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Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
LandCaRe 2020
Temporal downscaling of heavy precipitation
andsome general thoughts about downscaling
Ralf Lindau
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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)
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
Effects on pdf
Problem: Additive noise creates negative rain values
Original pdf Downscaled pdf
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
Multiplicative noise
Solution: Multiplicative noise instead of additive noise
cxx
cxx
02
01
002
001
cxxx
cxxx
Original Downscaled (down – org) / (down + org)
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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]
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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, ...
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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 ;-)
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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 (*).
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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)
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
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