10 IMSC, August 2007, Beijing Page 1 Construction and applications of 2-d digital filters for separating regional spatial scales Hans von Storch 12, Frauke Feser 1 and Matthias Zahn 12 1 Institute for coastal research, GKSS research center, Germany 2 Meteorological Institute, University of Hamburg, Germany

10 IMSC, August 2007, Beijing Page 2 Construction and applications of 2-d digital filters for separating spatial scales Hans von Storch, Frauke Feser and Matthias Zahn Institute for Coastal Research GKSS, Geesthacht, Germany An algorithm for deriving 2-d isotropic digital filters, which approximates a given response function, has been derived. These filters are used to examine the performance of regional atmospheric models on different spatial scales. One application is related to the identification of added value of simulations with such models as compared to driving global models, another for the identification of smaller storms, in particular typhoons and polar lows.

10 IMSC, August 2007, Beijing Page 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, August 2007, Beijing Page 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 with the „response function“ The filtered function f* may be written as Response function of a filter

10 IMSC, August 2007, Beijing Page 5 Fourier-filtering along a limited segment

10 IMSC, August 2007, Beijing Page 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,

10 IMSC, August 2007, Beijing Page 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, August 2007, Beijing Page 8 We construct an 2d-filter, which approximate a given response function with -  (k)  0 for 0≤ k ≤ k 1 and k 2 ≤ k ≤ k max, and -  (k)  1 for k 1 ≤k≤k 2 : -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, August 2007, Beijing Page 9 Determining the weights  k 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, August 2007, Beijing Page 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-passBand-passHigh-pass Grid points

10 IMSC, August 2007, Beijing Page 11 Response Functions Low-passBand-passHigh-pass Wave number k Wave number l Response functions associated with the filter weights of the last figure.

10 IMSC, August 2007, Beijing Page 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, August 2007, Beijing Page 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, August 2007, Beijing Page 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, August 2007, Beijing Page 15 Weather chart, , 1:00 Woetmann Nielsen, N., Om forudsigelighed af polare lavtryk, Vejret, 20, 37–48, in Danish, 1998 band-pass filtered Full field Example: detection of polar low

10 IMSC, August 2007, Beijing Page 16 Automatic detection of polar low locations (no tracks yet)

10 IMSC, August 2007, Beijing Page 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 over the driving large-scale analysis  Characterizing meso-scale structures of “small” cyclones  Automatic detection of meso-scale features, such as polar lows. Summary