**The Geodynamics 101 series serves to showcase the diversity of research topics and/or methods in the geodynamics community in an understandable manner. In this week’s Geodynamics 101 post, Marcel Thielmann, Senior Researcher at the University of Bayreuth, discusses the possible mechanisms behind the ductile deformation at great depths that causes deep earthquakes. **

Earthquakes are one of the expressions of plate tectonics that everybody seems to be familiar with. When I started studying geophysics, people used to ask me what exactly I was studying. As soon as I mentioned earthquakes, I usually got a knowing nod and no further questions were asked (the same goes for volcanoes, but that’s a topic for another day).

Most earthquakes occur at the boundaries of tectonic plates, where rock breaks due to the forces originating from the plates’ relative movement. In 1928, Kiyoo Wadati discovered earthquakes that occurred at depths larger than 60 km, which were previously thought to be impossible. Today, we know that these earthquakes are not that extraordinary: about one out of four earthquakes observed on Earth occurs at depths larger than 60 km. At this depth, the pressure inside the Earth reaches values of about 3 GPa and more. Laboratory experiments have shown that at this pressure, rocks do not deform by breaking, but rather by ductile creep, like putty. This kind of deformation should not produce any earthquakes. So, 90 years after their discovery, the question still remains: **What causes deep earthquakes?**

As rocks get transported to larger depths, the minerals making them up can experience phase transformations. Due to these transformations, two things may happen: (1) Previously stored water in the minerals is released. This release may trigger earthquakes due to the released water acting against the pressure of the surrounding rock in a mechanism called *dehydration embrittlement (Green and Houston, 1995; Frohlich, 2006). *(2) The phase transition renders a fine-grained rock that is easier to deform. If enough of this weak material is produced, rock failure occurs in a process called *transformational faulting (Green and Houston,1995; Ferrand et al.,2017)*. Besides these two mechanisms, a third one called *thermal runaway *has been thrown into the ring *(Hobbs et al., 1986; Ogawa, 1987)*. This mechanism is a result of *shear heating*, which describes the generation of heat inside a deforming rock. If heat generation is faster than its transport, temperatures inside the rock will continue to increase and ultimately result in its destabilization, thus causing an earthquake.

While most of the observed deep earthquakes occur in *subduction zones*, where one tectonic plate descends beneath another, there are some that occur far from them. One such earthquake hit the Wind River range in Wyoming with a magnitude of M_{W} 4.7 in 2013 (*Frohlich et al., 2015; Craig and Heyburn, 2015*). This earthquake is not only enigmatic due to its depth of 75 km (making it the second deepest earthquake in such a stable continental region), but also because the Wind River area is considered to be “seismically quiet”. The location of the earthquake is far away from any plate boundary, with the closest tectonic feature being the Yellowstone supervolcano more than 200 km away. Since it occurred, the cause of this earthquake has been a matter of debate, with some scientists preferring a purely brittle origin (*Craig and Heyburn, 2015)*, while others argue for a ductile mechanism (*Prieto et al., 2017*).

*Dehydration embrittlement *seems to be an unlikely candidate, since the earthquake is located far away from any subduction zone. How could fluids get down to those depths if not by subduction? *Transformational faulting* also seems to be unlikely, since this would require a phase transition to take place. The Wind River earthquake occurred in the continental mantle lithosphere, where we would not expect any major phase transitions. *Thermal runaway* may be a candidate, but studies have shown that very high stresses are required to make this mechanism work, stresses that are very hard to achieve in the Earth.

However, there may be a way out: *grain size assisted thermal runaway.* Oh no, yet another one you might think. But fear not, this mechanism is essentially the same as the “classical” thermal runaway, just with the effect of small grains included. The consequences of this effect are by no means small however, as it significantly reduces the stresses required for thermal runaway. Indeed, numerical models of this process at the conditions of the Wind River earthquake indicate that it may indeed be a viable mechanism to have generated this earthquake (*Thielmann, 2018*). However, these models also show that rock deformation has to be sufficiently fast (about 100 times faster than what is commonly assumed) in order to allow for earthquake generation.

So now we have shifted the question from “How could fluids get down to those depths if not by subduction?” to “How could we deform that fast at those depths?” Here, seismology may come to the rescue: tomographic models of the north-western United States show that the Wind River earthquake lies directly at the transition between two regions with strongly varying seismic wave speeds (*Shen et al., 2013; Wang et al., 2016*). Fast wave speeds are commonly seen as an indicator for cold material, while slow wave speeds indicate warm material. 3D seismic tomographies such as the one from Shen et al. (2013) show that the 2013 Wind River earthquake occurred in a region where the continental lithosphere may be detaching in the form of a drip (*Wang et al., 2016*). In such tectonic environments, deformation rates may reach the values needed to initiate *grain size assisted thermal runaway* (*Lorinczi and Houseman, 2009*).

Does this now answer all questions we have on the Wind River earthquake and deep earthquakes in general? Certainly not. The example given above was just a single instance of where the combined information from seismology, laboratory experiments and numerical modelling may help us find an answer. We still have to keep in mind G.E.P. Box’s famous expression „Essentially, all models are wrong, but some are useful“. It is certain that deep earthquakes contain a wealth of information that remains to be unlocked. The following quote by Heidi Houston (*2015*) points the way:

Integration of seismological, laboratory, and modelling effort is needed to bridge the stubborn gap between source properties, which are extracted under strong assumptions and possess substantial intrinsic variability, and physical mechanisms of rupture generation, which are as yet neither well understood nor well constrained.(H. Houston)

References Craig, T. J., and R. Heyburn (2015), An enigmatic earthquake in the continental mantle lithosphere of stable North America,]]>Earth Plan. Sc. Lett.,425, 12–23, doi:10.1016/j.epsl.2015.05.048. Ferrand, T., N. Hilairet, S. Incel, D. Deldicque, L. Labrousse, J. Gasc, J. Renner, Y. Wang, H. W. Green II, and A. Schubnel (2017), Dehydration-driven stress transfer triggers intermediate-depth earthquakes,Nat. Commun.,8, 15247, doi:10.1038/ncomms15247. Frohlich, C. (2006),Deep Earthquakes, Cambridge University Press. Frohlich, C., W. Gan, and R. B. Herrmann (2015), Two Deep Earthquakes in Wyoming,Seismological Research Letters,86(3), 810–818, doi:10.1785/0220140197. Green, H. W., and H. Houston (1995), The Mechanics of Deep Earthquakes,Annu. Rev. Earth. Planet. Sci.,23, 169–213. Hobbs, B. E., A. Ord, and C. Teyssier (1986), Earthquakes in the Ductile Regime,Pure Appl. Geophys., 124(1-2), 309–336. Houston, H. (2015), 4.13 - Deep Earthquakes, inTreatise on Geophysics (Second Edition), edited by G. Schubert, pp. 329–354, Elsevier, Oxford. Lorinczi, P., and G. A. Houseman (2009), Lithospheric gravitational instability beneath the Southeast Carpathians,Tectonophysics,474, 322–336, doi:10.1016/j.tecto.2008.05.024. Ogawa, M. (1987), Shear instability in a viscoelastic material as the cause of deep focus earthquakes,J. Geophys. Res.,92, 13,801–13,810. Prieto, G. A., B. Froment, C. Yu, P. Poli, and R. Abercrombie (2017), Earthquake rupture below the brittle-ductile transition in continental lithospheric mantle,Sci. Adv.,3(3), e1602642, doi:10.1126/sciadv.1602642. Shen, W., M. H. Ritzwoller, and V. Schulte Pelkum (2013), A 3‐D model of the crust and uppermost mantle beneath the Central and Western US by joint inversion of receiver functions and surface wave dispersion,J. Geophys. Res.,118(1), 262–276, doi:10.1029/2012JB009602. Thielmann, M. (2018), Grain size assisted thermal runaway as a nucleation mechanism for continental mantle earthquakes: Impact of complex rheologies,Tectonophysics,746, 611–623, doi:10.1016/j.tecto.2017.08.038. Wang, X., D. Zhao, and J. Li (2016), The 2013 Wyoming upper mantle earthquakes: Tomography and tectonic implications,J. Geophys. Res.,121(9), 6797–6808, doi:10.1002/2016JB013118.

Earth’s mantle is the planet’s engine. The loss of heat from the interior to space drives Earth’s tectonic processes, mountain building and orogeny, volcanism, and the core dynamo generating Earth’s magnetic field. But perhaps less appreciated is that the mantle also plays a critical role in shaping the state of the atmosphere. This link between surface and interior evolution is not just important for studying the Earth, but also the other rocky planets in our solar system, and rocky exoplanets. Factors that make a planet, like Earth, a suitable home for life, such as the presence of liquid water oceans, weathering processes that provide critical nutrients to the oceans, and a temperate climate are all directly influenced by deep interior processes (Foley & Driscoll, 2016). Likewise, a complex interaction between life, atmospheric chemistry, weathering, volcanism, and sediment burial led to the rise of oxygen on Earth, which is both critical for some forms of life and a signature of the presence of life (Claire et al, 2006; Kump & Barley, 2007; Lyons et al, 2014). Thus, unravelling the factors that allowed Earth to develop into a planet teaming with life, whether those same factors are likely to be present on other rocky planets, and whether potential biosignatures, like atmospheric oxygen, are likely to arise on exoplanets if life is present, all require considering the role of the mantle.

The abundance of atmospheric gases is determined by the balance between their sources and sinks, and the mantle acts as an important source and sink for many gases: volcanism releases volatiles locked in rocks to the surface, while subduction brings volatiles from the surface back into the interior. One of the most critical for habitability is CO_{2}, which controls the climate state. On Earth, the cycling of CO_{2 }between surface, interior, and atmosphere involve a stabilizing feedback that acts to buffer climate (Kasting & Catling, 2003). CO_{2 }is drawn out of the atmosphere by weathering of silicate rocks and the formation of carbonate minerals on the seafloor, which are then subducted to return carbon to the mantle (Figure 2). Critically, the rate of silicate weathering increases with increasing surface temperature or atmospheric CO_{2}. Thus when the climate warms the weathering rate increases, acting to cool the climate down, and when the climate is cool the weathering rate decreases, allowing outgassing of CO_{2 }by volcanism to warm the climate up (Walker et al, 1981).

However, this feedback can fail in two ways: first, rates of CO_{2}outgassing must be high enough to keep the climate from plunging into a globally glaciated or snowball state (Kadoya & Tajika, 2014); second, there must be sufficiently high rates of physical erosion to remove weathered rock and bring fresh rock into the near-surface weathering zone (Foley, 2015). The mantle plays an important role in both CO_{2}outgassing and surface erosion rates. The CO_{2}outgassing rate is determined by the rate of volcanism, mantle carbon content, and oxidation state, while erosion rates are controlled by rates of tectonic uplift and mountain building over geologic timescales.

However, there are many aspects of how the mantle influences CO_{2 }outgassing and weathering rates that are still poorly understood, and exciting avenues of future research. First-order constraints on rates of volcanism and outgassing, and how they change over time, are straightforward to calculate from both simple box models of planetary thermal evolution or 2- and 3-D mantle convection calculations (Noack et al, 2017; Tosi et al, 2017). As planets cool over time, volcanic outgassing rates decline and eventually become low enough for frozen, snowball climates to develop. Factors that keep a planet’s mantle warmer for longer, such as higher rates of radiogenic heat production or tidal heating, will thus act to prolong the lifetime of habitable surface conditions (Foley & Smye, 2018; Valencia et al, 2018). Yet there are still important uncertainties, in particular on how carbon is carried into, and circulates within, the mantle that are key avenues for future research. Moreover, the connection between mantle dynamics, mountain building, erosion, and weathering rate is still poorly understood. Erosion rates are high when topographic gradients are large, as in mountainous regions. Mountain building is most likely connected to surface plate speed and the vigor of mantle convection, however just what form this connection takes is not known. How mantle convection and plate tectonics leads to the formation of topography, and hence influences weathering and erosion, is a critical area of research for understanding the controls on long-term climate evolution.

Ultimately one of the most important questions driving future research in planetary evolution and exoplanets, and which geodynamicists should be a central part of answering, is how Earth-like a planet needs to be in order to sustain volatile cycles that allow for the development of life and for biosignatures, such as oxygen, to accumulate in the atmosphere once life has developed (Tasker et al, 2017). Exoplanets come in a wide range of sizes (see Figure 3): planets up to about 4 Earth masses are found to be rocky, while beyond this limit planets are volatile-rich like Neptune (Rogers, 2015), and likely compositions as well (Hinkel & Unterborn, 2018). These planets could display a range of different surface tectonic modes, including plate tectonics, stagnant lids, or some intermediary style of tectonics. Oxidation states could be different, influencing the type of gases outgassed by volcanism. Instead of outgassing predominantly CO_{2}, a planet with a more reduced mantle could outgas mostly CO or CH_{4}. Likewise, different crustal compositions could alter weathering processes and the stability of volatiles as they are recycled into the interior at subduction zones or by crustal foundering. Exploring these issues will require interdisciplinary research including geochemists, mineral physicists, and geodynamicists, as well as biogeochemists, climate scientists, and astronomers. With future space telescopes poised to image exoplanet atmospheres, research on the role of the planetary interior in shaping the surface environment and atmosphere has never been so relevant.

Claire, M. W., Catling, D. C., & Zahnle, K. J. (2006). Biogeochemical modelling of the rise in atmospheric oxygen.Geobiology,4(4), 239-269. Foley, B. J., & Driscoll, P. E. (2016). Whole planet coupling between climate, mantle, and core: Implications for rocky planet evolution.Geochemistry, Geophysics, Geosystems,17(5), 1885-1914. Foley, B. J., & Smye, A. J. (2018). Carbon Cycling and Habitability of Earth-Sized Stagnant Lid Planets.Astrobiology,18(7), 873-896. Hinkel, N. R., & Unterborn, C. T. (2018). The Star–Planet Connection. I. Using Stellar Composition to Observationally Constrain Planetary Mineralogy for the 10 Closest Stars.The Astrophysical Journal,853(1), 83. Kadoya, S., & Tajika, E. (2014). Conditions for oceans on Earth-like planets orbiting within the habitable zone: importance of volcanic CO_{2}degassing.The Astrophysical Journal,790(2), 107. Kasting, J. F., & Catling, D. (2003). Evolution of a habitable planet.Annual Review of Astronomy and Astrophysics,41(1), 429-463. Kump, L. R., & Barley, M. E. (2007). Increased subaerial volcanism and the rise of atmospheric oxygen 2.5 billion years ago.Nature,448(7157), 1033. Lyons, T. W., Reinhard, C. T., & Planavsky, N. J. (2014). The rise of oxygen in Earth’s early ocean and atmosphere.Nature,506(7488), 307. Noack, L., Rivoldini, A., & Van Hoolst, T. (2017). Volcanism and outgassing of stagnant-lid planets: Implications for the habitable zone.Physics of the Earth and Planetary Interiors,269, 40-57. Rogers, L. A. (2015). Most 1.6 Earth-radius planets are not rocky.The Astrophysical Journal,801(1), 41. Tasker, E., Tan, J., Heng, K., Kane, S., Spiegel, D., Brasser, R., ... & Houser, C. (2017). The language of exoplanet ranking metrics needs to change.Nature astronomy,1, 0042. Tosi, Nicola, Mareike Godolt, Barbara Stracke, Thomas Ruedas, John Lee Grenfell, Dennis Höning, Athanasia Nikolaou, A-C. Plesa, Doris Breuer, and Tilman Spohn. "The habitability of a stagnant-lid Earth."Astronomy & Astrophysics605 (2017): A71. Valencia, D., Tan, V. Y. Y., & Zajac, Z. (2018). Habitability from Tidally Induced Tectonics.The Astrophysical Journal,857(2), 106. Walker, J. C., Hays, P. B., & Kasting, J. F. (1981). A negative feedback mechanism for the long‐term stabilization of Earth's surface temperature.Journal of Geophysical Research: Oceans,86(C10), 9776-9782.

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**In this week’s Geodynamics 101 post, Juliane Dannberg, Assistant Professor at the University of Florida, outlines the role of mantle melt generation and transport in geodynamics.**

Mantle melting and magma transport are important influences on the dynamics and chemical evolution of the Earth’s interior. All of Earth’s oceanic crust and depleted oceanic lithosphere is generated through melting at mid-ocean ridges, the main process to introduce chemical heterogeneity into the mantle. Mechanically, melt and fluids play an important role in the dynamics of plate boundaries like mid-ocean ridges and subduction zones, and in the formation of oceanic islands. Specifically, partially molten regions are weaker than solid rock, and because of that, mechanisms that localise melt often also localise deformation.

Many mantle convection models include the effects of melting and melt transport to a first order. They allow for the generation of oceanic crust and lithosphere when mantle material approaches the surface. This is typically done by using a thermodynamic model to determine where melting occurs, removing melt as soon as it it forms, and distributing it further up or at the surface of the model as basaltic crust. The residuum that is left behind becomes the depleted, harzburgitic lithosphere.

But considering the addition of melt, a low-viscosity fluid phase, in models of mantle dynamics is not only important for geochemical considerations. It also introduces new length and time scales and new dynamics. Rather than talking about numerical methods, my goal for this article is to give a short overview of the physics of magma dynamics, and to build intuition on how magma generation and transport can affect geodynamic processes.

In mantle convection models, the driving force, buoyancy, is usually balanced by viscous stresses.

Consequently, the length scales of convective features are usually determined by how fast boundary layers can grow, by the density differences causing material to be buoyant, and by the mantle viscosity. When fluids are added to this system, new forces become important^{1}:

(1) **Darcy drag**. Melt can segregate from where it forms by flowing through the pore spaces of the solid host rock. We can think of the system as a sponge-like solid rock matrix that can be squeezed and stretched, saturated with liquid melt. As melt starts to flow, the traction on the interface between the melt and the mineral grains opposes this motion. The more pore space there is, the better it is connected, and the lower the viscosity of the melt, the easier it becomes for the for the melt to flow through.

(2) **Viscous compaction**. When melt is generated, or when melt is about to flow into a space that had no melt before, the pore space of the solid rock has to dilate so that melt can flow in. In other words, magma is pushing the mineral grains of the rock apart. Conversely, when melt flows out of an existing pore space, it pulls the grains together and the solid rock needs to compact to fill this volume. This means that the forces that drive the flow of melt have to overcome the resistance of the solid rock to volumetric deformation, governed by the host rock’s compaction viscosity.

So instead of the Stokes equation used in mantle convection models, magma dynamics is described by a more complicated force balance with additional terms (McKenzie, 1984; Scott & Stevenson, 1984,1986; Bercovici, Ricard et al., 2001, 2003; Sramek et al, 2007):

This equation includes a number of material properties that control magma dynamics. The *melt* *viscosity* η_{melt} and the *permeability* k express how easy it is for melt to flow through the pore space of the rock. The larger the pore space, and the better the connection between the pores, the larger the permeability. Because the rock is saturated with melt, the amount of pore space, or *porosity* ϕ, is equivalent to the fraction of melt. Consequently, the more melt is present, the larger the permeability. Δv is the difference between solid velocity and melt velocity and can be expressed using Darcy’s law:

In words, melt segregation relative to the motion of the solid is driven by differences in the *melt* or *fluid pressure* p_{f} relative to the hydrostatic pressure in the melt, ρ_{melt} **g**.

The viscous stresses are controlled by the *shear viscosity* η and the *compaction viscosity* ξ. Both properties depend on the melt fraction: The more melt is present (or, the larger the pore space of the rock), the easier it becomes to deform the solid rock matrix. This dependence is quite strong, and as soon as the porosity reaches 20–30% (the *disaggregation threshold*), the solid rock will break apart and will no longer form a connected matrix. Instead, the mineral grains will be suspended within the melt. Buoyancy is controlled by the density difference between solid and melt Δρ and the gravity **g**.

Depending on the importance of each of these forces in a given setting, the transport of melt can occur in different ways (Sramek et al., 2007). Buoyancy is caused by the lower density of the melt compared to the solid rock, so the more melt is present, the stronger the buoyancy forces. If viscous compaction forces are small (which corresponds to rocks that are easy to deform), buoyancy is balanced by Darcy drag (the so-called *Darcy equilibrium*; Sramek et al., 2007). In this case, melt ascends pervasively through the pores of the host rock, and its segregation velocity is limited by the rock’s permeability. Accordingly, the more melt is present, the faster it can ascend. This mode of melt transport may be representative of melting zones below mid-ocean ridges, where the amount of melt present in the pores of the host rock is small.

Conversely, if Darcy drag is small, then buoyancy is balanced by viscous compaction (the so-called *viscogravitational equilibrium*; Sramek et al., 2007). Melt can only flow if the solid rock deforms, and in this case the viscosity of the rock limits how fast melt can rise. If the compaction viscosity is comparable to or smaller than the shear viscosity, the host rock expands and melt can flow through the pores (Scott, 1988). But if the bulk viscosity is much larger than the shear viscosity, melt is forced to stay in the pocket of host rock it was created in over a long time scale. Instead of segregating, it will ascend together with the solid in form of a diapir (Scott, 1988). Because the Darcy drag becomes smaller the more melt is present (corresponding to a larger permeability), this mode of melt transport may occur near the top of plume heads (Fig. 1; Dannberg & Heister, 2016). Within the plume, high temperatures reduce the viscosity, but above the plume head, ambient mantle viscosities are larger and may limit melt segregation.

The equations also introduce a new length scale that controls the size of features that emerge in magma dynamics. This scale is controlled by the ratio of viscous forces and Darcy forces:

and is called the *compaction length*. It controls how far dynamic pressure differences between melt and solid, the compaction pressures, are transferred in partially molten rocks. This is important for the buoyant ascent of melt, which is hindered by the viscous resistance of the solid rock matrix to this compaction (Spiegelman 1993a, Spiegelman 1993b, Katz, 2015).

Variations in porosity in the direction of gravity cause disturbances in the compaction pressure (Fig. 2): For example, if the porosity decreases in the direction of flow, the corresponding decrease in permeability makes it more difficult for melt to flow through and causes a decreasing flux of melt upwards into the low-porosity region. This negative gradient in melt flux leads to melt pressures being larger than solid pressures (a positive compaction pressure) at the location of the porosity perturbation, pushing the grains of the solid rock apart. This means that more melt can flow into this region and porosity increases. At the upper end of this perturbation, this same process continues to act, drawing more melt in.

At the lower end of the high-porosity region, the opposite happens: Because permeability increases in the direction of flow, the melt flux increases in upwards direction, causing a negative compaction pressure. Mineral grains are pulled together, and melt is expelled. Because the process continues in upwards and downwards direction, variations in the amount of melt will develop into magmatic waves (or *solitary waves*). At the front of the wave, the positive compaction pressure draws in more melt, and at the back of the wave, the negative compaction pressure pulls the mineral grains together as melt flows out. The length scale of these waves is on the order of the compaction length. In one dimension, this process can also be illustrated by the flow of fluid through a viscously deformable pipe (Scott, 1988). Imagine a part of the pipe that locally has a larger radius than the rest of the pipe and that moves upwards. To accommodate the arrival of more liquid, the pipe has to expand in front of the perturbation. In the same way, the pipe contracts behind the perturbation, reverting to its original radius.

There are many more mechanisms that can cause melt to localise. When partially molten rock is sheared, melt is drawn into thin, melt-rich bands in between larger melt-poor regions. Channels of high porosity can form when reactive melting is driven by the flux of magma along a solubility gradient. A good overview over these processes and sources of further information is given in the lectures notes by Katz, 2015.

All of these processes may be important for the flow of melt in subduction zones, at mid-ocean ridges, and for hotspot volcanism. Therefore I believe it is essential to better understand magma dynamics if we want to answer questions such as: How much of the mantle melt reaches the surface? What is the reason for the location and spacing of volcanoes? How are plate boundaries generated and maintained? I hope this article helps to understand this topic better and inspires you to consider melt generation and transport in this bigger context.

^{1}These are the two most important forces in the mantle, but there are other forces that may impact deformation of partially molten rock. I have made a number of assumptions here, for example that the melt viscosity is much smaller than the solid velocity, that surface tension is negligible, and that deformation of the solid rock is predominantly viscous.

References: Bercovici, D. et al., 2001. A two‐phase model for compaction and damage: 1. General theory. J. Geophys. Res. Solid Earth, 106(B5), pp.8887-8906. Bercovici, D. and Ricard, Y., 2003. Energetics of a two-phase model of lithospheric damage, shear localization and plate-boundary formation. Geophys. J. Int., 152(3), pp.581-596. Dannberg, J. and Heister, T., 2016. Compressible magma/mantle dynamics: 3-D, adaptive simulations in ASPECT. Geophys. J. Int., 207(3), pp.1343-1366. Dannberg, J. et al., 2019. A new formulation for coupled magma/mantle dynamics. Geophys. J. Int., 219(1), pp.94-107. Katz, R.F., 2015. An introduction to coupled magma/mantle dynamics. http://foalab.earth.ox.ac.uk/files/IntroMagmaLectures.pdf McKenzie, D., 1984. The generation and compaction of partially molten rock. J. Petrol., 25(3), pp.713-765. Scott, D.R. and Stevenson, D.J., 1984. Magma solitons. Geophys. Res. Lett., 11(11), pp.1161-1164. Scott, D.R. and Stevenson, D.J., 1986. Magma ascent by porous flow. J. Geophys. Res. Solid Earth, 91(B9), pp.9283-9296. Scott, D.R., 1988. The competition between percolation and circulation in a deformable porous medium. J. Geophys. Res. Solid Earth, 93(B6), pp.6451-6462. Spiegelman, M., 1993a. Flow in deformable porous media. Part 1 Simple analysis. J. Fluid Mech., 247, pp.17-38. Spiegelman, M., 1993b. Flow in deformable porous media. Part 2 numerical analysis–the relationship between shock waves and solitary waves. J. Fluid Mech., 247, pp.39-63. Šrámek, O. et al., 2007. Simultaneous melting and compaction in deformable two-phase media. Geophys. J. Int., 168(3), pp.964-982.]]>

Why not share your knowledge on correct code management and version control with the community? Maybe you could spark a discussion about disabilities or gender equality in geosciences. Or maybe you and your colleagues can devise the perfect session for your niche research topic, such that finding a session won’t be a problem any more! Don’t hesitate: anyone can do it!

What? You don’t believe me? Well, we have been hosting the Geodynamics 101 short courses (inspired by our blog posts) for 2 years now and they have been a great success! And trust me, I didn’t know anything about organising short courses at EGU either before I embarked on this journey. Turns out: it’s pretty straightforward.

For a little transparency and insight into organising the courses, here is a little summary of the feedback from the two courses we hosted at EGU 2019:

• Geodynamics 101A: Numerical methods

• Geodynamics 101B: Large-scale dynamic processes

And we’ll also discuss our plans for the short course next year!

We decided to have 2 short courses at EGU 2019, because of the feedback we got at EGU 2018 about people desiring more info on the applications of numerical methods in geodynamics. Your wish is our command, so we delivered. Both courses were attended well, but the first course (about numerical methods) was more popular by far with people queueing and standing even outside of the room. So, we will ask for a bigger room at EGU 2020.

People from many different disciplines attended the courses. Surprisingly a large amount of geodynamicists were present! Most of them (you?) attended to get a quick refresher in (the broad area) of geodynamics and to see how you can do geodynamics outreach. Our course was particularly popular with geologists, and there were also some hydrologists, meteorologists, and computer scientists attending!

People gave the course a high rating with an average grade of 4.4/5 for Geodynamics 101A and Geodynamics 101B. The highest ratings were given by the geodynamicists attending the short courses. Hm… maybe because you are our friends?

The main feedback was that people would like to know how to actually run codes and see some more hands-on examples. So, next year, we will be back with a new and improved version of the “Geodynamics 101: Numerical methods” course which will include some demos of running a code.

Maybe I will see you there and who knows…? Maybe I will be at your session or short course!

]]>**“The present is the key to the past” is a oft-used phrase in the context of understanding our planet’s complex evolution. But this perspective can also be flipped, reflected, and reframed. In this Geodynamics 101 post, Lorenzo Colli, Research Assistant Professor at the University of Houston, USA, showcases some of the recent advances in modelling mantle convection. **

Mantle convection is the fundamental process that drives a large part of the geologic activity at the Earth’s surface. Indeed, mantle convection can be framed as a *dynamical* theory that complements and expands the *kinematic* theory of plate tectonics: on the one hand it aims to describe and quantify the forces that cause tectonic processes; on the other, it provides an explanation for features – such as hotspot volcanism, chains of seamounts, large igneous provinces and anomalous non-isostatic topography – that aren’t accounted for by plate tectonics.

*Mantle convection is both very simple and very complicated*. In its essence, it is simply thermal convection: hot (and lighter) material goes up, cold (and denser) material goes down. We can describe thermal convection using classical equations of fluid dynamics, which are based on well-founded physical principles: the continuity equation enforces conservation of mass; the Navier-Stokes equation deals with conservation of momentum; and the heat equation embodies conservation of energy. Moreover, given the extremely large viscosity of the Earth’s mantle and the low rates of deformation, inertia and turbulence are utterly negligible and the Navier-Stokes equation can be simplified accordingly. One incredible consequence is that the flow field only depends on an instantaneous force balance, not on its past states, and it is thus time reversible. And when I say incredible, I really mean it: it looks like a magic trick. Check it out yourself.

This is as simple as it gets, in the sense that from here onward every additional aspect of mantle convection results in a more complex system: 3D variations in rheology and composition; phase transitions, melting and, more generally, the thermodynamics of mantle minerals; the feedbacks between deep Earth dynamics and surface processes. Each of these additional aspects results in a system that is harder and costlier to solve numerically, so much so that numerical models need to compromise, including some but excluding others, or giving up dimensionality, domain size or the ability to advance in time. More importantly, most of these aspects are so-called *subgrid-scale processes*: they deal with the macroscopic effect of some microscopic process that cannot be modelled at the same scale as the macroscopic flow and is too costly to model at the appropriate scale. Consequently, it needs to be parametrized. To make matters worse, some of these microscopic processes are not understood sufficiently well to begin with: the parametrizations are not formally derived from first-principle physics but are long-range extrapolations of semi-empirical laws. The end result is that it is possible to generate more complex – thus, in this regard, more Earth-like – models of mantle convection at the cost of an increase in tunable parameters. *But what parameters give a truly better model? How can we test it?*

Meteorologists face similar issues with their models of atmospheric circulation. For example, processes related to turbulence, clouds and rainfall need to be parametrized. Early weather forecast models were… less than ideal. But meteorologists can compare every day their model predictions with what actually occurs, thus objectively and quantitatively assessing what works and what doesn’t. As a result, during the last 40 years weather predictions have improved steadily (Bauer et al., 2015). Current models are better at using available information (what is technically called data assimilation; more on this later) and have parametrizations that better represent the physics of the underlying processes.

We could do the same, *in theory*. We can initialize a mantle convection model with some *best estimate* for the present-day state of the Earth’s mantle and let it run forward into the future, with the explicit aim of forecasting its future evolution. But mantle convection evolves over millions of years instead of days, thus making future predictions impractical. Another option would be to initialize a mantle convection model in the distant past and run it forward, thus making predictions-in-the-past. But in this case we *really* don’t know the state of the mantle in the past. And as mantle convection is a chaotic process, even a small error in the initial condition quickly grows into a completely different model trajectory (Bello et al., 2014). One can mitigate this chaotic divergence by using data assimilation and imposing surface velocities as reconstructed by a kinematic model of past plate motions (Bunge et al., 1998), which indeed tends to bring the modelled evolution closer to the true one (Colli et al., 2015). But it would take hundreds of millions of years of error-free plate motions to eliminate the influence of the unknown initial condition.

As I mentioned before, the flow field is time reversible, so one can try to start from the present-day state and integrate the governing equations backward in time. But while the flow field is time reversible, the temperature field is not. Heat diffusion is physically irreversible and mathematically unstable when solved back in time. Plainly said, the temperature field blows up. Heat diffusion needs to be turned off [1], thus keeping only heat advection. This approach, aptly called *backward advection* (Steinberger and O’Connell, 1997), is limited to only a few tens of millions of years in the past (Conrad and Gurnis, 2003; Moucha and Forte, 2011): the errors induced by neglecting heat diffusion add up and the recovered “initial condition”, when integrated forward in time (or should I say, *back to the future*), doesn’t land back at the desired present-day state, following instead a divergent trajectory.

As all the simple approaches turn out to be either unfeasible or unsatisfactory, we need to turn our attention to more sophisticated ones. One option is to be more clever about data assimilation, for example using a *Kalman filter* (Bocher et al., 2016; 2018). This methodology allow for the combining of the physics of the system, as embodied by the numerical model, with observational data, while at the same time taking into account their relative uncertainties. A different approach is given by posing a formal inverse problem aimed at finding the “optimal” initial condition that evolves into the known (best-estimate) present-day state of the mantle. This inverse problem can be solved using the *adjoint method* (Bunge et al., 2003; Liu and Gurnis, 2008), a rather elegant mathematical technique that exploits the physics of the system to compute the sensitivity of the final condition to variations in the initial condition. Both methodologies are computationally very expensive. Like, many millions of CPU-hours expensive. But they allow for *explicit predictions of the past history of mantle flow* (Spasojevic & Gurnis, 2012; Colli et al., 2018), which can then be compared with evidence of past flow states as preserved by the geologic record, for example in the form of regional- and continental-scale unconformities (Friedrich et al., 2018) and planation surfaces (Guillocheau et al., 2018). The past history of the Earth thus holds the key to significantly advance our understanding of mantle dynamics by allowing us to test and improve our models of mantle convection.

1.Or even to be reversed in sign, to make the time-reversed heat equation unconditionally stable.

]]>**Subduction zones are ubiquitous features on Earth, and an integral part of plate tectonics. They are known to have a very important role in modulating climate on Earth, and are believed to have played an essential part in making the Earth’s surface habitable, a role that extends to present-day. This week, Antoniette Greta Grima writes about the ongoing debate on how subduction zones form and persist for millions of years, consuming oceanic lithosphere and transporting water and other volatiles to the Earth’s mantle.**

Before we can start thinking about how subduction zones form, we need to be clear on what we mean by the term **subduction zone**. In the most generic sense, this term has been described by White et al. (1970) as “an abruptly descending or formerly descended elongate body of lithosphere, together with an existing envelope of plate deformation”. In simple words this defines subduction zones as places where pieces of the Earth’s lithosphere bend downwards into the Earth’s interior. This definition however, does not take into account the spatio-temporal aspect of subduction zone formation. It also, does not differentiate between temporary, episodic lithosphere ‘peeling’ or ‘drips’, thought to precede the modern-day ocean plate-tectonic regime (see van Hunen and Moyen, 2012; Crameri et al., 2018; Foley, 2018, and references therein) and the rigid self-sustaining subduction, which we see on the present-day Earth (Gurnis et al., 2004).

A **self-sustaining subduction zone** is one where the total buried, **rigid slab length** extends deep into the **upper mantle** and is accompanied at the surface by **back-arc spreading** (Gurnis et al., 2004). The latter is an important surface observable indicating that the slab has overcome the resistive forces impeding its subduction and is falling quasi-vertically through the mantle. Gurnis et al. (2004) go on to say that if one or the other of these defining criteria is missing then subduction is forced rather than self-sustaining. Forced or induced subduction (Stern, 2004; Leng and Gurnis, 2011; Stern and Gerya, 2017; Baes et al., 2018) is described by Gurnis et al. (2004) as a juvenile, early stage system, that cannot be described as a fully fledged subduction zone. These forced subduction zones are characterised by incipient margins, short trench-arc distance, narrow trenches and a volcanically inactive island arc and/or trench. Furthermore, although these juvenile systems might be seismically active they will lack a well defined Benioff Zone. Examples of forced subduction include the Puysegur-Fiordland subduction along the Macquarie Ridge Complex, the Mussau Trench on the eastern boundary of the Caroline plate and the Yap Trench south of the Marianas, amongst others. On the other hand, Cenozoic (<66 Ma) subductions, shown in figure 2, are by this definition self-sustaining and mature subduction zones. These subduction zones including the Izu-Bonin-Mariana, Tonga-Kermadec and Aleutians subduction zones, are characterised by their extensive and well defined trenches (see figure 2) (Gurnis et al., 2004). However, despite their common categorization subduction zones can originate through various mechanisms and from very different tectonic settings.

We know from the geological record that the formation of subduction zones is an ongoing process, with nearly half of the present day active subduction zones initiating during the Cenozoic (<66 Ma) (see Gurnis et al., 2004; Dymkova and Gerya, 2013; Crameri et al., 2018, and references therein). However, it is less clear how subduction zones originate, nucleate and propagate to pristine oceanic basins.

Crameri et al. (2018, and references therein) list a number of mechanisms, some which are shown in figure 3, that may work together to weaken and break the lithosphere including:

- Meteorite impact
- Sediment loading
- Major episode of delamination
- Small scale convection in the sub-lithospheric mantle
- Interaction of thermo-chemical plume with the overlying lithosphere
- Plate bending via surface topographic variations
- Addition of water or melt to the lithosphere
- Pre-existing transform fault or oceanic plateau
- Shear heating
- Grain size reduction

Some of these mechanisms, particularly those listed at the beginning of the list are more appropriate to early Earth conditions while others, such as inherited weaknesses or fracture zones, transform faults and extinct spreading ridges are considered to be prime tectonic settings for subduction zone formation in the Cenozoic (<66 Ma) (Gurnis et al., 2004). As the oceanic lithosphere grows denser with age, it develops heterogeneity which facilitates its sinking into the mantle to form new subduction zones. However, it is important to keep in mind that without inherited, pre-existing weaknesses, it is extremely difficult to form subduction zones at passive margins. This is because as the oceanic lithosphere cools and becomes denser, it also becomes stronger and therefore harder to bend into the mantle that underlies it (Gurnis et al., 2004; Duarte et al., 2016). Gurnis et al. (2004) note that the formation of new subduction zones alters the force balance on the plate and suggest that the strength of the lithosphere during bending is potentially the largest resisting component in the development of new subduction zones. Once that resistance to bending is overcome, either through the negative buoyancy of the subducting plate and/or through the tectonic forces acting on it, a shear zone extending through the plate develops (Gurnis et al., 2004; Leng and Gurnis, 2011). This eventually leads to plate failure and subduction zone formation.

From our knowledge of the geological record, observations of on-going subduction, and numerical modelling (Baes et al., 2011; Leng and Gurnis, 2011; Baes et al., 2018; Beaussier et al., 2018) we think that subduction zone initiation primarily occurs through the following:

An intra-oceanic setting refers to a subduction zone forming right within the oceanic plate itself. Proposed weakening mechanisms include weakening of the lithosphere due to melt and/or hydration (e.g., Crameri et al., 2018; Foley, 2018, and references therein), localised lithospheric shear heating (Thielmann and Kaus, 2012) and density variations within oceanic plate due to age heterogeneities, where its older and denser portions flounder and sink (Duarte et al., 2016). Another mechanism proposed by Baes et al. (2018) suggests that intra-oceanic subduction can also be induced by mantle suction flow. These authors suggest that mantle suction flow stemming from either slab remnants and/or from slabs of active subduction zones can act on pre-existing zones of weakness, such as STEP (subduction-transfer edge propagate) faults to trigger a new subduction zone, thus facilitating spontaneous subduction initiation (e.g. figure 3) (Stern, 2004). The Sandwich and the Tonga-Kermadec subduction zones are often cited as prime examples of intra-oceanic subduction zone formation due to mantle suction forces (Baes et al., 2018). Ueda et al. (2008) and Gerya et al. (2015) also suggest that thermochemical plumes can break the lithosphere and initiate self-sustaining subduction, provided that the overlying lithosphere is weakened through the presence of volatiles and melt (e.g. figure 3). This mechanism can explain the Venusian corona and could have facilitated the initation of plate tectonics on Earth (Ueda et al., 2008; Gerya et al., 2015). Similarly Burov and Cloetingh (2010) suggest that in the absence of plate tectonics, mantle lithospheric interaction through plume-like instabilities, can induce spontaneous downwelling of both continental and oceanic lithosphere.

Subduction polarity reversal describes a process where the trench jumps from the subducting plate to the overriding one, flipping its polarity in the process (see figure 3). This can result from the arrival at the trench of continental lithosphere (McKenzie, 1969) or young positively buoyant lithosphere (Crameri and Tackley, 2015). Subduction polarity reversal is often invoked to explain and justify the two juxtaposed Wadati-Benioff zones and their opposite polarities, in the Solomon Island Region (Cooper and Taylor, 1985). Indications of a polarity reversal are also exhibited below the Alpine and Apennine Belts (Vignaroli et al., 2008). Furthermore, Crameri and Tackley (2014) also suggest that the continental connection between South America and the Antarctic peninsula has been severed through a subduction polarity reversal, resulting in the lateral detachment of the South Sandwich subduction zone.

Subduction zones can also initiate at ancient, inherited zones of weakness such as old fracture zones, transform faults, extinct subduction boundaries and extinct spreading ridges (Gurnis et al., 2004). Gurnis et al. (2004) suggest that the Izu-Bonin-Mariana subduction zone initiated at a fracture zone, while the Tonga-Kermadec subduction initiated at an extinct subduction boundary. The same study also proposes that the incipient Puysegur-Fiordland subduction zone nucleated at an extinct spreading centre.

In conclusion, we can say that subduction zone formation is a complex and multi layered process that can stem from a variety of tectonic settings. However, it is clear that our planet’s current convection style, mode of surface recycling and its ability to sustain life are interlinked with subduction zone formation. Therefore, to understand better how subduction zones form is to better understand what makes the Earth the planet it is today.

]]>References:Baes, M., Govers, R., and Wortel, R. (2011). Subduction initiation along the inherited weakness zone at the edge of a slab: Insights from numerical models. Geophysical Journal International, 184(3):991–1008. Baes, M., Sobolev, S. V., and Quinteros, J. (2018). Subduction initiation in mid-ocean induced by mantle suction flow. Geophysical Journal International, 215(3):1515–1522. Beaussier, S. J., Gerya, T. V., and Burg, J.-p. (2018). 3D numerical modelling of the Wilson cycle: structural inheritance of alternating subduction polarity. Fifty years of the Wilson Cycle concept in plate tectonics, page First published online. Burov, E. and Cloetingh, S. (2010). Plume-like upper mantle instabilities drive subduction initiation. Geophys. Res. Lett., 37(3). Cooper, P. and Taylor, B. (1985). Polarity reversal in the Solomon Island Arc. Nature, 313(6003):47–48. Crameri, F., Conrad, C. P., Mont ́esi, L., and Lithgow-Bertelloni, C. R. (2018). The dynamic life of an oceanic plate. Tectonophysics. Crameri, F. and Tackley, P. J. (2014). Spontaneous development of arcuate single-sided subduction in global 3-D mantle convection models with a free surface. Journal of Geophysical Research: Solid Earth, 119(7):5921–5942. Crameri, F. and Tackley, P. J. (2015). Parameters controlling dynamically self-consistent plate tectonics and single-sided subduction in global models of mantle convection. Journal of Geophysical Research: Solid Earth, 3(55):1–27. Duarte, J. C., Schellart, W. P., and Rosas, F. M. (2016). The future of Earth’s oceans: Consequences of subduction initiation in the Atlantic and implications for supercontinent formation. Geological Magazine, 155(1):45–58. Dymkova, D. and Gerya, T. (2013). Porous fluid flow enables oceanic subduction initiation on Earth. Geophysical Research Letters, 40(21):5671–5676. Foley, B. J. (2018). The dependence of planetary tectonics on mantle thermal state : applications to early Earth evolution. 376. Gerya, T. V., Stern, R. J., Baes, M., Sobolev, S. V., and Whattam, S. A. (2015). Plate tectonics on the Earth triggered by plume-induced subduction initiation. Nature, 527(7577):221–225. Gurnis, M., Hall, C., and Lavier, L. (2004). Evolving force balance during incipient subduction. Geochemistry, Geophysics, Geosystems, 5(7). Leng, W. and Gurnis, M. (2011). Dynamics of subduction initiation with different evolutionary pathways. Geochemistry, Geophysics, Geosystems, 12(12). McKenzie, D. P. (1969). Speculations on the Consequences and Causes of Plate Motions. Geophys. J. R. Astron. Soc., 18(1):1–32. Stern, R. J. (2004). Subduction initiation: spontaneous and induced. Earth and Planetary Science Letters, 226(3-4):275–292. Stern, R. J. and Gerya, T. (2017). Subduction initiation in nature and models: A review. Tectonophysics. Thielmann, M. and Kaus, B. J. (2012). Shear heating induced lithospheric-scale localization: Does it result in subduction? Earth Planet. Sci. Lett., 359-360:1–13. Ueda, K., Gerya, T., and Sobolev, S. V. (2008). Subduction initiation by thermal-chemical plumes: Numerical studies. Phys. Earth Planet. Inter., 171(1-4):296–312. van Hunen, J. and Moyen, J.-F. (2012). Archean Subduction: Fact or Fiction? Annual Review of Earth and Planetary Sciences, 40(1):195–219. Vignaroli, G., Faccenna, C., Jolivet, L., Piromallo, C., and Rossetti, F. (2008). Subduction polarity reversal at the junction between the Western Alps and the Northern Apennines, Italy. Tectonophysics, 450(1-4):34–50. Waldron, J. W., Schofield, D. I., Brendan Murphy, J., and Thomas, C. W. (2014). How was the iapetus ocean infected with subduction? Geology, 42(12):1095–1098. White, D. A., Roeder, D. H., Nelson, T. H., and Crowell, J. C. (1970). Subduction. Geological Society of America Bulletin, 81(October):3431–3432. W.K, H. and E.H., C. (2003). Earth’s Dynamic Systems. Prentice Hall; 10 edition.

Tomography… wait, isn’t that what happens in your CAT scan? Although the general public might associate tomography with medical imaging, Earth scientists are well aware that ‘seismic tomography’ has enabled us to peer deeper, and with more clarity, into the Earth’s interior (Fig. 1). What are some of the ways we can download and display tomography to inform our scientific discoveries? Why has seismic tomography been a valuable tool for plate reconstructions? And what are some new approaches for incorporating seismic tomography within plate tectonic models?

**Downloading and displaying seismic tomography**

Seismic tomography is a technique for imaging the Earth’s interior in 3-D using seismic waves. For complete beginners, IRIS (Incorporated Research Institutions for Seismology) has an excellent introduction that compares seismic tomography to medical CT scans.

A dizzying number of new, high quality seismic tomographic models are being published every year. For example, the IRIS EMC-EarthModels catalogue currently contains 64 diverse tomographic models that cover most of the Earth, from global to regional scales. From my personal count, at least seven of these models have been added in the past half year – about one new model a month. Aside from the IRIS catalog, a plethora of other tomographic models are also publicly-available from journal data suppositories, personal webpages, or by an e-mail request to the author.

Downloading a tomographic model is just the first step. If one does not have access to custom workflows and scripts to display tomography, consider visiting an online tomography viewer. I have listed a few of these websites at the end of this blog post. Of these websites, a personal favourite of mine is the Hades Underworld Explorer built by Douwe van Hinsbergen and colleagues at Utrecht University, which uses a familiar Google Maps user interface. By simply dragging a left and right pin on the map, a user can display a global tomographic section in real time. The displayed tomographic section can be displayed in either a polar or Cartesian view and exported to a .svg file. Another tool I have found useful are tomographic ‘vote maps’, which provide indications of lower mantle slab imaging robustness by comparison of multiple tomographic models (Shephard et al., 2017). Vote maps can be downloaded from the original paper above or from the SubMachine website (Hosseini et al. (2018); see more in the website list below).

**Using tomography for plate tectonic reconstructions**

Tomography has played an increasing role in plate tectonic studies over the past decades. A major reason is because classical plate tectonic inputs (e.g. seafloor magnetic anomalies, palaeomagnetism, magmatism, geology) are independent from the seismological inputs for tomographic images. This means that tomography can be used to augment or test classic plate reconstructions in a relatively independent fashion. For example, classical plate tectonic models can be tested by searching tomography for slab-like anomalies below or near predicted subduction zone locations. These ‘conventional’ plate modelling workflows have challenges at convergent margins, however, when the geological record has been significantly destroyed from subduction. In these cases, the plate modeller is forced to describe details of past plate kinematics using an overly sparse geological record.

**A ‘tomographic plate modelling’ workflow** (Fig. 2) was proposed by Wu et al. (2016) that essentially reversed the conventional plate modelling workflow. In this method, slabs are mapped from tomography and unfolded (i.e. retro-deformed) (Fig. 2a). The unfolded slabs are then populated into a seafloor spreading-based global plate model. Plate motions are assigned in a hierarchical fashion depending on available kinematic constraints (Fig. 2b). The plate modelling will result in either a single unique plate reconstruction, or several families of possible plate models (Fig. 2c). The final plate models (Fig. 2c) are fully-kinematic and make testable geological predictions for magmatic histories, palaeolatitudes and other geological events (e.g. collisions). These predictions can then be systematically compared against remnant geology (Fig. 2d), which are independent from the tomographic inputs (Fig. 2a).

The proposed** 3D slab mapping workflow **of Wu et al. (2016) assumed that the most robust feature of tomographic slabs is likely the slab center. The slab mapping workflow involved manual picking of a mid-slab ‘curve’ along hundreds (and sometimes thousands!) of variably oriented 2D cross-sections using software GOCAD (Figs. 3a, b). A 3-D triangulated mid-slab surface is then constructed from the mid-slab curves (Fig. 3c). Inspired by 3D seismic interpretation techniques from petroleum geoscience, the tomographic velocities can be extracted along the mid-slab surface for further tectonic analysis (Fig. 3d).

For relatively undeformed upper mantle slabs, a pre-subduction slab size and shape can be estimated by unfolding the mid-slab surface to a spherical Earth model, minimizing distortions and changes to surface area (Fig. 3e). Interestingly, the slab unfolding algorithm can also be applied to shoe design, where there is a need to flatten shoe materials to build cut patterns (Bennis et al., 1991). The three-dimensional slab mapping within GOCAD allows a self-consistent 3-D Earth model of the mapped slabs to be developed and maintained. This had advantages for East Asia (Wu et al., 2016), where many slabs have apparently subducted in close proximity to each other (Fig. 1).

**Web resources for displaying tomography**

Hades Underworld Explorer : http://www.atlas-of-the-underworld.org/hades-underworld-explorer/

Seismic Tomography Globe : http://dagik.org/misc/gst/user-guide/index.html

SubMachine : https://www.earth.ox.ac.uk/~smachine/cgi/index.php

]]>ReferencesBennis, C., Vezien, J.-M., Iglesias, G., 1991. Piecewise surface flattening for non-distorted texture mapping. Proceedings of the 18th annual conference on Computer graphics and interactive techniques 25, 237-246. Hosseini, K. , Matthews, K. J., Sigloch, K. , Shephard, G. E., Domeier, M. and Tsekhmistrenko, M., 2018. SubMachine: Web-Based tools for exploring seismic tomography and other models of Earth's deep interior. Geochemistry, Geophysics, Geosystems, 19. Li, C., van der Hilst, R.D., Engdahl, E.R., Burdick, S., 2008. A new global model for P wave speed variations in Earth's mantle. Geochemistry, Geophysics, Geosystems 9, Q05018. Shephard, G.E., Matthews, K.J., Hosseini, K., Domeier, M., 2017. On the consistency of seismically imaged lower mantle slabs. Scientific Reports 7, 10976. Wu, J., Suppe, J., 2018. Proto-South China Sea Plate Tectonics Using Subducted Slab Constraints from Tomography. Journal of Earth Science 29, 1304-1318. Wu, J., Suppe, J., Lu, R., Kanda, R., 2016. Philippine Sea and East Asian plate tectonics since 52 Ma constrained by new subducted slab reconstruction methods. Journal of Geophysical Research: Solid Earth 121, 4670-4741

One integral part of doing estimations on parameters is an uncertainty analysis. The aim of a general inverse problem is to find the value of a parameter, but it is often very helpful to indicate the measure of certainty. For example in the last figure of my previous post, the measurement values at the surface are more strongly correlated to the upper most blocks. Therefore, the result of an inversion in this set up will most likely be more accurate for these parameters, compared to the bottom blocks.

In linear deterministic inversion, the eigenvalues of the matrix system provide an indication of the resolvability of parameters (as discussed in the aforementioned work by Andrew Curtis). There are classes of methods to compute exact parameter uncertainty in the solution.

From what I know, for non-linear models, uncertainty analysis is limited to the computation of second derivatives of the misfit functional in parameter space. The second derivatives of X (the misfit function) are directly related to the standard deviations of the parameters. Thus, by computing all the second derivatives of X, a non-linear inverse problem can still be interrogated for its uncertainty. However, the problem with this is its linearisation; linearising the model and computing derivatives may not be truly how the model reacts in model space. Also, for strongly non-linear models many trade-offs (correlations) exist which influence the final solution, and these correlations may very strongly depending on the model to be inverted.

Enter reverend Thomas Bayes

This part-time mathematician (he only ever published one mathematical work) from the 18th century formulated the Bayes’ Theorem for the first time, which combines knowledge on parameters. The mathematics behind it can be easily retrieved from our most beloved/hated Wikipedia, so I can avoid getting to caught up in it. What is important is that it allows us to combine two misfit functions or probabilities. Misfits and probabilities are directly interchangeable; a high probability of a model fitting our observations corresponds to a low misfit (and there are actual formulas linking the two). Combining two misfits allows us to accurately combine our pre-existing (or commonly: prior) knowledge on the Earth with the results of an experiment. The benefits of this are two-fold: we can use arbitrarily complex prior knowledge and by using prior knowledge that is *bounded* (in parameter space) we can still invert underdetermined problems without extra regularisation. In fact, the prior knowledge *acts* as regularisation.

Let’s give our friend Bayes a shot at our non-linear 1D bread. We have to come up with our prior knowledge of the bread, and because we did not need that before I’m just going to conjure something up! We suddenly find the remains of a packaging of 500 grams of flour

This is turning in quite the detective story!

However, the kitchen counter that has been worked on is also royally covered in flour. Therefore, we estimate that probably this pack was used; about 400 grams of it, with an uncertainty (standard deviation) of 25 grams. Mathematically we can formulate our prior knowledge as a Gaussian distribution with the aforementioned standard deviation and combine this with our misfit of the inverse problem (often called the likelihood). The result is given here:

One success and one failure!

First, we successfully combined the two pieces of information to make an inverse problem that is no longer non-unique (which was a happy coincidence of the prior: it is not guaranteed). However, we failed to make the problem more tractable in terms of computational requirements. To get the result of our combined misfit, we still have to do a systematic grid search, or at least arrive at a (local) minimum using gradient descent methods.

We can do the same in 2D. We combine our likelihood (original inverse problem) with rather exotic prior information, an annulus in model space, to illustrate the power of Bayes’ theorem. The used misfit functions and results are shown here:

This might also illustrate the need for non-linear uncertainty analysis. Trade-offs at the maxima in model space (last figure, at the intersection of the circle and lines) distinctly show two correlation directions, which might not be fully accounted for by using only second derivative approximations.

Despite this ‘non-progress’ of still requiring a grid search even after applying probability theory, we can go one step further by combining the application of Bayesian inference with the expertise of other fields in *appraising* inference problems…

Up to now, using a probabilistic (Bayesian) approach has only (apparently) made our life more difficult! Instead of one function, we now have to perform a grid search over the prior *and* the original problem. That doesn’t seem like a good deal. However, a much used technique in statistics deals with exactly the kind of problems we are facing here: given a very irregular and high dimensional function

How do we extract interesting information (preferably without completely blowing our budget on supercomputers)?

Let’s first say that with interesting information I mean minima (not necessarily restricted to global minima), correlations, and possibly other statistical properties (for our uncertainty analysis). One answer to this question was first applied in Los Alamos around 1950. The researches at the famous institute developed a method to simulate equations of state, which has become known as the Metropolis-Hastings algorithm. The algorithm is able to draw samples from a complicated probability distribution. It became part of a class of methods called Markov Chain Monte Carlo (MCMC) methods, which are often referred to as samplers (technically they would be a subset of all available samplers).

The reason that the Metropolis-Hastings algorithm (and MCMC algorithms in general) is useful, is that a complicated distribution (e.g. the annulus such as in our last figure) does not easily allow us to generate points proportional to its misfit. These methods overcome this difficulty by starting at a certain point in model space and traversing a random path through it – *jumping around* – but visiting regions only proportional to the misfit. So far, we have only considered directly finding optima to misfit functions, but by generating samples from a probability distribution proportional to the misfit functions, we can readily compute these minima by calculating statistical modes. Uncertainty analysis subsequently comes virtually for free, as we can calculate any statistical property from the sample set.

I won’t try to illustrate any particular MCMC sampler in detail. Nowadays many great tools for visualising MCMC samplers exist. This blog by Alex Rogozhnikov does a beautiful job of both introducing MCMC methods (in general, not just for inversion) and illustrating the Metropolis Hastings Random Walk algorithm as well as the Hamiltonian Monte Carlo algorithm. Hamiltonian Monte Carlo also incorporates gradients of the misfit function, thereby even accelerating the MCMC sampling. Another great tool is this applet by Chi Feng. Different target distributions (misfit functions) can be sampled here by different algorithms.

The field of geophysics has been using these methods for quite some time (Malcom Sambridge writes in 2002 in a very interesting read that the first uses were 30 years ago), but they are becoming increasingly popular. However, strongly non-linear inversions and big numerical simulations are still very expensive to treat probabilistically, and success in inverting such a problem is strongly dependent on the appropriate choice of MCMC sampler.

In the third part of this blog we saw how to combine any non-linear statistical model, and how to sample these complex functions using MCMC samplers. The resulting sample sets can be used to do an inversion and compute statistical moments of the inverse problem.

If you reached this point while reading most of the text, you have very impressively worked your way through a huge amount of inverse theory! Inverse theory is a very diverse and large field, with many ways of approaching a problem. What’s discussed here is, to my knowledge, only a subset of what’s being used ‘in the wild’. These ramblings of a aspiring seismologist might sound uninteresting to the geodynamiscists at the other side of the geophysics field. Inverse methods seem to be not nearly discussed as much in geodynamics as they are in seismology. Maybe it’s the terminology that differs, and that all these concepts are well known and studied under different names and you recognise some of the methods. Otherwise, I hope I have given an insight in the wonderful and sometimes ludicrous mathematical world of (some) seismologists.

Interested in playing around with inversion yourself? You can find a toy code about baking bread here.

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The idea of inversion is to literally invert the forward problem. For the aforementioned problems our knowns become unknowns, and vice versa. We wish to infer physical system parameters from measurements. Note that in our forward problem there are multiple classes of knowns; we have the forcing parameters, the material properties and the boundary conditions. All of these parameters could be inverted for in an inversion, but it is typically one only inverts for one class. Most of the time, medium parameters are the target of inversions. As we will go through the examples, I will gradually introduce some inversion methods. Every new method is marked in **blue**.

Now, let’s consider our first example: the recipe for a bread. Let’s say we have a **0.5 kilogram bread**. For our first case (case **1a**), we will assume that the amount of water is ideal. In this case, we have *one* free variable to estimate (the used amount of flour), from *one* observable (the resulting amount of bread). We have an analytical relationship that is both **invertible** and **one-to-one**. Therefore, we can use the** direct inversion ** of the relationship to compute the amount of flour. The process would go like this:

1. Obtain the forward expression G(m) = d;

2. Solve this formula for G^-1 (d) = m;

3. Find m by plugging d into the inverse function.

Applying this direct inversion shows us that **312.5 grams of flour** must have been used.

The two properties of the analytical relationship (**invertible** and **one-to one**) are very important for our choice of this inversion method. If our relationship was sufficiently complex, we couldn’t solve the formula analytically (though the limitation might then lie by the implementer 😉 ). If a function is not one-to-one, then two different inputs could have the same output, so we cannot analytically invert for a **unique** solution.

In large linear forward models (as often obtained in finite difference and finite element implementations), we have a matrix formula similar to

A **m** = **d**.

In these systems, **m** is a vector of model parameters, and **d** a vector of observables. If the *matrix is invertible*, we can directly obtain the model parameters from our observables. The invertibility of the matrix is a very important concept, which I will come back to in a bit.

Let’s see what happens when the first condition is not satisfied. I tried creating a non-linear function that at least *I* could not invert. Someone might be able to do it, but that is not the aim of this exercise. This new relationship is given here:

The actual forward model is included in the script at the bottom of the post. When we cannot analytically solve the forward model, one approach would be to iteratively optimise the input parameters such that the end result becomes as close as possible to the observed real world. But, before we talk about how to propose inputs, we need a way to compare the outputs of our model to the observables!

When we talk about our bread, we see one end-product. It is very easy to compare the result of our simulation to the real world. But what happens when we make a few breads, or, similarly, when we have a lot of temperature observations in our volcanic model? One (common) way to combine observables and predictions into a single measure of discrepancy is by defining a misfit formula which takes into account all observations and predictions, and returns a single – you guessed it – **misfit**.

The choice of misfit directly influences which data is deemed more important in an inversion scheme, and many choices exist. One of the most intuitive is the L2-norm (or L2 misfit), which calculates the vector distance between the predicted data (from our forward model) with the observed data as if in Euclidean space. For our bread it would simply be

X = | predicted bread – observed bread |

Let’s try to create a misfit function for our 1D bread model. I use the relationship in the included scripts. Again, we have 500 grams of bread. By calculating the quantity | G(m) – 500 |, we can now make a graph of how the misfit varies as we change the amount of flour. I created a figure which shows the observed and predicted value:

and a figure which shows the actual misfit at each value of m

Looking at the two previous figures may result in some questions and insights to the reader. First and foremost, it should be obvious that this problem does **not** have a unique solution. Different amounts of flour give exactly the same end result! It is thus impossible to say with certainty how much flour was used.

We have also recast our inversion problem as an optimisation function. Instead of thinking about fitting observations we can now think of our problem as a minimization of some function X (the misfit). This is very important in inversion.

Iterative optimizations schemes such as **gradient descent methods (and the plethora of methods derived from it) work in the following fashion:**

1. Pick a starting point based on the best prior knowledge (e.g., a 500 gram bread could have been logically baked using 500 grams of flour);

2. Calculate the misfit (X) at our initial guess;

3. If X is not low enough (compared to some arbitrary criterion):

• Compute the gradient of X;

• Do a step in the direction of the steepest gradient;

• Recompute X;

• Repeat from 2.

4. If X is low enough:

• Finalise optimisation, with the last point as solution.

These methods are **heavily influenced by the starting point** of the inversion scheme. If I would have started on the left side of the recipe-domain (e.g. 100 grams of flour), I might well have ended up in a different solution. Gradient descents often get ‘stuck’ in local solutions, which might not even be the optimal one! We will revisit this non-uniqueness in the 2D problem, and give some strategies to mitigate creating more than one solution. Extra material can be found here.

One thing that often bugged me about the aforementioned gradient descent methods is the seemingly complicated approach for such simple problems. Anyone looking at the figures could have said

Well, duh, there’s 3 solutions, here, here and here!

Why care about such an complicated way to only get one of them?

The important realisation to make here is that I have precomputed all possible solution for this forward model in the 0 – 700 grams range. This precomputation on a 1D domain was very simple; at a regular interval, compute the predicted value of baked bread. Following this, I could have also programmed my Python routine to extract all the values with a sufficiently low misfit as solutions. This is the basis of a ** grid search**.

Let’s perform a grid search on our second model (**1b**). Let’s find all predictions with 500 grams of bread as the end result, plus-minus 50 grams. This is the result:

The original 312.5 grams of flour as input is part of the solution set. However, the model actually has infinitely many solutions (extending beyond the range of the current grid search)! The reason that a grid search might not be effective is the inherent computational burden. When the forward model is sufficiently expensive in numerical simulation, exploring a model space completely with adequate resolution might take very long. This burden increases with model dimension; if more model parameters are present, the relevant model space to irrelevant model space becomes very small. This is known as the **curse of dimensionality** (very well explained in Tarantola’s textbook).

Another reason one might want to avoid grid searches is our inability to appropriately process the results. Performing a 5 dimensional or higher grid searches is sometimes possible on computational clusters, but visualizing and interpreting the resulting data is very hard for humans. This is partly why many supercomputing centers have in-house departments for data visualisation, as it is a very involved task to visualise complex data well.

Now: towards solving our physical inversions!

One big problem in inversion is non-uniqueness: the same result can be obtained from different inputs. The go-to way to combat this is to add extra information of any form to the forward problem. In our bread recipe we could think of adding extra observables to our problem, such as the consistency of the bread, its taste, color, etc. Another option could be to add constraints on the parameters, such as using the minimum amount of ingredients. This is akin to asking the question: given this amount of bread, how much flour and water was *minimally* used to make it?

Diffusion type problems are notorious for their non-uniqueness. Many different subsurface heat conduction distributions might result in the observations (imagine differently inclined volcanic conduits). An often used method of regularisation (not limited to diffusion type studies!) is spatial smoothing. This method requires that among equally likely solutions, the smoothest solutions are favoured, for it is more ‘natural’ to have smoothly varying parameters. Of course, in many geoscientific settings one would definitely expect sharp contrasts. However, in ‘underdetermined’ problems (i.e., you do not have enough observations to constrain a unique solution), we favour Occam’s Razor and say

The simplest solution must be assumed

When dealing with more parameters than observables (non-uniqueness) in linear models it is interesting to regard the forward problem again. If one would parameterize our volcanic model using 9 parameters for the subsurface and combine that with the 3 measurements at the surface, the result would be an underdetermined inverse problem.

This forward model (the Laplace equation) can be discretised by using, for example, finite differences. The resulting matrix equation would be A**m** = **d**, with A a 3 x 9 matrix, **m** a 9 dimensional vector and **d** a 3 dimensional vector. As one might recall from linear algebra classes, for a matrix to have an inverse, it has to be square. This matrix system is not square, and therefore not invertible!

Aaaaahhh! But don’t panic: there is a solution

By adding either prior information on the parameters, smoothing, or extra datapoints (e.g., taking extra measurements in wells) we can make the 3 x 9 system a perfect 9 x 9 system. By doing this, we condition our system such that it is invertible. However, many times we end up overdetermining our system which could result in a 20 x 9 system, for example. Note that although neither the underdetermined nor the overdetermined systems have an exact matrix inverse, both *do* have pseudo-inverses. For underdetermined systems, I have not found these to be particularly helpful (but some geophysicists do consider them). Overdetermined matrix systems on the other hand have a very interesting pseudo-inverse: the **least squares solution**. Finding the least squares solution in linear problems is the same as minimising the L2 norm! **Here, two views on inversion come together: solving a specific matrix equation is the same as minimising some objective functional** (at least in the linear case). Other concepts from linear algebra play important roles in linear and weakly non-linear inversions. For example, matrix decompositions offer information on how a system is probed with available data, and may provide insights on experimental geophysical survey design to optimise coverage *(see “Theory of Model-Based Geophysical Survey and Experimental Design Part A – Linear Problems” by Andrew Curtis)*.

I would say it is common practice for many geophysicists to pose an inverse problem that is typically underdetermined, and keep adding regularization until the problem is solvable in terms of matrix inversions. I do not necessarily advocate such an approach, but it has its advantages towards more agnostic approaches, as we will see in the post on probabilistic inversion next week!

We’ve seen how the forward model determines our inversion problem, and how many measurements can be combined into a single measure of fit (the misfit). Up to now, three inversion strategies have been introduced:

• **Direct inversion**: analytically find a solution to the forward problem. This method is limited to very specific simple cases, but of course yields near perfect results.

• **Gradient descent methods**: a very widely used class of algorithms that iteratively update solutions based on derivatives of the misfit function. Their drawbacks are mostly getting stuck in local minima, and medium computational cost.

• **Grid searches**: a method that searches the entire parameter space systematically. Although they can map all the features of the inverse problem (by design), they are often much too computationally expensive.

What might be even more important, is that we have seen how to reduce the amount of possible solutions from infinitely many to at least a tractable amount using regularisation. There is only one fundamental piece still missing… Stay tuned for the last blog post in this series for the reveal of this mysterious missing ingredient!

Interested in playing around with inversion yourself? You can find a toy code about baking bread here.

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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.)

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

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.

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:

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:

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

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:

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

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?

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