Download - Construction and applications of 2-d digital filters for separating regional spatial scales
10 IMSC, 20-24 August 2007, BeijingPage 1
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 Institute, University of Hamburg, Germany
10 IMSC, 20-24 August 2007, BeijingPage 3
Talk structure:
Construction of a 2D-filter
Filter applications:
- added value in RCM simulations
- characterization of typhoon patterns
- detection of polar lows
10 IMSC, 20-24 August 2007, BeijingPage 4
With certain constants k. 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
])()([)()(1
0*
K
kk kxfkxfxfxf
)2cos()2sin()(2/
10 kxbkxaaxf k
n
kk
with the „response function“
The filtered function f* may be written as
)]2cos()2sin([)(2/
100
* kxbkxaaxf k
n
kkk
K
llk lk
10 )2cos(2
Response function of a filter
10 IMSC, 20-24 August 2007, BeijingPage 5
Fourier-filtering along a limited segment
10 IMSC, 20-24 August 2007, BeijingPage 6
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
10 IMSC, 20-24 August 2007, BeijingPage 7
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.
10 IMSC, 20-24 August 2007, BeijingPage 8
We construct an 2d-filter, which approximate a given response function with
- (k) 0 for 0≤ k ≤ k1 and k2 ≤ k ≤ kmax, and
- (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.
The construction of a 2d digital filter
10 IMSC, 20-24 August 2007, BeijingPage 9
Determining the weights k
K
llk nlk
10 )/2cos(2
When more „responses“ k are specified than filter weights, then an underspecified set of linear equations with the weights k, k=0...K combine the unknown weights and these responses:
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 k. We treat this as a matter of trial- and error.
10 IMSC, 20-24 August 2007, BeijingPage 10
Filters were chosen with N = 8 points, so that their spatial extension is
(2N + 1)x(2N + 1) = 17x17 points.
Filter Weights / Footprint
Low-pass Band-pass High-pass
Grid points Grid points Grid points
Gri
d p
oin
ts
Gri
d p
oin
ts
Gri
d p
oin
ts
10 IMSC, 20-24 August 2007, BeijingPage 11
Response Functions
Low-pass Band-pass High-pass
Wave number k Wave number k Wave number k
Wave n
um
ber
l
Wave n
um
ber
l
Wave n
um
ber
l
Response functions associated with the filter weights of the last figure.
10 IMSC, 20-24 August 2007, BeijingPage 12
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
10 IMSC, 20-24 August 2007, BeijingPage 13
Positive values show added value
of the regional model.
95% significant deviations are
marked by a *.
Pattern correlations (%)
PCC
DWD and NCEP
PCC improvement/ deterioration
RCM Nudge
PCC improvement/ deterioration
RCM Standard
10 IMSC, 20-24 August 2007, BeijingPage 14
Spatially filtered air
pressure field
Left sides: NCEP re-
analysis after interpolation on
50 km grid; Right sides:
RCM simulation on 50 km grid.
large scales retained, (diameter ≥600 km)
medium scales retained (≤360 km; ≥180 km)
small scales retained (≤ 180 km)
Typhoon Winnie (1997): filtered SLP fields
10 IMSC, 20-24 August 2007, BeijingPage 15
Weather chart, 18.1.1998, 1:00
Woe
tma
n n Niel sen
, N
., O
m f
oru
dsig
elig
he d
af
pola
re la
vtry
k ,
Vej
r et,
20 ,
37–
48,
in D
anis
h,
1 99
8
band-pass filtered
Full field
Example: detection of polar low
10 IMSC, 20-24 August 2007, BeijingPage 16
Automatic detection of polar low locations (no tracks yet)
10 IMSC, 20-24 August 2007, BeijingPage 17
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
Determining added value of RCM simulations overthe driving large-scale analysis
Characterizing meso-scale structures of “small” cyclones
Automatic detection of meso-scale features, such aspolar lows.
Summary