Geo479/579: Geostatistics

Ch16. Modeling the Sample Variogram

Necessity of Modeling Variograms Sample variograms do not provide all of the

separation distances (h) and the corresponding semivariances needed by the kriging system

It is necessary to have a model that enables computing a variogram value for any possible separation distance

Outline of the Chapter Constraints that must be satisfied by a variogram

model Several basic variogram models Combining basic models to build a general model

of the sample variogram in one direction Modeling a geometric and zonal anisotropy in two

or three dimensions with different combinations of the basic models

Consistency between axes of anisotropy and those of the data coordinate system

Restrictions on the Variogram Models If we wish the ordinary kriging system has one,

and only one solution, we must ensure that matrix K satisfies a condition called “positive definiteness”

Note that K contains Cov between samples and between samples and the point to be estimated

k

˜ C 00 ˜ C 01 ˜ C 0n

˜ C 10 ˜ C 11 ˜ C 1n

˜ C n 0 ˜ C n1 ˜ C nn

Restrictions..

DC w

D w C

1

~

~

0 1 1

1 ~

~

1 ~

~

1-

0

101

1

111

nnnnn

n

C

C

w

w

CC

CC

Positive Definiteness

Equation 16.2 is the same as Eq 12.6, so the positive definite condition in Eq 16.2 can be seen as a guarantee that the variance of any random var which is a linear combination of other random var will be positive

w tKw wiw j˜ C ij

j0

n

i0

n

0 (16.2)

Var{ wi Vi

i1

n

} wiw jCov(Vi,V j ) (16.3, j1

n

i1

n

12.6 P283)

Positive Definiteness …

One such random variable is the error model The positive definite condition guarantees that

the estimation error will have a positive variance

(12.9) )1(2~

2~~=)}({

(12.8) ~

2~~)}({

(16.4)

10

11 1

20

011 1

20

01

0

n

iii

n

iiij

n

i

n

jji

i

n

iiij

n

i

n

jji

n

iii

wCwCwwxRVar

CwCwwxRVar

VVwR

Positive Definite Variogram Models We can use a few functions that are known to

satisfy the positive definite condition Any linear combination of positive definite

variogram models are also positive definite models Basic models presented in this section are positive

definite and simple

Positive Definite Variogram Models.. Variogram models can be divided into two types,

those that reach a plateau and those that do not Transition models are those that reach a plateau.

The plateau is the sill, and the corresponding lag distance is the range.

Some of the transition models reach their sill asymptotically. Their range is arbitrarily defined to be the lag distance at which 95% of the sill is reached

In this section, all sills are standardized to 1

Variogram Models.. 1. Nugget Effect Model (C0)

2. Spherical Model

It has a linear behavior at small h near 0 but flattens out at larger h, and reaches the sill at a

(h) 1.5(

h

a) 0.5

h

a

3

if h a

1 otherwise

otherwise 1

0 if 0)(0

hh

3. The Exponential Model

It is linear at small h near 0, but it rises more steeply and then flattens out more gradually than the spherical model

It reaches its sill asymptotically. Its practical range, a, corresponds to 95% of the sill

(h) 1 exp(3h

a)

Variogram Models..

4. The Gaussian Model

It is used to model extremely continuous phenomena

It reaches its sill asymptotically. Its practical range corresponds to 95% of the sill

5. The Linear Model

(h) 1 exp(3h2

a2 )

Basic Variogram Models..

(h) | h |

In fitting the model to a sample variogram, it is often helpful to remember that the tangent at the origin reaches the sill at about 2/3 of the range.

Models in One Direction The choice of basic models usually depends on

the behavior of the sample variogram near the origin:

- Parabolic behavior Gaussian

- Linear behavior Spherical or Exponential

Models in One Direction.. Combination of models: nested structures

E.g. for a sample variogam that does not reach a stable sill but has a parabolic behavior near the origin, we may combine a Gaussian and a linear model

(h) | wi |i(h)i 1

n

Models in One Direction.. Principle of parsimony: complex models are not

necessarily better than simple models The physical explanation of the phenomenon is

important Of the parameters, a and C0 are easy to decide.

Picking the weight for each model often requires a “trial and error” approach, and all weights must sum to the sill

Models of Anisotropy Geometric anisotropy: range changes with

direction while sill remains constant (Fig 16.3 a, p378)

Zonal anisotropy: sill changes with direction while range remains constant (Fig 16.3 b)

To deal with changes of range and sill with direction, we need to identify the anisotropy axes, using variogram surface maps or knowledge of the phenomenon

Geometric anisotropy Zonal anisotropy

Models of Anisotropy.. We need to combine directional variogram models

into a single model that is consistent in all directions. To do this, we need to define a transformation that reduces all directional variograms to a common model with a standardized range of 1

The trick is to transform the separation distance so that the standardized model will provide a variogram value that is identical to any of the directional models for that separation distance

sill1=sill2, a1=1, a2=a; Semivariance1(h/a) = Semivariance2(h)

1(ha ) a (h) or 1(h) a (ah)

Let h1 ha then 1(h1) a (h)

Extension to 2 and 3 Dimensions

Two-dimensions:

Three-dimensions:

221

11

)()(

)(),()(

y

y

x

x

a

h

ah

yx

h

hhhh

2221

11

)()()(

)(),,()(

z

z

y

y

x

x

ah

a

h

ah

zyx

h

hhhhh

(h) | w1 |1(h1)

h1 ( hx

ax)2 (

hy

ay)2 ( hz

az)2

Geometric Anisotropy -

One Structure

Three directional variograms have the same sill but different ranges, each with a single structure

The nugget effect is isotropic in all three directions. The other two structures are isotropic between the x and y directions, but anisotropic between the z direction and the x,y directions

Geometric Anisotropy -

Multi Structures

Geometric Anisotropy – Nugget and Two Structures..

(h) w00(h)

(h) | w1 |1(h1)

h1 ( hx

ax,1)2 (

hy

ay,1)2 ( hz

az ,1)2

Nugget effect:

The second structure:

Geometric Anisotropy – Nugget and Two Structures..

(h) | w2 |1(h2)

h2 ( hx

ax,2)2 (

hy

ay,2)2 ( hz

az ,2)2

(h) w00(h) w11(h1) w21(h2)

Note : w0 w1 w2 sill

The third structure:

Combination:

Geometric Anisotropy Summary

For each nested structure, the directional models must all be the same type, i.e. all spherical, or all exponential, or etc.

All directional models must have identical sill and wi

The model types can differ between nested structures, e.g. the first is Gaussion, while the second can be spherical

(h) | wi |i(h)i 1

n

The directional models along the x and y directions have the same sill but different ranges. The directional model in the z direction has a shorter range and larger sill than those for x and y

Zonal ad Geometric Anisotropy

Zonal and Geometric Anisotropy.. First structure: Geometric anisotropy

Second structure: Zonal anisotropy

The complete model:

2221

111

)()()(

)(||)h(

z

z

y

y

x

x

ah

a

h

ahh

hw

An isotropic model along x and y directions with a sill of and a range of 11w

z

z

a

hh

hw

2

212 )()h(

)()()( 212111 hwhwh

with a sill of and exists only in the hz direction

w2

Representing the method of reducing directional variogram models by matrix notation:

hn Th where T

1ax

0 0

0 1ay

0

0 0 1az

h1

1ax

0 0

0 1ay

0

0 0 1az

hx

hy

hz

, h2

0 0 0

0 0 0

0 0 1az

hx

hy

hz

Matrix Notation

(Eq 16.25) (Eq 16.27)

vector

Check the final model. E.g. It should return w1 at range ax in direction x

Eq 16.25, 16.27

(ax,0,0) w1, (0,ay ,0) w1, and (0,0,az) w1 w2

Matrix Notation …

Coordinate Transformation by Rotation

Anisotropy axes often do not coincide with the axes of the data coordinate system, i.e. x, y, z directions

The orientation of of the anisotropy is controlled by some feature in the data, while the orientation of the coornate system is arbitrary

Coordinate Transformation by Rotation

In this case, the components (hx,hy,hz) of the separation vector h in the data coordinate system will have different values when referenced in the coordinate system coincident with the anisotropy axes

Thus, it is necessary to transform the vector from the data coordinate system to the coordinate system coincident with the anisotropy axes before evaluating the variogram model

Clockwise when look in the positive direction of an axis

where h is the vector in the data coordinate system h’ is the same vector transformed to the anisotropic

coordinate system, while R is the transformation matrix

Case of two-dimension:

Case of three-dimension:

Rhh

cossin

sincosR

Coordinate Transformation by RotationCoordinate Transformation by Rotation

R cos cos sin cos sin sin cos 0

cos sin sin sin cos

Transformed reduced vector

T is the reduced distance matrix (Eq 16.31, p386), and the vector h must be defined in the anisotropic coordinate system

TRhhn

Coordinate Transformation by Rotation ..

The Linear Model of Coregionalization

It helps modeling the auto- and cross-variograms of two or more variables.

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 (h), V (h), and UV (h) are auto- and cross-variogram models for U and V, respectively

0(h), 1(h), and m (h)

u,v,and w

are basic variogram models

are coefficients, which can be negative

(16.40)

The Linear Model of Coregionalization ..

The auto- and cross-variograms can be rewritten in a matrix form as combinations of each basic model

Combinations of the first basic model,

11 = row1* col1 12 = row1* col2 21 = row2* col1 22 = row2* col2

U ,0(h) UV ,0(h)

VU ,0(h) V ,0(h)

u0 w0

w0 v0

0(h) 0

0 0(h)

0(h)

The Linear Model of Coregionalization ..

Combinations of the second basic model,

Combinations of the mth basic model,

1(h)

U ,1(h) UV ,1(h)

VU ,1(h) V ,1(h)

u1 w1

w1 v1

1(h) 0

0 1(h)

)h(0

0)h(

)h()h(

)h()h(

,,

,,

m

m

mm

mm

mVmVU

mUVmU

vw

wu

m (h)

The Linear Model of Coregionalization ..

To ensure the linear models in 16.4 are positive definite, coefficients u,v,and w need to be positive definite

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)

Positive Definiteness

Equation 16.2 guarantees that the variance of any random var, which is a linear combination of other random var, will be positive

One such random variable is the error model

w tKw wiw j˜ C ij

j0

n

i0

n

0 (16.2)

Var{ wi Vi

i1

n

} wiw jCov(Vi,V j ) (16.3, j1

n

i1

n

12.6 P283)

R0 wi Vi

i1

n

V0 (16.4)

Modeling the Walker Lake Sample Variograms

V (h) 22,000 40,000Sph1(h'1 ) 45,000Sph1(h'2 )

22,000 w0, 40,000 w1, 45,000 w2, w0 w1 w2 sill

h' = T R h

h'1 =h'x,1

h'y,1

1

250

01

30

cos(14) sin(14)

sin(14) cos(14)

hx

hy

h'2 =h'x,2

h'y,2

1

500

01

150

cos(14) sin(14)

sin(14) cos(14)

hx

hy

N76E

minor

N14W

major

Omni

Ch12, p316

Modeling the Walker Lake Sample Variograms …

V (h'x ) 22,000 40,000Sph25(h'x ) 45,000Sph50(h'x )

V (h'y ) 22,000 40,000Sph30(h'y ) 45,000Sph150(h'y )

U (h'x ) 440,000 70,000Sph25(h'x ) 95,000Sph50(h'x )

U (h'y ) 440,000 70,000Sph30(h'y ) 95,000Sph150(h'y )

UV (h'x ) 47,000 50,000Sph25(h'x ) 40,000Sph50(h'x )

UV (h'y ) 47,000 50,000Sph30(h'y ) 40,000Sph150(h'y )

The linear model ofco-regionalizationis positive definite

det(nugget) 22,000 47,000

47,000 440,000

7,471,000 0

U ,i(h) UV ,i(h)

VU ,i(h) V ,i(h)

ui wi

wi v i

i(h) 0

0 i(h)

u j > 0 and v j > 0, for all j = 0, ..., m

u j v j > wj wj, for all j = 0, ..., m

det( firstst ructure) 40,000 50,000

50,000 70,000

300,000,000 0

det(third structure) 45,000 40,000

40,000 95,000

2,675,000,000 0

The Random Function Model and Error Variance Ch9 gives a formula for the variance of a weighted

linear combination (Eq 9.14, p216):

Var{ wi i1

n

Vi} i1

n

wiw j j1

n

Cov{ViV j} (12.6)