# Construction and applications of 2-d digital filters for

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Construction and applications of 2-d digital filters for

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 Page 1 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 Page 3 10 IMSC, 20-24 August 2007, Beijing 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 Page 4 10 IMSC, 20-24 August 2007, Beijing Fourier-filtering along a limited segment Page 5 10 IMSC, 20-24 August 2007, Beijing 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 Page 6 10 IMSC, 20-24 August 2007, Beijing 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. Page 7 10 IMSC, 20-24 August 2007, Beijing 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. Page 8 10 IMSC, 20-24 August 2007, Beijing 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. Page 9 10 IMSC, 20-24 August 2007, Beijing 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. Page 10 10 IMSC, 20-24 August 2007, Beijing 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. Page 11 10 IMSC, 20-24 August 2007, Beijing 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 Page 12 10 IMSC, 20-24 August 2007, Beijing 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 *. Page 13 10 IMSC, 20-24 August 2007, Beijing 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. Page 14 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 Page 15 10 IMSC, 20-24 August 2007, Beijing Automatic detection of polar low locations (no tracks yet) Page 16 10 IMSC, 20-24 August 2007, Beijing 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 Page 17 ( 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