I have for quite some time had an issue with the Mogi-model as soon as it is used outside of a central volcano. At a central volcano it makes sense to use a point sourced half sphere as a model for predicting amount of magma intruding into the system from given GPS-coordinates. If only the crust is thick enough and if you know the approximate depth of the magma reservoir you will get pretty good and reliable numbers. If the crust is thinner you should though allow for a semi-elastic bottom, or in other words, allow for a slightly dented full sphere.
Problem is that we often are dabbling with volcanic systems that lack a central volcano such as rifting fissure volcanoes. As most of you know those are pretty much the standard type in Iceland.
A while ago I was pondering about a recent paper by Sturkel (2013) about Heklas volcanic plumbing in which he used a point source half sphere. Only problem is that Hekla is a rifting fissure that gotten stuck so that it masquerades as a central volcano. This has a few interesting effects that I will get back to another time, let us just say that I got interested in other shapes than point sourced half spheres.
Hekla is in a way easy since we fairly well know how the fissure swarm is oriented, but in other volcanoes we really do not know the shape and orientation of the underlying fissure swarms, or in the case of El Hierro how the faultlines are shaped. All that the literature gives us is that El Hierro has 3 arms and make up roughly into a Mercedes-star without the ring around.
Prediction stress field
Instead I realized that I could use the data points from the GPS and make a prediction field over the sum of motions recognized by the publicly available GPSs on El Hierro (IGN-set). Since I did not have access to the raw data points I tried my best to get x,y,z,(t*)-coordinates out of the image for today, so it is bound to be a bit fuzzy. But with many GPSs the fuzziness is really quite acceptable and would not have made a great deal of difference (I hope).
Problem with El Hierro is that parts of the island is seismically dead due to alternating layers of old and cold basaltic intrusion, basaltic crustal under-layer, sedimentary mid-layer, hardened basalt top-layer, and variously hot intrusions. So, the magmatic intrusions are very hard to pinpoint from a depth perspective. That also makes the Mogi-model quite unreliable for this purpose and at El Hierro.
As magma moves into a system it creates stress, mainly on the top of the system. So instead of chasing the magma I used the GPS data to hunt where the stress would be the highest based on the various motion patterns over the time period, and then summing it. In short, creating a prediction stress field.
From that we know that the magma is lurking about below the stress field, and we get a general shape of how the intrusion(s) is/are shaped and how they are located. And since this is done in a 4D model we also get a rough estimate at the depth of the magma over time.
Another neat little feature of this is that we also get a probabilistic view of where the stress is highest in the model, and therefore we can deduce that an eruption should be most likely somewhere on top of the predicted stress field. This though should be seen from the point that El Hierro has unique weaknesses built into it. So, it is statistics, view it as such, but the most likely spot for a new eruption would be on top of the stress field with the center of attention slightly west of Pinar.
A stress field like this should conform to known features along a fissure line to be accurate, if the area has been subjected to previous intrusions there could be volcanic vents roughly forming the same shape as the predicted stress field, at least if it has erupted previously.
The stress field forms a roughly oblong shape that is slightly curved westwards in the middle and with an estimated average depth range at 4 – 10 km. Where a Mogi-model would have given a rough estimate of 0.15km3 the stress field equates to a maximum of 0.5km3 of magma.
We find that the shape of the stress field (represented by an actual banana) spans from Taibique in the North to La Restinga in the south and is half-moon shaped westwards. We also find that a spot near Pinar has suffered the greatest predicted strain and that the strain value roughly equates to 100 – 130mm of uplift.
It is here important to point out that the stress field continues outside of the boundaries here given, it is just that outside of this area it is below the cut-off value in the computer model.
Most important, we find that the banana covers more cones and vents than is found outside of the banana.
Why a banana in the picture? Well, first of all, my skills with Photoshop is really meagre so I could not import the image of the stress field from Mathematica to Photoshop, but mostly it is because I think a banana is way more cool than a banana-shaped blue blob.
*= Time-coordinate so that time can be used as the fourth dimension. After that it is just 4-position integers all the way (except that it is 4-way path integers)