Download - variance for the error covariance
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variance for the error covariance
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horizontal scale for the error covariance
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vertical scale for the error covariance
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vertical scale profile for the error covariance
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projection of stream function onto balanced temperature
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projection of streamfunction onto velocity potential
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projection of streamfunction onto balanced ln(ps)
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GSI User orientation meeting4-6 Jan 2005
Minimization and Preconditioning
John C. DerberNOAA/NWS/NCEP/EMC
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GSI User orientation meeting4-6 Jan 2005
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
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GSI User orientation meeting4-6 Jan 2005
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
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GSI User orientation meeting4-6 Jan 2005
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-1xhatsave).
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GSI User orientation meeting4-6 Jan 2005
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
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GSI User orientation meeting4-6 Jan 2005
Inner iteration - algorithm• J = xTB-1x + (Hx-o)TO-1(Hx-o) (assume linear)• define y = B-1x • J = xTy + (Hx-o)TO-1(Hx-o) • Grad Jx = B-1x +HTO-1(Hx-o) = y + HTO-1(Hx-o)• Grad Jy = x + BHTO-1(Hx-o) = B Grad Jx
• 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)
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GSI User orientation meeting4-6 Jan 2005
Inner iteration - algorithmSpecific algorithmx0=y0=0Iterate over n
Grad xn = yn-1 + HTO-1(Hxn-1-o) Grad yn = B Grad xn
Dir xn = Grad yn + β Dir xn-1
Dir yn = Grad xn + β Dir yn-1
xn = xn-1 + α Dir xn (Update xhatsave as well) yn = yn-1 + α Dir yn (Update yhatsave as well)
Until max iteration or gradient sufficiently minimized
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GSI User orientation meeting4-6 Jan 2005
Inner iteration - algorithm
• intall routine calculate HTO-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
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GSI User orientation meeting4-6 Jan 2005
Inner iteration – algorithm Estimation of β
• β = (Grad xn-Gradxn-1)T Grad yn/ (Grad xn-Gradxn-1)T Dir xn
• Calculation performed in dprodx• Also produces (Grad xn)T Grad yn
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GSI User orientation meeting4-6 Jan 2005
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
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GSI User orientation meeting4-6 Jan 2005
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)