### Abstract:

A nested, nondivergent barotropic numerical weather prediction model for forecasting tropical cyclone motion out to 48 h is initialized at time t = 0 by assimilating data from the preceding 24 h. The assimilation scheme finds the generalized inverse of the model and data for −24 ≤ t ≤ 0. That is, the inverse estimate of the streamfunction is a weighted least-squares best fit to the initial conditions at t = −24, to the data at t = −12 and t = 0, and to the dynamics and the boundary conditions in the interval −24 ≤ t ≤ 0. In particular, the dynamics are imposed only as a weak constraint.
The inverse estimate satisfies the Euler-Lagrange equations for a least-squares penalty functional; these nonlinear equations are solved using an iterative technique that yields a sequence of linear Euler-Lagrange equations. A representer expansion produces explicit expressions for the reduced penalty functional, which may be shown to be a χ2 variable with as many degrees of freedom as them are data. The representer expansion also yields expressions for the posterior covariance of the various residuals.
The inverse method was tested on ten cases from four different tropical cyclones observed in the South China Sea during the 1990 Tropical Cyclone Motion Program (TCM-90). Predictions of cyclone tracks out to 48 h were compared with the Australian Weather Bureau’s operational barotropic model which has no data assimilation procedure, and with a simple nudging scheme. In the latter, a model integration from t = −24 to t = 0 is nudged toward target analyses at t = −12 and at t = 0.
Compared with the no-data assimilation and nudging forecasts, the inverse method yielded reductions in mean track forecast error of about 14% and 10%, respectively, at 24 h, reducing to 10% and 7% at 48 h. The results were quite consistent, with the inverse method providing the smallest mean position error in all ten cases.
In addition, all the numerical predictions were compared with a CLIPER (climatology and persistence) scheme. The inverse scheme yielded the lowest mean errors at all times t = 12, 24, 36, and 48 h, with improvements over CLIPER being 26% at 24 h and 29% at 48 h.