Current data assimilation systems, including the Weather Research and Forecasting Data Assimilation system (WRFDA; Barker et al., 2012), do not consider wind direction errors when assimilating wind observations. Despite being collected in the form of speed (sp) and direction (dir), wind observations are transformed to their longitudinal and latitudinal components, u and v, prior to assimilation. The observation errors of u and v, which may be estimated from the wind speed error, are used in the assimilation. For example, the default observation error table used in WRFDA assumes a constant 3.6 m s^{−1} for u and v at all levels for aircraft data, following the operational practice at National Centers for Environmental Prediction (NCEP) (Kalnay et al., 1996). Although the dir errors can impact the uncertainty of u and v, they cannot be considered during the assimilation process, and therefore have no independent influence on the assimilation results.
In addition, the conventional method assumes the u and v observation errors to be uncorrelated to simplify the observation error covariance matrix to a diagonal matrix. Since u and v components are not observed in the real world, but are derived from observations of sp and dir, not only are u and v observation errors correlated, but they are both directly impacted by the dir observation errors. For observing systems directly observing sp and dir (e.g. surface stations), it is safe to assume the sp observation errors are not correlated with dir observation errors. For example, most operational 10-m synoptic observation stations use a rotating cup anemometer to measure sp, while dir is observed independently by a vane, thus isolating independent and uncorrelated characteristics of the respective observing systems. Newer sonic anemometers observe sp, and then calculate the dir angle from the vector difference before reporting time-averaged values for sp and dir. In this case, the errors are correlated; however, the dir error, despite being a function of sp, behaves differently (e.g. van der Molen et al., 2004), so treating sp and dir independently would still be beneficial when considering the error statistics during data assimilation.
Aircraft-based observing systems derive sp and dir observations from both air speed and heading information. The observed wind vector is the difference between the ground track vector (i.e. aircraft motion with respect to Earth) and the aircraft track vector. The ground track vector is typically determined by a very accurate satellite-based global positioning system (GPS). The aircraft track vector is calculated from the true air speed (TAS) and the heading angle, where the TAS is the difference between the dynamic and static pressure. The heading angle is typically determined by a laser gyro heading system. The pitot tube system that determines TAS, and the heading system that determines the track angle, are considered two unrelated observing systems. Both systems provide information to calculate the wind observation, which is then broken down into u and v vector components. Once the observed sp and dir from the aircraft are converted into u and v, both the original sp error and the original dir error are imbedded in the u and v components, and are no longer recoverable.
This derivation, which uses the vector difference between ground speed and air speed, allows for some dir error, which originates from the heading instrumentation, to impact the sp magnitude. Therefore, based on the specific type of observation, it is theoretically possible for sp and dir errors to be correlated, but the degree of correlation is much less than the direct influence sp and dir errors have on the u and v error when u and v are converted from sp and dir.
A new method of directly assimilating sp and dir observations is proposed in this article and is referred to as asm_sd. It takes into account both sp and dir observation error. It ensures that the analysed sp and dir will be consistent with the background sp and dir, as well as the observed sp and dir. The new formulation and results are compared with the conventional method, which is referred to as asm_uv in this article. In section 2, both asm_uv and asm_sd are introduced using the WRFDA framework. Section 3 presents the differences between the asm_uv and asm_sd analyses using a highly simplified approach and WRFDA single observation pair experiments. Idealised WRFDA experiments are presented in section 4. Conclusions and plans for future work are discussed in section 5.
WRFDA employs the incremental formulation of Courtier et al. (1994). The analysis increment minimises a cost function,
where the superscripts −1 and T denote the inverse and adjoint, respectively, of a matrix or linear operator; δx=x^{a}−x^{b} is the analysis increment, where x^{a} is the analysis and x^{b} is the background; B is the background error covariance; d=y−H(x^{b}) is the innovation vector; y is the observation vector; H is the non-linear observation operator; H is the linearised H; and R is the observation error covariance. To minimise eq. (1), WRFDA computes the gradient of J,
where H^{T}, the adjoint of H, is needed. A more detailed description of WRFDA can be found in Barker et al. (2004) and Huang et al. (2009). Here, we only discuss the items related to wind observations.
The model state vector, x, contains, among other variables, u and v,
and the observation vector, y, contains the wind observations. In the conventional data assimilation system (asm_uv), wind observations, despite largely being collected in the format of dir^{o} and sp^{o}, are transformed to u^{o} and v^{o} before being assimilated; therefore, y can be written as
In the new formulation (asm_sd), wind observations are assimilated in their original form of dir^{o} and sp^{o}, and y is written as
Observation operators H, H and H^{T} provide mapping between the model state space and observation space, and can be expressed as a series of operators
where subscript t denotes variable transforms and subscript i denotes interpolation. The difference between the asm_uv formulation and the asm_sd formulation is in the transform-related operators. For asm_uv, H_{t}, H_{t} and are all identity operators:
For asm_sd, the non-linear observation operator, H_{t}, which computes sp and dir from u and v, is expressed as
where n is an integer, which is chosen to confine the dir part of the innovation vector d to within [–180°,+180°]. The tangent linear operator, H_{t}, which computes perturbations in sp and dir from perturbations in u and v, is expressed as
The adjoint operator H_{t}^{T}, which computes the gradients in u and v from the gradients in sp and dir, is expressed as
The errors of most observation types are assumed to be uncorrelated; thus, the observation error covariance, R, becomes a diagonal matrix with observational error variance on the diagonal. In the asm_uv formulation,
where and are the observational error variances in u and v, respectively. Despite observations being collected in sp and dir, it is common practice to ignore the dir error, and apply a single sp error to both u and v. In the asm_sd formulation, the observation error covariance, R, is expressed as
where and are observation error variances in sp and dir, respectively. In contrast to the approximations and assumptions made for and , estimations of and may be obtained directly (Gao et al., 2012).
To illustrate the difference between the two assimilation methods, we start with a scalar analysis by reducing the dimension of x and y to 1, and assuming the observation and background are collocated. The solution of eq. (1) is easily obtained by
where all the variables are scalar and otherwise defined the same as their vector equivalent.
From eq. (12), we see that the analysis is a weighted average of the background and the observation. The weights are determined by the ratio between the background error variance (B) and the observational error variance (R). In a special case, when R/B =1, the analysis is simply midway between the background and the observation (i.e. x^{a}=(x^{b}+y)/2).
To further simplify the illustration, we assume u, v, dir and sp can be analysed independently. We will return to this assumption at the end of this subsection. We use eq. (12) to derive analyses for all of them separately. For asm_uv,
and for asm_sd,
where superscripts a, b and o denote analysis, background and observation, and subscripts uv and sd denote solutions by asm_uv and asm_sd, respectively.
The analyses are shown in Fig. 1. Starting from the same background (BG) wind vector and observation (OBS) wind vector, we obtain two very different analysis wind vectors using asm_uv and asm_sd. To derive the wind analysis using asm_uv, eq. (13) is used to calculate u^{a} and v^{a}, and the final result is the wind vector ANA_{uv}. Likewise, to produce the wind analysis using asm_sd, eq. (14) is used to calculate sp^{a} and dir^{a}, and the final result is the wind vector ANA_{sd}.
It is evident that asm_uv and asm_sd could lead to very different analyses. The analysis vectors differ both in directions and in magnitude. The asm_uv produces (u^{a}, v^{a}), which is consistent with (u^{b}, v^{b}) and (u^{o}, v^{o}), while the asm_sd produces (sp^{a}, dir^{a}), which is consistent with (sp^{b}, dir^{b}) and (sp^{o}, dir^{o}). In the Appendix, it is shown that under certain conditions, the analysis error covariance of asm_sd equals that of asm_uv; however, as the wind observations are made with sp and dir, we believe the asm_sd analysis is more reasonable.
Under the assumptions stated above, it is easy to show that the analysis wind vector obtained by asm_sd is always longer than the analysis wind vector obtained by asm_uv when the background wind vector and the observation wind vector do not point in the same direction. The analysis wind speed, as a function of u^{b}, v^{b}, u^{o}and v^{o}, for both asm_uv and asm_sd, can be written as
An example of is shown in Fig. 2. The background wind vector is chosen to be (u^{b}, v^{b})=(25.0, 5.0), which can also be expressed as (sp^{b}, dir^{b})=(25.5, 258.7). The observation wind vectors are varied across the range (−100:100, −100:100), which were produced by 80,802 [2×(201×201)] experimental runs. The units used for u, v and sp are m s^{−1} and for dir is ° (degree). The ratio is less than 1 with the exception of the case when the observation wind vector is exactly along the background wind vector, and in this case, the ratio equals 1. Thus, asm_uv produces a weaker wind analyses than asm_sd. In the extreme case of the observed wind vector being exactly opposed to the background wind vector, (u^{o}, v^{o})=(−25.0, −5.0), the asm_uv wind analysis vector has 0 magnitude (length), while the asm_sd wind analysis vector has the same magnitude (length; 25.5) as the background wind vector or the observed wind vector.
In this particular subsection of the study, an assumption is made that the background error and observational error are equal. In most operational data assimilation systems, the background errors are not typically expressed in terms of u and v. In WRFDA, they are estimated from model data in the form of a stream function and velocity potential. On the other hand, observational errors are expressed in terms of u and v (asm_uv). With the new option (asm_sd), the observational errors are given in terms of sp and dir. In both situations, it is not straightforward to make the background errors equal to observation errors. Additional experiments, discussed in the next subsection, do not make this same assumption.
The comparisons presented above are based on a univariate analysis performed independently on u, v, sp and dir. Operational data assimilation systems would not analyse u and v separately, so it is highly idealised to make this assumption with asm_uv. It is also questionable to assume the observational errors in u and v are uncorrelated, although this assumption has been made in all variational data assimilation systems. It is certainly unreasonable to make the uncorrelation assumption about the background error correlation. Similar arguments can be made for asm_sd, and the independent analysis of sp and dir. The motivation for this assumption was based on the simplicity of an idealised case. This assumption is not made in the following subsection and we will illustrate that the results from WRFDA single observation pair experiments are consistent with those we obtained above using the highly idealised approach.
In this subsection, we reproduce qualitatively similar results as the previous subsection by comparing asm_uv and asm_sd without making the same idealised assumptions as discussed above. Single observation pair experiments are performed. For each experiment, a single observation pair, (u^{o}, v^{o}) or (sp^{o}, dir^{o}), is introduced at a model grid point, which has background values (u^{b}, v^{b}) or (sp^{b}, dir^{b}). The experiment is very similar, but not identical, to the common practice of a single observation experiment in data assimilation.
The results from three experiments, with (u^{o}, v^{o})=(−15.0, 6.0) or (sp^{o}, dir^{o})=(16.2, 111.8) and (u^{b}, v^{b})=(25.0, 5.0) or (sp^{b}, dir^{b})=(25.5, 258.7), are shown in Fig. 3. The two wind vectors for background and observation are labelled BG and OBS, respectively. The WRFDA default background error, in terms of stream function and velocity potential, are employed for all experiments presented in this section. In the asm_uv experiment, we use 1.5 m s^{−1} for both u and v observational errors. The analysis wind vector ANA_{uv} indicates that the background error in u and v should be reasonably close to the observational error in u and v.
In the first asm_sd experiment, we use 1.5 m s^{−1} for the sp observation error and 10° for dir observation error; the latter closely follows the estimated observational errors in Gao et al. (2012). The analysis vector ANA_{sd} (solid line) indicates that the background sp error should be similar to the observational sp error, but the background dir error is much smaller than the observational dir error. In the second asm_sd experiment, a smaller observational dir error of 2.9° was employed. It is noted that there is an existing approach (Andersson et al., 2000) to show background errors in observation space, which can help to compare and tune the background and observations errors. The analysis vector ANA_{sd} (dot-dashed line) rotated left considerably with a dir 185.2°, which is halfway between the OBSdir of 111.8° and BGdir 258.7° indicating that the background dir error is close to the observational dir error. As discussed in the previous sections, the observational dir error should influence the analysis, and it is easily demonstrated in this test with asm_sd.
It is also evident from Fig. 3 that the same background and observation wind vectors produce a larger magnitude (longer) analysis wind vector when using asm_sd versus asm_uv. The same 80,802 [2×(201×201)] experiments from Fig. 2 were repeated using the WRFDA single observation pair configuration, and the results are presented in Fig. 4, which are strikingly similar to those in Fig. 2. The shift of the minimum point simply indicates that the observation errors are not equal to the background errors. Thus, we conclude that even without the assumptions made in the previous subsection, asm_sd produces a stronger wind analysis than asm_uv.
Through the assimilation of the single wind observation pair at 559 hPa (−15.0 m s^{−1}, 6.0 m s^{−1}) or (16.2 m s^{−1}, 111.8°), the OBS vector in Fig. 3, WRFDA produces a 3-D analysis increments of u and v, as well as other variables with its multivariate formulation, using either asm_uv or asm_sd. The wind analysis increments for asm_uv (left), and asm_sd (right), are shown in Fig. 5. Due to horizontal correlation of the background errors, the differences are quite large across the entire model domain, as well as at the observation point, which is consistent with Fig. 3. The increment maps in Fig. 5 from asm_uv (5a and 5c) resemble the classic textbook figures (e.g. Daley, 1991) for responses to a single u innovation. This is consistent with Fig. 3, as the v innovation (1.0 m s^{−1}) is much smaller than the u innovation (40.0 m s^{−1}). The increment maps in Fig. 5 from asm_sd (5b and 5d) can be considered a combination of responses to both the u and v innovations, which are comparable in size (cf. Figs. 3 and 5).
The wind increment vectors are also shown in Fig. 5e and 5f. At the observation location, the increment vector in Fig. 5e is consistent with the vector difference between ANA_{uv} and BG in Fig. 3. Likewise, the increment vector in Fig. 5f is consistent with the vector difference between ANA_{sd} (solid) and BG in Fig. 3. In summary, with an identical background field (i.e. not just a single model point), the same wind observation pair could produce very different analysis wind increments when using asm_uv or asm_sd.
In the previous section, the difference between asm_uv and asm_sd was demonstrated using both simple examples and WRFDA single observation pair experiments. The primary goal of data assimilation in the context of numerical weather prediction is to improve the analysis, which leads to a more accurate forecast. In this section, we compare our new formulation (asm_sd) with the conventional formulation (asm_uv) using an observing system simulation experiment (OSSE) framework.
OSSE studies are typically conducted to assess the potential impact of a proposed future observing system (Masutani et al., 2010). The standard OSSE framework consists of four components or steps:
In this study, we follow the OSSE protocol with the exception of the last step, where instead of assessing the simulated observation impact, we compare the analyses and forecasts to the reference atmosphere to evaluate the performance of the new formulation (asm_sd).
The weather research and forecasting (WRF) modelling system (Skamarock et al., 2008) is used over a domain covering North America and the surrounding oceans (Fig. 6). For the ‘nature run’, a domain with 6-km grid spacing and 57 vertical levels with a model top at 50 hPa are employed. Initial and lateral boundary conditions were obtained from the NCEP final analysis (FNL). The physics configuration is identical to that used by Huang et al. (2009) with the exception of the cumulus parameterisation, which in this case was Kain–Fritsch (Kain, 2004). The reference atmosphere is produced by a 5-day continuous WRF run initialised at 0000 UTC 15 December 2011.
To assess the realism of the nature run, the 500-hPa synoptic flow (e.g. height, temperature and wind vectors) at forecast hour 72 from the nature run is compared to the FNL analysis in Fig. 6, which is valid 0000 UTC 18 December 2011. This is the last cycle time for the outputs of synthetic observations, as well as the assimilation experiments. The nature run provides a solid basis for the following OSSE by successfully simulating the development and propagation of the main synoptic system, where the closed isohypse west of Baja, Mexico, and the deep trough over the eastern United States verified well against the FNL analysis.
Since we are examining assimilation methods for wind observations, and not observing systems, only wind observations are generated by adding the observational errors to the nature run reference atmosphere every 6 h over a uniform network with a horizontal resolution of 180 km (i.e. every 30th model grid point, Fig. 7), and across 12 standard pressure levels. From the reference atmosphere, we extract sp and dir at observation locations, and add observation errors to simulate observations, sp^{o} and dir^{o}, over the entire network. Once the simulated observations are obtained, they are converted to u^{o} and v^{o}. The observation errors for sp, dir, u and v (denoted as δsp^{o}, δdir^{o}, δu^{o} and δv^{o}) are then computed by comparing the simulated observations against the direct respective variables within the reference atmosphere.
The sp and dir observation errors are assumed to be uncorrelated and Gaussian with standard deviations estimated for aircraft data by Gao et al. (2012). Previous studies on aircraft and retrieved wind observations (e.g. Leotta and Long, 1989; Gao et al., 2012) have shown that sp error can be characterised by a Gaussian distribution, and dir errors can be characterised with a Gaussian-like distribution. It should be noted that the representativeness error from the observations of sp and dir are not specially considered in this research. In reality, and particularly for near-surface winds, the representativeness error may cause correlation between dir and sp, and may account for a major part of the total observation error. The error correlation between dir and sp, as well as the distribution of wind dir error, needs further investigation.
The observation error variances are shown in Fig. 8a. The sp and dir observation error variances, (δsp^{o})^{2} and (δdir^{o})^{2}, are close to those used for observation error generation, where (δsp^{o})^{2} varies only slightly around the mean value (2.5)^{2} m^{2}s^{−2} and (δdir^{o})^{2} changes with height with a maximum at the surface and a minimum at 250 hPa. The u and v observation error variances, (δu^{o})^{2} and (δv^{o})^{2}, change with height, but the variations are difficult to directly relate to (δsp^{o})^{2} and (δdir^{o})^{2}. To verify the error estimations, (δu^{o})^{2} and (δv^{o})^{2} are also computed from (δsp^{o})^{2} and (δdir^{o})^{2} using the following equations, which can be derived from eq. (A5) in the appendix:
The estimated (δu^{o})^{2} and (δv^{o})^{2} using eq. (16) and eq. (17) are shown in Fig. 8b (labelled as u_linear and v_linear, respectively), and are very close, although not identical, to those shown in Fig. 8a. As shown in eq. (16) and eq. (17), (δu^{o})^{2} and (δv^{o})^{2} are combinations of (δsp^{o})^{2} and (δdir^{o})^{2}, and are weighted by the background u^{2} and v^{2}.
Without the contribution from (δdir^{o})^{2}, both (δu^{o})^{2} and (δv^{o})^{2} would be smaller than (δsp^{o})^{2}, and since the upper air mean flow of this experiment configuration is more zonal, (δu^{o})^{2} would be larger than (δv^{o})^{2} at the higher levels. In Fig. 8b, (δu^{o})^{2}, the first term of eq. (16), is labelled u_sp, and (δv^{o})^{2}, the first term of eq. (17) is labelled v_sp. The last term in both eq. (16) and eq. (17) appears as u_dir and v_dir, respectively, in Fig. 8b. The contributions from (δdir^{o})^{2} to (δu^{o})^{2} and (δv^{o})^{2} are larger than those from (δsp^{o})^{2} in this experimental configuration. These contributions are proportional to (δdir^{o})^{2}, as well as u^{2} and v^{2}.
To ensure that the observations extracted from the nature run would be sufficiently different to the background, a background field with a resolution of 18 km is taken from the 24-h forecast of a no-physics run, which is initialised at 0000 UTC 15 December 2011 on the same domain as the nature run. By following this protocol, we can maximise the difference between the nature run and the background, which will enable a more accurate quantification of the forecast skill improvement between the two assimilation methods.
Beginning with the 24-h forecast, valid 0000 UTC 16 December 2011, WRFDA is run in cycling mode for nine cycles (i.e. 2 d) for the parallel data assimilation experiments. The 6-h forecast from each cycle serves as the background for the analysis of the subsequent cycle. The National Meteorological Center (NMC) method (Parrish and Derber, 1992) is employed to generate the background error covariances using past monthly statistics of differences between WRF 24-h and 12-h daily forecasts. No additional quality control procedures are applied to the observations for either experiment. Finally, a 48-h forecast is produced from each cycle of the parallel runs using identical forecast model configurations. Lateral boundary conditions for the forecasts were obtained from NCEP GFS global forecasts.
For verification, we consider the native 6-km grid of the nature run to be ‘truth’, and the parallel forecasts are compared against this grid at each point. Fig. 9 presents the average RMSE of all analyses and 48-h forecasts over the nine data assimilation cycles. The 0-h RMSE is the verification of the analyses, while the 6-h RMSE is the verification of the background.
The asm_sd method produced significantly better verification scores for the sp (Fig. 9a) and dir (Fig. 9b) analyses and forecasts compared to the asm_uv method. The improvements of sp and dir were 16% and 12%, respectively, at the analysis, and 9% and 4%, respectively, for the mean forecast period. Notable improvements from asm_sd over asm_uv were also found for u forecasts, which are shown in Fig. 9c; however, the dominant zonal flow hinders the ability to highlight differences between asm_sd and asm_uv when comparing v forecasts.
In this study, we present a new method (asm_sd) for assimilating wind observations in their observed form of speed and direction. Comparisons between asm_sd and the conventional method of assimilating u and v (i.e. asm_uv) are made using simplified arguments and graphic illustrations, followed by single wind observation pair experiments using WRFDA. Finally, idealised experiments using the OSSE framework were employed to compare asm_sd to asm_uv.
Using a highly simplified configuration (i.e. Figs. 1 and 2), we have shown how asm_sd and asm_uv can produce very different analyses from the same background and the same observation. Using the WRFDA single observation pair experiments, qualitatively similar results are also produced (Figs. 3 and 4). We have also quantified the difference between asm_sd and asm_uv in the 3-D analysis increment fields, while the same 3-D background field and the same single observation pair are used (Fig. 5).
The formulation of asm_sd ensures that the analysis vector falls between the background and observation in terms of sp and dir. In the asm_sd framework, the analysis can be influenced by the wind direction observation error, which likely plays an important role in most data assimilation systems.
The asm_sd and asm_uv methods are further compared in a clean and more realistic framework, using simulated wind observations. Because the ‘truth’ was defined by the nature run, observation errors (Fig. 8) were computed and used for the data assimilation experiments. The results, presented in Fig. 9, show that the asm_sd method produces a more accurate analysis, and, as a result, can also lead to a better forecast when compared to the asm_uv method. Additional tests are being performed on real observations, and the results will be presented in a subsequent article.
Because the scope of this article focuses on the differences between asm_sd and asm_uv in the variational formulation using idealised configurations, the quality control aspect of the data assimilation has not been directly addressed. However, for experiments using real observations, the asm_sd methodology has the ability to accept or reject observations based on sp and dir innovations and observation error.
It is well understood that large directional errors in wind observations are correlated to lighter wind speeds for both in-situ observations (e.g. Dobson et al., 1980; Schwartz and Benjamin, 1995; Gao et al., 2012), as well as remotely sensed wind observations (e.g. Plant, 2000; Ebuchi et al., 2002). The observational error for sp is positively correlated to sp, yet the observational error for dir is negatively correlated to sp. The new method may be further refined to account for these aspects, as it already deals with sp and dir directly. For the purposes of this OSSE framework, the sp and dir observation errors are assumed to be uncorrelated and with a Gaussian distribution. It is of interest to investigate the probability distribution of wind direction error and its relation to the speed error in future research.
The authors thank Chris Snyder, Fuqing Zhang, Xin Zhang, Xiaoyan Zhang, Dongmei Xu and Peter Childs for fruitful discussions. We are also very grateful for the invaluable comments and suggestions from three anonymous reviewers. The National Center for Atmospheric Research is sponsored by the National Science Foundation. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and are not necessarily the views of the National Science Foundation.
The assumption is made that wind speed and direction observations (sp^{o}, dir^{o}) and their errors (δsp^{o}, δdir^{o}) are given. The u- and v-wind observations (u^{o}, v^{o}), and their observation errors (δu^{o}, δv^{o}), are derived from (sp^{o}, dir^{o}) and (δsp^{o}, δdir^{o}). For simplicity, we consider one observation pair (sp^{o}, dir^{o}) taken at the analysis grid point, and we assume the analysis variables are u and v.
For asm_sd (i.e. direct assimilation of sp^{o} and dir^{o}), the analysis error covariance A in the (u,v) space can be expressed as
where B is the background error covariance in (u,v) space, H is the observation operator to convert u and v to sp and dir, respectively, as shown in eq. (8), H^{T} is the adjoint of H, and
which is the observation error covariance in (sp, dir) space. For asm_uv (i.e. assimilation of u^{o} and v^{o}), the observation operator is a unit matrix, and the analysis error covariance can be expressed as,
where the observation error covariances in (u,v) space are defined as
It is easily shown that the analysis error covariance in (A1) equals that of (A3) as
and
where H^{−1} is the inverse of H,
In other words, if the u and v errors are linearly calculated from sp and dir errors, respectively, the two methods proposed in this article should produce analyses with the same analysis error covariance, although the analyses can be very different. It should be stressed that in the above derivation, no assumptions are made on the cross error correlation terms. In the WRFDA implementation of asm_uv and asm_sd, the cross error correlation terms are assumed to be zero, and minor differences between the two WRFDA analysis error covariances do exist.
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