How many geophysicists does it take to guess what is inside this can?


Perception and reality do not always match: and a single point of view seldom captures the multifaceted truth of nature.

What can be seen, is only a ridiculously small and partial portion of the reality; as geophysicists, we can become very modest about our perceptions. At least professionally we try to combine different observations, points of view, projections.

The famous image of the cylinder projected on two walls as a yellow circle and blue rectangle doesn’t surprise us very much: I could not find who the author is, so I redrew it with the famous-to-be can of the title. The shapes of the shadows comes from the projection, but their colours comes from the walls: in a sense, from the data acquisition.



We have to say that this would be a very trivial shadow puppetry, compared to the elaborate art expressions developed as ancestors of cinema hundreds of years ago (

Moreover shadows and perception were already associated by Plato, 2500 years ago, in the Allegory of the cave. People who have been chained to the wall of a cave all their lives, watching shadows projected on one wall by the light of a fire: these shadows are their only reality. For Plato, the philosopher is a prisoner freed from the cave, who goes out and perceives the reality.

A geophysicist, if chained to the wall of Plato’s cave, maybe would not strive to get out: “the pay is good, the food is not bad and that in the cave at least it doesn’t rain”. But certainly he would try to understand what the projected object is, trying at least to see the shadows on two walls, like in the figure. He will start thinking of illumination of the objects: he would know that multiple projections allow to identify the shape of the projected object.

But the point here is to understand what is inside the can: just because, as the saying goes, we don’t want to open a can of worms.

Projecting is still a good idea, but we can’t use the light of a fire: we need something that can go through the matter, like X-ray photons, so that the shadow of the object is not completely dark. One projection will be one radiography: a can of sardines could look like this.


A single projection will be ambiguous: in a radiography you can see that bones are broken because you know how they should look like, and because they have been illuminated from the right angle. But if you are trying to understand the grains and pores in a rock sample, one single projection won’t be very useful.

To reconstruct an image, and to understand the reality, we need many projections, many points of views. To see how the projections of an empty can look like, we can use the integral transform conceived in 1917 by Johan Radon. If you Radon transform the can, to look inside it from different angles, you get something like the image below: the typical bone-looking colormap of the first line of images shows in white what absorbs more. It is the inverse of the colormap below.


Johan Radon derived also the formula for the inverse transform: when we have the projections, we can get the tomographic reconstructions, and see inside the can.

Unfortunately the airport security refused to lend me their machine to scan the can, and I have no real projections to show and to invert. But fortunately I know that the can is empty and that it would look like the synthetic projections above.

But how many projections would we need to see that it is empty? When objects are complex, many points of views are needed, and the viewpoints have to be different. If you look from the same point of view, you will see the same incomplete partial view.

Sometimes our stored knowledge about the world allows to understand the nature of the unknown object from a partial view, from a single projection, that we interpret. From the single, fake projection of the sardines shown above, I could conclude that it is a sardine box. And when you tell me that the can in empty, I can mention partial sardine saturation, or a paleo-sardine can. And for my manager who paid to drill the can and open it, I can say that a simulation run after the well showed that the sardines flew away in 2009.


This is not only about geophysics: refusing to consider other viewpoints doesn’t bring us closer to the reality. And maybe the world would be a better place if people knew that they see only one side.




While writing this I thought about discussing whether a CT scan could identify the content of the cans of the 1961 artwork of Piero Manzoni: but this would have forced me to use some explicit words.




Appendix: the transform that can transform a zebra into a donkey

The transformation of a donkey into a zebra requires some painting: if you have been to Tijuana you must have seen the Tijuana Zebras, or Zonkeys (just type it in google). The same strategy was used in the Gaza zoo more recently.

Here we want to perform the inverse transform, from a zebra to a donkey: or just remove the fake stripes from a painted donkey.

The Radon transform is widely used in seismic processing, since it identifies linear events, lines and stripes. Three projections of the zebra are shown in the figure: they are cumulative transform, the Radon is the result when the ray gets out of the object, and for each angle you get one line only, that is collected as a column in the 360 degree Radon transform at the bottom of the figure.


The good thing is that the Radon transform projects the stripes and focuses them as it does with linear events. You can use it as we use linear tau-p and remove the stripes from the zebra, transform it into a donkey.


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