variance for the error covariance

horizontal scale for the error covariance

vertical scale for the error covariance

vertical scale profile for the error covariance

projection of stream function onto balanced temperature

projection of streamfunction onto velocity potential

projection of streamfunction onto balanced ln(ps)

4-6 Jan 2005GSI User orientation meeting Minimization and Preconditioning John C. Derber NOAA/NWS/NCEP/EMC

4-6 Jan 2005GSI User orientation meeting Basic structure Outer iteration – Most of outer iteration in routine glbsoi – Accounts for small nonlinearities which are complex to include and changes in quality control (especially for radiances) – If linear and no changes in QC should be same as doing more inner iterations Inner iteration – Solves partially linearized version of objective function – Mostly in routine pcgsoi

4-6 Jan 2005GSI User orientation meeting Outer iteration Currently background error cannot change in outer iteration (due to preconditioning in inner interation) For the current problem, we find that 2 outer iterations appears to work reasonably well except for precip. Often we run three outer iterations so that we can see the fit to the obs. at the end of the second outer iteration

4-6 Jan 2005GSI User orientation meeting Outer iteration Major operations within outer iteration – Read current solution (read_wrf_…, read_guess) – Read in statistics (prewgt, first outer iteration only) – Call setup routines (setuprhsall) – Call pcgsoi (solve inner iteration) – Write solution Difference from background kept in xhatsave and yhatsave (= B -1 xhatsave).

4-6 Jan 2005GSI User orientation meeting Inner iteration Solves partially linearized sub-problem Uses preconditioned conjugate gradient Stepsize calculation exact for linear terms and approximate for nonlinear term Current nonlinear terms (if not using variational QC) – SSM/I wind speeds – Precipitation – Penalizing q for supersaturation and negative values

4-6 Jan 2005GSI User orientation meeting Inner iteration - algorithm J = x T B -1 x + (Hx-o) T O -1 (Hx-o) (assume linear) define y = B -1 x J = x T y + (Hx-o) T O -1 (Hx-o) Grad J x = B -1 x +H T O -1 (Hx-o) = y + H T O -1 (Hx-o) Grad J y = x + BH T O -1 (Hx-o) = B Grad J x Solve for both x and y using preconditioned conjugate gradient (where the x solution is preconditioned by B and the solution for y is preconditioned by B -1 )

4-6 Jan 2005GSI User orientation meeting Inner iteration - algorithm Specific algorithm x 0 =y 0 =0 Iterate over n Grad x n = y n-1 + H T O -1 (Hx n-1 -o) Grad y n = B Grad x n Dir x n = Grad y n + β Dir x n-1 Dir y n = Grad x n + β Dir y n-1 x n = x n-1 + α Dir x n (Update xhatsave as well) y n = y n-1 + α Dir y n (Update yhatsave as well) Until max iteration or gradient sufficiently minimized

4-6 Jan 2005GSI User orientation meeting Inner iteration - algorithm intall routine calculate H T O -1 (Hx-o) bkerror multiplies by B dprod x calculates β and magnitude of gradient stpcalc calculates stepsize Note that consistency between x and y allows this algorithm to work properly

4-6 Jan 2005GSI User orientation meeting Inner iteration – algorithm Estimation of β β = (Grad x n -Gradx n-1 ) T Grad y n / (Grad x n -Gradx n-1 ) T Dir x n Calculation performed in dprodx Also produces (Grad x n ) T Grad y n

4-6 Jan 2005GSI User orientation meeting Inner iteration – algorithm Estimation of α (the stepsize) The stepsize is estimated through estimating the ratio of contributions for each term α = ∑a ∕ ∑b The a’s and b’s can be estimated exactly for the linear terms. For nonlinear terms, the a’s and b’s are estimated by fitting a quadratic using 3 points around an estimate of the stepsize The estimate for the nonlinear terms is reestimated using the stepsize for the first estimate

4-6 Jan 2005GSI User orientation meeting Inner iteration – u,v Analysis variables are streamfunction and velocity potential u,v needed for int routines u,v updated along with other variables by calculating derivatives of streamfunction and velocity potential components of dir x and creating a dir x (u,v)