variable, and thus here to the residual variance of the Pu-FIDLER regression; hence the difference between FIDLER > 5,000 and FIDLER < 5,000 sills,
An enlargement of the variogram for FIDLER > 5,000 is also shown in
Figure 7.
We can fit it to a spherical model with a range "a" of 600
feet (the distance at which correlations vanish):
y(0) = 0
y(h) = 0.065 + 0.025 [3/2 (h/600) - 1/2 (h/600)3}
(12)
0 < |h] < 600 feet
y(h) = 0.09
In| < 600 feet
The "apparent nugget effect" of 0.065 is the sum of the FIDLER data
nugget effect aon = (1.287)2 x 0.0127 = 0.02 and the Pu data nugget
effect, which by Pu variogram extrapolation is found to be 0.045 (wariogram not shown here). Thus T(x)--the true value--has no nugget effect of
its own and its variogram is simply the spherical model with a sill at
0.025.
Though blurred by noise, T(x) does show a structure up to a
distance of 600 feet and it is worth exploiting it.
The variogram of T(x) is not well determined when FIDLER < 5,000, espe~
cially at short distances, as a consequence of much sparser sampling
(number of pairs typically less than 15).
For simplicity, we will assume
that the variogram in this case is the same as above, except for an
upward shift of 0.135 = 0.225 - 0.09 (Figure 7).
It is convenient to
think of this shift as due to a nonsystematic uncertainty of variance
0.135, attached to correction terms when FIDLER < 5,000.
Then, all data
may be processed by BLUEPACK in the same manner:
the uncertainty variance
is just added to the appropriate diagonal terms of the kriging system
matrix.
COMING BACK TO ARITHMETIC SCALE
We now have all the elements to carry out the estimation of Pov) --or
equivalently P(v), since these are equal~~that is, of the mean concentration in the log scale.
How can we come back to the arithmetic scale?
An Estimate of the Geometric Mean
*
P (v)
.
One way is to simply use the inverse transform 10
to estimate the
.
geometric mean 10
*
:
P(v)
By the unbiasedness property of kriging, we know
that the error P (v) - P(v) has mean zero.
symmetric distribution, then
If moreover this error has a
384
r-
: