak shifts towards lower altitudes as its concentration increases, providing valuable information about the profile shape that helps reduce the associated uncertainties in the SO₂ retrievals.

Accordingly, we calculated mean O₃ profiles for different O₃ column densities, summarized in Table . For consistency, mean air density and temperature profiles were computed using the same approach. These mean profiles were then used as input for radiative transfer simulations, covering a wide range of conditions (see Table ). In total, approximately 60×106 total radiance spectra and 180×103 O₃ radiance spectra were produced using a high-performance computer (HPC). Note that we considered the same absorption cross-section datasets as previously (see Sect. ) for ozone and sulfur dioxide . However, these spectral data were pre-convolved here with a 0.05 nm box-car function, sampled at 0.065 nm and interpolated to the mean temperatures. Afterwards, the simulated radiances were exploited to calculate the SO₂ optical depths and Jacobians. These data were convolved with the TROPOMI IRSF (varying as a function of the detector row) and corrected for the solar-I0 effect , before being incorporated into a comprehensive look-up table to analyze TROPOMI BD2 measurements.

The retrieval of SO₂ LH and VCD in BD2 is fundamentally the same as in BD3. The core of the algorithm is explained in more detail by and here we only describe the new developments applied to LUT-COBRA.

First, one needs to determine the parameters related to the reflection (i.e., albedo and surface height), geometry (i.e., SZA, VZA, RAA), and O₃ amount corresponding to a measured spectrum. These pieces of information are essential for the retrievals. Indeed, during the process, a sub-LUT depending only on the SO₂ LH and VCD, and representative of the measurements, is extracted by linear interpolation. The observation angles can usually be derived from the TROPOMI Level-1 files. For both the albedo and surface height, we assumed a Lambertian equivalent reflector (LER), similar to , and characterized by effective parameters. This model provides a simplified representation of the complex mechanisms of reflection by the surface, clouds, and aerosols. In BD3, the effective height is computed from the surface and cloud altitudes weighted by the cloud fraction, based on data from the TROPOMI Level-2 product. The effective albedo, on the other hand, is calculated from the reflectance at 340 nm, a spectral region free from any absorption by sulfur dioxide or ozone to ensure a correct computation. Regarding the O₃ input, the corresponding operational offline TROPOMI Level-2 product is utilized. In addition, as outlined in Sect. , a priori values on the sulfur dioxide plume height and column density have to be provided to LUT-COBRA. The SO₂ LH a priori is set to 7 km, or LER height + 2 km if the LER height is higher than 5 km, and the prior VCD is defined from the SOD and AMF found in the operational TROPOMI SO₂ column product, assuming an altitude of 7 km. However, all these input data are not available in BD2 and must therefore be read from BD3. It is important to remind that the two bands do not share the same number of across-track positions (as explained in Sect. ), and there is also a shift in the flight direction (along-track positions). We performed this BD3-to-BD2 transition by directly interpolating the BD3 data into BD2 using the coordinates (latitudes and longitudes) provided in TROPOMI Level-1 files. This approach proved to be a good approximation, while simplifying our script and optimizing the computation time.

Once all BD2-interpolated data are obtained, the covariance matrix needed for the retrieval (Eq. ) is calculated for each row. A set of SO₂-free radiance spectra measured arbitrary for 300 pixels along the flight direction is selected for a given orbit. The SO₂ VCDs and corresponding retrieval errors from the TROPOMI Level-2 product are used to filter the pixels, discarding those with SO₂ VCDs > 2.5 × VCD errors. Note that to ensure a robust construction of the covariance matrix, at least 100 spectra must be included. However, we observed that the covariance matrix, as described in Eq. (), sometimes becomes ill-conditioned in BD2, resulting in unphysical values in the inverse matrix for some wavelengths, and thereby compromising the retrievals. To address this issue, we performed an eigen-decomposition of the covariance matrix, and refined the inverse of the covariance matrix by discarding eigenvalues of S that were too small, as 9S=∑i=1nλiviviT→S-1=∑i=1k1λiviviT, where λi and vi corresponds to the eigenvalue and eigenvector of S, n is the number of wavelengths, k denotes the threshold beyond which S-1 presents erroneous values. This correction, described in more detail by , ensures that the inverse matrix is calculated correctly and avoids inconsistent retrieval errors. Tests were performed to identify the threshold minimizing the errors, and a value of 10⁻⁷ on the eigenvalues was found, corresponding approximately to the precision of single-precision floating-point residuals. The procedure was thus applied systematically to all spectra, with typically less than 30 % of eigenvalues discarded to stabilize the inversion, depending on the conditioning of the covariance matrix.

The data were subsequently provided to LUT-COBRA, initiating the fitting process. Note that our analysis is restricted to measurements with SZAs below 65°, in order to avoid conditions of strong ozone absorption that could complicate the retrievals. It should also be mentioned that the algorithm may return values outside the predefined grids (see Table ) for some iterations. In such cases, the SO₂ height is set to surface height + 1 km or grid maximum - 1 km for the next iteration, depending on whether it falls below or above the limits. Similarly, if the derived SO₂ column density lies outside the VCD grid, the algorithm reverts to the a priori value for the next iteration.

In this section, we analyze the SO₂ LHs and VCDs retrieved from various TROPOMI BD2 observations of volcanic eruptions, as summarized in Table . These results are compared with BD3 estimates from LUT-COBRA and the operational product , as well as with SO₂ measurements from IASI and MLS .