Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Kriging Connection between Stepwise Kriging and Data Construction and Stepwise Kriging of Victorian

Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010

Kriging

Connection between Stepwise Kriging and Data Construction

andStepwise Kriging of Victorian LWP

Ralf Lindau


Two cases of Downscaling

Two principle cases:

Data consists of averages (1 h rain sum 30 min rain sum).

Downscaling should produce averages of smaller scale.

The variance of each scale should be increased by a certain amount.

The pdf should contain more extremes.

Data consists of point measurements (DWD rain stations rain map of Germany)

Downscling should produce synthetic data in observation gaps.

The variance and pdf should remain constant.


Kriging approach

= min

Suppose three available observations x1, x2, x3 (old)Kriged new value is 1x1 +2x2 + 3x3

Its covariance to the old data point x1 is:

[x1 (1x1 + 2x2 +3x3)]

= 1 [x1x1] + 2 [x1x2] + 3 [x1x3]

This covariance should be equal to the covariancebetween prediction point P0 and observation point P1 which is:

[x0x1]


Stepwise Kriging

The covariances of a new kriging point to all old observation points are correct by definition.

However, the explained variance is smaller than 1 (normalized case).

This leads to an underestimation of the correlation.

Thus:

Do not use the kriging technique several times in series for all intermediate points.

But:

1. Predict only a single point

2. Correct its variance by adding noise

3. Consider in the next step the predicted value as an old one.


Data Construction

Stepwise data construction with correct mutual correlations.

Construct n time series at n locations so that the spatial correlation between

all locations are „correct“ (known covariance matrix as input needed (as usual)).

Use weighted averages of uncorrelated normalized time series xa, xb, xc, ...

for the production of x1, x2, x3, ...


Construction Recipe Time series at data point 1:

axax 111

Time series at data point 2:

ba xaxax 22212

Correlation between data point 1 and 2:

2111

22112111

222111

2112

aa

xxaaxxaa

xaxaxa

xxr

baaa

baa

Variance at data point 2

222

221

2222221

221

22221

22

2

1

aa

xxaxxaaxxa

xaxa

xx

bbbaaa

ba

Time series at data point 3:

cba xaxaxax 3332313


3111

33323111

3113

aa

xaxaxaxa

xxr

cbaa


32223121

3332312221

3223

aaaa

xaxaxaxaxa

xxr

cbaba

Variance at data point 3

2

332

322

31

2333231

331

aaa

xaxaxa

xx

cba


Construction vs. Kriging

cba

ba

a

xaxaxaxxaxax

xax

3332313

22212

111

c

b

a

xcxcxcxxcxcx

xcx

332321313

221212

111

Construction Kriging

n random time series xa, xb, xc,... are used.

Coefficients aij determine the weights of each random time series.

The correct variance is finally achieved by adding noise from an additional random

time series.

Both methods produce successively data points x1, x2, x3, ...

n random time series xa, xb, xc,... are used.

Coefficients cij determine the weights of each already produced data point.

The correct variance is finally achieved by adding noise from an additional random

time series.


The transformation

Thus, the only difference is:

Data Construction uses weighted averages of n random time series,Stepwise Kriging uses weighted averages of n already produced data points.

But each already produced data point is in the end itself a weighted average of the used random time series.

Does a fixed transformation between Stepwise Kriging and Data Construction exist?

Yes:

1

1

,,ji

n njnj

jnjnji

jj

ijij cc

ca

c

ac


Kriging is faster?So, Data Construction and Stepwise

Kriging are essentially equal, leading to identical results.

However,

Construction needs information from all already included random time series. This is time consuming (modern talking: expensive).

Kriging takes most information from the immediate surrounding. We might disregard far away data points. This might save computing time. (modtalk: cheap)

Data Construction

Stepwise Kriging


Is NNW Kriging applicable?

Far away data points will have negligible weights.

This allows us to stop the calculation at a certain radius and save computer time.

An interruption is not only desirable, but necessary (if larger fields should be constructed)

Why?

To determine the last of 200 x 200 = 40000 data points, the covariance matrix is

as large as 39999 x 39999 (nearly 1,600,000,000 numbers). This is difficult to solve.

When to stop?

Normal: If no data point with a positive weight is able to reduce the error further.

Stepwise: Small radius correlations to larger distances become wrong.

So, allow negative weights again.


NNW Kriging vs DS Kriging

Kriging error

n

iiiii

n

i

n

jjiji

n

iii xxxxxxxx

11 11000 2

Potential change of kriging error (aasumption: all old wights remain constant)

n

innnnnnniinnn xxxxxxxx

111

2111

2111101 22

So far: No-Negative-Weights Kriging1. Choose that data point with the best potential error reduction (above equation).2. Calculate weights and error exactly (solve matrix).3. Include more and more data points by repeating the procedure.4. If no error reduction with a positive weight is possible, one new data point is ready.5. Repeat everything for all grid points.

Now: Deep-Search Kriging1. Choose those 15 data points with the best potential error reduction (above equation).2. Calculate weights and error exactly (solve 15 matrices) and determine in this way the actually best.3. Include more and more data points by repeating the procedure.4. Ensure that all weights remain between -1 and 1.5. If no further error reduction is possible, or 15 data points are included, one new data point is ready.6. Repeat everything for all grid points.


Weight constraint

Instable solution (huge positivenext to huge negative weights,if negative weights are allowed. Constraint (-1;1) necessary.


Three versions

No-negative-weights Kriging conventional Lindau Kriging

Stepwise Kriging noise is added to each new point

each new point is considered as old

search radius enlarged

Deep-search Kriging conventional, but with an enlarged search radius


Conventional KrigingOriginal Data

Used Data

Pseudo-measurements

NoNegWeightsKriging

Inp

ut

DeepSearchKriging

Ou

tpu

t


Stepwise Kriging

Original Data

Difference toOriginal

Step-kriged

Step-kriged pdf-corrected(by cumulative freq.)


Auto-correlation

Notpdf-corrected

pdf-corrected

0 : Original correlations : Input to kriging routine+ : Output of kriging routine


Cumulus Clouds


Summary

The two methods Data Construction and Stepwise Kriging produce identical results.

From only a few meaurements Stepwise kriging is able to produce full clouds fields having the correct autocorrelation.

Pdf can be corrected afterwards, without distroying the nice structure.

Documents

Diplomanden-Doktoranden-Seminar Bonn – 18. Januar 2010 Kriging Connection between Stepwise Kriging and Data Construction and Stepwise Kriging of Victorian