## 1. Introduction

Most methods of data assimilation employ a background error convariance matrix that spreads the influence of the observation in space, but also filters analysis increments through the cross-variable correlations. Usually a simple linear dynamical balance relationship (see Daley 1991) is used to obtain these cross-variable correlations. However, a linear balance constraint is often insufficient to prevent the development of spurious energy on the fastest time scales when the numerical forecast is produced. Thus a separate filtering procedure called initialization is done to filter the fastest time scales from the analysis. With four-dimensional variational methods, balance constraints applied during the assimilation process can ensure that the resulting analysis is sufficiently balanced so as to avoid a separate initialization (e.g., Thépaut and Courtier 1991; Polavarapu et al. 2000; Gauthier and Thépaut 2001). However, for applications that cannot afford a four-dimensional variational scheme, a separate initialization procedure may still be needed to reduce spinup effects and to provide a smooth background for quality controlling data for the next analysis.

In addition to generating forecasts, analyses are also used as proxies for the atmosphere in climate studies. For this application, it is important that a time series of analyses be smooth and not subject to sudden impulses. For this reason, the method of incremental analysis updating (IAU) was introduced by Bloom et al. (1996) whereby analysis increments are spread over a time interval. Despite a superficial similarity to the process of “nudging,” Bloom et al. (1996) showed that IAU filters analysis increments only, and not the background state (as nudging does). [For a detailed analysis of the mechanism of nudging, see Bao and Errico (1997).]

Another method of filtering analyses involves an explicit digital filter (Lynch and Huang 1992). If atmospheric motions are largely balanced and there is a clear frequency separation between balanced (slow) and unbalanced (fast) motions, then initialization can be accomplished by simply filtering the fast motions. For flows on a midlatitude *f* plane, a frequency gap is present. Unfortunately, it is difficult to find such a frequency gap for motions on a sphere, in the Tropics (Daley 1991), at mesoscales or in the mesosphere (Koshyk et al. 1999). Nevertheless, because it is simple, effective, and easy to implement, the digital filter is used at many operational forecast centers as a constraint in 4D variational schemes (Gauthier and Thépaut 2001) or as a separate initialization procedure (Puri et al. 1998; Fillion et al. 1995; Lynch et al. 1998; Zupanski and Zupanski 1995).

While Bloom et al. (1996) demonstrated the filtering aspects of IAU, the relationship to explicit digital filtering was not made clear. The goal of this work is to theoretically examine the similarities and differences between these two methods in both linear and nonlinear contexts. By doing so, advantages and shortcomings of the two methods become clearer. For example, it will be shown that IAU is identical to an incremental digital filtering (IDF) in the linear case for identical weight coefficients. Based on this equivalence, it is then possible to deduce that the typical (constant) weights used in IAU damp slow waves too heavily compared to more general weights used in IDF implementations. Then it is very simple to sharpen the IAU response by applying windowing functions as is typically done with digital filters. Differences in the two methods also lead to a better understanding of the properties of the initialized fields obtained with the two methods.

The proof that IAU and IDF are equivalent in the linear case is provided in section 2. The case of weakly nonlinear models is considered in section 3. A brief summary and discussion of the results is presented in section 4.

## 2. Linear analysis of IAU and IDF

To make the connection between IAU and IDF, it is simplest to start with the linear case, that is, when the assimilating model is linear and adiabatic (no physical processes). The extension for both nonlinear models with physical processes will be made subsequently.

### a. IAU

This section follows Bloom et al. (1996, hereafter BTDL96) except that the IAU interval is assumed to be [−*τ*/2, *τ*/2], with the analysis time being *t* = 0. In BTDL96, the IAU interval was [0, *τ*] with the analysis time being *t* = *τ*/2. Although this is a rather modest change to the calculations of BTDL96, it greatly facilitates the comparison of the filtering due to IAU with IDF.

**x**is the discrete model state of dimension

*p,*

*A*is a constant

*p*×

*p*matrix that represents the linear dynamics of the system, and

**f**(

*t*) is a state-independent forcing of dimension

*p.*In this work, this forcing will come from analysis increments output from an assimilation system. By repeatedly inserting the analysis increments during a finite time interval (Fig. 1) centered on the analysis time, an implicit filtering of increments is done (BTDL96). This process of inserting analysis increments into the model during its run is reminiscent of the “nudging” or dynamic relaxation (Daley 1991) procedure. However, as demonstrated by BTDL96 and Bao and Errico (1997), nudging is a rather different procedure since the forcing term depends on the state, with the result that the background state is also filtered. The filtering of the background state (e.g., the removal of diurnal or tidal signals) is highly undesirable. In general, NWP centers around the world filter only analysis increments rather than analyses (model states) in order to preserve the physical signals in the background state. The filtering of increments can be done in many ways such as nonlinear normal-mode initialization (NNMI; Ballish et al. 1992), digital filtering, or within a variational assimilation algorithm (Gauthier and Thépaut 2001).

*t*= 0 and is composed of the background state (or trial field)

**x**

^{b}

_{0}

**x**

^{a}

_{0}

**x**

^{a}

_{0}

**x**

^{b}

_{0}

**x**

^{a}

_{0}

*t*= 0. Thus, the solution to (1) and (2) for an intermittent assimilation is

**x**

^{INT}

*t*

*e*

^{A(t+τ/2)}

**x**

*e*

^{At}

**x**

^{a}

_{0}

*t*

**f**(

*t*) = Δ

**x**

^{a}

_{0}

*δ*(

*t*) and substituting this into (3).

*g*(

*t*) = 0 for

*t*>

*τ*/2. The IAU solution is then

*A*

**u**

_{j}

*λ*

_{j}

**u**

_{j}

*j*

*p*

*λ*

_{j}is the complex eigenvalue,

*λ*

_{j}=

*σ*

_{j}+

*iω*

_{j},

*σ*

_{j}is the growth (decay) rate of mode

*j,*and

*ω*

_{j}is the mode frequency. The matrix

*A*is assumed to include both normal and nonnormal modes. If 𝗘 is the matrix of eigenvectors, we can multiply (5) on the left-hand side by 𝗘

^{−1}to project vectors in physical space into mode space:where we have definedand used the fact that

*A*= 𝗘

*D*𝗘

^{−1}. In other words, for each eigenmode

*j,*we have that

*x̂*

^{INT}

_{j}

*e*

^{λj(t+τ/2)}

*b̂*

_{j}

*e*

^{λjt}

*d̂*

_{j}

*t*=

*τ*/2. This function measures the impact on the IAU analysis increments compared to an intermittent assimilation. The complex response function

*R*(

*λ,*

*τ*/2) is formed from the ratio of the second terms in (11) and (12):

*R*

*λ,*

*τ*

*γ*

*λ,*

*τ*

*t*= 0. The response function depends on the choice of the time-dependent weights,

*g*(

*t*). In fact,

*γ*(

*λ,*

*τ*/2) is similar to a finite Fourier transform of the weight function. This is readily apparent on discretizing the time domain using

*t*

_{n}=

*n*Δ

*t*and

*τ*/2 =

*N*Δ

*t.*Then,We can expand the complex eigenvalue as

*λ*=

*σ*+

*iω*and introduce

*θ*=

*ω*Δ

*t*and

*α*=

*σ*Δ

*t*so thatwhere

*g̃*

_{n}=

*g*

_{n}Δ

*t*/

*g*

### b. IDF

*τ*], using a discrete time interval of Δ

*t*:(see Lynch and Huang 1992). The coefficients

*h*

_{n}are not yet defined, but they determine the response of the filter. The transfer (or response) function describes the impact of the digital filter on a single wave in frequency space:Here

*θ*=

*ω*Δ

*t*is the digital frequency. In Fig. 1, we see that the digital filter (DF) span that produces a filtered state at

*t*=

*τ*/2 corresponds to the interval, (0,

*τ*). Although the analysis increment is available at

*t*= 0, the filtered state is produced at

*t*=

*τ*/2. To produce an initialized state at

*t*

_{0}requires a backward model integration starting from

*t*

_{0}, so that the filter span is centered on

*t*

_{0}. With irreversible physical processes such as diffusion or precipitation, these processes must be omitted from the backward integration (Huang and Lynch 1993). However, for NWP purposes, the filtered state is only required at the next analysis time (here

*t*=

*τ*), so no backward integration is necessary (see, e.g., Fillion et al. 1995). Fillion et al. (1995) refer to this process as “finalization,” and it is this process that we consider here. The integration from

*t*=

*τ*/2 to

*t*=

*τ*is performed after the filtering is complete and produces a forecast, or a trial field for the next assimilation cycle.

**x**

^{b}

_{0}

**x**

^{b}

_{0}

**x**

^{a}

_{0}

*τ*/2 to the unfiltered background at

*τ*/2:

**x**

^{F}

*τ*

**x**

^{b}

*τ*

*F*

**x**

^{b}

_{0}

**x**

^{a}

_{0}

*F*

**x**

^{b}

_{0}

*F*(

**x**) represents the filtering process. Because the model is linear in this section, and the filtering process is linear, this process is equivalent to applying a digital filter to a time series of increments, Δ

**x**

_{j},

*j*= 0, … , 2

*N*where we discretized the time interval using

*τ*/2 =

*N*Δ

*t*:with the initial condition Δ

**x**(0) = Δ

**x**

^{a}

_{0}

**x**

^{b}(

*τ*/2) +

*F*[Δ

**x**(0)].

**x**

*t*

*e*

^{At}

**x**

^{a}

_{0}

*t*

**x**

^{b}

_{0}

**x**

^{a}

_{0}

*x̂*

^{INT}

_{j}

*e*

^{λjt}

*b̂*

^{0}

_{j}

*e*

^{λjt}

*d̂*

_{j}

*b̂*

^{0}

_{j}

^{−1}

**x**

^{b}

_{0}

*t*= −

*τ*/2 so the solution includes the integration of the background from [−

*τ*/2, 0].

*x̂*

^{IDF}

_{j}

*e*

^{λjt}

*b̂*

^{0}

_{j}

*e*

^{λjt}

*d̂*

_{j}

*γ*

^{DF}

*R*

^{IDF}(

*λ*) =

*γ*

^{DF}. Thus the IAU and IDF responses are identical when (13) equals (23) for

*t*=

*τ*/2. Now, for the same time discretization used for IAU, (23) can be approximated asEquation (25) is the DF response function in mode space while (17) is the response function in frequency space. On comparing (25) and (17), it is clear that each dynamical mode of the model is temporally filtered in the same way.

Comparing (25) to (15) reveals that the IAU and IDF responses are identical when *h*_{n} = *g̃*_{n}, that is, when their weights are identical. There are several implications of this equivalence. BTDL86 noted that the amplitude response is the same for both growing and decaying waves with the same *e*-folding time for IAU. Thus, IDF also has the same response for growing and decaying waves with the same *e*-folding time. Since the model dynamics are involved, the filtering is model and flow dependent for both methods. Finally, the response of IAU can easily be controlled and examined in the linear context for arbitrary weights, *g̃*_{n}. In section 2c, we compare the response of constant weight filters typically used for IAU with the response of more general filters typically used for IDF schemes.

### c. Changing the IAU response

*T*

_{c}is the cutoff period of 6 h,

*θ*

_{c}is the digital frequency at

*T*

_{c}, and

*w*

_{n}is a Lanczos window used to reduce Gibbs effects arising from the truncation of Fourier series. The parameters

*N*= 13, Δ

*t*= 1 h, and

*T*

_{c}= 6 h correspond to the filter in use at the Canadian Meteorological Centre for operational weather forecasting (Fillion et al. 1995) with the Global Environmental Multiscale (GEM) model (Côté et al. 1998a,b). In contrast, our motivation is to employ data assimilation with a climate model, the Canadian Middle Atmosphere Model (CMAM; Beagley et al. 1997). For this model a shorter time step of Δ

*t*= 300 s is used. To compare with the IAU interval of BTDL96 of 6 h, a 6-h DF span is chosen. This results in a 73-point filter (i.e.,

*N*= 36). In Fig. 2, we compare the response of these two digital filters to the IAU response for constant weights for the case of neutral waves (

*σ*= 0,

*g̃*

_{n}= 1/ 73,

*τ*= 6 h). In Fig. 2, we see from the right panel that the filter span for the GEM filter is 12 h, while the CMAM filter span is 6 h, as is the IAU interval. With the coefficients defined by (26), the IDFs are weighted toward the center of the DF span or IAU interval. The GEM coefficients are heavily weighted toward the middle coefficient because the filter span is much larger than the cutoff period. The CMAM filter has coefficients of more similar magnitude since the filter spans and cutoff periods are equal. Of course the IAU weights are exactly equal. The problem is that the response function (left panel, dash-dot curve) exhibits oscillations for wave periods shorter than 6 h. The IDFs here have minimized the oscillatory response by employing a (Lanczos) window. A more severe problem with the IAU response is the excessive damping of slower waves. (The response is less than 1 for waves with periods longer than 12 h.) In contrast, the IDF responses are much closer to 1 for slow waves. As a result, IAU will damp these slow waves more than the IDFs (with the parameters chosen here) will. Clearly, it would be beneficial to sharpen the response function for IAU. With the equivalence to IDF established, this is easily accomplished by simply choosing the coefficients,

*g̃*

_{n}, similar to those used in the IDFs. In particular, the IAU response can be improved by applying a windowing function to its weights. Moreover, because the high-frequency response may be amplified when the system is unstable [see Eq. (12)], applying a window to adjust the IAU response may be necessary. In practice, however, fast waves tend to decay rapidly, so the response will not be amplified, although in the mesosphere, this may not be the case.

From (26) it is clear that the IDF response is determined by three parameters: *N,* Δ*t,* and *θ*_{c}, the number of points in the filter, the time step (or equivalently the filter span 2*N*Δ*t*), and the cutoff frequency. For IAU with constant weights, only *N* and Δ*t* are relevant. Figure 3 shows that there is little impact of increasing *N* (and decreasing Δ*t*) on the IAU response for a fixed filter span of 6 h. In practical implementations, the model time step is likely fixed so Fig. 3b shows the impact of increasing the number of points (or filter span) for a fixed time step of 10 min. Clearly shorter spans result in less damping of long waves and in less filtering overall.

## 3. Nonlinear models

**x**

_{n+1}

*M*

_{n}

**x**

_{n}

*M*

_{n}(

**x**

_{n}) so they are not indicated by a separate forcing term. The background or trial field is assumed to be available at either

*t*

_{−N}or

*t*

_{0}and is denoted

**x**

^{b}

_{−N}

**x**

^{b}

_{0}

### a. IAU

**x**

_{n+1}

*M*

_{n}

**x**

_{n}

*g̃*

_{n+1}

**x**

^{a}

_{0}

*N,*

*N*]. At the initial time,

*t*

_{−N}, the state is

**x**

_{−N}

**x**

^{b}

_{−N}

*g̃*

_{−N}

**x**

^{a}

_{0}

_{−N}=

*dM*/

*d*

**x**(

**x**

^{b}

_{−N}

*t*

_{N}, the state is approximately given by

### b. IDF

*τ*], with the time-filtered analysis increment produced at

*t*=

*τ*/2. Thus the IDF runs from time steps [0, 2

*N*] with the filtered state being produced at

*t*

_{N}. An incremental digital filter involves three steps. First the filter is run from the analysis,

**x̃**

_{0}=

**x**

_{0}+ Δ

**x**

^{a}

_{0}

**x**

_{0}=

**x**

^{b}

_{0}

**x**

_{0}=

**x**

^{b}

_{0}

*N*:The term in round brackets is the difference between the perturbed and unperturbed trajectories.

**x̃**

_{n}

**x**

_{n}

_{n−1}

_{n−2}

_{0}

**x**

^{a}

_{0}

### c. Comparing IAU and IDF

Comparing (30) and (34) reveals that the background state in both cases is unaltered. Just as in the linear case, IAU and IDF act to filter only analysis increments. Even if the filter coefficients are identical, IAU and IDF do not produce the same initialized state at *t*_{N} when the models are nonlinear, or even weakly nonlinear, as assumed here. For identical filter coefficients, the initialized state will be different because the sequence of states will differ for the two different time spans of application. IAU evolves the weighted analysis increment from the point of insertion (*t*_{j}) until *t*_{N} while IDF evolves the analysis increment from *t*_{0} to *t*_{N−j} for *j* ∈ [−*N,* *N*] (see Fig. 4).

With IAU, the initialized state is composed of the background state at *t*_{N} plus 2*N* + 1 contributions. Each contribution represents the the evolution of the weighted analysis increment, *h*_{j}Δ**x**^{a} for time step *j,* from *t*_{j} to *t*_{N} using the TLM. On the other hand, with the IDF, the weighted analysis increment is also evolved using the TLM but from time *t*_{0} through to *t*_{N−j}. Both IAU and IDF will keep all frequencies in the background (including tidal and diurnal signals) and are thus superior to a digital filtering of full fields.

*h*

_{0}. Because the coefficients are designed to sum to 1, then

*h*

_{0}= 1. In this case, (30) and (34) are identical and equal to

**x**

_{N}

**x**

^{b}

_{N}

_{N−1}

_{N−2}

_{0}

**x**

^{a}

_{0}

*h*

_{0}= 1, may be useful for debugging purposes during implementation, although this test (of reproducing the result of no filter) alone is insufficient to verify proper implementation of either scheme.

The approximate equivalence between IAU and IDF raises some interesting questions about what constitutes an analysis. This question was first raised by BTDL96 with respect to IAU. However, with a filter or IDF, it is clear that the analysis exists only after the filtering is complete, at the midpoint of the filter span (*t* = *τ*/2). The first half of the span is left unfiltered, which results in a discontinuity of time series at the initialization time (midspan). In Fig. 4 it is also clear that for IAU, the filtering is not *complete* until time *t* = *τ*/2. Because IAU is obtained by simply adding state-independent forcing terms to the model tendency equations, the states obtained between [−*τ*/2, *τ*/2] are readily available. For example, Schubert et al. (1993) use fields halfway through the IAU interval as the analysis. However, from the point of view of filtering, it is not clear that these states should be used for climate diagnostics.

## 4. Summary and discussion

For time-independent linear models, IAU is shown to be equivalent to IDF for identical filter coefficients. The IAU and IDF response is shown to depend on the model dynamics and thus not only on wave frequency but also its growth rate. The similarity of the two methods is useful to better understand how both methods work to smooth the discontinuous process of inserting analysis increments into a numerical model. In particular, it is shown that the IAU response for constant weights damps slower waves too much. By using a windowing function, the weights could be made comparable to digital filters with much sharper response functions. Moreover, without a windowing function, in IAU certain fast waves are insufficiently damped, and these can grow in time if they are also unstable.

For weakly nonlinear models with physical processes, the IAU and IDF filter analysis increments only. With equal filter coefficients, the filtered states will differ because, for a response at *t* = *τ*/2, IAU adds increments to model time steps [−*τ*/2, *τ*/2] while the IDF samples increments between [0, *τ*]. Consequently, the *j*th increment is evolved from *t*_{j} to *t*_{N} for IAU but from *t*_{0} to *t*_{N−j} for IDF, where *j* ∈ [−*N,* *N*].

If the tangent linear model is stationary for time scales of up to 6 h (the assumed IAU span), then the shift in IDF and IAU insertion times will be irrelevant. However, forecast models are unlikely to be stationary on these scales. Even global models with coarse resolution will contain diurnal and tidal signals that vary over 6 h. If the analysis increment is primarily composed of large-scale components, the shift in time between the IAU and IDF insertions may again be argued to be of little importance. In the tropospheric midlatitudes where a geostrophic-type balance is applied by many assimilation schemes, this argument may be plausible. However, in the Tropics, or in the mesosphere where this type of geostrophic balance is inappropriate, the analysis increment will contain much information on small scales. Moreover, if the analysis increment were primarily on large scales and balanced, there would be no need for a separate initialization procedure. Thus it is not feasible to argue that the analysis increment is primarily composed of large scales.

One difference between IAU and IDF is that the IAU solution can be nearly temporally continuous with an appropriate choice of weights while the IDF will result in discontinuities at the midpoint of the filter span. Thus the IAU may be preferable to the IDF for applications in which temporal continuity is important, such as the production of reanalyses for climate diagnostics. However, for both IAU and IDF, the analyses are not complete until the filtering is complete. Thus analyses are not available at time intervals shorter than (typically) 6 h.

In terms of computational cost, IAU involves 2*N* model time steps to produce an initialized state at *t*_{N}. IDF requires 2*N* steps for the filter but the filter must be run twice, once for the background state, and once for the analysis. Thus it is twice as expensive as IAU, if both IAU and IDF can be run as a single job. If each model run is a separate job submission, then the IAU will require a single job while the IDF will require two job submissions. Again, IAU will result in faster job turnaround.

While we have shown here that IAU and IDF are not identical for identical filter coefficients in the weakly nonlinear case, the nature of the difference can only be determined through implementations of both methods in practical applications. It is left to future studies to determine the importance of these differences.

Financial support for this work is from the “Modelling of Global Chemistry for Climate” project which is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS), and the Canadian Space Agency (CSA). Financial support was also received from the Meteorological Service of Canada through its Climate Research Network. We thank Ron Errico and the reviewers for providing comments which helped to improve this manuscript.

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