Lost Weekend…

Photograph from Wikipemedia Commons. Menengai Caldera in Kenya, one of the largest calderas on the planet.

How to kill a weekend.

As some of you have observed, last week I asked for anyone running across a caldera size and eruption volume to give me a quick shout here on the forums. Ostensibly, I was going to compile a spreadsheet in order to look at Hagstrum’s hotspot list compared to large caldera locations. Despite Carl’s disdain for the Antipode Impact idea, I think Hagstrum’s hotspot list is still pretty good, and it collates several other lists and weeds out some of the less than accepted ones.

While trudging through the calderas that were readily supplied, grabbing what info I could and trying to stay focused on DRE, the question of DRE again came up again in discussions. It wasn’t an actual argument or disagreement, but it did give me enough doubt in my data to seek other sources. Along the way, I found “Sulfur dioxide initiates global climate change in four ways” by Peter L. Ward. Well, to be truthful, I didn’t find that first, I found his table that supports his paper. I had to dig around to find the paper. I HIGHLY recommend the table. It is awesome. While the focus is on SO2 and climate change, they include the names of the tephra deposits that go with specific eruptions. Not all, but quite a few.

From his table, and with the re-worked VolcanoCafe user provided data, I came up with this (distraction#1) :

The first thing I would like to point out, is that it’s a log-log plot. The formula is a bit cantankerous to work with in Excel or on a calculator. (uses 10 raised to a power from a function that then has a logarithm in it.) The log-log plot was the only way to make it come out halfway usable. This formula was derived with DPlot, and in order to minimize the sigma fight (which I lost, quite readily) I left the individual points in place so that you can see just how far the estimate can be off. In one incarnation, I came up with the estimated value being within 0.77 of the actual value, 75% of the time. At this point I needed a beer and would continue later.

Moving back to the plot, and poking around in the text of the paper, I found that Professor Yukio Hayakawa of Gunma University (Japan) had compiled a list of large eruptions covering the last 2000 years. I had to go find that. Unfortunately, the list cuts off at 1999 with the eruption of Hudson in Chile. Distraction #2 involved updating the list with everything that happened since. While using his calculation of eruption magnitude, I decided to look back at how some of the calculations compared to fresher data from GVP. The paper uses M=log(m) -7, where m is the erupted mass in kg.

That’s actually a pretty handy formula. It sort of tracks with the VEI range, (but it’s not VEI, that’s different) Eyjafjallajökull comes in at 4.62, Merapi at 4.55, and Sarychev Peak at 5.04 when using GVP combined lava and tephra (DRE) volumes.

Photograph from Wikimedia Commons. The Somma caldera of Mt Aso in Japan.

I did find a problem with the data though… it wasn’t lining up with GVP info very well. In general, it was running 1.13 times the Hayakawa data when redone with GVP info. Then I ran into the problem of GVP not having anything more than a guesstimate for the VEI of some of the volcanoes with no tephra or magma volumes listed. (and these were pretty recent eruptions) Since Hayakawa used a lower cutoff of M=3.8, anything less than a VEI-4 would not get that high. (VEI=3 yeilds an M of 3.43). Ehh… give up and go find something to gnaw on. I did find out that my stepson had retribution against the Pelicans. I had skipped the King Mackerel fishing since I was “in the groove” with the data. The bait fish they were using had a tendency to attract the Pelicans attention but was so swift that it would be gone by the time the bird got to it.

Referring to Carl’s “Did you notice the erupting Supervolcano?” post, you will note that in the reference, it doesn’t state what the size of the Tondano Caldera eruption was. Being focused primarily on the geothermal energy capability of the system, that is understandable. Using the outline from Figure 5 of the paper, and applying our handy formula, we can get a ballpark estimate of how much “stuff” was involved. At roughly 30km by 9km, it comes in at 197km³… give or take. Solid VEI-7, but the calculation has a sigma of 351km³ so it could quite easily have been large enough to be withing spitting distance of VEI-8. (900km³ is within 2 sigma, and VEI-8 is 1000km³) 

[Editors remark (Carl): I actually was a bit more devious than that. For this caldera I have a bit more data. Through drill core samples I know how much of the caldera is infilled with original ash and later ash. That gave me the actual depth of the original caldera bottom. One should recognize the difference between a subsided caldera and a blow out large caldera event. The first one gently drops with lost material, the other ejects more material due to explosion, in this case when the ocean hit the magma inside the magma chamber. I then calculated the amount of DRE by size. To get a low enough number I did not assume that there was anything ontop, ie. that the volcano was flat with the surrounding landscape. I then got a 918 km^3 of ejected DRE. Size is not everything as I discovered, depth is equally important. Add a couple of the known active volcanoes before the large caldera event and you are comfortably at the 1000 cubic range for a comparatively small caldera. I then did a sanity check against known ash depths for the layer across distance, and fount it to be within the ballpark.]

Okay, back to the data. In 2009, Deligne, Coles, and Sparks put out a paper entitled “Recurrence rates of large explosive volcanic eruptions”. Yet another kick arse piece of work. In it, they use Extreme Value Theory to attack the problem of recurrence rates of large eruptions. Now that is something that I can appreciate. Extreme Value Theory deals with the failings of the Gaussian curve… out there in the tail, the realm of the infamous Black Swan that I am always yammering about exists. I have to go back and read that paper. Anyway, they mentioned Hayakawa’s list, and then using those methods, took the list back to the last 10000 years. Hmm… what can we do with that? I have the Greenland Temperature from the ice core data available, so I plotted it. It didn’t look that interesting until I ran an integral of the M value, then detrended it. That brings out the relative change in the sum that is going on without the actual data trend obscuring it. Plotted against the temperature, it look… “interesting”

There are a couple of peaks that seem coincidental, but for the most part, not a flipping thing there. I found it interesting that there was a peak in activity about 3527 BC and over all, volcanic activity has been declining ever since. I don’t know why that is. That’s just what it looks like. Being a glutton for punishment, and since it was “just sitting there,” I ran a couple of correlation routines on it to see if anything was present, but not obvious. Pearon’s correlation coefficient of 0.0111. Okay, I didn’t really expect a linear correlation. Spearman’s rho is supposed to be able to detect non-linear relationships, and I expected a higher score. I got 0.0017. What? It’s worse? “Wow.”

I have, on this computer, a program called “Formulize” by Eureqa. It’s free, unless you want to use a server farm. You can set it up and run it on your on PC and it will churn through whatever data you feed it and try to find a formula that relates the data sets. It’s the ultimate “beat the data with a stick” program. It can yield garbage… (generally if you feed it garbage) but it’s pretty good at coming up with something. So, I turned it loose. It turns out, that if you have a delay of 1405 days, it can roughly predict the temperature in Greenland from the running detrended integral of the Volcanic activity with a correlation coefficient of 0.7177. (Actually pretty good considering where we started out from) I calculated a sigma for the function based on what the formula predicted and what the actual data was.

That… was distraction 3.

What’s it all mean? Beats me. Greenland is just one point on the globe. There seems to be a 1405 and 1422 day delay relationship in the data, or about 3.8 years. Formulize also ground on a 4.13 and 4.44 year offset for a while. It was quite fun watching it dance back and forth with the delay. Make of it what you will.

And now the all important caveat: I am not a Geologist or trained in any of the fields that have been touched on in this post. My specialty is electronics and cross correlating threats… if you must know. (such as the 230 knot Shvall torpedo tested by Iran having been designed for 533mm torpedo tubes postulated as a design criteria… and the the Kilo class sub launched from Bandar Abass last week or so, having six 533 mm tubes. And that’s all from published data in various sources on the web.) But.. I don’t do that anymore. Volcanoes will have to do.

What to take away from this post, something that can be used by my fellow volcanophiles, is the first plot. You can find a hole in the ground in Google Earth and do a ballpark estimate of how much material may have come out of it when it initially formed. Remember that it may not have all happened at once.

Several thousand years of activity can produce the same effect.

Enjoy!

GEOLURKING

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Sulfur dioxide initiates global climate change in four ways – Ward (2009)
http://tetontectonics.org/Climate/SO2InitiatesClimateChange.pdf
And the table:
http://www.tetontectonics.org/Climate/Ward2009TableS1.pdf

Hayakawa Paleovolcanology Laboratory
http://www.edu.gunma-u.ac.jp/~hayakawa/English.html

Recurrence rates of large explosive volcanic eruptions – Deligne, Coles, and Sparks (2010)
http://www.globalvolcanomodel.org/documents/Deligne%20et%20al%20(2010).pdf
Data Set
ftp://ftp.agu.org/apend/jb/2009jb006554/2009jb006554-ds01.pdf

Did you notice the erupting Supervolcano?

This idylic scene is from Lake Tondano situated within the 20 by 30km Tondano Caldera.

Some volcanoes just can’t catch a break. Imagine for a little while that you are a bona fidé supervolcano. You are the largest of your type on the planet, you are highly active, and by gosh you have shown what you are capable of. In a perfect world your 20 by 30 caldera explosion should have put the world into awe, and the 1 000 cubic kilometer of DRE you ejected in the form of pumicious tuff covers an entire sub-continent. Yepp, you really did reach the small highly exclusive club of VEI-8 volcanoes. You smirk at your little sibling Monte Sommas antics with Vesuvius. Your Vesuvius event left a 3.5 by 5 km God honest caldera on its own. To top it off you have a huge underground reservoir of liquid acid that would seriously alter the planets weather if you felt like discharging it. You are also perfectly located to have a maximum kill ratio. So, you wake up and stretch your arms and start a double eruption from two different sub-volcanoes just to celebrate the new day. You have your largest eruption in recorded history. Then you look around to see the fearful faces of the residents as they offer up motorcycles in your name, you expect volcanologists doing somersaults as they play lip banjo, and literally thousands of blog pages glorifying your power and shear awesomeness. What do you find? Yawning people and a cockerel trying to wake up a pig sty. You find that for being an erupting supervolcano you are a massive PR failure. One single small earthquake at Yellowstone outperforms you in publicity.

Tondano

Compund satellite image/map of the Tondano area courtesy of JPL.

The quarternary volcano of Tondano in northern Sulawesi (Indonesia) had its massive caldera event about 2.5 to 2 million years ago. Technically it is a somma type volcano, complete with the remnants of Pangalombian, a former stratovolcano that disappeared in a Vesuvian VEI-7 total caldera event.

Parts of the Pangalombian caldera were later covered by the now dormant Tompaso volcano that ejected large amounts of basaltic andesites in a long series of VEI-6 eruptions.

Todays Tondano is known for having acidic maar eruptions inside the caldera, a couple of mud volcano events during recorded history and no less than 4 active volcanoes, Lokon-Empung, Mahawu, Sempu and Soputan. Quite often Lokon-Empung and Soputan have tandem eruptions.

Lokon-Empung

Lokon-Empung is a double coned strato-volcano located at the northern rim of Tondano. Lokon is a flat topped probably dormant volcano that no longer exhibits a crater on top and Empung is a historically active volcano that last erupted 1775. From 1829 onwards the site of no less than 25 eruptions has been Tompaluan, a smaller double crater situated in the saddle pass between Empung and Lokon. It has erupted since 2011 in tandem eruption with Soputan. The tandem eruption before that occurred on the 13^th of May in 2000.

The current ongoing eruption is slowly working its way to becoming a VEI-3 eruption. But it has so far mainly been consisting of small explosive ash eruptions so it takes time to reach that level.

Soputan

This small stratovolcano is located on the southern rim of the Tondano caldera. It is part of trending line of ring dyke vents that formed in consequent eruptions ending with the formation of Soputan stratovolcano. It normally erupts from either the flanking vent of Aeseput or through the unusually large summit crater that pretty much has the same width as the top of the stratovolcano. This is of course a sign of a very young volcano with a highly potent vent system.

The current eruption consists of ejections of small to moderate explosive ash plumes. The ash columns according to the Darwin VAAC have been up towards 12.1km, with several slightly smaller columns reaching 9km height during the last few weeks. Smaller explosive ash plumes have been pretty much ongoing for the last 3 months now. This eruption is quickly ramping up to becoming a VEI-4, and is as such the largest sub-aerial eruption since Grimsvötn 2011 and that is without even counting in Lokon-Empung into the picture.

The system

As any volcano capable of a large caldera event Tondano has a large and intricate internal plumbing. It is believed that there is a very large reformed magma chamber at depth. As pressure increases in that magma chamber when new hot magma arrives it is believed that the magma either goes up into the caldera as emplacements, and that those sometimes cause maar explosions or reheats the very active thermal fields contained within the caldera. Or that the magma is pushed up into smaller sub-chambers under the active rim volcanoes. When that happens eruptions normally follows very rapidly. A sign of the rim volcanoes being systemically interconnected somehow is that Soputan and Lokon-Empung on many occasions has had eruption interspaced with mere hours.

As any Somma volcano the Tondano caldera is highly intricate and complex, and still it is surprisingly badly researched. The only good material is an Icelandic funded study on the possibility for hydrothermal energy plants in the region. Yepp, the Icelanders are going international with their knowhow.

Why it won’t happen

Image by Andreas / AFP – Getty Images. This image shows how relatively close the volcano is to villages, the height of the ash column and at the same time that the base of the ash column is equally wide to the width of the top of the volcano of Soputan.

For those who dream dark dreams about enormously destructive eruptions Tondano is a bad bet. Why? Tondano has it all really, large magmatic influx, steady inflation, a large central chamber, active volcanism. Pretty much everything that it should need for a VEI-8 eruption. Except for 3 small things, it does not any longer have the amount of water necessary to drive an eruption like that. As many of you know water is a large part of large caldera events. When Tondano went massively caldera it was situated pretty much at ocean level, so as the final large eruption (probably a large VEI-7) happened and the top of the caldera slumped inwards the ocean roared in and what is probably the largest steam explosion happened. Think of it as hundreds of Krakatau eruptions happening at the same time, and you have the picture. As time has passed the land has been lifted due to tectonic uplift.

Second thing is that the magma before the massive caldera event was highly crystallized rhyolites. After the eruption the magmas have been predominantly alkali basalt-andesites.

And the third reason is that Tondano is very well vented as long as the rim volcanoes are connected to the central magma chamber. As soon as the pressure gets above a certain level the magma squirts into the sub-chambers and the volcano on top erupts.

To put it simply, Tondano is a champagne bottle with 5 bottlenecks. The cork is well fastened on top of the actual central chamber, so it cannot erupt that way. Then it has one volcano with the cork slammed back fairly well (Sempu), but that is not fully dormant. One that has the cork put lightly back on (Mahawu) and two bottlenecks that haven’t seen a cork for hundreds of years.

Basically, the pressure is almost constantly being released by Lokon-Empung and Soputan, and if that is not enough Mahawu erupts too. Last time Mahawu erupted with another of the volcanoes it was Lokon-Empung in 1958. Currently even if pressure got really high the only thing that would happen is that all 4 volcanoes would go off.

The only risk for anything really interesting happening would be if one or two of the vents got blocked off. Even then no caldera event would happen, but the likelihood of a Vesuvius event would increase a lot. Currently the candidates for that is either Lokon-Empung or Soputan. Soputan seems to have a very wide bore caliber vent so it could probably release the pressure without exploding from the face of the planet. But Lokon-Empung has evolved quite a lot more, and as it has grown older the vent has narrowed down considerably. If Lokon-Empung was subjected to high pressure it would probably not be able to handle the stress and subsequently go off with a VEI-6+. This is though not likely at current geological timescale.

The only real risk is that a magmatic emplacement will happen in, or around the large reservoir of sulphuric acid (water with a ph of 2). I think anybody can imagine how un-nice a maar event, or even worse, a phreatic explosion, would be if it happened to cubic kilometers of liquid acid. First of all it would make northern Sulawesi uninhabitable and kill off large portions of all life there. And a phreatic explosion would severely hamper the world weather for quite some time. Not a nice thought is it, an acid caldera event. I would decidedly not want to be around if that happens.

Tondano today

Lokon-Empung belching out a 3km ash column.

For being a highly active volcanic region with at best medium risk of fatalities the volcano is surprisingly badly monitored and highly under-studied. Almost all I have written is from one study alone, and that was produced by Orkustöfnun as a part of the geothermal engineering program. Interestingly that report predates the recent article in Nature about a new tectonic plate forming next to Sulawesi. You can clearly see the rift fault in one of the maps in the PDF. Nature seem to have done a bad background check on their paper before publication.

In reality if we look beyond the doom and gloom prophecies of a large caldera event volcano the risk is the bad monitoring. The area is heavily populated and an unexpected VEI-4 eruption at a flanking vent, or lahars, or pyroclastic flows will kill people, potentially a lot of people.

A thought

When a volcano of this size erupts and the world’s volcanologists, volcano-bloggers, and generally the large number of volcano aficionados yawn and continue to look at other less interesting volcanoes that is not even erupting, then something is a bit wrong. I happily admit that it took me almost a week before I actually got around researching the volcano. Then my jaw dropped and I started doing somersaults while playing lip banjo. It is just the sad truth that there are more well known supervolcanoes in the predominantly white western world that steal all the attention.

While we sit and moan about there being no interesting eruption we did not even reflect as we read that two more volcanoes in Indonesia erupted simultaneously 30km from each other. The only comments about it was that people rode their motorcycles inside an ash cloud to get to and from work (Lokon-Empung), and that a rooster cackled at a video of Soputan barfing up a 9km ash column. Then we went back to looking at out Katlas, Heklas, and the rest of the non-erupting volcanoes. Indonesian volcanoes could do with a good PR-Agency.

CARL

http://www.os.is/gogn/unu-gtp-report/UNU-GTP-2010-03.pdf

Name that Volcano Riddle!

Resident of Siglufjördur looking rather wooden eyed after yet another sleepless night with earthquakes.

The week has been calm eruptionwise, but fairly frisky earthquakewise in Iceland. The transform faultline north of Siglufjördur kicked into high gear with continous medium sized earthquake swarms.

The active faultline is one of two major faults making out two of the boundaries for a microplate that is seismically locked at its southeastern corner under Theistareykjarbunga volcano. Theistarykjarbunga is as most readers of this blog know one of the two volcanoes in Iceland and the world competing for the title of having had the largest flood basalt eruption during the last 10 000 years, the other one being Bárdarbungas Thjórsahraun eruption.

Image by IMO. The earthquake band of the fifth most powerfull earthquake swarm during the last 12 years.

The earthquake swarm has so far had a couple of hundred earthquakes in them, 13 between 3 to 4, and 5 earthquakes from 4 and upwards to 4.3M. This is a fairly vigorous, but not unheard of amount of energy released from an Icelandic earthquake swarm. It is though the 5th largest during the last 12 years. There is currently no other signs that this will lead into anything too exciting in the near future.

RIDDLE

This week we will try a new version of the friday volcanic riddle game. It is a “Name that Volcano Riddle!” by commenter Suzie. One point to be had.

The French footie fans looked on in horror as their European opposites ran riot round the capital! They asked themselves “What does this unruly orange mob mean to us?”

VOLCANOSPORTS

During the week I tried to come up with some extremely extreme sports that you can do if you have a volcano handy. My favourite was a surfing down a Hawaiian lava stream on a ceramic surfboard. The idea though was not as novel as I thought. Apparantly it is big too skate down scoria cones.

TGIF!
CARL

The Santorini Midget Lava Blob!

When Santorini erupts I will pack a particle filter mask, a mining helmet, goggles, a backpack full of water-bottles and book a room at Katikies hotell. Imagine sitting in this infinity-pool looking at the distant view of Nea Kameni as it farts out a slow and nice VEI-2. I will so do it.

Sometimes there is news about volcanoes that get blown out of all proportion. The island volcano of Santorini has unassumingly been erupting with small eruptions for the last 2 000 years or so. It is though mainly known for its VEI-7 Thera eruption 3 600 years ago.

Why do I mention the small ones? Well, a large volcano takes a lot of time to recuperate after a large eruption. First we need to understand what the large eruption was and what it did to the plumbing of the volcano. The Thera eruption started as a very large normal eruption that emptied out the magma chamber at rapid pace. In the end the roof of the chamber fell in on itself and water poured in. That in turn caused a very large steam explosion blowing away a sizeable chunk of the Island. One should remember that this is by far not the largest such eruption of Santorini, but thing is that they are far apart.

The effect of the collapse and the subsequent hydro magmatic explosion is that it left the volcano without a magma chamber. Where the chamber used to be there was only rubble. For the first 1 600 years after the eruption there was not even a chance for even a small normal eruption, any magma pushing up was deposited straight into the water filled rubble. In the end a solid roof started to build, and the cycle turned into the pattern we have today of small island building eruptions. Those eruptions are centered on the Nea Kameni Island with a few exceptions.

A few days ago Nature Geoscience published a paper on the current activity at Santorini. The paper itself was rather factual. Nea Kameni uplifted 14 centimeters from January 2011 to April 2012. That equals to 10 to 20 million cubic meters of magma. Sounds like a lot of magma does it not? We are after all talking about one of the largest active volcanoes around.

Well, the worlds combined press services though it was a lot. They picked up on it with war headlines.

Now let us look at it critically. The Italian volcano of Campi Flegrei inflated a bit in the 70s. It did 270 centimeters in less than half the time of Santorinis 14 centimeters without showing any other signs of erupting. After inflating 270 centimeters it went back to sleep, goes to show that it takes quite a lot for a supervolcano to go super if nothing else.

Now back to the magma chamber at Santorini and its size. The reason for Santorini having small and slow VEI-2s and a couple of VEI-3s and nothing bigger is that the chamber is still too small and weak to be able to withstand the pressure and volumes of a large eruption.

So, how about all that magma, it must be dangerous? No, not at all. Why? Because it is only 0,01 to 0,02 cubic kilometers of magma. If all of that jumped out of the volcano in one good eruption we are talking about a small VEI-3. Only problem is that all of it will not come up. Normally only one tenth to one twentieth of the magma comes to the surface from the chamber during an eruption.  Bummer for all those who dream about gloom and doom.

So, taking that fact into account we are looking for 0,0005 to 0,002 cubic kilometers of lava coming out of the volcano. That is a midsized VEI-1 to a small VEI-2. Quite normal for Santorini really, the island has had a lot of them.

Gosh darn, who stole my end of civilization?

CARL

A bit about VEI, and a numeric tool for amateurs.

Just to keep things in perspective.

Previously, Carl has pointed out just how lacking the VEI scale is.

VEI is based off of the total quantity of material that came out of the hole. In this respect, not a bad scale… but VEI means “Volcanic Explosivity Index.” What about the less energetic eruptions? Say, Kilauea? How about a volcano that makes a big show at first then oozes magma for a year afterwards? Eruptive sequences generally include the entire time that stuff is coming out of the ground, and in order to compare one eruption with another, one of the best comparison is by how much came out. VEI will be with us for some time, but it helps if you have some context as to what it means.

Eyjafjallajökull was out paced by Grímsvötn in about a day. Grímsvötn is a true monster, and fortunately for us, all it did was one of it’s lesser burps.

Here is a plot of volcanic plume height over a period of hours, and how much material, in “Dense Rock Equivalent” (DRE) that equates to. The formula was derived from Mastin et al, who essentially did an update on Sparks’ equation. The purpose was to get an estimate of the eruptive rate of a volcano based on sporadic or sparse information… such as only having plume height data of a remote volcano off in the middle of nowhere.

Image by GeoLurking. Click for larger image.

As you can see, for a sustained plume height at the indicated level, the mass adds up over time. As time goes on, eventually different levels of VEI are reached.

You can also see how Grímsvötn blew the doors off of Eyjafjallajökull’s “puny” eruption.

In “A multidisciplinary effort to assign realistic source parameters to models of volcanic
ash-cloud transport and dispersion during eruptions”  L.G. Mastin et al there is a formula that allows a computation of the mass ejection rate based on plume height.

http://www.geo.mtu.edu/~raman/papers/MastinetalJVGR09.pdf

It’s geared mainly towards volcanoes located in remote areas (such as Cleveland) and proves to be a handy tool if you put in a little work.

VAAC reports are probably the most handy reports available for any given eruption. VAAC is most concerned about keeping aircraft from plummeting out of the sky, so they try to stay on top of the hazard. This also means that their reports, though good, are more focused on the threat than the volcano. What the volcano is actually doing is little concern for them… what it did do is the most important. This means that once a plume is lofted into the air, the max elevation of the plume sort of remains fixed until it dissipates. The threat envelope will move around with the cloud… and generally the max elevation will remain mostly fixed.

If you are lucky, the VAAC report will state somewhere in the warning what the plume height is over the volcano. That is the data that a volcanophile will keep track of. That gives you the current state of the eruption.

Taking the time stamps for each report, along with the height of the plume over the volcano, and adding in the heights and time stamps from what ever geological agency reports, you can get a pretty decent record of the activity levels, and make a rough estimation (using Mastin et al) of the total amount ejected.

You do this by interpolating the rate from one data point to the next. You could connect the dots using a straight linear trend, or you could use some sort of poly curve or cubic spline (what ever your spreadsheet or data fitting program is capable of). From this curve, you need to get the interpolated increments down to one second intervals. Once you have interpolated timestamps and the estimated column heights at those moments in time, apply the Mastin formula to determine the DRE rate.

Then you just sum those rates in order to fabricate the total amount erupted to that point in time.

It may sound complicated, but it’s pretty straight forward.

From Mastin et al

H = 2 * V^0.241

Solving for V

V = (H/2)^(1/0.241)

V = Rate in m³/s
H = Height in km.

And.. a very important caveat… the formula has an error envelope of a factor of four. That’s pretty large, but it gets you in the ballpark for eruption estimates.

A sample run:

This is about a fictional volcano.  Since it’s my construction, I choose to name it Mt. Gibbons.  (I’m a Billy Gibbons fan) On 1/15/2012  at 12:00, Mt. Gibbons erupted to an initial altitude of 15 km.  A compilation of VAAC reports and eyewitness reports from the Soso MS Volcano Observatory, show this for Mt Gibbons activity (plume height)

Calculating the number of seconds since the start and re-plotting, we get

Now we apply the Mastin et al derived equation, this gives us the rate of the eruption.

This is where I cheat.  Using the built in integrated function of Dplot, I have it calculate the integral.  You can do something similar with your spreadsheet program if you calcuate the linear trend between each point, then put together a running sum of those calculated points.

So.. as you can see, Mt Gibbons, starting to wane in activity at around the 24th of Febuary, will probably come in as a solid VEI-4. One thing that is very important, is to remember that the reference document, Mastin et al, clearly stated that there is an error factor of about four in the equation. In other words, this will get you into the ballpark, but it’s not full proof.

Enjoy!

GEOLURKING