diplomanden-doktoranden-seminar bonn – 29. juni 2008 surrogates and kriging part i: kriging ralf...

Diplomanden-Doktoranden-Seminar Bonn – 29. Juni 2008

Surrogates and Kriging

Part I: Kriging

Ralf Lindau


Contents

• Stepwise Kriging (the future)

• Simple Kriging (the past) avoiding negative weights

• Victorian LWP

• Stefanian Permeability


Summary of 18.5.2009

Distingish between two types:

1. downscaling of averages (true downscaling)

2. downscaling of point measurements (interpolation)

Example for average downscaling:

Precipitation from 60 to 30 min

Three methods for point downscaling:

1. Linear interpolation plus noise

2. Stepwise kriging

3. Stepwise spatio-temporal data construction


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.


Sequential vs Stepwise

J.A. Vargas-Guzman, T.-C. Jim Yeh, 1999:

Sequential kriging and cokriging: Two powerful geostatistical approaches

Stochastic Environmental Research and Risk Assessment, 13, 416-435.

„The sequential estimator is shown to produce the best, unbiased linear estimate, and to provide the same estimates and variances as classic simple kriging or cokriging with the simultaneous use of the entire data set. However, by using data sets sequentially, this new algorithm alleviates numerical difficulties associated with the classical kriging or cokriging techniques when a large amount of data are used.“

The expression Sequential Kriging means here:

Decomposition of a huge matrix into several smaller matrices, which can be solved easier.

The presented method of Stepwise Kriging means something else:

Add the information and correct the variance stepwise, data point by data point.


Stepwise Kriging

We proudly present:

The algorithm is readily programmed and awaits eagerly to work.


So far the future.

Now, the past: two examples for simple kriging.

1. Clouds (Victor)

2. Soil (Stefan)

Most important difference to OK:

No „trend“: no gradient, only (normalized) anomalies.

No negative weights: don‘t panic, no more discussion about that.


Negative weights in literature

Deutsch, C.V., 1996: Correcting for Negative Weights in Ordinary Kriging, Computers &

Geociences, 22, (7), 765-773.

„Negative weights in ordinary kriging (OK) arise when data close to the location being estimated screen outlying data. Depending on the variogram and the amount of screening, the negative weights can be significant; there is nothing in the OK algorithm that alerts the kriging system about the zero threshold for weights. Negative weights, when interpreted as probabilities for constructing a local conditional distribution, are nonphysical. Also, negative weights when applied to high data values may lead to negative and nonphysical estimates. In these situations the negative weights in ordinary kriging must be corrected.

An algorithm is presented to reset negative kriging weights, and compensating positive weights to zero. The sum of the remaining nonzero weights is restandardized to 1.0 to ensure unbiasedness. The situations when this correction is appropriate are described and a number of examples are given.“

Deutsch presented a steamroller method to avoid negative kriging weights.

My method is slightly ;-) more elegant.


Autocorrelation

Difficulties arise for cumulus clouds, where only a small number of non-zero observations may occur.

In this case the autocorrelation function is not derivable.

Solution: Kriging is based on the mean autocorrelation for all cumulus cases.

0.47

0.02

0.211.07


Victorian LWP„measurement“ „measurement“

krig error errorkrig


Concluding the total variance

Lines have a smaller variance than the

entire field.

Why?

Divide a field into several lines.

Then the total variance is equal to:

internal:

the mean variance inside the lines

plus external:

the variance of the lines‘ means

Technically, there are more short distances

in lines than in the entire area. Multiply the

relative frequencies of distances with their

corresponding variance. Then the expected

total variances (lines/area) can be concluded.


Stefanian Permeability

Original Kriged (the truth)

0.697

0.378

0.319

Semivariogram

Org

Krig


Nice effects of kriging

Half of the variance is attributed to errors.

This error variance can be interpreted in two ways:

• Pure inaccuracy of measurements. (Observation error)• Point measuremnts are less representative for the entire grid box.

(Small scale variability)

For both cases the reduction of variance ís necessary.

The kriged field is superior compared to the the original.

It can be called „Truth for this spatial resolution“


Only the edge

Original Kriged

0.839

0.553

0.286


Random 31%

Original Kriged

0.670

0.425

0.245


True, Edge, 31%

True

Edge

31%

0.697 – 0.319 = 0.378

0.839 – 0.553 = 0.286

0.670 – 0.425 = 0.245

Total – Error = Remain


Summary

It is the difficulty to calculate total and error variance from a reduced data amount, which defines mainly how much variability is remaining in the obtained kriging result.

Strong smoothing may occur. But the main reasons are underestimation of the total variance or overestimation of the error variance. Not the kriging method itself.

Stepwise kriging is ready to go.

diplomanden-doktoranden-seminar bonn – 29. juni 2008 surrogates and kriging part i: kriging ralf...

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