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Presentation skills – 2. Speech

Presentation skills – 2. Speech

Presenting: some people love it, some people hate it. I firmly place myself in the first category and apparently, this presentation joy translates itself into being a good – and confident – speaker. Over the years, quite a few people have asked me for my secrets to presenting (which – immediate full disclosure – I do not have) and this is the result: a running series on the EGU GD Blog that covers my own personal tips and experience in the hope that it will help someone (you?) become a better and – more importantly – more confident speaker. Last time, we discussed your presentation voice. In this second instalment, I discuss everything related to how you speak.

1. Get rid of ‘uh’

Counting the number of times a speaker says ‘uh’ during a presentation is a fun game, but ideally you would like your audience to focus on the non-uh segments of your talk. Therefore, getting rid of ‘uh’ (or any other filler word for that matter) is important. I have two main tips to get rid of ‘uh’:

Write down your speech and practice (but don’t hold on to it religiously)

Practice. Practice. And practice it again. Maybe a few more times. Almost… no: practice it again.
I am being serious here. If you know exactly what you want to say, you won’t hesitate and fill that moment of hesitation with a prolonged uuuuuhhh. The added benefit of writing down your presentation and practising it religiously is that it will help you with timing your presentation as well. I also find it helpful to read through it (instead of practising it out loud) when I am in a situation that doesn’t allow me to go into full presentation mode (on the plane to AGU for example). However, make sure to practise your presentation out loud even though you wrote it all down: thinking speed (or reading in your head) and talking speed are not the same!

If you write down your presentation, and you know exactly what you want to say, you have to take care to evade another (new) pitfall for saying ‘uh’: now that you know exactly what you want to say and how to say it most efficiently, you start saying ‘uh’ when you can’t remember the exact wording. Let it go. Writing down your speech helps you to clarify the vocabulary needed for your speech, but if you don’t say the exact sentences, just go with something else. You will have a well thought out speech anyway. Just go with the flow and try not to say ‘uh’.

The second main tip for getting rid of ‘uh’ is to

Realise that it is okay to stay silent for a while

If you forget the word you wanted to say and you need some time to think, you can take a break. You can stay silent. You don’t need to fill up the silence with ‘uh’. In fact, a break often seems more natural. Realise that you forgot something, don’t panic, take a breath, take a break (don’t eat a KitKat at this point in your presentation), and then continue when you know what to say again. Even if you don’t forget the exact words or phrasings, taking a breath and pausing in your narrative can be helpful for your audience to take a breath as well. It will seem as if your presentation is relaxed: you are not rushing through 50 slides in 12 minutes. You are prepared, you are in control, you can even take a break to take a breath.

2. Speed

A lot of (conference) presentations will have a fixed time. At the big conferences, like EGU and AGU, you get 12 minutes and not a second more or less. Well, of course you can talk longer than 12 minutes, but this will result in less (if any) time for questions.

I don’t think the conveners will kill you, but don’t pin me down on it

And on top of that, everyone (well, me at the very least) will be annoyed at you for not sticking to the time.

So: sticking to your time limit is important!

But how can you actually do this? Well, there are a few important factors:
1. Preparation: know exactly what you want to say (we will cover this more in a later instalment of this series)
2. The speed at which you speak.

We will be discussing the latter point in this blog entry. For me (and many other people), I know I can stick to the rule of “one slide per minute”, but I always have a little buffer in that I count the title slide as a slide as well. So, my 12-minute long presentation would have 12 slides in total (including the title slides). This actually spreads my 12 minutes over 11 scientific slides, so I can talk a little bit longer about each slide. It also gives me piece of mind to know that I have a bit of extra time. However, the speed at which you talk might be completely different. Therefore, the most important rule about timing your presentations is:

Knowing how fast you (will) speak

I always practice my short presentations a lot. If they are 30 minutes or longer, I like to freewheel with the one slide per minute rule. But for shorter presentations, I require a lot of practice. I always time every presentation attempt and make a point of finishing each attempt (even if the first part goes badly). Otherwise you run the risk of rehearsing the first part of your presentation very well, and kind of forgetting about the second part. When I time my presentation during practice, I always speak too long. For a 12 minute presentation, I usually end up at the 13.5 minute mark. However, I know that when I speak in front of an audience, I (subconsciously?) speed up my speech, so when I time 13.5 minutes, I know that my actual presentation will be a perfect 12 minutes.

The only way to figure out how you change or start to behave in front of an audience is by simply giving a lot of presentations. Try to do that and figure out whether you increase or decrease the speed of your speech during your talk. Take note and remember it for the next time you time your presentation. In the end, presenting with skill and confidence is all about knowing yourself.

3. Articulation and accent

There are as many accents to be heard at a conference as there are scientists talking. Everyone has there own accent, articulation, (presentation) voice, etc. This means that

You should not feel self-conscious about your accent

Some accents are stronger than others and may be more difficult for others to follow. Native speakers are by no means necessarily better speakers and depending on whom you ask, their accent might also not be better than anyone else’s.
Of course your accent might become an issue if people can’t understand you. You can try and consider the following things to make yourself understandable for a big audience:
1. Articulate well.
2. Adapt the speed at which you talk

Some languages are apparently faster than others. French is quite fast for example, whereas (British) English is a slower language. You have to take this into account when switching languages. If you match the pace of the language you are speaking, your accent will be less noticeable, because you avoid any ingrained rythm patterns that are language specific. Then you might still have your accent shine through in your pronunciation of the words, but it will not shine through in the rhythm of your speech.
In addition, you can consider asking a native speaker for help if you are unsure of how to pronounce certain words. Listening or watching many English/American/Australian tv series/films/youtube will also help with your pronunciation.

And that, ladies and gentlemen, is about everything I have to say on the matter of speech. You should now have full control over your presentation voice and all the actual words you are going to say. Next time, we go one step further and discuss your posture during the presentation and your movements.

An industrial placement as a geodynamicist

An industrial placement as a geodynamicist

After years of trying to get a PhD, publishing papers, networking with professors, and trying to land that one, elusive, permanent job in science, it can be quite easy to forget that you actually do have career options outside of academia. To get a little taste of this, Nico Schliffke, PhD student in geodynamics at Durham University, tries out the industry life for a few weeks!

When coming close to the final stages of a PhD life, many students reconsider whether they want to stay in academia or prefer to step over to industry or other non-academic jobs. This is surely not a simple decision to take, as it could strongly shape your future. In this blog post, I would like to report my industrial placement experience during my PhD and share a few thoughts on the topic.

The taste of industry life was an opportunity I had within the frame of my PhD project. Split into two terms, I spent four weeks at a medium-sized company developing optical imaging techniques (both software and equipment) to measure flow fields and deformation. The branch I worked in was “digital image correlation” (DIC) which measures strain on given surfaces purely by comparing successive images on an object (see figure below). This technique is used in both industry (crash tests, quality assessments, etc.) as well as in academia (analogue experiments, wind tunnels, engineering..), and has the substantial advantage of measuring physical properties precisely, without using any materials or affecting dynamical processes. DIC is not directly related to or used in my PhD (I do numerical modelling of subduction zones and continental collision), but surprisingly enough I was able to contribute more than expected – but more on that later.

Basic principle of ‘digital image correlation’. A pattern on a digital image is traced through time on successive images to calculate displacements and strain rates.
Credit: LaVision

The specific project I worked on was inspired by the analogue tectonics lab at GFZ Potsdam, that uses DIC measuring systems to quantify and measure the deformation of their sandbox experiments. Typical earthquake experiments like the figure below span periods of a few minutes to several days during which individual earthquakes occur in a couple of milliseconds. The experiment is continuously recorded by cameras to both monitor deformation visually and quantify deformation by using the optical imaging technique developed by my host company. To resolve the deformation related to individual earthquakes, high imaging rates are required which in turn produce a vast amount of data (up to 2TB per experiment). However, only a small fraction (max. 5%) of the entire dataset is of interest, as there is hardly any deformation during interseismic periods. The project I was involved in tried to tackle the issue of unnecessarily cluttered hard discs: the recording frequency should be linked to a measurable characteristic within the experiment, e.g. displacement velocities in these specific experiments, and controlled by the DIC software.

Setup of the analogue experiment to model earthquakes in subduction zones (courtesy of Michael Rudolf). Cameras above the experiment measure deformation and strain rates by tracking patterns on the surface created by the contrast of black rubber and white sugar.

My general task during the internship was to develop this idea and the required software. We finally developed a ‘live-extensometer’ to calculate displacements between two given points of an image during recording and link its values to the camera’s recording frequency. Therefore, restricting high imaging rates to large (and fast) displacements of earthquakes should result in reducing the total amount of data acquired for a typical earthquake experiment by 95%. However, we needed an actual experiment to verify this. So, I met up with the team at GFZ to test the developed feature.

The main experiment the GFZ team had in mind is sketched in the figure above: a conveyor belt modelling a subducting slab continuously creates strain in the ‘orogenic wedge’ which is released by earthquakes leading to surface deformation. Cameras above the experiment monitor the surface while software computes strain rates and displacement (see figure below). The developed feature of changing frequencies during the experiment depending on slip rates was included and worked surprisingly well. Yet freshly programmed software is seldom perfect: minor issues and bugs crept up during the experiments. My final contribution during the internship was to report these problems back to the company to be fixed.

Displacement measured by ‘digital image correlation’ during an earthquake lasting ~5 ms (courtesy of Mathias Rosenau).

My geodynamical background allowed me to contribute to various fields within the company and resulted in various individual tasks throughout the internship: coding experience helped with discussing ideal software implementations and testing the latest implemented software on small (physical) experiments. My knowledge of various deformation mechanisms and geosciences in general, with its numerous subdisciplines and methods, provided a solid base for searching further applications for the developed software within academia, but also in industry. Last but not least, pursuing my own large project (my PhD) strongly facilitated discussing possible future development steps.

The atmosphere at the company in general was very pleasant and similar to what I experienced at the university: relaxed handling, pared with discussion how to improve products or use of new techniques that might be applicable to a problem. To stay competitive, the company needs to further develop their products which requires a large amount of research, developments and innovative ideas. Meetings to discuss further improvements of certain products were thus scheduled on a (nearly) daily basis. On the one hand this adds pressure to get work done as quickly as possible, but working on a project as a team with many numerous areas of expertise is also highly exciting.

This internship help reveal the variability of possible jobs that geodynamicists can have in industry besides the ‘classical’ companies linked to exploration, tunnel engineering or geological surveys. The skill set acquired in a geodynamical PhD (coding, modelling, combining numerics, physics, and geosciences) makes a very flexible and adaptive employee which is attractive to companies who are so specialised, that there is (nearly) no classical education at university level. Jobs at small to medium-sized companies are often harder to find, but it’s just as difficult for the companies to find suitable candidates for their open positions. Hence, it may be worth searching in-depth for a suitable job, if you are considering stepping out of academia and maybe even out of geoscience as well.

If PhD students are hesitant whether to stay in academia or change into industry, I would advise to do such a short internship with a company to get a taste of ‘the other side’. During a PhD, we get to know academic life thoroughly but industry mostly remains alien. Besides giving a good impression of daily life at a company and how you can contribute, an industry internship might also widen your perspective of which areas might be relevant to you, your methodology and your PhD topic. In total, this internship was definitely a valuable experience for me and will help when deciding: academia or industry?


Here are a few links for more information:
Host company
Digital Image Correlation
TecLab at GFZ Potsdam
Previous EGU blog post interviews of former geoscientists


Thoughts on geological modelling: an analogue perspective

Thoughts on geological modelling: an analogue perspective

In geodynamics we study the dynamics of the Earth (and other planets). We ground our studies in as much data as possible, however we are constrained by the fact that pretty much all direct information we can collect from the interior of the Earth only shows its present-day state. The surface rock record gives us a glimpse into the past dynamics and evolution of our planet, but this record gets sparser as we go back in time. This is why it is common to use modelling in geodynamics to fill this gap of knowledge. There are different types of modelling, and this week João Duarte writes about the importance of analogue modelling. 

João Duarte. Researcher at Instituto Dom Luiz and Invited Professor at the Geology Department, Faculty of Sciences of the University of Lisbon. Adjunct Researcher at Monash University.

The first time I went to EGU, in 2004, I presented a poster with some new marine geology data and a few sets of analogue models. I was doing accretionary wedges in sandboxes. At the time, I was in the third year of my bachelor’s degree and I was completely overwhelmed by the magnitude of the conference. It was incredible to see the faces of all those scientists that I used to read articles from. But one thing impressed me the most. Earth Sciences were consolidating as a modern, theoretical based science. The efforts in trying to develop an integrated dynamic theory of plate tectonics as part of mantle convection were obvious. The new emergent numerical models looked incredible, with all those colours, complex rheologies and stunning visualization that allowed us to “see” stresses, temperature gradients and non-linear viscosities. I was amazed.

Analogue modelling was at a relative peak in 2004, however it was also anticipated by some that it would quickly disappear (and indeed several analogue labs have closed since). It was with this mindset, that I later did the experiments for my PhD, which I finished in 2012 (Duarte et al., 2011). But I was fortunate. My supervisors, Filipe Rosas and Pedro Terrinha, took me to state-of-art labs, namely Toronto and Montpellier (lead at the time by Sandy Cruden and Jacques Malavieille, respectively), and I started to develop a passion for this kind of models. When I moved to Monash for a post-doc position, in 2011, this turned out to be a great advantage. There, modelers such as Wouter Schellart, Louis Moresi, Fabio Capitanio, David Boutelier and Sandy Cruden (yes, I met Sandy again at Monash) were using analogue models to benchmark numerical models. Why? Because many times, even though numerical models produce spectacular results, they might not be physically consistent. And there is only one way to get rid of this, which is to make sure that whatever numerical code we are using can reproduce simple experiments that we can run in a lab. The classical example is the sinking of a negatively buoyant sphere in a viscous medium.

Sandbox analogue model of an accretionary wedge. Part of the same experiment as shown in the header figure. Here, a sliced section cut after wetting, is shown. University of Lisbon, 2009. Experiments published in Duarte et al. (2011).

That was what we were doing at Monash. I worked with Wouter Schellart in the development of subduction experiments with an overriding plate, which were advancing step by step in both analogue and numerical schemes (see e.g., Duarte et al., 2013 and Chen et al., 2015, 2016 for the analogue models, and Schellart and Moresi, 2013 for numerical equivalents). The tricky bit was, we wanted to have self-consistent dynamic experiments in which we were ascribing the forces (negative buoyancy of the slab, the viscosity of the upper mantle, etc) and let the kinematics (i.e. the velocities) to be an emergent phenomenon. So, no lateral push or active kinematic boundaries were applied to the plates. This is because we now recognize that in general, it is the slab pull at subduction zones that majorly drives the plates and not the other way around. Therefore, if we want to investigate the fundamental physics and dynamics of subduction zones we need to use self-consistent models (both analogue and numerical). In order to carry out these models, we had to develop a new rheology for the subduction interface, which is a complex problem, both in the analogue and the numerical approaches (Duarte et al. 2013, 2014, 2015). But this is another very long story that would lead to a publication by itself.

Analogue models of subduction with an overriding plate and an interplate rheology. Monash University, 2012. Adapted from Duarte et al. (2013)

But what is analogue modelling all about? Basically, analogue models are scaled models that we can develop in the laboratory using analogue materials (such as sand), and that at the scale that we are doing our models have similar physical properties to those of natural materials (such as brittle rocks). But, as it is widely known, under certain circumstances (at large time and space scales), rocks behave like fluids, and for that we use analogue fluids, such as silicone putties, glucose and honey. We can also use fluids to simulate the interaction between subduction zones and mantle plumes in a fluid reservoir (see below figures and links to videos of scaled experiments using three different fluids to study slab-plume interaction; Meriaux et al., 2015a, 2015b, 2016). These are generally called geodynamic analogue models.

End of a slab-plume experiment in the upper mantle (see below). The tank is partially filled with glucose. The slab (laying at the analogue 660 discontinuity) is made of silicone mixed with iron powder. The plume is made of a water solution of glucose dyed with a red colorant. And that’s me on the left. Monash University, 2014.

I usually consider two main branches of analogue models. The first, which is the one mostly used by geologists, was started by Sir James Hall (1761 – 1832), that squeezed layers of clay to reproduce the patterns of folded rocks that he had observed in nature. This method was later improved by King Hubbert (1937), who laid the ground for the development of the field by developing a theory of scaling of analogue models applied to geological processes.

The other branch is probably as old as humans. It began when we started to manipulate objects and using them to understand basic empirical laws, such as the one that objects always fall. When Galileo was using small spheres in inclined surfaces to extract the physical laws that describe the movement of bodies, from rocks to planets, he was in a certain way using analogue models. He understood that many laws are scale invariant. Still today, these techniques are widely used by physicist and engineers when understanding for example the aerodynamics of airplanes, the stability of bridges, the dynamics of rivers or the resistance of dams. They use scaled models that reproduce at suitable laboratory scales the objects and processes that they are investigating.

What we did at Monash, was a mixture of both approaches. Though, we were less interested in exactly reproducing nature from a purely geometric and kinematic point of view, but we were more interested in understanding the physics of the object we were investigating: subduction zones. Therefore, we had to guarantee that we were using the correct dynamical approach in order to be able to extract generic physical empirical laws, hoping that these laws would provide us some insight on the dynamics of natural subduction zones. These empirical laws could readily be incorporated in numerical models, which would then help exploring more efficiently the space of the controlling parameters in the system.

Slab-Plume interaction in the upper mantle. Experiments published in Meriaux et al. (2015a, 2015b).

I want to finish with a question that I believe concerns all of us: are there still advantages in using analogue models? Yes, I believe so! One of the most important advantages is that analogue models are always three-dimensional and high-resolution. Furthermore, they allow a good tracking of the strain and to understand how it occurs in discontinuous mediums, for example when investigating the localization of deformation or the propagation of cracks. Numerical schemes still struggle with these problems. It is very difficult to have an efficient code that can deal simultaneously with very high resolution and large-scale three-dimensional problems, as it is required to investigate the process of subduction. Nevertheless, numerical models are of great help when it comes to track stresses, and model complex rheologies and temperature gradients. To sum up: nowadays, we recognize that certain problems can only be tackled using self-consistent dynamic models that model the whole system in three-dimensions, capturing different scales. For this, the combination of analogue and numerical models is still one of the most powerful tools we have. An interesting example of a field in which a combined approach is being used is the fascinating investigations on the seismic cycle (for example, see here).

Links to videos:

VIDEO 1: https://www.youtube.com/watch?v=U1TXC2XPbFA&feature=youtu.be
(Subduction with an overriding plate and an interplate rheology. Duarte et al., 2013)

VIDEO 2: https://www.youtube.com/watch?v=n5P2TzS6h_0&feature=youtu.be
(Slab-plume interaction at mantle scale. Side-view of the experiment on the top, and top-view of the experiment on the bottom. Meriaux et al., 2016)
References:

Chen, Z., Schellart, W.P., Strak, V., Duarte, J.C., 2016. Does subduction-induced mantle flow drive backarc extension? Earth and Planetary Science Letters 441, 200-210. https://doi.org/10.1016/j.epsl.2016.02.027

Chen, Z., Schellart, W.P., Duarte, J.C., 2015. Overriding plate deformation and variability of forearc deformation during subduction: Insight from geodynamic models and application to the Calabria subduction zone. Geochemistry, Geophysics, Geosystems 16, 3697–3715. DOI: 10.1002/2015GC005958

Duarte, J.C., Schellart, W.P., Cruden, A.R., 2015. How weak is the subduction zone interface? Geophysical Research Letters 41, 1-10. DOI: 10.1002/2014GL062876

Duarte, J.C., Schellart, W.P., Cruden, A.R., 2014. Rheology of petrolatum – paraffin oil mixtures: applications to analogue modelling of geological processes. Journal of Structural Geology 63, 1-11. https://doi.org/10.1016/j.jsg.2014.02.004

Duarte, J.C., Schellart, W.P., Cruden, A.R., 2013. Three-dimensional dynamic laboratory models of subduction with an overriding plate and variable interplate rheology. Geophysical Journal International 195, 47-66. https://doi.org/10.1093/gji/ggt257

Duarte, J.C., F.M. Rosas P., Terrinha, M-A Gutscher, J. Malavieille, Sónia Silva, L. Matias, 2011. Thrust–wrench interference tectonics in the Gulf of Cadiz (Africa–Iberia plate boundary in the North-East Atlantic): Insights from analog models. Marine Geology 289, 135–149. https://doi.org/10.1016/j.margeo.2011.09.014

Hubbert, M.K., 1937. Theory of scale models as applied to the study of geologic structures. GSA Bulletin 48, 1459-1520. https://doi.org/10.1130/GSAB-48-1459

Meriaux, C., Meriaux, A-S., Schellart, W.P., Duarte, J.C., Duarte, S.S., Chen, Z., 2016. Mantle plumes in the vicinity of subduction zones. Earth and Planetary Science Letters 454, 166-177. https://doi.org/10.1016/j.epsl.2016.09.001

Mériaux, C.A., Duarte, J.C., Schellart, W.P., Mériaux, A-S., 2015. A two-way interaction between the Hainan plume and the Manila subduction zone. Geophysical Research Letters 42, 5796–5802. DOI: 10.1002/2015GL064313

Meriaux, C.A., Duarte, J.C., Duarte, S., Chen, Z., Rosas, F.M., Mata, J., Schellart, W.P., and Terrinha, P. 2015. Capture of the Canary mantle plume material by the Gibraltar arc mantle wedge during slab rollback. Geophysical Journal International 201, 1717-1721. https://doi.org/10.1093/gji/ggv120

Schellart, W.P., Moresi, L., 2013. A new driving mechanism for backarc extension and backarc shortening through slab sinking induced toroidal and poloidal mantle flow: Results from dynamic subduction models with an overriding plate. Journal of Geophysical Research: Solid Earth 118, 3221-3248. https://doi.org/10.1002/jgrb.50173

Inversion 101 to 201 – Part 3: Accounting for uncertainty – Bayes and friends

Inversion 101 to 201 – Part 3: Accounting for uncertainty – Bayes and friends

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! You can find the first post here. This week, we have our final post in the series, where Lars discusses probabilistic inversion.

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.

Bayes’ rule

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.

One probabilistic regularised bread, please

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:

Prior and original misfit

Combined misfit

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:

Original misfit for a bread of 500 grams

Prior knowledge misfit

Combined misfit

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…

Jump around!

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.


Summary of probabilistic inversions

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.


Some final thoughts

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.