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Geological inversion

Geological interpretation is the act of reasoning backwards. Geologists do what mathematicians and geophysicists call inverse problems all the time, but in their heads! 

For instance, if I give you a rock, it is easy to determine its density, a property that may be of interest to you. All you need is a theory (gravity) and a method (say an Archimedes-like displacement of water). But if I give you density as a starting point, it is considerably harder — indeed, it’s impossible — to reason backwards to reconstruct the rock from which it came. This is an inverse problem, and attempting to solve it is called inversion.

The difficulty of inverse problems is not due to some lack of knowledge about constituents and physical properties, but lies in all the different possible assemblies of grains, minerals, and fluids that could give the same result. Given such a lowly amount of data, we aren’t adequately equipped to describe the rock any further without bringing in a priori information — the rock’s appearance perhaps, or knowledge of where it was collected. Such problems are called under-determined: the number of unknowns exceeds the number of relationships or equations we can write down. We can’t possibly solve them exactly, but it is still appealing to describe them approximately.


The opposite of an inverse problem is a forward problem. It uses straight-up reasoning or deduction starting with a model or theory and going towards a specific observation. The goal is to construct a test that can be compared with real data to confirm (or not) a theory you started with. Given this sedimentary environment, what sedimentary features do I expect to see? How will grain size vary laterally?

Another example: the classic forward model of a seismic trace links the unknowable earth (on the left in the figure) to a 1D record of lithology, to the speed of sound in those rocks, to acoustic reflectivity, and so to a seismogram (on the right). Seismic inversion is the inverse problem of deducing rocks from seismic. 

Compared to interpretation these are easy problems, not because they are uncomplicated, but because there is a unique path between cause (model) and effect (data). Forward problems shouldn’t be overlooked just because they are easy, however. Indeed, it is because they are solvable that we can use them to validate an interpretation or teach us what to look for.


…and backwards

In this language, geological interpretation falls into the class of hard inverse problems — equivalent to moving to the left in the figure above. It begins with specific measurements, then we use inductive reasoning (informed guesses, basically) to detect patterns, regularities, and anomalies, and so formulate a hypothesis that we can explore. The output of a geological interpretation is a model full of assumptions. If the assumptions you brought to the problem are reasonable, the model can be considered meaningful. If you haven’t considered and tested your assumptions, you haven’t subscribed to reason.

Some are not aware, some choose to ignore, and some forget that works of geoscience are problems of extreme complexity. The only way we can cope with complexity is to make certain assumptions that make inverse problems solvable. If all you do is say, ‘here is my interpretation,’ you will be unconvincing. But if instead you ask, ‘have I convinced you that my assumptions are reasonable?’, it entirely changes the impact of your interpretation. It becomes a shared entity that anyone can look at and examine. 

Advice for a prospective geologist

Don’t neglect your math