LandCaRe 2020
Temporal downscaling
of heavy precipitation
and
some 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 Dx.
( x i  D x )( x j  D x )  x i x j  D x D x
It results:
The original covariance xixj
plus the added variance DxDx
This is valid for each scale as
xi and xj have an arbitrary time
lag.
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
Original
x 1  x 0  cx 0
x1  x 0  c
x 2  x 0  cx 0
x2  x0  c
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
Konstante
Varianzreduktion
um den
BeobFehler
Linear Interpolation
Once Dr Lindau wrote four pages on
that topic with the following summary:
Linear Interpolation underestimates
the variance of each scale by a
quarter of the variance found in the
smallest resolved scale.
Thus, the correct spatial structure can
be 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 l1x1 + l2x2 + l3x3
Its covariance to the old data point x1 is:
[x1 (l1x1 + l2x2 + l3x3)]
= l1 [x1x1] + l2 [x1x2] + l3 [x1x3]
This covariance should be
equal to the covariance
between 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] + b22[x2x2]
= b12+ b22
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+ c22+ 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:
time=1
....
time=10000
For each pair of
locations a given
correlation is
constructed.
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008
Check of correlation properties
Input
Output
Correlation for one example point (16,11)
to all others. Correlation is well reproduced.
The remaining 1599 checks will be shown
next time ;-)
Difference
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
Diplomanden-Doktoranden-Seminar Bonn – 18. Mai 2008