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Shielding calculations: sector averages vs. series of points

May 20, 2009

Q: Regarding your comment on the shielding calculator page:

“This procedure means that in the field you should have approximated the horizon by a series of points with straight lines between them. Note that this is not the same as approximating the horizon by the average elevation angle in a series of equal sectors: the latter procedure is inappropriate because the relationship between rise angle and cosmic-ray shielding is nonlinear, and it will underestimate the actual shielding for heavily shielded sites.”

I have been using a procedure, based on Dunne et al (1999, eq 6), of measuring angles in equal sectors using a clinometer. I record the angle at the azimuth, compute the shielding for that sector, sum over all sectors, then divide by the number of sectors. I think this is the method you warn against. Yet, I obtain very nearly the same result as you even in a well-shielded setting.

A: Most of the time for small shielding this will give a similar result. However, the sector-averaging scheme is, as noted above, incorrect because of the nonlinearity of the angular dependence of the cosmic-ray flux. This becomes significant when you are averaging out large changes in angle. For example, if you were standing in the middle of Torres del Paine and the horizon in one of your sectors started at 0, went up to 70, down to 0, up to 70, down to 0, etc. and so on, and you took it to be 35 all the way across, you would get a very wrong answer.

The way to think about it in math is to take the worst-case scenario where your sector has width a (radians), half of your sector has a shielding elevation of theta, and half of your sector has zero shielding. In this case, the fraction cosmic-ray flux obstructed calculated correctly would be:

\frac{a/2}{2\pi} \sin^{3.3}{(\theta)}

and calculated (incorrectly) using an average angle for the entire sector would be

\frac{a}{2\pi} \sin^{3.3}{(\theta/2)}

This shows the difference between these two formulae for a sector
angle of 90 degrees, and various values of theta:


I encourage people to use the points-on-the-horizon method and not the sector-average method because: 1) it is in fact somewhat more physically correct, 2) it is a less subjective measurement and does not require you to estimate or average anything in your head in the field, and 3) it preserves the most information about the actual horizon, thus making it easier to incorporate future improvements in the calculation scheme. For example, if we were to find later that the angular dependence is more nonlinear than we thought, you would not be sure how accurately you could recalculate your old results if all you had recorded were the averages.

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