Advanced Data Assimilation
Methods
10 December 2012
Thematic Outline of Basic Concepts
• What is 4D-Var, and how does it differ from and
extend upon 3D-Var?
• What is an Extended Kalman Filter, and how does it
extend upon least-squares minimization methods?
• What is an Ensemble Kalman Filter, and what are its
strengths relative to other data assimilation methods?
4D-Var
• Generalization of 3D-Var to allow for the continuous
assimilation of all available observations over some
assimilation interval.
• Recall: cost function for 3D-Var…
  x  x Bx  x   Hx y RHx y 
T
J x 
b
(background)
T
b
(observations, defining
analysis increment)
multidimensional, each weighted by the inverse of their
respective error covariance matrices
4D-Var
• For 4D-Var, the cost function has a similar form…
  xt   x t Bxt   x t 
J xt0  
T
0
b
0
0
b
0
t0
(background valid at the start of
the assimilation window)
m

i 0
Hx  y Hx  y 
T
i
i
i
Ri
(assimilate m observations,
one at a time)
• This cost function is equivalent to 3D-Var if
observations are valid only at a single time t0.
i
4D-Var
• Each observation is assimilated at the time that it is
taken, thus the xi is the model state vector valid at
the time of the observation.
• Contrast to xt0, the model state vector valid at the
start of the assimilation window. If all observations
are at t = t0, then xi= xt0 (i.e., collapses to 3D-Var).
4D-Var
• Minimization of the cost function proceeds similar to
in 3D-Var, albeit with some differences…
– Inherent temporal evolution: each observation time is
associated with a new xi and Ri.
– Each observation assimilation requires forward integration
of the model from to to tm (a/k/a ti).
– Post-assimilation, an adjoint operator must be used to
backward integrate to finalize the updated analysis.
• 4D-Var is thus more computationally expensive than
is 3D-Var, limiting its operational acceptance.
4D-Var
• Adjoint model: tangent linear version (M) of a nonlinear model operator (M).
– Defined as the transpose of the model operator (MiT)
– Adjoint model is specific to the forecast model used for DA
and to conduct the subsequent forecast simulations.
– Thus, every time the model code is upgraded, the adjoint
code must also be updated, not a trivial process!
4D-Var
• Cost function minimization also requires that the
sequence of model states xi (the modified forecast
model state xf) provides valid solutions to the model.
• In other words, there must be dynamical consistency
between xi and the model dynamical equations.
• Mathematically,

i, xi  M 0i x
(over all i, xi must be representative of the model forecast from t=0 to t=ti)
4D-Var
4D-Var
• 3D-Var: minimize cost function to optimally combine
background (dashed line) and observations (stars) at
one assimilation time
• 4D-Var: estimate the analysis state vector xa (at t=t0)
producing a model solution M that minimizes the
cost function.
– Defines an initial condition that produces a forecast that
minimizes total cost throughout assimilation window.
– Cost function includes both the background (JB) and each
observation (all of the JO terms at each time ti).
4D-Var
• In the depicted example, the corrected and previous
forecasts do not significantly differ at tn.
– Note: the previous forecast is the one that defines the
background at t0 and continues forward in time.
• However, the details between t0 and tn differ,
providing a better fit to the observations as manifest
through the temporal evolution of the forecast.
4D-Var
• Start: background estimate, nominally from a prior
model forecast.
• Assimilate observations at background time (3D-Var).
• Integrate model forward to next observation time.
• Assimilate observation to help correct forecast state.
4D-Var
• Repeat the previous two steps over all observations.
• Use adjoint model to go backward to the background
time to complete assimilation.
• Result: a “corrected” analysis and short-term forecast
(~3- to ~6-h) evolution.
• Implicit: balance condition; consistency with model.
Extended Kalman Filter (EKF)
• A specific application of least squares minimization
that emphasizes time dependency in the calculation
of the background error covariance matrix.
Time-Independent
Time-Dependent
B = background error covariance matrix
A = analysis error covariance matrix
Pf = background error covariance matrix
Pa = analysis error covariance matrix
• Since Pf often comes from a short-range model
forecast, it is sometimes referred to as a forecast
error covariance matrix.
Extended Kalman Filter
Extended Kalman Filter
• First, obtain a first guess estimate by integrating the
model forward from a previous analysis.
 
x f t 1  M t t 1 xa t 
• Next, compute the forecast error covariance matrix
at t=t+1 for use when constructing the next analysis.
P f t  1  M t t 1 P a t M
T
t t 1
 Qt 
M = Jacobian of M, i.e., ∂M/∂x
Q = model forecast error covariance matrix at t=t
Extended Kalman Filter
• The model forecast error covariance matrix Q(t) is
used as a starting point.
• Added to this is the forward-propagation (in time) of
the analysis error covariance matrix Pa(t)
– Conducted using the tangent linear version of the model
M and its adjoint MT.
– Requires knowledge of Pa at the very first time; for later
times, we are able to compute it (more shortly).
Extended Kalman Filter
• Forecast time t+1 now becomes the analysis time t.
• Next, we compute the Kalman gain matrix K(t), the
analog to the weighting matrix K used before.
K
BH
T
T
HBH  R
P f t Ht 
T
K
Ht P f t Ht   R t 
T
R = observation error covariance matrix
B = time-independent background error covariance matrix
Pf = time-dependent background error covariance matrix
HHT = transformation to observation space
H = Jacobian of forward observation operator H (∂H/∂x)
Extended Kalman Filter
• Both K(t) and K are identical apart from the timedependency manifest via B versus Pf.
• As before, they represent the ratio of the
transformed background error covariance matrix to
the total (obs + background) error covariance matrix.
• Once K(t) has been computed, it is used as the
weighting in obtaining a new analysis state estimate.
– Determines multidimensional spread of observational info.
Extended Kalman Filter
• Next, obtain the new analysis state estimate.


xa t   x f t   Kt  yt   H t x f t 
• As before, this represents the background estimate
plus an optimally-weighted innovation (posed in
observation space).
– Minimizes the variance in xa by optimally weighting xf and
y based upon their known error characteristics.
– Difference from before: K(t) is computed directly from the
model (flow-dependent) at each time (time-dependent)!
Extended Kalman Filter
• Finally, compute the analysis error covariance matrix.


Pa t   P f t  I  Kt H t 
• Recall: simple, 1-D analog for analysis error variance
– Background variance weighted by 1-weighting factor, or…
 a2   b2 1  k 
σa2  Pa, 1  I (identity matrix), k KH (weighting with obs. operator)
Extended Kalman Filter
• Having created a new analysis, the model can then
be integrated forward in time to the next time.
• The above-computed analysis error covariance is
then used to compute the forecast error covariance
at the next time.
• And on and on from there…
EKF vs. 4D-Var
• Evolution of the background error covariance.
–
–
EKF: explicitly computed over each assimilation cycle
4D-Var: manifest via different H(xi) – yi values for each
observation (e.g., different innovations for each time as
model forecast evolves, giving time-dependency)
• Characteristics of the model.
–
EKF: model is imperfect
•
•
Forecast errors due to the forward propagation of analysis errors
and errors in the model itself (Q ≠ 0).
Need to have some way to describe Q (difficult) to obtain the
best possible analysis!
EKF vs. 4D-Var
–
4D-Var: model is perfect
•
Forecast errors due to the forward propagation of analysis errors
only (cost function analogy to Q = 0).
• Method of data assimilation.
– EKF: intermittent/sequential, least squares minimization
– 4D-Var: continuous, cost function minimization
• Computational expense.
–
EKF is more computationally expensive than is 4D-Var
due to added computations at each assimilation time.
Ensemble Kalman Filter (EnKF)
•
As the name implies, we start with an ensemble of
atmospheric state analyses.
•
How is such an ensemble initially obtained?
–
–
Basis: a uniform background (e.g., from a global model)
Ensemble: constructed by perturbing the common
background state estimate.
•
•
Typically random perturbations drawn from some estimate of
the background error covariance matrix B.
May also incorporate the assimilation of observations with
perturbations of magnitude smaller than the expected obs error.
Ensemble Kalman Filter
•
Each member of the ensemble is then integrated
forward in time.
•
The ensemble of short-range forecasts is used as
the background for the next analysis.
–
–
•
Observations are assimilated (without any perturbations)
to correct the forecast to the observations.
If the simulation has lateral boundaries, the LBCs must be
perturbed at each time on the outermost domain.
The cycle then repeats from here.
Ensemble Kalman Filter
•
Assimilation is accomplished using the EKF
framework from before with slight modifications.
•
Definitions…
–
Ensemble mean forecast state:
•
•
–
1 K
x f   x f ,i
K i 1
i = individual ensemble member
K = total number of ensemble members
Forecast error covariance matrix:
T





x

x
x

x
f  f ,i
f  

 f ,i


 
ˆP f  i 1 
K 1
K
(of the sample of xf)
Ensemble Kalman Filter
•
This equation replaces our earlier equation for Pf.
–
Provides a direct estimate of Q and how analysis errors
have propagated forward without needing an adjoint!
•
Represents the average departure of each ensemble
forecast state from the ensemble mean forecast
state (i.e., not explicitly the analysis or true state).
•
Pˆ f
: an estimate of Pf from a finite ensemble.
Ensemble Kalman Filter
•
Similar to the EKF method,
ˆ 
K
ˆP t H t T
f
(Kalman gain matrix)
T
H t Pˆ f t H t   R t 


ˆ t  y t   H t x f t 
x a t   x f t   K
•
(new analysis state)
From the new analysis state, the analysis error
covariance Pˆ a may be computed similarly to Pˆ .
f
Ensemble Kalman Filter
•
Analysis error covariance:
T





x

x
x

x
a  a ,i
a 

 a ,i


 
i 1 
K
Pˆ a 
•
K 1
This highlights the strengths of the EnKF: internal
calculation of Pa and Pf based upon the ensemble’s
flow-dependent error characteristics!
Ensemble Kalman Filter
initial analyses
integrate forward
in time
estimate Pf
compute K and
perform DA
obtain new analyses
repeat
Ensemble Kalman Filter vs. 4D-Var
•
Computation of Pf does not require the use of an
adjoint model.
–
–
By contrast, even EKF uses one (see Eqn. 6.26b earlier).
This eliminates the assumptions (linearity, etc.) and
additional code associated with the adjoint method.
•
Both methods are computationally demanding.
•
EnKF provides an elegant way to determine the
temporally- and spatially-varying structure in the
background error covariance from the ensemble.
Ensemble Kalman Filter
shaded: normalized covariance of SLP
contour: sea-level pressure
Emphasis is on the five dots (observation locations).
Ensemble Kalman Filter
•
How does the ensemble estimate of sea level
pressure vary as a function of the observations at
these locations?
–
–
•
Recall: sensitivity metric defined in an earlier lecture
Can use covariance information to illustrate this!
Covariance is largest in data-sparse areas and/or
along sharp gradients.
–
Forecast metric at the observation locations is most
sensitive to SLP uncertainty in the shaded regions.