Geo479/579: Geostatistics
Ch17. Cokriging
Data sets often contain more than one variable of interest
These variables are usually spatially cross-correlated
Introduction
A method for estimation that minimizes the variance of the estimation error by exploiting the cross-correlation between several variables
Cross-correlated information contained in the secondary variable should help reduce the variance of the estimation errors
The Cokriging System
When is the secondary variable useful in estimates?
Primary variable of interest is under sampled then the only information we have is the cross correlated information
The Cokriging System
The Cokriging System
The cokriging estimate is a linear combination of both primary and secondary data values
01 1
ˆ (17.1)n m
i i j ji j
u a u b v
This is the Equation used in Ordinary Kriging pg 279
n
jj vwv
1
ˆ
The development of the cokriging system is identical to the development of ordinary kriging system
Estimation Error R can be defined as
This is a modification of the error estimation in Ordinary Kriging( pg 279) iii vvr ˆ
The Cokriging System
Using matrix notation we can write
w = { a1, a2, a3,…an, b1, b2, b3,…bm}
Z = { U1, U2…..Ui, V1,….Vj}
The Cokriging System
0
0 0 0
(17.4)
2 2
2 (17.5)
tz
n n m m
i j i j i j i ji j i j
n m n
i j i j i ii j i
m
j jj
Var R w C w
a a Cov U U b b Cov VV
a b Cov U V a Cov U U
b Cov V U Cov U U
Using Equation 9.14 (p216), 12.6 (p283), 16.3 (p372) we can write
The Cokriging System
This is similar to Chapter 16
)283P 6.12 (16.3, ),(}{
(16.2) 0~
Kww
1 11
0 0
t
n
i
n
jjiji
n
iii
n
i
n
jijji
VVCovwwVwVar
Cww
0 1 1
1 1
1 1
ˆ
(17.6)
n m
i i j ji j
n m
i i j ji j
n m
U i V ji j
E U E aU b V
a E U b E V
m a m b
1 1
1 0 (17.7)n m
i ji j
a and b
Note: Other nonbias conditions are also possible
1) Unbiasedness condition
The Cokriging System
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
It is similar to unbiasedness in Ordinary Kriging (p281)
2) Minimizing error variance
1
1
min
. .
1
0
n
ii
m
jj
Var R
s t
a
b
1 21 1
2 ( 1) 2 ( ) (17.8)n m
tz i j
i j
Var R w C w a b
Lagrangean Relaxation:
The Cokriging System
1 21 1
2 ( 1) 2 ( ) (17.8)n m
tz i j
i j
Var R w C w a b
Lagrangean Relaxation:
Original Lagrange parameter:
The Cokriging System
˜ 2R = ˜ 2 wiw j
j1
n
i1
n
˜ C ij 2 wi
i1
n
˜ C i0 2( wi
i1
n
1)
(12.9)
1 01 1
2 01 1
1
1
1,...,
1,...,
1
0 (17.9)
n m
i i j i i j ji i
n m
i i j i i j ji i
n
ii
m
ii
a Cov U U b Cov VU Cov U U for j n
a Cov U V b Cov VV Cov U V for j m
a
b
Equating n+m+2 partial derivatives of Var{R} to zero, we get the following system of equations
The Cokriging System
This is similar to minimizing the varianves of error in Ordinary Kriging
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)
Positive definiteness must hold for the set of auto- and cross-variograms (Eq16.44, p391).
U (h) u00(h) + u11(h) ... umm (h)
V (h) v00(h) +v11(h) ... vmm (h)
UV (h) w00(h) + w11(h) ... wmm(h)
u j > 0 and v j > 0, for all j = 0, ..., m
u j v j > w j w j , for all j = 0, ..., m
U , j (h) UV , j
(h)
VU , j (h) V , j (h)
u j w j
w j v j
j (h) 0
0 j (h)
The Cokriging System
If the primary and secondary variables both exist at all data locations and the auto- and cross-variograms are proportional to the same basic model then the cokriging estimates will be identical to the ordinary kriging estimates
The Cokriging System
A Cokriging Example
1 1 1 2
1 1 1 2
1 1 1 2
( ) 440,000 70,000 ( ) 95,000 ( )
( ) 22,000 40,000 ( ) 45,000 ( )
( ) 47,000 50,000 ( ) 40,000 ( ) (17.11)
U
V
VU
h Sph h Sph h
h Sph h Sph h
h Sph h Sph h
Compares cokriging and ordinary kriging
Undersampled variable U is estimated using 275 U & 470 V sample data for cokriging and only the 275 U data for ordinary kriging
1 1 1 2
1 1 1 2
1 1 1 2
( ) 440,000 70,000 ( ) 95,000 ( )
( ) 22,000 40,000 ( ) 45,000 ( )
( ) 47,000 50,000 ( ) 40,000 ( ) (17.11)
U
V
VU
h Sph h Sph h
h Sph h Sph h
h Sph h Sph h
A Case Study
,1
1,1
10
cos(14) sin(14)25 (17.12)1 sin(14) cos(14)
030
x x
y y
h hh
h h
,2
2,2
10
cos(14) sin(14)50 (17.13)1 sin(14) cos(14)
0150
x x
y y
h hh
h h
12 22
1 ,1 ,1
12 22
2 ,2 ,2 (17.14)
x y
x y
h h h
h h h
Ordinary kriging
275 U values
Using eq 17.11 for the variogram model
Cokriging
275 U and 470 V values
Using eq 17.11 for the variogram model
Two non-bias conditions
1) uses the initial conditions
2) uses only one nonbias condition
1 1
1n m
i ji j
a b
01 1
ˆ ˆ ˆ( ) (17.16)n m
i i j j V Ui j
U aU b V m m
0 1 1
1 1
1 1
1 1
ˆ ˆ ˆ( )
ˆ ˆ
(17.17)
n m
i i j j V Ui j
n m
i i j j V Ui j
n m
U i U ji j
n m
U i ji j
E U E aU b V m m
a E U b E V E m E m
m a m b
m a b
In the alternate unbiased condition, the unknown U value is now estimated as a weighted linear combination of nearby V values adjusted by a constant so that their mean is equal to the mean of the U values
Negative estimates occur because of the nonbias
condition 1
0m
jj
b
Cokriging with two nonbias conditions is less than satisfactory
A physical process with both negative and positive weighting scheme is difficult to imagine
Cokriging with one nonbias condition considerably improved the spread of errors and bias
Though we had to calculate global means of U and V
( ) ( ) ( )UV UV UVC h h
If the spatial continuity is modeled using semivariograms then they can be converted to covariance values for cokriging matrices by following equation:
If we want an estimate over a local area A, there are two options:
1) Average of point estimations within A
The Cokriging System
2) Replace all the covariance terms and
in point cokriging system, with average
covariance values and
0 iCov U U
0 jCov U V
A iCov U U A jCov U V
The Cokriging System
0 0 1 0 01 1
(17.10)n m
i i j ji j
Var R Cov U U a Cov U U b Cov V U
With the unbiasedness conditions, we can calculate error variance as follows
The Cokriging System