Geoscience Reference
In-Depth Information
ordinary polynomial trend surface analysis. The 2-D power spectrum and autocor-
relation function are important tools even when there are no spatial periodicities.
Box 7.2: 2-D Harmonic Analysis
The inverse Fourier transform
A
(
p
,
q
) of a 2-D array of gridded (
m
n
) data,
X
A
(
i
,
j
)
where
I
p
.
mn
P
m
1
0
P
n
1
satisfies:
Ap
ðÞ
¼
i
p
m
jq
n
;
q
1
0
X
A
i
ðÞ
;
j
2
ˀ
I
þ
¼
exp
i
¼
j
¼
. Without loss of
generality, it can be assumed that
X
A
is zero. Any value
A
(
p
,
q
) consists of a real
part Re (
A
p,q
) and an imaginary part Im (
A
p,q
). Any wave can be represented by the
continuous function:
Xp
mn
P
m
1
0
P
n
1
ip
m
þ
jq
n
1
Also,
X
A
i; ðÞ
¼
0
A
p
ðÞ
;
q
2
ˀ
I
exp
i
¼
j
¼
Re
A
p
,
q
cos 2
Im
A
p
,
q
sin
ip
m
þ
jq
n
ð
;
q
;
u
;
v
Þ ¼
ˀ
where
u
and
v
are geographical coordinates as before. The square of
amplitude is given by:
P
(
p
,
q
)
ip
m
þ
jq
n
2
ˀ
Re
2
(
A
p
,
q
)+Im
2
(
A
p
,
q
). The 2-D autocovariance
¼
.
X
(
p
,
q
;
u
,
v
) functions
for a block of values (
p
,
q
)formaso-calledharmonic trend surface with:
Y
(
u
,
v
)
ðÞ¼
P
m
1
p¼
0
P
n
1
ip
jq
n
q¼
0
P
p
ðÞ
;
q
ˀ
I
m
þ
satisfies:
Cr
;
s
exp
2
¼
P
p
P
p
X
(
p
,
q
;
u
,
v
).
7.4.1 Virginia Gold Mine Example
A map of average gold values for the southwestern portion of the Virginia Mine is
shown in Fig.
7.26
. Every contoured value is the average of all individual values in
the surrounding 100 ft.
100 ft. area. Moving average values for a 50-ft. grid were
subjected to various types of statistical analysis (Krige
1966
; Krige and
Ueckermann
1963
; Whitten
1966
). Agterberg (
1974
) took (18
18) values for a
900
900 ft. area in the southeast corner of Fig.
7.26
. Some gaps in the array in the
array were filled in by using interpolation values. In order to make use of the Fast
Fourier Transform method (Cooley and Tukey
1965
), the array was enlarged to size
(32
32). The 2-D spectrum
P
(
p
,
q
) and autocorrelation function are shown in
Fig.
7.27
. Both maps are symmetrical with respect to the central point (origin) and
only part of the lower half is shown.
In the 2-D power spectrum of Fig.
7.27
the relatively large values of
P
(
p
,
q
) are
located within a block around the origin. This indicates that the second degree
harmonic trend surface provides a reasonable approximation in this application.
Values of the autocorrelation function for points less than 100 ft. from the origin are
biased because original values for the 50-ft. grid are averages for overlapping cells
measuring 100 ft. on a side.
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