Geoscience Reference
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Fig. 8.2 Variation of preferred paleocurrent direction along straight line coinciding with surface
outcrop of Member C, Bjorne Formation (Triassic, Melville Island, Northwest Territories, Canada);
systematic change in azimuth is represented by trend line (Source: Agterberg 1974 ,Fig.12)
x , because azimuths are periodic with period 360 . The constant C is arbitrary. A
value can be assigned to it by inserting specific values for u and v into the equation.
For example, if u
0.7018. It can be used to calculate a set of
values for u forming a sequence of values for x ; the corresponding values for
v follow from the original equation for the linear trend surface. The resulting curves
(1) and (2) are shown in Fig. 8.3b . If the value of C is changed, the curves (1) and
(2) become displaced in the x
¼
0 and x
¼
0, C
¼
54 .4 direction. In this way, a set of curves is
created that represents the paleocurrent direction for all points in the area.
Suppose that the paleocurrents were flowing in directions perpendicular to the
average topographic contours at the time of sedimentation of the sand. If curve
(1) of Fig. 8.3b is moved in the 144 4 direction over a distance that corresponds to
90 in x , the result represents a set of directions which are perpendicular to the
paleocurrent trends. Four of these curves which may represent the shape of the
paleodelta are shown in Fig. 8.4c . These contours, which are labeled a , b , c , and
d satisfy the preceding equation for different values of C and with x replaced by
( x +90 )or( x
¼
90 ). Definite values cannot be assigned to these contours because
x represents a direction and not a vector with both direction and magnitude.
In trend surface analysis, the linear trend surface (also see Fig. 8.4a ) had explained
sum of squares ES S
78 %. The complete quadratic and cubic surfaces has ESS of
80 % and 84 %, respectively. Analysis of variance for the step from linear to quadratic
surface resulted in
¼
F 3
ð
80 78
Þ= 3
04. This would correspond to F 0.60
(3,37) showing that the improvement in fit is not statistically significant. It is tacitly
assumed that the residuals are not autocorrelated as suggested by their scatter around
the line of Fig. 8.1 . Consequently, the linear trend surface as shown in Fig. 8.4a is
acceptable in this situation. The 95 % confidence interval for this linear surface is
shown in Fig. 8.4b . This is a so-called half-confidence interval with values equal to
p
ð
;
37
Þ ¼
Þ= 37 ¼
1
:
ð
100 80
Y
with F
Y ¼
X 1
0
k
0
s 2
2.84 and s 2
s 2
ð
þ
1
Þ
F
¼
F 0.95 (3,40)
¼
X
X
X k with
residual variance s 2
¼
380 square degrees.
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