Minimizationalgorithmsaresingularcomponentsinfour-dimensionalvariationaldataassimilation(4DVar).Inthispaper,theconvergenceandapplicationoftheconjugategradientalgorithm(CGA),whichisbasedontheLanczositerativealgorithmandtheHessianmatrixderivedfromtangentlinearandadjointmodelsusinganon-hydrostaticframework,areinvestigatedinthe4DVarminimization.First,theinfluenceoftheGram-SchmidtorthogonalizationoftheLanczosvectorontheconvergenceoftheLanczosalgorithmisstudied.TheresultsshowthattheLanczosalgorithmwithoutorthogonalizationfailstoconvergeaftertheninthiterationinthe4DVarminimization,whiletheorthogonalizedLanczosalgorithmconvergesstably.Second,theconvergenceandcomputationalefficiencyoftheCGAandquasi-Newtonmethodinbatchcyclingassimilationexperimentsarecomparedonthe4DVarplatformoftheGlobal/RegionalAssimilationandPredictionSystem(GRAPES).TheCGAis40%morecomputationallyefficientthanthequasi-Newtonmethod,althoughtheequivalentanalysisresultscanbeobtainedbyusingeithertheCGAorthequasi-Newtonmethod.Thus,theCGAbasedonLanczositerationsisbetterforsolvingtheoptimizationproblemsintheGRAPES4DVarsystem.
