# 2+2=5 in geophysics. Anomalies, uncertainties and pataphysics: just blame it to the wind

Anomalies

“In all attempts made until today to prove that 2+2=4, nobody has ever considered the wind velocity” writes Raymond Queneau in a short writing called A Few Brief Remarks on the Aerodynamic Properties of Addition.

Queneau, was a member of the «College de ‘Pataphysique », a collective of avant-garde writers and artists, created in 1948. ‘Pataphysics is a surrealist, scientific (maybe pseudo-scientific) discipline with the characteristic of resisting to being described with a single simple definition. It is the science of imaginary solutions: these solutions do not have the generality of scientific theories. The ‘pataphysicist is more modest and cautious than an exact scientist, he is satisfied with particular solutions. ‘Pataphysics has been described as the science of exceptions.

And this sounds quite familiar: some geophysicists spend their lives looking for anomalies. Which are indeed exceptions, ‘deviations from a uniform, predictable field’ (Sheriff, 1991). Anomalies are interpreted to find interesting and valuable objects: the gravity anomaly of a golden egg is shown in the following figure. The egg is modelled as two halves of ellipsoid with different radius dimensions. Gold is 19.30g/cm3, egg 1.03g/cm3.

Some of the definitions of anomaly quickly become rather ‘pataphysical. The anomaly is the difference between the observation of the reality and what we would observe if the reality was more uniform and less interesting than it is. This could have been written by Boris Vian («Je m’applique volontiers à penser aux choses auxquelles je pense que les autres ne penseront pas»).

Uncertainties

Geophysics and ‘pataphysics share the interest for anomalies; but they also can have similar relationships with the exceptions to the rules. Often we invoke exceptions to explain why results are different from predictions.

The problem is that often, to a question like “how much is 2+2?” we should answer: “ehm, well, between 3 and 5”. Maybe the poor precision of this answer would be a problem, when decisions have to be taken based on our predictions. But certainly neglecting the uncertainty doesn’t solve the problem: and the chronic overconfidence, with the resulting underestimation of the uncertainty, can be one of the reasons for deceiving estimations.

Sometimes the model simplification brings us the wrong answer and makes us miss that, in our case, 2+2=5. It only looks absurd: if the first 2 is actually a rounded 2.4, and the second is a rounded 2.3, the sum is closer to 5 than to 4.

The equation 2+2=5 attracted the interests of writers and thinkers (Dostoevsky, Orwell, Queneau, and let me include Radiohead), but also mathematicians, like Houston Euler, who says:

2+2=5, but only for large quantities of “2”…

A posteriori, 2+2 can be 5, 4 or 3,

just tell me how much you want it to be

When it turns out, after drilling, that 2+2 doesn’t give a simple 4, some people use excuses that are similar to the aerodynamic explanation of Queneau: the addition can be performed well only in calm weather, as the wind displaces the integer numbers, and can make fall a 1, very light number, into a calculation where it does not belong, or the wind blows away one of the two and the little cross and produces an absurd 2=4.

Geophysicists have to be a bit more serious, and in order to blame the wind we need a bit more of argumentation. Looking for culprits for the prediction failure: we can blame the acquisition parameters; we can blame some sort of near-surface wizardry; or anisotropy, always useful scapegoat. Or, the easiest, we can always blame the noise: on land, the wind noise can be a serious issue, reducing the signal-to-noise ratio. And the wind is responsible also for the swell noise at sea.

Explanations and exceptions

Besides finding someone to blame, once we know the result we can investigate and find explanations: and we can find exceptions to the rules. In the 1984 Orwell interrogation room for thought-criminals, we could even explain to O’Brien why 2+2=5.

Normally nothing should be wrong in the calculations, because 2+2 must to give a 4. And physical laws do not have many exceptions: probably we have to add another term to the equation, remove an approximation from the model to make it less approximate.

Sometimes, the answer is the least obvious. You could find out that you data do not match the topography: and discover that the wind has moved the topography.

Your old elevation model is not accurate anymore, because the wind has moved the dunes.

The sand is pushed up the dune’s windward face, then down the steeper slip face: there is a large mass transfer, and with the strong winds of a single sand storm the dune can move by several meters.

The barchan dune, which forms with a constant wind direction and has a crescent shape, can move by tens of meters per year. An example of a moving barchan dune is drawn in the figure below. You can see its changing shape, and you can appreciate that it moves faster than the small linear dune on its side. This is a drawing, inspired by Google Earth satellite images.

. If you want to see more, you can read a paper on moving dunes here (http://arxiv.org/pdf/1301.1290.pdf).

The wind moves the dunes, but also creates waves at sea: the wind transfers energy to the waves because of the pressure difference due to the sheltering effect of the wave crest, which creates a higher pressure on the rear face of the wave. So the wind is indirectly responsible for the swell noise.