© crown copyright met office does the order of the horizontal and vertical transforms matter in the...
TRANSCRIPT
© Crown copyright Met Office
Does the order of the horizontal and vertical transforms matter in the representation of an operational static covariance model in global atmospheric DA? Marek Wlasak and Mike Cullen
Outline
Give the argument for and properties of swapping the order of the vertical and horizontal transforms in the static covariance model.
Show benefits of swapped order
Summarise
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Argument and properties for swapping the order of the horizontal and vertical transforms
Static covariances are still important
• Hybrid DA methods rely on a static covariance model (Bs) to give sufficient degrees of freedom.
• Static part still contributes to over 50% of the whole forecast error. This is not going to change much in the near future unless there is a huge increase in the number of ensemble members used. (Reference: Hidden Error Variance Theory, Bishop 2013)
• So modelling of static covariances is important.
Rationale for conserving the properties of the training data
• We need to calibrate a covariance model with training data from an ensemble properly representing the forecast error.
• Ideally we would like to get to a position where the covariance model conserves some key properties and the implied background error covariance faithfully represents the training data.
• This philosophy should in the long run help in bootstrapping to provide a good covariance file.
• It also means that the emphasis moves away from tuning a covariance model, to getting the training data right.
Conservation of variance
• The total kinetic energy on each vertical level is determined by :
• the power spectra of the stream-function ψ’ and velocity potential χ’. ( It holds variance and gradient information.)
• If the power spectra of ψ’, χ’ are conserved so will the total kinetic energy.
• This will have a knock on effect on horizontal length scales of the balanced pressure and temperature.
Background covariance structure.
• TvThTp (swapped)
• Latitude dependence built into parameter transform only.
• Global variances and power spectra of model variables conserved.
• ThTvTp (current)
• Latitude dependence built into vertical and parameter transform.
• Power spectra not conserved for model variables.
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Results: Background error covariance comparison
Swapped order
ConserveKE
Current orderObserved (above) andImplied u (below)
Current orderObserved (above) andImplied T (below)
Not conservedfor each modelLevel.
Current orderObserved (above) andImplied P (below)
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Trial results
Non-hybrid trials
Two trials were run with UM at N320 and VAR at N216 and N108.
The main difference is seen with an increase in the size of the theta and pressure increments, due to the increase in associated background errors.
There is also an increase horizontal length scale in pressure and theta.
Non-hybrid trials:Summer Case+ 0.75
Severe hit on fit toanalyses
Model moreactive
Non-hybrid trials:Winter Case
Fit to obs+ 1.2
Summary:
•It is clear that swapping the transform order has a big effect on results when tested without the flow dependence of the hybrid. Neutral results have so far been obtained in hybrid trials.
• The biggest benefit is that the background error is now represented by the training data. It is expected that an improved choice of training data, consistent with the hybrid error modes, will help.
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Additional slides.
Overview of the current generation of static forecast error covariance model in the global domain. (1)
Vertical transform
Calibration Statistics
U’
V’
P’
Θ’
ρ’
qT’
Ψ’
Χ’
Ap’
μ’
Horizontal transform
A priori structure fromphysical equations etc
Parameter transform
in/out of uncorrelated
variables
Covariance Model
Linear balance -----> Vertical RegressionHydrostatic balance _____________________ |New humidity transform| -----------------------------------Eqn of state
Vertical modes (Eigenvectors generalised)Eigenvalues
Spherical harmonicbasis SQRT(Power
spectra) multiplied divided
T transform
U transform
B = U UT
Tp
Up
Alternate generation of static forecast error covariance model in the global domain
Calibration Statistics
U’
V’
P’
Θ’
ρ’
qT’
Ψ’
Χ’
Ap’
μ’
A priori structure fromPhysical equations etc
Parameter transform
in/out of uncorrelated
variables
Covariance Model
Spherical harmonicbasis
SQRT(Vertical Covariancefor each total wavenumber)or inverse
T transform
U transformB= U UT
Linear balance -----> Vertical RegressionHydrostatic balance _________________Eqn of state New humidity transform| ---------------------------
Background covariance structure.
• The static covariance model is an approximation to the true forecast error covariance B and is constructed by making a number of prior assumptions.
• One choice is the order of the horizontal Uh and vertical transforms (as they do not commute). i.e. (whether the vertical transform is applied in spectral space or not.)
Ie. B1 = UpUvUh (UpUvUh)T /= UpUhUhv (UpUhUhv)T = B2
Where :
• Up is the parameter transform
• Uh is the horizontal transform
• Uv is the vertical transform is grid-point space.
• Uhv is the vertical transform is done in spectral space.
Hybrid trials
The benefit is not seenin hybrid trials
Fit to observationsneutral
Overview of the FUTURE generation of static forecast error covariance model in the global domain. (1)
Calibration Statistics
U’
V’
P’
Θ’
ρ’
qT’
Ψ’
Χ’
Ap’
μ’
Horizontal /vertical transform combined
A priori structure fromPhysical equations etc
Parameter transform
in/out of uncorrelated
variables
Covariance Model
PV transform Ekman balance?
_____________________ |Additional balances? | -----------------------------------
Spherical harmonicbasis SQRT(Vertical Covariance
for each total wavenumber)or inverse.
T transform
U transformB= U UT
Adaptive grid1D/3D
Monge –Ampère Eq
Basic overview.
•Code comprises of a number of small Fortran programs being called from ksh scripts
•Most important script for the user is the Launcher script.• Expect all switches that the non-developer uses to be accessed from there.
•Glue between modules• Environment variables some from Launcher• NetCDF files – including its header• Small text files that give the paths to groups of NetCDF files.
Basic overview.
•Code comprises of a number of small Fortran programs being called from ksh scripts
•Most important script for the user is the Launcher script.• Expect all switches that the non-developer uses to be accessed from there.
•Glue between modules• Environment variables some from Launcher• NetCDF files – including its header• Small text files that give the paths to groups of NetCDF files.
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Future Directions in Global
Overview of the current generation of static forecast error covariance model in the global domain. (1)
Vertical transform
Calibration Statistics
U’
V’
P’
Θ’
ρ’
qT’
Ψ’
Χ’
Ap’
μ’
Horizontal transform
A priori structure fromPhysical equations etc
Parameter transform
in/out of uncorrelated
variables
Covariance Model
Linear balance -----> Vertical RegressionHydrostatic balance _____________________ |New humidity transform| -----------------------------------Eqn of state
Vertical modes (Eigenvectors generalised)Eigenvalues
Spherical harmonicbasis SQRT(Power
spectra) multiplied divided.
T transform
U transform
B = U UT
Tp
Up
To show how vertical structure is a function of horizontal total wave-number.
Look at the vertical covariances of control variables for specific horizontal total wavenumber from 300 samples of unpacked ECMWF forecast differences.
Vertical covariances as a function of total wave-number : from 300 samples of unpacked EC data.
PSIWavenumber1
PsiWavenumber 5
PsiWavenumber 10
PsiWavenumber 30
PsiWavenumber 100
Psiwavenumber
200
Psiwavenumber 300
Chiwavenumber 1
Chi
wavenumber 5
Chiwavenumber
10
Chiwavenumber
30
Chiwavenumber
100
Csiwavenumber
200
Csiwavenumber
300
Unbalancedpressure
Wavenumber
0
Unbalancedpressure
Wavenumber 1
Unbalanced pressure
Wavenumber
5
Unbalancedpressure
Wavenumber
10
Unbalancedpressure
Wavenumber
30
Unbalanced pressure
Wavenumber
100
Unbalancedpressure
Wavenumber 200
Unbalanced pressure
Wavenumber
300
New humidity control variable
Wavenumber
0
New humidity control variable
Wavenumber 1
New humidity control variable
Wavenumber 5
New humidity control variable
Wavenumber
10
New humidity control variable
Wavenumber
30
New humidity control variable
Wavenumber
100
New humidity control variable
Wavenumber
200
New humidity control variable
Wavenumber 300