1. Introduction
Digital filtering has been shown to be an effective means of initializing data for numerical weather prediction. In Lynch and Huang (1992, hereafter LH92) a simple filter was applied to a sequence of values centered on the initial time t = 0; these values were generated by two short adiabatic model integrations, one forward and one backward from t = 0. It was found that a total span of 6 h was required for effective elimination of high-frequency noise. Huang and Lynch (1993, hereafter HL93) showed, by means of an optimal filter, that equally effective noise control could be achieved with a span of only 3 h. They also described a means of incorporating diabatic effects: the backward adiabatic integration (of length 1.5 h) is followed by a diabatic forecast of twice the length, and the values generated by the diabatic integration are processed by the filter.
In Lynch and Huang (1994, hereafter LH94), recursive filters were applied to the initialization problem. Several schemes were investigated; in one, designated RAD—recursive adiabatic-diabatic—in LH94, the backward adiabatic integration was processed with a recursive filter, and the terminal output value was used to initiate a forward diabatic integration, to which recursive filtering was again applied. The application of filtering to both the reverse and forward integrations is also possible using a nonrecursive filter. In this note we examine the advantages of this idea.
2. A simple conceptual model
The initialization scheme described in HL93 is illustrated schematically in Fig. 1a. The thin line represents the reverse adiabatic integration of duration T. This is not filtered; its terminal value, depicted by an open circle, is used to initiate a forward diabatic integration (the thick line) that is then subjected to filtering. This integration must be of duration 2T to ensure that the output of the symmetric filter is valid at t = 0. The modified scheme, to be analyzed here, is illustrated in Fig. 1b. The reverse adiabatic integration (thin curve) is filtered, yielding an output valid at t = −T/2. This value, depicted by the black spot, is used to initiate a forward diabatic integration (thick curve) of duration T, centered on the initial time t = 0.
Considering the adiabatic component of the solution, we see that the HL93 scheme involves a single application of a filter of span 2T; the modified scheme involves two applications of a filter of span T. Comparison between the schemes can be made by considering the relationship between the magnitude of the response function, |H2T(ω)|, of the filter with the longer time span and the squared response function |HT(ω)|2 of the filter with the shorter time span.
3. Some examples of filters
In Fig. 2, two response functions are shown. The ordinate is the filter attenuation, δ = 20 log|H(θ)| (dB). The solid curve is the response of a Dolph–Chebyshev filter (denote it H2T) of total span 2T = 3 h; the stop-band edge is τs = 3 h and M = 18. The dashed curve is the squared response of a Dolph–Chebyshev filter (
If a semi-Lagrangian advection scheme is used, relatively long time steps are possible. We consider a comparison between filters H2T and
4. Choosing the filter parameters
Methods of ensuring that singly and doubly applied filters have the same level of damping for high frequencies will now be described. Let unprimed and primed quantities refer to the filters with the longer and shorter time spans, respectively.
5. Application to a forecast model
The new initialization scheme has been evaluated by application to the limited-area spectral model ALADIN (partially described in Bubnová et al. 1995). This model is run quasi-operationally at Météo-France and coupled with the global spectral NWP model ARPEGE. Horizontal representation of the variables is achieved by double Fourier series. A nonhydrostatic version of the model has been developed, but the results presented below are for the hydrostatic version (application to the nonhydrostatic version would simply involve filtering of the two additional prognostic variables). The grid-size here is approximately 18.3 km and there are 27 vertical levels. The time step is Δt = 450 s. It was found to be necessary to use a smaller time step, Δt = 225 s, for the backward adiabatic integration to ensure stability. The geographical area covered may be seen later in Fig. 7.
ALADIN does not have its own analysis scheme. The initialized analysis produced by the global spectral model ARPEGE is transformed to the resolution required for ALADIN. This transformation introduces spurious noise that must be removed by initialization on the limited domain. Four forecasts were performed starting from analyses valid at 0000 UTC 21 January 1996. One was from the transformed analysis without any subsequent initialization. The other three forecasts followed digital filtering initialization in three versions.
A Lanczos filter with span 6 h and a cutoff τc = 6 h. A backward integration of 3 h was followed by a 6-h diabatic forecast. Filtering was applied once only, to the forward integration.
A Dolph–Chebyshev filter with span 4.5 h and stop-band edge τs = 3 h. Filtering was again applied once only, to the forward integration.
A Dolph–Chebyshev filter with span 2.25 h and stop-band edge τs = 3 h. Filtering was applied twice, to both the backward and forward integrations. This is the new scheme.
The frequency responses of the filters used in the three schemes are shown in Fig. 5. Note that for the new scheme it is the square of the response function that is plotted since the filter is applied twice. The Lanczos filter is precisely that used in LH92 and is known to be effective. Note that the abscissa is the period τ. It is clear that the responses of the three filtering schemes are very similar. The time spans used here are longer than those considered in section 3 and allow a better damping of high frequencies, with a ripple ratio lower than 0.05 instead of 0.10 for Dolph–Chebyshev filters. At the same time, the attenuation of periods longer than 12 h remains small. Note that the constraints on the filters are not as strong in this case as might be required for a global model (as discussed in Fillion et al. 1995) or a limited-area model with its own assimilation cycle.
The evolution of the absolute tendency of surface pressure averaged over the forecast domain is shown in Fig. 6. The excessive noise present in the uninitialized forecast (solid curve) is successfully removed by all the filtering schemes, and they appear to be equally effective in this regard. There is a small amount of residual noise but it is not of sufficient amplitude to cause concern.
The average absolute surface pressure tendency is an indicator of the level of noise in the external gravity wave components. It is also necessary to study the impact of filtering on spurious internal gravity wave energy. For this purpose, the 500-hPa vertical velocity at the initial time was examined for the four forecasts (results not shown). It was found that large amplitude, small-scale noise in the uninitialized run, particularly in the region of the Alps, was effectively removed by all the filtering schemes.
The changes induced by the three filtering schemes were examined and compared. It was found that the new scheme consistently led to smaller changes than either of the schemes involving single filtering. The impact on the 500-hPa temperature analysis is shown in Fig. 7. Figures 7a,b,c show, respectively, the increments due to filtering schemes 1, 2, and 3. It is clear that the changes brought about by the new scheme are significantly smaller than those of the alternatives. Notwithstanding this, the noise reduction is equally effective. It is reasonable to ascribe the reduction in initialization increments for the new scheme to the reduced diabatic discrepancy associated with the shorter span since, as we have seen, the filter frequency response is virtually identical for the three schemes.
The above results demonstrate that a significant gain in efficiency can be made by means of the new scheme. This scheme results in smaller increments to the initial fields but is equally effective in reducing high frequency noise in the forecast. The new scheme with Dolph–Chebyshev filters has been implemented in the operational ALADIN-LACE suite (focusing on central and eastern Europe) and in the preoperational ALADIN-FRANCE suite at Météo-France. It has also been implemented in the global system ARPEGE, in the framework of incremental initialization, so as to further reduce the diabatic discrepancy and preserve tidal modes (as suggested in LH94). In order to reasonably damp the large-scale gravity components introduced by analysis, a more selective Dolph–Chebyshev filter has been used (with τs = 5 h as the stop-band edge).
REFERENCES
Bubnová, R., G. Hello, P. Bénard, and J.-F. Geleyn, 1995: Integration of the fully elastic equations cast in the hydrostatic pressure terrain-following coordinate in the framework of the ARPEGE/Aladin NWP system. Mon. Wea. Rev.,123, 515–535.
Fillion, L., H. L. Mitchell, H. Richie, and A. Staniforth, 1995: The impact of a digital filter finalization technique in a global data assimilation system. Tellus,47A, 304-323.
Huang, X.-Y., and P. Lynch, 1993: Diabatic digital-filtering initialization: Application to the HIRLAM model. Mon. Wea. Rev.,121, 589–603.
Lynch, P., 1997: The Dolph–Chebyshev window: A simple optimal filter. Mon. Wea. Rev.,125, 655–660.
——, and X.-Y. Huang, 1992: Initialization of the HIRLAM model using a digital filter. Mon. Wea. Rev.,120, 1019–1034.
——, and ——, 1994: Diabatic initialization using recursive filters Tellus,46A, 583–597.
Schematic illustration of original and modified filtering procedures. Thin lines indicate backward adiabatic integrations; thick lines indicate forward diabatic integrations. (a) Scheme of Huang and Lynch (1993): adiabatic step is not filtered. (b) Modified scheme: both steps are filtered.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
Frequency response (dB) for Dolph filters with ripple ratio r ≈ 0.1. Solid line: filter order N = 2M + 1 = 37; δ = 20 log |H(θ)| plotted. Dashed line: filter order N = 2M + 1 = 19; δ = 20 log |H(θ)|2 plotted.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
Amplitude response for Dolph filters as in Fig. 2 above. Solid line: filter order N = 2M + 1 = 37; |H(θ)| plotted. Dashed line: filter order N = 2M + 1 = 19; |H(θ)|2 plotted.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
Frequency response (dB) for Dolph filters with ripple ratio r ≈ 0.1. Solid line: filter order N = 2M + 1 = 13; δ = 20 log |H(θ)| plotted. Dashed line: filter order N = 2M + 1 = 7; δ = 20 log |H(θ)|2 plotted.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
The frequency response of the filters used in the three schemes described in the text. Solid line—scheme 1, Lanczos filter applied once; dashed line—scheme 2, Dolph–Chebyshev filter applied once; dotted line—scheme 3, Dolph–Chebyshev filter applied twice. Note that for scheme 3 the square of the response function is plotted.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
Evolution of the mean absolute surface pressure tendency for the first six forecast hours, starting from uninitialized data (solid) and from data filtered with schemes 1 (dotted), 2 (dashed), and 3 (dot–dashed).
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2
Impact on the 500-hPa temperature analysis: (a), (b), and (c) show the increments due to filtering schemes 1, 2, and 3. The contour interval is 0.2 K.
Citation: Monthly Weather Review 125, 8; 10.1175/1520-0493(1997)125<1976:ITEOAD>2.0.CO;2