Transcript whitaker_ranmatrices

Methods for dealing with spurious
covariances arising from small
samples in ensemble data assimilation
Jeff Whitaker [email protected]
NOAA Earth System Research Lab, Boulder
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what is ensemble data assimilation?
what are the consequences of sampling error?
covariance localization.
alternatives to covariance localization.
Ensemble data assimilation
• Parallel forecast and analysis cycles
• Background-errors estimated from sample covariances, depend
on weather situation.
Ensemble Kalman Filter
Ensemble Kalman Filter
k ensemble members from a forecast model
ensemble (sample) mean
Ensemble Kalman Filter
k ensemble members from a forecast model
ensemble (sample) mean
background-error (sample) covariance
Ensemble Kalman Filter
k ensemble members from a forecast model
ensemble (sample) mean
background-error (sample) covariance
analysis-error covariance
Consequences
of Sampling
Error
Mis-specification of background-error
covariance
Effect of localization in a
simplied GCM (1)
2-layer PE model on a sphere
46 observations over the globe
Effect of localization in a
simplied GCM (2)
Effect of localization in a
simplied GCM (3)
Covariance localization
increases rank of Pb
• If the ensemble has k members, then Pb
describes nonzero uncertainty only in a kdimensional subspace .
• Analysis only adjusted in this subspace.
• If the system is high-dimensionally unstable
(if it has more than k positive Lyapunov
exponents) then forecast errors will grow in
directions not accounted for by the ensemble,
and these errors will not be corrected by the
analysis.
Alternative to localization
• Localizing covariances works because it
increases the dimensionality….
• So, one can instead compute updates in
local regions where error dynamics
evolves in a lower-dimensional
subspace (< k).
• (LETKF - Hunt et al, 2007)
Two EnKF approaches
• Serial approach - for each observation,
update each model variable (tapering the
influence of the observation to zero at a
specified distance). Used in NCAR DART.
• Local approach - update each model variable
one at a time, using all observations within a
specified radius (increasing R with distance
between observation and model variable) we use this approach since it scales well on
massively parallel computers
Outstanding issues
• Both methods assume a priori that
covariance is maximized at the
observation location - problematic for
non-local and time-lagged obs.
• Both methods are flow-independent
(assume same degree of locality for
every situation).
• Localization can destroy balance.
Localization and Balance
Analysis of single zonal wind observation, using idealized
nondivergent and geostrophically balanced covariances.
Control imbalance by time-filtering first-guess forecast.
Flow Dependent Localization
(Hodyss and Bishop, QJR)
Stable flow error correlations
km
Unstable flow error correlations
km
Ensembles give flow dependent noisy correlations
Flow Dependent Localization
Stable flow error correlations
Fixed moderation
Current ensemble DA techniques
reduce noise by multiplying ensemble
correlation function by fixed
moderation function (green line).
km
Unstable flow error correlations
Resulting correlations (blue line) are
too thin when true correlation is broad
and too noisy when true correlation is Fixed moderation
thin.
km
Today’s fixed moderation functions limit adaptivity
Flow Dependent Localization
Stable flow error correlations
SENCORP moderation
Smoothed ENsemble
Correlations Raised to a Power
(SENCORP) moderation
functions provide flow adaptive
moderations functions.
km
Unstable flow error correlations
SENCORP moderation
km
SENCORP moderation functions adapt
“SENCORP” Recipe
1. Smooth Pb = P1b
2. Element-wise
cube of P1b =
P2b
3. Normalized
matrix product of
P2b with itself =
P3b
4. Use elementwise square of
P3b to compute
K.
Hierarchical Ensemble Filter
• Proposed by Jeff Anderson (NCAR).
• Evolve K coupled N-member ensemble filters.
• Use differences between sample covariances
to design a situation-dependent localization
function.
• asymptotes to optimally localized N member
ensemble (not K*N).
Conclusions
• Localization (tapering the impact of
observations with distance from analysis grid
point) makes ensemble data assimilation
feasible with large NWP models.
• Both model errors and localization make filter
performance suboptimal. Right now model
error is the bigger problem, but improvements
in localization are needed.