Geoscience Reference
In-Depth Information
Box 12.4: Hermite Polynomials
p
j
j
ð
2
u
j
2
j
ðÞ
2
2
u
j
u
j
4
Hermite polynomials
H
j
(
u
) satisfy
H
j
u
ðÞ
¼
þ
2
!
j
ðÞ
2
3
u
j
6
,wh e
j
(
t
)
þ ...
¼
j
(
j
1)(
j
2)
...
(
j
t
+ 1). For
example:
3
!
u
2
u
3
u
4
6
u
2
+3;
H
0
(
u
)
¼
1;
H
1
(
u
)
¼
u
;
H
2
(
u
)
¼
1;
H
3
(
u
)
¼
3u;
H
4
(
u
)
¼
10
u
3
+15
u
.If
Z
1
and
Z
2
are two standard normal random variables
with zero means, unit variances and correlation coefficient
u
5
H
5
(
u
)
¼
ˁ
, their bivariate
½
ð
Þ
=
21
ˁ
ð
Þ
½
ð
Þ
exp
z
1
2
ˁz
1
z
2
þz
2
2
exp
½
z
1
þz
2
p
1
ˁ
density function is:
ˆ
ð
z
1
;
z
2
Þ
¼
¼
2
ˀ
2
ˀ
2
h
i
representing the Mehler identity. Other properties
X
j
¼1
ˁ
j
H
j
zðÞ
1
þ
H
j
zðÞ
j
where
X
is a normal random variable with mean
include
E
{
H
j
(
X
)}
¼
ʼ
ʼ
.
X
2
.
If
f
(
x
1
) and
f
(
x
2
) are of the same type, their frequency distribution is permanent.
Examples include the lognormal and logbinomial distributions. Permanence of
these two distributions was first established by Matheron (
1974
,
1980
). It means
that blocks of different sizes have the same type of frequency distribution; for
example, all can be lognormals with the same mean but different logarithmic
variances.
The regression of
X
1
on
X
2
,or
E
(
X
1
|
X
2
)
When
f
(
x
1
) is known,
f
(
x
2
) can be derived in combination with
E
(
X
1
|
X
2
)
¼
¼
X
2
, results in the following two
relations between block variances:
2
XðÞ
¼
σ
2
X
1
2
Xð ; σ
2
XðÞ
¼
ˁ
2
2
XðÞ
σ
ð
X
2
Þ þ σ
x
σ
with
0 representing the product-moment correlation coefficient of
X
1
and
X
2
.
The first part of this equation can be rewritten as
ˁ
x
>
2
(
V
). Suppose
that
v
is contained within a larger volume
v'
that, in turn, is contained within
V
.
Then,
2
(
v
,
V
)
2
(
v
)+
σ
¼
σ
σ
2
(
v
0
,
V
) These results are general in that they do not
only apply to original element concentration values but also to transformed element
concentration values; e.g.
E
(ln
X
1
|ln
X
2
)
2
(
v
,
V
)
2
(
v, v
0
)+
σ
¼
σ
σ
ln
X
2
. The resulting logarithmic
variance relationship for average values in blocks of three different sizes was
originally discovered by Krige (
1951
) and was called “Krige's formula for the
This result also can be derived as follows. Using Hermite polynomials the
lognormal
¼
random variable
X
1
can
be
written
as
X
1
¼
ʼ
exp
j
1
H
j
(
Z
1
)/
j
!. From results for correlation between
X
1
and
X
2
described in Box
12.5
it follows that
X
2
satisfies the same expression when
σ
1
is
replaced by
1
)
¼
ʼ
∑
j
¼ 0
σ
(
σ
1
Z
1
½
σ
2
2
.
It is possible to use the relation
σ
2
, and
σ
2
¼
σ
1
ˁ
2
(ln
X
2
) to estimate the correla-
tion coefficient
ˁ
x
. The following example of application was described in
Agterberg (
1977
). Probability indices for occurrence of large copper deposits in
2
(ln
X
1
)
2
σ
¼
ˁ
x
σ
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