1. No category

'Wavelets': what and why ?
"!$# % &')( # %&')( +* &,-/.
1. The background error covariance matrix
(background background)
Bayes' theorem (Gaussian unbiased errors):
P x\y
P x P y\x
1
B
B
1
P x x x T B x x 2
1
1
P y\x y h x
T R y h x
2
B is the error covariance matrix
The structure functions
• The columns (or rows) of B
• Bx does a convolution by the structure functions
• Can be found in a single obs. test in Var.:
analysis inc.
x
where
K
single obs.
then
y
K y
T
y
H
T
xl
y R
HxB
K
l1
HxB
T
BH HBH
HBH
R y
y
2 1
y
B 2
kk
Blk Bkk
Blk
1
2
0 0 0 0 1 0 0 0
Bkk
'Wavelets': what and why ?
(1)
kth element
2. How is B computed?
A. Kalman filtering (1st choice)
B t PA t t T t
t
Mt t PA t t Mt t
Q t t I K t t H t t B t
t Resolution
432 325 38 4 c.v.
216 163 38 4 c.v.
96 73 38 4 c.v.
I
ra
p
m
Matrix elements
4 6 1014
2 9 1013
1 1 1012
ab
c
i
t
c
le
B. Estimate covariance statistics fully (2nd choice)
B would like x
settle for x
I
x x
T
x xt
x48 x24 (NMC e.g.)
ra
p
m
ab
c
i
t
c
le
'Wavelets': what and why ?
(2)
C. Background error covariance modelling (third choice)
•
•
•
•
Manageable number of parameters
Capture essential features only
Accept approximations
Exploit symmetries
Assimilate using control parameters
x
Uv
• Control parameters are uncorrelated, Bv v v T I
• Covariance model contained inside transformation, U
T
B UU
• Also helps to precondition the problem
x -space
v -space
• Assume U has form
U
T
z
UpUvUh
U
1 ThTvTp
Tv
Uv
'Wavelets': what and why ?
Th
Uh
(3)
l m
3. Implied covariances (for a given parameter)
B
Bp
UU
T
1/2 M
HORIZONTAL
1/2 1
h Fh
T
UvUhUh Uv
T-transform
VERTICAL
T
Z
U-transform
Z 1M1/2 Fh
1/2
h
UT -transform
T
M1/2 Z
1/2 T
h Fh
General assumptions/remarks:
• Climatological error statistics used synoptically
• Errors in parameters are uncorrelated
• Horizontal structure functions are homogeneous and
isotropic
• Horizontal length scales independent of height
• Order of transformations:
• vertical covs. f , but f l m
• horizontal covs. f , but f z (explicitly)
• Implied structure functions are nearly separable
• Bp 0 0 z z0 f r g z
• (exactly separable if Uv independent of horiz. pos.)
• Problems in capturing tropopause height and
baroclinic structures
'Wavelets': what and why ?
(4)
4. Real structure functions are not separable
Horizontal scales are associated with corresponding vertical scales
(from Ingleby, 2001)
'Wavelets': what and why ?
(5)
Further evidence ...
Horizontal correlation length for T is shorter than that for
T
h
h
(from Ingleby, 2001)
If short (long) horizontal lengths are associated with short (long)
vertical lengths then,
T
v
v
which is true under hydrostatic balance,
d
RT
d ln p
g
'Wavelets': what and why ?
(6)
5. Innovative 'quick' fixes
• Use alternative parameters
• E.g. PV/anti-PV to improve representation of balance in
parameter transform (Cullen/Payne/Wlasak/Roulstone/
Bannister)
• Geostrophic co-ordinate transform
• Realistic treatment of fronts
v /f y
g
G
• xG
xR
•T ThTvTgTp
yR
u /f
g
• Errors of the day
• Extra fit to 'bred' modes
• Extra 'alpha' control variables, v
•J J
J J v ½ T E 1 • Replacement for vertical\horizontal covariance model
• 'WAVELET' (actually 'WAVEBAND')
TRANSFORM
'Wavelets': what and why ?
(7)
6. A waveband summation (WS) transform
The transform
Current U-transform
'New' WS U-transform
UpUvUh
Up
J
j 0 UvjUh
2
j
Vertical transform has position and scale dependence
J
Up
Uvj
Uh
2
j
z z l m
l m
z z l m l m z p z
p l m
p l m
p j
0
u
v
A
A
A
A
The wavebands
1.0
2
1
2
2
2
3
2
4
2
5
(a) Oversized last bandpass function
2
j
0.5
0.0
1.0
2
1
J
2
2
2
3
2
4
2
5
(b) Undersized last bandpass function
2
6
2
j
0.5
0.0
0.0
100.0
200.0
300.0
total wavenumber, n
400.0
'Wavelets': what and why ?
500.0
(8)
600.0
j
0
2
j
1
Are structure functions in the current transform separable?
B
UU
x
Uv
x r z
d Z z
UvUhv
d Z z M
1
x r z
1
T
1/2
h
dr
M
1/2
1/2
1
Z M
r
1/2
h
1
d Z z
M
x r z
1
d Z z
f z
g r
1/2 M
k v k k
r
1/2
h
r 1/2
1/2
h
k
r Z z
1/2
1
r Z z
1/2
h
k M
1
k Fh r k 1/2 Z z
1
M
z
dkFh r k If M
k Fh r k M
1/2
r z
dkFh r k r
1/2
h
1/2
h
dkFh r k M1/2 r
v
k Fh r k dzM1/2 r Z 1 z x r z
d Z 1 z
1/2
h
r dkFh r k Fh
Let x r z
x r z
1/2
dkFh r k UvjUh
2
j
h
k Fh r k The WS scheme?
J
Up
j 0
J
x r z
d Z
1
z
Mj
Z
2
j
1
z
"
j j ! 0
dkFh r k h
k
'Wavelets': what and why ?
(9)
n
2
j
n Fh r k Remarks on the waveband summation (WS) transform
• WS is the simplest of many proposed transforms
• Some others require multiple control vectors
• (current length
No. of bands)
• This WS transform reduces to current scheme if band
index dropped
• Implied structure functions are not separable
• (even if Uv independent of horiz. pos.)
• The
2
j
are not mutually orthogonal
• No exact inverse transform
• Need approximations when calibrating
• Increase scale resolution, decrease position resolution
• Other documents
• Andrews P., Covariance development strategy: a
three point plan ... (Var. working paper 20)
• Ingleby N.B., Options for improving and modelling
background errors ...
• Andrews P., Lorenc A., On introducing a multi-scale
waveband summation covariance model ... (Var.
working paper 19)
• Bannister R., ...a description of the proposed
waveband summation transformation (available via
www.met.rdg.ac.uk/~ross/DARC/DataAssim.html)
'Wavelets': what and why ?
(10)