*
pa(y) - 10° (v) + (M/2)
*
—
[Var P(x) - Var P (v)] x [207/20]
*
Var P(x) - Var P (v) in fact depends on the variogram only:
*
Var P(x) - Var P (v) =
2
ae v(x, - x5) =O, 2
»
d
Q
Uyf (v),
where the A's and the u's are the solutions to the kriging system (Equation 9).
Bringing pieces together, we get the final correct formula
*
2
2g
——
*
pe(y)
= 10" (v) + M[o,/2 ~ > ut (v)] x [20/20")
;
(14)
£
The first approximation we use is to set 10my yo™v) = 1, i.e., we neglect
the variation of the drift m(x) within a cell v.
By the convexity of
the exponential function 10", we have 10/10" >
However, if m(x)
varies slowly enough at the scale of v, this corrective term should be
close to 1.
Let us look at orders of magnitude.
For a linear drift in
IR*:
m(x,y) = bo + bix + boy, and averaging over a square of side L
yields:
10"/10m(v)
= 196? + bo) L/2
a-10°) a -
gqb2k
)/M2 biboL2.
With a maximum b;L = boL of 0.5--corresponding to a three-fold variation
of m(x) within 100 feet--we find that
107/10")
.
= 1.115
With b,L = boL = 0.3--two-fold variation within 100 feet--the ratio is
1.048.
So our results may be biased downward, especially near GZ where
the gradient is high.
But in any case, the bias should not exceed 1C
percent unless more than three-fold variation of m(x) occurs within 100
feet.
The second approximation is introduced in order to relate the arithmetic
mean concentration Pu(v) to the logarithmic mean concentration P(v).
Under the assumption that the cell v is small enough-~say, to ensure
¥(LY2) < Q.5--it may be shown that, to the first order:
Pu(v) = 10° () +M y(v,v)/2
Naturally in this formula, y refers to the variogram of the logarithmic
concentrations P(x).
A confidence interval for Pu(v) is then simply
obtained by multiplying the bounds of Equation_13 by the factor 10 My (vv) /2
This factor may be calculated using charts of y(v,v), which have been
computed for the common variogram models (e.g., see Matheron, 1971).
However,
a simple exact formula is available in the case of a linear
variogram y(h)
w{hl, when v is a square of side L:
386