# Determination of reasonable volcanic island height from area?

My islands (in the world I'm building) were formed from a volcanic hotspot. I want to build realistic islands, so I was wondering how tall volcanic islands are generally in relation to their area. Intuitively, there should be a rule, given that erosion of volcanic island mountains generally is accompanied by the island slowly subsiding in the sea, reducing the surface area and the height. Similarly, due to them mostly being shield volcanoes, the island's building from volcanism should have a similar area of lava correspond to the stacked up height of lava. So, is there a formula on how tall islands are based on their area? (+/- 3 percentage points should be fine)

For an example, here is Kauai, an old island, and Hawaii, a new one:

• 1,598.1 m tall vs 1,456 square km (Kauai)
• 4,207.2 m tall vs 10,430 square km (Hawaii)

If this is impossible, could you do an estimate for a single island chain (Emperor Seamounts/Hawaii)?

• your best bet is to find a few island of the approximate area you want, many other factors like the height of the seafloor and volcanic composition have much large effects. – John Aug 21 '18 at 0:33
• @John does the height of the seafloor have an effect? I would think that the cross section would look kinda like a cone so that the surface area to the height would be possible to model. About volcanic composition, I am mainly looking at basaltic lava, a lot like the Hawaiian islands. – JavaScriptCoder Aug 21 '18 at 0:38
• The seafloor has a big effect because it alters how much things like coral and subsidence can effect it. Other factors like how productive the volcano is and how fast the seafloor is moving will also have much larger effects. – John Aug 21 '18 at 0:41
• I've edited to have it focus on the Hawaiian islands and Emperor Seamounts. Would that be more answerable? – JavaScriptCoder Aug 21 '18 at 0:41
• Not really, you can find out the average slope (5-10%) of a shield volcano but that won't tell you much since it a large variance and many islands are made of more than one volcano/dike. Also erosion can drastically alter the height. – John Aug 21 '18 at 0:47

# For Hawaii, $H\propto\sqrt{A}$.

I looked at the islands on the Hawaiian-Emperor Seamount Chain with surface volcanoes, and plotted their surface area vs. their maximum elevation, using data from Wikipedia. I checked that the topography of each island is indeed dominated by the central volcano or volcanoes, although this is a bit complicated on some of the larger islands, as there are multiple active cones.

I left out a number of islands, atolls, seamounts, and guyots. The islands I used in my dataset are the main volcanic islands; the few extinct islands are extremely small. Seamounts and guyots are underwater, and therefore aren't helpful. Atolls, of course, are old and too eroded to give us any helpful data.

I then plotted the points logarithmically and fit a power-law model to the data, using this code by Aziz Alto. Power-laws are great in general, and are often the physicist's method of choice, so I figured it would be a decent starting point.

I found that $H\propto A^{0.54}$, or, essentially, $H\propto \sqrt{A}$, where $H$ is the maximum height of the volcano and $A$ is the area of the island. In particular, the best-fit equation I found was $$H=34.67\left(\frac{A}{\text{km}^2}\right)^{0.54}\text{ meters}$$ The $R^2$ statistic is rather good; $R^2=0.796$.

I doubt this relation has a physical meaning; I'm interpreting it as merely a statistical fit that will help you in your calculations and worldbuilding. Although the fact that the exponent is close to $1/2$ is nice, and maybe indicative of some physical law, I'm inclined to doubt it Maybe there's underlying physics; maybe there isn't. But I suppose it's a good enough solution, for your intents and purposes.

Here's the data again, with the best-fit model plotted:

For this chain, a good approximation is $H\propto\sqrt{A}$. Since the islands are roughly circular and $A\propto r^2$, $H\propto r$, which is something of an interesting result. It does seem to agree with a model of shield volcanoes as having straight sides of constant slope - although that is still a simplistic picture.

### Other chains

I decided to test this model on other chains. In particular, you asked about the Canary Islands and the Azores. For the Canary Islands, I found a reasonably poor fit ($R^2=0.0375$) for a best-fit power-law model of $H\propto A^{0.13}$. The distribution of heights for the seven main islands appears to be somewhat random, which is unhelpful. A possible reason for the discrepancy could be that the Hawaiian-Emperor Seamount Chain is composed largely of shield volcanoes, and this trend may not apply to other types, like stratovolcanoes.

The Azores were interest, yielding $H\propto A^{0.28}$, with $R^2=0.368$. If we remove an outlier, Pico Island, we get $H\propto A^{0.20}$, with $R^2=0.541$. This is better than the Canary Islands, but still not as good as Hawaii. I do notice that all three of these exponents are lower than Hawaii's, which is interesting. I suspect it's partially because of the lack of a clear power-law trend, from what I've seen.

### Notes

The Big Island of Hawaii does lower the power-law model a bit, but not by a huge amount (the exponent becomes $0.60$). It also lowers the proportionality constant somewhat.

• Is there a reason you left out most of the islands? the hawain chain has close to a hundred islands. – John Aug 21 '18 at 1:06
• @John I've only gotten to look at some of the data so far - more on the way. – HDE 226868 Aug 21 '18 at 1:06
• Still getting my vote a statistical approximation of the existing islands is about hte best possible answer.Although an error range would be a nice addition. – John Aug 21 '18 at 1:10
• @John I can get you the errors at some point in the future. I've looked through the other islands in the chain, and determined that the others are either atolls or underwater, meaning that they're not useful for the models here - in other words, they aren't surface volcanoes that we can use. – HDE 226868 Aug 21 '18 at 1:15
• Nice answer! I'll accept in 24 hrs if no other better answers appear. – JavaScriptCoder Aug 21 '18 at 1:17

A shield volcano is characterized by gentle upper slopes (about 5 degrees) with somewhat steeper lower slopes (about 10 degrees). The shield volcanoes are almost entirely composed of relatively thin lava flows built up over a central vent.

So, use those 5 and 10 degree slope angles and you should be able to generate realistic heights for any volcano you want based on basal diameter.

• that will ive you the base of the volcano not the island. – John Aug 21 '18 at 0:49
• Not... sure I understand the distinction you're making... if we're talking about the Hawaiian islands or anything like them, there isn't any difference. The island is the volcano, and vice versa. – Morris The Cat Aug 21 '18 at 0:54
• No the island is what is left of the volcano/s, The hawaiian chains are great examples most of the islands are made of more than one volcano, and all are in different stages of the island building process. Some are very far into the aroll stage, in which they may have virtually no height while still having the same area. – John Aug 21 '18 at 1:01
• Ahh, got it. I see what you're saying now. – Morris The Cat Aug 21 '18 at 1:03

I direct you to look at these references: https://en.wikipedia.org/wiki/Mount_Vesuvius, your better bet might be to look at this, since it was an actual island instead of Vesuvious. https://en.wikipedia.org/wiki/Krakatoa. Then there's the beautifully new island off the coast of Japan which is still growing, but I've had my eye on it since it first appeared some years ago. https://en.wikipedia.org/wiki/Nishinoshima_(Ogasawara) This one is of great interest to me particularly as judging by the placement of a former under water caldera, this one has the potential to actually be a permanent island unlike some newly formed volcanic islands. That and merging with an already existing island also contributes to it's chances of survival instead of merely falling back into the ocean as so many do. I'm neither a mathmatist nor a volcanologist so I can't give specific info. I hope this helps.

• Hi @VaniJoy, welcome to WB.SE! We hope you enjoy your stay here, and if you haven't already, take a look at the tour and the help center for our current policies and guidelines. Thanks for the info about this, however I was looking for an equation and HDE was able to generate one. – JavaScriptCoder Aug 23 '18 at 1:02