Page 123 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology - Chapter 5

mathematical development of Kendall and Moran (1963). An older text by Uspensky
(1937) derives the more general elliptical case used here.
Assume the target being sought is an ellipse whose dimensions are given by the
major semiaxis u and minor semiaxis b. (If the target is circular, then u = b = r,
the radius of the circle.) The search pattern consists of a series of parallel traverses
spaced a distance D apart (Fig. 5-1 a). The probability that a target (smaller than
the spacing between lines) will be intersected by a line is
D
p=-
7TD
where P is the perimeter of the elliptical target. The equation for the perimeter of
,
an ellipse is P = 2 7 ~ d m where u and b are the major and minor semiaxes.
Substituting,
2TIpqz- 2JqF
-
-
’= ~TD D (5.2)
We can define a quantity Q as the numerator of Equation (5.2); that is, Q =
24(u* + b2)/2. With this simplification, the probability of intersecting an elliptical
target with one line in a set of parallel search lines can be written as
p=- Q
D (5.3)
In the specific case of a circular target, u and b are both equal to the radius, so Q
can be replaced by twice the radius:
2r
p=- (5.4)
D
At the other extreme, one axis of the ellipse may be so short that the target
becomes a randomly oriented line. This geometric relationship is known as Bwffon’s
problem, which specifies the probability that a needle of length 8, when dropped at
random on a set of ruled lines having a spacing D, will fall across one of the lines.
The probability is
28
p=- (5.5)
7TD
where 4? is the length of the target.
A similar geometric relationship, known as Laplace’s problem, also pertains to
the probabilities in systematic searches. Laplace’s problem specifies the probability
that a needle of length 8, when dropped on a board covered with a set of rectangles,
will lie entirely within a single rectangle. A variant gives the probability that a coin
tossed onto a chessboard will fall entirely within one square. In exploration, the
complementary probabilities are of interest, i.e., that a randomly located target
will be intersected one or more times by a set of lines, such as seismic traverses,
arranged in a rectangular grid (Fig. 5-1 b).
The general equation is

where D1 is the spacing between one set of parallel seismic traverses and DZ is the
spacing between the perpendicular set of traverses. In the specific instance of a

296