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