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Don’t neglect your math

Doing mathematics is like exercising. Do a little bit every day and you stay in shape, either intellectually (in the case of math) or physically (in the case of exercising). Neglect it and your muscles (intellectual or physical) fade away.

Geophysics is a hard science. By that I mean that it is a science based on ‘hard’ facts, but also that it can be difficult. We all struggled through tough math and physics classes at university to get our degrees. But once we were in the working world, especially if we became seismic interpreters, we tended to leave the details to the specialists. Indeed, picking up a copy of Geophysics and trying to read every article is a daunting task. And I do not expect that every exploration geophysicist should be able to understand the latest implementation of Green’s functions in anisotropic depth imaging. However, I do think that an appreciation of some of the fundamental applied mathematical ideas in our profession can go a long way towards enhancing your enjoyment and appreciation of your day-to-day job.

Two examples

Let me illustrate my point with two equations. Let us start with:

$$ d = Gm $$

where d is our data, a set of n geophysical observations, m is our model, a set of k model parameters, and G is a linearized relationship that relates the observations to the parameters. This ubiquitous equation can be found in every area of geophysics, from seismology through potential fields to electromagnetic theory. The simplicity of the way I have written the equation hides the fact that d is usually written as an n-dimensional vector, m as a k-dimensional vector, and G as an n row by k column matrix. 

Solving the equation is a little more difficult. Since n is usually greater than k, the solution can be written:

$$ m = (G^\mathrm{T} G + \lambda I)^{-1} G^\mathrm{T} d = C^{-1} h $$

where C is the autocorrelation matrix found by multiplying the G matrix by its transpose \(G^\mathrm{T}\) (and adding a little pre-whitening by multiplying the value λ by I, the k by k identity matrix), and h is the zero-lag cross-correlation vector, found by multiplying the transpose of the G matrix by the data vector. Again, this equation, sometimes called the Normal Equation, is ubiquitous in geophysics. It is the basis of deconvolution, AVO attribute analysis, post- and pre-stack inversion, refraction and reflection statics, and so on. So, what lesson should we take away from these equations?

My advice

The way that you react to these equations tells me a lot about you as a geophysicist. If you are thinking: ‘what’s the big deal, I use those types of equations every day,’ you probably don’t need my advice. If you are thinking: ‘yes, I saw those equations once in a class, but haven’t thought about them for years,’ perhaps I can inspire you to look at them again. On the other hand, if you are thinking: ‘why would I ever need to use those boring-looking equations,’ you are a tougher challenge! I would recommend starting with these equations and really trying to understand them (perhaps you will need to dust off your linear algebra, and I recommend the book by Gilbert Strang). Then, pick up a copy of Geophysics, or any geophysics textbook, and see how many of the equations can be expressed in the same way. Or, take some quantitative industry training courses and see what the mathematics is really telling you about your data.

I guarantee it will be good for you!


References

Strang, G (2009). Introduction to Linear Algebra. Wellesley–Cambridge Press, 574 pages; math.mit.edu/linearalgebra

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