Variational assimilation of IASI SO₂ plume height and total column retrievals in the 2010 eruption of Eyjafjallajökull using the SILAM v5.3 chemistry transport model

This study focuses on two new aspects of inverse modelling of volcanic emissions. First, we derive an observation operator for satellite retrievals of plume height, and second, we solve the inverse problem using an algorithm based on the 4D-Var data assimilation method. The approach is first tested in a twin experiment with simulated observations and further evaluated by assimilating IASI SO2 plume height and total column retrievals in a source term inversion for the 2010 eruption of Eyjafjallajökull. The inversion resulted in temporal and vertical reconstruction of the SO2 emissions during 1–20 May 2010 with formal vertical and temporal resolutions of 500 m and 12 h.The plume height observation operator is based on simultaneous assimilation of the plume height and total column retrievals. The plume height is taken to represent the vertical centre of mass, which is transformed into the first moment of mass (centre of mass times total mass). This makes the observation operator linear and simple to implement. The necessary modifications to the observation error covariance matrix are derived.Regularization by truncated iteration is investigated as a simple and efficient regularization method for the 4D-Var-based inversion. In the twin experiments, the truncated iteration was found to perform similarly to the commonly used Tikhonov regularization, which in turn is equivalent to a Gaussian a priori source term. However, the truncated iteration allows the level of regularization to be determined a posteriori without repeating the inversion.In the twin experiments, assimilating the plume height retrievals resulted in a 5–20 % improvement in root mean squared error but simultaneously introduced a 10–20 % low bias on the total emission depending on assumed emission profile. The results are consistent with those obtained with real data. For Eyjafjallajökull, comparisons with observations showed that assimilating the plume height retrievals reduced the overestimation of injection height during individual periods of 1–3 days, but for most of the simulated 20 days, the injection height was constrained by meteorological conditions, and assimilation of the plume height retrievals had only small impact. The a posteriori source term for Eyjafjallajökull consisted of 0.29 Tg (with total column and plume height retrievals) or 0.33 Tg (with total column retrievals only) erupted SO2 of which 95 % was injected below 11 or 12 km, respectively.

Variational assimilation of IASI SO₂ plume height and total-column retrievals in the 2010 eruption of Eyjafjallajökull using the SILAM v5.3 chemistry transport model

This study focuses on two new aspects on inverse modelling of volcanic emissions. First, we derive an observation operator for satellite retrievals of plume height, and second, we solve the inverse problem using the 4D-Var method. The approach is demonstrated by assimilating IASI SO2 plume height and total column retrievals in a source term inversion for the 2010 eruption of Eyjafjallajökull. The inversion resulted in temporal and vertical reconstruction of the SO2 emissions during the 1–20 May, 2010 with formal vertical and temporal resolutions of 500 m and 12 hours. The plume height observation operator is based on simultaneous assimilation of the plume height and total column retrievals. The plume height is taken to represent the vertical centre of mass, which is transformed into the first moment of mass. This makes the observation operator linear and simple to implement. The necessary modifications to the observation error covariance matrix are derived. Regularisation by truncated iteration is investigated as a simple and efficient regularisation method for the 4D-Var based inversion. In an experiment with synthetic observations, the truncated iteration was found to perform similarly to the commonly used Tikhonov regularisation. However, the truncated iteration allows the amount of regularisation to be varied a posteriori, without repeating the inversion. For inverting the Eyjafjallajökull SO2 emission at the temporal and vertical resolution used in this study, the 4D-Var method required about 70 % less computational effort than commonly used methods based on performing a separate model simulation for each degree of freedom in the estimated source term. Compared to the inversion using only total column retrievals, assimilating the plume height resulted in a vertical emission profile more closely matching the ash plume heights observed by radar. The a posteriori source term gave an estimate of 0.29 Tg erupted SO2 of which 95 % was injected below 11 km.

Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs

We investigate the convergence rate of the quantics-TT (QTT) stochas- tic collocation tensor approximations to solutions of multiparametric elliptic PDEs and construct efficient iterative methods for solving arising high-dimensional parameter- dependent algebraic systems of equations. Such PDEs arise, for example, in the para- metric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated Karhunen-Loève expansion. We consider both the case of additive and log-additive dependence on the multivariate parameter. The local-global versions of the QTT-rank estimates for the system matrix in terms of the parameter space dimension is proven. Similar rank bounds are observed in numerics for the solutions of the discrete linear system. We propose QTT-truncated iteration based on the construction of solution-adaptive preconditioner that provides robust conver- gence in both additive and log-additive cases. Various numerical tests indicate that the numerical complexity scales almost linearly in the dimension of parametric space M.