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The control of gravity waves
in data assimilation
27 April 1999
By Adrian
Simmons
European Centre for Medium-Range Weather
Forecasts
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4 . Digital filtering
The method of digital filtering provides an approach to initialization that is conceptually simple and easier to implement than non-linear normal-mode initialization. It involves generating a sequence of model fields and then applying a filter to the resulting time series for each model grid-point and variable or each model spectral coefficient. The filter is chosen to reduce the amplitudes of high-frequency components of the time series to acceptable levels. The forecast is then run from an appropriate point within the filtered time series.
4.1 Adiabatic, non-recursive filtering
We consider first the simplest case of adiabatic, non-recursive filtering. We denote the uninitialized analysis by
. A forward adiabatic integration is carried out for 
timesteps to generate the set of values:
|
where
denotes the model state after 
timesteps. Also, a backward adiabatic integration of the same length
is carried out to generate values:
|
The filtered initial state, or initialized analysis, is then given generally by:
|
(68) |
where
|
(69) |
The original application of digital filtering for initialization by Lynch and Huang (1992) used a modification of a basic filter defined by:
|
(70) |
This represents the discrete equivalent of the filter of a continuous function that leaves low-frequency components unchanged but removes high-frequency components completely, multiplying a fourier component
by 
, where
|
(71) |
The modified "Lanczos" filter was defined by:
|
(72) |
The term
|
in (72) provides what is known as a Lanczos window.
The filters whose coefficients are given by (70) and (72) have the property of leaving the phase of a sinusoidal wave unchanged (apart from a possible 180o shift) while reducing the amplitude of the wave. The reduction in amplitude is shown as a function of wave period in Fig. 18 . The calculation is for a cut-off period (
) of six hours, a span (
) also of six hours and a timestep (
) of 15 minutes. The solid line denotes the amplitude response for
the continuous filter 
. The basic
filter (dashed line) exhibits the familiar Gibbs oscillations, which are
greatly reduced by application of the Lanczos filter (dotted line). The
Lanczos filter gives less attenuation of longer-period waves, but weaker
filtering of waves with periods just shorter than six hours.The impact on waves of unit input amplitude and periods of 3, 6, 12 and 24 hours is shown in Fig. 19 . The amplitudes of the 6-, 12- and 24-hour period waves are reduced less when the Lanczos window is included. The phase of the 3-hour wave is reversed by the basic filter. It has smaller amplitude and no phase reversal with the Lanczos window.
An example of the noise reduction found when
and Huang (1992) applied the method to a version of the HIRLAM model is presented in Fig. 20 . The Lanczos filter was used with six-hour cutoff and span, and the model was run with a six-minute timestep. Fig. 20 shows that digital filtering initialization reduced noise more effectively than the normal-mode initialization scheme developed for HIRLAM. Other diagnostics confirmed the success of the digital filtering approach.
Figure 18 . Amplitude of the filtered wave as a function of wave period for an input sinusoidal wave of unit amplitude. The idealized continuous filter has a 6-hour cutoff, and the corresponding basic discrete filter and basic filter modified by a Lanczos window are shown for a 15-minute timestep and 6-hour span, following Lynch and Huang (1992).
Figure 19 . The response of waves of periods 3, 6, 12 and 24 hours to the basic filter (upper) and the filter modified by the Lanczos window (lower).
Figure 20 . Evolution of the mean absolute surface pressure tendency (hPa /3h) in 24-hour forecasts starting from an uninitialized analysis (solid line) and from analyses initialized using non-linear normal-mode initialization (dotted) and non-recursive digital filtering (dashed), from Lynch and Huang(1992).
4.2 Diabatic, recursive filtering
The backward integration used to generate the values
in the approach described in the preceding subsection cannot
be carried out using parametrizations of irreversible physical processes.
Use of a recursive filter offers one way to use the digital filtering
method for diabatic initialization (Lynch
and Huang, 1994).Consider the sequence of values
of model variables from consecutive timesteps of a forecast starting
from the uninitialized analysis 
, possibly
including diabatic and frictional processes. A recursive filter of order
is defined in general by the 
values 
and the 
values 
in the following expression for the filtered value 
at step 
:
|
(73) |
The process is started by applying lower-order filters to compute values of
for 
. A non-recursive implementation of this filter is set out in Appendix
C.We illustrate the case of the Second-order Quick-Start filter presented by Lynch and Huang (1994). This has
and:
|
(74) |
with
|
(75) |
and
|
(76) |
If we define:
|
(77) |
and
|
(78) |
the filter coefficients are given by:
|
(79) |
|
(80) |
|
(81) |
and
|
(82) |
The upper panel of Fig. 21 shows the input and filtered waves for a 12-hour input period, a cutoff frequency (
) of three hours and a timestep of 15 minutes. The filter causes
little reduction in amplitude of this wave, but introduces a delay or phase-lag
of a little more than half an hour. A very similar delay can be seen in
the lower panel of Fig. 21 for waves with longer periods.
Figure 21 . The input wave and filtered output wave for 12-hour input period and a second-order recursive filter (upper) and the filtered output for 12-, 24- and 48-hour input periods (lower).
The filtered response to input waves of periods 12, 3, 1 and 0.5 hours are shown in Fig. 22 , for 15- and 1-minute timesteps. The amplitude of the wave with 3-hour period is reduced by about 30%, and it too is delayed by a little over half an hour. Waves with shorter periods are damped considerably over the first hour or so. That with half-hour period is soon damped completely in the case of the 15-minute timestep.
The rapid initial damping of short-period waves and the uniformity of the phase-lags for the longer-period waves means that an initialized forecast may be successfully launched by applying the second-order filter for a span of an hour or two, and then setting the model's clock back by half an hour or so to account for the delay, before extending the forecast from the end-point of the filtered sequence. The effect of applying this procedure on the level of noise in a forecast using the HIRLAM system can be seen by comparing the dashed and solid curves in Fig. 23 . The procedure is evidently successful, and is sufficient for most forecasting purposes. It does not, however, provide initialized conditions at the analysis time or for the following hour or so, such as may be useful for diagnostic purposes or physical initialization. The alternative of simply assuming that the filtered value at the end of the span applies at the time of the uninitialized analysis is shown by the dotted curve in Fig. 23 . This introduces a phase shift of about an hour, the effect of which can be reduced by adopting the incremental approach presented in 2.5.
Figure 22 . The filtered output wave for 12-, 3-, 1- and 0.5-hour input periods and a second-order recursive filter with 15-minute timestep (upper) and 1-minute timestep (lower).
Figure 23 . Evolution of the mean absolute surface pressure tendency (hPa /3h) over six-hour forecasts starting from an uninitialized analysis (solid line) and from using two types of recursive digital filtering schemes, from Lynch and Huang(1994).
4.3 Diabatic, non-recursive filtering
Another approach to diabatic initialization is to carry out an initial backward adiabatic integration over a time interval
followed by a forward diabatic integration over an interval
, and then to apply non-recursive filtering to the sequence
of values from the diabatic integration. Lynch et al.(1997) describe a particularly
efficient variation of this approach that has been used to initialize both
limited-area and (in incremental form) global models at Météo-France.The principal cost of digital filtering initialization is that of the model integration. Improved efficiency arises partly from using a Dolph-Chebyshev filter (Lynch, 1997) that requires a shorter span to achieve the same degree of noise reduction as the Lanczos filter. A further gain comes from filtering the results of the backward adiabatic integration. This yields a filtered value at a time
prior to the analysis time, and this provides initial conditions
for a diabatic forecast over an interval 
. The initialized analysis is derived from a non-recursive filtering
of this diabatic forecast.Fig. 24 shows that similar levels of noise reduction are obtained by using the Lanczos and (shorter-span) Dolph-Chebyshev filters without filtering the backward adiabatic forecast, and by using the Dolph-Chebyshev filter on both the backward adiabatic and (half-length) forward diabatic integrations.
A disadvantage of this general approach to diabatic initialization is that the initialized fields are subject to error due to changes brought about by diabatic processes over the first half of the forward integration that is filtered. This error is reduced by using filters that need a shorter span to be effective, and by the filtering of the backward integration, which enables the length of the diabatic forecast to be halved.
Figure 24 . Evolution of the mean absolute surface pressure tendency (hPa /3h) for the first six hours of a forecast starting from an uninitialized analysis (solid line) and from analyses initialized using three non-recursive diabatic digital filtering schemes, from Lynch et al.(1997).
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07.06.2002 |
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