GNGTS – Atti del 19° Convegno Nazionale / 03.02
L. Sambuelli and C. Strobbia
Diget, Politecnico di Torino
THE BUFFON’S NEEDLE PROBLEM AND THE DESIGN
OF A GEOPHYSICAL SURVEY
A correct procedure to plan a geophysical survey to detect the anomaly caused
by a body within an area could be outlined according to the following steps:
1. Modeling the response of the body;
2. Estimate the noise level;
3. Calculate the area of the anomaly above the noise level;
4. Choose a sampling interval along the profile direction (x) and choose the
profile spacing (y).
When a further processing, such as inversion, is likely to be performed, these
latter two quantities can be selected according to two strategies: sampling very
closely so that no further measurements are needed; sampling coarsely performing
a further refinement, at the needed sampling interval, in restricted areas. Depending
on the target, on the methodology, on the instruments, it can be convenient the one
or the other solution. It is anyway easy to see that would be quite useful, when the
second strategy has to be choose, to have an idea on how much coarse the
sampling could be or what could be the risk of missing the target. In other words to
know, given two orthogonal dimensions of the expected anomaly, the probability of
sampling it in a given number of points with a given profile spacing. This problem can
be reduced to find the probability that a rectangular (sides L x l and diagonal D) coin,
tossed on a floor, tiled with equi-spaced lines (d = line spacing = profile spacing),
covers one or more lines for one or more segments of a given length S. It is in fact
clear that the intercepted segment can then be sampled as many times as are
needed according to the necessity of recognizing the presence of the anomaly.
The aforementioned problem is a variation of two well known questions posed
some century ago to mathematicians involved in the foundations of probability theory:
the “franc carreau” and “the needle” problems. They have been faced to and solved
by Buffon (1777) and Laplace (1812) in the XVIII and XIX century. The studies about
this kind of problems of geometrical probability gave rise to a new branch of
mathematics: the Integral Geometry (see Kendall and Moran,1963; Santalò,1976).
Within this frame of studies, many authors focused their attention to the solution of
practical problems in various fields such as military (Walters, 1947), mineralogy
(Zhuravskii, 1932), biology (Boyarskii, 1955), ecology (Mc. Intire, 1958), cosmology
(Pepper,1929) and generally in the field of searching for objects (Koopman, 1956).
Coming closer to the geophysical applications it can be cited the paper of Shurygin
(1966) dealing with optimal drilling grid to find an elliptic layer and, with a more
mathematical point of view, the one of Duma and Stoka (1993) where the formulas
useful to calculate the probability of an elliptic coin of intersecting a rectangle-tiled
floor are given together with the particular case of line-tiled floor.
To proof the correctness of the results a Monte Carlo simulation on a synthetic
magnetic map (Fig. 1) has been performed obtaining, (Tab. 1) for different ratios
between sides, segment length and profile interval, an agreement lower than 1‰ with
the analytical formulas.
GNGTS – Atti del 19° Convegno Nazionale / 03.02
Fig. 1 - Magnetic anomaly map with the rectangle to be sampled in evidence (yellow).
Tab. 1 - Comparison between analytical and simulated results.
d
4
d>D
2
d<D
4
d>D
2
d<D
4
d>D
S
L, l, S
P theor.
P Monte Carlo
1.5
S<l<L
0.4934
0.4933
0.9050
0.9050
0.1231
0.1231
0.2464
0.2462
0.0280
0.0279
2.5
3
l<S<L
l<L<S
2
d<D
0.0559
(L = 2.9 ; l = 1.7; D = 3.36; number of trials 10^7)
0.0558
REFERENCES
Boyarskii Ya.,1955: “Statistical methods in experimental medical research”. Medgiz, Moskow (in
Russian).
Buffon G. L.,1777: “Essay d’arithmétique morale”. Supplèment à l’Histoire Naturelle, Vol.4, Paris.
de Laplace P. S.,1812: “Théorie analytique des probabilités”. Courcier, Paris.
Duma A. and Stoka M.,1993: “Hitting probabilities for random ellipses and ellipsoids”. J. Appl. Prob.,
30, 971-974.
Kendall M. G., Moran P. A. P.,1963: “ Geometrical probability”. C. Griffin & Co., London.
Koopman B. O.,1956: “The theory of search. I. Kinematic bases”. Operations Res., 4, 324-346.
McIntire G. A.,1958: “Estimation of plant density using line transect”. Ecology, 41, 319-330.
Pepper E. D.,1929: “On density distribution in stellar space”. Proc. London Math. Soc., (2). 29, 98-110.
Santalò L. A.,1976: “Integral geometry and geometric probability”. Addison-Wesley.
Shurygin A. M.,1966: “Computation of optimal drill-hole networks for searches of elliptic layers of
definite size”. Razvedka i Okhrana Nedr., 12, 16-20 (in Russian).
Walters G.,1947: “The distribution of projected areas of fragments”. Proc. Cam. Phil. Soc., 43, 342347.
Zhuravskii M.,1932: “Mineralogical analysis of a microsection from a probability-theoretic point of
view”. Trudy in-ta Mekhanobrabotki (in Russian).