Geoscience Reference
In-Depth Information
x_1 = sum(sin(data_radians_1));
y_1 = sum(cos(data_radians_1));
mean_radians_1 = atan(x_1/y_1);
mean_degrees_1 = 180*mean_radians_1/pi;
mean_degrees_1 = mean_degrees_1 + 180;
Rm_1 = 1/length(data_degrees_1) .*(x_1.^2+y_1.^2).^0.5
Rm_1 =
0.8901
h e mean resultant length in our example is 0.8901. h e critical
Rm
(ʱ=0.05,
n
=40) is 0.273 (Table 10.1 from Mardia 1972). Since this value is lower than
the
Rm
from the data we reject the null hypothesis and conclude that there is
a preferred single direction, which is
theta_1 = 180 * atan(x_1/y_1) / pi
theta_1 =
43.2357
h e negative signs of the sine and cosine, however, suggest that the true
result is in the third sector (180-270°), and the correct result is therefore
180+43.2357=223.2357.
10.7 Test for the Dif erence between Two Sets of Directions
Let us consider two sets of measurements in two i les
directional_1.txt
and
directional_2.txt
. We wish to compare the two sets of directions and test the
hypothesis that these are signii cantly dif erent. We use the Watson-William
test to test the similarity between two mean directions
where ʺ is the concentration parameter,
R
A
and
R
B
are the resultant lengths
of samples
A
and
B
, respectively, and
R
T
is the resultant lengths of the
combined samples (Watson and Williams 1956, Mardia and Jupp 2000). h e
concentration parameter can be obtained from tables using
R
T
(Batschelet
1965, Gumbel et al. 1953, Table 10.2). h e calculated
F
is compared with
critical values from the standard
F
tables (Section 3.8). h e two mean
directions are not signii cantly dif erent if the calculated
F
-value is less than
the critical
F
-value, which depends on the degrees of freedom ʦ
a
=1 and
ʦ
b
=
n
-2, and also on the signii cance level ʱ. Both samples must follow a von
