Characterizing the emission height of sulfur dioxide (SO2) from volcanic eruptions yields information about the strength of volcanic activity, and is crucial for the assessment of possible climate impacts and validation of satellite retrievals of SO2. Sensors such as the Ozone Monitoring Instrument (OMI) on the polar-orbiting Aura satellite provide accurate maps of the spatial distribution of volcanic SO2, but provide limited information on its vertical distribution. The goal of this work is to explore the possible use of a trajectory model in reconstructing both the temporal activity and injection altitude of volcanic SO2 from OMI column measurements observed far from the volcano. Using observations from the November 2006 eruption of Nyamuragira, back trajectories are run and statistical analyses are computed based on the distance of closest approach to the volcano. These statistical analyses provide information about the emission height time series of SO2 injection from that eruption. It is found that the eruption begins first injecting SO2 into the upper troposphere, between 13 km and 17 km, on November 28th 2006. This is then followed by a slow decay in injection altitude, down to 6 km, over subsequent days. The emission height profile is used to generate an optimal reconstruction based on forward trajectories and compared to OMI SO2 observations. The inferred altitude of the Nyamuragira SO2cloud is also compared to the altitude of sulfate aerosols detected in aerosol backscatter vertical profiles from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) instrument aboard the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO).
 The explosive power of volcanoes is capable of launching gases and aerosols upwards to altitudes greater than 30 km. This results in the nearly instantaneous injection of large and highly concentrated ash, gas, and aerosol clouds into the atmosphere, which has many important consequences for aviation and on longer time scales, for climate. Another area in which volcanic emissions are important is their potential use for tracking upper atmosphere dynamical processes, since volcanoes act as a point source of atmospheric tracers whose spatial distribution can give clues about upper atmospheric dynamics and can be used as tests of model transport. The time dependence and emission altitude of volcanic activity may also provide clues about the underlying geophysical processes at work in volcanoes. A better understanding of all of these problems is the motivation behind the research described in this paper.
 The aviation community has a particular interest in the detection and tracking of the large ash clouds produced from volcanic eruptions as they can cause serious damage to airplane engines [Dunn and Wade, 1994] and have caused engine failure, “flame out,” in mid-flight. There are also concerns about the effects of SO2 clouds on aircraft, but their impacts are currently not well understood. Since volcanic eruptions often emit collocated SO2 and ash clouds, SO2 can in some cases be used to monitor the dispersion of ash in the atmosphere [Carn et al., 2009]. The transport-derived method for height estimation we describe in this paper is computationally fast, especially on parallel platforms, making it feasible for near-realtime applications.
 The injection of large amounts of gas and aerosols in the atmosphere by volcanic eruptions also has a long-term impact on the global atmospheric radiation budget. The influence of atmospheric processes or constituents on the radiation budget is often defined in terms of their potential heating or cooling perturbations, frequently referred to as their radiative forcing potential. While it is well known that aerosols have a negative (or cooling) radiative forcing potential, there are large uncertainties [Intergovernmental Panel on Climate Change (IPCC), 2007]. Such uncertainties are partly due to an insufficient understanding of volcanic aerosols, primarily sulfate aerosols formed from volcanic SO2 and water vapor [IPCC, 2007]. While the 1991 eruption of Mount Pinatubo provided strong evidence of the volcanic aerosol influence on the radiation budget, few eruptive cases of this magnitude have been observed since the satellite observations in the early 1980s. The lack of observations of highly explosive volcanic eruptions has made it difficult to estimate the radiative forcing from them [IPCC, 2007].
 There are two different types of aerosols that play an important role in the radiation budget: ash and sulfate aerosols. While ash, tropospheric sulfate aerosols, and stratospheric sulfate aerosols all produce a net cooling effect, the atmospheric lifetime of ash and tropospheric sulfate aerosols is relatively short (1 week and 1–3 weeks, respectively) when compared to the stratospheric sulfate aerosol lifetime (1–3 years) [Robock, 2000]. The radiative forcing produced by ash and tropospheric sulfate aerosols is therefore rather short-lived, while the climate record shows that the impact from stratospheric sulfate aerosols can last for several years [Dutton and Bodhaine, 2001]. The stratospheric sulfate aerosols produced from the 1991 eruption of Mt Pinatubo lowered the globally averaged temperature by about ∼0.5°C [Robock, 2000].
 Knowledge of the altitude of the SO2 is important for all of these issues. It is clearly necessary to know the height of ash and gas clouds for effective aircraft flight planning. If the emissions are directly injected into the stratosphere, or enter the stratosphere by slower vertical transport, their longer stratospheric lifetime leads to more significant radiative forcing effects than if they remain in the troposphere [Robock, 2000]. Using volcanic emissions to investigate upper tropospheric dynamics or to evaluate the quality of model winds also requires some information about the height of the SO2 measurements.
 Current satellite measurements of SO2 have poor vertical resolution. While recent methods have demonstrated the ability to retrieve effective SO2 altitudes from OMI measurements [Yang et al. 2010], the retrieval resolves a single effective altitude for the entire SO2 column. Transport based estimates of SO2 altitude provide similar information but do not constrain the altitude in such a manner, which can be useful for validation purposes.
 Measurements from the Ozone Monitoring Instrument (OMI) that are used in this research are able to retrieve high horizontal resolution (13 km × 24 km at nadir, along the x cross-track) global maps of SO2 on a daily basis. These measurements are given in Dobson Units (DU, 1 DU = 2.69 × 1016 molecules/cm2). The standard OMI SO2 retrievals, however, give no information about the height distribution of the SO2 within the column. The main goal of the research presented here addresses this problem by attempting to infer the injection height of SO2 from a statistical approach based on the way in which its horizontal distribution subsequently evolves in time. The dependence of the wind speed and direction on altitude is the basis for this method.
 The horizontal transport path taken by a cloud of gas or aerosols is dependent on the speed and direction of the wind, which in turn is dependent on altitude. This transport can be modeled using meteorological winds and temperatures to compute trajectories for volcanic emissions. In cases where the injection height is unknown, trajectories can be run for a range of heights. In this analysis, trajectories are initialized at various heights, at locations where the SO2 cloud was measured, and driven backward in time. The relative number of trajectories that successfully track back to within a certain minimum distance from the volcanic source will depend on the assumed altitude of the trajectory and are used to derive the injection height of a cloud of volcanic SO2 and the time when this injection occurred. A detailed description of this method is outlined in section four.
 There have been several recent studies that attempt to infer the injection height of volcanic emissions using a trajectory model and meteorological winds [Prata et al., 2007; Eckhardt et al., 2008; Kristiansen et al., 2010; Krotkov et al., 2010]. Obtaining quantitative results is generally problematic. For example, small wind errors can be amplified due to the fundamental chaotic nature of atmospheric transport which causes trajectories to diverge exponentially on average [Pierce and Fairlie, 1993]. To some extent, these issues can be mitigated by using a robust statistical approach, such as the one described in this study. Some other studies have used inversion methods that can also give quantitative results and error estimates for the SO2 vertical profile [Eckhardt et al., 2008; Kristiansen et al., 2010].
Schoeberl et al.  demonstrated the ability to resolve the height of the 1991 Cerro Hudson SO2 cloud with a forward trajectory analysis. This type of forward analysis was furthered in the work of Allen et al.  with the modeling of an aerosol cloud observed by TOMS. The authors note that such type of modeling is limited by an “uncertainty principle” in the sense that model precision improves with time, while accuracy degrades with time because of accumulating trajectory errors. Other studies, such as Prata et al. , have also used forward trajectories to infer the heights of trace gases by visual inspection of the trajectories that most accurately mimic the observed transport, but this yields only qualitative information.
 Backward (or simply “back”) trajectory modeling is another type of analysis frequently used in trace gas transport modeling. Back trajectory modeling is often used as an inversion in order to determine the source of trace gas measurements. However, a recent study by Scheifinger and Kaisera  showed that these Trajectory Statistical Methods (TSMs), from a backward trajectory analysis yield results that greatly vary in accuracy.
 A recent inversion method approach [Eckhardt et al., 2008; Kristiansen et al., 2010] was applied to measurements of a cloud of volcanic SO2 to determine its initial concentration and injection height. In their dispersion model, an amount of SO2 was initialized at several heights above the volcano. The model advected the SO2 forward in time (taking into account the decay of SO2 by reaction with OH) and total column amounts of the SO2were output at the same time and spatial resolution as satellite observations. The inversion-method optimizes a fit with the observations that leads to a most probable emission height at the volcano as well as an estimate of the most probable amount of SO2. Krotkov et al.  used a forward and backward trajectory analysis to similarly estimate the height and emission time of an SO2 cloud. The emission time and height was estimated by looking at the back trajectories that arrived at the volcano.
 The work presented here has a similar goal but focuses on evaluating possible transport SO2 paths with a statistical method that is not strongly dependent on the exact concentration of the SO2. This approach is useful in comparing measurements and methods from different platforms or cases where the SO2 retrievals from different satellites do not agree. The method used in this work is based on the likelihood of transport paths that link the volcano to observations of SO2. It does not constrain or make any assumptions about the time of eruption of the volcano; the times of SO2 release are inferred from the back trajectories.
 This paper is organized as follows: section 2 is a brief overview of the OMI and CALIPSO measurements used and the physical basis of the measurements, and presents an overview of the wind reanalysis and trajectory model used to model the transport. In section 3, the details of the 2006 November–December eruption of the Nyamuragira volcano located in the Virunga Mountains of the Democratic Republic of the Congo are given. OMI maps of SO2 are shown and the meteorological conditions are briefly described. A detailed description of the backward trajectory analysis is given in section 4 and the results of this analysis are discussed in section 5. Section 6 is a summary and suggestions for future work.
2. Overview of Measurements and Instrumentation
 The goal of the work presented here is to use meteorological reanalysis winds and temperatures to model the transport of a measured volcanic SO2 plume in order to infer its injection height and time. The overall method uses data from several sources, including SO2measurements from the Ozone Monitoring Instrument (OMI) and winds and temperatures from the National Centers for Environmental Prediction (NCEP) Meteorological Reanalysis. The injection height results will be compared to aerosol backscatter measurements from the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP), a lidar aerosol profiler. This section gives an overview of these instruments, describing the physical basis of the measurements, their sampling frequency, and the limitations of each. Included here is a discussion on the remote sensing of column SO2 concentrations in the atmosphere and the ability to infer sulfate aerosol heights from CALIOP. The computational methods used to compute the trajectories and the statistical methods used to analyze them are discussed in subsection 2.3.
2.1. OMI Measurements of SO2
 The Ozone Monitoring Instrument (OMI) is a passive UV-Visible nadir mapping spectrometer aboard the Aura satellite. OMI measures the earth's radiance in the 270–500 nm wavelength range. Along its orbit, OMI measures in pushbroom mode by imaging a 2600 km × 13 km region of the earth onto the surface of a CCD (charged-couple device). The sampling across the 2600 km swath is not linear; the measurements taken in the nadir path have a higher spatial resolution than the regions at the edge of the swath. More information about OMI instrumental specifications is provided by P. F. Levelt and R. Noordhoek (OMI ATBD Volume 1: OMI Instrument, Level 0–1b processor, Calibration & Operations. Version 1. 01, 2002,http://www.knmi.nl/omi/research/instrument/inst_docs.php).
 OMI takes measurements for the sunlit segments of its orbit. The measurements produced from this segment of an orbit are referred to as a data swath. These swaths are 2600 km across and generally stretch from pole to pole. The sampling time difference between two adjacent swaths is roughly 100 min. The retrieval algorithms used to derive column SO2 amounts from OMI irradiance measurements are described in Krotkov et al.  and Yang et al. . In order to derive the SO2 amounts, the height of the SO2, or center of mass altitude (CMA), within the column must be assumed. OMI SO2 retrievals are produced under four different height assumptions: Planetary Boundary Layer (PBL) SO2 Column (CMA = 0.9 km), Lower Tropospheric (TRL) SO2 Column (CMA = 2.5 km), Middle Tropospheric (TRM) SO2 Column (CMA = 7.5 km), and Upper Tropospheric and Stratospheric (STL) SO2 Column (CMA = 17 km). The PBL SO2 column is derived from the Band Residual Difference Algorithm while the other three column retrievals are produced from the Linear Fit algorithm. The main difference in these data products is the assumed SO2 height used in the retrieval. These different SO2 column retrievals generally differ from each other by a scaling factor (except in regions of cloud coverage) [Krotkov et al., 2006; Yang et al., 2007]. While the error from an improper knowledge of height may vary, it is limited to about 20% [Yang et al., 2007]. This study primarily uses the Middle Tropospheric SO2 data product for trajectory initialization.
2.2. CALIOP/CALIPSO Aerosol Measurements
 CALIOP is a satellite-borne lidar aboard the Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) that uses a pulsed-laser to measure curtain-like profiles of the distribution and properties of aerosols in the atmosphere. Of the various CALIOP data products, this research uses the 532-nm Total Attenuated Backscatter (TAB) to detect the vertical distribution of aerosols in the atmosphere [Thomason et al., 2007]. Therefore, these measurements can give information into the vertical location and distribution of sulfate aerosols formed from the reaction of SO2 with OH. The sulfate aerosol conversion process occurs within the SO2 cloud, so CALIOP measurements can indicate the height of the SO2 cloud [Carn et al., 2007].
 Both the Aura and CALIPSO satellites are part of NASA's A-train satellite constellation, a series of satellites following a similar orbital pattern (http://atrain.nasa.gov/). This configuration allows CALIOP and OMI to sample overlapping regions at similar times, with equator crossing times of 1:30 pm and 1:38 pm, respectively. However, the ability to find OMI and CALIOP measurements coincident both spatially and temporally is limited. While CALIOP makes continuous measurements, the OMI SO2products are only available for the sunlight (or daytime) part of the orbit. CALIOP measurements from the sunlit portions of the orbit tend to be much noisier. This creates a bit of a problem since sulfate aerosol clouds themselves tend to be optically thin as they can have an e-folding time of about 35 days in the stratosphere [Bluth et al., 1992]. Thus, sulfate aerosols tend to produce a weak signal in the CALIOP measurements and can be very difficult to observe in the daytime parts of the orbit. We will generally be limited to using nighttime CALIOP observations collocated with observations of SO2 with a longer atmospheric age. For our purposes, a more realistic measurement time difference between OMI and CALIOP is roughly 12 h. The limited number of measurements of sulfate aerosols from CALIOP are generally insufficient for a robust statistical analysis. CALIPSO observations, as curtain profiles, also have a poor horizontal resolution. However, it does yield valuable information about the vertical profile of volcanic emissions directly, and provides a number of opportunities to validate the trajectory results.
2.3. Lagrangian Modeling of Atmospheric Transport
 The work presented here uses the Goddard Space Flight Center (GSFC) trajectory model developed at the NASA/GSFC [Schoeberl and Sparling, 1995] to compute trajectories initialized at the location of the OMI measurements and tracked backward in time using winds from the National Centers for Environmental Prediction (NCEP) and National Center for Atmospheric Research (NCAR) Reanalysis [Kalnay et al., 1996]. The NCEP global reanalysis produces fields with a 2.5° × 2.5° spatial resolution at the standard 17 vertical pressure levels, and a 6-h temporal resolution.
 Because air parcels are tracked in the middle/upper troposphere for less than one week, the motion is assumed to be adiabatic, and the trajectories are computed on surfaces of constant potential temperature, θ, by interpolating the winds to the given θ surface. Comparisons with kinematic trajectories computed using the omega field for vertical transport did not lead to any important differences and did not change the main conclusions of this study.
 When using trajectories to study transport, it is important to analyze them using statistical ensembles. In certain regions of the flow (i.e., near hyperbolic points), air parcels will separate exponentially fast. These are points where small changes or errors in the air parcel position will amplify rapidly with time and individual trajectories will be highly uncertain. The best way to deal with this is to construct ensembles or clusters of air parcels and to consider the statistical properties of the ensemble. While the accuracy of an individual trajectory can be poor in these regions, the statistical properties of groups of air parcels are much more robust. However, there are cases where even ensemble statistics can be uncertain. Errors can arise due to the errors in the winds when, for example, a region of strong shear appears in the wrong place in the analysis. In this case, the overall transport of a group of air parcels will be incorrect and the air parcels will not track back to the source. We hope to minimize these errors by modeling transport over a range of different advection times and re-sampling the emissions at different times.
3. The Eruption of Nyamuragira: Spatial Distribution of SO2 and Prevailing Meteorological Conditions
3.1. SO2 Observations From OMI
 On November 27th, 2006 the first signals of SO2 from the Nyamuragira eruption were measured by OMI, although large amounts of SO2 were not observed until the morning of Nov. 28th (Figure 1). The eruption lasted for several days, releasing large plumes of SO2 into the atmosphere. After injection, the plumes drifted northwestward and became entrained in the subtropical jet on December 1, and from December 3–6 the strong wind shear associated with the jet stretched and dispersed the cloud. By December 7th much of the SO2 was either well mixed into the atmosphere or had converted into sulfate aerosol particles, and no longer detectable by OMI.
 While the daily column SO2 maps clearly show that the volcanic activity begins around November 28, these maps provide limited information about the volcanic activity throughout the eruption. Understanding quantities like the production of SO2 throughout the eruptive period is difficult using SO2 maps alone since the OMI sampling pattern only provides a daily measurement of SO2, at the location of the volcano. One goal of this work is to explore the possibility of gaining information about the evolution of this eruption by means of a trajectory analysis that tracks emissions back to the source.
 A study of the December 1981 eruption of Nyamuragira by Krueger et al.  outlined a straightforward method that uses total observed SO2 measurements to compute the daily total amount of SO2 emitted from a volcano. This method decomposes the total observed SO2 into new SO2 versus old SO2:
Where SO2(n)obs is the total SO2 observed on the nth day, SO2(n)new is the new SO2 produced from the volcano since the previous day and SO2(n)loss is the amount of SO2 lost since the previous day.
 If the SO2 is assumed to decay at some daily loss rate, f, and some terms are rearranged, then this equation can be rewritten as:
Solving this equation with a range of loss rates produces a series of SO2 production curves. The correct loss rate is found by applying two constraints on the resulting curves: that SO2(n)new amounts shouldn't be negative and that SO2(n)new should be zero over a period when the volcano is known to have stopped producing SO2. Figure 2 shows this analysis applied to the four different OMI SO2 products previously discussed. The loss rate, f, was found by constraining the SO2 production, or SO2(n)new, to be near zero over the period Dec. 6th–Dec. 7th 2006. Although the details of the curves from the different height assumed products differ, they all suggest that the peak intensity of this eruption occurred rapidly between Nov. 28th–30th and was followed by a slow decay in the emission intensity. The eruption appears to end on the morning of Dec. 4th. It should be noted that adjusting the loss rates generally induces an offset along the y axis, this can be seen in Krueger et al. .
3.2. Meteorological Conditions
 Observations of SO2 shown in Figure 1 suggest that the strong westerly transport of the SO2 cloud was due to entrainment in the subtropical jet. The NCEP operational analysis wind speeds of the 500mb and 300mb surfaces are plotted in Figure 3, showing the location of the subtropical jet over this time period and the strong westerly upper level flow across Northern Africa.
4. Methods: Trajectory Modeling and Statistical Analysis
4.1. Modeling Volcanic SO2 Transport
 The ability to resolve significant differences in horizontal transport from differences in initialized height will depend on the vertical wind shear along the transport path. If the wind speed and direction vary sufficiently with altitude, then air parcel paths on different θ levels will be very different; these differences in the horizontal transport, when compared to the motion of the SO2 observed from satellite observations, can be used to infer its vertical location. Figure 4 shows an example of mean forward trajectory paths and an OMI composite SO2 map. Comparing these results, it is clear that only certain levels correctly describe the SO2 transport. This demonstrates qualitatively the idea that the altitude can be inferred from horizontal motion, but a more quantitative approach, based on tracking OMI observations back in time, will be used to infer both the injection height and the injection time of volcanic SO2.
4.2. Backward Trajectory Analysis
 Back trajectories are initialized at the location and sampling time of the OMI measurements and traced back in time. Because the OMI measurements are column integrated, we represent this column by placing trajectories at six vertical levels for each measurement; at θ = [330 K, 340 K, 350 K, 360 K, 370 K, 380 K] corresponding roughly to altitudes [6 km, 9 km, 13 km, 14 km, 15 km, 17 km], at the location of the volcano. Note that the altitude of a θ level at the location of the measurement and above the volcano will generally be different since constant θ surfaces are not constant in altitude. Backward trajectories are computed over a 7∼10 day period. The θ levels of those that get within a prescribed minimum distance to the volcano are considered to have most accurately followed the actual transport of the measured SO2. In this analysis, a total of roughly 53,000 trajectories met this condition. Only the subset of OMI measurements which indicate high levels of SO2concentration are used (i.e., in the upper ∼10% of the clouds measurements on a given day) under the assumption that higher values have undergone less mixing and dilution and therefore convey more information about their origin. The measurement's sampling time and the central point of the measurement's field-of-view are used to establish the trajectory initialization.
 To expect any back trajectory to arrive at the exact location of a point source is of course unrealistic, so we use the concept of the “distance of closest approach” to the volcano to decide whether an air parcel likely originated at the volcano. The θ levels from trajectories with the smallest distance of closest approach are candidates for the height of the SO2.
4.3. The Distance of Closest Approach
 When discussing trajectories it is often useful to describe the points that make up a trajectory in terms of their advection time. The trajectories are outputted with the same temporal resolution as the meteorological fields, Δt; meaning that the elapsed time between adjacent points along a trajectory is fixed (the same cannot be said for the elapsed distance). Thus, we define the distance between a trajectory point and the volcano as r(θ, t), where the potential temperature, θ, and advection time, t, are parameters described by the trajectory. The calculation is isentropic, thus θ is constant along the trajectory.
 For every trajectory, there is a point where an air parcel comes closest to the volcano. We denote this distance of closest approach by r*(θ, t*), the time of closest approach as t*, and the potential temperature of closest approach θ.A given trajectory is considered to have successfully arrived at the volcano if r* is less than some minimal arrival distance R. Defining a value for R can be tricky; if too large, then all trajectories meet the criteria, if it is too small then no trajectories will. To understand what a reasonable assignment of R may be, we consider a hypothetical scenario where a trajectory is initialized at an OMI measurement and crosses over the exact location of the volcano, and refer to this as an “ideal trajectory.” Ideally, for such a trajectory r* = 0; but the temporal and spatial resolution of the trajectory will not allow this to be realized. In the limit where the temporal resolution Δt→0, the ideal distance of closest approach r*→0. However, our trajectory step-size is finite. As noted insection 2.3, the NCEP reanalysis winds have a 6-h temporal resolution, so trajectory locations are saved only at 6-h intervals and we must consider that even an ideal trajectory has a 6-h temporal resolution, and even in this ideal case there will be a range of possible values for r*. In order to determine what this range may be, we need to look at the stepping distance, or spatial resolution, of an air parcel as it approaches the volcano.
 The spatial resolution of a trajectory is a function of both its temporal resolution (generally fixed) and the local wind speed; the variation of local wind speed along the transport path causes the spatial resolution to vary. An air parcel advected by a constant velocity field would have a constant spatial resolution. Air parcels traveling in a region of slow moving winds will have a higher spatial resolution than those traveling in faster winds. For fast moving winds, r* may be rather large; while slower winds allow much smaller values of r*. Thus, the minimum distance between the ideal trajectory and the volcano is also constrained by the winds in the region of the volcano. Figure 5 shows the stepping distance for a trajectory traveling (at any altitude) in the region of the volcano. Figure 5 estimates P(r*), the PDF (Probability Distribution Function) of r*, that results solely from finite resolution.
 P(r*) shows that the variable wind speed at the volcano leads to many possible minimum arrival distances. Using the information from Figure 5 we can make an estimate of what our minimum arrival distance, R, should be. We could select R based on the median of the PDF. However, this would mean that there is a 50% chance that even an ideal trajectory will not arrive within R. This would also discriminate against those θ-levels with faster wind speeds. Realistically, we would like to have a near 100% probability of r* < R for a perfect trajectory. To accommodate this condition, we choose R = 700 km, which means that there is roughly a ∼90% chance of r* < R.
 This choice of R can also be motivated by considering Figure 6, which shows P(r*) for all trajectories used in this study. This PDF shows that a distinct minimum occurs at 780 km; it also shows a steep drop-off as r* approaches 780 km. This suggests that those trajectories that are actually at the correct altitude manage to get well within the cutoff and those not at the correct altitude do not. If no information were conveyed by the r* statistic, then P(r*) would be consistent with a uniform distribution in that the likelihood of an air parcel arriving in a neighborhood around the volcano is no greater than arriving in a neighborhood around any other point. Using this experiment as motivation, we conservatively define that an air parcel has successfully arrived back at the volcano if r*(θ, t*) < 700 km.
 There will similarly be limitations imposed by the spatial resolution of the wind fields, but there is no simple way to estimate how the spatial resolution of the winds impacts the results. A well-resolved large-scale wind feature can nevertheless generate spatial structure in a tracer field or patterns in trajectories that are well below the spatial resolution of the winds due to the time dependence of the wind. Filamentary structures in trajectory simulations are common [see, e.g.,Appenzeller and Holton, 1997]. The extent to which this occurs depends on the flow dynamics, for example the scale collapse of tracer structure across a strong wind shear. A full examination of the impact of spatial resolution on the method presented in this paper will appear in a future report.
 We define t* as the time of closest approach, i.e., the time when r = r* for a given trajectory. For the backward trajectories, the time t* is the arrival time for an air parcel that travels from the location of an SO2 measurement back to the volcano. Alternatively, we can regard t* as the emission time for an air parcel to travel from the volcano to the location of the measurement, since the transport time due to the mean flow between the two points is the same going backward or forward. In the following it may be more convenient to refer to t* as the trajectory arrival time or SO2 emission time as appropriate. It is assumed that these two definitions are equivalent; issues related to irreversible mixing by the unresolved scales will not be considered here.
4.4. PDFs of Air Parcel Arrival Time and θ Level for a Single Day of Measurements
 The condition r* < 700 km gives us a rule to test whether or not a trajectory has correctly described the SO2 transport. Applying this rule to an ensemble of trajectories creates a subset whose statistical properties can be examined to understand the most probable point and time of origination of later SO2 measurements, and subsequently, statistics to describe the volcanic activity.
 From the subset of trajectories with r* < 700 km, we compute the PDF of the air parcel arrival times at the volcano, denoted by P(t*). Figure 7 shows an example of P(t*) computed from SO2 observed on December 2nd, 2006. This distribution describes the likelihood that the SO2 observed on December 2nd was emitted from the volcano at time t*, irrespective of arrival altitude. We can extract more information from this distribution if we consider the conditional distribution of θ for air parcels with emission times in a given bin, i.e., in the range t* to t* + Δt*. This conditional distribution, P(θ|t*), is the distribution of arrival heights at the volcano for air parcels emitted from t* to t* + Δt*. This joint probability of arrival time and height at the volcano, P(θ, t*) is the fraction of transport paths that arrive at the volcano at time t* on level θ for observations of SO2 on a single day. The most probable value of θ is θ*, thus this estimates the emission time and height of the observed SO2.
 The conditional probability of any two events can be found from Bayes' theorem. Given the events A and B, the conditional probability of P(A|B) is defined as:
where P(A, B) is the joint probability of occurrence of both A and B, P(B) is The probability of occurrence B, and P(A|B) is the probability of A, given that B has occurred (J. Joyce, Bayes' Theorem, Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/bayes-theorem).
 Using the quantities P(t*) and P(θ|t*), the joint probability P(θ, t*), the probability of an air parcel arriving at the volcano at time t and level θ, is defined as:
4.5. Daily SO2Observations and Air Parcel Re-sampling
 It could be argued that the fact that a back trajectory gets close to the volcano at some point is not by itself evidence of a correct path, since an air parcel might take a path that is clearly not along the path emissions are observed to take. In other words, if the ending location of a trajectory happens to be near the volcano, but the path taken does not follow the emission path, then that trajectory is not valid. The way to deal with this is to track OMI observations made on a series of successive days. This has the effect of re-sampling the same air parcel along its path so that those trajectories that actually follow the SO2 plume back to the volcano reinforce each other in the total ensemble.
 Observations from a single day are unique in that they each sample a different part of the SO2plume (assuming negligible satellite aliasing). Thus, trajectories from these observations on a single day provide non-overlapping information. If we think about this in regard to our t* distribution, it is equivalent to stating that no two trajectory members of this single day ensemble originated from the same OMI observation and at the same altitude. We would like to take advantage of several days of OMI observations to build larger and more robust distributions, P(t*) and P(θ, t*). But in doing this, we lose the uniqueness of the trajectory information in our distributions since successive days observations will resample certain parts of the cloud. If we consider the motion of the SO2 cloud as it moves away from the volcano, it is clear that the measurements on a given day will resample some of the same portions of the SO2 cloud that were sampled on the previous day (although we expect the amount of SO2 to decrease along its path due to losses from mixing and conversion processes). In addition, measurements on the later day will also sample any SO2 that was released from the volcano since the previous day. This means that a distribution comprised of two successive days of observations will include those parts of the SO2 cloud that persisted long enough to be sampled on both days. This reinforces the statistics because if an air parcel is sampled twice, for example, then the two air parcels should have the same values of r* and t*.
 There are two questions to consider when combining measurements from observations on different days: (1) For a given day of observations, which OMI measurements are used to initialize trajectories? And then, (2) how are distributions from several days combined into one overall distribution?
 We answer the first question by initializing trajectories for measurements that show a high concentration of SO2 (i.e., in the upper ∼10%) within a daily set of observations. This means that the concentration threshold used to select these high concentration measurements varies daily. This is motivated by the desire to keep the total number of trajectories initialized on each day within the same order of magnitude. This is important because when we combine the distributions from individual days, we don't want to under or over represent the statistics derived from any particular day's observations. Such problems can arise when defining a constant high concentration threshold to apply across several days.
 Using the statistics outlined in the previous section, the distributions P(t*) and P(θ, t*) for observations initialized on each day are computed. We then define the full distributions (t*) and (θ, t*) as the sum of the distributions for each day, weighted by the total number of members in each distribution:
Where NT is the total number of trajectories with r* < R (that reach the volcano), starting on day T, P(t*)T is the arrival time distribution for trajectories initialized on day T, and P (θ; t*)T is the distribution of arrival height at time t* for trajectories initialized on day T. In equations (5) and (6), the summation runs from November 30th 2006, represented as day of year (DOY) T = 334, until December 7th 2006, T = 341.
 With the formalism of the statistical analysis established, we apply these results to the set of OMI measurements that observed the Nyamuragira SO2 cloud over the period of November 30th 2006 to December 7th 2006.
5.1. The PDFs of Volcanic Activity
Figure 8 shows (t*), the PDF of air parcel arrival times at the volcano; constructed using a 12-h bin size. This distribution shows a distinct peak, corresponding to the most probable (i.e., most frequently observed) air parcel arrival time, on Nov. 28th. Note that the peak is asymmetric; few air parcels are emitted from the volcano before this peak time, while the proceeding days see a steady decline in air parcel arrivals. As discussed earlier, the time t* can be regarded as the emission time of an air parcel that arrived at a measurement at the SO2 observation time. For an ensemble of trajectories initialized at many measurement locations, the PDF (t*) is the probability of SO2 emission from the volcano between time t* and t* + 12 h. This makes it clear that (t*) can be directly compared with the derived SO2 production curves from section 3.1 (Figure 2).
 The SO2 production curves and SO2 injection PDF show very similar characteristics. Both have a peak that occurs at the beginning of the eruption period, followed by a decaying intensity. (t*) suggests that the volcanic eruption of SO2 occurred over the period Nov. 26th to Dec 3rd, with a peak intensity occurring on Nov. 28th. The SO2 production curve in Figure 10 shows that the eruption occurred over the period of Nov. 28th to Dec. 5th, with the peak production occurring on Nov. 30th. These two curves show a similar trend of activity, but are offset by about 1 day. This difference is perhaps not surprising, given that the two methods used are very different: one is based only on transport characteristics, without regard to SO2 amounts, while the other is based on the SO2 and completely neglects the transport.
 We next consider the joint distribution (θ, t*) which is the joint distribution of emission height and time as defined in section 4.5. (θ, t*) is the fraction of back trajectories that arrive at the volcano between t* and t* + 12 h, at level θ. As before, (θ, t*) can be interpreted as the fraction of trajectories that start at the volcano at time t* at level θ and end at an SO2 measurement, in other words (θ, t*) is the joint distribution of emission height and time, giving a profile of the volcanic intensity throughout the eruption period. The distribution P(θ, t*) is a major result of this research.
 The joint PDF (θ,t*) in Figure 9 shows a decay in the emission height throughout the period of volcanic activity. Each symbol on the plot is shaded to correspond to the fraction of air parcels that were injected into the atmosphere at level θat time t*. For example, the plot shows that most air parcels that arrive on the second-half of Nov. 27th are between the 350 K and 370 K (13 km and 15 km) surfaces. This emission altitude decreases over the next few hours, to the 340 K layer (∼09 km) on first half of day Nov. 29th. By Dec 1st, the primary arrival height is the 330 K layer (∼06 km). It is important to note that the emission height of the SO2 from this volcano is in the altitude range of airline traffic, which is roughly 9 km.
(θ, t*) also gives information about the dynamics of the volcano. At the beginning of the eruption, SO2 was injected high, on the 370 K θ surface. Over the next two days the injection altitude gradually decreased to 340 K, and by the next day, the injection altitude had decreased to 330 K. The decrease of the injection altitude over the eruption period suggests that the volcano's strength is greatest at the beginning of the eruption.
5.2. Building the PDFs and Solution Convergence
 The PDFs in section 5.1were constructed across time, as a combination of the individual PDFs produced from the daily observations made over the period Nov. 30th - Dec 7th (although the eruption actually began on Nov. 27th, measurements of SO2 were not significantly far enough from the volcano [>700 km] until Nov. 30th). Here we explore how the joint distribution evolves across the temporal summation.
 First, we examine how the PDF (t*) evolves as the sum runs from T = 334 to 341, November 30th to December 7th, as shown in Figure 10. Over the course of the first 5 days, from Nov. 30th to Dec. 4th, the structure of (t*) changes with the addition of another day's PDF. This PDF is expected to evolve since the volcano is actively erupting SO2 over this time period; new information is being added from newly produced/observed SO2. The eruption ends after Dec. 4th, so the addition of observations after Dec 4th does not add new information to (t*) in the sense that these trajectories are initialized from SO2 emitted at times that have already been accounted for. It can be seen that the addition of the PDFs from the Dec. 5th to 7th measurements has little effect since the structure of (t*) generally converges as the sum is taken to Dec. 7th. This is despite that fact that N339→N341 (i.e., the sum of trajectories from Dec. 5th to Dec. 7th) accounts for roughly ∼35% of the total number of trajectories.
 The evolution of the joint PDF (θ, t*) attempts to resolve two quantities (emission height and time) and issues of convergence are more subtle because the older trajectories have larger errors that introduce noise into the joint distribution. The first two days of this PDF show results that are fairly temporally resolved, while the height and overall structure of the PDF is still unclear. With the addition of measurements up until Dec. 5th, the PDF begins to show structure. As T then approaches Dec. 7th, the structure of the PDF clearly shows a height profile that decreases with time. The addition of data from these later days also causes some decay in the overall height/time resolution. Observations of SO2 from these later days come from SO2 air parcels with a larger age, so we should expect the PDFs from these days to be much less resolved in arrival time.
5.3. Reconstruction of the SO2 Cloud
 The height-time information from (θ, t*) was used to produce a forward trajectory reconstruction of the cloud in order to check the consistency of the results and to see if the cloud could be reproduced. Using (θ, t*) for a forward trajectory cloud modeling is straightforward. Since the PDF (θ, t*) sums to 1, multiplication by a total number of forward trajectory air parcels, N, describes how these N air parcels should be distributed in height and time in order to reproduce the SO2 emissions. (θ, t) is the fraction of total trajectories that track back to the volcano at time t and level θ, and so is the best estimate of the time and altitude of the SO2 emission. Launching forward trajectories from the volcano initialized according to (θ, t) can now be used to reconstruct the spatial distribution of the SO2 at a later time and comparing with the observed distribution as a further check on the method.
Figure 11 shows the results of the forward reconstruction for the morning of December 5th 2006. Much of the structure observed in the OMI SO2 cloud is also observed in the trajectory reconstruction. In particular, the stretching feature observed over India was successfully reproduced by the reconstruction. However, it appears that the tail end of the cloud is less resolved.
5.4. Comparison With CALIPSO
 Measurements of CALIPSO Total Attenuated Backscatter (532 nm) displayed in Figure 12 show what appears to be sulfate aerosols located at roughly 14∼16 km. The trajectory reconstruction predicts that the cloud should be at 350 K (∼13 km). There is a large time difference between the CALIPSO measurement and the Trajectory Reconstruction / OMI SO2 measurement, about 12 h. However, OMI measurements from December 6th confirm that the CALIPSO measurements are sampling the correct section of the cloud and that it occurs at an altitude consistent with what was inferred from the trajectories. The difference between the height seen by CALIPSO and the height from the trajectory analysis may be due to errors in the wind reanalysis, or radiative heating by sulfate aerosol absorption of solar radiation could cause the sulfate cloud to rise.
6. Summary and Discussion
 The injection height and time of volcanic SO2 was found by analyzing horizontal transport characteristics of OMI column measurements of SO2. A set of vertically stacked trajectories were initialized at OMI SO2 measurement locations and driven backward in time using an isentropic trajectory model. Trajectories initialized on those θ surfaces that successfully arrived at the volcano were considered to have correctly described the height of the measured SO2 cloud. The analysis performed used several consecutive days of observations, and the PDF of SO2emission time, t*, and the joint PDF of the emission time and height were estimated. The PDF of emission times was compared to an observation-derived curve of volcanic SO2 production. The structure of the curves was very similar, but the distribution of emission times (t*) had a ∼1 day offset. The joint PDF, (θ, t*), suggests that the height of SO2 injection lowers as the eruption continues. The last few days of this distribution appear rather noisy, which is most likely due to the lack of air parcels initialized for this part of the cloud. The joint PDF was then used to drive a forward trajectory model in an attempt to reconstruct the SO2 observations seen by OMI. This reconstruction reproduced distinct stretching characteristics seen in the OMI measurements. The results of this forward trajectory model were compared with CALIPSO observations of sulfate aerosols and the height of the SO2 cloud agreed within ±1 km of the observed sulfate layer.
 The ability and nature of a backward trajectory analysis to resolve the height of the volcanic eruption was also analyzed. By combining the trajectory analysis results made from observations of the same SO2 cloud, across varying ages, the most probable arrival height and time can be resolved. To a certain extent, this allows us to overcome the “uncertainty principle” noted by Allen et al. , where trajectories with a greater age accumulate uncertainties that impose limitations on the ability to resolve height. Nevertheless, our method is still limited by some uncertainties, such as the definition of a cutoff value for the distance of closest approach. Future studies will explore sensitivities of this method to the spatial and temporal resolutions of the winds.
 The methods presented here may be useful in investigations of volcanic activity and in assessments of the possible mechanisms by which sulfate aerosols enter the stratosphere. This work also has promising applications in aviation as a possible means of providing information crucial to forecasting the location of volcanic emissions for use in flight planning.
 The authors wish to thank Mark Schoeberl for his work on the trajectory model used in this study. This study was supported by NASA under grant NNX07AD87G and by the U.S. National Science Foundation under grant EAR 0910795.