Great geophysicists #6: Robert Hooke

Robert Hooke was born near Freshwater on the Isle of Wight, UK, on 28 July 1635, and died on 13 March 1703 in London. At 18, he was awarded a chorister scholarship at Oxford, where he studied physics under Robert Boyle, 8 years his senior. 

Hooke's famous law tells us how things deform and, along with Newton, Hooke is thus a parent of the wave equation. The derivation starts by equating the force due to acceleration (of a vibrating particle, say), and the force due to elastic deformation:

where m is mass, x is displacement, the two dots denote the second derivative with respect to time (a.k.a. acceleration), and k is the spring constant. This powerful insight, which allows us to compute a particle's motion at a given time, was first made by d'Alembert in about 1742. It is the founding principle of seismic rock physics.

Hooke the geologist

Like most scientists of the 17th century, Hooke was no specialist. One of his best known works was Micrographia, first published in 1665. The microscope was invented in the late 1500s, but Hooke was one of the first people to meticulously document and beautifully draw his observations. His book was a smash hit by all accounts, inspiring wonder in everyone who read it (Samuel Pepys, for example). Among other things, Hooke described samples of petrified wood, forams, ammonites, and crystals of quartz in a flint nodule (left). Hooke also wrote about the chalk formations in the cliffs near his home town.

Hooke went on to help Wren rebuild London after the great fire of 1666, and achieved great respect for this work too. So esteemed is he that Newton was apparently rather jealous of him, and one historian has referred to him as 'England's Leonardo'. He never married, and lived in his Oxford college all his adult life, and is buried in Bishopsgate, London. As one of the fathers of geophysics, we salute him.

The painting of Hooke, by Rita Greer, is licensed under a Free Art License. It's a interpretation based on descriptions of him ("his chin sharp, and forehead large"); amazingly, there are no known contemporary images of him. Hear more about this.

You can read more about the relationship between Hooke's law and seismic waves in Bill Goodway's and Evan's chapters in 52 Things You Should Know About Geophysics. Download their chapters for free!

L is for Lambda

Hooke's law says that the force F exerted by a spring depends only on its displacement x from equilibrium, and the spring constant k of the spring:

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We can think of k—and experience it—as stiffness. The spring constant is a property of the spring. In a sense, it is the spring. Rocks are like springs, in that they have some elasticity. We'd like to know the spring constant of our rocks, because it can help us predict useful things like porosity. 

Hooke's law is the basis for elasticity theory, in which we express the law as

stress [force per unit area] is equal to strain [deformation] times a constant

This time the constant of proportionality is called the elastic modulus. And there isn't just one of them. Why more complicated? Well, rocks are like springs, but they are three dimensional.

In three dimensions, assuming isotropy, the shear modulus μ plays the role of the spring constant for shear waves. But for compressional waves we need λ+2μ, a quantity called the P-wave modulus. So λ is one part of the term that tells us how rocks get squished by P-waves.

These mysterious quantities λ and µ are Lamé's first and second parameters. They are intrinsic properties of all materials, including rocks. Like all elastic moduli, they have units of force per unit area, or pascals [Pa].

So what is λ?

Matt and I have spent several hours discussing how to describe lambda. Unlike Young's modulus E, or Poisson's ratio ν, our friend λ does not have a simple physical description. Young's modulus just determines how much longer something gets when I stretch it. Poisson's ratio tells how much fatter something gets if I squeeze it. But lambda... what is lambda?

• λ is sometimes called incompressibility, a name best avoided because it's sometimes also used for the bulk modulus, K.  
• If we apply stress σ₁ along the 1 direction to this linearly elastic isotropic cube (right), then λ represents the 'spring constant' that scales the strain ε along the directions perpendicular to the applied stress.
• The derivation of Hooke's law in 3D requires tensors, which we're not getting into here. The point is that λ and μ help give the simplest form of the equations (right, shown for one dimension).

The significance of elastic properties is that they determine how a material is temporarily deformed by a passing seismic wave. Shear waves propagate by orthogonal displacements relative to the propagation direction—this deformation is determined by µ. In contrast, P-waves propagate by displacements parallel to the propagation direction, and this deformation is inversely proportional to M, which is 2µ + λ. 

Lambda rears its head in seismic petrophysics, AVO inversion, and is the first letter in the acronym of Bill Goodway's popular LMR inversion method (Goodway, 2001). Even though it is fundamental to seismic, there's no doubt that λ is not intuitively understood by most geoscientists. Have you ever tried to explain lambda to someone? What description of λ do you find useful? I'm open to suggestions. 

Goodway, B., 2001, AVO and Lame' constants for rock parameterization and fluid detection: CSEG Recorder, 26, no. 6, 39-60.

Your child is dense for her age

Alan Cohen, veteran geophysicist and Chief Scientist at RSI, secured the role of provacateur by posting this question on the rock physics group on LinkedIn. He has shown that the simplest concepts are worthy of debate.

From a group of 1973 members, 44 comments ensued over the 23 days since he posted it. This has got to be a record for this community (trust me I've checked). It turns out the community is polarized, and heated emotions surround the topic. The responses that emerged are a fascinating narrative of niche and tacit assumptions seldomly articulated.

Any two will do

Why are two dimensions used, instead of one, three, four, or more? Well for one, it is hard to look at scatter plots in 3D. More fundamentally, a key learning from the wave equation and continuum mechanics is that, given any two elastic properties, any other two can be computed. In other words, for any seismically elastic material, there are two degrees of freedom. Two parameters to describe it.

• P- and S-wave velocities
• P-impedance and S-impedance
• Acoustic and elastic impedance
• R₀ and G, the normal-incidence reflectivity and the AVO gradient
• Lamé's parameters, λ and μ 

Each pair has its time and place, and as far as I can tell there are reasons that you might want to re-parameterize like this:

1. one set of parameters contains discriminating evidence, not visible in other sets;
2. one set of parameters is a more intuitive or more physical description of the rock—it is easier to understand;
3. measurement errors and uncertainties can be elucidated better for one of the choices. 

Something missing from this thread, though, is the utility of empirical templates to makes sense of the data, whichever domain is adopted.

Measurements with a backdrop

In child development, body mass index (BMI) is plotted versus age to characterize a child's physical properties using the backdrop of an empirically derived template sampled from a large population. It is not so interesting to say, "13 year old Miranda has a BMI of 27", it is much more telling to learn that Miranda is above the 95th percentile for her age. But BMI, which is defined as weight divided by height squared, in not particularity intuitive. If kids were rocks, we'd submerge them Archimedes style into a bathtub, measure their volume, and determine their density. That would be the ultimate description. "Whoa, your child is dense for her age!" 

We do the same things with rocks. We algebraically manipulate measured variables in various ways to show trends, correlations, or clustering. So this notion of a template is very important, albeit local in scope. Just as a BMI template for Icelandic children might not be relevant for the pygmies in Paupa New Guinea, rock physics templates are seldom transferrable outside their respective geographic regions. 

For reference see the rock physics cheatsheet.

AVO* is free!

The two-bit experiment is over! We tried charging $2 for one of our apps, AVO*, as a sort of techno-socio-geological experiment, and the results are in: our apps want to be free. Here are our download figures, as of this morning: 

You also need to know when these apps came out. I threw some of the key statistics into SubSurfWiki and here's how they stack up when you account for how long they've been available:

It is clear that AVO* has performed quite poorly compared to its peers! The retention rate (installs/downloads) is 100% — the price tag buys you loyalty and even a higher perceived value perhaps? But the hit in adoption is too much to take. 

There are other factors: quality, relevance, usefulness, ease-of-use. It's hard to be objective, but I think AVO* is our highest quality app. It certainly has the most functionality, hence this experiment. It is rather niche: many geological interpreters may have no use for it. But it is certainly no more niche than Elastic*, and has about four times the functionality. On the downside, it needs an internet connection for most of its juicy bits.

In all, I think that we might have expected 200 installs for the app by now, from about 400–500 downloads. I conclude that charging $2 has slowed down its adoption by a factor of ten, and hereby declare it free for everyone. It deserves to be free! If you were one of the awesome early adopters that paid a toonie for it, I have only this to say to you: we love you.

So, if you have an Android device, scan the code or otherwise hurry to the Android Market!

What we did over the summer holidays

The half-life of a link is hilariously brief, so here is an attempt to bring some new life back into the depleted viewership of our summer-time blogging. Keep in mind that you can search for any of the articles on our blog using the search tool, shown here, or sign up for email updates lower down on the side bar, for hands-free, automated Agile goodness every time we post something new.  

Well worth showing off, 4 July: This post was a demonstration of the presentation tool Prezi applied to pseudo-digital geoscience data. Geoscience is inherently visual and scale dependant, so we strive to work and communicate in a helicoptery way. I used Prezi to navigate a poster presentation on sharing geo-knowledge beyond the experts. 

Geophysical stamps—Geophone, 15 July: Instalment 3 of Matt's vintage German postage stamps was a tribute to the geophone. This post prompted a few readers to interject with suggestions and technical corrections. We strive for an interactive, dynamic and malleable blog, and their comments certainly improved the post. It was a reminder to be ready to react when you realize someone is actually reading your stuff. 

Petrophysics cheatsheet, 25 July and its companion post: Born out a desire to make a general quick reference for well logs, we published the Petrophysics cheatsheet, the fourth in our series of cheatsheets. In this companion post, you can read why petrophysics is hard. It sits in a middle ground between drilling operations, geoscience, and reservoir engineering, and ironically petrophysical measurements seldom measure the properties we are actually interested in. Wireline data is riddled with many service providers and tool options, data formats, as well as historical and exhaustive naming conventions.

How to cheat at spot the difference, 3 Aug: Edward Tufte says, "to clarify, add detail". Get all your data into one view to assist your audience in making a comparison. In this two-part post Matt demonstrated the power of visual crossplotting using two examples: a satelite photo of a pyroclastic flow, and a subsurface horizon with seismic attributes overlain. Directly mapping partially varying properties is better than data abstractions (graphs, tables, numbers, etc). Richer images convey more information and he showed us how to cheat at spot the difference using simple image processing techniques.

Digital rocks and accountability, 10 Aug: At the First International Workshop in Rock Physics, I blogged about two exciting talks on the first day of the conference on the promise of digital rock physics and how applied scientists should strive to be better in their work. Atul Gawande's ternary space of complexity could serve as tool for mapping out geoscience investigations. Try it out on your next problem and ask your teammates to expose the problem as they see it.

Wherefore art thou, Expert?, 24 Aug: Stemming from a LinkedIn debate on the role of service companies in educating and empowering their customers, Matt reflected on the role of the bewildered generalist in today's upstream industry. Information systems have changed, perfection is a myth and domain expertise runs too deep. Generalists can stop worrying about not knowing enough, specialists can builder shallower and more accesible tools, and service companies can serve instead of sell. 

Pseudogeophysics, 31 Aug: Delusion, skeptisicm, and how to crack a nut. This post drew comments about copyright control and the cost of lost opportunity; make sure to read the comments section of this post.

So yeah, now go catch up on your reading.