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Tuesday
Jan292013

## Great geophysicists #7: Leonhard Euler

Leonhard Euler (pronounced 'oiler') was born on 15 April 1707 in Basel, Switzerland, but spent most of his life in Berlin and St Petersburg, where he died on 18 September 1783. Has was blind from the age of 50, but took this handicap stoically—when he lost sight in his right eye at 28 he said, "Now I will have less distraction".

It's hard to list Euler's contributions to the toolbox we call seismic geophysics—he worked on so many problems in maths and physics. For example, much of the notation we use today was invented or at least popularized by him: (x), e, i, π. He reconciled Newton's and Liebnitz's versions of calculus, making huge advances in solving difficult real-world equations. But he made some particularly relevant advances that resonate still:

• Leonardo and Galileo both worked on mechanical stress distribution in beams, but didn't have the luxuries of calculus or Hooke's law. Daniel Bernoulli and Euler developed an isotropic elastic beam theory, and eventually convinced people you could actually build things using their insights.
• Euler's equations of fluid dynamics pre-date the more complicated (i.e. realistic) Navier–Stokes equations. Nonetheless, this work continued into vibrating strings, getting Euler (and Bernoulli) close to a general solution of the wave equation. They missed the mark, however, leaving it to Jean-Baptiste le Rond d'Alembert
• optics (also wave behaviour). Though many of Euler's ideas about dispersion and lenses turned out to be incorrect (e.g. Pedersen 2008, DOI 10.1162/posc.2008.16.4.392), Euler did at least progress the idea that light is a wave, helping scientists move away from Newton's corpuscular theory.

The moment of Euler's death was described by the Marquis de Condorcet in a eulogy:

He had full possession of his faculties and apparently all of his strength... after having enjoyed some calculations on his blackboard concerning the laws of ascending motion for aerostatic machines... [he] spoke of Herschel's planet and the mathematics concerning its orbit and a little while later he had his grandson come and play with him and took a few cups of tea, when all of a sudden the pipe that he was smoking slipped from his hand and he ceased to calculate and live.

"He ceased to calculate," I love that.

Tuesday
Nov272012

## 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:

$\large \dpi{120} m\ddot{x} = -kx$

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.

Monday
Sep102012

## Great geophysicists #5: Huygens

Christiaan Huygens was a Dutch physicist. He was born in The Hague on 14 April 1629, and died there on 8 July 1695. It's fun to imagine these times: he was a little older than Newton (born 1643), a little younger than Fermat (1601), and about the same age as Hooke (1635). He lived in England and France and must have met these men.

It's also fun to imagine the intellectual wonder life must have held for a wealthy, educated person in these protolithic Enlightenment years. Everyone, it seems, was a polymath: Huygens made substantial contributions to probability, mechanics, astronomy, optics, and horology. He was the first to describe Saturn's rings. He invented the pendulum clock.

Then again, he also tried to build a combustion engine that ran on gunpowder.

Geophysicists (and most other physicists) know him for his work on wave theory, which prevailed over Newton's corpuscles—at least until quantum theory. In his Treatise on Light, Huygens described a model for light waves that predicted the effects of reflection and refraction. Interference has to wait 38 years till Fresnel. He even explained birefringence, the anisotropy that gives rise to the double-refraction in calcite.

The model that we call the Huygens–Fresnel principle consists of spherical waves emanating from every point in a light source, such as a candle's flame. The sum of these manifold wavefronts predicts the distribution of the wave everywhere and at all times in the future. It's a sort of infinitesimal calculus for waves. I bet Newton secretly wished he'd thought of it.

Tuesday
Aug142012

## Great geophysicists #4: Fermat

This Friday is Pierre de Fermat's 411th birthday. The great mathematician was born on 17 August 1601 in Beaumont-de-Lomagne, France, and died on 12 January 1665 in Castres, at the age of 63. While not a geophysicist sensu stricto, Fermat made a vast number of important discoveries that we use every day, including the principle of least time, and the foundations of probability theory.

Fermat built on Heron of Alexandria's idea that light takes the shortest path, proposing instead that light takes the path of least time. These ideas might seem equivalent, but think about anisotropic and inhomogenous media. Fermat continued by deriving Snell's law. Let's see how that works.

We start by computing the time taken along a path:

$\fn_cm t = \frac{\sqrt{a^2 + x^2}}{v_1} + \frac{\sqrt{b^2 + y^2}}{v_2}$

Then we differentiate with respect to space. This effectively gives us the slope of the graph of time vs distance.

$\fn_cm \frac{\mathrm{d}t}{\mathrm{d}x} = \frac{x}{v_1 \sqrt{a^2 + x^2}} - \frac{y}{v_2 \sqrt{b^2 + y^2}}$

We want to minimize the time taken, which happens at the minimum on the time vs distance graph. At the minimum, the derivative is zero. The result is instantly recognizable as Snell's law:

$\fn_cm 0 = \frac{\sin\theta_1}{v_1} - \frac{\sin\theta_2}{v_2}$

### Maupertuis's generalization

The principle is a core component of the principle of least action in classical mechanics, first proposed by Pierre Louis Maupertuis (1698–1759), another Frenchman. Indeed, it was Fermat's handling of Snell's law that Maupertuis objected to: he didn't like Fermat giving preference to least time over least distance.

Maupertuis's generalization of Fermat's principle was an important step. By the application of the calculus of variations, one can derive the equations of motion for any system. These are the equations at the heart of Newton's laws and Hooke's law, which underlie all of the physics of the seismic experiment. So, you know, quite useful.

### Probably very clever

It's so hard to appreciate fundamental discoveries in hindsight. Together with Blaise Pascal, he solved basic problems in practical gambling that seem quite straightforward today. For example, Antoine Gombaud, the Chevalier de Méré, asked Pascal: why is it a good idea to bet on getting a 1 in four dice rolls, but not on a double-1 in twenty-four? But at the time, when no-one had thought about analysing problems in terms of permutations and combinations before, the solutions were revolutionary. And profitable.

For setting Snell's law on a firm theoretical footing, and introducing probability into the world, we say Pierre de Fermat (pictured here) is indeed a father of geophysics.

Friday
Mar232012

## The spectrum of the spectrum

A few weeks ago, I wrote about the notches we see in the spectrums of thin beds, and how they lead to the mysterious quefrency domain. Today I want to delve a bit deeper, borrowing from an article I wrote in 2006.

### Why the funny name?

During the Cold War, the United States government was quite concerned with knowing when and where nuclear tests were happening. One method they used was seismic monitoring. To discriminate between detonations and earthquakes, a group of mathematicians from Bell Labs proposed detecting and timing echoes in the seismic recordings. These echoes gave rise to periodic but cryptic notches in the spectrum, the spacing of which was inversely proportional to the timing of the echoes. This is exactly analogous to the seismic response of a thin-bed.

To measure notch spacing, Bogert, Healy and Tukey (1963) invented the cepstrum (an anagram of spectrum and therefore usually pronounced kepstrum). The cepstrum is defined as the Fourier transform of the natural logarithm of the Fourier transform of the signal: in essence, the spectrum of the spectrum. To distinguish this new domain from time, to which is it dimensionally equivalent, they coined several new terms. For example, frequency is transformed to quefrency, phase to saphe, filtering to liftering, even analysis to alanysis.

Today, cepstral analysis is employed extensively in linguistic analysis, especially in connection with voice synthesis. This is because, as I wrote about last time, voiced human speech (consisting of vowel-type sounds that use the vocal chords) has a very different time–frequency signature from unvoiced speech; the difference is easy to quantify with the cepstrum.

### What is the cepstrum?

To describe the key properties of the cepstrum, we must look at two fundamental consequences of Fourier theory:

1. convolution in time is equivalent to multiplication in frequency
2. the spectrum of an echo contains periodic peaks and notches

Let us look at these in turn. A noise-free seismic trace s can be represented in the time t domain by the convolution of a wavelet w and reflectivity series r thus

$s(t) = w(t) \ast r(t)$

Then, in the frequency f domain

$S(f) = W(f) \times R(f)$

In other words, convolution in time becomes multiplication in frequency. The cepstrum is defined as the Fourier transform of the log of the spectrum. Thus, taking logs of the complex moduli

$\ln|S| = \ln|W| + \ln|R|$

Since the Fourier transform F is a linear operation, the cepstrum is

$\mathcal{F}(\ln|S|) = \mathcal{F}(\ln|W|) + \mathcal{F}(\ln|R|)$

We can see that the spectrums of the wavelet and reflectivity series are additively combined in the cepstrum. I have tried to show this relationship graphically below. The rows are domains. The columns are the components w, r, and s. Clearly, these thin beds are resolved by this wavelet, but they might not be in the presence of low frequencies and noise. Spectral and cepstral analysis—and alanysis—can help us cut through the seismic and get at the geology.

Time series (top), spectra (middle), and cepstra (bottom) for a wavelet (left), a reflectivity series containing three 10-ms thin-beds (middle), and the corresponding synthetic trace (right). The band-limited wavelet has a featureless cepstrum, whereas the reflectivity series clearly shows two sets of harmonic peaks, corresponding to the thin- beds (each 10 ms thick) and the thicker composite package.

References

Bogert, B, Healy, M and Tukey, J (1963). The quefrency alanysis of time series for echoes: cepstrum, pseudo-autocovariance, cross- cepstrum, and saphe-cracking. Proceedings of the Symposium on Time Series Analysis, Wiley, 1963.

Hall, M (2006). Predicting stratigraphy with cepstral decomposition. The Leading Edge 25 (2), February 2006 (Special issue on spectral decomposition). doi:10.1190/1.2172313

Greenhouse George image is public domain.

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