This research is supported by NOAA Grant NA37WA0361 and NSF Grant ATM-9812729. The authors would like to thank Dr. E. Kalnay for her persistent encouragement on this study. We also thank Dr. Claude Lemaréchal from INRIA for providing to us the software for his bundle method. The original bundle algorithm was modified by Profs. Navon and Nazareth to fit the test problems.
Betts, A. K., 1986: A new convective adjustment scheme. Part I: Observational and theoretical basis. Quart. J. Roy. Meteor. Soc.,112, 677–691.
Broyden, C. G., 1970: The convergence of a class of double rank minimization algorithms. Parts I and II. J. Inst. Maths. Appl.,6, 76–90.
Fletcher, R., 1970: A new approach to variable metric algorithms. Comput. J.,13, 312–322.
Goldfarb, D., 1970: A family of variable metric methods derived by variational means. Math. Comput.,24, 23–26.
Hiriart-Urruty, J.-B., and C. Lemaréchal, 1993: Convex Analysis and Minimization Algorithms. II: Advanced Theory and Bundle Methods. Vol. 306, Springer-Verlag, 346 pp.
Kiwiel, K. C., 1985: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, Vol. 1133, Springer-Verlag, 362 pp.
Kuo, Y.-H., X. Zou, and Y.-R. Guo, 1996: Variational assimilation of precipitable water using a nonhydrostatic mesoscale adjoint model. Part I: Moisture retrieval and sensitivity experiments. Mon. Wea. Rev.,124, 122–147.
Le Dimet, F. X., and O. Talagrand, 1986: Variational algorithms for analysis and assimilation of Meteorological observations: Theoretical aspects. Tellus,38A, 97–110.
Lemaréchal, C., 1977: Bundle methods in nonsmooth optimization. Proceeding of the IIASA Series, C. Lemaréchal and R. Mifflin, Eds., Pergamon Press, 79–103.
——, 1978: Nonsmooth optimization and descent methods. International Institute for Appl. Sys. Analysis, Laxenburg, Austria, 25 pp.
——, 1989: Nondifferentiable optimization. Optimization, G. L. Nemhauser et al., Eds., Handbooks in ORGMS, Vol. 1, Elsevier Science, 529–572.
——, and C. Sagastizabal 1997: Variable metric bundle methods: From conceptual to implementable forms. Math. Programming,76, 393–410.
Liu, D. C., and J. Nocedal, 1989: On the limited memory BFGS method for large scale optimization. Math. Programming,45, 503–528.
Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convection adjustment. J. Atmos. Sci.,21, 361–385.
Rabier, F., J.-N. Thepaut, and P. Courtier, 1998: Extended assimilation and forecast experiments with a four-dimensional variational assimilation system. Quart. J. Roy. Meteor. Soc.,124, 1861–1887.
Shanno, D. F., 1970: Conditioning of quasi-Newton methods for function minimization. Math. Comput.,24, 647–657.
Shor, N. Z., 1985: Minimization Methods for Nondifferentiable Functions. Springer-Verlag, 162 pp. (Translated from Russian by K. C. Kiwiel and A. Ruszczynski.).
Thépaut, J. N., and P. Courtier, 1991: 4-dimensional data assimilation using the adjoint of a multilevel primitive equation model. Quart. J. Roy. Meteor. Soc.,117, 1225–1254.
Tsuyuki, T., 1997: Variational data assimilation in the tropics using precipitation data. Part III: Assimilation of SSM/I precipitation rates. Mon. Wea. Rev.,125, 1447–1464.
Zou, X., 1997: Tangent linear and adjoint of “on–off” processes and their feasibility for use in 4-dimensional variational data assimilation. Tellus,49A, 3–31.
——, and Y.-H. Kuo, 1996: Rainfall assimilation through an optimal control of initial and boundary conditions in a limited-area mesoscale model. Mon. Wea. Rev.,124, 2859–2882.
——, I. M. Navon, M. Berger, P. K. H. Phua, T. Schlick, and F. X. LeDimet, 1993: Numerical experience with limited-memory quasi-Newton and truncated-Newton methods. SIAM J. Optim.,3, 582–608.
Zupanski, D., 1993: The effects of discontinuities in the Betts–Miller cumulus convection scheme on four-dimensional variational data assimilation in a quasi-operational forecasting environment. Tellus,45A, 511–524.
——, and F. Mesinger, 1995: Four-dimensional variational assimilation of precipitation data. Mon. Wea. Rev.,123, 1112–1127.