construction and applications of 2-d digital filters for separating regional spatial scales

10 IMSC, 20-24 August 2007, Beijing

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


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


Fourier-filtering along a limited segment


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


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.


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


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.


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


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.


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


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


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


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


Automatic detection of polar low locations (no tracks yet)


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

construction and applications of 2-d digital filters for separating regional spatial scales

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