flow dependent jb in a global grid-point 3d-var...flow dependent jb in a global grid-point 3d-var r....
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![Page 1: Flow Dependent Jb in a global grid-point 3D-var...Flow Dependent Jb in a global grid-point 3D-var R. J. Purser, W.-S. Wu, M. Pondeca, R. Treadon, D. Parrish, D. Kleist, J.C. Derber](https://reader030.vdocument.in/reader030/viewer/2022040914/5e8ba246be1ac032ab52fa14/html5/thumbnails/1.jpg)
Flow Dependent Jb in a global grid-point 3D-var
R. J. Purser, W.-S. Wu, M. Pondeca, R. Treadon, D. Parrish, D. Kleist, J.C.
DerberEnvironmental Modeling Center
NCEP/NWS/NOAA
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Overview• Most current implementations of global
variational analysis systems use spectrally defined background error covariances and analysis variables
• The improved specification of the background error covariances is an ongoing goal of data assimilation groups.
• NOAA/NWS/NCEP/EMC is attempting to improve our system by defining the background error covariance and analysis variables in grid-space
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Analysis variable and Background error definition
• Spectral– Simple definitions of
background error covariance straightforward
– Consistent with spectral model
– Poles easy to handle– Computational cost?
• Grid space– Inhomogeneous –
anisotropic background errors less complicated (but still not trivial)
– Local definition of errors– Easy to distinguish between
land-sea, tropics, mid-latitudes, etc.
– Consistency between global & regional systems
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Grid-point space background • Two major considerations
– How to computational perform the background error computations
• Recursive filters
– How to define the appropriate background errors
• Ongoing research
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Recursive filters• Closely related to diffusion operator methods
(Derber and Rosati, 1989 and Weaver and Courtier, 2001)
• Most recent references– Purser, Wu, Parrish and Roberts, 2003: Numerical
Aspects of the application of recursive filters to variational statistical analysis, Part I and II, Mon. Wea. Rev.
– Wu, Purser and Parrish, 2002: Three-Dimensional variational analysis with spatially inhomogeneous covariances, Mon. Wea. Rev.
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Recursive Filters• 3-D representations from simple 1-D filters• General 1-D form – two steps
∑=
−+=n
j
jijii qpq1
αβ
∑=
++=n
j
jijii sqs1
αβ
j
n
j
a∑=
−1 = 1
β
Advancing step
Backing step
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Recursive Filters• First order (n=1)
• Note recursive filter is self-adjoint• Produces a quasi-Gaussian filter response
11( −= +)− iii qpq αα11( ++)−= iii sqs αα
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1-D Response
n=1 n=2 n=4
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2-D isotropic application• Successive application in x then y direction
n=1 n=2
n=4 n=1x4
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Laplacian of 2-D
n=2 n=4
n=6 Gaussian
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Comments on Recursive Filters• 4th Order minimum• Non-Gaussian isotropic shapes created by
adding Gaussians• Fat-tailed error covariances
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Inhomogeneous anisotropic covariances on a globe
• A bit more complicated.– Smoothing must be done not only along x-y-z
directions but along other directions as well –triad – hexad algorithms.
– Transitions can create additional problems (solvable but at a cost)
– polar problem
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Anisotropic Model
qlf=20% Isotropic Model
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Anisotropic Model
qlf=20% Isotropic Model
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Transition problem3 color triad
no bridging functionone iteration
transition scale= .250
4 color triad2nd degree bridging function
one iterationtransition scale= .250
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Incorporation in NCEP global system
• Version of NCEP global system developed identical to current experimental spectral system except:– Background term estimated using recursive filters– Modified error statistics (due to recursive filter form)– Balance equation– Minimization algorithm– No divergence tendency equation constraint (not
currently used in spectral version)
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Incorporation in NCEP global system
• Initial version attempting to produce similar to spectral version.
• Inner loop performed completely on linear Gaussian grid.
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Background Term• Initially assumes Background of the form:
• where includes the vertical component of the recursive filter and the balance relationships. This part of the background term is incorporated into the definition of the analysis variables
• and represents three horizontal applications of the recursive filters
2/132 )( 12/
VHH BBBBBB H
T
V ++=2/1
VB
321HHH BBB ++
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Background Term• The length scales used in and in the
balance equation are calculated using the NMC method
• The length scales in are ½ those in and are ¼ those in this gives fat-tail distribution
• Vertical error terms are defined for each latitude• Horizontally homogeneous in physical space not
grid space
1HB
2/1VB
2HB
1HB
3HB
1HB
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Background Term• Balance equation
– function of latitude
ψAT =
ψχ C=
ψDsp =
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Polar recursive filter problem• Define two polar stereographic grids over north
and south poles• Interpolate values from Gaussian grid to polar
grids ( using blending region)• Perform recursive filter on three grids (2 polar + 1
rest of globe• recombine using adjoint of interpolations and
blending
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Minimization changes• Since difficult to square-root
use Derber and Rosati (1989) minimization algorithm (only requires B not B1/2)
• Requires saving 2 N component vectors (x, B-1x)
321HHH BBB ++
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Single point ob45N180W T1000hPa
SSI GSI
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Single point ob45N180W T1000hPa
SSI GSI
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Single point ob45N180W T1000hPa
GSISSI
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T analysis increment GSI~500hPa
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T analysis increment SSI~500hPa
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T GSI-SSI~500hPa
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T GSI-SSI 180W
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Final comments• NCEP is exploring the grid point form of the
background error covariance (GMAO collaboration) to allow local definition of background error structure.– Need to complete modifications for parallel processing
efficiency– Test current version to replace spectral version– Modification to work on regional and global domains– Inclusion of situation dependent covariances
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Final comments• Initial experiments (with some bugs) showed
promising results (especially in the tropics)• Defining the local background error structure
remains an ongoing research problem. – Possibilities – background structure based, ensemble
based, etc.– Estimation and validation of statistics based on
innovations• Expectation is that improved specification of
background error will result in largest enhancement of analysis quality.