1
Ensemble forecasting and data assimilation: two problems with the same solution?
Eugenia Kalnay Department of Meteorology and Chaos Group University of Maryland, College Park, MD, 20742
perturbations and/or model deficiencies, 1. Introduction the forecasts are not able to track the verifying truth, and remain relatively Until 1991, operational NWP centers close to each other. In this case the used to integrate a single control ensemble is not helpful to the forecast starting from the analysis, forecasters at all, since the lack of which is the best estimate of the state of ensemble spread would give them the atmosphere at the initial time. In unjustified confidence in the erroneous December 1992, both NCEP and forecast. Nevertheless, for NWP ECMWF started running ensembles of development, the “bad” ensemble is still forecasts from slightly perturbed initial very useful: after the verification time conditions (Molteni and Palmer, 1993, arrives, it clearly indicates the presence Buizza, 1997, Toth and Kalnay, 1993, of a deficiency in the forecasting Tracton and Kalnay, 1993, Toth and system . A single “deterministic” forecast, Kalnay, 1997). by contrast, would not be able to distinguish between a deficiency in the Ensemble forecasting provides human system and errors in the initial forecasters with a range of possible conditions as the cause of failure. solutions, whose average is generally more accurate than the single Ideally, the initial perturbations should deterministic forecast (see, e.g., Fig. 4), sample well the analysis “errors of the and whose spread gives information day” and the spread among the about the forecast errors. It also ensemble members should be similar to provides a quantitative basis for that of the forecast errors. The two probabilistic forecasting. essential problems in the design of an ensemble forecasting system are how to Schematic Fig. 1 shows the essential create effective initial perturbations, and components of an ensemble: a control how to handle model deficiencies, which forecast started from the analysis, make the ensemble forecas t spread forecasts started from two perturbations smaller than the forecast error. to the analysis (in this case equal and opposite), the ensemble average, and In this paper we present a brief historic the “truth”, or forecast verification, which review of ensemble forecasting, current becomes available later. The first methods to create perturbations. We schematic shows an example of a “good point out that the promising approach of ensemble” in which “truth” looks like a ensemble Kalman Filtering for data member of the ensemble. In this case, assimilation can solve, at the same time, the ensemble average is closer to the the problems of obtaining optimal initial truth, due to nonlinear filtering of errors, ensemble perturbations, and possibly and the ensemble spread is related to estimating the impact of model errors. the forecast error. The second We also discuss the problem of coupled schematic is an example of a “bad systems with instabilities that have very ensemble”: due to poor initial different time scales. 2
POSITIVE Good ensemble PERTURBATION CONTROL
ENSEMBLE AVERAGE
TRUTH
NEGATIVE PERTURBATION
POSITIVE Bad ensemble PERTURBATION
CONTROL AVERAGE
NEGATIVE PERTURBATION
TRUTH
Figure 1: Schematic of the essential components of an ensemble of forecasts: The analysis (a cross) which constitutes the initial conditions for the control forecast (in green); the initial perturbations (a thick dot) around the analysis, which in this case are chosen to be equal and opposite; the perturbed forecasts (in black); the ensemble average (in blue); and the verifying analysis or truth (in red). The first schematic is that of a “good ensemble” in which the truth is a plausible member of the ensemble. The second is an example of a bad ensemble, quite different from the truth, pointing to the presence of a problem in the forecasting system such as deficiencies in the analysis, in the ensemble perturbations and/or in the model. 3
2. Ensemble forecasting methods number of integrations (of the order of 8) it is possible to approximate this It should be noted that human important advantage of an infinite forecasters have always performed ensemble. The estimate of the analysis subjective ensemble forecasting by error covariance was constant in time, checking forecasts from previous days, so the MCF method did not include the and comparing forecasts from different effects of “errors of the day”. The MCF centers, approaches similar to lagged method is shown schematically in Fig. forecasting and multiple systems 2a. Errico and Baumhefner (1987) forecasting. The consistency among applied this method to realistic global these forecasts at a given verification models, using perturbations that time provided a level of confidence in represented a realistic (but constant) the forecasts, confidence that changed estimation of the errors in the initial from day to day and from region to conditions. Hollingsworth (1980) showed region. that random errors in the initial conditions took too long to spin-up into 2.1 Early methods growing “errors of the day”, making MCF an inefficient approach for ensemble Epstein (1969), introduced the idea of forecasting. Stochastic-Dynamic forecasting (SDF), and pointed out that it could be also Hoffman and Kalnay (1983) suggested used in the analysis cycle to provide the as an alternative to MCF, the Lagged forecast error covariance. Epstein Averaged Forecasting (LAF) method, in designed SDF as a shortcut to estimate which forecasts from earlier analyses the true probability distribution of the were included in the ensemble forecast uncertainty, given by the (schematic Fig. 2b). Since the ensemble Liouville equation (Ehrendorfer, 2002), members are forecasts of different which Epstein approximated running a “ages” they should be averaged with huge (500) number of perturbed (Monte weights estimated from their average Carlo) integrations for the 3-variable forecast errors. Hoffman and Kalnay Lorenz (1963) model. However, since found that compared to MCF, LAF SDF involves the integration of forecast resulted in a better prediction of skill (a equations for each element of the stronger relationship between ensemble covariance matrix, this method is still not spread and error), presumably because computationally feasible for models with LAF includes the effect of variable large number of degrees of freedom. “errors of the day”. The main disadvantage of LAF, that the “older” Leith (1974) suggested using directly a forecasts are less accurate, was Monte Carlo Forecasting approach corrected by the Scaled LAF (SLAF) (MCF), where random perturbations approach of Ebisuzaki and Kalnay sampling the estimated analysis error (1991), in which the LAF perturbations covariance are added to the initial (difference between the forecast and the conditions. He noted that in an infinitely current analysis) are scaled by their large ensemble, the average forecast “age”, so that all the SLAF perturbations error variance at long time leads have errors of similar magnitude. They converges to the climatological error also suggested that the scaled variance, whereas the error variance of perturbations should be both added and individual forecasts converges to twice subtracted from the analysis, thus the climatological error variance. Leith increasing the ensemble size and the showed that with just a relatively small probability of “encompassing” the true 4
solution within the ensemble. SLAF can impact of perturbed boundary conditions be easily implemented in both global (Hou et al, 2001). and regional models, including the
X MCF
0 tf t
X LAF
-3dt -2dt -dt 0 tf t
Fig. 2: Schematic of the Monte Carlo forecasting and Lagged Average Forecasting methods. 5
2.2 Operational Ensemble Forecasting 1993, 1997). Bred Vectors (like the methods leading Lyapunov Vectors) are independent of the norm and represent In December 1992 two methods to the shapes of the instabilities growing create perturbations became operational upon the evolving flow. In areas where at NCEP and at ECMWF. They are the evolving flow is very unstable (and based on bred vectors and singular where forecast errors grow fast), the vectors respectively, and like LAF, they BVs tend to align themselves along very include “errors of the day”. These and low dimensional subspaces (the locally other methods that have since become most unstable perturbations, Patil et al, operational or are under consideration in 2001). An example of such situation is operational centers are briefly shown in Figs. 3, where the forecast discussed. More details are given in uncertainty in a 2.5 day forecast of a Kalnay (2003). storm is very large, but the subspace of the ensemble uncertainty lies within a a. Singular Vectors (SVs) one-dimensional space. In this extreme (but not uncommon) case, a single Singular vectors are the linear observation at 500hPa would be able to perturbations of a control forecast that identify the best solution! grow fastest within a certain time interval (Lorenz, 1965), known as The bred vectors are the differences “optimization period”, using a specific between the forecasts (the forecast norm to measure their size. SVs are perturbations). In unstable areas of fast strongly sensitive to the length of the growth, they tend to have shapes that interval and to the choice of norm are independent of the forecast length (Ahlquist, 2000). Ehrendorfer and or the norm, and depend only on the Tribbia (1997) showed that if the initial verification time. This suggests that norm used to derive the singular vectors forecast errors, to the extent that they is the analysis error covariance norm, reflect instabilities of the background then the initial singular vectors evolve flow, should have shapes similar to bred into the eigenvectors of the forecast vectors, and this has been confirmed error covariance at the end of the with model simulations (Corazza et al, optimization period. This indicates that if 2002). the analysis error covariance is known, then singular vectors based on this NCEP implemented an ensemble specific norm are the ideal system based on breeding in 1992. perturbations. Later, the US Navy, the National Centre for Medium Range Weather Forecasting ECMWF implemented an ensemble in India, and the South African system with initial perturbations based Meteorological Weather Service on singular vectors using a total energy implemented similar ensemble norm (Molteni and Palmer, 1993, forecasting systems. The Japanese Molteni et al, 1996, Buizza et al, 1997, Meteorological Agency ensemble Palmer et al 1998). forecasting system is also based on breeding, but imposing a partial global b. Bred Vectors (BVs) orthogonalization among the bred vectors, thus reducing the tendency of Breeding is a nonlinear generalization of the bred vectors to converge towards a the method to obtain leading Lyapunov low dimensional space of the most vectors, which are the sustained fastest unstable directions (Kyouda and growing perturbations (Toth and Kalnay, Kusunoki, 2002). 6
Fig. 3: “Spaghetti plots” showing a 2.5 day ensemble forecast verifying on 95/10/21. Each 5640gpm contour at 500 hPa corresponds to one ensemble forecast, and the dotted line is the verifying analysis. Note that the uncertainty in the location of the center of the predicted storm in SE US is very large, but that the differences among forecasts of the storm lie on a 1-dimensional space (red line). 7
achieving the goals of ensemble c. Multiple data assimilation systems forecasting. However, since they both introduce perturbations in the best Houtekamer et al (1996) developed a estimate of the initial conditions and the system based on running an ensemble model, which are in the control forecast, of data assimilation systems using it can be expected that the individual perturbed observations, implemented in perturbed forecasts should be worse the Canadian Weather Service. Hamill than the control. A typical ensemble et al (2000) showed that in a quasi- average for a season (Figure 4) shows geostrophic system, a multiple data that, indeed, the individual perturbed assimilation system performs better than forecasts have less skill than the the singular vectors and the breeding unperturbed control. Nevertheless, the approaches. With respect to the ensemble average is an improvement computational cost, the multiple data over the control, especially after the assimilation system and the singular perturbations grow into a nonlinear vector approach are comparable, regime that tends to filter out some of whereas breeding is essentially cost- the errors. free. An alternative to the introduction of d. Perturbed physical perturbations is the use of multiple parameterizations systems developed independently at different centers. In principle, an The methods discussed above only ensemble of forecasts from different include perturbations in the initial operational or research centers, each conditions, assuming that the error aiming to be the best and choosing growth due to model deficiencies is different competitive approaches, should small compared to that due to unstable sample well the uncertainty in our growth of initial errors. In addition, knowledge of both the models and the several groups have introduced initial conditions. It has long been known changes in the physical parameter- that the ensemble average of multiple izations to allow for the inclusion of center forecasts is significantly better model uncertainty (Houtekamer et al, than even the best individual forecasting 1996, Stensrud et al, 2000). Buizza et al system (e.g., Kalnay and Ham, 1989, (1999) developed a perturbation Fritsch et al, 2000, Arribas et al, 2004). approach that introduces a stochastic This has also been shown to be true for perturbation of the impact of subgrid regional models (Hou et al, 2001). scale physical parameterizations by Krishnamurti (1999) introduced the simply multiplying the time derivative of concept of “superensemble”, using the “physics” by a random number linear regression and past forecasts of normally distributed with mean 1 and different systems as predictors to standard deviation 0.2. minimize the ensemble average prediction errors. This results in e. Multiple system ensembles remarkable forecast improvements (e.g., Fig. 5). This method is also called “poor Both the perturbations of the initial person’s” method to reflect that it does conditions and of the subgrid scale not require running a forecasting physical parameterizations have been system. shown to be successful towards
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Anomaly Correlation for the winter of 1997/98 (from 00Z) Controls (T126 and T62) and 10 perturbed ensemble forecasts
1
0.8 Control (deterministic) forecasts, T126 and T62 Individual perturbed 0.6 forecasts Average of 10 perturbed forecasts
0.4 anomaly correlation
0.2
0 0 5 10 15 forecast day
Fig. 4: Anomaly correlation of the ensembles during the winter of 1997/1998 (data courtesy Jae Schemm, of NCEP).
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Fig. 5: Mean typhoon track and intensity errors for the west Pacific, 1998-2000 for several global models and a superensemble trained on the first half of the same season (Kumar et al, 2003). The forecasts are verified on the second half of the season. 10
f. Other methods Until recently, the problem has been that This field is changing quickly, and we do not know A , which changes improvements and changes to the substantially from day to day and from operational systems are under region to region due to instabilities of the development. For example, ECMWF background flow. These instabilities, has implemented changes in the length associated with the “errors of the day”, of the optimization period for the SVs, a are not taken into account in data combination of initial and final or assimilation systems, except for 4D -Var evolved SVs (which are more similar to and Kalman Filtering, methods that are BVs), and the introduction of a computationally very expensive. 4D-Var stochastic element in the physical has been implemented at ECMWF and parameterizations, all of which at MeteoFrance (Rabier et al, 2000, contributed to improvements in the Andersson et al, 2004), with some ensemble performance. NCEP is simplifications such as reducing the considering the implementation of the resolution of the analysis in order to Ensemble Transform Kalman Filter reduce the computational cost (from (Bishop et al, 2001) to replace breeding ~40km in the forecast model to ~120km (see also section 3). A recent in the assimilation model). Versions of comparison of the ensemble 4D-Var are also under development in performance of the Canadian, US and other centers. ECMWF systems (Toth et al., 2004) suggests that the ECMWF ensembles The original formulations of Kalman based on singular vectors behave well Filtering (Kalman, 1960) and Extended beyond the optimization period, at which Kalman Filtering (Ghil and Malanotte- time the model advantages of the Rizzoli, 1991, Cohn, 1997) are ECMWF system are also paramount. prohibitive in practice because they The NCEP bred vectors are better at would require the equivalent of N model shorter ranges, and the multiple integrations, where N is the number of analyses Canadian method also seems degrees of freedom (d.o.f.) of the model, to perform well. which is of the order of a 10^6 or more. Considerable work has been done on finding simplifying assumptions to 3. Ensemble Kalman Filtering for reduce the cost of KF (e.g., Fisher, data assimilation and ensemble 1998, Fisher et al, 2003), but so far they forecasting. have been successful only under special circumstances. As indicated before (e.g., Ehrendorfer and Tribbia, 1997), “perfect” initial Ensemble Kalman Filtering (EnKF) was perturbations for ensemble forecasting introduced by Evensen (1994) as a should sample well the analysis errors. more efficient approach to Kalman Filtering. Recent developments suggest The ideal initial perturbations d xi should have a covariance that that in the near future it will be feasible represents the analysis error to determine simultaneously the covariance A : covariance matrix A and the ideal perturbations d xi . These methods take 1 K advantage of the fact that the size of ddxxAT » (1.1). K -1å ii ensembles needed to represent the i=1 analysis and forecast error covariances 11 is of order O(100), much smaller than method known as sequential the number of degrees of freedom of the assimilation of observations. This is model. When EnKF is performed locally done using a localization of the error in physical space (Ott et al, 2002), the covariance in the horizontal and in the number is smaller than 100. vertical, to avoid spurious long-distance correlations due to sampling. There are two basic approaches to EnKF. In the first one (known as In the Local Ensemble Kalman Filter “perturbed observations”) an ensemble (LEKF) method (Ott et al, 2002, 2004, of data assimilations is carried out using Szunyogh et al, 2004a, b), the the same observations to which random Ensemble Kalman Filter problem is have been added. The ensemble is solved locally in physical space. For used to estimate the forecast error each grid point a local 3D volume of the covariance needed in the Kalman Filter order of 1000km by 1000km and a few (Evensen, 1994, Houtekamer and vertical layers is used to perform the Mitchell, 1998, Hamill and Snyder, analysis. The Kalman Filter equations 2000). This approach has already been are solved exactly on this subspace shown to be very competitive with the locally spanned by the global ensemble operational 3D-Var, an important members, using all the observations milestone, given that 3D-Var has the available within the volume. This benefit of years of improved localization in space results in a further developments (Houtekamer et al, 2004). reduction of the number of necessary ensemble members, so that matrix The second approach is the class of operations are done in a very low square root filters (Tippett et al, 2002), dimensional space. The analysis is and does not require perturbing the carried out independently at each grid observations. Several groups have point, with essentially perfectly parallel recently independently developed computations. square root filters (Bishop et al, 2001, Anderson, 2001, Whitaker and Hamill, In order to visualize the main advantage 2002, Ott et al, 2002). The ensemble of Ensemble Kalman Filtering (EnKF), forecasts are used to obtain a we compare it with the 3D-Var approach background error covariance at the time currently operational in most centers, in of the analysis. The full Kalman Filter which the analysis error covariance is equations and the new observations are estimated as an average over many then used to obtain the analysis cases. The top schematic Fig. 6 shows increment (difference between the how the 3D-Var analysis maximizes the analysis and the forecast) and the joint probability defined by the analysis error covariance, all lying within observations error covariance and the the subspace of the forecast ensemble. background error covariance, but does After this is completed, the new initial not know about “errors of the day”. The bottom schematic shows how in the analysis perturbations d xi are obtained by solving the square root problem of EnKF, the ensemble perturbations equation (1.1), and the analysis cycle defines a low-dimensional subspace can continue. (10-100), and the KF analysis maximizes the joint probability within Several of the Ensemble Kalman Filters this subspace. Because all the further reduce the computational cost by computations are performed within this assimilating the observations one at a subspace, the rank of the matrices involved is low, and the Kalman Filter time, for the whole physical domain, a equations providing the analysis and 12 analysis error covariance are solved analysis, as shown with experiments directly, not iteratively. done with the Lorenz (1996) model (Fig. 8, from Ott et al, 2004). When Fig. 7 shows an example of EnKF and performed globally, the number of 3D-Var using a quasi-geostrophic data ensemble members required for the assimilation system (Morss et al, 2000, EnKF to converge is proportional to the Hamill and Snyder, 2000, Corazza et al, size of the model. When done locally, 2002). The colors show the background the number of ensemble members is (12 hour forecast) errors and the reduced from 27 to 8, and does not contours are the analysis corrections increase with the size of the model. In based on a given set of observations. addition, the analysis for different grid The top panel corresponds to 3D-Var, points can be carried out in parallel, using a background error covariance since they are independent from each constant in time. Since the system does other. The advantage of a local analysis not know about the “errors of the day”, in reducing the size of the required the corrections brought by the new ensemble is illustrated in Fig. 9, showing observations tend to be isotropic. The a schematic of an ensemble with three bottom panel shows that the Local independent unstable regions. In each Ensemble Kalman Filter (Ott et al, 2004, of the regions wave number 1 and 2 are Szunyogh et al, 2004b) is much more unstable. This situation is reminiscent of efficient in correcting the background the global atmosphere, containing errors with the same observations. The regions where baroclinic instabilities can large improvements that the LEKF develop independently. From a local makes on the analysis are also apparent point of view, the first two perturbations in the 3-day forecasts (not shown). are enough to represent all possible combinations of instabilities, whereas Performing the EnKF locally in space from a global point of view, the third substantially reduces the number of perturbation and many others are ensemble members required for the linearly independent.
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The 3D-Var Analysis doesn’t know about the errors of the day
Observations Background ~106-8 d.o.f. ~105-7 d.o.f.
R B
Ensemble Kalman Filter Analysis: the correction is computed within the low dim attractor
R Background ~106-8 d.o.f. Observations ~105-7 d.o.f. Errors of the day lying The 3D-Var Analysis doesn’t know on a very low-dim attractor about the errors of the day
Figure 6: Schematic of the analysis given the background forecast in a very large dimensional space, the background error covariance B (which in the case of 3D-Var is isotropic and constant in time), the vector of observations, in a very large dimensional space, with observations error covariance R. The analysis estimate of the true state of the atmosphere maximizes the joint probability distribution. Top: 3DVar. Bottom: EnKF, in which the ensemble forecast members define a low-dimensional subspace within which the analysis lies. Background error (collor) and 3D- Var anallysiis iincrement (conto14ur s)
Background error (collor) and LEKF anallysiis iincrements (contours)
Fig. 7: Simulation of data assimilation in a quasi-geostrophic model, assimilating potential vorticity observations at a particular day (June 15). The color represents the 12 hr forecast (background) error and the contours the analysis corrections. Top: 3D-Var. Bottom: Local Ensemble Kalman Filter. Figures courtesy of Matteo Corazza. 15 FULL ENSEMBLE KALMAN FILTER ANALYSIS ERROR AS A FUNCTION OF THE NUMBER OF ENSEMBLE MEMBERS
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4
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3
M=80
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RMS ANALYSIS ERROR M=40
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0 0 20 40 60 80 100 120 140 NUMBER OF ENSEMBLE MEMBERS LEKF ANALYSIS ERROR AS A FUNCTION OF THE NUMBER OF ENSEMBLE MEMBERS 4
3.5
3 M=120
2.5 M=80 2
RME ERROR 1.5 M=40 1
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0 4 5 6 7 8 9 10 11 12 Number of ensemble members
Fig. 8: Number of ensemble members required for convergence to the optimal solution in a Lorenz (1996) model. Top: using a full global Ensemble Kalman Filter. Bottom: using a Local Ensemble Kalman Filter. The size of the domain, M is 40, 80 or 120. Note that the Kaplan-Yorke dimension is about 27 for the 40 variable model and it increases linearly with size. 16
B1 B2 B3
A B C A B C A B C
Fig. 9: Schematic showing the advantages of performing a local rather than a global analysis. The domain in composed of three regions, each of which has instabilities with wave numbers 1 and 2, positive (solid) or negative(dashed). From a local point of view, the ensemble perturbations B1 and B2 are sufficient to represent all possible perturbations, whereas from a global point of view there are many independent perturbations.
Fig. 10: Evolution of the LEKF analysis error in surface pressure in hPa as a function of assimilation step (in units of 6 hr). The rms error of the observations is shown by the dashed line. Observations are made at 11% of the grid points, and the model has T62 (about 200km) resolution. (From Szunyogh et al, this volume) 17
Fig. 11: Global rms analysis error for temperatures (left, oC) and zonal winds (right, m/sec). The dashed line is the rms of observational errors. From left to right, the following percentage of the grid points have “rawinsonde” data: 100% (red), 11% (blue), 5% (green), 2% (purple). Since the grid resolution is about 200km, the blue is similar to the current rawinsonde density in the NH, and the purple to the SH and tropics. (From Szunyogh et al, this volume)
Fig. 12: Example of the trace of the 6hr forecast error covariance matrix from the LEKF. It indicates where errors are large or small: regions in purple do not need new observations because the uncertainties are small. This information can be used in adaptive observations.
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The LEKF has been tested in a “perfect observation operator H, another model” mode, using the operational important advantage (Takemasa NCEP Global Forecasting System at Miyoshi, 2004, pers. communication). 3) T62/28levels resolution, with excellent It can be easily extended to 4 results (Szunyogh et al, 2004b, this dimensions, so that observations are volume). Fig. 10 shows the evolution of assimilated correctly even if their time of the analysis of surface pressure, when observation falls in between analysis about 11% of the grid points (separated times (Sauer et al, 2004). This 4D LEKF by about 200km) have “rawinsonde” is performed at a low additional observations of surface pressure, and computational cost using the ensemble vertical profiles of temperature and to “transport” observational increments winds, with random errors of 1hPa, 1K from the time of observation to the and 1m/s on each wind component. analysis time. This is important since Similarly to the rapid reduction of global experience indicates that the ability to analysis rms error for surface pressure assimilate observations at the right time in this figure, the analysis errors for is the main advantage of 4D-Var (rather temperature and winds quickly converge than the evolution of the covariance). to values much smaller than the Another advantage of EnKF is the observational errors (not shown). Fig. 11 precise way it allows to perform shows the vertical rms analysis errors adaptive observations: Fig. 12 shows an for temperature and winds for several example of the trace of the background levels of observational density, from error covariance for a given day of the 11% of the grid points, a density similar LEKF data assimilation simulation. This to that of the rawinsondes in the NH provides a clear estimate of the forecast extratropics, to 2%, a density similar to uncertainty. This field, and the rawinsondes in the SH. The ability of the corresponding analysis uncertainty can LEKF to extract information through the be contrasted with the analysis knowledge of the “errors of the day” is increments as a “reality check”. The very encouraging. LEKF could eventually be computed alongside a satellite instrument, such as Although experiments with real a wind lidar, and provide an indication of observations have not yet been where there is a large analysis/forecast completed, the LEKF shows great error and the instrument should dwell promise: 1) it is very efficient, due to its longer, and where observations are not complete parallelism and need for currently needed. relatively few ensemble members, allowing for a direct rather than iterative 4. Summary and some remaining solution. It takes only 15 minutes to problems assimilate 1.5 million observations using 40 ensemble members on a cluster of In summary, ideal initial ensemble 25 dual processor 2.8 GH PCs; 2) like perturbations should span effectively the all EnKF methods, LEKF does not analysis error covariance, as in equation require the creation and maintenance of (1.1). Until recently, this was an the forecast model’s linear tangent or apparently impossible task, since the adjoint models, saving the very large analysis error covariance was unknown. effort that these models usually require. Methods such as breeding and singular It also avoids many required vectors were devised as shortcuts to simplifications, since the full nonlinear (1.1). However, Ensemble Kalman model is used for every operation. Filtering holds the promise of solving Similarly it does not require the both the data assimilation and the Jacobian or the adjoint of the ensemble forecasting problem 19 simultaneously, since both the lhs and variables with a relatively small number the rhs of this equation are obtained of parameters associated with model during the EnKF assimilation. errors, and using the observations to Furthermore, experience with the NCEP estimate the optimal value of their time- global model indicates that it is possible varying coefficients (e.g., Anderson, to carry out this advanced approach with 2001). present day supercomputers and without sacrificing the resolution of the Another important problem in ensemble analysis model. Additional advantages forecasting and data assimilation arises are that the linear tangent and adjoints with the presence of coupled instabilities codes of the forecast model or the with a wide range of time scales. For observation operators are not needed, example, atmospheric convection, and that atmospheric data can be baroclinic instabilities, and the ENSO assimilated at the time they were instabilities of the coupled ocean- observed. atmosphere system are characterized by time scales of minutes, days and One of the most important remaining months respectively. problems is that of model deficiencies, leading to bias in data assimilation (Dee This difficult problem of coupled and DaSilva, 1999), to systematic systems (Timmerman, 2002) has been forecast errors, and to ensemble attacked by allowing fast, small deficiencies such as indicated in Fig. 1b. amplitude modes to saturate (Toth and Obviously, the ultimate solution to this Kalnay, 1993, Aurell et al, 1996, Boffetta problem lies in the improvement of the et al, 1998). Toth and Kalnay (1993, models (e.g., Simmons and 1996) proposed that breeding with Hollingsworth, 2002), but until that stage rescaling amplitudes above the is reached, empirical approaches may saturation level could be used to filter be needed. fast, small amplitude instabilities such as convection, without the need to A successful approach to start eliminate or simplify their physical addressing the problem of errors and impact. Conversely, the choice of very uncertainties in the model is the use of small amplitudes for the rescaling can multiple models (Krishnamurti 1999, identify the bred vectors dominated by Hou et al, 2001, Fritsch, 2001, Kalnay convection. This conjecture was and Ham, 1989). Other approaches validated by Lorenz (1996), and Peña (DelSole and Hou, 1999, Kaas et al, and Kalnay (2004). 1999) rely on empirical statistical methods to modify the model and Unfortunately, In the case of the coupled reduce forecast errors. In another ocean-atmosphere system, the method, known as “dressing”, random “atmospheric weather noise” has larger, perturbations are added to the not smaller amplitudes than the slow ensemble forecasts in order to ENSO signals. Peña and Kalnay (2004) reproduce the observed error have shown that in this case one can covariance with an enhanced ensemble still take advantage of the longer time forecast spread (Roulston and Smith, scale of the slow modes. Breeding with 2003, Wang and Bishop, 2004). rescaling at longer intervals can identify the slower coupled mode (unlike linear It is possible that the Ensemble Kalman singular vectors or Lyapunov vectors, Filtering approach will also be able to which are linear, and select the fastest efficiently minimize model errors by mode). Cai et al (2002) performed augmenting the space of model breeding with the Cane-Zebiak model 20 and used the bred vectors for simulated Anderson, J. L., 2001: An ensemble ensemble forecasting and data adjustment filter for data assimilation. assimilation experiments, with very Mon. Wea. Rev., 129, 2884-2903. encouraging results. Yang et al (2004) also obtained good results using the Andersson, E, C. Cardinali, M. Fisher, operational NAS NSIPP coupled GCM, E. Hólm, L. Isaksen, Y. Trémolet and A. and very similar results were obtained Hollingsworth, 2004: Developments in with an NCEP coupled GCM. These ECMWF’s 4d-Var system. Paper J.1. “perfect model” breeding experiments AMS preprints of the 20th Conference were done using a rescaling interval of on Weather Analysis and one month and the Niño-3 SST Forecasting/16th Conference on anomalies as the rescaling norm. Numerical Weather Prediction. Results with the operational NSIPP coupled data assimilation system Arribas, A.; Robertson, KB; Mylne, KR, indicate that the growth rate and shape 2004: Test of a poor man's ensemble of coupled bred vectors in the Equatorial prediction system for short-range region is strongly related to the ENSO probability forecasting. Submitted forecast error. Aurell E., Boffetta G., Crisanti A., These results suggest that in coupled Paladin G. and Vulpinai A., 1996: systems that contain fast and slow Predictability in systems with many instabilities, it may not be possible to degrees of freedom. Phys. Rev. E, 52, use the same EnKF approach to do data 2337. assimilation for both the fast and the slow processes. As in breeding, it may Bishop, C. H., B. J. Etherton and S. J. be necessary to choose different time Majumdar, 2001: Adaptive sampling intervals for the analysis of fast and slow with the ensemble transform Kalman instabilities. filter. Part I: Theoretical aspects. Mon. Wea. 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