A Near-Real-Time Method for Estimating Volcanic Ash Emissions Using Satellite Retrievals

Abstract

We present a Bayesian inversion method for estimating volcanic ash emissions using satellite retrievals of ash column load and an atmospheric dispersion model. An a priori description of the emissions is used based on observations of the rise height of the volcanic plume and a stochastic model of the possible emissions. Satellite data are processed to give column loads where ash is detected and to give information on where we have high confidence that there is negligible ash. An atmospheric dispersion model is used to relate emissions and column loads. Gaussian distributions are assumed for the a priori emissions and for the errors in the satellite retrievals. The optimal emissions estimate is obtained by finding the peak of the a posteriori probability density under the constraint that the emissions are non-negative. We apply this inversion method within a framework designed for use during an eruption with the emission estimates (for any given emission time) being revised over time as more information becomes available. We demonstrate the approach for the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions. We apply the approach in two ways, using only the ash retrievals and using both the ash and clear sky retrievals. For Eyjafjallajökull we have compared with an independent dataset not used in the inversion and have found that the inversion-derived emissions lead to improved predictions.

1. Introduction

Volcanic ash clouds pose a hazard to aviation and there have been significant incidents where ash has affected aircraft [1,2]. In response to this hazard, the Volcanic Ash Advisory Centres (VAACs) were established to provide guidance to aviation on the location of volcanic ash. Satellite observations and atmospheric dispersion models play an important role in providing such guidance by enabling the ash to be detected and its evolution to be predicted.

For modelling volcanic ash dispersion, and especially when estimates of ash concentration (rather than just ash extent) are needed, estimates of the ash emissions are required. These are often characterised by the rise height of the volcanic plume, the ash mass release rate and the vertical distribution of the ash released, together with the time evolution of these quantities. These values should also be effective values, appropriate to the physics that is included in the dispersion model. For example, if there is a lot of near source fallout due to large particles or due to aggregation of small particles, and if these processes are not included in the dispersion model, then an estimate of the reduced effective mass release rate is needed.

Estimates of the rise height of the volcanic plume are often available. This is especially true in regions such as Iceland where volcanoes are well monitored with radar and web cameras [3,4]. The mass release rate is harder to determine. Techniques to assess this directly (e.g., infrasound, thermal imagery, microwave radar) are an active area of research [5,6] but are not considered here. Instead, estimates can be made using empirical relationships [7,8]. These relationships tend to take the form of an expression for the total mass eruption rate as a function of the plume rise height, based on data from past eruptions. However, they do not account for the effect of the wind strength on the plume rise [9] and do not estimate the vertical distribution of ash. The vertical distribution of ash is sometimes approximated by assuming it to be uniform. Such an approximation is rather crude but has the advantage of reducing the risk of predicting no ash where ash may be present [10]. Alternatively, plume rise models can be used. Plume rise models can estimate the mass emission rate and the vertical spread of the ash from the atmospheric conditions and the observed plume rise height [8,11,12,13].

Alternatively, satellite data can be combined with a dispersion model in an inversion calculation to estimate the volcanic ash emissions, including their height distribution and their evolution over time [14,15,16,17,18,19,20,21]. By considering the satellite data some distance away from the volcano vent, these estimates will be for the effective values after the occurrence of any near source processes that are not included in the dispersion model. Similar inversion techniques have also been used to estimate the SO${}_2$ emissions from volcanic eruptions [18,22,23,24,25].

This paper presents a Bayesian inversion modelling technique employed in InTEM (Inversion Technique for Emissions Modelling), the Met Office’s inversion modelling system, for estimating volcanic ash emissions. The approach is demonstrated using the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions. The method presented here is available for use at the London VAAC to assist in providing guidance. An early version of the approach is described in [26] with some later modifications presented in [27,28].

2. Materials and Methods

We have extended the InTEM system, which was originally designed to estimate emissions of greenhouse gases and ozone depleting substances [29,30,31], in order to estimate volcanic ash emissions. We use a Bayesian approach to combine a priori estimates of the emissions (and of the uncertainties in these estimates) with the satellite ash retrievals and with dispersion model simulations to provide improved a posteriori emission estimates.

We make assumptions about the a priori probability distribution of the emissions using local measurements of the eruption column height. Satellite retrievals of ash column load are produced using the approach of Francis et al. [32]. Ash cloud dispersion predictions, produced using NAME (Numerical Atmospheric-Dispersion Modelling Environment) [33], are used to relate emissions to column loads. Then the inversion process is used to make deductions about the a posteriori probability distribution of the emissions. In particular, InTEM estimates the volcanic ash source profile (i.e., the time- and height-varying ash release rate) corresponding to the peak of the a posteriori probability distribution. This source profile can be regarded as that which gives an optimal fit of both the dispersion model predictions to the satellite retrievals of ash column load and of the source profile’s emissions to the a priori emission estimates based on the observed plume rise height. What we mean by ‘optimal’ is determined by the Bayesian formalism, which automatically weights these two aspects appropriately to allow for estimated uncertainties in the satellite column loads and in the a priori emissions. This source profile can then be used in dispersion models to make forecasts. This type of approach follows that used in [14,15,17]. Note we do not account for uncertainties in the dispersion model or in the met data driving the dispersion model. This deserves further investigation, especially the uncertainties arising from uncertainties in the meteorology.

Within the Bayesian formalism we assume joint Gaussian distributions for simplicity, similarly to the approaches described in [22,34,35,36]. As a result, all the probability distributions are determined by the means, variances and covariances. However, with the Gaussian assumption, we require a method to ensure the emissions are non-negative as this is not imposed through the probabilistic assumptions. Eckhart et al. [22] and Stohl et al. [14] solved this problem by reducing the a priori uncertainty on the emissions from any troublesome negative sources until non-negative emissions were obtained. However this risks the a priori assumptions exerting too strong an influence on the results. Here we prefer to simply find the location of the peak of the a posteriori probability distribution under the constraint that the emissions are non-negative.

Note that we perform the satellite retrievals (themselves being a type of inversion) and the source inversion separately, rather than choosing the emissions that, using forward modelling of the atmospheric dispersion and the radiative transfer, fit the satellite measured radiances. As far as we are aware, this two-step method is in line with all other approaches to volcanic ash inversion. The method has logistical and computational-speed advantages, with the two steps being separately maintained and developed, and with the ability to take advantage of the linearity of the dispersion in the source inversion. The approach is also assisted by the nature of the satellite retrieval scheme used. This seeks to give results only where these are robust and does not seek to extract low-grade information from the satellite observations. It seems possible that a one-step approach might be more effective at making use of low-grade information at locations where currently the satellite retrievals give no information, but we do not explore that here.

Most published volcanic ash inversion techniques (see references in Section 1) produce the source estimates post eruption. Here we apply the approach as it would be used during a volcanic eruption with updated predictions being produced as more satellite data become available.

2.1. The Satellite Retrievals

We use data from the Spinning Enhanced Visible and InfraRed Imager (SEVIRI) on the Meteosat Second Generation geostationary satellite. At the latitude of Iceland (∼${65}^{\circ }$ N) the pixel size for the infrared channels used for ash detection and retrieval is of the order of 10 km. Scans are completed every 15 min.

Ash properties are estimated from SEVIRI data using the approach described in [32]. The approach first applies a detection algorithm to estimate which SEVIRI pixels contain detectable ash. The retrieval then applies an optimal estimation technique (a one-dimensional variational analysis) to each of these “ash pixels” which estimates the ash column load (L), the effective ash particle radius ($R_{\mathrm{eff}}$) and the atmospheric pressure at the ash layer ($p_{\mathrm{ash}}$), as well as the size of the errors in these quantities. The approach assumes an infinitesimally thin ash layer, which is why only a single pressure value is retrieved. If the layer has significant depth then the retrieved pressure is likely to be between the layer centre and the layer top, depending on the optical thickness of the ash. Note our inversion method does not currently use the information on $R_{\mathrm{eff}}$ and $p_{\mathrm{ash}}$.

Clear sky pixels are also diagnosed. A satellite pixel is classified as clear sky and assigned a column load of zero if the detection algorithm fails to detect ash and no cloud (which might obscure any ash) is identified based on the scheme described by Saunders and Kriebel [37]. The minimum detectable column load depends on the height of the ash and the atmospheric temperature profile. A conservative estimate of the minimum detectable level for the approach [32] is 0.5 g m${}^{-2}$. As a result, it is possible that pixels with less than 0.5 g m${}^{-2}$ of ash could be labelled as clear sky pixels. Hence, we assign an error estimate of 0.5 g m${}^{-2}$ to the column loads in the clear sky pixels.

Finally, those pixels which are not classified as either ash or clear sky pixels remain unclassified. These pixels might contain ash hidden by cloud or might not.

The ash column loads and associated errors are then regridded to a resolution of 0.5625${}^{\circ }$ longitude and 0.3750${}^{\circ }$ latitude (approximately 40 km) and hourly averages are computed. The purpose of regridding is both to reduce the data volume and to match the output resolution chosen for the NAME dispersion simulations. It also helps with the treatment of correlations in the satellite data. For Icelandic volcanic eruptions the domain of the regridded satellite data extends from 50${}^{\circ }$ W to 34${}^{\circ }$ E and from 35${}^{\circ }$ N to 80${}^{\circ }$ N. The column load for a 40 km grid cell is calculated by averaging the column loads for the ash and clear sky pixels in the cell (assuming zero for the clear sky pixels). The error estimate for the cell column load is evaluated by averaging the analysis errors for the ash and clear sky pixels. This method of combining the error estimates (interpreted as standard deviations) amounts in effect to assuming a high correlation between the errors in the pixels within the cell. It is important to assume some correlation between the pixels to avoid greatly underestimating the error for the cell as a whole due to the tendency of uncorrelated errors to cancel.

There remains the problem of representativity errors, which arise when the ash and clear sky pixels are not representative of all pixels. We treat this by only using the 40 km cells for which at least 50% of the pixels are ash pixels or for which 90% of the pixels are either ash or clear sky pixels. This also has the advantage of reducing the total number of 40 km cells used in the inversion and thus reduces the computational cost.

We then classify these 40 km grid cells as ash or clear sky cells according to whether the estimated column load is greater than zero or equal to zero. The purpose of this is to ensure that we can compare the effect of using only the ash cells (i.e., excluding the zero values) or using both the ash cells and the clear sky cells (i.e., including the zero values). For the ash cells detected during the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions, the average column load from the satellite data is 1.9 g m${}^{-2}$ and the average error estimate is 6.1 g m${}^{-2}$. Error estimates tend to be smaller in the high column load parts of the ash cloud.

For the inversion used to estimate the emissions, we will need to assume something about the correlation between the errors in the different cells. Following Stohl et al. [14], we take these as zero. This has the advantage of simplicity and reflects the idea that correlation decays with separation. While the sharp cut-off in the correlation (i.e., the high correlation assumed within the cells as discussed above and the zero correlation between cells) is not ideal, this is a pragmatic choice. This is similar conceptually to the approach that is sometimes used in data assimilation for numerical weather prediction, where satellite retrievals are ‘thinned’ (on a scale typically in the range 10–100 km) and zero correlation is assumed for the errors in the remaining retrievals [38]. If in the future we wish to exploit higher-resolution satellite information but avoid the satellite data, by virtue of the large quantity of data, having too much weight in the inversion relative to the a priori distribution, then a more sophisticated treatment of correlations would be needed.

2.2. The NAME Model

InTEM requires information on how erupted material is transported and dispersed from the volcano. This is provided by NAME, a Lagrangian stochastic atmospheric dispersion model. In NAME, model particles trajectories are calculated using winds from a numerical weather prediction (NWP) model. A stochastic component is added to each trajectory to account for atmospheric turbulence. Model particles carry a mass of the dispersing material (in our case volcanic ash) calculated from the mass emission rate. Each model particle also has a sedimentation velocity which is calculated from the diameter and density of the particles it represents. A density of 2300 kg m${}^{-3}$ is assumed for volcanic ash [10,39]. NAME also models wet deposition and non-sedimentation-driven dry deposition, such as that caused by diffusion to the surface [10].

We do not attempt to model the fraction of ash that, as a result of large grain sizes or the aggregation of smaller grains, falls out close to the volcano. Instead, NAME uses an effective source strength corresponding to the emission rate of the fine ash particles that undergo long-range transport [10]. NAME does not model aggregation.

In modelling particulates, each NAME model particle represents numerous real particles of a particular diameter. The particle diameters modelled are described by a particle size distribution. Here we use a particle size distribution based on data provided by Hobbs et al. [40]. The range of diameters in this size distribution is 0.1 $\mathsf{\mu }$m to 100 $\mathsf{\mu }$m and is not intended to represent all the ash particles from the volcano but just the fine ash particles that undergo long-range transport.

NAME can use NWP meteorological data from a range of NWP models. Here we use meteorological data from the Met Office’s Unified Model (MetUM) in its global configuration. This configuration has a horizontal resolution of about 25 km in mid latitudes and a vertical grid with 70 unevenly spaced levels with a spacing of 300–400 m in the mid-troposphere. We use data obtained from the model at three hourly intervals.

Because NAME’s ash concentration is linear in the emissions, the concentrations for a linear combination of sources can be obtained from those for the individual sources by combining the predicted concentrations. Here we use sources at different times and heights above the volcano with each source having a release rate of 1 g s${}^{-1}$, and we separately store the results due to each source. InTEM then attempts to select the best linear combination (defined using the Bayesian formalism described below) of these sources (the ‘source profile’). We note that, although the satellite data used in the inversion do not provide information on the vertical distribution of the ash, we expect the combination of the dispersion model and the satellite data to be able to constrain the height variation of the emissions because of the changes with height of the wind, and of the wind direction especially. The accuracy of the height distribution of the emissions (or indeed other aspects of the emissions) will, however, be affected by errors in the wind data. The individual sources are defined on a time–height grid with particle release locations for a given grid cell evenly distributed across the cell.

To compare with the one-hour averaged column loads estimated from the satellite data, the ash concentrations from NAME are averaged over 1 h and vertically integrated. The horizontal grid used for the NAME output is chosen to coincide with that of the regridded satellite retrievals.

The particle size distribution used in NAME covers diameters between 0.1 $\mathsf{\mu }$m and 100 $\mathsf{\mu }$m but the diameters detectable by the SEVIRI satellite are limited. The satellite retrieval assumes a log-normal size distribution which can extend beyond the size range to which the satellite is sensitive and provides an estimate of the ash mass in a diameter range of approximately 1 $\mathsf{\mu }$m–30 $\mathsf{\mu }$m [14,32,41]. Figure 1 shows the distribution used in NAME and where this overlaps with the range that the satellite ‘sees’. About 95% of the mass in the NAME emissions is detectable by the satellite retrievals. To allow for this, we output from NAME only the part of the column load due to the particles within the detectable range of 1 $\mathsf{\mu }$m–30 $\mathsf{\mu }$m.

For computational efficiency, particles are not followed beyond six days after release. It is expected that, by this time, most of the ash will have been removed from the atmosphere or widely dispersed [10]. Hence, we expect column loads due to particles older than six days to be small, at least for eruptions comparable in size to the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions. The six-day limit could of course be changed if required.

2.3. The Prior Source Distribution

As we use a Bayesian approach, we need to make assumptions about the probability distribution of the emissions, prior to using the satellite data to refine our assumptions. For this, we make use of the observed plume rise height. The prior assumptions should help regularise the problem, avoid overfitting to the satellite data, and ensure that the observations of plume rise are taken into account. If we just look for a best fit to the satellite data, then the resulting source profiles may have emissions at heights which are too high to be consistent with the observed plume rise height and its uncertainty. While we expect the variation of the wind (and especially the wind direction) with height to enable emission heights to be inferred from satellite data, this may not always be possible, and so it is important to try to make optimal use of all available information. Additionally, sometimes there may be little satellite data available, making the use of the a priori source description even more crucial.

In principle, the observations of plume rise could be regarded as more observations to be used in the inversion. The a priori description would then need to reflect the range of possible volcano behaviours prior to any observations of plume rise. With a wide range of observation types, this might have advantages. However, with the main sources of data being just the plume rise height and satellite data, it is easiest to treat these separately and include the former in the a priori description.

Our starting point in defining the a priori description is the default source specification that is currently used by the London VAAC [10,42]. This expresses the mass eruption rate in terms of the plume rise height using

$$M=140.84H^{4.15}$$

(1)

where M is the mass eruption rate in kg s${}^{-1}$ and H is the plume rise height above the volcano vent in km. This expression is based on an empirical power law best fit to many volcanic eruptions presented by Mastin et al. [7]. The emission profile is taken as uniform between the volcano summit and the plume rise height. While the true effective source profile is more likely to be weighted towards the top of the eruption column, this choice is made by the London VAAC to avoid the risk of predicting no ash where there is a significant risk [10]. Since we are only interested in the fine ash that undergoes long-range transport, the Mastin emission formulae is multiplied by a constant $c_f$ representing the distal fine ash fraction, i.e., the mass fraction of the ash that survives the near source fall out of particles larger than 100 $\mathsf{\mu }$m and the near source fall out of smaller particles due to aggregation or due to wet deposition associated with the eruption. The effective mass emission rate m of fine ash which reaches the distal ash cloud is then given by $m=c_{f}M$. The factor $c_f$ is taken to be $0.05$ [10,42].

For the inversion calculation, we need a probabilistic description which describes the uncertainties in the emissions. It is not sufficient to simply take the London VAAC emission profile as the mean emission and add uncertainty information to that because, for example, this gives zero mean emissions above the observed rise height and, if there is a risk that the rise height observation is an underestimate, the mean emission should be greater than zero at these heights. Moreover, it is desirable to include estimates of the correlations between the emissions in the various time–height grid cells. This is because, for example, if the rise height is higher than estimated or the mass eruption rate is greater than that given by the Mastin relationship, this is expected to have an impact on all the emissions. If the correlations are ignored in the probabilistic description, then the implied variance of the overall emission will be too small for a given variance in each grid cell due to the tendency of the fluctuations from the mean in the individual cells to cancel. Either the a priori description will overconstrain the overall emission or it will allow too much uncertainty in individual grid cells leading to very noisy emission profiles. Neglecting correlations would also mean that the uncertainty in the shape of the emission profile will be intrinsically linked to the uncertainty in the total emission which is undesirable. Including correlations should also result in an approach that gives results independent of the resolution of the height–time grid as the resolution increases.

In order to estimate the means, variances and covariances of the emissions in the various height–time grid cells, we propose a stochastic model of the source emission profile. In principle, this could be used to generate many realisations of the profile from which the mean and variance of the emission in each cell and the covariance between the emissions in each pair of cells can be calculated. However, it turns out that the model is susceptible to analytic study and (provided $H_b\left(t\right)$ and $\delta H\left(t\right)$, defined below, are piecewise constant) we can calculate the required quantities with the aid of some numerical integrals, with the benefit of reduced computational cost and a more accurate calculation.

We assume that the plume rise height $H\left(t\right)$ has mean $H_b\left(t\right)$, our best estimate of the rise height as assessed from the local observations, and varies in the range $H_b\left(t\right)\pm \delta H\left(t\right)$ on a time scale $T_H$, where $\delta H$ is an estimate of the error in the rise height. We choose hard limits on H to avoid any potential difficulties with negative H such as might occur with Gaussian processes and, to make this easy to model, we treat the time dependence as involving random jumps to new values which are uniformly distributed in $H_b\pm \delta H$, with exponentially distributed waiting times between jumps and mean waiting time $T_H$. Between jumps, $(H-H_b)/\delta H$ remains constant. For a given rise height H we assume an effective mean emission rate per unit height (in kg s${}^{-1}$ km${}^{-1}$ for H in km) given by

$$\frac{m}{H}=140.84c_f{H}_b^{3.15}\left(1+3.15\frac{H-H_b}{H_b}\right)$$

(2)

where we have based this on (1) but have linearised about $H=H_b$ to keep the mean total emission per unit height in the bulk of the column (up to $H_b-\delta H$) equal to the default value used by the London VAAC. The emission rate per unit height in a single realisation is then modelled by multiplying this by two random factors. The first factor, $1+r\left(t\right)$, which is independent of height, accounts for the variation in the mass emission rate in time which is not accounted for by (2). We take r to be exponentially correlated in time with mean zero, standard deviation ${\sigma }_r$, and time scale $T_r$. The second factor, $1+s(t,z)$, accounts for variations in the shape of the emission profile but does not alter the total mass emitted. s is taken to be $q-\frac{1}{H}{\int }_0^{H}qdz$ where q has standard deviation ${\sigma }_q$ and is exponentially correlated in t and $z/H$ with correlation time scale $T_q$ and (non-dimensional) correlation height scale $L_q$. The relationship between s and q is defined so that s has zero vertical integral and hence does not alter the total mass emitted. We also assume q decorrelates completely in time whenever there is a jump in $(H-H_b)/\delta H$ as this is interpreted as reflecting a significant change in volcano behaviour. Unlike the time series of H, there is no need to consider whether r and q are Gaussian processes or not because this does not affect the first and second moments of the emissions.

Details of the method used to obtain values for the means, variances and covariances of the emission rates integrated over each grid cell using only numerical integrals and without actually simulating an ensemble of realisations of the stochastic model are given in [27]. The results obtained using this analytic method were compared with results using an ensemble of realisations of the emission profile and were found to agree within the sampling error expected for a finite number of realisations. The means, variances and covariances are expressed using units of g s${}^{-1}$ for the emission rates for consistency with the nominal release rate of 1 g s${}^{-1}$ used in the NAME runs.

Of course there are many aspects of this stochastic model that are greatly simplified compared to reality and there is a lot of uncertainty in the values to assign to the model parameters (discussed below). However we believe that including correlations in this way has the potential to give significant benefits in achieving a better balance between the a priori constraint on the total emission and on the individual cell emissions, as well as in decoupling the uncertainty in the emission profile shape from the uncertainty in the total emission. Parametrizing the various components of the uncertainty separately also makes clear how the assumptions relate to the physical situation and makes explicit where things could be improved. This should enable the improvement of the approach through further studies (e.g., detailed simulations or observations of the eruption column for a range of conditions or experience in applying the approach to a wide range of eruptions). The inclusion of the correlations between the cell emissions should also help to achieve greater benefit when there are few satellite observations, by allowing the observations to inform the emissions from cells that are not directly linked (via the NAME model) to the observations as well as informing the emissions from the cells that are so linked. In contrast, when there are substantial quantities of satellite data, we expect results to be insensitive to the details of the a priori emission description. There might be benefit in assuming an a priori distribution for the uncertain parameters and attempting to refine this as part of the inversion. However, we have not attempted that here.

Our approach can be regarded as lying somewhere between an approach that neglects the correlations in the a priori emissions (e.g., [14]) and an approach that uses high correlations or (similarly but not quite equivalently) that uses an emission model with a small number of degrees of freedom (e.g., [17,19,20,21]). In designing the a priori model we have adopted a conservative choice in staying close to the default London VAAC emission specification, but there may be value in moving away from this in future (e.g., using plume rise models or choosing a non-uniform vertical distribution for the mean emissions).

We now discuss the choice of parameters for the stochastic model. Many volcanoes, including those in Iceland, are observed by various methods such as radar and webcams, and these observations can be used to estimate the plume rise [3,43]. These estimates are subject to an uncertainty (e.g., resulting from the radar’s discrete scanning elevation angles) which can itself be estimated. Here we assume values of $H_b$ and $\delta H$ can be estimated as a function of time. We take the time scale $T_H$ to be 12 h, similar to the observed rate of change of plume height in the Eyjafjallajökull 2010 eruption, using the idea that the plume height and the errors in the plume height are likely to evolve on comparable time scales. For very short eruptions this value will be too large, but the value is then less important than for sustained eruptions like Eyjafjallajökull.

Mastin et al. [7] indicate that in 50% of cases the mass emission rate falls within a factor of four either side of (1). However, it is likely that this error estimate is at least partly due to errors in the measured rise height which we are treating separately via $\delta H$, and so this error estimate may be too large for use in estimating ${\sigma }_r$ in our model. On the other hand, the error in the effective mass emission rate is likely to be increased by the uncertainty in the distal fine ash fraction. On balance, we judge that ${\sigma }_r=1$ is a reasonable estimate. This allows emissions of several times the value calculated from Equation (2) at $H=H_b+\delta H$ and also allows very small values, reflecting the large uncertainties. $T_r$ is also taken to be 12 h, based on the idea that the relationship between mass emission rate and the plume height is likely to evolve on a similar time scale to the plume height due to changes in volcano behaviour, or on a time scale related to changes in meteorology which influence the relationship between the emission rate and rise height [12,13].

To allow for substantial departures from the uniform height profile of the emission we take ${\sigma }_q=1$ and we take $T_q=3$ hours and $L_q=0.3$ to allow substantial variations in the vertical profile shape on time scales that are a little faster than the total emission time scale and on height scales significantly smaller than the plume rise height.

2.4. The Inversion Technique

The inversion calculation requires a way to predict the column loads (at the locations of the satellite retrievals) in terms of the emissions. For a given set of emissions, we will call the model predictions corresponding to the satellite observations the ‘model observations’. We wish to compare these with the satellite retrievals. Suppose there are k satellite retrievals of column load at space-time locations $(x_i,y_i,t_i)$, $i=1,\cdots ,k$. If we consider the ith satellite retrieval, the corresponding model observation $o_{mi}$ for a given set of emissions can be expressed in terms of the emissions and the NAME column loads obtained from individual emission cells as

$$o_{mi}=\sum _{j=1}^n{N}_{i,j}{e}_j$$

(3)

where $N_{i,j}$ is the NAME prediction of column load at $(x_i,y_i,t_i)$ for a 1 g s${}^{-1}$ emission rate in the jth source cell and $e_j$ is the effective emission rate from the jth source cell in units of g s${}^{-1}$ (for $j=1,2,\cdots ,n$ where n is the total number of model sources). This can be expressed as the matrix equation

$${\mathbf{o}}_m=\mathbf{Ne}$$

(4)

where $\mathbf{N}$ is the NAME source dilution matrix, i.e., the $k\times n$ matrix of the model column loads ($N_{i,j}$), $\mathbf{e}$ is the vector of source strengths and ${\mathbf{o}}_m$ is the vector of model observations. The analogous vector of satellite retrievals will be indicated by ${\mathbf{o}}_a$ and the vector of a priori mean emissions by ${\mathbf{e}}_{ap}$.

The goal of the inversion technique is to find emissions $\mathbf{e}$ which result in a fit of both the model observations to the satellite retrievals, i.e., $\mathbf{Ne}\approx {\mathbf{o}}_a$, as well as of the emissions to the a priori mean emissions, i.e., $\mathbf{e}\approx {\mathbf{e}}_{ap}$. The optimal choice uses information on the uncertainty in the satellite retrievals and in the a priori values to decide on the relative priority for fitting $\mathbf{Ne}\approx {\mathbf{o}}_a$ and $\mathbf{e}\approx {\mathbf{e}}_{ap}$. We do this by maximising the probability density of the emissions given the satellite retrievals, $p\left(\mathbf{e}\right|{\mathbf{o}}_a)$, which Bayes theorem relates to the probability density of the satellite retrievals given the emissions, $p\left({\mathbf{o}}_a\right|\mathbf{e})$, and the (a priori) probability densities for the emissions and the satellite retrievals, $p\left(\mathbf{e}\right)$ and $p\left({\mathbf{o}}_a\right)$.

With our Gaussian assumption, the probability density $p\left({\mathbf{o}}_a\right|\mathbf{e})$ of ${\mathbf{o}}_a$ given $\mathbf{e}$ is proportional to

$$exp\left[-\frac{1}{2}{\left(\mathbf{Ne}-{\mathbf{o}}_a\right)}^{\mathrm{T}}{\mathbf{R}}^{-1}\left(\mathbf{Ne}-{\mathbf{o}}_a\right)\right]$$

(5)

and the a priori probability density $p\left(\mathbf{e}\right)$ of $\mathbf{e}$ is proportional to

$$exp\left[-\frac{1}{2}{\left(\mathbf{e}-{\mathbf{e}}_{ap}\right)}^{\mathrm{T}}{\mathbf{B}}^{-1}\left(\mathbf{e}-{\mathbf{e}}_{ap}\right)\right].$$

(6)

Here $\mathbf{R}$ is the error covariance matrix for the satellite retrievals and $\mathbf{B}$ is the error covariance matrix for the a priori source profile. In principle, $\mathbf{R}$ could be regarded as including the errors in the transport model. However, such errors are likely to have a linear dependence on $\mathbf{e}$ and a complicated correlation structure (e.g., a typical error might involve the bodily displacement of the ash cloud) which is difficult to treat well except through using ensembles of meteorology. Hence, we do not attempt to account for transport model errors here. $\mathbf{R}$ is a $k\times k$ matrix with the satellite retrieval variances on the diagonal and zeros off the diagonal in line with the assumptions described in Section 2.1. Similarly, the matrix $\mathbf{B}$ is filled with the variances and covariances of the a priori emissions derived from the model in Section 2.3 to give an $n\times n$ (non-diagonal) matrix. Combining $p\left({\mathbf{o}}_a\right|\mathbf{e})$ and $p\left(\mathbf{e}\right)$ by Bayes theorem gives an expression proportional to $p\left(\mathbf{e}\right|{\mathbf{o}}_a)$. Taking the log of this expression allows the problem to be re-expressed in terms of minimising the cost function

$$J\left(\mathbf{e}\right)={\left(\mathbf{Ne}-{\mathbf{o}}_a\right)}^{\mathrm{T}}{\mathbf{R}}^{-1}\left(\mathbf{Ne}-{\mathbf{o}}_a\right)+{\left(\mathbf{e}-{\mathbf{e}}_{ap}\right)}^{\mathrm{T}}{\mathbf{B}}^{-1}\left(\mathbf{e}-{\mathbf{e}}_{ap}\right).$$

(7)

As discussed above, our best estimate of the emissions is taken to be the $\mathbf{e}$ which minimises J under the constraint that the emissions are non-negative.

To find the cost function minimum we use the ‘non-negative least squares’ (NNLS) algorithm of Lawson and Hanson [44]. This is similar to an approach presented in [45] and is an ‘active set method’ in that it iterates over choices of which components of $\mathbf{e}$ have the non-negative constraint active, i.e., which components have value zero. To apply this approach the cost function needs to be expressed in the form

$$J\left(\mathbf{e}\right)={|\mathbf{L}\mathbf{e}-\mathbf{q}|}^2={\mathbf{e}}^{\mathrm{T}}{\mathbf{L}}^{\mathrm{T}}\mathbf{L}\mathbf{e}-2{\mathbf{q}}^{\mathrm{T}}\mathbf{L}\mathbf{e}+{\left|\mathbf{q}\right|}^2$$

(8)

for some matrix $\mathbf{L}$ and vector $\mathbf{q}$. This can be achieved (up to the addition of a term independent of $\mathbf{e}$) by expanding the cost function in the form

$$J\left(\mathbf{e}\right)={\mathbf{e}}^{\mathrm{T}}\left({\mathbf{N}}^{\mathrm{T}}{\mathbf{R}}^{-1}\mathbf{N}+{\mathbf{B}}^{-1}\right)\mathbf{e}-2\left({\mathbf{o}}_a^{\mathrm{T}}{\mathbf{R}}^{-1}\mathbf{N}+{\mathbf{e}}_{ap}^{\mathrm{T}}{\mathbf{B}}^{-1}\right)\mathbf{e}+{\mathbf{o}}_a^{\mathrm{T}}{\mathbf{R}}^{-1}{\mathbf{o}}_a+{\mathbf{e}}_{ap}^{\mathrm{T}}{\mathbf{B}}^{-1}{\mathbf{e}}_{ap},$$

(9)

equating the quadratic and linear terms, and solving for $\mathbf{L}$ using Cholesky decomposition. We note that after some preliminary calculations to compute the coefficients in the quadratic function (9), the size of the matrices and vectors, and hence the computational cost, depends only on the number of source cells and is independent of the, usually much larger, number of satellite retrievals.

An advantage of assuming joint Gaussian distributions is that, for a linear forward model $\mathbf{e}\mapsto {\mathbf{o}}_m=\mathbf{N}\mathbf{e}$ as assumed here, the a posteriori distribution is also Gaussian. The a posteriori error covariance matrix for $\mathbf{e}$, $\mathbf{A}$, is then given by the inverse of the coefficient of the quadratic term in (9),

$$\mathbf{A}={\left({\mathbf{N}}^{\mathrm{T}}{\mathbf{R}}^{-1}\mathbf{N}+{\mathbf{B}}^{-1}\right)}^{-1}.$$

(10)

Of course the Gaussian assumption is not exact because it implies a possibility of negative emissions, and here it has been supplemented by a truncation of the negative values. Hence, there is some uncertainty in the precise interpretation of $\mathbf{A}$. As a result, it is more appropriate to regard (10) as giving a general indication of uncertainty.

2.5. Solving the Inverse Problem during an Eruption

Our approach is designed for use during an eruption with results being updated as more data become available. By default, we produce source profile estimates every six hours (using satellite retrievals made for times up to 00:00, 06:00, 12:00 and 18:00 UTC each day). We denote these data cut-off times by $t_i$, $i=1,2,$ etc., and denote the eruption start time by $t_0$.

The procedure used to estimate the source profile up to time $t_i$ is as follows:

1.

Run NAME with the latest available analysis meteorology valid until time $t_i$, using all the source components that can emit up to time $t_i$ and tracking the particles from any given source up to the earlier of $t_i$ and six days after release. For $i>1$ the runs use stored results from $t_{i-1}$ in order to reduce cost.

2.

Estimate the means and covariances of the a priori emissions from these source components using the observed plume rise heights from $t_0$ to $t_i$.

3.

Run InTEM to estimate the a posteriori source profile given the satellite retrievals from $t_0$ to $t_i$. Note in particular that, for $i>1$, we do not use the InTEM a posteriori source profile from $t_{i-1}$, nor do we consider only satellite data for times between $t_{i-1}$ and $t_i$. Instead, the inversion starts afresh from the a priori source profile and accounts for all the satellite data up to $t_i$.

This is repeated every 6 h.

Generally, the NAME runs dominate the cost of the calculation. However, at each update the NAME runs only need to be advanced by 6 h for 6 days’ worth of sources, whereas the inversion calculation (i.e., the calculation of the a priori emission statistics, the cost function coefficients and the optimal emissions) needs (at least with our current setup) to cover the whole eruption up to the current time $t_i$. Hence, for a long eruption such as the 2010 Eyjafjallajökull eruption, the cost of the inversion can become comparable to the cost of the NAME runs. Towards the end of the Eyjafjallajökull eruption, the time taken using 4 cores on a linux server is about 19 min for the NAME simulations and 13 min for the inversion calculation. This does not include the cost of performing the satellite retrievals and of producing the met data to drive the dispersion model. These timings are for the configuration used in producing the results below for the case with ash and clear sky satellite data.

The time resolution of the source profile must exactly divide the source profile update interval (6 h). For computational speed, we choose, as default, the source discretisation to be 3 h in time and 4 km in height. Some tests of the sensitivity of inversion results to the source resolution for the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions are presented in [28].

3. Results and Discussion

In this section we demonstrate the approach by showing results for the 2010 Eyjafjallajökull and 2011 Grímsvötn Icelandic eruptions.

3.1. Eyjafjallajökull

Eyjafjallajökull is located at 63.63${}^{\circ }$ N, 19.62${}^{\circ }$ W and has a vent height of 1666 m asl. It erupted on the 14th of April 2010 at 09:00 UTC with the eruption lasting 39 days until 23 May 2010 at 18:00 UTC. Gudmundsson et al. [46] have described the eruption as having four phases: 14–18 April, the first explosive phase; 18 April–4 May, the low discharge mainly effusive phase; 5–17 May, the second explosive phase; and 18–22 May, the declining phase. The northwesterly winds [47] and the duration of the eruption resulted in substantial disruption to aviation in Europe.

The plume rise heights were provided in real time to the London VAAC by the Icelandic Met Office. They are based on data from the radar at Keflavík airport [3] and other local observations. The heights used here to calculate the a priori emissions are a post eruption revision [10] that is still based on local Icelandic observations and does not incorporate any satellite data. These heights are shown in Figure 2. An uncertainty of $\delta H$ = 2000 m is assumed. This allows for a reasonable amount of variability in the a priori model and is broadly consistent with the discrete radar scanning angles used [3]. Satellite ash and clear sky retrievals were obtained for the period between the start of the eruption and 29/5/2010 23:00 UTC. This period extends to some time after the end of the eruption in order to include any detections of ash remaining in the atmosphere. The total number of ash and clear sky 40 km cells in the satellite data which are relevant to the inversion (i.e., which are reached by model particles from one of the source elements) is 739,469 of which 40,728 are ash cells and the rest are clear skies.

Figure 3 and Figure 4 show four of the inversion source profiles produced for the eruption using ash-only retrievals and using ash and clear sky retrievals respectively, together with the mean a priori source. Each profile makes use of six more days of satellite retrievals compared to the previous profile, with the satellite data cut-off times chosen so as to focus on the changes occurring as more data are introduced during phases three and four of the eruption. More ash satellite retrievals are available in these two phases than earlier in the eruption. The source profiles show some significant changes relative to the a priori mean emissions. In addition, the emission estimates evolve for a period of time as more data become available; this can be seen by looking at the emissions near the end of one plot and noting how they have changed in the next. From looking at the plots corresponding to all satellite data cut-off times (of which only a sample is shown in Figure 3 and Figure 4), it is clear that the emission estimates are mostly converged by four days after the emission occurred (in the sense that results do not show significant changes later on when more satellite data become available to inform the emissions), although there are a few instances where convergence takes five or six days. This is probably because, for this eruption, the ash is sufficiently diluted after this time that the sensitivity of the satellite retrievals to the ash is low. With ash-only data there are some significant reductions in emissions (for example, near the start of the eruption) and in fact the total emission over the entire eruption is reduced (see below). There are also some periods with only small changes from the a priori emissions. Generally, these periods have weaker emissions, which lead to ash column loads that are harder to detect by satellite. When both ash and clear sky retrievals are used, the reductions in emissions are substantially larger.

The a posteriori estimate of the total emitted (effective) mass for the whole eruption is $1.09\times {10}^{13}$ g using ash-only satellite retrievals and $4.63\times {10}^{12}$ g using ash and clear sky retrievals. These values are significantly reduced from the a priori estimate of $1.85\times {10}^{13}$ g. We note that the a posteriori mass estimates are similar to those obtained by the inversion method of Stohl et al. [14] ($8.3\pm 4.2\times {10}^{12}$ g). Using Gudmundsson et al.’s [46] estimate of the total emitted airborne mass (excluding lava) of $3.8\pm 1.0\times {10}^{14}$ g yields a distal fine ash fraction in the range 1–4%.

Example ash clouds for 6th May calculated with NAME using these emissions are shown in Figure 5 and Figure 6. Using the inversion-derived source profiles, the ash column loads on 6th May are reduced significantly from those obtained from the a priori source. Additionally, the column loads on the 6th change significantly as more data become available between 6 and 12 May, but they show little change as subsequent data are included. Of course, the results with later data included would not be available on the 6th; however, comparison with these results gives an indication of the errors relative to a post-event analysis and these results would feed through to predictions after the 6th as the plume is advected. The column loads obtained using ash and clear sky satellite data are generally lower than those obtained using ash-only data.

Table 1. Statistical comparison between modelled column loads and satellite ash retrievals for the Eyjafjallajökull 2010 eruption at the locations of the 40,728 ash-only retrievals * made up to 29 May 2010 23:00 UTC.

	Emissions Used in the Model
	A Priori	A Posteriori Using Ash	A Posteriori Using Ash and
		Only Satellite Data ${}^{†}$	Clear Sky Satellite Data ${}^{†}$
mean error ${}^{‡}$ (g m${}^{-2}$)	−0.52	−1.10	−1.41
mean absolute error (g m${}^{-2}$)	2.01	1.33	1.53
rms error (g m${}^{-2}$)	4.22	1.87	2.06
correlation	0.24	0.47	0.41
$\sqrt{{\chi }^2}$	0.81	0.25	0.27

* mean and rms of the ash-only column load retrievals are 2.06 and 2.58 g m${}^{-2}$. ${}^{†}$ satellite retrievals up to 29 May 2010 23:00 UTC are used in the inversion. ${}^{‡}$ model minus observations.

compares model and satellite derived column loads, with the model values calculated using a priori and a posteriori emissions. The statistics presented include only (space-time) locations with retrieved ash column loads (clear sky retrievals are excluded) and which can be reached by model particles from at least one of the source elements. Without this restriction, the comparison would involve many comparisons of zero retrievals with zero or small model predictions which leads to much smaller error statistics but is not very informative. However, to give an indication of what happens outside these locations, we note that the mass fraction of satellite ash retrievals outside locations that model particles can reach is 6.2% of the total ash retrievals, while the mass fraction of model ash in locations of clear sky satellite data (when the model is run with inverted sources using ash and clear sky satellite data) is 5.9% of the total model ash in locations of ash and clear sky satellite data. Using ash-only data in the inversion, the mean absolute error, rms error and correlation all improve substantially relative to the results for the a priori emissions. In contrast, the mean error (bias) deteriorates. With both ash and clear sky data in the inversion, all the error statistics are slightly worse than with ash-only data. This is perhaps expected as the inversion is not now targeting just the locations considered in the statistics. However, except for the mean error, the statistics remain substantially better than with the a priori emissions. The bias may indicate that the inversion has reduced the emissions too much, especially when the clear sky data are used. However, compared to the rms error, the mean error is more strongly affected by the smaller errors which are likely to arise in the low concentration tails of the ash cloud where both model and satellite column loads are very uncertain. The table also contains the reduced ${\chi }^2$ statistic, defined as ${\chi }^2={\left(\mathbf{Ne}-{\mathbf{o}}_a\right)}^{\mathrm{T}}{\mathbf{R}}^{-1}\left(\mathbf{Ne}-{\mathbf{o}}_a\right)/(k-n)$ with $k=dim{\mathbf{o}}_a$ and $n=dim\mathbf{e}$. This is less than 1 even for the a priori emissions and reduces for the a posteriori emissions, suggesting that the errors in the satellite retrievals may have been overestimated.

The ash cloud predictions obtained using the source profile from the inversion calculations and the NAME model were compared with the dataset of observations considered by Webster et al. [10]. This dataset consists of a variety of concentration data from research aircraft, ground based lidars and sunphotometers, and balloon-borne instruments, which were not used in the inversion. These measurements are estimated to be within a factor of about two of the true values and the observations were selected with the aim of obtaining local peak values. However, we note that the sampling strategy is likely to have had biases. For example, there was naturally more interest in sampling ash than clean air, and, for the aircraft, high and potentially hazardous concentrations of ash had to be avoided. Additionally, the measurements did not sample all phases of the eruption equally. These aspects are discussed in detail in [10]. The data were measured over the period 16 April 2010 to 18 May 2010, with the bulk of the data being over the period 4 May 2010 to 18 May 2010, and with the measurement locations being over the UK, Germany and surrounding seas (Table 1 in [10]).

In comparing the predictions and observations, the same approach was adopted as that used in [10] for assessing predictions made without the aid of an inversion calculation. Three methods were used involving (i) model predictions averaged over deep vertical layers (FL000 to FL200, FL200 to FL350 and FL350 to FL550, where FL indicates the flight level which is approximately the height in hundreds of feet but is defined in pressure), (ii) model predictions averaged over thin layers of depth 25FL, and (iii) a hybrid scheme which used the maximum of the thin layer predictions over the thick layers. A peak-to-mean factor was then applied to estimate the peak values. This factor was 20 for the deep layer method and 10 for the thin layer and hybrid methods. Webster et al. [10] used (among other approaches) the Mastin et al. [7] relationship (1) together with the assumption of a uniform emission profile with height and the assumption that 5% of the mass survives into the distal ash cloud in order to estimate the effective source; here the effective source estimated from the inversion is used instead.

As in [10], we have calculated some statistics on the agreement between model predictions and observations. The number of observation–prediction pairs which were in “agreement” was assessed in two ways. The first takes account only of the estimated observations errors, with agreement declared when values are within a factor or two. The second makes some allowance for the uncertainty range due to errors in the ash cloud position. The uncertainty range was assessed as the range of predictions across grid boxes up to two grid-boxes away in each horizontal direction (40 km resolution) and, for the 25FL layer scheme only, one grid-box away in the vertical (25FL resolution), and agreement was declared if the uncertainty ranges for the observation and prediction overlapped. We have also followed [10] in calculating the geometric mean bias and geometric standard deviation, defined as $exp\left(\mu \right)$ and $exp\left(\sigma \right)$, where $\mu$ and $\sigma$ are the mean and standard deviation of the error in ${log}_e(max(\mathrm{concentration},20\mathsf{\mu }{\mathrm{gm}}^{-3}))$, where the limit value of 20 $\mathsf{\mu }$g m${}^{-3}$ is designed to circumvent issues with noise at low concentrations.

Table 2. Statistical comparison of modelled ${}^{‡}$ and observed ^§ peak concentrations for the Eyjafjallajökull 2010 eruption.

Observations	Model	% in	% of over	% of under	Geom.	Geom.
Used in	Scheme	Agreement	Predictions	Predictions	Mean	s.d.
Inversion					Bias
Ash only	Deep layer	43 * 69 ${}^{†}$	31 * 15 ${}^{†}$	25 * 15 ${}^{†}$	1.04	3.80
	25FL layer	45 * 80 ${}^{†}$	23 * 2 ${}^{†}$	32 * 18 ${}^{†}$	0.80	4.02
	Hybrid	42 * 70 ${}^{†}$	36 * 18 ${}^{†}$	23 * 12 ${}^{†}$	1.18	3.88
Ash and	Deep layer	40 * 55 ${}^{†}$	8 * 4 ${}^{†}$	52 * 40 ${}^{†}$	0.47	3.42
clear sky	25FL layer	33 * 59 ${}^{†}$	6 * 1 ${}^{†}$	61 * 40 ${}^{†}$	0.42	3.58
	Hybrid	40 * 60 ${}^{†}$	12 * 7 ${}^{†}$	48 * 34 ${}^{†}$	0.56	3.65

${}^{‡}$ with a posteriori emissions using satellite retrievals up to 29 May 2010 23:00 UTC. ^§ the observations used are those considered in [10]. * assessed using a factor of 2 uncertainty in the observations but no model uncertainty. ${}^{†}$ assessed using uncertainty in both the observations and in the model predictions.

shows results using the emissions derived from ash-only observations and emissions derived from both ash and clear sky observations. In both cases, we have used all the observations available (up to 29 May 2010 23:00 UTC). Using the ash-only observations, the fraction of values in agreement is improved relative to the results found in Table 3 in [10] without an inversion model, substantially so when no allowance is made for ash cloud position errors. We see a better balance between the fraction of over- and underpredictions than in Table 3 in [10] with a reduced tendency to underpredict, with the exception of the hybrid method where we see little change. The geometric mean bias is closer to 1 than in Table 3 in [10] (except for the hybrid method where there is again little change) and the geometric standard deviation is smaller. The general improvement in bias and the reduced tendency to under predict could of course also have been obtained by increasing the peak-to-mean ratio assumed. However the large peak-to-mean ratio used in [10] is likely to be partly a result of the model ash being too widely spread as a result of the ash being released with a wide spread in the vertical at the source. This vertical spread at the source both dilutes the ash at the source and, in combination with changes in wind direction with height, increases the ash dispersion in the horizontal. Hence the improvement (i.e., reduction) in the tendency to under predict without the need to increase the peak-to-mean ratio is to be expected if the inversion improves the height profile of the emissions. As noted above, the a posteriori estimate of the total emitted (effective) mass for the whole eruption is significantly reduced from the a priori estimate. The reduction in mass emitted and the reduced tendency to underpredict suggests that the height–time distribution of the emissions has been improved by the inversion, with the emissions more focussed at the right heights and times. We hope this will reduce any overpredictions outside the actual ash cloud or in its low concentration edges, but this is hard to assess from the measurements available.

When we use both the ash and clear sky observations, the geometric standard deviation decreases further, but the results show quite a strong tendency to underpredict with a worse geometric mean bias than both the ash-only inversion results and the results without inversion modelling in Table 3 in [10]. The geometric error $exp\left(\sqrt{{\sigma }^2+{\mu }^2}\right)$ increases relative to the ash-only results, although it is still lower than the value without inversion modelling from Table 3 in [10]. While this bias could be corrected by increasing the assumed peak-to-mean factor, this is undesirable because a more correct source profile should lead to a lower optimal peak-to-mean ratio, as discussed above. Although the reduction in the geometric standard deviation is encouraging, it may be that errors in the ash transport are leading to the modelled ash cloud overlapping with more clear sky satellite observations than it should, leading to emissions being reduced too much (consistent with the bias reported in Table 1). This possibility is investigated in more detail below in connection with the Grímsvötn eruption.

Some limited experiments with higher-resolution source discretizations showed little difference in the validation statistics against these observations.

3.2. Grímsvötn

Grímsvötn is located at 64.42${}^{\circ }$ N, 17.33${}^{\circ }$ W and has a vent height of 1719 m asl. It erupted on the 21 May 2011 at 19:13 UTC with the eruption lasting for about three days until 25 May 2011 at 02:30 UTC. Satellite ash and clear sky retrievals were obtained for the period between the start of the eruption and 31 May 2011 00:00 UTC. As for Eyjafjallajökull, we use retrievals up to some time after the end of the eruption, allowing ash remaining in the atmosphere and detected by the retrievals to be used in the inversion. The total number of ash and clear sky 40 km cells in the satellite data which are relevant to the inversion (i.e., which are reached by model particles from one of the source elements) is 86,746 of which 3267 are ash cells and the rest are clear sky.

As for Eyjafjallajökull, plume rise heights were provided in real time to the London VAAC by the Icelandic Met Office. They are shown in Figure 7. An uncertainty of $\delta H=2000$ m is again assumed.

A plume rise of 20 km was reported near the start of the eruption. At this time wind shear led to most of the upper material being transported to the north and the lower material to the south [18,48]. The a priori emissions therefore lead to predictions of ash to the north and south of Iceland. The satellite retrievals however show very little ash north of Iceland [18,48]. It is thought that this is because the upper part of the plume contained mainly SO${}_2$ and the ash was mostly in the lower part. It is an interesting challenge for the inversion system to try to correct for these large errors in the a priori ash emission profile.

Figure 8 and Figure 9 show how the emission profile from the inversion evolves as more data become available. Five emission profiles are shown together with the a priori mean profile. The mass released is reduced substantially from the a priori emissions, and this reduction is considerably larger when the clear sky retrievals are included. The height of the ash release profile is also reduced, substantially so when the clear sky data are used, in line with the analysis of [18,48]. The time period over which the source estimates evolve as more data become available is similar to that seen for Eyjafjallajökull. We also see a period where the inversion has little impact, as for Eyjafjallajökull.

The total emitted (effective) mass for the whole eruption is $3.20\times {10}^{13}$ g for the a priori source, $3.54\times {10}^{12}$ g for the inversion source using ash-only retrievals, and $2.85\times {10}^{11}$ g using ash and clear sky retrievals. This shows that the inversion gives substantial reductions in emissions which are much larger than for the Eyjafjallajökull eruption. The estimate using ash and clear sky retrievals is comparable to the estimate of $4.9\times {10}^{11}$ g made from satellite data in [18].

Table 3. Statistical comparison between modelled column loads and satellite ash retrievals for the Grímsvötn 2011 eruption at the locations of the 3267 ash-only retrievals * made up to 31 May 2011 00:00 UTC.

	Emissions Used in the Model
	A Priori	A Posteriori Using Ash	A Posteriori Using Ash and
		Only Satellite Data ${}^{†}$	Clear Sky Satellite Data ${}^{†}$
mean error ${}^{‡}$ (g m${}^{-2}$)	15.8	−1.04	−1.63
mean absolute error (g m${}^{-2}$)	17.2	1.26	1.68
rms error (g m${}^{-2}$)	50.4	1.75	2.15
correlation	0.27	0.46	0.25
$\sqrt{{\chi }^2}$	5.26	0.22	0.26

* mean and rms of the ash-only column load retrievals are 1.99 and 2.42 g m${}^{-2}$. ${}^{†}$ satellite retrievals up to 31 May 2011 00:00 UTC are used in the inversion. ${}^{‡}$ model minus observations.

compares model- and satellite-derived column loads, using the same approach as used in Table 1 for Eyjafjallajökull. The general trends in the statistics are similar to those seen for Eyjafjallajökull, although the improvement over the a priori results is more dramatic (because the a priori overprediction is much bigger) and the underprediction when the clear sky data are used is somewhat larger. In the same way as for Eyjafjallajökull, we note that the mass fraction of satellite ash retrievals outside locations that model particles can reach is 6.1% of the total ash retrievals, while the mass fraction of model ash in locations of clear sky satellite data (when the model is run with inverted sources using ash and clear sky satellite data) is 7.7% of the total model ash in locations of ash and clear sky satellite data. The reduced ${\chi }^2$ statistic is much greater than 1 for the a priori emissions but, for the a posteriori emissions, it reduces to similar values to those found for Eyjafjallajökull.

Figure 10b–f shows the NAME ash cloud for the hour between 23 May 2011 23:00 UTC and 24 May 2011 00:00 UTC, calculated using various emissions. We have chosen this time to illustrate how using the clear sky data can be beneficial or can result in emissions being reduced too much when there are small errors in the modelled plume position. Results are shown for the a priori emissions and for the inversion derived emissions using both ash-only and ash and clear sky satellite retrievals with two different observation cut-off times. The satellite retrievals are also shown for comparison in Figure 10a.

The ash clouds in Figure 10c–f using the inversion-derived source profiles show a large reduction in the amount of ash seen to the north of Iceland relative to that in the a priori ash cloud in Figure 10b, in line with the conclusions in [18,48]. With ash-only data the northerly ash cloud is much reduced, although still too prominent. However, when the clear sky data are included, the northerly ash cloud is all but completely removed. The latter behaviour seems correct in relation to the satellite retrievals. Note that the retrievals north of Norway in Figure 10a are likely to be a false detection. These retrievals are a long way north for a geostationary satellite and they are not supported by detections at nearby times.

To the south of Iceland, the column loads are of the right order of magnitude (in comparison to the retrievals) when the ash-only data are used (Figure 10c,e). When the clear sky data are included (Figure 10d,f) the column load magnitude is again about right when using only data up to 24 May 2011 00:00 UTC. However, when later data are included in the inversion, the ash cloud, although still present, is reduced too much. It seems possible that this is an example of an ash cloud being reduced too much by being slightly in the wrong place and intersecting some clear sky retrievals. The ash cloud is narrow and slightly too far south and, although there are no nearby clear sky retrievals at this time, this is not the case a short while later. From about 24 May 2011 08:00 UTC onwards, the satellite data show the ash moving from an area to the north of Scotland across the North Sea with clear skies to the south (Figure 10g). The a priori ash cloud at these times (Figure 10h) remains slightly too far to the south and intersects the clear sky retrievals. It seems likely that this is the reason that the ash cloud is reduced too much by the inversion calculation.

To further examine the errors associated with the inversion, the total emission for an example day, 21 May 2011, is shown in Figure 11. The a priori and a posteriori estimates are presented together with their error bars, showing the changes as more satellite data become available over time. The error bars show one standard deviation, calculated by adding the relevant components of B or A and taking the square root. We note that this is only an approximation because of the non-negative emissions constraint, as discussed at the end of Section 2.4.

The results in Figure 11 need careful interpretation. For example, the use of a log scale to show a wide range of emission estimates means that the lower end of the error bar can be very sensitive to the standard deviation when this is of the same order as the mean. However, the results give broad support to our probabilistic assumptions in that, when more data are used, the emission estimates are consistent with the estimates and error bars obtained using less data, being neither a long way outside the error bars nor too close to the previous estimates. In principle, the parameters of the a priori model could be adjusted using this sort of information but we have not attempted that here.

4. Conclusions

We have presented a method for estimating volcanic ash emissions during a volcanic eruption. The method uses an a priori model of the emissions using estimates of the eruption height from local observations. This is combined with satellite retrievals of ash column loads and cloud free locations from the SEVIRI instrument and with dispersion information from the NAME model. This calculation method is embedded in a suitable rolling framework to update results as more data become available.

The a priori model is based on a stochastic representation of possible evolutions for the volcano emissions which yields first- and second-order moments of the emissions, including correlations between emissions at different heights and times. The ash detection and retrieval method of Francis et al. [32] is used for the satellite retrievals. This method is designed to be conservative in order to avoid false alarms, with the aim of achieving a robust performance. A Bayesian probabilistic approach is used to combine the evidence. Gaussian assumptions are made for simplicity, but, to avoid problems with negative emissions, the optimal estimate of the emissions is calculated as the location of the peak of the pdf under the constraint that all the emissions are non-negative. This optimisation problem is solved using the NNLS method of Lawson and Hanson [44].

In principle, the approach can provide an automated method of using satellite information to inform the dispersion predictions. However, because the testing so far is limited to two eruptions and because there is likely to be other evidence that is not used in the approach (e.g., other satellite products, ground or aircraft-based observations), we recommend using the results only in combination with a subjective assessment. The inversion results, together with other available information and expertise, should provide valuable input into the production of advice on volcanic ash to the aviation industry.

The approach has been demonstrated for the 2010 Eyjafjallajökull and 2011 Grímsvötn eruptions. The optimal emission estimates change as more satellite retrievals become available and provide more information to constrain the emissions. Generally the emission estimates for a given time converge within about 4 days of the time of the emission, although there are a few instances where this takes five or six days. Often however there is significant movement away from the a priori value towards the converged final result at much earlier times. For both eruptions the inversion results in some substantial changes in the emission estimates and the ash cloud predictions relative to the a priori values. As expected, the agreement with the satellite retrievals improves, as quantified by, e.g., the rms difference. The inversion improves the agreement in the ash concentration predictions for Eyjafjallajökull when compared to the (independent) measurement data used in [10]. For Grímsvötn we do not have any independent data readily available. However, the reduction in the height and magnitude of the emissions is consistent with the analysis of the eruption in [18,48].

From this set of data for just two eruptions, it is unclear to what extent the use of clear sky observations in the way they have been used here will be beneficial in general. While there are some instances showing clear benefits, there are also some occasions where too much ash is removed because of errors in predictions of the ash cloud position that lead to the ash cloud intersecting clear sky observations. It seems likely, however, that greater benefits would be obtained if an approach to allow for errors in ash cloud position (arising from errors in meteorology) was added to the inversion approach (currently there is no allowance for transport errors). This is a significant challenge but could perhaps be achieved by using ensembles of meteorological predictions and/or by adjusting the modelled concentration field (in addition to adjusting the source properties).