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 Japan
Scope 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 only use forecast perturbations and observations to increase ensemble spreads.
Covariance inflation methods 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 CI. The nature of these two methods is quite puzzling and it seems that there are not any similar methods in prior CI. � ���� = �� � + (1 − �)� � � ���� = (1 − α)� � + α� � After this talk, I hope you will know more than 10 other CI methods!
Questions on covariance inflation 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 inconsistency. Can we find equivalently prior CI for each 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?
RTPP vs. RTPS 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.
1. Background: Decomposition form of data assimilation
Kalman filter theory �� � − �� � � � = � � + � � � � �� � � � + � �� �� � � � = � � − � � � � �� � � � + � Assume we have a decomposition (ensemble, modulated ensemble, hybrid, …): � � = � � � �� Then: �� � = � � � + � �� � �� � � �� � �� � �� � � � = � � � + � �� � �� � � �� � �� where: �� � = � � − � � � � = �� � � = � − �� �
More compact form ⁄ � � � � = � �� � By setting: � ⁄ � � = � �� � � We have: � � �� � �� � = � � � + � � �� � � �� � � ⁄ � � �� � � � = � � � � = � � � + � � �� � Here we replace the equation for � � with the equation for � � using the positive symmetric square root � � : � � = � � � ��
Singular value decomposition � � � = � � � � � � � � � ��� � � , where �, � are the Here � ≤ min (�, �) is the rank of � dimensions of observation and ensemble space, respectively. Then the scaled background error covariance � ��/� �� � � � � ��/� is given by � � � � �� = � � � � � � � � � � � � � � � � � � � � � � � � � � = � � � ��� ��� ���
Impact of assimilation � � � = � � � � � � � = � � � � � � � � � � � � �� ��� ��� � � � � = � � � � � � � � � = � � � � � � � � � � � � � ��� ��� � � � �� = � � � � � � � � � � � � � � � � �� = � � � � � � � � � � � � � � ��� ��� � and � � � That means the component-wise scaling factors are � � � � , respectively.
Does this make sense? � � � � = � � � � = � � � � � � � � � � � � � � �� ��� ��� � � � �� = � � � � � � � � � � � � � � � � �� = � � � � � � � � � � � � � � ��� ��� If we assume that � = � and observations are of the same kind � = with observation error variance � �� , we will have � � �� /� �� �� � � where � � is the background error variance associated with the mode � � . And we recover DA for a scalar variable: �� + � �� ) � � � � � � = � � �� � � /(� � � � �� � �� ( �� + � �� � � � �� = � � � � � ⁄ � � � �
Perturbation response � � � � = � � � � � � � � � = � � � � � � � � � � � � � ��� ��� This defines response of analysis singular values as a function of background singular values: 1 + � � ⁄ � � = �� = � Two important properties: 1. Order-preserving: the orders of singular values are preserved in assimilation, i.e. if � � ≥ � � then � � � � ≥ � � � � 2. Upper-bound: analysis singular values are bounded from above by 1.
2. Inflation function
Impact of ensemble transform matrix in ensemble space � � = � � � � Model space: � �� = � � � �� Ensemble space: � = � � � �. � That means for each jth row vector: � �. Or if �� � and �� � represent background and analysis perturbations in ensemble space: � �� � = � � �� � = � � � � � , �� � � � ��� � � , �� � = � � � � , �� � ∀� ≤ � In other words:
Idea of inflation functions � f � � � � f � � � � � � � � � = � � � � ���� = � � � � � � � � � � � �� �� ��� ��� � � � � = � � � � � � � � � � � ���� = � � � f � � � � � � � � � ��� ��� Not any function can be an inflation function. It has to satisfy the following conditions.
Definition 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: 1. The functional condition: analysis perturbations have to increase at all modes. 2. The no-observation condition: when no observations are available, inflated analysis perturbations have to be identical to background perturbations. 3. The order-preserving condition: the response function � � = �f �(�) has to be an increasing function.
The functional condition � �� � = � � � � � , �� � � � ��� � �� ���� = � f � � � � , �� � � � ��� < � � , �� ���� > ≥ < � � , �� � > ∀� ≤ � This requires: Equivalently: f � � ≥ � � ∀� ≤ � Extending to the whole domain, we require: f � ≥ � ∀� ∈ [0,1]
The no-observation condition � �� � = � � � � � , �� � � � ��� � �� ���� = � f � � � � , �� � � � ��� This condition requires f 1 = 1 when all � � = 1 . To ensure continuity of inflation impact on analysis perturbations when spectra change continuously, we require f 1 ⟶ 1 whenever all � � ⟶ 1 . This condition involves a function that inflation functions converge to when the spectrum � � ���,� converges to the spectrum of the identity matrix.
The order-preserving condition � �� � = � � � � � , �� � � � ��� � �� ���� = � f � � � � , �� � � � ��� This condition requires the inflated analysis singular values � � f(� � ) have to form an increasing function when considered as a function of � � .
Examples f ���� � = 1 No assimilation: f ����� � = � No inflation: �� � ⁄ � � + 1 1 f ����� � = Prior multiplicative inflation: � − 1 f ������ � = Posterior multiplicative inflation: �� Here � is a fixed inflation factor.
Derivation of the inflation function corresponding to prior MI � � → �� � If we inflate: � � → �� � This is equivalent to: � � → � � � �� � � �� � ⁄ � � = � � � + � Then from: � �� � We have: ⁄ = � � � � � � �� � � � �� � ⁄ � ���� = �� � � + �� � �� � � �� � ⁄ + � ��� = 1/ This can be obtained by changing to � � � � � . In other words: 1/� + � � �� � ⁄ � � + 1 1 f ����� � = � − 1
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