varenkf localisation for an unstructured grid
TRANSCRIPT
VarEnKF Localisation for an Unstructured Grid
Andreas Rhodin
German Weather Service
Internationsl Symposium on Data Assimilation, Munich 2014
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 1 / 24
1 DWD Global Forecast systemICON Model - Icosahedral Grid with local RefinementsGlobal Data Assimilation Setup
2 VarEnKF BasicsLocalisationTransformed Representation
3 Wavelet TransformationsDiscrete Wavelet transformation (DWT)Transformed Covariance MatricesCharacteristics of Wavelet TransformationsLocalisation in Wavelet Representation
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 2 / 24
DWD Global Forecast system
Current setup:I 20 km GME modelI provides boundary conditions to 7 km COSMO-EU
F provides boundary conditions to 2.8 km COSMO-DE
I 3 hourly cycling 3D-VarI experimental: 40 km LETKF with 40 members
PlanI 13 km ICON model (end of 2014)I 7 km local refinement (to replace current COSMO-EU)I provide boundary conditions to 2.8 km COSMO-DE
F with 40 member 2.8 km LETKF
I VarEnKF using 30 km LETKF with 40 membersI embedded 1D-Var mode for evaluation, QC, FG for nonlinear processes
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 3 / 24
ICON Model - Icosahedral Griddeveloped by DWD, MPI for Meteorology
main goals:I Unified modelling system for NWP and climate predictionI better conservation propertiesI flexible grid nesting in order to replace both GME and COSMO-EUI nonhydrostatic dynamical core for capability of seamless predictionI scalability and efficiency on multi-core systems
Grid generation is based on the icosahedron (unstructured grid):
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 4 / 24
ICON Model - Icosahedral Grid with local Refinements.
Effective grid spacing(distance betweengrid points):
∆x ≈ 5050n 2k
km
Example:
R3B7: n=3, k=7
1st subdivision by factor of 37 subdivisions by factor of 2
Grid spacing 13: km
Deterministic forecast will run with 13 km resolution
Shall become operational end of 2014
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 5 / 24
Global Data Assimilation SystemStatus: 3D-Var (operationally) + LETKF (experimental)
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME detGME det
LETKF
QC obs
obs
3D−Var
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 6 / 24
Global Data Assimilation SystemCurrently implemented: VarEnKF
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME ens
GME detGME det
LETKF
QC obs
obs
VarEnKF
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 7 / 24
VarEnKF Basics
Idea: Use the (flow dependent) ensemble background error correlationsin the deterministic variational analysis system.
P(b)3DVar → αP
(b)3DVar + β P
(b)EnKF
Goal:Provide flow dependence to the 3D-Var P(b)
Representation:
P(b)EnKF = XXT
Localisation:
Required to suppress noise of the sample covariance matrix.
P(b)EnKF = Cx ◦ XXT
Schur produxt ◦ with localisation matrix Cx
ensures that correlations become zero for distances > rloc .
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 8 / 24
Multiscale Localisation
Localisation length scales rloc should be different for differentsituations:
I rainy situationsI no rain / high pressure situations
I synoptic scale phenomenaI smaller (convective) scale phenomena
May become important for grid refinement areas
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 9 / 24
Subsequent 1d-examples for raw and localised sample covariancesare taken from the COSMO-DE (regular) 2.8 km grid
for illustrative purposes
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 10 / 24
Raw Sample covariance and correlation matrix
Temperature covariances and correlationsalong a latitude line (COSMO-DE level 29)
provided by a 40 member ensemble without localisation
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 11 / 24
Transformed Representation
Localisation in physical space is an ad hoc approach:I Cx ◦ XXT : Correlations at large distances are small.
Alternative: Localize in transformed representations:Z = Tr(X), X = Tr−1(Z)
Localisation in spectral space:I Cz ◦ ZZT : Correlations between different scales are small.I Corresponds to spatial smoothing of the correlation functions.
(FFT has been used for zonal averaging of NMC correlations)
Localisation in wavelet space:I Cz ◦ ZZT : Correlations between different scales
and at large distances are small.I Wavelet transformations can be regarded as transformations to a
hierarchy of grids.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 12 / 24
Discrete Wavelet transformation (DWT)
Fast hierarchical transform with operation count O(n).
coefficients (λ) of the gridded function
λ4,1 λ4,2 λ4,3 λ4,4 λ4,5 λ4,6 λ4,7 λ4,8 λ4,9 λ4,10 λ4,11 λ4,12 λ4,13 λ4,14 λ4,15 λ4,16
low pass high pass↓ ↓
λ3,1 λ3,2 λ3,3 λ3,4 λ3,5 λ3,6 λ3,7 λ3,8 γ3,1 γ3,2 γ3,3 γ3,4 γ3,5 γ3,6 γ3,7 γ3,8
low pass high pass↓ ↓
λ2,1 λ2,2 λ2,3 λ2,4 γ2,1 γ2,2 γ2,3 γ2,4 γ3,1 γ3,2 γ3,3 γ3,4 γ3,5 γ3,6 γ3,7 γ3,8
low pass high pass↓ ↓
λ1,1 λ1,2 γ1,1 γ1,2 γ2,1 γ2,2 γ2,3 γ2,4 γ3,1 γ3,2 γ3,3 γ3,4 γ3,5 γ3,6 γ3,7 γ3,8
low high↓ ↓
λ0,1 γ0,1 γ1,1 γ1,2 γ2,1 γ2,2 γ2,3 γ2,4 γ3,1 γ3,2 γ3,3 γ3,4 γ3,5 γ3,6 γ3,7 γ3,8
wavelet (γ) and scaling function (λ) coefficients
Coefficients of the transformed vector correspond to the average value(λ) and to the deviations (γ) on different scales.
Inverse transform is a fast hierarchical transform as well.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 13 / 24
Wavelet transformed covariance matrices
Wavelet transformations applyindependently to each row andcolumn of a matrix.
Covariances of phenomena at thesame scale are represented by thediagonal blocks.
Covariances of phenomena atdifferent scale are represented byoff-diagonal blocks.
Covariances of phenomena atnearby locations are representedby coefficients in the vicinity ofthe ‘branches’ (diagonals of theblocks).
Only these coefficients areconsiderably different from zero.
0
ψ
ψ
ψψ
ψ ψ ψ ψϕϕ
0 0 1 2 3
0
1
2
3
Block structure of wavelet transformedmatrices
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 14 / 24
Transformed Covariance Matrices
Sample corre-lation matrix inwavelet repre-sentation
.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 15 / 24
Characteristics of Wavelet Transformations
Example: Daubechies-8 wavelet basis functionsThe wavelet (ψ) and scaling (ϕ)basis functions are implicitlydefined by the high and low-passfilter coefficients.
Direct wavelet transform W:
f =∑
k λ0,k ϕ0,k +∑
j,m γj,m ψj,m
Inverse Wavelet transform W−1:
λj,k = 〈f , ϕ̂j,k〉γj,m = 〈f , ψ̂j,m〉
For orthogonal wavelettransformations the direct basisfunctions (ψ,ϕ) and duals (ψ̂, ϕ̂)are the same.
Basis functions (ϕ, ψ) for the orthogo-nal Daubechies-8 wavelet transform.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 16 / 24
Characteristics of Wavelet Transformations
The wavelet transform is not unique (as FFT) but many different choicesfor low pass and high pass filter functions exist.
Different desiriable properties cannot be achieved at the same time:
Compact support (for fast transform)
Smoothness (for scale separation)
Orthogonality or bi-orthogonality
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 17 / 24
Characteristics of Wavelet Transformations
Our Choice (so far)
Use Frames:Number of variables in transformed representation is larger.allows smoother basis functions.
Use the Lifting Scheme:allows to define wavelets with desired properties on unstructured grids,splits the wavelet transform into a seqhence of sinmpe operationswhich can easyly be inverted.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 18 / 24
Lifting Scheme / Frame
This kind of wavelet transform reduces to a very simple hierarchicalalgorithm:
1 Low pass filter: Average the field (to a coarser grid)
2 High pass filter: Re-interpolate the field to the fine grid and
3 subtract the result from the original one to get the low high passfiltered component.
Iterate the algorithm with the low pass filtered field.
This algorithm can be easyly applied to the ICON gridwhich is generated by a hierachy of grid refinements.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 19 / 24
Localisation in Wavelet RepresentationGoal: only keep the entries close to the diagonalsand the ’branches’ of the wavelet transformed cor-relation matrix:
Close to the diagonals:spatial correlations between phenomena ofthe same scale at slightly differentlocations.
I localise as usual with (now scaledependent) prescribed localisationradius.
On the branches: correlations betweendifferent scales (at the same location)
I perform a similar localisation inbetween scales
Localisation in the 2 ’directions’ seperate.
Same approach as used in (Buehner MWR 2012)but for a different wavelet basis.
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 20 / 24
Sample Covariance Localised in Wavelet Representation
Temperature covariances and correlationsleft: Raw sample
right: reconstructed after localisation in wavelet representation
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 21 / 24
Conclusions
The wavelet lifting scheme provides an efficient method for scaleselective localisation on the unstructured ICON grid.
Thank you for your attention
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 22 / 24
Sample Covariances
Ensemble x = xmodel − xtrue, true covariance: B = E{x xT
}Unbiased estimate: sample covariance
S =1
N − 1
N∑m=1
(x (m) − x̄
)(x (m) − x̄
)T, E {S} = B
Variance of sample covariance coefficients (Gaussian errors)
σ2ij ≡ E
{(Sij − Bij)
2}
=1
N − 1
[BiiBjj + (Bij)
2]
Error of off-diagonal coefficients dominated by diagonal elements!
EnDA systems rely on Covariances and thus requireI large ensemble sizes (40 . . . 200)I methods to suppress correlations known to be small (localisation)
Andreas Rhodin (DWD) VarEnKF for ICON 2014-02-26 24 / 24