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An Observation Operator for the Variational Assimilation of Vortex ...
An Observation Operator for the Variational Assimilation of Vortex Position and Intensity Jeffrey D. Kepert Centre for Australian Weather and Climate Research, Bureau of Meteorology, 700 Collins St, Melbourne Vic 3000. Email: J.Kepert@bom.gov.au Fifth WMO Symposium on the Assimilation of Observations for Meteorology, Oceanography and Hydrology, Melbourne, 5 - 9 October 2009. 1. Introduction Tropical cyclone forecasters produce manual estimates of vortex position and intensity ev- ery few hours, based mainly on satellite imagery interpretation with assistance from radar, scat- terometer and aircraft reconnaissance when available. The volume of such estimates is small (a few thousand per year), but the high impact of tropical cyclones and the difﬁculties with util- ising many forms of conventional and satellite data within the cyclone core implies that these estimates are of high importance. In particular, track is one of the most critical parameters in tropical cyclone forecasting, and good forecasts of track clearly require that the initial position be accurate. Cyclone position estimates are currently either ignored by assimilation systems or used for the generation of synthetic observations, due to the limitations of current assimila- tion systems. Neither approach is completely satisfactory. Problems with the use of synthetic observations include that they will usually have correlated errors, that they may not represent the structure of a particular storm (including any asymmetries), and that is an essentially ad hoc procedure that can require careful tuning for good performance. The disadvantages of the alternative, ignoring the data, are obvious. To directly assimilate position estimates, an observation operator is required to estimate the vortex position and intensity from the background ﬁeld. If the assimilation system is variational, this operator must be differentiable, since its tangent linear and adjoint are also required. Thus algorithms which search directly for the lowest pressure or maximum vorticity are unsuitable for variational assimilation. They have, however, been demonstrated in EnKF-based assimilation (Chen and Snyder 2007), who showed that assimilation of cyclone position could also improve the depiction of cyclone structure in the analysis. The cyclone centre has been deﬁned in terms of wind, pressure or vorticity. Several objec- tive techniques have been developed to deﬁne the centre from observations. Willoughby and Chelmow (1982) presented a variational technique that minimised the distance from a set of lines drawn perpendicular to aircraft observations of wind direction. Marks et al. (1992) de- ﬁned the centre as that which maximised the circulation around the centre of the storm. By the circulation theorem, this is equivalent to a vorticity-based deﬁnition. In contrast to these wind-based deﬁnitions, Kepert (2005) used pressure observations from reconnaissance aircraft 106.1 1000 (a) Pressure (hPa) 980 960 940 920 0 20 40 60 80 100 1000 (b) Pressure (hPa) 980 Figure 1: Dropsonde observations of surface pres- 960 sure in the core of Hurricane Georges on 19 Sept 940 1998, plotted against radius according to (a) the U.S. National Hurricane Center best track and (b) the 920 0 20 40 60 80 100 track found by the method of Kepert (2005). From Radius (km) Kepert (2006). or dropsondes to ﬁnd the centre, by minimising a cost function n (pi − pH )2 J(a) = (1) i=1 σpi 2 2 Here, pi are the n pressure observations, σpi are the error variances of those observations, and pH is the parametric tropical cyclone pressure ﬁeld of Holland (1980), which may be written pH = pe − ∆p 1 − exp{−(rm /r)b} , (2) where pe is the environmental pressure, ∆p is the pressure drop to the centre of the storm, rm is the radius of maximum winds, r is the radius and b controls the extent of the outer cir- culation of the storm, with 1 < b < 2.5. The control vector for the minimisation of J is a = (xt , yt , ut, vt , ∆p, rm , b) where (xt , yt ) are the coordinates of the cyclone centre and (ut , vt ) is its motion, used for calculating r from the observation locations in the above formulae. For example, dropsonde observations of surface pressure in Hurricane Georges taken over about 8 hours on 19 Sept 1998 are plotted against radius according to the ofﬁcial “best-track” from the US National Hurricane Center in Fig 1a. A large amount of scatter is apparent, par- ticularly between 20 and 40 km radius. The same data were used to estimate the position and motion of the storm by minimising (1), and are plotted against radius from that track in Fig 1b. Clearly they fall much closer to a single curve. In this case, the two tracks differ by about 10 km. This method relies on the use of a parametric proﬁle to represent the tropical cyclone. Such proﬁles have a long history of successful use and can represent most real tropical cyclones to reasonable accuracy. The advantages of deﬁning the centre in terms of pressure rather than wind are discussed by Kepert (2005). While this method was developed for use on observations, it may equally well be applied to NWP ﬁelds. In this case, the data are all valid at the same time, so the control vector for the ﬁt omits the storm motion (ut , vt ). The cyclone centre is found by iteratively minimising the cost function (1). If one iteration of the minimiser is differentiable, then the derivative of many iterations follows by the chain 106.2 0.5 ∂x /∂x 0 t −0.5 −5 0 5 0.5 ∂∆p/∂x 0 −0.5 −5 0 5 0.5 /∂x max 0 ∂r −0.5 −5 0 5 0.2 ∂b/∂x 0 −0.2 −5 0 5 Figure 2: The components of the adjoint of the Location / rmax centre-ﬁnder for a univariate one-dimensional state. rule. The steepest descent and conjugate gradient methods are examples of minimisation algo- rithms where one iteration is differentiable. The examples in this paper use conjugate gradient minimisation. In summary, the method takes a model state x and estimates the storm position and param- eterised structure from the pressure ﬁeld part of x, H(x) = (xt , yt , ∆p, rm , b). Each iteration of the minimiser may be written an+1 = Hn+1 (x, an ) = Hn+1 (x, Hn (x, Hn−1 (x, . . . H1 (x, a0 ) . . .))) (3) Each iteration is differentiable, so the derivative Hn+1 of Hn+1 is obtained from that of Hn by the chain rule. Figure 2 shows the components of the adjoint HT in a simple one-dimensional setting. The stopping criterion for the minimisation is not a differentiable operation. However, in practice, when Hn has nearly stopped changing per iteration, so has its derivative Hn . In prac- tise the stopping criterion seems to not be a problem, although a more rigorous approach might be to run for a ﬁxed large number of iterations instead of using a convergence criterion. The accuracy of the calculation of H was checked by comparison to ﬁnite-difference estimates. The use of an iterative observation operator might seem to be undesirable for variational assimilation because of the computational cost. However, in this case it is acceptable, since the volume of observations is relatively small. Also, the iterations to calculate H are performed in the outer loop of the assimilation, while the inner loop uses only the linearization H. This is consistent with normal assimilation practice and ensures that the inner loop is minimising a true quadratic form, as well as limiting the cost of the centre-ﬁnder iterations. 2. Assimilation Tests The centre-ﬁnder has been implemented in a simple 1-dimensional variational assimilation system. This system is univariate, analyses pressure on a 1-dimensional domain with cyclic boundary conditions. The background error spatial correlations are homogeneous and Gaussian, with length scale 1, and the background error variance is 0.22 . The analysis is in rescaled 106.3 0 −0.2 −0.4 Pressure −0.6 −0.8 a −1 b H(x ) H(x ) Observation Figure 3: Assimilation result when the cyclone is in the wrong spot. The thin line is the background, the thick line the analysis. The diamond shows the loca- −15 −10 −5 0 5 10 15 tion and intensity of H(xb ), the triangle similarly for Distance H(xa ) and the circle similarly for the observation. spectral space, truncated at half-resolution, and closely follows Lorenc (1997) apart from the restriction to univariate one dimensional analysis. The inner loop is fully linear and 15 iterations are used. The outer loop has three iterations. A single observation of cyclone location, intensity, size and structure (xt , ∆p, rm , b) is as- similated. The observation error matrix R is assumed diagonal, with observation error standard deviations of (0.2, 1, 0.5, 1) respectively. These can be adjusted to give more or less weight to particular elements of the observation. The background xb will for simplicity be taken to be a Holland (1980) parametric proﬁle. Fig 3 shows a sample analysis, where the observed position is 2 units from the background position, but the background intensity is correct. The analysed storm is slightly weaker than observed, but the position is very close due to the small value of the observation error for this component. Notice how the analysis increment is positive to the left of the observation, and negative to the right. In contrast, a single pressure observation at the observed cyclone centre would have produced negative increments everywhere. The structure of the analysis increment is generated by the adjoint of the observation operator HT , and is related to that shown in Fig 3. Fig 4 shows another example, where the cyclone in the background is in the observed po- sition but is too weak. The analysis successfully deepens the storm. Fig 5 shows an example that successfully combines both changes. However, if the assimilation is required to move the storm too far, then a double centre can occur, as shown in Fig 6. Such “double vortices” are also a well-known problem with vortex bogus schemes. This case would likely produce a fore- cast failure, and one approach could be to tune the quality control to ignore the observation. A somewhat better alternative might be to pre-ﬁlter the background to remove or weaken the vor- tex prior to assimilation, as is often done with vortex bogus schemes. Perhaps the best solution would be to assimilate frequently enough that “bad” background ﬁelds are rare. 3. Discussion A differentiable observation operator for cyclone position has been developed, and demon- strated in a highly idealised variational assimilation system. It is expected that the operator will 106.4 0 −0.5 Pressure b −1 H(x ) −1.5 H(xa) −2 Observation −15 −10 −5 0 5 10 15 Figure 4: As for Fig 3 except that the observation Distance differs in intensity from the background. 0 −0.5 Pressure b −1 H(x ) −1.5 H(xa) −2 Observation −15 −10 −5 0 5 10 15 Figure 5: As for Fig 3 except that the observation Distance and background differ in intensity and location. also work in full 3D and 4D variational systems, although this remains to be tested. The advan- tages and disadvantages of the scheme compared to bogussing also need investigation. Possible advantages are that less tuning will be required, and that the problem of correlated observation error may be more easily handled. If implemented in 4D-Var, observations could be made available at several points within the assimilation time window. This approach could possibly reduce the problem of double-vortices arising from large position changes. Tropical cyclones are known to possess asymmetries re- lated to their motion, the so-called beta-gyres (e.g. Elsberry 1995), and some bogussing schemes attempt to bogus in an asymmetric vortex to account for these (e.g. Davidson and Weber 2000). In principle, the model adjoint should cause asymmetries consistent with the motion to develop in the analysis when vortex positions are inserted at several times during a 4D-Var assimila- tion. The beta-gyres are relatively large-scale and are reasonably well described by linearised 106.5 0 −0.2 −0.4 −0.6 Pressure −0.8 −1 H(xb) Observation H(xa) −1.2 −15 −10 −5 0 5 10 15 Figure 6: As for Fig 3 except that the observed lo- Distance cation is further from the background. dynamics, so this expectation is perhaps not as optimistic as it might seem. The observation operator is quite nonlinear, and our experience has been that several iter- ations of the assimilation outer loop are necessary for good performance. While the operator requires an iteration and is therefore relatively expensive to calculate, the small number of ob- servations means that this is not a severe problem. The operator should also be useful for adjoint-based sensitivity studies. Errico (1997) de- scribes how to use the adjoint of a model to determine the sensitivity of some differentiable metric of the forecast to the initial condition. Examples of metrics could be the distance of the cyclone from some location, or the direction of movement of the cyclone. As the centre-ﬁnder is differentiable, so are these metrics. Errico mentions cyclone position as an example of a po- tentially very useful metric for which it is difﬁcult to construct a differentiable operator. This difﬁculty has now been overcome, although conﬁrmatory testing is still required. Assuming that these tests are successful, the centre-ﬁnder is expected to be a better sensitivity analysis tool for tropical cyclone motion than the “steering ﬂow” concept that has been previously used (e.g. Wu et al. 2009). References Chen, Y. and C. Snyder, 2007: Assimilating vortex position with an Ensemble Kalman Filter. Mon. Wea. Rev., 135, 1828–1845. Davidson, N. E. and H. C. Weber, 2000: The BMRC high-resolution tropical cyclone prediction system: TC-LAPS . Mon. Wea. Rev., 128, 1245–1265. Elsberry, R. L., 1995: Tropical cyclone motion. Global Perspectives On Tropical Cyclones, Elsberry, R. L., Ed., World Meteorological Organisation, 106–197. Errico, R. M., 1997: What is an adjoint model? Bull. Amer. Meteor. Soc., 78, 2577–2591. Holland, G. J., 1980: An analytic model of the wind and pressue proﬁles in hurricanes. Mon. Wea. Rev., 108, 1212–1218. Kepert, J. D., 2005: Objective analysis of tropical cyclone location and motion from high density obser- vations. Mon. Wea. 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