1. Introduction
To eliminate spurious high-frequency oscillations, the initial data for numerical weather prediction models must be modified to reduce gravity wave components to a realistic level. This process is called initialization. Of the many methods of initialization that have been developed, one of the simplest is based on digital filtering (Lynch 1990). In Lynch and Huang (1992, hereafter LH92) an adiabatic initialization is performed by carrying out two short model integrations, one forward and one backward from the initial time. For each model variable at each grid point and level, this produces a sequence of values centered on the initial time. Each sequence is processed with a simple low-pass filter, and the initialized data comprises the resulting values. In LH92 a filter based on the Fourier transform of an ideal frequency response function, modified by a Lanczos window (defined below) was used. It was found that a filter span of 6 h was required to achieve adequate suppression of spurious oscillations.
The generalization of the filtering procedure to account for diabatic effects was demonstrated in Huang and Lynch (1993, hereafter HL93). The idea is to integrate the model adiabatically backward for half the span and use the terminal values so obtained as initial data for a forward diabatic integration over the total span. The sequences of values produced by the forward integration are centered on the initial time, and they may be low-pass filtered to produce the initialized data. This paper also showed how a more efficient initialization is possible using an optimal filter: the total filter span was reduced from 6 to 3 h.
The theoretical background for the design of optimal filters is the Chebyshev alternation theorem (Oppenheim and Schafer 1989). The construction of the optimal filter involves the solution of a nonlinear minimization problem using an iterative procedure called the Remez exchange algorithm (Esch 1990). In this note we describe a simple filter based on the Dolph–Chebyshev window, which gives results similar to those achieved with the optimal filter but which is much simpler to implement.
2. The Dolph–Chebyshev window
3. Design of low-pass filter
4. Comparison with other window functions
For initialization, we wish to keep the filter span as short as possible to minimize computation. However, if the filter is to be used as a constraint in four-dimensional data assimilation, the time span is set to the period over which data is to be assimilated. As the span increases and, with it, the order N, closer approximation to the ideal square-wave frequency response should be possible. For the Fourier expansion, this is so; but the amplitude of the Gibbs oscillations does not diminish with increasing order of truncation, so windowing is still necessary. For the optimal filter discussed below, excellent approximation to the ideal is attainable for higher order: it is possible to limit the approximation error in the pass band 0 ≤ θ ≤ θp. As the Dolph function is monotonic in the range 0 ≤ θ ≤ θs it is not possible to guarantee a response in the pass band whose flatness increases sufficiently quickly with increasing order. Thus, the Dolph function may be unsuitable for direct use as a filter; however, it can be used in the same way as other windows, in combination with the truncated Fourier transform of the response function for an ideal low-pass filter, to control the Gibbs oscillations and achieve a high accuracy of approximation to the ideal.
5. Comparison with optimal filter
An optimal filter has the smallest maximum approximation errors in the pass and stop bands for a prescribed transition band. Once the filter order N, pass-band edge θp, and stop-band edge θs are given, the optimal filter coefficients may be calculated by an iterative numerical procedure. The examples below were generated using the code in McClellan et al. (1973). In Fig. 5 we compare the optimal filter response for four values of the pass-band edge, τp = 4 h, 6 h, 8 h, and 10 h with the Dolph filter. The fixed parameters are T = 3 h, Δt = 0.5 h, and τs = 3 h. Thus M = 3, θs ≈ 1.05, and θp varies from about 0.75 to about 0.3. The response of the optimal filter approaches that of the Dolph filter (with the same values of M and θs) as τp increases (or θp decreases): for τp = 10 h, the two curves are indistinguishable on the plot.
In HL93 the filter parameters used for the optimal filter were T = 3 h, Δt = 360 s, τs = 3 h, and τp = 15h (so M = 15, θs ≈ 0.2, and θp ≈ 0.04). The response of this filter (see Fig. 12 of HL93) was compared to the Dolph filter with the same values of M and θs: the results (not shown here) were, for practical purposes, identical.
To further illustrate the close similarity between the Dolph and optimal filters, Table 1 presents the filter coefficients {hn: 0 ≤ n ≤ M} for two filters. In each case the total span is T = 3 h and the time step Δt = 300 s, so that M = 18 and N = 37. The stop-band edge is τs = 3 h for each filter and the pass-band edge for the optimal filter is τp = 15 h. It can be seen from Table 1 that the coefficients agree to three significant figures: for practical purposes the two filters may be considered to be essentially equivalent.
The optimal filter is more general than the Dolph filter: it can be designed to have multiple pass and stop bands and may have ripples in the pass bands. The Dolph window cannot replicate this behavior, as it is monotone in the interval [0, θs]. But for the parameter values of interest here the Dolph filter gives comparable results. Since the optimal filter is, by construction, the best possible solution to minimizing the maximum deviation from the ideal in the pass and stop bands, the Dolph filter shares this property provided the equivalence holds. In the appendix it is proved that the Dolph window is, in fact, an optimal filter whose pass-band edge, θp, is the solution of the equation W(θ) = 1 − r. Note the essential distinction: for the general optimal filter, θp can be freely chosen; for the Dolph window, it is determined by the other parameters. The algorithm for the optimal filter is complex, involving about one thousand lines of code; calculation of the Dolph filter coefficients is simplicity: the Chebyshev polynomials are easily generated from the recurrence relation, and the coefficients follow immediately from (3).
6. Application to initialization
In view of the practical indistinguishability of the Dolph filter and the optimal filter for the parameter values chosen in HL93 (see Table 1), we may expect that the results obtained by initializing with a Dolph filter would be virtually identical to those reported in that paper. Another application will be described here. Richardson (1922) calculated the pressure tendency using observations valid at 0700 UTC 20 May 1910. Richardson’s data tables have been extended using original sources, and a model based on his formulation of the primitive equations has been written (Lynch 1994). For the unmodified data, the initial pressure tendency at a central point calculated using the model was 145 hPa per 6 h, in agreement with Richardson’s value. When an initialization was performed using a Lanczos windowed filter with time step Δt = 300 s, cutoff period τc = 6 h, and span T = 6 h (as in LH92), the tendency was drastically reduced to a value of −2.3 hPa per 6h. The same data was initialized using a Dolph filter with a 3-h span and stop-band edge τs = 3 h; the filter coefficients are shown in Table 1. The initial pressure tendency was, in this case, further reduced to a value of −0.9 hPa per 6 h. Richardson reported observations for the date and time in question showing that the barometer was almost steady in the region of the central point. Thus, the value produced with the Dolph filter is the more realistic result.
7. Conclusions
The Dolph–Chebyshev window has properties similar to those of an optimal filter: it has been shown to be optimal for an appropriate choice of parameters. It has an explicit analytical expression, making it especially easy to implement. Its effectiveness has been demonstrated by application to the forecast first made by Richardson. The initial pressure tendency was reduced from an unrealistic to a reasonable level. The use of the filter is not restricted to initialization; it may also be applied as a weak constraint in four-dimensional data assimilation. Preliminary four-dimensional variational assimilation experiments with the HIRLAM model have been made using the Dolph window as a weak constraint on high-frequency components. The procedure was successful in controlling gravity waves. A comparison with the normal mode method is planned; this work will be reported elsewhere.
Acknowledgments
It is a pleasure to thank the following for helpful discussions and comments on an earlier draft: Jean-François Geleyn, Dominique Giard, Nils Gustafsson, Xiang-Yu Huang, Jan Erik Haugen, Erland Källén, Aidan McDonald, and two anonymous reviewers.
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APPENDIX
Optimality of the Dolph Filter
The Dolph window W(θ) = rT2M [x0 cos(θ/2)] has M zeros in θs < θ < π. There are M + 1 extreme points in θs ≤ θ ≤ π (these include θs and π) for which W(θ) = ±r. If we define θp such that W(θp) = 1 − r, the point θp is a further extreme point, bringing the total to M + 2. Thus, W(θ) fulfils the conditions of the alternation theorem and must be the unique optimal solution with the pass-band edge given by θp.
Frequency response (dB) for Dolph filter with ripple ratio r = 0.1 for filter orders N = 2M + 1 with M = 2, 4, 8, and 16.
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Frequency response (dB) for Dolph filter with stop-band edge τs = 3 h (θs ≈ 1.05) for filter orders N = 2M + 1 with M = 2, 3, and 4.
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Frequency response for low-pass filter with parameter values T = 24 h, Δt = 0.5 h, and τc = 6 h (M = 24 and θc ≈ 0.5) modified by a Dolph window with stop-band edge θs ∈ {π/M, 2π/M, 4π/M} or τs ∈{T, T/2, T/4}.
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Frequency response for low-pass filter with parameter values T = 24 h, Δt = 0.5 h, and τc = 6 h (M = 24 and θc ≈ 0.5) modified by uniform, Lanczos, Hamming, and Dolph windows.
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Frequency response (dB) for Dolph filter with M = 3 and τs = 3 h, and for optimal filters for τp = 4 h, 6 h, 8 h, and 10 h. Other parameters: T = 3 h, Δt = 0.5 h, and τs = 3 h.
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Evolution of the noise parameter N1 (10−6 s−1) for the first 3 h of forecasts from uninitialized data (solid), and data initialized using the Lanczos filter (dashed) and the Dolph filter (dotted).
Citation: Monthly Weather Review 125, 4; 10.1175/1520-0493(1997)125<0655:TDCWAS>2.0.CO;2
Filter coefficients for a Dolph filter and for an optimal filter with τp = 15 h. In both cases the stop-band edge is τs = 3 h, the span T = 3 h and Δt = 300 s, so M = 18 and N = 37.