geo479/579: geostatistics ch12. ordinary kriging (1)

33
Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

Upload: andrew-watkins

Post on 20-Jan-2016

259 views

Category:

Documents


12 download

TRANSCRIPT

Page 1: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

Geo479/579: Geostatistics

Ch12. Ordinary Kriging (1)

Page 2: 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

Page 3: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 4: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 5: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 6: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 7: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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}

Page 8: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 9: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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}

Page 10: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 11: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 12: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 13: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 14: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)

Page 15: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 16: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 17: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 18: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

( ˜ 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

Page 19: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 20: Geo479/579: Geostatistics Ch12. Ordinary Kriging (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)

Page 21: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)

Page 22: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)

Page 23: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 24: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)

Page 25: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)

Page 26: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

An Example of Ordinary Kriging

Page 27: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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

Page 28: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

˜ 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

Page 29: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

˜ C (h) 10e 0.3|h |

C0 0,

a 10,

C1 10

Page 30: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)
Page 31: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

DC 1

Page 32: Geo479/579: Geostatistics Ch12. Ordinary Kriging (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

Page 33: Geo479/579: Geostatistics Ch12. Ordinary Kriging (1)

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)