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illustrated on copper in exploratory drill-holes originally drilled from the surface into
the Whalesback deposit. Use of 2-D harmonic analysis is also illustrated by applica-
tion to density of gold and copper deposits in the Abitibi area on the Canadian Shield,
east-central Ontario. An advantage of using double Fourier series instead of ordinary
polynomials in trend surface analysis is that many geological features are to some
extent characterized by similarity of patterns along equidistant straight lines. Period-
icities of this type are accentuated by harmonic analysis.
Keywords Trend surface analysis 2-D and 3-D polynomials Kriging Harmonic
trend analysis Arbuckle formation Mount Albert peridotite intrusion
Whalesback copper deposit Matinenda uranium deposits Lingan mine sulphur
in coal Harmonic trend analysis
7.1
2-D and 3-D Polynomial Trend Analysis
Trend analysis is a relatively simple technique that is useful when (1) the trend and the
residuals (observed values - trend values) can be interpreted from a spatial geoscience
point of view, and (2) the number of observations is not very large so that interpolations
and extrapolations must be based on relatively few data. In practical applications, care
should be taken not to rely too heavily on statistical significance tests to decide of
polynomial degree (analysis of variance) or on 95 % confidence belts that can be
calculated for trend surfaces. These statistical tests would produce exact results only if
the residuals are uncorrelated (Krumbein and Graybill
1965
). If the residuals them-
selves show systematic patterns of variation, they are clearly not uncorrelated. Then the
test statistics would be severely biased. However, often the residuals can bemodeled as
a zero-mean, weakly stationary random variable. In that situation the trend surface
(or hypersurface in 3-D) is unbiased (Agterberg
1964
;Watson
1971
). Trend surfaces
are fitted to data by using the method of least squares that is also used in multiple
regression analysis. Consequently, it is also an application of the general linear model.
Suppose that the geographic co-ordinates of a point on a map are written as
u
and
v
for the east-west and north-south directions, respectively (in 3-D applications,
w
is
added for the vertical direction). An observation at a point
k
can be written as
Y
k
¼
Y
(
u
k
,
v
k
) to indicate that it represents a specific value assumed by the variable
Y
(
u
,
v
).
One can write:
Y
k
¼
R
(
u
k
,
v
k
)
the residual at point
k. T
k
is a specific value of the variable
T
(
u
,
v
)with:
T
(
u
,
v
)
T
k
+
R
k
where
T
k
¼
T
(
u
k
,
v
k
) represents the trend and
R
k
¼
¼
b
00
+
b
10
u
+
b
01
v
+
b
20
u
2
+
b
11
uv
+
b
02
v
2
+
+
b
pq
u
p
v
q
.Ingeneral,
u
and
v
form a
rectangular co-ordinate system. However, latitudes and longitudes also have been
used (Vistelius and Hurst
1964
).
In specific applications,
p
+
q
...
r
where
r
denotes the degree of the trend surface.
Depending on the value of
r
, a trend surface is called linear (
r
¼
1), quadratic
(
r
5). Higher degree trend surfaces
also can be used. For example, Whitten (
1970
) employed octic (
r
¼
2), cubic (
r
¼
3), quartic (
r
¼
4), or quintic (
r
¼
8) surfaces.
Coons et al. (
1967
) fitted polynomial surfaces of the 13th degree. A trend surface of
degree
r
has
m
¼
¼
½
(
r
+ 1)(
r
+ 2) coefficients
b
pq
. These can only be estimated if the
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