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Scale-dependent localization in ensemble-variational data assimilation: Application in global and convective-scale systems Jean-François Carona, Étienne Arbogastb, Mark Buehnera, Thibaut Montmerleb, Yann Michelb and Benjamin Ménétrierb
aECCC bCNRM/Météo-France
Outline
1. An approach to improve spa2al covariance localiza2on in EnVar : scale-‐dependent localiza2on (SDL)
2. Applica2on in two EnVar-‐based DA systems a) A simplified version of ECCC’s global opera2onal system b) Meteo-‐France AROME convec2ve-‐scale system (R&D version)
Localisation
• Spa2al covariance localiza2on is essen2al to obtain useful analyses with “small” ensembles (a 256-‐member ensemble is s2ll "small"!).
• Currently, ECCC's EnVar uses simple localiza2on of ensemble covariances, similar to EnKF: single length scale in both horizontal and ver2cal localiza2ons based on Gaspari and Cohn (1999) 5th order piecewise ra2onal func2on.
• Comparing various NWP studies, seems that the best amount of horizontal localiza2on depends on applica2on/resolu2on:
o convec2ve-‐scale assimila2on: ~10km o mesoscale assimila2on: ~100km o global-‐scale assimila2on: ~1000km – 3000km (2800km at ECCC)
A one-‐size-‐fits-‐all approach for localiza2on does not seem appropriated for analysing a wide range of scales.
Scale-dependent localisation (SDL)
Defini2on: Simultaneously apply appropriate (i.e. different) localiza2on to different range of scales.
• The approach can be applied to both horizontal and ver2cal localiza2on but this presenta2on will only focus on horizontal-‐scale-‐dependent horizontal localiza2on.
• Pros: • Seems appropriated for mul2-‐scale analysis. • In limited-‐area: Could avoid the need of mul2-‐step or large-‐scale
blending approaches.
• Cons: • Adds more parameters to tuned. • Increases the cost of the analysis step (at least in our formula2on).
Horizontal Scale Decomposition
Large scale
Medium scale
Small scale
2000 km 10000 km 500 km
Filter response func2ons for decomposing with respect to 3 horizontal scale ranges
Horizontal Scale Decomposition
Full Large scale
Small scale Medium scale
Perturba2ons for ensemble member #001 – Temperature at ~700hPa
0
-‐2
+2
0
-‐2
+2
Horizontal Scale Decomposition
Homogeneous horizontal correla2on
length scales
Large scale
Medium scale
Small scale
Full
6-‐h temperature perturba2on from 256-‐
member EnKF hP
a
km
Horizontal scale-‐dependent localiza2on leads to (implicit)…
ver2cal-‐level-‐dependent horizontal localiza2on
Horizontal Scale Decomposition
Waveband integrated variances
Large scale Medium scale
Small scale
6-‐h perturba2on from 256-‐member EnKF
hPa
Horizontal scale-‐dependent localiza2on leads to (implicit)… variable-‐dependent horizontal
localiza2on
T
All the scales
log(q)
• The original ensemble covariances with localiza2on
LeeB !Tkk
kL ∑=
The basic idea (from Buehner and Shlyaeva, 2015, Tellus)
k: member index
Scale-dependent localisation (SDL)
• With a scale-‐decompose ensemble
LeeB !∑∑∑=1 2
2,1,j j
Tjk
kjkL
• With scale-‐dependent localiza2on
kjjk eFe =,F: filtering step j: scale index ∑=
jjkk ,ee
2,11 2
2,1, jjj j
Tjk
kjkSDL LeeB !∑∑∑=
EnVar with B1/2 preconditioning (ECCC)
• Analysis increment computed from control vector (B1/2 precondi2oning) using:
( )∑∑=Δk j
kjjk ξLex 2/1, !
( )∑=Δk
kk ξLex 2/1!
• Varying amounts of smoothing applied to same set of amplitudes for a given member
Current (one-‐size-‐fits-‐all) Approach
Scale-‐dependent Approach (as in Buehner and Shlyaeva, 2015, Tellus)
where ek,j is scale j of normalized member k perturba2on
k: member index j: scale index
k: member index
Page 11
• Analysis increment computed from control vector (B precondi2oning) using:
( )∑ Δ=Δk
ikko )( xeLex !!
• Direct applica2on to the above formula2on
Current (one-‐size-‐fits-‐all) Approach
Scale-‐dependent Approach
j: scale index
k: member index
EnVar with B preconditioning (Meteo-France)
( )∑∑∑ Δ=Δ1 2
2,2,11, )(j j k
ijkjjjko xeLex !!
• Reformula2on using B1/2 BT/2
( )∑∑=Δk j
kjjko ξLex 2/1, ! ∑ Δ=
jijk
Tjk )( ,2/ xeLξ !
Application in a global EnVar system: ECCC’s Global Determinisitic
Prediction System • 256 ensemble member @ 50km • Localiza2on in spectral space • Determini2c forecast @ 25 km
Scale-dependent covariance localization
Normalized temperature increments (correlation-like) at 700 hPa resulting from various B matrices.
Bnmc
Bens No hLoc Bens Std hLoc
Bens SD hLoc
hLoc: 1500km / 4000km / 10000km
700 hPa T observation at the center of Hurricane Gonzalo (October 2014) hLoc: 2800km
0
-‐1
+1
0
-‐1
+1
0
-‐1
+1
0
-‐1
+1
Impact in single observa3on DA experiments
Scale-dependent covariance localization Impact in single observa3on DA experiments
Bens No hLoc
Normalized temperature increments (correlation-like) at 700 hPa resulting from various B matrices.
Bnmc
Bens Std hLoc
Bens SD hLoc
hLoc: 1500km / 4000km / 10000km
700 hPa T observation at the center of a High Pressure
hLoc: 2800km
0
-‐1
+1
0
-‐1
+1
0
-‐1
+1
0
-‐1
+1
Scale-dependent covariance localization
• 2.5-month trialling (June-August 2014) in our global NWP system.
1) Control experiment with hLoc = 2800 km, vLoc = 2 units of ln(p)
2) Scale-‐Dependent experiment with a 3 horizontal-‐scale decomposi2on I. Small scale uses hLoc = 1500 km II. Medium scale uses hLoc = 2400 km III. Large scale with uses = 3300 km
• 3DEnVar with 100% Bens used in both experiments
Ad hoc values!
Same vLoc (2 units of ln(p)) for every horizontal-‐scale
Forecast impact
Scale-dependent covariance localization Forecast impact
T+24h Northern E-T
Ø Control
U
Z T
RH U
Z T
RH
Ø Scale-‐Dependent
T+24h Southern E-T
Scale-dependent covariance localization Forecast impact
T+72h Northern E-T
Ø Control
U
Z T
RH U
Z T
RH
Ø Scale-‐Dependent
T+72h Southern E-T
Scale-dependent covariance localization Forecast impact
T+120h Northern E-T
Ø Control
U
Z T
RH U
Z T
RH
Ø Scale-‐Dependent
T+120h Southern E-T
Scale-dependent covariance localization Forecast impact
T+168h Northern E-T
Ø Control
U
Z T
RH U
Z T
RH
Ø Scale-‐Dependent
T+168h Southern E-T
Scale-dependent covariance localization Forecast impact
Time series Northern E-T Time series Southern E-T
Std Dev for U at 250 hPa
Std Dev for U at 250 hPa
Ø Control Ø Scale-‐Dependent
Scale-dependent covariance localization Forecast impact
• 2 new 1.5-month trialling (June-July 2014) with a single localization approach (still using 3DEnVar with 100% Bens)
1) hLoc = 2400 km (the value used for medium scale in SD hLoc)
2) hLoc = 3300 km (the value used for large scale in SD hLoc)
Is it possible to do as good as SDL with a single localiza2on approach? Awer all, perhaps our one-‐size-‐fits-‐all horizontal localiza2on radius of 2800 km is not op2mal...
Scale-dependent covariance localization Forecast impact
Time series Northern E-T Time series Southern E-T
Std Dev for U at 250 hPa
Std Dev for U at 250 hPa
Ø Control (hLoc = 2800 km) Ø hLoc = 2400 km
Scale-dependent covariance localization Forecast impact
Time series Northern E-T Time series Southern E-T
Std Dev for U at 250 hPa
Std Dev for U at 250 hPa
Ø Control (hLoc = 2800 km) Ø hLoc = 3300 km
Scale-dependent covariance localization Impact on dynamical balance – Rota2onal Part
It is well know that localiza2on can disrupt the dynamical balance of the analysis increments. Does the SDL increase or decrease this problem?
rrrrrrrrrrrrr u
yf
xv
yu
yv
xu
xv
yu
yv
xu
xv
yu
yv
xuf ʹ′
∂
∂−⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
ʹ′∂
∂
ʹ′∂−
∂
ʹ′∂
∂
ʹ′∂+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂
∂
ʹ′∂−
∂
∂
∂
ʹ′∂+⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
ʹ′∂
∂
∂−
∂
ʹ′∂
∂
∂+ʹ′=Φʹ′∇ 22 ζ
Wind Mass
• Rotational part: Charney’s (1955) nonlinear balance equation
Balance diagnos2cs as in Caron and Fillion (2010; MWR)
One conclusion from Caron and Fillion: Both horizontal and ver2cal localiza2on have significant deleterious effect on the rota2onal balance with the largest detrimental impact coming from the ver2cal localiza2on.
Scale-dependent covariance localization Impact on dynamical balance – Rota2onal Part
Ver2cal profile of average normalized departure from (n-‐l) balance
1.6 0.4
hLoc 2800 km SD hLoc
0.8 1.2
800
600
400
200
hPa
1000 1.6 0.4 0.8 1.2
Northern E-T Southern E-T
□ hLoc 3300 km ○ hLoc 2400 km
hLoc 2800 km SD hLoc
□ hLoc 3300 km ○ hLoc 2400 km
800
600
400
200
1000
Application in a convective-scale EnVar system:
Meteo-France’s AROME R&D version • 25 ensemble member @ 3.8km • Localiza2on in spectral or gridpoint space • Determini2c forecast @ 3.8 km
(Adhoc) scale-decomposition for AROME
Temperatutre perturba2ons at ~950 hPa for member #1. 3h forecast valid on 06 February 2016 at 00 UTC
Pseudo-single obs - Frontal case
SP 250km
Pseudo-single obs - Frontal case
SP-‐SDL 080/250/500km
Pseudo-single obs - Convective case
SP 250km
Pseudo-single obs - Convective case
SP-‐SDL 080/250/500km
Performed DA and forecast cycles
• 2 weeks trialling (February 2016) were used testing many horizontal localization lengtscale for both SDL and ‘one-size-fits-all’ approaches.
• Extension to 1 month for both the control experiment and the best performing SDL configurations
• Spectral and gridpoint (recursive fitler) localization were used/compared. • The same horizontal-scale decomposition (3 wave band) was used in all
the SDL experiments. • 3-hourly DA cycle • 30-hour forecasts issued from 00, 06, 12 and18 UTC
Verification against aircraft
SP 075/150/300km vs SP 250km
2 weeks
Verification against aircraft
SP 075/150/300km vs SP 250km
2 weeks
Verification against aircraft
SP 075/150/300km vs SP 250km
2 weeks
Verification against aircraft
SP 075/150/300km vs SP 250km
2 weeks
Verification against aircraft
SP 075/150/300km vs SP 250km
2 weeks
Verification against aircraft(+3 to +30h, 3h)
SP 075/150/300km vs SP 250km
2 weeks
SP 075/150/300km vs SP 250km
Verification against aircraft (+1 to +10h, 1h)
2 weeks
SP 300km vs SP 250km
Verification against aircraft (+3 to +30h, 3h)
2 weeks
SP 200km vs SP 250km
Verification against aircraft (+3 to +30h, 3h)
2 weeks
SP 150km vs SP 250km
Verification against aircraft (+3 to +30h, 3h)
2 weeks
Full verification (+3 to +30h, 3h)
SP 075/150/300km vs SP 250km
1 month
Full verification (+3 to +30h, 3h)
RF 075/150/300km vs RF 250km
1 month
Impact on balance
RF-‐SDL RF SP-‐SDL SP
Summary and conclusion
• SDL is feasible and straightforward to implement in EnVar, but more expensive than using single-scale localization.
– In the SDL experiments reported here: 3x to 3.5x more expensive
• Results using a horizontal-scale-dependent horizontal localization indicate small forecast improvements in both systems examined. However, the time-scales over which the SDL method impacted the forecasts are completely different:
– Improvements up to day 5 were noticed in the global system, – In the convective-scale system, they were limited mostly in the first 9 hours of the forecasts
• In terms of dynamical balance, SDL seems to alleviate somewhat the imbalance generated by the localization.
• Finding the optimal SDL setup is not straightforward. – No objective approach has been found useful so far.
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