geo479/579: geostatistics ch12. ordinary kriging (1)
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Geo479/579: Geostatistics
Ch12. Ordinary Kriging (1)
Ordinary Kriging
Objective of the Ordinary Kriging (OK) Best: minimize the variance of the errors Linear: weighted linear combinations of the data Unbiased: mean error equals to zero Estimation
Ordinary Kriging Since the actual error values are unknown, the
random function model are used instead A model tells us the possible values of a random
variable, and the frequency of these values The model enables us to express the error, its
mean, and its variance If normal, we only need two parameters to define
the model, and
˜ m R
˜ 2R
Unbiased Estimates
In ordinary kriging, we use a probability model in which the bias and the error variance can be calculated
We then choose weights for the nearby samples that ensure that the average error for our model is exactly 0, and the modeled error variance is minimized
˜ m R
˜ 2R
n
jj vwv
1
ˆ
The Random Function and Unbiasedness A weighted linear combination of the nearby
samples
Error of ith estimate =
Average error = 0
This is not useful because we do not know the actual
iii vvr ˆ
k
i
k
iiii vv
kr
kmr
1 1
ˆ11
v i
n
jj vwv
1
ˆ
The Random Function and Unbiasedness …
Solution to error problem involves conceptualizing the unknown value as the outcome of a random process and solving for a conceptual model
For every unknown value, a stationary random function model is used that consists of several random variables
One random variable for the value at each sample locations, and one for the unknown value at the point of interest
n
jj vwv
1
ˆ
The Random Function and Unbiasedness …
Each random variable has the expected value of Each pair of random variables has a joint
distribution that depends only on the separation between them, not their locations
The covariance between pairs of random variables separated by a distance h, is
˜ C v (h)
E{V}
The Random Function and Unbiasedness …
Our estimate is also a random variable since it is a weighted linear combination of the random variables at sample locations
The estimation error is also a random variable
The error at is an outcome of the random variable
0 01
( ) ( ) ( )n
i ii
R x w V x V x
R(x0) ˆ V (x0) V (x0)
x0
R(x0)
n
iii xVwxV
10 )()(ˆ
The Random Function and Unbiasedness …
For an unbiased estimation
E{R(x0)} E{ wi
i1
n
V (x i) V (x0)}
wi
i1
n
E{V (x i)} E{V (x0)}
wi
i1
n
E{V} E{V}
If stationary
E{R(x0)} 0
E{V} wi
i1
n
E{V}
The Random Function and Unbiasedness …
We set error at as 0:
x0
E{R(x0)} E{V} wi
i1
n
E{V}
E{V} wi
i1
n
E{V}
wi
i1
n
1
E{R(x0)} E{V} wi
i1
n
E{V} 0
The Random Function Model and Error Variance
The error variance
We will not go very far because we do not know
R2
1
k(ri mR )2
i1
k
1
k[( ˆ v i v i
i1
k
) 1
k i1
k
( ˆ v i v i)]2
v i
Unbiased Estimates …
The random function model (Ch9) allows us to express the variance of a weighted linear combination of random variables
We then develop ordinary kriging by minimizing the error variance
Refer to the “Example of the Use of a Probabilistic Model” in Chapter 9
˜ 2R
The Random Function Model and Error Variance …
We will turn to random function models
0 01
( ) ( ) ( )n
i ii
R x w V x V x
R(x0) ˆ V (x0) V (x0)
n
iii xVwxV
10 )()(ˆ
The Random Function Model and Error Variance …
Ch9 gives a formula for the variance of a weighted linear combination (Eq 9.14, p216):
}{}{
111
ji
n
j
ji
n
i
i
n
i
i VVCovwwVwVar
(12.6)
The Random Function Model and Error Variance …
We now express the variance of the error as the variance of a weighted linear combination of other random variables
Var{R(x0)} E[{ ˆ V (x0) V (x0)}2]
Var{ ˆ V (x0)} 2Cov{ ˆ V (x0),V (x0)}Var{V (x0)}
Var{ ˆ V (x0)} Var{ wi
i1
n
Vi} wiw j
j1
n
i1
n
˜ C ij , Var{V (x0)} ˜ 2
Stationarity condition
The Random Function Model and Error Variance …
~
},{
)()()(
)()()(
) ,()}(),(ˆ{
01
01
01
01
01
01
01
00
i
n
iii
n
ii
i
n
iii
n
ii
i
n
iii
n
ii
i
n
ii
CwVVCovw
VEVEwVVEw
VEVwEVVwE
VVwCovxVxVCov
The Random Function Model and Error Variance
If we have , , and , we can estimate the To solve
Var{R(x0)} ˜ R2 ˜ 2 wiw j
j1
n
i1
n
˜ C ij 2 wi
i1
n
˜ C i0 (12.8)
˜ 2
˜ C ij
˜ C i0
wi
( ˜ R2 )
w1
0,
( ˜ R2 )
w2
0,
( ˜ R2 )
w3
0,...,
( ˜ R2 )
wn
0,
wi
( ˜ R2 )
w1
0,
if ˜ R2 w1
2 3w1, ( ˜ R
2 )
w1
(w12 3w1)
w1
2w1 3
( ˜ R2 )
w1
0, 2w1 3 0, w1 =1.5
The Random Function Model and Error Variance
Minimizing the variance of error requires to set n partial first derivatives to 0. This produces a system of n simultaneous linear equations with n unknowns
In our case, we have n unknowns for the n sample locations, but n+1 equations. The one extra equation is the unbiasedness condition
w1w2,...,wn
wi
i1
n
1
The Lagrange Parameter
To avoid this awkward problem, we introduce another unknown into the equation, , the Lagrange parameter, without affecting the equality
˜ 2R = ˜ 2 wiw j
j1
n
i1
n
˜ C ij 2 wi
i1
n
˜ C i0 2( wi
i1
n
1)
(12.9)
Minimization of the Error Variance The set of weights that minimize the error variance
under the unbiasedness condition satisfies the following n+1 equations - ordinary kriging system:
R2
wi
0 w j˜ C ij ˜ C i0
j1
n
i 1,,n
R2
0 wi
i1
n
1
(12.11)
(12.12)
Minimization of the Error Variance The ordinary kriging system expressed in matrix
˜ C 11 ˜ C 1n 1
˜ C n1 ˜ C nn 1
1 1 0
w1
wn
˜ C 10
˜ C n 0
1
C w D
w C-1 D (12.14)
(12.13)
Ordinary Kriging Variance Calculate the minimized error variance by using
the resulting to plug into equation (12.8)
˜ R2 ˜ 2 wi
j1
n
w j˜ C ij
i1
n
2 wi˜ C i0
i1
n
˜ 2 ( wi˜ C i0
i1
n
) ˜ 2 w'D
iw
Ordinary Kriging Using or
ij ˜ 2 ˜ C ij , ˜ ij ˜ C ij / ˜ 2
w j
j1
n
˜ ij ˜ i0 i 1,,n wiw j ˜ ij wi ˜ i0i1
n
j1
n
i1
n
˜ R2 ˜ 2 wiw j (
j1
n
i1
n
˜ 2 ˜ ij ) 2 wi( ˜ 2 ˜ i0)i1
n
wi
i1
n
˜ i0
Refer to Ch9
(12.20)
Ordinary Kriging Using or …
w j
j1
n
˜ ij ˜ i0 i 1,,n wiw j ˜ ij wi ˜ i0i1
n
j1
n
i1
n
˜ R2 ˜ 2 ˜ 2 wiw j
j1
n
i1
n
˜ ij 2 ˜ 2 wi ˜ i0i1
n
˜ 2{1 ( wi ˜ i0i1
n
)} (12.22)
An Example of Ordinary Kriging
We can compute and based on data in order to solve
w j˜ C ij ˜ C i0
j1
n
i 1,,n
wi
i1
n
1
(12.11)
(12.12)
˜ C i0
˜ C ij
w j
˜ C (h) {C0 C1
C1 exp( 3 | h |
a)
if | h |0
if | h | 0
˜ (h) {0
C0 C1(1 exp( 3 | h |
a))
if | h |0
if | h | 0
nugget effect, range, sill
C0
a
C0 C1
˜ C (h) 10e 0.3|h |
C0 0,
a 10,
C1 10
DC 1
Estimation
˜ v 0 wiv i
i1
n
(0.173)(477) (0.318)(696)
(0.129)(227) (0.086)(646)
(0.151)(606) 0.057)(791)
(0.086)(783) 592.7ppm
Error Variance
2
10
22
96.8
907.0)18.0)(086.0(
)68.0)(057.0()34.1)(151.0(
)58.0)(086.0()89.0)(129.0(
)39.3)(318.0()61.2)(173.0(10
)(~~
ppm
Cwn
iiiR
(12.15)
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