Abstract
 Top of page
 Abstract
 1. Introduction
 2. Spinup, nocost smoothing and ‘running in place’ in EnKF
 3. Results
 4. Discussion
 Acknowledgments
 References
Ensemble Kalman Filter (EnKF) may have a longer spinup time to reach its asymptotic level of accuracy than the corresponding spinup time in variational methods (3DVar or 4DVar). During the spinup EnKF has to fulfill two independent requirements, namely that the ensemble mean be close to the true state, and that the ensemble perturbations represent the ‘errors of the day’. As a result, there are cases, such as radar observations of a severe storm, or regional forecast of a hurricane, where EnKF may spinup too slowly to be useful. A heuristic scheme is proposed to accelerate the spinup of EnKF by applying a nocost Ensemble Kalman Smoother, and using the observations more than once in each assimilation window during spinup in order to maximize the initial extraction of information. The performance of this scheme is tested with the Local Ensemble Transform Kalman Filter (LETKF) implemented in a quasigeostrophic model, which requires a very long spinup time when initialized from random initial perturbations from a uniform distribution. Results show that with the new ‘running in place’ (RIP) scheme the LETKF spins up and converges to the optimal level of error faster than 3DVar or 4DVar, even in the absence of any prior information. Additional computations (2 to 12 iterations for each assimilation window) are only required during the initial spinup, since the scheme naturally returns to the original LETKF after spinup is achieved. RIP also accelerates spinup when the initial perturbations are drawn from a welltuned 3DVar backgrounderror covariance, rather than being uniform noise, and fewer iterations and RIP cycles are required than in the case without such prior information. Copyright © 2010 Royal Meteorological Society
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Spinup, nocost smoothing and ‘running in place’ in EnKF
 3. Results
 4. Discussion
 Acknowledgments
 References
The relative advantages and disadvantages of fourdimensional variational data assimilation (4DVar), already operational in several numerical forecasting centres, and Ensemble Kalman Filter (EnKF), a newer approach that does not require the adjoint of the model, are the focus of considerable current research (e.g. Lorenc, 2003; Kalnay et al, 2007a,b; Gustafsson, 2007; WWRP/THORPEX Workshop, 2008).
One area where 4DVar may have an advantage over EnKF is in the initial spinup, since there is evidence that 4DVar converges faster than EnKF to its asymptotic level of accuracy. For example, Caya et al(2005) compared 4DVar and EnKF for a storm simulating the development in a sounding corresponding to 0000 UTC on 25 May 1999. They found that
Overall, both assimilation schemes perform well and are able to recover the supercell with comparable accuracy, given radialvelocity and reflectivity observations where rain was present. 4DVar produces generally better analyses than EnKF given observations limited to a period of 10 min (or three volume scans), particularly for the wind components. In contrast, EnKF typically produces better analyses than 4DVar after several assimilation cycles, especially for model variables not functionally related to the observations.
In other words, for the severe storm problem, EnKF eventually yields better results than 4DVar, presumably because of the assumptions made in the 4DVar backgrounderror covariance, but during the crucial initial time of storm development, when radar data start to become available, EnKF provides a worse analysis. For a global shallowwater model, which is only mildly chaotic, Zupanski et al(2006) found that initial perturbations that had horizontally correlated errors converged faster and to a lower level of error than perturbations created with white noise. In agreement with these results, Liu (2007) found using the SPEEDY global primitiveequations model that perturbations obtained from differences between randomly chosen states (which are naturally balanced and have horizontal correlations of the order of the Rossby radius of deformation) spun up faster than white noise perturbations.
Yang et al(2009) compared 4DVar and the Local Ensemble Transform Kalman Filter (LETKF; Hunt et al, 2007) within a quasigeostrophic (QG) channel model (Rotunno and Bao, 1996). They found that, if the LETKF is initialized from randomly chosen fields with uniform distribution perturbations, it takes more than 100 days before it converges to the optimal level of error. If, on the other hand, the ensemble mean is initialized from an existing 3DVar analysis, which is already close to the true state, using the same random perturbations, the LETKF converges to its optimal level very quickly, within about 3 to 5 days. However, with a welltuned backgrounderror covariance, 3DVar and 4DVar converge fast without needing a good initial guess. This has also been observed for severe storm simulations (Caya et al, 2005), especially when using real radar observations (Jidong Gao, 2008, personal communication). It is not surprising that EnKF spins up more slowly than 3DVar or 4DVar because, in order to be optimal, the ensemble has to satisfy two independent requirements, namely that the mean be close to the true state of the system, and that the ensemble perturbations represent the characteristics of the ‘errors of the day’ in order to estimate the evolving backgrounderror covariance B. In both 3DVar and 4DVar, by contrast, B is tuned a priori and assumed to be constant.
Within a global operational system it is feasible to initialize the EnKF from a state close enough to the optimal analysis, such as an existent 3DVar analysis, with balanced perturbations drawn from a 3DVar error covariance, so that spinup may not be a serious problem. However, there are other situations, such as the storm development discussed above, where radar information is not available before the storm starts, so that no information is available to guide the EnKF in the spinup towards the optimal analysis. The system may start from an unperturbed state without precipitation, and if a severe storm develops within a few minutes and the EnKF takes considerable real time to spin up from the observations, it will not give reliable forecasts until later in the storm evolution, and thus give results that are less useful for severe storm forecasting than 4DVar or even 3DVar. Similarly, a regional model initialized from a global analysis at lower resolution may take too long to spin up when confronted with mesoscale observations.
In this note we propose a new method to accelerate the spinup of the EnKF by ‘running in place’ (RIP) during the spinup phase and using the observations more than once in order to extract maximum initial information. We find that it is possible to accelerate the convergence of the EnKF so that (in real time) it spins up even faster than 3DVar or 4DVar without losing accuracy after spinup and without requiring prior information such as a ‘well tuned’ initial backgrounderror covariance. After spinning up, RIP is automatically turned off and the system returns to the standard EnKF formulation. Section 2 contains a brief theoretical motivation and discussion of the method, results are presented in section 3 and a discussion is given in section 4.
2. Spinup, nocost smoothing and ‘running in place’ in EnKF
 Top of page
 Abstract
 1. Introduction
 2. Spinup, nocost smoothing and ‘running in place’ in EnKF
 3. Results
 4. Discussion
 Acknowledgments
 References
In this section we briefly review the conditions that justify the rule that in Kalman Filter data should be used once and then discarded. We then suggest that this rule is not strictly valid during spinup, when the initial covariance is still influencing the results, or when the statistics of the ‘errors of the day’ suddenly change due to strong nonlinearity, as during the initial development of a severe storm. During these transition periods, the ensemble perturbations are not representative of the ‘errors of the day’ and extracting information from observations using them only once is not efficient.
After the cost function in (1) is minimized, finding the analysis and its corresponding covariance , a similar relationship holds for the analysis at t_{n}, and another constant c′:
 (3)
Equating the terms in (3) that are linear and quadratic in x, the linear Kalman Filter equations for a perfect model are obtained.
EnKF, like the Kalman Filter, also provides a maximum likelihood analysis, except that the background and analysiserror covariances are estimated from an ensemble of K generally nonlinear forecasts:
 (4)
where is a perturbation matrix whose kth column is the background (forecast) perturbation and is the most likely forecast state, i.e. the ensemble average. Similar equations to Kalman Filter are valid for the analysis mean and the analysis error covariance . Thus, EnKF, like the original Kalman Filter, is a sequential data assimilation (DA) system where, after the new data are used at the analysis time, they should be discarded (Ide et al, 1997), but this is true only if the previous analysis and the new background are not only the most likely states given the past observations, but they also have already ‘forgotten’ the choice of initial ensemble and carry the proper perturbation structures corresponding to the true dynamic instabilities. In other words, if the system has converged after the initial spinup, all the information from past observations is already included in the background and the data can be discarded after the new analysis is computed. In contrast, 4DVar is a smoother that best fits all the observations (even asynoptic data) within an assimilation window. We note that EnKF can be extended to four dimensions as in 4DVar, allowing for the assimilation at the right time of asynoptic observations made between two analyses (e.g. Hunt et al, 2004, 2007), but, being a filter, the EnKF analysis is only obtained at the end of the assimilation window.
In summary, after the initial spinup, all the information from past observations is already included in the background field, so that the observations should be used only once and then discarded. However, there is no theoretical reason why this constraint should also be applied when EnKF is ‘coldstarted’, and the initial ensemble is not representative of the most likely state and its uncertainty, since during spinup the background term still ‘remembers’ the arbitrarily chosen initial ensemble. In practical applications, the rule of using the data only once is usually applied even during spinup (e.g. Zupanski et al, 2006), and depending on the initial ensemble, a slow EnKF spinup can then be observed.
In this note we suggest that when a quick EnKF spinup (in real time) is needed in order to make useful shortrange forecasts for fast weather instabilities, the initial observations can be used more than once in order to extract more initial information from them, and that this procedure can lead to a much faster spinup of the initial ensemble. This RIP algorithm is made possible by the use of a ‘nocost’ Ensemble Kalman Smoother (EnKS; Kalnay et al, 2007b; Yang et al, 2009).
Since a linear combination of ensemble trajectories within an assimilation window is also a model trajectory, a linear combination that is close to the truth at one time within the window should remain close to the truth over the entire window (at least as close as model errors allow). As illustrated in Figure 1, the dashed line indicates the model trajectory constructed by the weights derived at t_{n}: this trajectory ends at the analysis mean as derived from (5a) with the initial state derived from (6a). Since the weights contain the observation information within the assimilation window, the smoothed analysis (the cross in Figure 1) at t_{n−1}, is expected to be more accurate than the analysis mean (the end of the first dashed line in Figure 1) because it knows the ‘future’ observations. This argument (B. Hunt, 2009, personal communication) indicates that the weights used in constructing the analysis ensemble mean, although determined at the end of the assimilation window, should be valid throughout the window. A similar argument suggests that the ensemble analysis perturbation weights obtained using Bayes' theorem are also valid throughout the assimilation window [t_{n−1}, t_{n}] (Hunt et al, 2007). As other properties of EnKF, this one may be affected by localization.
The nocost EnKS is easy to implement if the weights that transform the ensemble forecasts into the ensemble analysis are explicitly computed and available, as is the case in the LETKF. The analysis ensemble members at time t_{n} are each a weighted average (linear combination) of the ensemble forecasts valid at t_{n} (Hunt et al, 2007). Since the ensemble analysis estimates the linear combination of the trajectories that best fits the observations within an assimilation window, not just at the end of the interval, the nocost EnKS valid at the beginning of the window is obtained by simply applying the same weights obtained at analysis time t_{n} to the initial ensemble at t_{n−1}.
The nocost EnKS was tested by Yang et al(2009) on the QG model of Rotunno and Bao (1996). Figure 2 compares the analysis error of the LETKF with that obtained using the nocost EnKS, and shows that, indeed, the nocost ensemble Kalman smoother at t_{n−1} is more accurate than the analysis ensemble valid at t_{n−1}, as could be expected from the fact that the smoothed ensemble at the beginning of the window has benefited from the information provided by the ‘future’ observations within the window [t_{n−1}, t_{n}]. Although the nocost smoothing improves the accuracy of the initial analysis at t_{n−1}, it does not improve the final analysis at t_{n}, since the forecasts started from the new initial analysis ensemble will end as the final analysis ensemble (at least in a linear sense; Figure 1 and also the Appendix of Yang et al, 2009).
With the nocost EnKS it is thus possible to go backwards in time within an assimilation window, and then advance with the regular EnKF using the initial observations repeatedly in order to extract maximum information from them. During spinup, when the prior (background field and backgrounderror covariance) are not representative of the true state and the ‘errors of the day’, this procedure improves the quality (likelihood) of the initial ensemble mean faster, and leads the ensemblebased backgrounderror covariance to be more representative of the true forecasterror statistics.
As indicated above, EnKF requires the choice of an initial prior ensemble at t_{0} with covariance . We have tested the RIP algorithm using three different initial ensembles, all with the same randomly chosen ensemble mean but with different distributions of the initial random perturbations: (1) a uniform distribution; (2) a Gaussian distribution and (3) perturbations drawn from a carefully optimized 3DVar error covariance. Cases (1) and (2) include no prior information and each state variable is independently perturbed; case (3) contains the same prior information used for the 3DVar experiments. The 4DVar experiments make use of the same 3DVar covariance with an optimal rescaling. In case (1), the random perturbations are uniformly distributed between −0.05 and 0.05 and in case (2), the Gaussian perturbations have a zero mean with a standard deviation of 0.05.
The RIP algorithm that we have tested (not necessarily optimal) is as follows: at t_{0} we integrate the initial ensemble to t_{1}. Then the RIP loop with n = 1, is:
(a) Perform a standard EnKF analysis and obtain the analysis weights at t_{n}, saving the mean square observations minus forecast,
computed by the EnKF.
(b) Apply the nocost smoother to obtain the smoothed analysis ensemble at t_{n−1} by using the same analysis weights obtained at t_{n}.
(c) Perturb the smoothed analysis ensemble with a small amount of random Gaussian perturbations, a method similar to additive inflation. These added perturbations have two purposes: they avoid the problem of otherwise reaching the same final analysis at t_{n} as in the previous iteration (Figure 1), and they allow the ensemble perturbations to evolve into fastgrowing directions that may not have been included in the unperturbed ensemble subspace‡.
(d) Integrate the perturbed smoothed ensemble to t_{n}. If the forecast fit to the observations is smaller than in the previous iteration according to a criterion such as
 (7)
then go to (a) and perform another iteration. If not, let t_{n−1}t_{n} and proceed to the next assimilation window. In the results presented here, we have used ε = 0.05 as the criterion for relative improvement.
(e) If no additional iteration beyond the first one is needed, the RIP analysis is the same as the standard EnKF. When the system converges, no additional iterations are needed, so that if several assimilation cycles take place without invoking a second iteration, the RIP can be switched off and the system returns to a normal EnKF. As expected, we observed that the results are slightly degraded if the RIP continues to be executed after convergence due to overfitting the observations. In the results presented here, we switched off RIP after five cycles without invoking a second iteration.
3. Results
 Top of page
 Abstract
 1. Introduction
 2. Spinup, nocost smoothing and ‘running in place’ in EnKF
 3. Results
 4. Discussion
 Acknowledgments
 References
The LETKF with the RIP method was implemented in the Rotunno and Bao (1996) QG model. The DA experiments are performed with a 12 h analysis cycle. The analysis is validated every 12 h against the truth simulation, a long nature run of this QG model. The validation is done through the RMS analysis error, defined as the domainaveraged RMS difference of the model variables (potential vorticity and temperature) between the analysis and truth. The ‘rawinsonde’ observations are vertical profiles of zonal and meridional wind components and temperature, generated by adding random Gaussian errors on the truth. Details of the QG DA setup can be found in Yang et al(2009). As indicated in section 2, step (c), we also added to the smoothed analysis ensemble Gaussian perturbations with size 0.01, small compared to the amplitude of the model natural variability and to the observation errors.
In this section we compare several DA methods started from the same randomly chosen mean. We measure the (realtime) spinup by the number of cycles required to reduce the RMS error in potential temperature, which starts from a nondimensional value of 0.76, to a level of 0.038, i.e. 5% of the initial analysis error (grey line in Figures 3(a), (b), (c)). The results, including both spinup time and asymptotic level of analysis error are also summarized in Table I
Table 1. Comparison of the spinup time (number of DA cycles to reduce the initial RMS error in potential temperature to 5% of the original value) and the asymptotic RMS error for LETKF ensembles with and without RIP, and fixing the number of RIP iterations to 10 rather than determining them adaptively.  LETKF (1) Random Uniform Initial Ensemble  LETKF (2) Random Gaussian Initial Ensemble  LETKF (3) B3DVar Initial Ensemble  LETKF Random Initial Ensemble  Variational 

 No RIP  With RIP  No RIP  With RIP  No RIP  With RIP  Fixed 10 RIP iterations  3DVar/ B3DVar  4DVar/0.05B 3DVar 


Spinup time  141  47  57  37  53  39  37  44  54 
RMS error (×10^{−2})  0.51  0.51  0.51  0.51  0.50  0.50  1.26  1.24  0.53 
Figures 3(a), (b) and (c) show the RMS error of the analysis obtained during the spinup, using several methods over 200 analysis cycles of 12 h each (corresponding to a total of 100 days). In Figure 3(a) we compare the number of cycles required for spinup for the LETKF with initial random perturbations uniformly distributed, with and without using RIP (black), with 3DVar and 4DVar (grey). As indicated before, all the experiments started from the same randomly chosen mean state. 3DVar (dashed grey line) takes about 60 cycles to spin up, and 4DVar (full grey line) takes about 80 cycles, but converges to a much lower RMS error than 3DVar. The standard LETKF (full black line) using the observations once and discarding them takes much longer, a total of 170 cycles. It is interesting that the LETKF devotes the first 120 cycles essentially to create ensemble perturbations representative of the ‘errors of the day’, with little reduction in the analysis mean error, and only then, between 120 and 170 cycles, does the LETKF converge rather quickly to the asymptotic level of error (the analysis accuracy that the LETKF with optimized parameters is able to reach). After spinup, the LETKF and 4DVar have a similar asymptotic RMS error but significant daytoday differences. The LETKF with RIP for Case 1 takes about 80 cycles to reach the asymptotic level of error, with the same error level and number of cycles as 4DVar.
Figure 3(b) compares the spinup of the LETKF starting from a random uniform ensemble (Case 1), with and without RIP, as in Figure 3(a), with the LETKF started from perturbations drawn from the 3DVar error covariance (Case 3), i.e. where each ensemble perturbation is a column of the matrix . Here E is an M × K matrix whose columns are random Gaussian numbers such that EE^{T} ≈ I, M is the dimension of the model and K is the number of ensemble members. It is apparent from Figure 3(b) that, as suggested by both Anderson (2008, personal communication) and Zupanski et al(2006), when the initial ensemble is drawn from the 3DVar covariance matrix, the spinup is much faster than when started from random, uniformly distributed perturbations. We can view Figure 3(b) as a comparison between the worst and best choices of the initial ensemble perturbations one could make, since in the first case we make no use of prior information and assume perturbations are nonGaussian, and in the second we use besttuned 3DVar prior information. Nevertheless it is remarkable that, even in the case of faster spinup, the application of the RIP algorithm is able to accelerate the spinup even further.
Figure 3(c) compares the spinup of the LETKF starting from 3DVar covariance ensemble (Case 3) with and without RIP, as in Figure 3(b), and a Gaussian initial ensemble (Case 2), without any a priori information. Initializing from perturbations with 3DVar structures spins up faster than using uncorrelated Gaussian perturbations, but such difference disappears when RIP is applied and similar results are obtained. Given that an optimally tuned 3DVar covariance matrix may not be always available for ensemblebased DA systems, the use of RIP appears to be an attractive alternative.
In an additional experiment in which the LETKF RIP algorithm was forced to always perform 10 iterations (not shown), the LETKF showed an even faster spindown but it converged to a higher level of error, close to that of 3DVar (Table I). This is not surprising, since once the system is close to the maximum likelihood solution, as indicated by the theoretical arguments discussed above, observations should be used only once and then discarded. By performing 10 iterations even after the system spun up, the EnKF analysis fits the data too closely and this increases the analysis errors.
Finally, Figure 4 shows the number of iterations needed to accelerate the spinup in the RIP algorithm when started with random initial ensemble perturbations and with the 3DVar initial perturbations. One iteration in the figure corresponds to the normal LETKF case, i.e. when a second iteration would give a relative improvement in the fit of the forecast to the observations of less than ε = 0.05 (7), and thus it is not used. For the B3DVar initial ensemble (Case 3) only 2 to 6 iterations are needed during the spinup, but the other two ensembles without prior information need 11 iterations at cycle 19. The last second iteration is executed at cycle 65, 46 and 41 for ensembles (1), (2) and (3) respectively. After RIP is turned off, the analysis accuracy for the three ensembles is essentially identical (Table I).
We found that using a lower value of ε = 0.01 (not shown) leads to a faster initial reduction of errors but requires a large number of iterations. Values of ε within a range 0.02–0.05 gave optimal results, leading to a spindown of the initial errors similar to 3DVar and faster than 4DVar, and converging to an error level at least as good as that of 4DVar.