Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 LandCaRe 2020 Temporal downscaling of heavy...

Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008

LandCaRe 2020

Temporal downscaling of heavy precipitation

andsome general thoughts about downscaling

Ralf Lindau


The task

Soil erosion model within the LandCaRe „model chain“

needs rain input with a temporal resolution of 30 min.

CLM output is available hourly.

Downscaling technique is needed.

First step: All model grid boxes with more than 20 mm

daily precipitation are extracted from CLM output.


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.


Principle of average downscaling

Two coarse averages xi and xj are altered by a random x.

xxxxxxxx jiji ))((

It results:The original covariance xixj plus the added variance xx

This is valid for each scale asxi and xj have an arbitrary timelag.


Determination of the variance to be added

The original (1h) data variance is 4.379 mm2/h2

Averaging over 2,4,8 hours reduces the variance.

A linear fit enables us to estimate the potential variance for 0.5 h time resolution: 5.457 mm2/h2


Effects on semi-variogram

The total variance (horizontal lines) is increased (as desired) from 4.379 to 5.455 (mm/h)2

This increase (as desired) is added equally to each scale (see dashed line for difference)


Effects on pdf

Problem: Additive noise creates negative rain values

Original pdf Downscaled pdf


Multiplicative noise

Solution: Multiplicative noise instead of additive noise

cxx

cxx

02

01

002

001

cxxx

cxxx

Original Downscaled (down – org) / (down + org)


Point downscaling

Well done.

But what about the second type of downscaling?

(Production of synthetic data in observation gaps)

Why did kriging perform such a marvelous job?

Do you remember?

You don‘t.


Kriging of Rain

Beispiel: Regen vom 01.01.1996 bis 07.01.1996

DWD Original Ergebnis Varianzeigenschaften

DWD Original

Ergebnis

BeobFehler:0.037 mm2/d2

KonstanteVarianz-reduktionum denBeobFehler


Linear Interpolation

Once Dr Lindau wrote four pages onthat topic with the following summary:

Linear Interpolation underestimatesthe variance of each scale by aquarter of the variance found in thesmallest resolved scale.

Thus, the correct spatial structure canbe obtained by just adding a constant, known amount of variance.


Kriging

In the considered case kriging worked well because the variance not resolved by DWD stations was small. (Two 10 km separated stations measure a fairly similar daily precipitation.)

However, if kriging is used as interpolation tool to estimate many virtuel data points between a few observations, it will underestimate the intermediate spatial variance.


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.


Recapitulation

So far I presented two techniques:

1. Simple linear interpolation plus adding noise (quarter of small scale variance)

2. Stepwise Kriging

In the following I will present a third one:

3. Stepwise data construction


Stepwise 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 x1, x2, x3, ...

for the production of xa, xb, xc, ...


Construction Recipe (1)

1. Time series at data point a: xa = a1x1 a1 = 1

2. Time series at data point b:

xb = b1x1 + b2x2

Correlation to a: rab= [xaxb]

= [a1x1 (b1x1 + b2x2)]

= a1b1 [x1x1] + a2b2 [x1x2]

= a1b1

Variance at b: 1 = [xbxb]

= [(b1x1 + b2x2)2]

= b12 [x1x1] + 2b1b2[x1x2] + b2

2[x2x2]

= b12+ b2

2


Construction Recipe (2)3. Time series at data point c:

xc = c1x1 + c2x2 + c3x3

Correlation to a: rac = [xaxc]

= [a1x1 (c1x1 + c2x2 + c3x3)]

= a1c1

Correlation to b: rbc = [xbxc]

= [(b1x1+ b2x2)(c1x1 + c2x2 + c3x3)]

= b1c1 + b2c2

Variance at c: 1 = [xcxc]

= [(c1x1 + c2x2 + c3x3)2]

= c12+ c2

2+ c32


Construction examples

10000 of such fields are produced.

Each of them can be considered as time step.

Statistical property is:

For each pair of locations a given correlation is constructed.

time=1 .... time=10000


Check of correlation properties

Input Output

Difference

Correlation for one example point (16,11)to all others. Correlation is well reproduced.

The remaining 1599 checks will be shown next time ;-)


Spatial correlation of individual fields

So far the spatial correlation between two points obtained by averaging in time.

For some processes only a single field is available.

In such cases spatial correlations are obtained by averaging over data pairs of equal distance.

The method produces fields with varying spatial structure; however in average (+) it is correct.

If a single strictly correct field is desired, the best of the 10000 produced can be selected (*).


The future (bright) ;-)Future steps:

Use mutually uncorrelated but internal correlated time series, so that a realistic temporal development results.

Use others than gaussian distributed time series. The pdf of the used underlying time series determine the pdf of the obtained fields.

In this way any desired output-pdf could be created.

Allow to include observations by prescribing a few members of the basic time series. In this way the method would be able to reproduce also the Victorian mean and not only the structure (the correct position of highs and lows)


Summary

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

Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008 LandCaRe 2020 Temporal downscaling of heavy...

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