Analysis and design covariance inflation methods from functional - - PowerPoint PPT Presentation
Analysis and design covariance inflation methods from functional viewpoint Le Duc 13 , Kazuo Saito 23 , and Daisuke Hotta 3 1 Japan Agency for Marine-Earth Sciences and Technology 2 AORI, University of Tokyo 3 Meteorology Research Institute of
Le Duc13, Kazuo Saito23, and Daisuke Hotta3
1Japan Agency for Marine-Earth Sciences and Technology 2AORI, University of Tokyo 3Meteorology Research Institute of Japan
Additive inflation and physics-based model uncertainty schemes are out of the scope of this study. Here, we focus on the covariance inflation (CI) methods applied at each data assimilation cycle that
use forecast perturbations and observations to increase ensemble spreads.
Prior multiplicative inflation (MI): belongs to prior CI
→
Posterior MI (inspired by prior MI): belongs to posterior CI
→
Relaxation schemes RTPP and RTPS: belongs to posterior
it seems that there are not any similar methods in prior CI.
= (1 − α) + α = + (1 − ) After this talk, I hope you will know more than 10 other CI methods!
Inflating is somehow equivalent to inflating , but posterior CI cannot say explicitly how is inflated. Because of this, posterior CI leaves mean analysis intact and only inflate analysis perturbations , leading
posterior CI method? How can we explain the relaxation forms of RTPP and RTPS in posterior CI? Is there any other posterior CI method? If they exist, do they have relaxation forms? Can they have their counter- parts in prior CI?
Given the similar relaxation forms of RTPP and RTPS, one may expect that perturbations produced by both methods should not have large differences. But this is far from the true. Whereas RTPS tends to generate analysis perturbations with small spreads, thus leading to under-dispersive ensemble forecasts, RTPP tends to be less under- dispersive with analysis spreads comparable to analysis errors (Bowler et al., 2017). No adequate interpretation has been given yet as to why we have such different behavior, and we will address this issue.
Then: = = + +
−
= − +
= + where: = − = − = Assume we have a decomposition (ensemble, modulated ensemble, hybrid, …):
By setting: We have: =
⁄
= = +
⁄
= +
using the positive symmetric square root :
Then the scaled background error covariance // is given by
, where , are the dimensions
and ensemble space, respectively.
That means the component-wise scaling factors are
, respectively.
If we assume that = and observations are of the same kind with observation error variance , we will have
=
where
associated with the mode . And we recover DA for a scalar variable:
/( + )
(
⁄
+
This defines response of analysis singular values as a function of background singular values:
1 + ⁄ Two important properties:
the
are preserved in assimilation, i.e. if ≥ then ≥
analysis singular values are bounded from above by 1.
Model space: Or if and represent background and analysis perturbations in ensemble space: Ensemble space: = = .
= .
= = ,
, = , ∀ ≤
Not any function can be an inflation function. It has to satisfy the following conditions.
f
to increase at all modes.
are available, inflated analysis perturbations have to be identical to background perturbations.
= f () has to be an increasing function. Given any spectrum , ∈ [0,1] of , we associate with this spectrum a continuous function, which we coin the term an inflation function, f: [0,1] → [0, ∞) that maps to its inflated value f and satisfies the following three conditions:
This requires: = ,
,
f ≥ ∀ ≤ Extending to the whole domain, we require: Equivalently: f ≥ ∀ ∈ [0,1]
This condition requires f 1 = 1 when all = 1 . To ensure continuity
inflation impact
analysis perturbations when spectra change continuously, we require f 1 ⟶ 1 whenever all ⟶ 1. = ,
,
converge to when the spectrum , converges to the spectrum of the identity matrix.
This condition requires the inflated analysis singular values f() have to form an increasing function when considered as a function of . = ,
,
Here is a fixed inflation factor. No assimilation: f = 1 No inflation: f = Prior multiplicative inflation: f = 1 + 1 − 1
⁄
Posterior multiplicative inflation: f =
If we inflate: → This is equivalent to: →
This can be
by changing
= 1/
1/ +
. In other words:
= +
= +
⁄ +
We have: f = 1 + 1 − 1
⁄
It is easy to check the three conditions: inflation functions lie above the line f = and go through the point (1,1), response functions are increasing functions.
For the first time, it shows clearly that the prior and posterior MI are two different methods. Both do not satisfy the no-observation condition.
Prior inflation functions help to explain for the unusual form of the inflation function of the prior MI method.
g 1 + g
g 1 + g
Calculate the analysis increment at the jth point: Or to avoid calculation with − vectors in the null space of , we use the fact that .
= ∑
, .
rewrite:
.
< , . >
= f
, .
= . − 1 − f ) , .
If a, do not depend on the th point: f = + .
= f
, .
+ .
The functional and order-preserving conditions impose two conditions for linear inflation functions: f = + ≥ 0 + ≥ 1
If is fixed, the no-observation condition entails f 1 = = 1, which is f . Therefore, to obtain non-trivial solutions, we need to parameterize as a function of certain variables. f = These variables should characterize the spectrum , so that the no-observation condition when all = 1 can be described. There are various candidates in this case. We can list here the trace of : ∑ or ̅ = ∑ /, the determinant of : ∏ , the spectrum norm
∑
.
We will mainly work with two quantities: / and ̅, which are both equal to 1 when no observations are assimilated. f = Here we choose ̅ to parameterize . Thus, we replace the function f = by the new form f = α̅ + where α, are two fixed hyper-parameters and a linear function, the simplest one, is chosen for the form of . The no-observation condition now yields the relation f 1 = α + = 1. And we have: f = α̅ + 1 − α = 1 − α(1 − ̅ Do you know what it is?
When = 1, we recover the diagonal ETKF However, it is better to think that the generalized case defines the family of Diagonal ETKF. f = α̅ + 1 − α = 1 − α(1 − ̅ = (̅ + 1 − ) = ̅
Note we relax the functional condition by a weaker one: Norms of analysis perturbations have to increase
.
If is fixed, the no-observation implies = 1, which is the function f . Therefore, like in constant inflation functions, cannot be fixed and has to be a function of certain variables. f = Here we first choose / to parameterize . Again, a linear function is chosen for the form of and we have f = (α
with two fixed hyper- parameters α, .
The no-observation condition now yields the relation f 1 = α + = 1. And we have: f = (α
f = α
1 + α(
In other words: = 1 + α(
Can you guess what it is?
From this, the norm of is given by = 1 + α(
= + (1 − ) = (1 − α) + α And we recover the well-known RTPS method
RTPS is not the unique realization of multiplicative inflation functions: f = 1 + α(
However, now it is more difficult to interpret the meaning of the resulting inflation functions. f = 1 + α(1 ̅ − 1 f = 1 + α(1 − f = 1 + α(1 − ̅
This raises the question whether the use of / , and therefore the RTPS, is truly important or not. Clearly, it is not necessary to use / if we want to obtain the minimum RMSE.
Clearly, MIFLM with α = 2.3 had the same skill as RTPS with α = 0.9. The forecast performances would have been better if we had increased the values of α. f = 1 + α(1 − ̅
Applying the no-observation condition, we must have f 1 = + = 1. If is a fixed parameter, this entails that is also a fixed parameter given by = 1 − . Therefore, we obtain: f = + f = + (1 − ) In other words: Can you guess what it is? = + (1 − ) RTPP!
The figures point out that RTPP are distinctively different from RTPS despite the apparent resemblance in relaxation
produce under-dispersive analysis ensemble since RTPS sets an upper bound for magnitudes of the most active processes (large ) in inflated analysis perturbations.
The family of fixed linear inflations functions is quite restrictive, covering only a very small part of potential linear inflation functions. A way to extend the repertoire
f are two inflation functions and the range of f belongs to [0,1] then the composite map f ∘ f is also an inflation function. This suggests that generalized linear inflation functions should have the following form: f = + = (α̅ + ) + (α̅ + ),
The no-observation condition dictates the first constraint f 1 = α + + α + = 1. The second constraint is determined by the converged form of f when all converge to 1. For generalized linear inflation functions, it is reasonable to choose f as its converged form. This means we impose the constraint ⟶ 0 when ̅ ⟶ 1, or equivalently α + =
f = + = (α̅ + ) + (α̅ + ), f = 1 + α(1 − ̅) + α(1 − ̅)
The best performance was obtained with α + α = 3.3, better forecasts might have been obtained if α and α + α were set to higher values.
Compared with the case of linear inflation functions, we have an additional term ∑
f = + +
.
= f
, .
, .
+ .
− , .
− .
.
= ( + ). − . + .
For full quadratic inflation functions, the additional step compared to linear inflation functions is with . To
we need to run the same program that produces the analysis times. However, the fact that we still need to run another program to generate analysis perturbations make this less practical. Therefore, it is better to set equal to zero so that analysis perturbations are not needed to be computed explicitly.
= ( + ) − +
f = +
= ( + ) −
The functional and order-preserving conditions impose four conditions for quadratic inflation functions without first-order terms: ≥ 0 + ≥ 1 f = + If ≥ 1/2, then c ≥ 1/4 c ⁄ ≤ 8
If a and c are fixed, the no-observation condition yields f 1 = a + = 1. Thus, similar to the family of fixed linear inflation functions we have the family of fixed quadratic inflation functions: In other words: Can you guess what it is? f = + f = + (1 − ), = −
The four conditions together imply the inequality ≤ 0.5. When = 0.5, the parabola + (1 − ) is tangent to the line f = at = 1. For this critical value of , we recover the well-known DEnKF (Sakov and Oke, 2008). It is quite surprising to see that the functional approach to inflation problem leads to a new interpretation for the DEnKF scheme. = − = − 1 2
Again, the family of fixed quadratic inflation functions is quite restrictive, and extension to parameter-varying quadratic inflation functions is straightforward. Now we choose the quadratic inflation function associated with DEnKF as the converged function when all converge to 1. By applying the same mathematical arguments we have the family of generalized quadratic inflation functions with two hyper-parameters α, α f = + = (α̅ + ) + (α̅ + ). f = 0.5 + α(1 − ̅ + 0.5 + α(1 − ̅
Using the same arguments with quadratic inflation functions, it can be shown that the cubic form implies We can reduce the computational cost by setting = 0, but we still need to compute and . This shows that inflation using high-order polynomials consume more computational
polynomials not appealing in practice. However, the more important reason is that high-order polynomials do not necessarily yield better inflation than linear
quadratic polynomials.
f = + + + .
= ( + ). − . + ( + ). − .
between prior inflation functions and inflation functions.
CI methods.
including multiplicative inflation, the RTPS, RTPP and DEnKF.
family of EnKF: diagonal ETKF.
perturbations based on the Kalman gain without a need for ensemble transform matrices.