Constraining the relationships between aerosol height, aerosol optical depth and total column trace gas measurements using remote sensing and models

Proper quantification of the aerosol vertical height is essential to constrain the atmospheric distribution and lifetime of aerosols, as well as their impact on the environment. We use globally distributed, daily averaged measurements of aerosol stereo heights of fire aerosols from MISR to understand the aerosol distribution. We also connect these results with a simple 10 plume rise model and a new multi-linear regression model approach based on daily measurements of NO2 from OMI and CO from MOPITT to understand and model the global aerosol vertical height profile over biomass burning regions. First, plumes associated with the local dry-burning season at mid to high latitudes frequently have a significant fraction lofted into the free troposphere, and in some cases even the stratosphere. Second, plumes mainly associated with less polluted regions in developing countries and heavily forested areas tend to stay closer to the ground, although they are not always uniformly 15 distributed throughout the boundary layer. Third, plumes associated with more serious loadings of pollution (such as in Africa, Southeast Asia and Northeast China) tend to have a significant amount of smoke transported uniformly through the planetary boundary layer and up to around 3 km. Fourth, the regression model approach yields a better ability to reproduce the measured heights as compared to the plume rise model approach. This improvement is based on a removal of the negative bias observed from the plume model approach, as well as a better ability to work under more heavily polluted conditions. However, over 20 many regions, both approaches fail, requiring deeper work to understand the physical, chemical, and dynamical reasons underlying the failure over these regions.


Geography
Around the world, biomass burning and deforestation have undergone tremendous changes in the past few decades, with current extremes making the news in many places throughout the world. To better interpret the land use conditions in the 125 biomass burning areas, we apply global land-cover type data of 18 different vegetation types, as measured in 2015 in Fig. 1.
We specifically focus on those areas where the land type has undergone known significant changes from forest to agriculture, or from forest or agriculture to urban, as demonstrated in the black boxes in Fig. 1.
Considering MISR daily plume heights (where the 1.1km pixels are first averaged to 10km x 10km grids) throughout the globe, we have determined that the respective average and standard deviation of the plume height over the three and a half 130 years of MISR daily measurements (from January 1 2008 through June 30 2011) are 1.37km and 0.72km. However, over our regions of interest, we find that we are able to capture the large bulk of the standard deviation globally, as demonstrated in Table S1.
The geographical data yields us a few conclusions about those regions which have the largest contribution to the biomass burning height variation. First, they are distributed in the middle and low latitudes (between the Tropic of Cancer and the 135 Tropic of Capricorn) and/or high latitudes (near the Arctic Circle). Second, they tend to occur in regions of more rapid economic growth, and/or in regions which are experiencing the most rapid change in land surface temperature.

MOPITT Carbon Monoxide (CO) Measurements
Carbon monoxide (CO) is a colorless and odorless gas that plays a major role in moderating the chemistry of the Earth's atmosphere as well as having a deleterious effect on human health. One of the world's major sources of CO is emissions from 140 biomass burning. For these reasons, we obtain measurements of CO from the MOPITT satellite (an instrument mounted on NASA's Terra satellite), which has collected data since March 2000. MOPITT's resolution is 22 km at nadir and observes the Earth in swaths that are 640 km wide.
In specific we use all version 7, Level 2 daily data from January 1, 2008 through June 30, 2011, combining data from both infrared channels (Deeter et al., 2013;Worden et al., 2010). This data has been demonstrated to provides an estimate of the 145 global distribution of CO in the troposphere (Worden et al., 2010). In reality, due to orbital conditions, aerosols, and clouds, there is not entire coverage over all of our areas of interest each day. Therefore, we first average all MOPITT data to 1 o x1 o , and secondly, we only use those values which subsequently have measurements from both MOPITT and MISR at the same time for our inter-comparisons.

OMI Nitrogen dioxide (NO2) Measurements 150
Another chemical species co-emitted by biomass burning with aerosols and CO is NO2 (Seinfeld and Pandis, 2006). For this reason, we also use the daily average total column loading of NO2 as measured by the Ozone Monitoring Instrument https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License. (OMI). In specific we use version 3 Level 2 measurements taken from the Aura satellite (Boersma et al., 2007;Lamsal et al., 2011;Levelt et al., 2006), which detects the radiance spectra from 60 across-track pixels with ground pixel sizes ranging from 13kmx24km at nadir to about 13kmx150km at the outermost part of the swath. 155 One advantage of the OMI NO2 column measurements is that they can often be observed under relatively cloudy or smoky conditions (Lin et al., 2014). Another advantage is that the atmospheric lifetime of NO2 is only a few hours, and therefore the temporal-spatial distribution of the NO2 column measurements is highly correlated with wildfire sources (Lin et al. 2019;Lan et al. 2019). NO2 has another interesting property in that its production/emissions is a strong function of the temperature at which the fires are burning, since NO2 is formed based on the air temperature (Seinfeld and Pandis, 2006). 160

Plume Rise Model
Although emissions from biomass burning are similar to those from urban combustion sources, with the major difference being the much higher burning temperature. This ensures that a significant amount of the emissions from biomass burning will be transported upwards due to the positive buoyancy generated by the fire. Due to the confluence of both local and non-local dynamical forcing in-situ, the ultimate height reached by these emissions is a complex function of the local fire energy and 165 both the local and large-scale meteorology at the time of combustion. While the aerosol particles are immediately transported horizontally by the large-scale winds, their vertical rise will only stop once their local buoyancy has reached equilibrium, and any dynamical motion has degraded back to the background conditions (Freitas et al., 2007;Val Martin et al., 2018).
To approximate this rise, we use a simple plume rise model (Briggs, 1965) to generate the final injection height of the biomass burning emissions based on the buoyancy and horizontal velocity of the plume and various atmospheric conditions. 170 Although this model is based on an empirical formula mathematically, it is essentially a thermodynamic approximation (Cohen et al., 2018) which costs much less computationally as well as being quite efficient when the biomass burning source covers a large area.
In theory, if such an approach was successful, and it was given appropriate environmental data, it should be able to reproduce the heights derived from the MISR multi-angle measurements. For this reason, we use a 1-D plume rise model to 175 independently predict the position and height of each measured MISR plume at each 10km x10km grid which is found to have measurements. To initialize the model, we require meteorological data as well as MODIS hot-spot information.

NCEP Reanalysis Data
NCEP and NCAR produce an analysis/prediction system to produce a meteorological field analysis of the 6-hourly state of the atmosphere from 1948 to the present. The measurements incorporated into this approach include ground based, in-situ, 180 and remotely sensed sources, while the modeling aspect is based on state-of-the-art meteorological models. In specific, we obtain daily data for each day which we have MISR data, from reanalysis version 1 (Kalnay et al., 1996) .Specifically we use https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License. the data required for us to run the plume rise model: the vertical temperature and pressure distributions, the surface air temperature, and the initial vertical velocity of the smoke emissions (dP/dt). We then compute the vertical temperature gradient (dT/dz) and the vertical velocity (dz/dt). 185

Regression Model
Linear regression is a simple method by which one can relate the impact that a set of orthogonal inputs have in terms or reproducing measured environmental values. It does not imply causation, merely which the behavior acts in a similar manner.
However, when looking to describe whether or not a new variable has a significant amount of correlation with a given phenomenon, it can be found to be very useful. 190 In this case, we are interested to see if the loadings of NO2 and CO are related to the heights of the fires. There is a strong physical case to be made here, since both are directly emitted by the fires themselves. Furthermore, the underlying causes of these substances are different: NO2 is a function of the fire temperature, while CO is a function of the Oxygen availability.
Furthermore, these are proxies for radiatively active substances such as soot and ozone.
For our work, we have decided to apply a simple linear regression model of the wind speed, FRP, CO, NO2, and the ratio of 195 NO2/CO. This is because the traditional plume rise models always include wind speed and FRP in their representations, so we wanted to specifically include as many different representations of the co-emitted gasses as well, as given in Equations 1-7.
We calculate all of the correlation coefficients (R 2 >0.2) between the different models and the MISR measurements, 205 ensuring that (P<0.05). Finally, we analyze both the magnitude of the regression coefficient as well as the magnitude of the various best-fit terms. These models are then used to reproduce the aerosol heights and are ultimately compared with both the plume model and the actual measurements. https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License.

Discussion and Results
We approach this problem with additional measurements compared to what are normally made so that we can have a 210 deeper insight into how these somewhat related species have on height to which aerosols from biomass burning rise in the atmosphere. Due to the fact that there are additional processes in-situ which can lead to heating, cooling, and other changes to the dynamics, it is essential that we establish any first-effect relationships, and then work more deeply as a community to address them in turn.
First, we to enforce consistency, we impose a condition that for all days analyzed, we must have data present from all of 215 the data sources: MISR, MODIS, MOPITT and OMI. On this basis, we explore the relationships between the two basic data sets (MISR and MODIS) and the source regions, as well as injecting additional information from MOPITT and OMI datasets without bias. Second, since these datasets make measurements with different assumptions, we also will reduce our bias in our inputs measurements as a function of clouds, different burning conditions, radiation feedbacks, and other actual atmospheric effects. We hope that this will help us to more clearly clarify the actual atmospheric phenomena responsible for the vertical 220 transport, which a more conventional plume rise model may not be able to account for.

Characteristics of MISR, OMI and MOPITT species
We use a PDF analysis to look at the distribution of the daily fire-constrained aggregated Measurements from MISR from each region over the entire dataset from 2008 to 2011 in Fig. 2. The statistical mean and standard deviation over each region are given in Table S1. We determine that the height of measurements ranges from 0 to 29 km, which is not only higher than 225 previous studies (Cohen et al., 2018;Val Martin et al., 2018), but also includes some extreme events which have made their way into the stratosphere. Due to the fact that first, the majority of the plumes are injected into the boundary layer or the lower free troposphere, second that we are not looking into the underlying physics of stratospheric injection (Pengfei Yu et al., 2019), and third that plumes tend to accumulate within layers of relative atmospheric stability, we therefore want to set an upper bound cutoff on the measured values that we will not consider after this point. Given the fact that the total percent of 230 measurements over 5000m is less than 7.9%, we therefore only look at data with an upper limit of 5 km for the remainder of this work.
We first observe that there are very different distributions of the measured heights over the different regions Fig. 2. The corresponding mean, standard deviation, and skewness of the heights over each respective region is given in Table S1. The average percentage of the data which has a measured height above 2 km (selected because it is always in the free troposphere) 235 is 15.0%, with the lowest in Central Canada of 41.7% and the highest in Midwest Africa of 0.8%. In terms of the amount of data measured with a height more than 4km, the average over the globe is 1.5%, while the range is as high as 6.6% in Central Canada and as low as 0.1% in Midwest Africa and Northern Australia. On the other end of the comparison, we also have some regions which are very polluted near the surface, while others show the vast majority of their heights are elevated off the https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License.
ground. Overall, we have a percentage of total plumes with a height below 200 m (the minimum rough level of the boundary 240 layer through the day) with a global mean of 2.3%, and a range from a minimum of 0.04% in Central Africa to 6.7% at Western Siberia. Given these results, we need more deeply understand the driving factors across all of these different regions, as well as the importance of biomass burning in terms of transporting aerosols through the boundary layer.
Second, we perform a comparison across the different daily time series of measured aerosol heights, CO column, and NO2 column as aggregated from January 1, 2008 to June 30, 2011 over all of the biomass burning regions Fig. S1. We consider 245 the burning season to be when we observe aerosol plumes and a peak in at least one of the CO and/or NO2 column measurements. This allows us to clearly demonstrate that the observed smoke peak is in fact due to burning of a significant amount of material. In these cases, the peak occurs from November to March in Central Africa, Midwest Africa; June to September in Central Canada, Eastern Europe, South America; April to July in Central Siberia; May to December in Southern Africa, Northern Australia; January to April in Northern Southeast Asia; March to September in Siberia and North China;250 April to September in West Siberia. In addition, the length of the peak burning time is also an important consideration which varies greatly across the different regions. The length of the total number of burning days from the three and a half years of data is an average is 108 days, with a minimum of 14 days in Eastern Siberia and a maximum of 388 days in Southern Africa.
Next, we look at the impact of FRP measurements and buoyancy in terms of the plume height distribution. In general, the higher the FRP, the higher the plumes should rise. However, these measurements seem to include a larger number of total 255 measurements into the lower free troposphere than previous plume rise model studies have been able to account for. From our measurements, we notice that the FRP (as computed on average over 1.1kmx1.1km grids where a fire exists) has a global mean of 37.7W/m 2 and a reginal minimum and maximum of 31.1W/m 2 (Siberia and North China) and 82.6W/m 2 (Central Canada) during the respective biomass burning seasons. Based on previous work, we would expect a general plume rise model to not be able to match the observed heights well under these conditions, since the FRP is too low (Cohen et al., 2018;Gonzalez-Alonso 260 et al., 2019). One possible explanation for this phenomenon is that the biomass burning occurring during the times of year where there is a negligible impact on the atmospheric loadings of NO2 and/or CO is significantly more energetic and therefore has a very different height profile, as compared to the times when the most emissions of NO2 and/or CO are produced. Another explanation is that there is additional forcing which are also playing a role in terms of the aerosol plume height rise that are independent of the FRP. Yet another possibility (Mims et al., 2010;Val Martin et al., 2018) is related to there being some type 265 of problem with the presentation of the nature of the land-surface itself, since fires occurring in heavily forested and agricultural areas are likely to have significantly different vertical distributions. Finally, it is possible that the intense aerosol loadings themselves are leading to absorption of a significant amount of the IR radiation which is in turn biasing the FRP measurements too low (Cohen et al., 2017;Cohen et al., 2018).
It is also possible that there are significant differences to be found in the non-linearity between FRP and the wind speed. 270 Interestingly, if the horizontal wind speed is quite high when the air passes over the fire source, it will cause turbulence and https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License. vortices, resulting in a lifting force. On the other hand, if the wind speed is too high, it will bend the plume's momentum and reduce the upward transference based on any initial vertical injection velocity. Furthermore, the wind speed may have different relationships with convection, which itself plays a dominant role in the rise of the plume. Given these effects, we do not directly consider wind-speed and the plume rise height independently, only within the confines of the plume rise model. 275 Since there are many underlying direct and indirect theoretical physical and chemical connections between the loadings of the CO and NO2 and the overall plume heights from MISR, we want to investigate this possibility more deeply. To make this comparison, we first looked at the entire time series, not only those periods during which the measured aerosol heights obviously had an impact on the atmospheric loadings of the CO and NO2. Next, we selected days which had data from all three measurement sources: MISR, MOPITT, and OMI. Furthermore, since we could not find such a paper in the literature, we have 280 decided to keep the relationship open and simple, without worry of over-constraining any relationship found. In theory, the injection height of the aerosol plume is related to the emission of smoke in the wildfires, since this is a function of the amount of heat released. Therefore, we would expect that higher emissions of CO and NO2 should correspond to higher heights of aerosols. However, the formation mechanisms of these two trace gasses is different, with CO a function of oxygen availability (and possibly surface wetness), while NO2 is a function of the temperature of the burning. Furthermore, very high co-emitted 285 levels of aerosols with the very high levels of trace gasses could also lead to a change in the vertical profile of the heating (Freitas et al., 2007).
To ensure that the variables are relatively independent, our analysis only considers only three mixtures of these species: the independent concentrations of CO, NO2, and the multiple of the two with each other. We then investigate how changes in the loadings of NO2 and/or CO are associated with changes in the height of the plume. Furthermore, we need to consider the 290 more extreme conditions in addition to the means, and are particularly interested in seeing how well loadings of the CO and NO2 can be used to model those conditions where the plume heights are extreme.
In all of the regions of the world, with the exception of the case of NO2 over Siberia and Northern China, we have a case where the mean value of the CO and NO2 measurements is higher over the set of points where the actual FRP measurements were made, than over the region as a whole Table 1. This is the point of this work, since we want to focus on the measured 295 values from MOPITT and OMI which correspond to the same spatial locations as the measured FRP. This makes sense, since the magnitude of emissions from fires is very large compared with the non-burning season and/or surrounding areas. However, the differences in the CO are in generally smaller than for NO2, which is further consistent with the fact that the lifetime of CO is much longer than that of NO2. Thankfully the case is well understood over Siberia and North China is because there are some known significant urban areas nearby to the burning regions. Furthermore, this exception occurs in winter, where we know 300 there is a significant enhancement of NO2 emissions due to the increase in urban biomass burning to offset the brutally cold winter conditions. https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License.
Over these fire-constrained points, we find that the variability of both CO and NO2 remains very low when computed on a point-by-point basis. On the other hand, over the entire regions, the variability of the point-by-point measurements of both NO2 and CO are much higher. This is in large part due to the rapid changes in different land-use types in different parts of the 305 regions of interest being studied (consistent with Cohen et al., 2018). These results are based on the statistics of more than 67000 daily MISR measurements. Therefore, for the remainder of the work, we only use the data for the NO2 and CO which are obtained at points where FRP measurements exist.
Note that the measurements and the results here are looking at the aerosol heights measured over small spatial and temporal domains. We are looking to analyze the impact of the initial plume rise, and any very rapid adjustments in the 310 atmosphere. The plume heights, both measured and modeled are not consistent with large-scale transport due to meteorology, factors enhancing the stability of a layer or changing the chemistry within a plume. They certainly are not receptive to a Lagrangian type of modeling effort, which is supposed to be focused on the air itself and in particular air at the large scale.
Therefore, the results given here show here are the best methods currently used to reproduce the spatial distribution of aerosol plumes produced by wildfires. 315

Plume Rise Model Applied to MISR and Meteorological Measurements
The annual average global total cumulative FRP from 2008 to 2011 is 209MW, based on more than 16000 measured MODIS fire hotspots. Overall, the measured FRP has been shown to be on the rise in recent years (Cohen 2018;Freeborn et al., 2014), although there is still a fundamental and significant amount of underestimation based on the current measurement techniques (Giglio et al., 2006). The plume rise model in theory should take these FRP values, and combine them with 320 knowledge of the vertical thermal stability and the wind speed, to approximate the height to which the plume ultimately rises at equilibrium with its environment.
However, in reality, direct and semi-direct effects are not considered when using the simple plume rise model, although they are known to be important (Tao et al., 2012). Therefore, a different approach which attempts to take these forcings and/or the underlying aerosol loadings into account may lead to a better representation of the plume height rise, if such a model can be 325 parameterized. Furthermore, the plume rise model relies on the atmospheric stability, and therefore does not take into account rainfall, changes in fire burning, in-situ chemical and physical production and removal, as well as the afore mentioned interactions radiatively between the aerosol and the atmospheric environment.
All of these shortcomings aside, the use of simple plume models is the current scientific standard approach, and therefore we will apply it here as well. This is done by first aggregating the daily statistics of the vertical aerosol height over all parts of 330 each region of interest Table 2 Next, we look at the difference from day-to-day at each of the sites which has satisfied the overall requirement. The mean daily difference between the plume rise model and the MISR measurements as a whole show a large amount of variation, with a global average of 0.44, a maximum of 1.13 km (in West Siberia), and a minimum of 0.04 km (in Argentina). Across all of the different regions we find that the plume rise model underestimates the plume height. Hence, these differences are not normally distributed, and are strongly indicative of a bias. In addition to this, we compute the RMS error Table 3 as a way of quantifying 340 overall how well the data. The RMS is found to be considerably larger than the difference of the means, indicating that a small number of extreme values are dominating the overall results, which were found to be 0.67 km, 0.88 km, 1.36 km, 0.40 km, and 0.85 km in the respective five areas.
To more carefully determine the extent of any bias, we analyze the PDF of the model and measurement results Fig. S2.
This approach yields a clear determination that the plume rise model consistently underestimates the measured injection 345 height, with the underestimate ranging from 6% (in Argentina) to 66% (in Southern Africa), and a global average of 33%.
However, if we constrain ourselves to those fires occurring only in heavily forested regions, the average underestimate is reduced considerably to 11%. On the other hand, if we look across Africa as a whole, we find that the underestimate is on average 52%, a finding which deviates more from the measured aerosol vertical distribution than previous global studies (Val Martin et al., 2018) as well as those over Southeast Asia (which previously has been considered one of the world's worst 350 performing regions for such plume rise models, such as Reid et al., 2013;Cohen, 2018).
Furthermore, even though the plume rise model leads to a low bias compared with the measured height, it is still not ideal for very low plumes which are found near the surface. The plume rise model tends to instead uniformly overestimate the amount of aerosol found in the upper parts of the boundary layer from 0.5km to 1.5km, while at the same time not providing any reliable amount of prediction for the cases where there is a considerable amount of aerosol under 0.5km. For example, the 355 plume rise model is sometimes a good fit for aerosol heights under 0.5 km such as in West Siberia and Eastern Europe (where 23.5% and 12.3% of the measurements are under 0.5km and 27% and 13.6% of the plume rise model heights are under 0.5km, respectively). However, in other locations, the plume rise model grossly overestimates the amount under 0.5km such as in Central Africa and East Siberia (where 3.6% and 17.9% of the measurements are under 0.5km and 20.5% and 51.0% of the plume rise model heights are under 0.5km, respectively). In the case of Argentina there is a slight underestimate of the 0.5km 360 heights by the plume rise model (49.4% of measurements and 30.1% of the plume rise model heights). One of the reasons for this is that in general the plume rise model tends to underestimate the results from 1.5km to 2.5km, and cannot reproduce results reliably at all above 2.5km. This is partly due to the effect that the FRP values are too low, and possibly due to other processes occurring in-situ which further lead to buoyancy and/or convection. https://doi.org/10.5194/acp-2019-1017 Preprint. Discussion started: 12 February 2020 c Author(s) 2020. CC BY 4.0 License.
A few special regions of interest have been observed when comparing the plume rise results with the measurements. In 365 Southern Africa plumes cover 9763-pixels or 19% of the total research area, and therefore are extremely representative of the overall atmospheric conditions. What is observed is that there is almost no aerosol (only 5.9%) present close to the ground (from 0km to 1km). The vast majority of the aerosols, 92.6%, are concentrated from 1km to 3km. Furthermore, we observe that the time series of both CO and NO2 loading is significantly higher than other regions Fig. S1. This finding is completely the opposite form the plume rise model result, which shows that most of the pollutants (97%) are concentrated in the range of 370 0-1km, while almost none (3%) is found from 1km to 3km. There are a few reasons for this finding. First of all, when both CO and NO2 loadings are high, the aerosol concentration and AOD will also be high, since they are co-emitted at roughly similar ratios from the fires. This in turn will both lead to a further underestimation of the FRP due to the outwelling infrared which is partially absorbed by the aerosols, as well as provide a further uplifting energy source due to the semi-direct effect (Tao et al., 2012;Guo et al., 2019). In other words, the assumptions underlying the plume rise model may not be completely relevant or 375 dominant over this region under these conditions. A second special region, which completely contrasts with Southern Africa is found in Argentina. In this region, a much smaller amount of the total research area is covered in plumes of 1063-pixels or 2.1%. A large amount of the total aerosol (83.8%) exists below 1km, while only a small amount (5.1%) is found above 2km. In this case, the plume rise model achieved its best match globally, with a large amount (92.2%) found below 1km and a small amount (0.35%) found above 2km. 380 Furthermore, the loadings of CO and NO2 are both considerably low as compared to other regions studied in this work. It is under these relatively lesser polluted, fewer and less intense fires, and lower density of total surface area burning, that the plume rise model can achieve its best results, implying that small-scale fires truly are dominant over this region, and that the model is reasonable to use in such a region. Although it is still obvious that even in this best result case, that the plume rise model is fundamentally biased towards the aerosol vertical distribution being too low, especially the amount into the free 385 troposphere.
As we have observed, the simple plume rise models based on Briggs, 1965, are useful under specific circumstances. This is especially the case when the atmosphere is relatively stable, the total loading of pollutants is not too large (i.e. there is fewer fire masking and less of the semi-direct effect to contend with), and where the density of fires is lower (and hence there is less overall buoyancy changing the atmosphere's dynamics). On top of this, more flat and uniform areas are less likely to have local 390 convection, further leading to an improvement of the effectiveness of the simple plume rise model. It is for these many reasons why we find that the simple plume rise model does not provide an ideal fit over many regions, and for this reason, we propose a simple statistical model as an alternative.