- 著者
- DUC Le SAITO Kazuo HOTTA Daisuke
- 出版者
- Meteorological Society of Japan
- 雑誌
- 気象集誌. 第2輯 (ISSN:00261165)
- 巻号頁・発行日
- pp.2020-022, (Released:2020-01-16)
- 被引用文献数
- 1
In the ensemble transform Kalman filter (ETKF), an ensemble transform matrix (ETM) is a matrix that maps background perturbations to analysis perturbations. All valid ETMs are shown to be the square roots of the analysis error covariance in ensemble space that preserve the analysis ensemble mean. ETKF chooses the positive symmetric square root Ts as its ETM, which is justified by the fact that Ts is the closest matrix to the identity I in the sense of the Frobenius norm. Besides this minimum norm property, Ts are observed to have the diagonally predominant property (DPP), i.e. the diagonal terms are at least an order of magnitude larger than the off-diagonal terms. To explain the DPP, firstly the minimum norm property has been proved. Although ETKF relies on this property to choose its ETM, this property has never been proved in the data assimilation literature. The extension of this proof to the scalar multiple of I reveals that Ts is a sum of a diagonal matrix D and a full matrix P whose Frobenius norms are proportional, respectively, to the mean and the standard deviation of the spectrum of Ts. In general cases, these norms are not much different but the fact that the number of non-zero elements of P is the square of ensemble size while that of D is the ensemble size causes the large difference in the orders of elements of P and D. However, the DPP is only an empirical fact and not an inherently mathematical property of Ts. There exist certain spectra of Ts that break the DPP but such spectra are rarely observed in practice since their occurrences require an unrealistic situation where background errors are larger than observation errors by at least two orders of magnitude in all modes in observation space.
- 著者
- DUC Le SAWADA Yohei
- 出版者
- 公益社団法人 日本気象学会
- 雑誌
- 気象集誌. 第2輯 (ISSN:00261165)
- 巻号頁・発行日
- pp.2024-003, (Released:2023-09-19)
It is well-known in rainfall ensemble forecasts that ensemble means suffer substantially from the diffusion effect resulting from the averaging operator. Therefore, ensemble means are rarely used in practice. The use of the arithmetic average to compute ensemble means is equivalent to the definition of ensemble means as centers of mass or barycenters of all ensemble members where each ensemble member is considered as a point in a high-dimensional Euclidean space. This study uses the limitation of ensemble means as evidence to support the viewpoint that the geometry of rainfall distributions is not the familiar Euclidean space, but a different space. The rigorously mathematical theory underlying this space has already been developed in the theory of optimal transport (OT) with various applications in data science. In the theory of OT, all distributions are required to have the same total mass. This requirement is rarely satisfied in rainfall ensemble forecasts. We, therefore, develop the geometry of rainfall distributions from an extension of OT called unbalanced OT. This geometry is associated with the Gaussian-Hellinger (GH) distance, defined as the optimal cost to push a source distribution to a destination distribution with penalties on the mass discrepancy between mass transportation and original mass distributions. Applications of the new geometry of rainfall distributions in practice are enabled by the fast and scalable Sinkhorn-Knopp algorithms, in which GH distances or GH barycenters can be approximated in real-time. In the new geometry, ensemble means are identified with GH barycenters, and the diffusion effect, as in the case of arithmetic means, is avoided. New ensemble means being placed side-by-side with deterministic forecasts provide useful information for forecasters in decision-making.