a. Model error

The atmospheric state at a given time t_n is represented by a spatially continuous vector field u(r, t_n) (e.g., the components of u might consist of the components of the wind vector, temperature, pressure, relative humidity, etc.). Here, r is a three-dimensional vector of position (the typical components of r are longitude, latitude, and altitude). For brevity, we introduce the notation u_n for u(r, t_n). With this notation, the evolution of the atmospheric state between times t_n−1 and t_n is

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where F is the evolution operator. A forecast model f mimics F in a finite-dimensional model space to predict the future atmospheric state. We assume that a suitable projection P from the infinite-dimensional atmospheric states to the finite-dimensional model states exists. We denote the finite-dimensional representation of the atmospheric state u_n by P(u_n). Since, in practice, the atmospheric state u_n−1 is not known, a forecast x_n of u_n is obtained by integrating the model from an estimate x_n−1 of u_n−1:

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The goal of data assimilation is to provide an estimate (analysis) x_n^a of P(u_n) using the model f and the observations collected up to time t_n. In what follows, we treat the model as a “black box,” that is, we consider the case where we have to infer information about the error in f from the knowledge of x_n−1, x_n and the observations. In a Kalman filter data assimilation scheme, the state estimate is updated sequentially by

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The background x_n^b represents our best estimate of P(u_n) at time t_n prior to the assimilation of the observations y_n^o and 𝗞 is the Kalman gain matrix. The observation operator h_n maps the model representation of the state to observables at analysis time t_n and is assumed to satisfy

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where the vector of Gaussian random variables ε_n, with mean 0 and covariance matrix 𝗥_n, represents the observation noise. In practice, h_n interpolates the model gridpoint variables to the location and time of the observations and converts the model variables to the observed quantities. In other words, when formulating h_n, the goal is to assume that h_n[P(u_n)] would differ from the observed quantity at the observation locations only by the noise ε_n. The use of the interpolation operator h_n in (5) is based on the implicit assumption that x_n is an accurate estimate of P(u_n). That is, the formulation of the Kalman filter we described so far is based on the assumption that the contribution of model error to the error in the background is zero. In practice, however, the atmospheric evolution operator and the model evolution operator differ from each other. This difference would lead to a systematic departure,

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of the forecast from the atmospheric state, even if the model was integrated from the P(u_n−1) finite-dimensional image of the true state (which is unknown in practice).

We obtain the equation that motivates our error models by rearranging Eq. (6),

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Our goal is to modify Eqs. (3)–(5), which assume that b_n is the null vector, so as to account for a nonzero b_n in the state estimation process.

b. Bias model I

The derivation of bias model I is based on the assumption that the model is subject to the same model error, regardless of whether it is started from the unknown P(u_n−1) or from its best available estimate . Under this assumption, Eq. (7) implies that in the presence of model error b_n, the best available estimate x_n^b of P(u_n) before the assimilation of the observations y_n^o at t_n is

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Thus the analysis can be obtained by replacing the forecast equation of the Kalman filter [Eq. (3)] by Eq. (8).

Equation (8) offers a practical way to account for the model error, only if one can find an efficient approach to estimate b_n. Our preferred approach is the method of state augmentation, which is based on the assumption that the evolution of the model error can be accurately modeled by a deterministic evolution equation:

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In this case, an estimate b_n^a of b_n can be obtained by applying the Kalman filter equations [Eqs. (3)–(4)] to the augmented state vector z = [x, b]^T. The simplest case arises when we can assume that b_n is a time-independent systematic error (bias), since in that case the evolution equation for the model error is

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c. Bias model II

Bias model II is designed to cope with the situation where the model error is due to differences between the attractor of the atmosphere and the attractor of the model dynamics. It is envisioned that P(u_n) lies on the projection into model space of the attractor of the true atmospheric dynamics, and that this projection differs from the attractor of the model. Furthermore, as a working hypothesis, it is further envisioned that there is a one-to-one transformation between orbits on the model space projection of the true attractor to orbits on the attractor of the model. In practice, for any point P(u_n) on the image of the true attractor there is a corresponding point P̂u_n) on the attractor of the model, and these evolve in time in accord with the presumed one-to-one transformation. In this error model scenario, our goal is to obtain an estimate of the transformation between P(u_n) and P̂u_n). This goal is motivated by the concern that starting the model from a state not on the model attractor (as one would seek to do if bias model I were applicable), no matter how close that state is to the true state, would trigger an unpredictable, and very likely negative, transient effect on the model forecast as the model orbit relaxes to its attractor. Thus, we replace the equation that defines the model error [Eq. (7)] with

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That is, we assume that the transformation between the true attractor image and the model attractor is a translation by the amount b_n.

Assume that we have a good estimate of P̂u_n−1). Then, is an estimate of P̂u_n) on the model attractor, while the model error b_n is a shift from the model attractor to the attractor of the atmosphere. Since our goal is to keep the state estimate on (or near) the model attractor, our choice for the forecast equation of the Kalman filter is the original Eq. (3) instead of Eq. (8) of bias model I.

One important consequence of choosing Eq. (3) as the forecast equation is that x_n^b is no longer our best estimate of P(u_n) before the assimilation of the observation. Our choice of defining h_n by an interpolation scheme, however, is based on the assumption that h_n operates on an estimate of P(u_n), which is x_n^b − b_n according to Eq. (7). Thus, in order to maintain the equality that defines h_n [Eq. (5)], we implement the observation operator as

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where ĥ_n is the function associated with the interpolation scheme. (With this notation, the observation operator used in the original Kalman filter and in bias model I is defined by h_n = ĥ_n.) In summary, although the background is not corrected for the estimated bias in bias model II, the observation is compared to the bias corrected model state in the analysis update equation. Similar to bias model I, an estimate of the model bias vector b_n can be obtained by the method of state augmentation. We note that the best estimate x_n^a − b_n^a of the true state P(u_n) is never computed in the data assimilation process when bias model II is used, but it can be easily obtained if needed (e.g., for reporting forecasts), as the data assimilation computes both x_n^a and b_n^a.

Estimation of the model error significantly increases the computational cost of the data assimilation algorithm. Even in the simplest case, when we consider only the estimation of a location dependent model bias, as we do in this paper, the dimension of the augmented state vector z is twice as large as the dimension of the state vector x. For bias model II, however, the dimension of the state vector can be reduced by estimating the error term in observation space instead of model grid space: since in a typical data assimilation problem the number of observations is at least an order of magnitude smaller than the number of model variables,¹ this strategy can reduce the number of components of the model error vector by at least an order of magnitude. Formally, this strategy involves replacing the definition of h by Eq. (12) with

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When the interpolation operator ĥ_n is linear, c_n = ĥ_n(b_n) and we should obtain the same analyses using either Eq. (12) or (13). The most important limitation of Eq. (13) is that it can be used only for a fixed observing network, otherwise the components of c_n cannot be defined consistently at the different analysis times. Another limitation is that because the correction is done in observation space, the information provided by x_n^a and the analysis of the bias term c_n^a is insufficient to obtain an estimate of the true state P(u_n). On the other hand, when ĥ_n is nonlinear, using Eq. (13) has the added advantage that, as we later illustrate for the surface pressure, it allows for accounting for the bias in selected components of the state vector instead of the bias in the full state vector. Estimating the bias only in selected components further reduces the number of bias parameters leading to an even bigger reduction of the computational cost.

d. Surface pressure bias

The surface pressure component of the background is typically subject to large bias, primarily due to the finite-resolution representation of the orography in the model. To be more precise, the surface pressure bias is expected to grow exponentially with the difference, Δz, between the model representation of the orography and the real orography:

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Here p_s is the atmospheric surface pressure, p_s^model is the surface pressure in the model, g = 9.8 m s⁻² is the gravitational acceleration, R = 287 J kg⁻¹ K⁻¹ is the gas constant for dry air, and T is the mean temperature in the |Δz| deep (hypothetical) atmospheric layer, that is, . Equation (14) is based on the assumption that the atmosphere in the |Δz| deep layer is in hydrostatic balance, which is well founded, since synoptic- and large-scale motions of the real atmosphere are hydrostatic to a good approximation. Hydrostatic balance is also built into the equations of all models that have resolutions lower than about 10 km.

If the bias from the surface pressure background was removed using bias model I, the surface pressure analysis would have a value consistent with the real orography. This would induce an adjustment process of the surface pressure, and of the other state variables through their dynamical relation to the surface pressure, to the model orography in the forecast phase of the next analysis cycle. In bias model II, in contrast, the surface pressure background, which reflects the model orography, is left unchanged. Thus, accounting for the surface pressure bias is an ideal test problem to investigate the efficiency of bias model II.

A careful treatment of the surface pressure is especially important in models that use σ = p/p_s^model as the vertical model coordinate. Since such models represent the atmospheric variables at constant levels of σ, the pressure p at a given model level is spatiotemporally varying, following the changes in the surface pressure. Since the vertical position of the observations is typically given by the pressure p at the location of the observations, in a σ-coordinate model, the application of the observation operator involves two steps: first, the pressure is computed at each σ level multiplying σ by the surface pressure, then, the model state is interpolated to the observational location in pressure units.

Since our goal is to account only for the surface pressure component of the background bias, we choose Eq. (13) to define the observation operator h for the surface pressure. Using Eq. (12) would change the pressure p at the sigma levels before the vertical interpolation to the observation location. This would have undesirable effects on variables other than the surface pressure, since the model levels are defined according to the biased model surface pressure. The alternative solution would be to apply Eq. (12) to the full-state vector, instead of the surface pressure component, because then the bias component of the other variables would adjust those variables to the bias-corrected surface pressure in the computation of the h. This approach would be computationally more expensive than the one we pursue as it would require the estimation of many more bias components.

To evaluate the efficiency of our surface pressure bias-correction strategy, we compare its performance to the more traditional approach of using Eq. (14) to make the correction. That is, Eq. (12) is replaced by

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for the surface pressure observations. This is approximately equivalent to choosing c^b according to Eq. (A2). This use of the hydrostatic balance equation is different from the approach we and others used in earlier implementations of ensemble-based data assimilation systems (e.g., Whitaker et al. 2004, 2008; Szunyogh et al. 2008). In those systems the adjustment was made to the surface pressure observations and the increased uncertainty introduced by this adjustment was taken into account by modifying the observational error for the adjusted observations.

e. Accounting for the surface pressure model bias in the LETKF

In this section we discuss how to incorporate the bias correction into the LETKF algorithm at time t_n, and so now we drop the subscript n. In the LETKF algorithm, the components of the analysis vector x^a are obtained model grid point by grid point. We introduce the notation x_ℓ for the local state vector, whose components are the L model variables at a given grid point. The LETKF obtains the local analysis x_ℓ^a by adding a linear combination of the background ensemble perturbations, x_ℓ^b(k) − x_ℓ^b for k = 1, … , K, to the background state x_ℓ^b, which is defined by the ensemble mean of the K-member background ensemble, x_ℓ^b(k) for k = 1, … , K. [Hereafter, we denote the ensemble mean of a vector over the K-member ensemble by dropping the superscript (k) that refers to the kth ensemble member.] That is,

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Here, 𝗫_ℓ^b is the L × K matrix of the background ensemble perturbations, whose kth column is x^b(k) − x^b. The analysis is obtained by computing the “weight vectors” w_ℓ^a based on the error statistics and the observations. The LETKF assimilates all observations that are thought to have useful information about the state at the grid point where the state is estimated. We use the notations M_ℓ for the number of observations selected for use at location ℓ, y_ℓ^o for the vector of selected observations, and 𝗥_ℓ for the associated M_ℓ × M_ℓ submatrix of 𝗥. We also introduce the following notation:

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We use the local components of y^b(k) to define the ensemble y_ℓ^b(k), k = 1, … , K, and the M_ℓ × K matrix 𝗬^b whose kth column is y^b(k) − y^b.² With this notations, the weight vector w_ℓ^a is computed by

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where

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The LETKF also computes the analysis error covariance matrix

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to obtain an estimate of the uncertainty in the analysis x_ℓ^a. Then, 𝗣_ℓ^a is used to compute the ensemble of “weight” vectors w^a(k), where k = 1, … , K, to obtain the analysis ensemble

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which represents the analysis uncertainty. Finally, we apply a multiplicative variance inflation factor ρ (ρ > 0) to the global analysis ensemble perturbations δx^a(k) = x^a(k) − x^a, for k = 1, … , K. The analysis ensemble that provides the initial conditions for the forecast step of the next analysis cycle is obtained by adding the inflated analysis perturbations

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to the mean analysis x^a. This variance inflation strategy is different from the one used in the original formulation of the LETKF by Hunt et al. (2007). In that paper, a multiplicative variance inflation was applied while computing P̃_ℓ^a, dividing the identity matrix 𝗜 by 1 + ρ (ρ > 0). We made this alteration to the algorithm, because this way can easily apply different variance inflation factor to the state vector components and the bias-correction components of the augmented state vector.

In this paper we use four different configurations of the LETKF:

1. We proceed with the computation as given by Eqs. (16)–(22). In these experiments we account for the model bias by tuning the variance inflation coefficient ρ.

2. We implement the hydrostatic-balance-based correction of the observation operator by using Eq. (15) to define the observation operator.

3. We incorporate bias model II and the method of state augmentation by replacing the state vector x with the augmented state vector z: at locations ℓ where no surface pressure observations are assimilated, the local augmented state vector z_ℓ is identical to x_ℓ, while at locations where surface pressure observations are assimilated, the L vector x_ℓ is replaced with the (L + M_ℓ^ps) vector z_ℓ. Here, M_ℓ^ps is the number of surface pressure observations assimilated at location ℓ. Also, at locations where no surface pressure observations are assimilated, the observation operator h in Eq. (17) is the interpolation operator ĥ, but at locations where surface pressure observations are assimilated, those components of h that map the model state vector to surface pressure are defined by Eq. (13). In this configuration, we apply different variance inflation coefficients to the surface pressure components of the state vector, to the other components of the state vector, and to the bias components.

4. We use an observation operator that combines the hydrostatic-balance-based correction and bias model II:

   

   

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a. Simulated observations

Observations are generated with 6-h frequency at 0000, 0600, 1200, and 1800 UTC. The horizontal locations of the observations are chosen to coincide with the horizontal locations of the grid points for the simulated atmosphere. This choice is made to avoid introducing interpolation errors into the knowledge of u_n. Observations are generated at only selected grid points by the following procedures: the locations of the surface pressure observations in a typical 0000 UTC operational NCEP data file are identified, then, observations are generated at grid points that are the closest to one of the observations. Only one observation per grid point is created when the grid point is the closest one for multiple observation locations. The density of the observations is the highest over the continents and the lowest over the oceans and in the two polar regions (Fig. 1). In addition to the surface pressure observations, observations of the horizontal wind vector and the virtual temperature are generated at seven pressure levels (at 925, 850, 700, 500, 300, 200, and 100 hPa). The observation noise has zero mean for all observed quantities and standard deviation of 1 hPa for the surface pressure, 1 K for the temperature, and 1 m s⁻¹ for the wind components.

b. Ensemble configuration

We obtain estimates of the atmospheric state (analysis) every 6 h; hence, f(x_n−1) is a 6-h forecast. We account for the model bias only in the 6-h surface pressure forecast, but we also hope to see a reduction of the bias in the 6-h forecast of the other variables due to the coupling between the surface pressure and the other variables through the dynamics. The initial ensemble of atmospheric states x₀^b(k), for k = 1, … , 60, at time t₀ is obtained by sampling a time series of states from a free run with the SPEEDY model. This generation of the initial forecast ensemble ensures that the ensemble members are on the model attractor.

In the experiments where we use bias model II to account for the model bias, the initial ensemble of the bias estimates is a realization of Gaussian random noise with zero mean and 1-hPa standard deviation. Then the augmented initial ensemble z₀^b(k) is formed by augmenting the initial atmospheric states by the initial bias estimates. We start our data assimilation cycle at time t₀ = 0000 UTC 1 January 2004 by assimilating the observations y₀^o into the initial ensemble z₀^b(k) to generate the analysis ensemble z₀^a(k).

c. Localization approach

In the LETKF, we have the freedom to choose any subset of the observations for the estimation of a given state variable. In a practical implementation, it is useful to define a local volume for each model grid point P and analyze all (local) state vector components at P simultaneously using all observations from the local volume. In our earlier studies (e.g., Szunyogh et al. 2005, 2008) we found that the computationally most efficient approach was to choose the volume of the local region to be as small as possible without degrading the accuracy of the analyses.³ We found by numerical experimentations (results not shown), that the optimal definition of the local region is 3 × 3 × 1 grid points when no adaptive bias correction is applied and 5 × 5 × 1 grid points when the state vector is augmented with the surface pressure bias-correction components.

d. Interpolation operator

The interpolation operator ĥ interpolates the model state to the observation locations in both the horizontal and vertical directions. The horizontal interpolation is implemented with a simple bilinear interpolation considering the model state vector components from the four grid points nearest to the given observational location. The vertical interpolation is somewhat more complicated because of the use of sigma as the model’s vertical coordinate: for each ensemble member, we first calculate the pressure at the sigma levels, multiplying sigma by the background surface pressure of the given ensemble member; the seven sigma levels define seven sigma layers, where the lowest layer is defined as the region between the surface (where sigma is 1) and the lowest sigma level; we find the sigma layer that contains the observation; finally we linearly interpolate using the logarithm of the pressure values at the bottom and top of the sigma layer. Since the surface pressure is part of the state vector, ĥ is nonlinear.

For the surface pressure observations, ĥ represents the horizontal interpolation of the model surface pressure to the observation locations. To obtain h for the surface pressure observation, we add the appropriate bias-correction term to the result of ĥ.

e. Verification method

To assess the effect of surface pressure bias correction on the analysis, we compute error statistics for the 6-h forecasts that provide the background. There are two reasons to verify the background instead of the analysis. First, since our goal is to account for the bias in the background, verifying the surface pressure provides a direct measure of our success. Second, the accuracy of the 6-h forecast is a better indicator of the quality of the analysis than the accuracy of the analysis itself, as it shows improvements only if the analysis is improved in a way that is consistent with the model dynamics. For instance, improving the analysis by increasing the projection of the initial state on free gravity waves would typically lead to a degradation or no change in the quality of the 6-h forecast.

We define all error statistics in observation space, that is, we compare the interpolated 6-h forecasts to the known state of the simulated atmosphere u_n at the observation locations. Thus, the 6-h forecast error e_n^b is defined by

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for the wind (u, υ) and the temperature (T) components of the state vector. For the surface pressure component (p_s) of the state vector, the forecast error is defined by replacing h(x_n^b) with the proper corrected observation operator [h(x_n^b, c_n), h(x_n^b, Δz), or h(x_n^b, Δz, c_n)] for the given configuration of the LETKF. Then, at a given time t_n and vertical level l we obtain the root-mean-square (rms) of the forecast error by computing the root-mean-square of those components of e_n^b that represent the given atmospheric variable at level l. For example, the rms of the temperature forecast error is computed by

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where the superscript T refers to the matrix transpose, M is the number of temperature observations at vertical level l, and e_n^b(T, l) is composed of those components of e_n^b that measure the temperature error at the observation locations at level l and time t_n. The time mean of the root-mean-square error is computed by

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where N₀ is the index of the analysis time t_N0 at which we start collecting the error statistics, N_f is the index of the analysis time t_Nf at which we stop collecting the error statistics, and N = N_f − N₀ + 1. In the computation of the error statistics, we do not consider the results from the first 15 days (N₀ = 61) to prevent transient effects from influencing the verification results. Since we run the experiments for 3 months, N_f = 364. Likewise, the bias is computed by

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where e_n^b (T, l, m), m = 1, … , M are the components of e_n^b(T, l). To measure the amplitude of the bias for vertical level ℓ we obtain the rms value of the bias, which is computed by

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a. The case of no bias correction

The rms error and the bias for the temperature [rms (T, l) and bias (T, l)] are shown in Figs. 2 and 3, while the same statistics for the zonal component of the wind vector [rms (u, l) and bias (u, l)] are shown in Figs. 4 and 5. All four error statistics take their minimum value where the covariance inflation factor ρ is about 0.25–0.3. This is also the range of the covariance inflation where the surface pressure bias has its minimum (Fig. 6).

In what follows, we use the results obtained at ρ = 0.25 as the baseline. In those configurations of the LETKF where we account for the surface pressure bias with bias model II the covariance inflation factor is 0.08 for the bias-correction term and for the surface pressure component of the state vector, and 0.25 for the other variables. These values were determined to be optimal by numerical experiments similar to those we presented here for the no bias-correction case.

As can be expected based on Eq. (A3), the spatial distribution of the surface pressure bias (Fig. 7) strongly resembles the spatial distribution of the orography difference between the model and the true orography (Fig. 8), with a strong anticorrelation between the two fields. (The correlation between the two fields is −0.76.)

b. The effect of bias correction on the surface pressure forecasts

The effect of the different bias-correction strategies on the surface pressure root-mean-square error is shown in Fig. 9. While all bias-correction strategies lead to large reduction of the surface pressure root-mean-square error, the strategy that is consistently the best is the one that combines bias model II with the hydrostatic-balance-based correction. In addition, the hydrostatic-balance-based correction is more efficient, on global average, in reducing the surface pressure error than bias model II alone.

The spatial distribution of the surface pressure bias for the different bias-correction strategies suggests a simple explanation for these results. Bias model II is efficient in correcting the bias at locations where the observation density is high (e.g., over North America, Eurasia, and Australia in Fig. 10), while the hydrostatic-balance-based correction can correct a reasonably large portion of the bias at all locations where the difference between the model orography and the true orography is large (Fig. 11). Combining the two bias-correction strategies, often leads to improvements which are larger than the sum of the improvements from the two approaches. Our example for this is the region of the Himalayas (Fig. 12). All three bias-correction strategies reduce the anticorrelation between the surface pressure bias and the orography difference: for the adaptive correction the correlation is −0.56, for the hydrostatic-balance-based correction the correlation is −0.31, and for the combined correction strategy the correlation is −0.20. This result suggests that all three strategies reduce that part of the surface pressure bias that is directly related to the reduced resolution representation of the orography in the model, and the one that does this the most efficiently is the strategy that combines bias model II and the hydrostatic-balance-based correction.

c. The effect of bias correction on the temperature and wind forecasts

The correction of the observation operator for the surface pressure model bias leads to a large improvement, not only in the surface pressure forecast, but also in the temperature and wind forecasts at higher altitudes. In particular, both the hydrostatic-balance-based correction and the adaptive correction lead to a reduction of the wind forecast bias (Fig. 13), and the root-mean-square error in the wind forecast (Fig. 14). Interestingly, the adaptive bias correction leads to much larger improvements than the hydrostatic-balance-based correction, and when the two approaches are combined the improvement is similar to that achieved by the adaptive correction alone. The situation is somewhat different for the temperature. This variable behaves similarly to the wind at and below the 300-hPa levels, but at the higher model levels the adaptive bias correction increases the bias (Fig. 15), and consequently, the root-mean square error (not shown). This behavior of the global bias is due to problems in the tropics (Fig. 16), while the temperature in the extratropics (not shown) behaves similarly to the wind.

A more careful examination of the cause of the increased temperature bias in the tropics reveals that the unusual behavior of the bias correction in the tropics is closely related to atmospheric tides. Atmospheric tides are temporal oscillations in one or more atmospheric variables whose periods are integer fractions of a solar or a lunar day. Atmospheric tides related to the solar day are primarily induced by the absorption of solar radiation by ozone in the stratosphere (Chapman and Lindzen 1970). The NCEP GFS, which has ozone as a prognostic variable, maintains both a diurnal and a semidiurnal tidal wave in the surface pressure in the tropics, but SPEEDY, which uses an empircal, seasonally varying function to define the absorption of solar radiation by the ozone in the stratosphere cannot simulate the tides (Fig. 17). This difference between the dynamics of the two models introduces a bias into the surface pressure background in the tropics.

The diurnal oscillation is sufficiently slow to be captured, with a reduced amplitude, by our bias model. Since bias model II forces the state estimate closer to the model attractor, the assimilation using this bias model dampens the diurnal tide in the analysis more than the assimilation that does not use adaptive bias correction (Fig. 18). This allows the background to stay closer to the model attractor in the entire atmospheric model column, which results in a smaller correction of the upper-level temperature in the direction of the observation.

This effect is illustrated by Fig. 19, which we obtain by computing the difference between the mean temperature in a free run with the SPEEDY model and a free run with the NCEP GFS for the 3-month period we investigate in this study. (To be precise, the free run with the NCEP GFS is the run that simulates the true evolution of the atmosphere, while the free run with the SPEEDY is started from the analysis at 0000 UTC 1 January 2004, which provided the initial condition for the NCEP GFS model run.) In essence, this field, which is shown by contours in Fig. 19, is a representation of the difference between the attractors of the two models. We also show in the same figure by color shades the 6-h forecast bias for the experiment that uses bias model II to correct for the surface pressure model bias. We find a close correspondence between the shapes of the two fields. Most importantly, in the upper troposphere in the tropics, both fields indicate a strong positive temperature bias. This supports our statement that bias model II leads to a larger temperature error in that region, because it allows the analysis, and the ensuing model forecast to shift in the direction of the model attractor.