Geo479/579: Geostatistics

Ch13. Block Kriging

Block Estimate

Requirements

An estimate of the average value of a variable within a prescribed local area

One method is to discritize the local area into many points and then average the individual point estimates to get the average over the area

This method is computationally expensive

Objective

See how the number of computations can be significantly reduced by constructing and solving only one kriging system for each block estimate

Block Kriging

Block Kriging

Block Kriging is similar to the point kriging The mean value of a random function over a

local area is simply the average (a linear combination) of all the point random variables contained in the local area

Where VA is a random variable corresponding to the mean value over an area A, and Vj are random variables corresponding to point values within A

Ajj

jA VA

V1

Equation 13.1

Point Kriging In point kriging, the covariance matrix D consists

of random variables at the sample locations and the location of interest

˜ 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)

Point Kriging

In point kriging, these are point-to-point covariances. For block kriging, these are point-to-block covariances (the block of interest)

˜ C 11 ˜ C 1n 1

˜ C n1 ˜ C nn 1

1 1 0

w1

wn

˜ C 1A

˜ C nA

1

C w D

w C-1 D

Block Kriging

Point-to-block covariances required for Block Kriging

˜ C iA Cov(VA ,Vi) E(VAVi) E(VA )E(Vi)

E1

| A |V j

j | jA

Vi

E

1

| A |V j

j | jA

E(Vi)

1

| A |{E(

j | jA

V jVi) E(V j )E(Vi)}

1

| A |Cov(V j

j | jA

Vi)

Block Kriging

The covariance between the random variable at the ith sample location and the random variable VA representing the average value over the area A is the same as the average of the point-to-point covariances between Vi and the random variables at all the points within A

Block Kriging

The Block Kriging System

The average covariance between a particular sample location and all of the points within A

C~

iA

1

ACij

~

j jA

Equation 13.3

Equation 13.4

Block Kriging

The Block Kriging Variance:

The value C is the average covariance between pairs of locations within A

~

OK

2

C~

AA ( wi

i1

n

C~

iA )

ijAii Ajj

AA CA

C

~

2

~ 1

Equation 13.5

Equation 13.6

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

)

iw

Block Estimates vs. the Averaging of Point Estimates

The average of the four point estimates is the same as the direct block estimate The average of the point kriging weights for a sample is the same as the block kriging weight for the sample

Figure 13.1

Varying the Grid of Point Locations within a Block

When using the Block Kriging approach - How to discretize the local area for block being estimated?• The grid of discretizing points should be

always regular• The spacing between points may be larger in

one direction than the other if the spatial continuity is anisotropic (Figure 13.2)

Discretizing Points The shaded block is approximated by six points located on a

2X3 grid. The closer spacing of the points in a north-south direction reflects a belief that there is less continuity in this direction than in the east-west direction. Despite the differences in the east-west and north-south spacing, the regularity of the grid ensures that each discretizing point accounts for the same area, as shown by the dashed line

Figure 13.2

Discretizing Points

Discretizing points < 16, Significant differences Discretizing points = > 16, Estimates are similar

Sufficient discretizing points number

2D block: 4x4 = 16, 3D block: 4x4x4 = 64

Table 13.2

Block Kriging vs. Inverse Distance Squared Block Estimates

A plus symbol denotes a positive estimation error while a minus symbol denotes negative estimation error

The relative magnitude of the error corresponds to the degree of shading indicated by the grey scale at the top of the figure

Block Kriging vs. Inverse Distance Squared Block Estimates

Figures 13.3, 13.4

Figure 13.4

Case Study

Comparison of summary statistics for Block Kriging and Inverse Distance Weighted

Inverse Distance Weighted has larger errors For Inverse Distance Weighted, there are several large

overestimation where relatively sparse sampling meets much denser sampling

Inverse Distance Weighted did not correctly handle the clustered samples, giving too much weight to the additional samples in the high-valued areas

Block Kriging showed some underestimation due to its smoothing effect and the positive skewness of the distribution of the true block values

Block Kriging vs. Point KrigingFigures 13.3, 13.5

Block Kriging vs. Inverse Distance Squared Block Estimates

Table 13.3Estimates

Table 13.4Errors

Block Kriging versus Point Kriging

Table 13.5 Errors