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Inversion 101 to 201 – Part 1: The forward problem

The Geodynamics 101 series serves to showcase the diversity of research topics and methods in the geodynamics community in an understandable manner. We welcome all researchers – PhD students to professors – to introduce their area of expertise in a lighthearted, entertaining manner and touch upon some of the outstanding questions and problems related to their fields. This time, Lars Gebraad, PhD student in seismology at ETH Zürich, Switzerland, talks about inversion and how it can be used in geodynamics! Due to the ambitious scope of this topic, we have a 3-part series on this with the next part scheduled for publication next week! This week, however, we start with the basics that many geodynamicists will be familiar with: the forward problem.

Inversion methods are at the core of many physical sciences, especially geosciences. The term itself is used mostly by geophysicists, but the techniques employed can be found throughout investigative sciences. Inversion allows us to infer our surroundings, the physical world, from observations. This is especially helpful in geosciences, where one often relies on observations of physical phenomena to infer properties of the (deep) otherwise inaccessible subsurface. Luckily, we have many, many algorithms at our disposal! (…cue computer scientists rejoicing in the distance.)

Inverse theory in a nutshell
Credit: xkcd.

The process of moving between data and physical parameters goes by different names in different fields. In geophysics we mostly use inversion or inference. Inversion is intricately related to optimization in control theory and some branches of machine learning. In this small series of blog posts, I will try to shed a light on how these methods work, and how they can be used in geophysics and in particular geodynamics! In the end, I will focus a little on Bayesian inference, as it is an increasingly popular method in geophysics with growing potential. Be warned though:

Math ahead!

(although as little as possible)

First, for people who are serious about learning more about inverse theory, I strongly suggest Albert Tarantola’s “Inverse Problem Theory and Model Parameter Estimation”. You can buy it or download it (for free!) somewhere around here: http://www.ipgp.fr/~tarantola/.

I will discuss the following topics in this blog series (roughly in order of appearance):

1. Direct inversion
2. Gradient descent methods
3. Grid searches
4. Regularization
5. Bayesian inference
6. Markov chain Monte Carlo methods

The forward ‘problem’

Rather backwards, I will start the first blog post with a component of inversion which is needed about halfway through an inversion algorithm. However, this piece determines the physics we are trying to investigate. Choosing the forward ‘problem’ or model is posing the question of how a physical system will evolve, typically in a deterministic setting. This can be any relationship. The forward problem is roughly the same as asking:

I start with these materials and those initial conditions, what will happen in this system?

In this post I will consider two examples of forward problems (one of which is chosen because I have applied it everyday recently. Hint: it is not the second example…)

1. The baking of a bread with known quantities of water and flour, but the mass of the end product is unknown. A recipe is available which predicts how much bread will be produced (let’s say under all ratios of flour to water).

2. Heat conduction through a volcanic zone where geological and thermal properties and base heat influx are known, but surface temperatures are unknown. A partial differential equation is available which predicts the surface temperature (heat diffusion, Laplace’s equation).

Both these systems are forward problems, as we have all the necessary ingredients to simulate the real world. Making a prediction from these ‘input’ variables is a forward model run, a forward simulation or simply a forward solution. These simulations are almost exclusively deterministic in nature, meaning that no randomness is present. Simulations that start exactly the same, will always yield the same, predictable result.

Modelling our bread

For the first system, the recipe might indicate that 500 grams of flour produce 800 grams of bread (leaving aside the water for a moment, making case 1a), and that the relationship is linear in flour. The forward model becomes:

flour * 800 / 500 = bread

This is a plot of that relationship:

Simple flour to bread relationship: a 1D forward model.
I think I am getting hungry now. Does anyone else have the same problem?

Of course, if we add way too much water to our nice bread recipe, we mess it up! So, the correct amount of water is also important for making our bread. Hence, our bread is influenced by both quantities, and the forward model is a function of 2 variables:

bread = g(flour, water)

Let’s call this case 1b. This relationship is depicted here:

Simple flour and water to bread relationship: a 2D forward model.
Yes, I am definitely getting hungry.

It is important to note that the distinction between these two cases creates two different physical systems, in terms of invertibility.

Modelling our volcanic zone (and other PDE’s)

Modelling a specific physical system is a more complicated task. Typically, physical phenomena can be described by set of (partial) differential equations. Other components that are required to predict the end result of a physical deterministic system are

1. The domain extent, suitable discretisation and parameter distribution in the area of investigation, and the relevant material properties (sometimes parameters) at every location in the medium
2. The boundary conditions, and when the system is time varying (transient), additional initial conditions
3. A simulation method

As an example, the relevant property that determines heat conduction is the heat conductivity at every location in the medium. The parameters control much of how the physical system behaves. The boundary and initial conditions, together with the simulation methods are very important parts of our forward model, but are not the focus of this blog post. Typically, simulation methods are numerical methods (e.g., finite differences, finite elements, etc.). The performance and accuracy of the numerical approximations are very important to inversion, as we’ll see later.
The result of the simulation of the volcanic heat diffusion is the temperature field throughout the medium. We might however be only interested in the temperature at real-world measurement points. We consider the following conceptual model:

Conceptual model of heat diffusion in a heterogeneous thermal conductivity model. Domain of interest is the black box. Under homogeneous heat influx, the deterministic simulation will result in a decreasing temperature in the three measurement points from left to right.

The resulting temperatures at the measurement locations will be different for each point, because they are influenced by the heterogeneous subsurface. Real-world observations would for example look something like this

‘Observations’ (made by me of course) from the volcanic model.

Now that you have a good grasp of the essence of the forward problem, I will discuss the inverse problem in the next blog post. Of course, the idea of inversion is to literally invert the forward problem, but if it were as easy as that, I wouldn’t be spending 3 blog posts on it, now would I?

CIDER summer school

And we’re back! After a refreshing holiday (or was it?), the EGU GD Blog Team is ready to provide you with amazing blog posts once more! Although holidays can be great, one thing that can be even more great is a good summer school. Yep, you heard that correctly! Let me convince you to apply for the CIDER Summer School program next year.

Let’s start with the basics. What the hell is CIDER? Well, CIDER stands for the Cooperative Institute for Dynamic Earth Research. One of it’s main focusses is the interdisciplinary training of early career scientists. To that end, they organise a summer school every year (usually in June/July) that lasts for 4 weeks.

4 weeks?!

Again, you heard that correctly. You are very good at listening!
The first two weeks of the summer school are dedicated to getting up to speed on the topic of the summer school by means of lectures, tutorials, a little field trip, etc. During the last two weeks you will work together in groups on a project of your choosing. The projects are determined during the first two weeks, when you figure out where the knowledge gaps are and you start making teams (no worries, nobody will be left out). You will come up with possible project topics yourself, so you can imagine that there can be quite some lobbying going on to make sure your team gets sufficient members to pursue your favourite project!

Together with your team of students and postdocs, you will confer with established experts in the field to make your project a success. After two weeks, you can probably show some reasonable first results during the final presentation in front of everyone.

If you want to continue working on your project with your team afterwards, you can even write a small proposal to CIDER to request some funding to meet up again and turn your project into a paper. Although they can’t reimburse intercontinental flights, it is still a pretty awesome opportunity!

The topic of the summer school changes every year and alternates between a ‘deep’ topic and a ‘shallow’ topic. I attended the CIDER 2017 summer school with the topic ‘Subduction zone structure and dynamics‘ – a shallow topic. This year (2018), the topic was ‘Relating Geophysical and Geochemical Heterogeneity in the Deep Earth‘ – clearly a deep topic. If you want to know more about this year’s summer school, our Blog Reporter Diogo wrote about it here. Students from all kinds of different disciplines are encouraged to apply: geology, geochemistry, seismology, geodynamics, mineral physics, etc. The more diversity the better, because you need to learn from each other!

More/actual reasons to apply

Now that we have all the details out of the way, I can properly start to convince you to apply! Did I already mention that the summer school is in an exotic place in California, USA? In 2017, the summer school was in Berkeley and this year it was in Santa Barbara. These locations are always fixed, with the ‘shallow’ topics being held in Berkeley, and the deep topics being held in Santa Barbara. Maybe this can act as your guide for finding out which kind of topic to ultimately pursue in your career.

Also, can you imagine? Four weeks, in beautiful, sunny California for ‘work’? Because, yes, it is work, technically, but it won’t feel like it. Actually, it’s kind of like being transported to one of those American high school / college movies. Does anyone else watch those? Nope, just me? Okay then. You will get the full American student experience, as you will sleep in an actual dorm with all your fellow students and go to the dining hall religiously for breakfast, lunch, and dinner each day and every day! Yes, also in the weekends, because it’s free and you’re a poor student! Minor side-effect is that you want be able to look at – let alone stomach – burgers, fries, pizzas, and hotdogs for at least a year, but it’s totally worth it for this all-American movie-like experience. Obviously, sharing a dorm with all your fellow students and complaining about the food will forge bonds that will last far longer than the duration of the summer school and you are guaranteed to have a lot of fun during the summer school also after the lectures.

Although the program is pretty packed, you will have free evenings (during which you might catch up on your actual work) and you will have some days off during the weekends. Of course, you can’t have all weekend days off, because it wouldn’t be a proper summer school experience if you don’t return completely exhausted, right? However, on your precious days off, you can go and explore beyond the campus and do some nice day trips to a nearby city or nature reserve. You can of course also use your free evenings and weekends to sample some of the night life of whatever Californian city you are staying in!

My CIDER 2017 experience

I thoroughly enjoyed my own CIDER experience in Berkeley, 2017. I learned loads of things about subduction zones and a lot of my knowledge was refreshed, specifically on geochemistry, mineral physics and geology. It was great fun to live on an American campus (I mean, I really did feel as if I’d stumbled into an American teen movie) and we did some pretty cool things besides the summer school! There was a lovely field trip to learn a bit more about rocks and it was also a great opportunity to see something of the landscape and enjoy incredible views over San Francisco. Of course, San Francisco itself was also visited during one of our days off and I finally saw the Golden Gate bridge up close and ate crab at Fisherman’s Wharf. Unforgettable experience. Best day of the summer school. I cannot recommend it enough! We also went out for dinner and drinks on occasion in the city centre of Berkeley and we even snuck in a visit to the musical ‘Monsoon Wedding’ at Berkely Rep.

After the summer school, our project group applied for funding to meet up again (I just couldn’t get enough of the American vibe) and lo and behold, we actually got the funding! So this spring, I found myself in Austin, Texas, to work on our project.

Howdy y’all!

It was pretty amazing to have an opportunity like that, and I can assure you that we also had lots of fun in Austin. I mean, it’s Texas, what did you expect? I was already over the moon by the fact that I had the possibility of spotting men wearing cowboy boots for real and not just for carnival!

All in all, I can thoroughly recommend the CIDER summer school as a great learning experience and opportunity for meeting fellow scientists interested in your topic of choice.

Next year, the topic will be ‘Volcanoes‘, so if you have any interest in that, be sure to apply! There is also always a one-day pre-AGU workshop, where you can get a little taste of the summer school, as the progress on the projects of the previous year is reported and lectures anticipating the coming topic are held.

So, are you going to apply to CIDER next year? I mean, who doesn’t lava volcanoes?!

Holiday recommendations – blog break summer 2018

Even dedicated workaholics such as the editors of your EGU GD Blog Team sometimes deserve a break! Let me clarify that by saying ‘an intentional break’ (because uploading every Wednesday is hard!). We will be ‘on holiday’ during August, so there won’t be any new blog posts then. But don’t worry: we will be back stronger than ever in September and we already have a lot of very good blog posts in the pipeline for you. To start the holidays properly and to get you in the holiday spirit as well, the EGU GD Blog Team shares their geodynamical holiday recommendations with you. Enjoy & relax!

Iris van Zelst – Edinburgh

Hutton’s Section with a very young me (in 2012) for scale

Go. To. Edinburgh. Seriously: Edinburgh is the place to be for anyone who has an affinity with the Earth sciences. In this beautiful, historic city, James Hutton – the founder of modern geology, who originated the idea of uniformitarianism – lived and died. Everywhere in the city you can find little reminders indicating this iconic scientist lived there. You could, for example, visit his grave, and hike to his geological section on Edinburgh’s Salisbury Crags. There are also little plaques spread around the city that mark significant James Hutton places and events. The city itself is also steeped in a mix of geology and history: Edinburgh Castle, situated on the impressive volcanic Castle Rock, boasts an 1100-year-old history and towers over the city. Directly across from the castle, connected by the charming Royal Mile is Holyrood Palace, where you can soak up even more history – Mary Queen of Scots lived here for a while. Nearby, there is Holyrood Park where you can find the group of hills that hosts Hutton’s Section and a 350 million year old volcano named Arthur’s Seat. Climb it when the weather is nice and you will have the most amazing view of Edinburgh. The whole park is perfect for day hikes and picknicks.
Even if you (or your travel buddy) are not that into Earth Sciences (or history), Edinburgh has plenty of other attractions. It is the perfect place for book and literature lovers with the large International Book Festival every August and a very rich literary history with iconic writers such as Walter Scott (Ivanhoe), Robert Louis Stevenson (Strange Case of Dr Jekyll and Mr Hyde; Treasure Island), Arthur Conan Doyle (Sherlock Holmes), and – more recently – J. K. Rowling (Harry Potter). Theatre fans will also love Edinburgh, particularly during August when it hosts the Edinburgh Festival Fringe – the largest arts festival in the world.
I totally should’ve booked a trip to Edinburgh this year… Learn from my mistakes and enjoy it in my stead!

The view of Edinburgh when you’re standing on top of Arthur’s Seat: a more than 300 million year old volcano. Pretty epic.
Picture by me in 2012 (also: proof that the weather can be good in Scotland!)

Luca Dal Zilio – Aeolian Islands

My recommendation? I vote for the Aeolian Islands! Smouldering volcanoes, bubbling mud baths and steaming fumaroles make these tiny islands north of Sicily a truly hot destination. This is the best place to practice the joys of “dolce far niente“: eat, sleep, and play. The Aeolian Arc is a volcanic structure, about 200 km long, located on the internal margin of the Calabrian-Peloritan Arc. The arc is formed by seven subaerial volcanic edifices (Alicudi, Filicudi, Salina, Lipari, Vulcano, Panarea, and Stromboli) and by several volcanic seamounts which roughly surround the Marsili Basin. The subduction-related volcanic activity showed the same eastward migration going from the Oligo-Miocene Sardinian Arc to the Pliocene Anchise-Ponza Arc and, at last, to the Pleistocene Aeolian Arc. My favourite island, Stromboli, is one of the few volcanoes on earth displaying continuous eruptive activity over a period longer than a few years or decades. I like Stromboli because it conforms perfectly to one’s childhood idea of a volcano, with its symmetrical, smoking silhouette rising from the sea. Most of this activity is of a very moderate size, consisting of brief and small bursts of glowing lava fragments to heights of rarely more than 150 m above the vents. Occasionally, there are periods of stronger, more continuous activity, with fountaining lasting several hours, violent ejection of blocks and large bombs, and, still more rarely, lava outflow. I can’t quite explain what made it so special to me. It may be because Stromboli itself is an island, and all the time during the hike I enjoyed splendid sea views (with a beer in my hand). It may be the all encompassing experience, where I could see, hear and literally feel the lava explosions. It was simply fantastic.

Credit: Flickr

Anne Glerum – Montenegro

In case you don’t make it to Montenegro/Serbia this summer, it’s fun in winter too. And yes, it’s fun in spring too – there’s snow, mountains and a younger me on a tiny sled. Photo courtesy of Cyriel de Grijs

My geo-holiday-destination: Montenegro!
A summer without beach-time is not a summer to me (already got one beach-day in this year, phew). Being Dutch, a proper holiday also requires some proper mountains – or hills at least. And no trip is complete without cultural and culinary highlights to explore.
Montenegro is a country that ticks all the boxes. Situated along the Adriatic Sea it hosts a score of picture-perfect beaches; quiet or taken over by the jet-set, intimate coves or long stretches of white sand, take your pick.
Further inland, you reach the Dinarides orogenic chain, the product of 150 My of contractional tectonics and later collapse during the Miocene. Traversing the chain into neighboring Serbia will lead you past complete ophiolite sequences, syn-orognic magma intrusions and major detachment zones of the extensional orogenic collapse.
Visit the centuries old fortified coastal cities of Budva or Kotor or one of the many churches and frescoed monasteries spread around the countryside. For more bodily sustenance, enjoy the fresh fish dishes, rich meats or the regional cheeses and yoghurts. Seasonal fruits are eaten for dessert or, even better, turned into wine and rakija. Ehm, why I am not going there again this year – this time in summer?

Not-so-sunny spring view from St. John’s fortress onto Kotor along the Bay of Kotor. Photo courtesy of Cyriel de Grijs

Diogo Lourenço – CIDER Summer School

This year, my favourite geodynamical destination is CIDER 2018! It’s far from holidays… but it’s really cool! For the last three weeks (one week to go), we have been intensely learning about heterogeneity in the Earth, and trying to understand it in an interdisciplinary perspective with contributions from geochemistry, geodynamics, and seismology. Quite an intense schedule and a lot of information to process, but I think we are all learning a lot, and hopefully in the future we will use more constraints coming from other fields into our own work. Oh, and did I mention that it is happening in Santa Barbara? Great Californian weather, beautiful coastal landscapes, barbecues by the beach, and swimming in the ocean, all sprinkled with scientific discussions! Quite the geodynamical destination, no?

Just had to cross the street from the KITP building where the conference is happening to take this photo…

Grace Shephard – Svalbard

Geoscientists are no strangers to travelling to exotic places and many of us take the opportunity to turn a work-related trip into potential holiday scouting. My suggested destination is most probably the northernmost point you can quite easily travel to on this planet – Svalbard.
Svalbard is an Arctic archipelago located around between 74-81°N latitude. It is sometimes confused with Spitsbergen, which is actually the name of the largest island where the main settlements, including Longyearbyen and Barentsburg, are situated. The islands are part of Norwegian sovereignty, though with some interesting taxation and military restrictions (the Svalbard Treaty of 1920 makes for some pretty interesting reading). Svalbard is host to a stream of tourists and scientific researchers year-round, and this week I will travel back to Longyearbyen as a lecturer for an Arctic tectonics, volcanism and geodynamics course at the University Centre in Svalbard (UNIS).
Geologically speaking, Svalbard makes for a very interesting destination. It offers a diverse range of rock ages and types; having experienced orogenic deformation events, widespread magmatism, and extensive sedimentary and glacial processes.
If you’re after a more usual tourist package amongst the draw cards are of course iconic polar bears (though please keep your distance), stumpy reindeer, arctic foxes, whales, birds and special flora. There are many glaciers – in fact around 60% of Svalbard is covered in ice – as well as fjords and mountains, former coal mining settlements… the list goes on. You are even spoilt for choice between midnight sun or midday darkness, depending on the time of year, so prioritise your activities wisely. Plus, did I mention those miles and miles of unvegetated, uninterrupted rock exposures to keep any geology enthusiast happy?… if you’re lucky you might come across some incredible fossil sites.

Itinerary recommendation, tried and tested: Whale watching and fjord cruising to a Russian mining ghost town (Pyramiden) followed by an important sampling of the world’s northernmost brewery.

Thirteen planets and counting

Apart from our own planet Earth, there are a lot of Peculiar Planets out there! In this series we take a look at a planetary body or system worthy of our geodynamic attention, and this week we move back to our own solar system. Many of us will clearly remember the downgrading of Pluto as a planet nearly 12 years ago to the month. In this informative and witty post, Laurent Montesi from the University of Maryland makes a case for reinstating Pluto of planetary status, plus a handful of others, or at least a review of definitions. Bring on Club Planet!

Laurent Montesi

A planet is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit. When Resolution 5A was passed by the International Astronomical Union (IAU) during the closing ceremony of its 2006 General Assembly, Pluto was “demoted” from the rank of the true planets to a dwarf planet. Children’s eyes filled with tears over the injustice made to “poor little Pluto”, textbooks were rewritten, and the Nine Pizzas that My Very Excellent Mother Just Served Us turned to Noodles.

I don’t really care.

I see where the IAU came from when crafting this definition, and to some extent, I agree with it. But it is not relevant to me. The thing is, I am not an astronomer. I recognize the authority of the IAU to name geological features on planets and other worlds, but I’m a geologist. Pluto, like many other solar system objects, has too much exciting geology to be ignored!

Figure 1: The five Dwarf Planets currently recognized by the IAU, proud members of Club Planet.

To me, a dwarf planet is first and foremost a planet, and what interests me in planets is their geological activity. If, as stated in part b of the IAU definition, an object is able to “overcome rigid body forces” (whatever that means), that should leave a geological trace. I don’t care if the planet cleared its neighbourhood or not.

So, I take the IAU definition as an invitation for Ceres, Pluto, Eris, Haumea, and Makemake to join the exclusive Club Planet (Figure 1). They all bring interesting geology to the Club. Look at the results of the Dawn and New Horizons missions! Ceres has mountains, fractures, oddly hexagonal craters, and a remarkable bright spot beckoning explorers to study its water-rich interior. Pluto has become a superstar of planetary exploration, with oceans of frozen nitrogen, diverse terrains, large rifts, perhaps a giant ice volcano, and the cutest heart tattoo in the solar system.

I’d like to go even further and open the door of the Club to many satellites (Figure 2). Our Moon is the doorway towards understanding the early evolution of terrestrial planets like the Earth. It taught us about giant impacts and magma oceans. If you want to find liquid water (and possibly life) today, go to Europa or Enceladus. If you are looking for alien plate tectonics, check out Ganymede and Europa. Are you searching for a thick atmosphere, rivers, and lakes? Welcome to Titan. Who has the most volcanic activity today? Please stand up, Io. What incredible rifts you have, Miranda and Charon! From a geological standpoint, satellites are as rich as any planet.

Figure 2: Knocking at the door of Club Planet are several of the satellites of the solar system: Earth’s Moon, the four Galilean Satellites Io, Europa, Ganymede, and Callisto, the large moons Titan and Triton, as well as numerous smaller, but geologically interesting satellites. They are led by Pluto’s moon Charon.

So, what actually is a planet? To the ancient Greeks, they were dots of light wandering against the rigid background of the night sky. These dots then turned out to be balls. Galileo saw four satellites around Jupiter, and in the redesigned solar system, planets could only orbit the Sun. Eventually, so many objects were found that it was decided that it mattered whether a planet “cleared their planetary neighbourhood” or not. Some objects were not enough of a bully to be regarded as a full planet, so they were called dwarfs. All along, astronomy guided our thinking about what is a planet and what is not.

Interestingly, the 2006 IAU definition merges astronomy and geophysics: what does it matter to an astronomer that the object has reached hydrostatic equilibrium? That is a geophysical criterion. Perhaps it matters in the sense that the interior is fluid enough that one should consider how dissipation influences orbital evolution. If that is the case, though, can tidal interaction with satellites be regarded separately?

I don’t know why the IAU was interested in hydrostatic equilibrium, or even if that is a valid question to consider, because, once again, I am not an astronomer. I’m a geologist. I study the geological activity and the interior evolution of… well… planets… and dwarf planets… and satellites… perhaps exoplanets one day… although not the ice giants and gas giants because, as far as I am concerned, they are different beasts altogether.

The fact is, the IAU definition does not help me. Perhaps there could be a geological definition of a planet, or whatever you want to call the various objects I am interested in. Perhaps the International Union of Geodesy and Geophysics (IUGG) — which, like the IAU, is a member of the International Science Council — could propose a definition more in line with my research interests, but as far as I know, there is no discussion of that.

In the meantime, resistance to the IAU definition is growing in our community. David Grinspoon and Alan Stern recently published a Perspective in The Washington Post1. Around twenty scientists got together to discuss a “Geophysical Planet Definition” at the start of the 2018 Lunar and Planetary Conference. One major point of agreement was that no one should feel obligated to follow the IAU’s definition (we are all rebels now), or any other definition.

At the 2017 Lunar and Planetary Conference, Kirby Runyon and coworkers proposed the following “Geophysical Planet Definition”2: A planet is a sub-stellar mass body that has never undergone nuclear fusion and that has sufficient self-gravitation to assume a spheroidal shape adequately described by a triaxial ellipsoid regardless of its orbital parameters. I find there is a lot to like with this proposal. For example, it would allow me to consider satellites as planets. If I focus on internal evolution, it doesn’t really matter what object my planet is orbiting. Of course, this influences the possibility of tidal heating, but I can regard that as an external energy flux, like the energy of accretion for impacts.

Interestingly, the draft “Geophysical Planet Definition” does not explicitly mention hydrostatic equilibrium. In the IAU definition, the hydrostatic equilibrium criterion implies that planets have a minimum size. It also assumes that the planet behaves as a fluid. In that case, what are we to do with the solid planets, like the Earth? We have evidence of frozen hydrostatic bulges, especially for the Moon. In other words, geological bodies can be strong enough to support a significant deviation from hydrostatic equilibrium. Hydrostatic equilibrium is not the best way to define a planet from a geological standpoint.

Figure 3: Ratio of relief scaled by planetary radius against mean radius based on best fitting triaxial ellipsoid for a variety of solar system objects, drawn following Melosh (2011). The maximum relief is controlled by friction for objects smaller than ~100 km in diameter and by strength for larger objects. Note that some objects like Mercury and Venus do not appear on this graph as they have no measurable flattening, due to their small rotation rate. Gas and ice giants appear to deviate from the trend of solid planets.

Where the IAU definition focuses on the driving force, it may instead be useful to focus on the strength of the planet. In his Planetary Surface Processes book, Jay Melosh discusses the relation between strength and gravity3. He concludes that for small bodies, relief (quantified as the difference between the maximum and minimum radius of an object, divided by the average radius) is independent of size, whereas it decreases inversely with the square of the average radius for larger solar system objects. In these larger objects, relief is limited by the strength of the body. The transition between these two trends is a planetary diameter of 200 to 400 km (Figure 3). This division leaves all of the objects for which we have evidence of geological activity driven by internal processes safely within the category of planets. Ancient planetesimals were probably big enough to be regarded as planets, and indeed, evidence for internal differentiation suggests that their interior was quite active.

So, in my view, a planet is simply a body large enough to have small relief as compared to its radius. This is evidence of relatively low internal strength, which allows geological activity to take place. I don’t need to consider where it orbits, and if it cleared its “planetary neighbourhood” or not, as that doesn’t affect geology. The pitfall of my very inclusive view of what is a planet is the consequentially large number of objects to consider, but variety is the spice of life. Why limit the diversity of geological activity to consider?

There can be subcategories, as Alan Stern actually advocated: gas giants, ice giants, terrestrial planets, dwarf planets, satellite planets, even exoplanets. From a geological standpoint, the ones I am least likely to study are actually the giant planets, whose activity is dominated by atmospheric processes. But feel free to consider them.

Perhaps I should leave the term “planets” to the astronomers, and advocate instead for a new term, “geological worlds”. What remains is, whichever classification you choose to adopt should be adapted to the research you do. For me, I want to embrace the geological diversity of our solar system.

```Further reading:

David Grinspoon and Alan Stern (2018), Yes, Pluto is planet, Speaking of Science – Perspective, Washington Post, May 7

Runyon, K.D., S.A. Stern, T.R. Lauer, W. Grundy, M.E. Summers, and K.N. Singer (2017), A Geophysical Planet Definition, Lunar and Planetary Science XLVIII, Abstract 1448

Jay Melosh (2011) Planetary Surface Processes, Chapter 3, ISBN 9780511977848

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