Construction and applications
of 2-d digital filters for
separating regional spatial scales
Hans von Storch12, Frauke Feser1
and Matthias Zahn12
1 Institute
for coastal research, GKSS research center, Germany
2Meteorological
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Institute, University of Hamburg, Germany
10 IMSC, 20-24 August 2007, Beijing
Talk structure:
Construction of a 2D-filter
Filter applications:
- added value in RCM simulations
- characterization of typhoon patterns
- detection of polar lows
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Response function of a filter
K
f * ( x) = α 0 f ( x) + ∑ α k [ f ( x − k∆) + f ( x + k∆)]
k =1
With certain constants αk. ∆ represents the grid distance.
Note that the filtering calculates a weighted average of the „base point“ f(x) and its K
neighbours to the right and to the left. At the interval ends this causes problems. However, in
case of regional modelling the interval ends are irrelevant because of the „sponge zones“.
When the digital filter is applied to a spectrally represented function
n/2
f ( x) = a0 + ∑ ak sin( 2πkx) + bk cos(2πkx)
k =1
The filtered function f* may be written as
n/2
f ( x) = γ 0 a0 + ∑ γ k [ak sin( 2πkx) + bk cos(2πkx)]
*
k =1
with the „response function“
K
γ k = α 0 + 2∑ α l cos(2πlk )
l =1
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Fourier-filtering along a limited segment
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For analytical
purposes, we are in
need for 2-d
isotropic filters,
which separate
large, medium and
small spatial scales
in a limited
(regional) gridded
field.
Feser, F., and H. von Storch,
2005: Spatial two-dimensional
discrete filters for limited area
model evaluation purposes.
Mon. Wea Rev. 133, 1774-1786
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The construction of a 2d digital filter
Filter should have ideally identical response
functions for all waves of same 2-d wavelength.
The filter array (footprint) must be quadratic and
symmetrical in respect to the meridional and to the
zonal, but also to both diagonals.
Thus only a few filter weights need to be
determined, the remaining ones are given by
symmetry.
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The construction of a 2d digital filter
We construct an 2d-filter, which approximate a given response
function with
- γ(k)≈ 0 for 0≤ k ≤ k1 and k2 ≤ k ≤ kmax, and
- γ(k)≈ 1 for k1≤k≤k2:
-This filter suppresses most of the variance on certain
frequency bands, while retaining almost all variance on others.
- The sum of large-scale, band-pass and small-scale
contributions do not add up to the original field.
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Determining the weights αk
When more „responses“ γk are specified than filter
weights, then an underspecified set of linear equations
with the weights αk, k=0...K combine the unknown weights
and these responses:
K
γ k = α 0 + 2∑ α l cos(2πlk / n)
l =1
Minimizing the error at the “too many” wave-numbers with
a specified response leads to a matrix problem, which can
be solved with conventional algebraic methods.
It remains the choice of the “too many” specified
responses γk. We treat this as a matter of trial- and error.
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Filter Weights / Footprint
Grid points
High-pass
Grid points
Band-pass
Grid points
Grid points
Low-pass
Grid points
Grid points
Filters were chosen with N = 8 points, so that their
spatial extension is
(2N + 1)x(2N + 1) = 17x17 points.
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Response Functions
Wave number k
High-pass
Wave number l
Band-pass
Wave number l
Wave number l
Low-pass
Wave number k
Wave number k
Response functions associated with the filter
weights of the last figure.
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Application of filters
Analysis of RCM output
• Determination of added value in an RCM driven
reconstruction of regional weather
• Characterization of simulation typhoon
patterns
• Detection of polar Lows in RCM simulations
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Pattern correlations (%)
PCC
DWD and NCEP
PCC improvement/
deterioration
RCM Nudge
Positive values
show added value
of the regional
model.
95% significant
deviations are
marked by a *.
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PCC improvement/
deterioration
RCM Standard
Typhoon Winnie (1997): filtered SLP fields
large scales
retained,
(diameter ≥600
km)
Spatially
filtered air
pressure field
medium scales
retained (≤360
km; ≥180 km)
Left sides:
NCEP reanalysis after
interpolation on
50 km grid;
Right sides:
RCM simulation
on 50 km grid.
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small scales
retained
(≤ 180 km)
10 IMSC, 20-24 August 2007, Beijing
Example: detection of polar low
band-pass filtered
Weather chart, 18.1.1998, 1:00
Full field
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Automatic detection of polar low
locations (no tracks yet)
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Summary
We have constructed a family of 2-d near-isotropic digital
filters suitable for analysing the output of RCMs (on a limited
grid).
-The filters approximate a given response function.
- The filters are not additive.
- We use the filters, in particular the band-pass filters,
routinely for
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(
Determining added value of RCM simulations over
the driving large-scale analysis
(
Characterizing meso-scale structures of “small” cyclones
(
Automatic detection of meso-scale features, such as
polar lows.
10 IMSC, 20-24 August 2007, Beijing