Don’t ask me for help unless
you have Googled it first. Anonymous
--------------------------
My last Seismos column in The Leading Edge was a slightly-modified version of my Mathematica Strat Column blog entry. It showed that an interactive stratigraphic column could be made using the Mathematica Manipulate function. At the SEG meeting in Houston last week, I spoke a minute with Jim Gaiser who was discovering all the great things that Manipulate can do.
It made me think that a
little discussion would be useful. Manipulate has the full power of M’s
analytical, numerical, and graphical behind it, and allows you to set
parameters via slider bars or other kinds of controllers. This may not
sound like much, but it is immensely powerful.
Consider for example an
equation with one coordinate (x) and 4 parameters (a,b,c,d): y = a
+ b x + c x^2 + d x^3. Given numerical values for (a,b,c,d) we can make a
plot in any graphics system, or we can fix (a,b,c) and plot a family of curves
associated with various values of d. Or we could let, say, c be the
parameter that takes on various values, giving another family of curves. It
would also be useful to make a 3D plot of the equation with axes of x and one
of the parameters, or maybe even a 3D surface movie as we cycle through a
second parameter. But none of these methods really lets us figure out the
effect of all 4 parameters. Manipulate
let's us make a single plot with default values for (a,b,c,d) and slider bars
to play around with other values of these parameters. Again it sounds
insignificant, but it is not. What we are actually doing is exploring a
multidimensional parameter space. Our example has 5 parameters, but there
could be 10, 20, or more. Clearly when you get into the hundreds or
thousands of parameters, an interactive tool like Manipulate will cease to be useful. But a universe of problems
have 15 parameters or less, including important things like Gassmann fluid
substitution.
For several years, Wolfram
Research (the makers of M) have maintained a contributed site of Manipulate examples from M users around
the world. The site is called The
Wolfram Demonstrations Project (Google it).
To get started, look for “Weibull Statistics for Fracture Data” or “Anisotropic
Elasticity” (see Figure 1). If you have an M license, the Demonstrations site
is a gold mine of source code. Without a
license, a demonstration will cycle through parameter selections. To get full manipulation capability for
demonstrations in a browser window without an M license, just download the free
Wolfram CDF Player. CDF (computable
document format) is analogous to the PDF format, but allows results to be
recomputed. This is a step toward
reproducible research, a subject of long interest to Stanford’s Jon Claerbout (Google
Claerbout reproducible research).
Manipulate
can be very useful for interpretation.
To give one example, consider the zero-dip 4-layer refraction problem.
The theory behind this problem can be found in many standard texts (e.g., Telford
et al., 1976, p.278-281). The equations give travel time as a function of
offset for the direct wave and refraction arrivals from base of layers 2, 3, and 4.
The equations are rather complicated and depend on several model
parameters, including thickness and velocity of each layer, the critical angles,
etc. Someone good in math can get it pretty quickly, but my geologists
would struggle to see how it works and, more importantly, how it relates to
field data.
In August of 2013, we did a
trial seismic experiment (Figure 2) on the lawn of Old Main, the original 1871
building at the University of Arkansas. Well, I went back to that data and manually
picked first breaks using SeismicUnix, creating a list of (time,offset) pairs
that were converted to CSV format in a spreadsheet. Bringing this
into M, allows us to make a Manipulate
widget that overlays theoretical refraction travel time curves on the observed
data points. Geology students can then adjust model parameters until an
acceptable fit is achieved (Figure 3).
In other words, we are
doing 7-parameter manual inversion using M’s Manipulate function. Unlike automatic inversion, students gain an
understanding of the role played by various parameters and can get results
despite minor data problems. On the
latter topic, note there is a section of time-shifted data between 80 and 150
ft, likely a trace header geometry bust. No problem, with this intuitive
interface and display we can work around the time shift, and fit the suspect
interval parallel to measured data.
From state survey geological maps and various building
excavations, it is known that Pennsylvanian shale and sandstone lie below
the soil layer at this site and deeper Miss. Fayetteville shale is found in outcrop 80 ft down
the hill. At some depth beneath our test site lies Pitkin limestone representing the major Miss/Penn unconformity. Such is the knowledge from 140
years of surface geology. With this small seismic experiment we can now estimate
Pitkin at 52 ft below the lawn of Old Main. Ya gotta love geophysics.
Suggested reading:
Telford, W. M.,
Geldart, L. P., Sheriff, R. E. and Keys, D. A. 1976. Applied Geophysics. Cambridge University Press.
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| Figure 1. Screen shot of Megan Fray’s elastic anisotropy Manipulate example on the Wolfram Demonstrations Project web site. The 3D plot can be interactively rotated for better inspection. |
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Figure 2. Early arrival plot of hammer-source field
seismic data from U Arkansas campus.
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Figure 3. Interactive 4-layer refraction interpretation
using Mathematica Manipulate function,
showing picked first arrivals (red dots) and theoretical travel time curves
(lines).
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