#### Decay Rate Correction

To estimate the decay rate (

k), we consider the problem in the context of first order reaction (loss) kinetics, that is, as a reactor model for the total cloud mass (

M) which was termed the “Delta-M" method by [

28]. Whenever the magnitude of the flux of

${\mathrm{SO}}_{2}$ is much less than the magnitude of the change in

${\mathrm{SO}}_{2}$ mass, the governing equation for the total mass becomes a simple decay model:

where

k is not necessarily fixed. To estimate

k at some instant we can choose to use one of two formulae:

referred to here as the percent method or using

referred to here as the log-difference method. For time series data, as the time interval (

$\mathsf{\Delta}t$) between measurements grows, discretizations of these formulae will diverge significantly. Additionally, in the presence of a detection threshold, which is certainly the case here, both formulae encounter significant problems. If the cloud mass drops below the detection threshold, the first formula will yield right-censored estimates of the decay rate as

$k=1/\mathsf{\Delta}t$, whereas the second formula will yield an infinite estimate as the mass decays apparently to zero in an finite time interval. For this reason, we work with the first formula to estimate

k and apply a type of continuity correction to correct for the right-censoring.

First, we generate valid samples of the decay rate:

for all

i where

${M}_{i+1}<{M}_{i}$ and

${M}_{i}$ is the

i-th total

${\mathrm{SO}}_{2}$ cloud mass in the sequence of plume images. As described above, the maximum calculable value of these samples by this method is

${k}_{i}^{max}=1/\mathsf{\Delta}{t}_{i}$ which occurs whenever the total cloud mass decays apparently to zero (

${M}_{i+1}=0$). According to the approximation of first-order linear reaction kinetics (upon which this estimate is based), the mass should never decay to exactly zero. If the decay constant were actually

$1/\mathsf{\Delta}{t}_{i}$, then the mass would only decay by a factor of

e in one time interval. However, due to the lower limit on detection sensitivity, it is common for small clouds to decay to apparently zero mass as little as one observation after first detection. In order to decay this much in one time interval, the decay constant must be sufficiently large for the concentration to drop below the detection limit. To derive a continuity correction, we consider the following. If the true decay rate (

k) was known and was constant over the interval

$\mathsf{\Delta}{t}_{i}$, then according to the solution of Equation (

A28), we would have:

which can be inverted to find an estimate (

$\widehat{k}$) of the true decay rate which corrects for the discretization as a function of the interval size:

This formula confirms that for an infinitesimal time interval (

$\mathsf{\Delta}{t}_{i}\to 0$), there is no correction and the original sample is accurate. For decay rate samples which approach the censoring value

${k}_{i}^{max}=1/\mathsf{\Delta}{t}_{i}$, the correction blows up, yielding an apparently infinite decay rate in cases where the cloud mass decays apparently to zero. To mitigate this effect (related to the existence of a detection threshold), we adopt a Taylor expansion of function

$f\left(\mathsf{\Delta}{t}_{i}\right)$ about the point

$\mathsf{\Delta}{t}_{i}=0$:

In general, fewer terms are needed for smaller values of the decay constant, or similarly, the closer the true

${\mathrm{SO}}_{2}$ lifetime (

$\tau =1/k$) is to the satellite repeat time interval (

$\mathsf{\Delta}{t}_{i}$), the fewer terms are needed and the correction need not be as severe. Of course, the number of required terms cannot be known precisely since the true lifetime is the unknown to be determined. Because the polynomial expansion is attempting to fit a singularity, this is only an issue as

${k}_{i}\to {k}_{i}^{max}$. Near this limit, we may estimate the appropriate number of terms from knowledge of the detection threshold as follows. In the limit

${k}_{i}\to {k}_{i}^{max}=1/\mathsf{\Delta}{t}_{i}$, Equation (

A34) becomes

where

${H}_{N+1}$ is the (

$N+1$)-st harmonic number. From this relation, the number of terms which minimizes the error between the true decay rate and the estimate is:

Using a continuous approximation of the harmonic numbers gives

where

$\gamma $ is the Euler–Mascheroni constant. Although the elapsed number of

${\mathrm{SO}}_{2}$ e-folding times in one repeat time interval (

$k\mathsf{\Delta}{t}_{i}$) is not known, we may substitute in a rough estimate. Among the instances where the mass decays to zero, we may consider that the larger preceding values of mass (

${M}_{i}$) require shorter lifetimes to decay below the detection limit one time interval later. From this reasoning, we may increase the number of terms for larger preceding cloud mass. Accordingly we derive the following estimate for the optimal number of terms:

where

$\u03f5$ is the mass detection limit. This expression is derived by estimating the repeat time interval as the time required for the maximum column to decay to the detection limit. Clearly this is a very coarse estimate, but it is used only to inform the number of terms in the decay rate correction, which yields a much better estimate. The cloud mass detection limit is defined here as

$\u03f5=\langle {X}_{z={z}_{th}}\rangle {A}_{pix}$ where

${A}_{pix}$ is the pixel area and

$\langle {X}_{z={z}_{th}}\rangle $ is the mean VCD from among all measurements with

${\mathrm{SO}}_{2}$ z-score near (within 0.5 of) an

${\mathrm{SO}}_{2}$ z-score detection threshold (

${z}_{th}=5$) with the angle brackets referring to a sample mean withing the scene. When no measurments from the interval fall near

${z}_{th}$, the detection limit is taken from a previous interval, and when there is no previous interval,

$\langle {X}_{z={z}_{th}}\rangle $ is set to the value 0.5 DU.

With a definite estimate of the appropriate number of terms, we can apply this formula to all valid samples

${k}_{i}$. Notably, in the limit

${k}_{i}\to 1/\mathsf{\Delta}{t}_{i}$, the correction this number of terms gives can be given with a better approximation of the harmonic numbers as

which can be seen as a correction to the log-difference method defined above.

Overall the decay rate estimation process can be summarized by four steps:

- Step 1:
Valid decay samples are computed as

for all

${M}_{i}>{M}_{i+1}$.

- Step 2:
The mass detection limit is estimated for a sequence of plume observations as

with the angle brackets referring to a sample mean over all pixels meeting the criteria.

- Step 3:
The number of terms needed for each correction is estimated as

- Step 4:
Each correction is performed:

Once a set of corrected samples (${\{\widehat{k}\}}_{i}$) is obtained, we can estimate the lognormal parameters (${\mu}_{k},{\sigma}_{k}$) representing, respectively, the mean and standard deviation of $lnk$. We estimate these by inverting the sample quantiles (25-th percentile, median, and 75-th percentile) to obtain an estimate of ${\mu}_{k}$ as the inverted median, and ${\sigma}_{k}$ as the average of the ${\sigma}_{k}$ estimates inverted from the 25-th and 75-th percentiles. This is a more robust estimation method than directly using the method of moments (computing the sample average and standard deviation).