*
*
Prob {p (v) - P(v) > of = 1/2 = Prob { ror © 5 yoP™) .
x
So for the symmetric case, 10° are a median unbiased estimator of the
geometric mean concentration 10
.
If in a cell v the point values
P(x) = log Pu(x) are symmetrically distributed about their mean P(v),
then 10° “) is also the median concentration in that cell.
Under Gaussian theory, it is easy to set a 95 percent confidence interval
for the geometric mean:
*
ro’ “yi0
2 0.
K
< 10 P(v) < 10 P* (v)
ok
(13)
*
where oF = E[P (v) - P(v)]2 is the kriging variance.
An Estimate of the Arithmetic Mean
Things are much more complicated if we insist that we want an unbiased
estimator of the mean concentration Pu(v) and a confidence interval for
it. The difficulty remains even if we adopt the (questionable) assump~
tion that point concentrations Pu(x) are lognormal; this for two reasons:
1.
We are dealing with a nonstationary phenomenon so that the distribu-
2.
We are dealing with mean block values, and theory shows that these
tion changes with location,
cannot be lognormal if the point values are lognormal.
So we will have to resort to approximations. We believe these are at
least as acceptable as the lognormal model itself.
Let us first derive an exact expression.
If P(x) = log Pu(x) is Gaussian
with mean m(x) and variance Var P(x), (Var P(x) = constant), then
E[Pu(x)] = 10™(*) + M Var P(x)/2
where the constant M = 1n10 = 2.3026 is introduced by the fact that we
E[Pu(v)] =
<n
use logarithms in base 10.
Now,
f E{Pu(x) ]dx = 10” x ro Var P(x)/2
v
where the upper bar denotes averaging over a cell v.
On the other hand,
(As usual, m(v) denotes the average of m(x) over v.)
mator of Pu(v) is therefore:
An unbiased esti-
e[io?() = igt(w) + M Var PACy)/2
385