The calculus of geology

Calculus is the tool for studying things that change. Even so, in the midst of the dynamic and heterogeneous earth, calculus is an under-practised and, around the water-cooler at least, under-celebrated workhorse. Maybe that's because people don't realize it's all around us. Let's change that. 

Derivatives of duration

We can plot the time f(x) that passes as a seismic wave travels though space x. This function is known to many geophysicists as the time-to-depth function. It is key for converting borehole measurements, effectively recorded using a measuring tape, to seismic measurements, recorded using a stop watch.

Now let's take the derivative of f(x) with repsect to x. The result is the slowness function (the reciprocal of interval velocity):

The time duration that a seismic wave travels over a small interval (one metre). This function is an actual sonic well log. Differentiating once again yields this curious spiky function:

Geophysicists will spot that this resembles a reflection coefficient series, which governs seismic amplitudes. This is actually a transmission coefficient function, but that small detail is beside the point. In this example, the creating a synthetic seismogram mimics the calculus of geology. 

If you are familiar with the integrated trace attribute, you will recognize that it is an attempt to compute geology by integrating reflectivity spikes. The only issue in this case, and it is a major issue, is that the seismic trace is bandlimited. It does not contain all the information about the earth's slowness. So the earth's geology remains elusive and blurry.

The derivative of slowness yields the reflection boundaries, the integral of slowness yields their position. So in geophysics speak, I wonder, is forward modeling akin to differentiation, and inverse modeling akin to integration? I find it fascinating that these three functions have essentially the same density of information, yet they look increasingly complicated when we take derivatives. 

What other functions do you come across that might benefit from the calculus treatment?

The sonic log used in this example is from the O-32-B/11-E-64 well onshore Nova Scotia, which is publically available but not easily accessible online.

Review: The Wave Watcher's Companion

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The Wave Watcher's Companion: From Ocean Waves to Light Waves via Shock Waves, Stadium Waves, and All the Rest of Life's Undulations
Gavin Pretor-Pinney, Perigee (USA), Bloomsbury (UK), July 2010, $22.95

This book was on my reading list, and then on my shelf, for ages. Now I wish I'd snapped it up and read it immediately. In my defence, the end of 2010 was a busy time for me, what with turning my career upside down and everything, but I'm sure there's a lesson there somewhere...

If you think of yourself as a geophysicist, stop reading this review and buy this book immediately. 

OK, now they've gone, we can look more closely. Gavin Pretor-Pinney is the chap behind The Cloud Appreciation Society, the author of The Cloudspotter's Guide, and co-creator of The Idler Magazine. He not a scientist, but a witty writer with a high curiosity index. The book reads like an extended blog post, or a chat in the pub. A really geeky chat. 

Geophysicists are naturally drawn to all things wavy, but the book touches on sedimentology too — from dunes to tsunamis to seiches. Indeed, the author prods at some interesting questions about what exactly waves are, and whether bedforms like dunes (right) qualify as waves or not. According to Andreas Baas, "it all depends on how loose is your definition of a wave." Pretor-Pinney likes to connect all possible dots, so he settles for a loose definition, backing it up with comparisons to tanks and traffic jams. 

The most eye-opening part for me was Chapter 6, The Fifth Wave, about shock waves. I never knew that there's a whole class of waves that don't obey the normal rules of wave motion: they don't obey the speed limits, they don't reflect or refract properly, and they can't even be bothered to interfere like normal (that is, linear) waves. Just one of those moments when you realize that everything you think you know is actually a gross simplification. I love those moments.

The book is a little light on explanation. Quite a few of the more interesting parts end a little abruptly with something like, "weird, huh?". But there are plenty of notes for keeners to follow up on, and the upside is the jaunty pace and adventurous mix of examples. This one goes on my 're-read some day' shelf. (I don't re-read books, but it's the thought that counts).

Figure excerpt from Pretor-Pinney's book, copyright of the author and Penguin Publishing USA. Considered fair use.

Interpreting spectral gamma-ray logs

Before you can start interpreting spectral gamma-ray logs (or, indeed, any kind of data), you need to ask about quality.

Calibrate your tool...

The main issues affecting the quality of the logs are tool calibration and drilling mud composition. I think there's a tendency to assume that delivered logs have been rigorously quality checked, but... they haven't. The only safe assumption is that nobody cares about your logs as much as you. (There is a huge opportunity for service companies here — but in my experience they tend to be focused on speed and quantity, not quality.)

Calibration is critical. The measurement device in the tool consists of a thallium-laced NaI crystal and a photomultiplier. Both of these components are sensitive to temperature, so calibration is especially important when the temperature of the tool is changing often. If the surface temperature is very different from the downhole—winter in Canada—calibrate often.

Drilling mud containing KCl (to improve borehole stability) increases the apparent potassium content of the formation, while barite acts as a gamma-ray absorber and reduces the count rates, especially in the low energies (potassium).

One of the key quality control indicators is negative readings on the uranium log. A few negative values are normal, but many zero-crossings may indicate that the tool was improperly calibrated. It is imperative to quality control all of the logs, for bad readings and pick-up effects, before doing any quantitative work.

...and your interpretation

Most interpretations of spectral-gamma ray logs focus on the relationships between the three elemental concentrations. In particular, Th/K and Th/U are often used for petrophysical interpretation and log correlation. In calculating these ratios, Schlumberger uses the following cut-offs: if uranium < 0.5 then uranium = 0.5; if potassium < 0.004 then potassium = 0.001 (according to my reference manual for the natural gamma tool).

In general, high K values may be caused by the presence of potassium feldspars or micas. Glauconite usually produces a spike in the K log. High Th values may be associated with the presence of heavy minerals, particularly in channel deposits. Increased Th values may also be associated with an increased input of terrigenous clays. Increases in U are frequently associated with the presence of organic matter. For example, according to the ODP, particularly high U concentrations (> 5 ppm) and low Th/U ratios (< 2) often occur in black shale deposits.

The logs here, from Kansas Geological Survey open file 90-27 by Macfarlane et al. shows a quite overt interpretive approach, with the Th/K log labelled with minerals (feldspar, mica, illite–smectite) and the Th/U log in uranium 'fixedness', a proxy for organic matter.

Sounds useful. But really, you can probably find just a paper to support just about any interpretation you want to make. Which isn't to say that spectral gamma-ray is no use — it's just not diagnostic on its own. You need to calibrate it to your own basin and your own stratigraphy. This means careful, preferably quantitative, comparison of core and logs. 

Further reading 

• Ross Crain's Petrophysical Handbook is a mine of well logging goodness
• Paul Glover's MSc course notes are very comprehensive on most well logging topics
• Fabricius et al. (2003). Geological Survey of Denmark and Greenland Bulletin 1, 349–366. A case stuudy from the Danish Central Graben. 

What is spectral gamma-ray?

The spectral gamma-ray log is a measure of the natural radiation in rocks. The amplitude of the signal from the gamma-ray tool, which is just a sensor with no active source, is proportional to the energy of the gamma-ray photons it encounters. Being able to differentiate between photons of different energies turns out to be very handy Compared to the ordinary gamma-ray log, which ignores the energies and only counts the photons, it's like seeing in colour instead of black and white.

Why do we care about gamma radiation?

First, what are gamma rays? Highly energetic photons: electromagnetic radiation with very short wavelengths. 

Being able to see different energies, or 'colours', means we can differentiate between the radioactive decay of different elements. Elements decay by radiating energy, and the 'colour' of that energy is characteristic of that element (actually, of each isotope). So, we can tell by looking at the energy of a photon if we are seeing a potassium atom (⁴⁰K) or a uranium atom (²³⁸U) decay. These are very different isotopes, with very different habits. We can do geology!

In fact, all sorts of radioisotopes occur naturally in the earth. By far the most abundant are potassium ⁴⁰K, thorium ²³²Th and uranium ²³⁸U. Of these, potassium is the most abundant in sedimentary rocks, but thorium and uranium are present in small quantities, and have particular sedimentological implications.

What exactly are we measuring?

Potassium ⁴⁰K decays to argon about 10% of the time, with γ-emission at 1.46 MeV (the other 90% of the time it decays to calcium). However, all of the decay in the ²³²Th and ²³⁸U decay series occurs by α- and β-particle decay, which don't always result in photon emission. The tool in fact measures γ-radiation from the decay of thallium ²⁰⁸Tl in the ²³²Th series (right), and from bismuth ²¹⁴Bi in the ²³⁸U series. The spectral gamma-ray tool must be calibrated to known samples to give concentrations of ²³²Th and ²³⁸U from its readings. Proper calibration is vital, and is temperature-sensitive (of note in Canada!).

The concentrations of the three elements are estimated from the spectral measure­ments. The concentration of potassium is usually measured in percent (%) or per mil (‰), or sometimes in kilograms per tonne, which is equivalent to per mil. The other two elements are measured in parts per million (ppm).

Here is the gamma-ray spectrum from a single sample from 509 m below the sea-floor at ODP Site 1201. The final spectrum (heavy black line) is shown after removing the background spectrum (gray region) and applying a three-point mean boxcar filter. The thin black line shows the raw spectrum. Vertical lines mark the interval boundaries defined by Peter Blum (an ODP scientist at Texas A&M). Prominent energy peaks relating to certain elements are identified at the top of the figure. The inset shows the spectrum for energies >1500 keV at an expanded scale. 

We wouldn't normally look at these spectra. Instead, the tool provides logs for K, Th, and U. Next time, I'll look at the logs.

Spectrum illustration by Wikipedia user Inductiveload, licensed GFDL; decay chain by Wikipedia user BatesIsBack, licensed CC-BY-SA.

Making images or making prospects?

Well-rounded geophysicists will have experience in each of the following three areas: acquisition, processing, and interpretation. Generally speaking, these three areas make up the seismic method, each requiring highly specified knowledge and tools. Historically, energy companies used to control the entire spectrum, owning the technology, the know-how and the risk, but that is no longer the case. Now, service companies do the acquisition and the processing. Interpretation is largely hosted within E & P companies, the ones who buy land and drill wells. Not only has it become unreasonable for a single geophysicist to be proficient across the board, but organizational structures constrain any particular technical viewpoint. 

Aligning with the industry's strategy, if you are a geophysicist, you likely fall into one of two camps: those who make images, or those who make prospects. One set of people to make the data, one set of people to do the interpretation.

This seems very un-scientific to me.

Where does science fit in?

Science, the standard approach of rational inquiry and accruing knowledge, is largely vacant from the applied geophysical business landscape. But, when science is used as a model, making images and making prospects are inseperable.

Can applied geophysics use scientific behaviour as a central anchor across disciplines?

There is a significant amount of science that is needed in the way that we produce observations, in the way that we make images. But the business landscape built on linear procedures leaves no wiggle room for additional testing and refinement. How do processors get better if they don't hear about their results? As a way of compensating, processing has deflected away from being a science of questioning, testing, and analysis, and moved more towards, well,... a process.

The sure-fire way to build knowledge and decrease uncertainty, is through experimentation and testing. In this sense this notion of selling 'solutions', is incompatible with scientific behavior. Science doesn't claim to give solutions, science doesn't claim to give answers, but it does promise to address uncertainty; to tell you what you know.

In studying the earth, we have to accept a lack of clarity in our data, but we must not accept mistakes, errors, or mediocrity due to shortcomings in our shared methodologies.

We need a new balance. We need more connectors across these organizational and disciplinary divides. That's where value will be made as industry encounters increasingly tougher problems. Will you be a connector? Will you be a subscriber to science?

Hall, M (2012). Do you know what you think you know? CSEG Recorder 37 (2), February 2012, p 26–30. Free to download from CSEG. 

Filters that distort vision

Almost two weeks ago, I had LASIK vision correction surgery. Although the recovery took longer than average, I am seeing better than I ever did before with glasses or contacts. Better than 20/20. Here's why.

Low order and high order refractive errors

Most people (like me) who have (had) poor vision fall short of pristine correction because lenses only correct low order refractive errors. Still, any correction gives a dramatic improvement to the naked eye; further refinements may be negligible or imperceptible. Higher order aberrations, caused by small scale structural irregularities of the cornea, can still affect one's refractive power by up to 20%, and they can only be corrected using customized surgical methods.

It occurs to me that researchers in optometry, astronomy, and seismology face a common challenge: how to accurately measure and subsequently correct for structural deformations in refractive media, and the abberrations in wavefronts caused by such higher-order irregularities. 

The filter is the physical model

Before surgery, a wavefront imaging camera was used to make detailed topographic maps of my corneas, and estimate point spread functions for each eye. The point spread function is a 2D convolution operator that fuzzies the otherwise clear. It shows how a ray is scattered and smeared across the retina. Above all, it is a filter that represents the physical eye.

Point spread function (similar to mine prior to LASIK) representing refractive errors of the cornea (top two rows), and corrected vision (bottom row). Point spread functions are filters that distort both the visual and seismic realms. The seismic example is a segment of inline 25, Blake Ridge 3D seismic survey, available from the Open Seismic Repository (OSR).Observations in optics and seismology alike are only models of the physical system, models that are constrained by the filters. We don't care about the filters per se, but they do get in the way of the underlying system. Luckily, the behaviour of any observation can be expressed as a combination of filters. In this way, knowing the nature of reality literally means quantifying the filters that cause distortion. Change the filter, change the view. Describe the filter, describe the system. 

The seismic experiment yields a filtered earth; a smeared reality. Seismic data processing is the analysis and subsequent removal of the filters that distort geological vision. 

This image was made using the custom filter manipulation tool in FIJI. The seismic data is available from OpendTect's Open Seismic Repository.

5 ways to kickstart an interpretation project

Last Friday, teams around the world started receiving external hard drives containing this year's datasets for the AAPG's Imperial Barrel Award (IBA for short). I competed in the IBA in 2008 when I was a graduate student at the University of Alberta. We were coached by the awesome Dr Murray Gingras (@MurrayGingras), we won the Canadian division, and we placed 4th in the global finals. I was the only geophysical specialist on the team alongside four geology graduate students.

Five things to do

Whether you are a staff geoscientist, a contractor, or competitor, it can help to do these things first:

1. Make a data availability map (preferably in QGIS or ArcGIS). A graphic and geospatial representation of what you have been given.
2. Make well scorecards: as a means to demonstrate not only that you have wells, but what information you have within the wells.
3. Make tables, diagrams, maps of data quality and confidence. Indicate if you have doubts about data origins, data quality, interpretability, etc.
4. Background search: The key word is search, not research. Use Mendeley to organize, tag, and search through the array of literature
5. Use Time-Scale Creator to make your own stratigraphic column. You can manipulate the vector graphic, and make it your own. Much better than copying an old published figure. But use it for reference.

All of these things can be done before assigning roles, before saying who needs to do what. All of this needs to be done before the geoscience and the prospecting can happen. To skirt around it is missing the real work, and being complacent. Instead of being a hammer looking for a nail, lay out your materials, get a sense of what you can build. This will enable educated conversations about how you can spend your geoscientific manpower, division of labour, resources, time, etc.

Read more, then go apply it 

In addition to these tips for launching out of the blocks, I have also selected and categorized blog posts that I think might be most relevant and useful. We hope they are helpful to all geoscientists, but especially for students. Visit the Agile blog highlights list on SubSurfWiki.

I wish a happy and exciting IBA competition to all participants, and their supporting university departments. If you are competing, say hi in the comments and tell us where you hail from. 

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: f (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.

Blurry vision and refractive power

I'm getting LASIK eye surgery today, so I've been preparing myself by learning about the eye's optics, and the surgical procedure that enhances handicapped eyes like my own. Unsurprisingly, there are some noteworthy parallels with seismic.

The eye as a gather

The human eye is akin to a common-depth point (CDP) gather. Both are like cameras constructed to focus rays at an imaging point. The retina, in the case of the eye; the reflection boundary in the case of the gather. In the eye, there are exactly four refracting interfaces at which light rays bend towards the midline and ultimately converge on the retina. In the earth, there an unknown number of interfaces, surely more than four.

Myopia, or near-sightedness, is the condition where images are focused just in front of the retina. Hyperopia, or far-sightedness, is the condition where the eyeball is too short and images would be focused behined the retina. The structure and density of the tissues in the eye have to be aligned just so, for perfect vision. If any combination of them are out of whack, you get blurry vision. Really blurry, in my case.

Characterizing blurry vision can be thought of as a two step process of measurement and validation. First, measurements of the refractive power of the eye are made with an autorefractor; quantifying the amount of first order correction needed. The correction is applied, verified, and fine-tuned by a qualitative visual assessment test. The measurement gets you close to the perfect correction; any residual adjustments may be negligible or imperceptible. And the patient, a subjective observer, is the final judge of clarity and quality of vision.

Four corrections

There are at least four ways to correct for common vision problems. Each is a different way to force the ray geometry:

• refract the light before it enters the eye (glasses),
• refract the light just above the cornea (contact lenses), 
• change the shape of the cornea using LASIK or PRK surgery, or 
• change the shape or structure of the lens (cataract surgery or implants). 

If the earth were an eye

Seismic processing is the act of measuring the refractive structure of the earth, and correcting for it's natural blurryness. Static correction, is done first in an effort to align the rays into a plane wave before it enters the 'eye'. Seismic velocity analysis is carried out on the rays, as a crude measurement of the earth's 'refractive power'. Migration, is the process of forcing geometries, mathematically instead of surgically, in order to rearrange ray paths to improve focusing. Generally speaking it's the same two-step process: measurement and validation. As with the eye, the quality of the final image is a perceptual one, coming down to subjective visual assessment. But unlike the eye, fortunately, multiple observers can share the same image, talk about it even. Changing the entire discussion about what acuity really means.

The process of vision correction goes sequentially from low order to high order. In the next post I will talk about higher order anomalies within the eye, that, once corrected, can cause super-human vision. Measurements and maps of how the eye sees show surgeons how to correct optical images. In the same vein, measurements and maps of how the seismic experiment sees, show geophysicists how to correct images in the seismic realm.

Rocks, pores and fluids

At an SEG seismic rock physics conference in China several years ago, I clearly remember a catch phrase used by one of the presenters, "It's all about rocks, pores, and fluids." He used it several times throughout his talk as an invocation for geophysicists to translate their seismic measurements of the earth into terms that are more appealing to others. Nobody cares about the VP/VS ratio in a reservoir. Even though I found the repetition slightly off-putting, he succeeded—the phrase stuck. It's all about rock, pores, and fluids.

Fast forward to the SEG IQ Earth Forum a few months ago. The message reared its head again, but in a different form. After dinner one evening, I was speaking with Ran Bachrach about advances in seismic rock physics technology: the glamour and the promise of the state-of-the-art. It was a topic right up his alley, but suprisingly, he seemed ambivalent and under-enthused. Which was unusual for him. "More often than not," he said, "we can get all the information we need from the triple combo." 

What is the triple combo? 

I felt embarrased that I had never heard of the term. Like I had been missing something this whole time. The triple combo is the standard set of measurements used in formation evaluation and wireline logging: gamma-ray, porosity, and resistivity. Simply put, the triple combo tells us about rocks, pores, and fluids. 

I find it curious that the very things we are interested in are impossible to measure directly. For example:

• A gamma-ray log measures naturally occuring radioactive minerals. We use this to make inferences about lithology.
• A neutron log measures Compton scattering in proportion to the number of hydrogen atoms. This is a proxy for pores.
• A resistivity log measures the conductivity of electrical current. We use this to tell us about fluid type and saturation.

Subsurface geotechnology isn't only about recording the earth's constituents in isolation. Some measurements, the sonic log for instance, are useful because of the fact that they are an aggregate of all three.

The well log is a section of the Thebaud_E-74 well available from the offshore Nova Scotia Play Fairway Analysis.

Update on 2013-01-18 00:32 by Evan Bianco

Part two of this post elaborates on the measurements that are an aggregate of all three.