1828 MONTHLY WEATHER REVIEW VOLUME 135 Assimilating Vortex Position with an Ensemble Kalman Filter YONGSHENG CHEN AND CHRIS SNYDER National Center for Atmospheric Research,* Boulder, Colorado (Manuscript received 14 July 2005, in final form 10 July 2006) ABSTRACT Observations of hurricane position, which in practice might be available from satellite or radar imagery, can be easily assimilated with an ensemble Kalman filter (EnKF) given an operator that computes the position of the vortex in the background forecast. The simple linear updating scheme used in the EnKF is effective for small displacements of forecasted vortices from the true position; this situation is operationally relevant since hurricane position is often available frequently in time. When displacements of the forecasted vortices are comparable to the vortex size, non-Gaussian effects become significant and the EnKF’s linear update begins to degrade. Simulations using a simple two-dimensional barotropic model demonstrate the potential of the technique and show that the track forecast initialized with the EnKF analysis is improved. The assimilation of observations of the vortex shape and intensity, along with position, extends the tech- nique’s effectiveness to larger displacements of the forecasted vortices than when assimilating position alone. 1. Introduction over, hurricane forecast models, even the simplest 2D barotropic model, are highly nonlinear. Small initial er- Hurricane track forecasts have improved 1%–2% rors in the vortex position, motion, and strength as well Ϫ1 yr over the last 3 decades (Franklin et al. 2003). as the environmental flows may lead to large discrep- These improvements have followed from refinements ancies in the short-term track and intensity forecast of both numerical models and their initial conditions, (Kurihara et al. 1993). the latter through improvements in satellite observa- Ensemble forecasts, which acknowledge the initial tions and their use, and through the addition of drop- uncertainties in the vortex and its environment, have sondes from special reconnaissance flights (Burpee et also improved track forecasts (e.g., Zhang and Krish- al. 1996). The current National Hurricane Center’s of- namurti 1997, 1999; Cheung and Chan 1999a,b; Aber- ficial track errors average 160 km for 24-h forecasts, son et al. 1998; Marchok et al. 2002; Weber 2003; Hem- increasing to 260 km at 48 h and 370 km at 72 h. ing et al. 2004; Vigh 2004). When the uncertainties in Despite the improvements in observations and data the initial conditions and in the model physics are prop- assimilation, initializing a realistic vortex in the correct erly considered, the ensemble mean is, on average, location and with the correct storm motion remains a closer to the real track than a single realization. In ad- challenge. Many operational numerical weather predic- dition, the ensemble forecasts directly approximate tion systems resort to vortex bogusing (Serrano and strike probabilities. However, ensemble forecasts suffer Undén 1994), though the analysis can be successful with similar difficulties in initializing the vortex. a high-resolution model and sophisticated use of re- Several methods exist to initialize a hurricane vortex motely sensed observations (Leidner et al. 2003). More- in its correct position. Vortex bogusing techniques in- sert an initial vortex into a large-scale background field. The initial bogus vortex can be specified by combining * The National Center for Atmospheric Research is sponsored empirical formulas or dynamical constraints with obser- by the National Science Foundation. vations (e.g., Lord 1991; Serrano and Undén 1994; Les- lie and Holland 1995; Krishnamurti et al. 1997; David- Corresponding author address: Dr. Yongsheng Chen, NCAR, son and Weber 2000; Zhu et al. 2002) or by transplant- P.O. Box 3000, Boulder, CO 80307-3000. ing a vortex with proper intensity and structure from a E-mail: [email protected] different simulation (e.g., Kurihara et al. 1993; Liu et al. DOI: 10.1175/MWR3351.1 © 2007 American Meteorological Society MWR3351 MAY 2007 C H E N A N D SNYDER 1829 1997). Track prediction can be substantially improved if schemes are thus plausible approaches to the assimila- a vortex bogusing scheme generates a well-balanced tion of hurricane position. initial vortex that induces fewer shocks in the forecast Using accurate and frequent position information, model (Bender et al. 1993; Kurihara et al. 1998). An we will show in a 2D barotropic flow that a satisfactory alternative approach is to take simulated observations ensemble analysis results from assimilating vortex po- of surface pressure or low-level winds from a bogus sitions with an EnKF and ignoring the non-Gaussianity vortex and assimilate them with a four-dimensional of the forecast errors associated with finite displace- variational scheme using a large-scale background fore- ments of the vortex. If the background forecast errors cast (Zou and Xiao 2000; Xiao et al. 2000; Pu and Braun for vortex position are comparable to the vortex size, 2001). The structure of the resulting vortex is then de- the linear EnKF update is noticeably suboptimal and termined in part by, and is potentially more consistent produces artifacts such as double vortices for members with, the dynamics of the assimilating model. The vor- whose positions are far from the mean. The barotropic tex relocation technique developed at the National experiments also demonstrate that assimilating addi- Centers for Environmental Prediction (NCEP) relo- tional information about the vortex, such as its shape or cates a misplaced hurricane vortex to the correct posi- intensity, extends the range of displacements for which tion in the background field prior to a three- the EnKF is effective, although it does not resolve in dimensional variational (3DVAR) analysis. Liu et al. any fundamental way the problems associated with a (2002) report improvements as large as 30% in hurri- linear analysis scheme when background errors are cane track forecasts for both global and regional mod- non-Gaussian. els at NCEP. The plan of the paper is as follows: In section 2, we briefly review the EnKF. Section 3 gives a simple ex- Another class of analysis schemes could also poten- ample of the EnKF update for an isolated vortex given tially be used to relocate a hurricane whose location observations of its position, shape, and intensity. Sec- was incorrect in the prior forecast. These share the idea tion 4 presents numerical results from a 2D barotropic that the analysis is a spatial distortion of the prior fore- model. The conclusions and discussion are summarized cast, so that the analysis variables become a displace- in section 5. ment field rather than the original state variables. Hoff- man et al. (1995) used variational analysis to separate the distortion, amplification, and residual errors from 2. The ensemble Kalman filter the total forecast errors. Ravela et al. (2004) also ap- Atmospheric data assimilation is essentially a prob- plied variational methods to analyze a displacement lem of estimating the state x of the dynamical system field and align coherent structures. More recently, Law- (the atmosphere) from noisy observations y taken on son and Hansen (2005) proposed a new error model the state. From the Bayesian probability point of view, that includes a non-Gaussian alignment error and a it is to obtain the conditional probability density func- Gaussian additive error, and they corrected two types | tion (pdf) p(xt yt , y1:tϪ1) of the state x at time t given the of errors separately with an ensemble Kalman filter observation yt taken at t and all previous observations (EnKF). ϭ y1:tϪ1 {y1, y2, ..., ytϪ1}. The maximum likelihood These existing analysis techniques are meant to estimate of the state xt is then the peak or mode of this “move” the vortex a distance that is significant com- pdf. When the pdf is symmetric and unimodal, the pared to the horizontal scale of the vortex. When the mode coincides with the conditional mean, and the error of the forecast position is comparable to vortex maximum likelihood estimate is the same as the mini- scale, the forecast errors for model variables are likely mum variance estimate. to be significantly non-Gaussian, and linear assimila- Bayes’ rule states tion schemes such as the EnKF may be far from opti- ͑ | ͒ ͑ | ͒ p yt xt , y1:tϪ1 p xt y1:tϪ1 mal. In fact, hurricane position estimates routinely pro- p͑x |y , y Ϫ ͒ ϭ . ͑1͒ t t 1:t 1 p͑ | ͒ duced by the National Hurricane Center using satellite, yt y1:tϪ1 radar, in situ aircraft, or synoptic observations are ac- If the pdf of the present observation given the present | curate relative to the size of vortex. These position es- state, p(yt xt), and the pdf of the state given previous | timates have errors of about 20 km for a mature hur- observations, p(xt y1:tϪ1), are known, the pdf of the ricane (Elsberry 1995). Given that position observa- state at the current time can be obtained easily given tions are accurate and frequent, there seems to be no the assumption that the observations at different times inherent reason that background forecasts should have are conditionally independent given the state. The large errors in vortex position. Linear assimilation Bayesian probabilistic estimation approach is theoreti- 1830 MONTHLY WEATHER REVIEW VOLUME 135 cally appealing but practically infeasible for real atmo- where Ne is the number of ensemble members and, in a spheric problems because of the computational cost. slight abuse of notation, overbars now denote an en- Under the assumptions of Gaussian error statistics, semble average. linear dynamics, and observation operators, the Kal- The EnKF also updates the deviation from the mean man filter (Kalman 1960; Kalman and Bucy 1961) is of each member as a counterpart to (2a). We use an equivalent to the Bayesian approach.