# Geophysical Assumptions

If you assume you make an ASS of yoU and ME.

That may be true in personal connections, but assumptions are necessary in physics. Particularly, geophysics needs all the assumptions it can get. The subsurface isn’t exactly nice, giving us data. We can only look from one side (mostly). The Earth itself filters our signal, so that the frequencies are attenuated dependent on the data. This messes up both the information and the resolution we can obtain.

Once the interpreter gets the stacked cube, not only does she get a seismic cube, but also a cube filled with assumptions. Today we will venture in the wonderful world of assumptions in seismic data. This will get messy, so it will only be for those that liked to play with mud as kids.

## High-frequency assumption

Generally, we see seismic waves as waves or as rays. Rays are just lines from the source to the wave-front. We use both concepts, because both concepts have very important properties that we need when looking at seismic data.

Waves are very good for getting the nitty gritty details, but evaluating wavefields globally is very computationally expensive. Of course, as computers got better, we can run these models faster, however, it’s still more expensive than rays. This is for the simple reason that we can evaluate rays with a triangle ruler, pencil and a piece of paper. Raytracing is great and it is even used in computer graphics to make games look nice.

Problem is, we make one very heavy assumption, when using rays, namely, the high-frequency assumption. If you have never heard about ray theory before, don’t despair. This is the same thing as tuning in seismic. We need high frequencies to resolve a tuning layer. The same physics applies to a theoretical ray, to get the ray nice thin and sharp, we have to assume that very high frequencies are available.

Coincidentally, they aren’t for two reasons. The subsurface filters out high frequencies due to attenuation being stronger in high frequencies. Additionally, once we perform stacking, high frequencies are often attenuated somewhat more, because small numerical imprecisions affect misalign a peak of a high frequency wiggle stronger than a low frequency.

## Born Approximation

In a stacked cube, we’d prefer only to have primary reflections. They are, after all, what represents the subsurface structure closest. Anything that isn’t a primary, we call multiples. We use a vast amount of multiple attenuation techniques to remove them from the data.

Once we get to a point, where the Born approximation is applied, hopefully all multiples are gone from the data. That is, because the Born approximation assumes single scattering. This approximation is usually baked into FWI.

## Isotropy

Going left would be so nice. Anisotropy at work.

Intuitively, our world is isotropic. We expect that a flashlight will work the same in all directions. However, anyone driving to work also knows that it takes much longer into the city in the mornings. Traffic jams make the driving velocity very anisotropic, as those leaving the city can go full speed, while you get a chance to exercise patience.

In the subsurface we have two contributors to anisotropy. One is aligned fractures, or more general aligned structural features. These cause the seismic wave to have a preferred direction of propagation, just like the morning traffic jam. But there’s more, anisotropy is well studied in crystalline structures. If you want to make your nerd-petrophysicists day talk to them about anisotropy.

A crystal can have an inherent preferred direction of propagation. Usually, that is irrelevant on seismic scales. When crystals align due to e.g. flow, that forms a macro pseudo-crystalline structure that inherits the properties of the micro crystal. Personally, I think this is amazing, what’s even greater, you can find these structures in nature. The mid-oceanic ridge flows in an aligned direction and with the mantle material being ductile, some crystals align in the direction of flow.

### Wrapping up

These thoughts formed in a fruitful discussion around machine learning in seismic data at the Edinburgh Time-Lapse Project (ETLP). What assumptions do you see used in geophysics?

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#### Jesper Dramsch

... is a geophysicist by heart. He works at the intersection of machine learning and geoscience. He is the founder of The Way of the Geophysicist and a deep learning enthusiast. Writing mostly about computational geoscience and interesting bits and pieces relevant to post-grad life.
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