The accurate determination of the location, height, and loading of sulfur dioxide (SO₂) plumes emitted by volcanic eruptions is essential for aviation safety. The SO₂ layer height is also one of the most critical parameters with respect to determining the impact on the climate. Retrievals of SO₂ plume height have been carried out using satellite UV backscatter measurements, but, until now, such algorithms are very time-consuming. We have developed an extremely fast yet accurate SO₂ layer height retrieval using the Full-Physics Inverse Learning Machine (FP_ILM) algorithm. This is the first time the algorithm has been applied to measurements from the TROPOMI instrument onboard the Sentinel-5 Precursor platform. In this paper, we demonstrate the ability of the FP_ILM algorithm to retrieve SO₂ plume layer heights in near-real-time applications with an accuracy of better than 2 km for SO₂ total columns larger than 20 DU. We present SO₂ layer height results for the volcanic eruptions of Sinabung in February 2018, Sierra Negra in June 2018, and Raikoke in June 2019, observed by TROPOMI.

The Copernicus Atmosphere Monitoring Service (CAMS), operated by the European Centre for Medium-Range Weather Forecasts on behalf of the European Commission, provides daily analyses and 5 d forecasts of atmospheric composition, including forecasts of volcanic sulfur dioxide (SO₂) in near real time. CAMS currently assimilates total column SO₂ products from the GOME-2 instruments on MetOp-B and MetOp-C and the TROPOMI instrument on Sentinel-5P, which give information about the location and strength of volcanic plumes. However, the operational TROPOMI and GOME-2 data do not provide any information about the height of the volcanic plumes, and therefore some prior assumptions need to be made in the CAMS data assimilation system about where to place the resulting SO₂ increments in the vertical. In the current operational CAMS configuration, the SO₂ increments are placed in the mid-troposphere, around 550 hPa or 5 km. While this gives good results for the majority of volcanic emissions, it will clearly be wrong for eruptions that inject SO₂ at very different altitudes, in particular exceptional events where part of the SO₂ plume reaches the stratosphere. A new algorithm, developed by the German Aerospace Centre (DLR) for GOME-2 and TROPOMI, optimized in the frame of the ESA-funded Sentinel-5P Innovation–SO₂ Layer Height Project, and known as the Full-Physics Inverse Learning Machine (FP_ILM) algorithm, retrieves SO₂ layer height from TROPOMI in near real time (NRT) in addition to the SO₂ column. CAMS is testing the assimilation of these products, making use of the NRT layer height information to place the SO₂ increments at a retrieved altitude. Assimilation tests with the TROPOMI SO₂ layer height data for the Raikoke eruption in June 2019 show that the resulting CAMS SO₂ plume heights agree better with IASI plume height data than operational CAMS runs without the TROPOMI SO₂ layer height information and show that making use of the additional layer height information leads to improved SO₂ forecasts. Including the layer height information leads to higher modelled total column SO₂ values in better agreement with the satellite observations. However, the plume area and SO₂ burden are generally also overestimated in the CAMS analysis when layer height data are used. The main reason for this overestimation is the coarse horizontal resolution used in the minimizations. By assimilating the SO₂ layer height data, the CAMS system can predict the overall location of the Raikoke SO₂ plume up to 5 d in advance for about 20 d after the initial eruption, which is better than with the operational CAMS configuration (without prior knowledge of the plume height) where the forecast skill is much more reduced for longer forecast lead times.

Precise knowledge of the location and height of the volcanic sulphur dioxide (SO₂) plume is essential for accurate determination of SO₂ emitted by volcanic eruptions. Current SO₂ plume height retrieval algorithms based on ultraviolet (UV) satellite measurements are very time-consuming and therefore not suitable for near-real-time applications. In this work we present a novel method called the full-physics inverse learning machine (FP-ILM) algorithm for extremely fast and accurate retrieval of the SO₂ plume height. FP-ILM creates a mapping between the spectral radiance and the geophysical parameters of interest using supervised learning methods. The FP-ILM combines smart sampling methods, dimensionality reduction techniques, and various linear and non-linear regression analysis schemes based on principal component analysis and neural networks. The computationally expensive operations in FP-ILM are the radiative transfer model computations of a training dataset and the determination of the inversion operator – these operations are performed off-line. The application of the resulting inversion operator to real measurements is extremely fast since it is based on calculations of simple regression functions. Retrieval of the SO₂ plume height is demonstrated for the volcanic eruptions of Mt. Kasatochi (in 2008) and Eyjafjallajökull (in 2010), measured by the GOME-2 (Global Ozone Monitoring Instrument – 2) UV instrument on-board MetOp-A.

This paper outlines the matrix exponential description of radiative transfer. The eigendecomposition method which serves as a basis for computing the matrix exponential and for representing the solution in a discrete ordinate setting is considered. The mathematical equivalence of the discrete ordinate method, the matrix operator method, and the matrix Riccati equations method is proved rigorously by means of the matrix exponential formalism. For optically thin layers, approximate solution methods relying on the Padé and Taylor series approximations to the matrix exponential, as well as on the matrix Riccati equations, are presented. For optically thick layers, the asymptotic theory with higher-order corrections is derived, and parameterizations of the asymptotic functions and constants for a water-cloud model with a Gamma size distribution are obtained