Comparison of Traditional and Hybrid Forms of Optimal Localisation for Mitigation of Sampling Error in Ensemble Kalman Filters

Rebecca Susanne Atkinson; Jonathan Flowerdew; Sue Hughes; Ian Roulstone

doi:10.16993/tellusa.35

Comparison of Traditional and Hybrid Forms of Optimal Localisation for Mitigation of Sampling Error in Ensemble Kalman Filters

Tellus

Volume 76 (2024): Issue 1

By: Rebecca Susanne Atkinson , Jonathan Flowerdew, Sue Hughes and Ian Roulstone

Open Access

|Apr 2024

Figures & Tables

Table 1

Types of Localisations. Bottom 3 rows show localisation application novel to this paper. Cost functions are given in terms of the parameter θ which represents the parameters of the distribution.

LOCALISATION	COST FUNCTION	FORM	BASED ON
Gaussian	NA	$\exp (- \frac{d_{i j}^{2}}{2 γ^{2}})$ Damps sample covariance towards 0.	Tuned width parameter γ. The distance between 2 points d_ij.
Optimal for a single covariance (OSTC)	$\int {(α \hat{b} - b)}^{2} P (\hat{b} \| θ) d \hat{b}$	$α = \frac{b_{i j}^{2}}{b_{i j}^{2} + σ_{\hat{b_{i j}}}^{2}}$ Damps sample sparse gain towards 0.	Signal: (mean) sparse gain. Noise: mean variance in sample sparse gain.
Optimal for a variable covariance (OVTC)	$\int P (θ) \int {(α \hat{b} - b)}^{2} P (\hat{b} \| θ) d \hat{b} d θ$	$α = \frac{{\bar{b}}^{2} + σ_{b}^{2}}{{\bar{b}}^{2} + σ_{b}^{2} + 〈 σ_{\hat{b}}^{2} 〉}$ Damps sample sparse gain towards 0.	Signal: mean and variance in sparse gain. Noise: mean variance in sample sparse gain.
hybrid Optimal for a variable covariance (HOVTC)	$\int P (θ) \int {(\bar{b} + α (\hat{b} - \bar{b}) - b)}^{2} P (\hat{b} \| θ) d \hat{b} d θ$	$α = \frac{σ_{b}^{2}}{σ_{b}^{2} + 〈 σ_{\hat{b}}^{2} 〉}$ Damps sample sparse gain towards mean sparse gain.	Signal: variance in sparse gain. Noise: mean variance in sample sparse gain.
hybrid tuned double Gaussian (HTDG)	NA	$\exp (- \frac{d_{i j}^{2}}{2 γ_{1}^{2}}) - \exp (- \frac{d_{i j}^{2}}{2 γ_{2}^{2}})$ Damps sample covariance towards mean covariance.	Tuned width parameters γ₁ and γ₂. The distance between 2 points d_ij.

Schematic of a single DA process using the EnKF.

Schematic showing the ensemble and observation generation, assimilation and error production for a single DA cycle.

The flow diagram of the experiments performed and reported in this paper.

**Plot showing the shape of each localisation and the statistics used to compute the localisation for Scenario 1**. The form of the localisations is shown for observation variance 0.1, 1.0 and 10.0. The shown tuned widths are tuned from assimilating a single observation.

**Plot showing the shape of each localisation and the statistics used to compute the localisation for Scenario 2**. The form of the localisations is shown for observation variance 0.1,1.0 and 10.0. The shown tuned widths are tuned from assimilating a single observation.

Table 2

RMS error results for Scenario 1.

SCENARIO 1 OBSERVATION ERROR VARIANCE 0.1
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	0.8633	0.8656	0.8647	0.8656	0.8634	0.8631	0.8633	0.8522	0.8557	0.8576	0.8558	1.0466
Spacing 33	0.5368	0.5525	0.5376	0.5366	0.5523	0.5402	0.5375	0.5804	2.4186	14.7818	0.5233	1.0462
Spacing 9	0.2440	0.2852	0.2468	0.4289	0.2855	0.2613	0.2449	1.8869	212.2495	38.626	0.2573	1.0461
SCENARIO 1 OBSERVATION ERROR VARIANCE 1.0
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	0.9480	0.9502	0.9523	0.9498	0.9490	0.9506	0.9487	0.9398	0.9410	0.9411	0.9417	1.0466
Spacing 33	0.7848	0.7947	0.7911	0.7860	0.7927	0.7901	0.7854	0.7698	0.7668	0.7682	0.7669	1.0462
Spacing 9	0.5633	0.5886	0.5701	0.5641	0.5870	0.5709	0.5642	1.1460	0.5931	4.6481	0.5493	1.0461
SCENARIO 1 OBSERVATION ERROR VARIANCE 10.0
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	1.0277	1.0283	1.0285	1.0283	1.0280	1.0282	1.0280	1.0252	1.0253	1.0253	1.0253	1.0466
Spacing 33	0.9977	0.9992	0.9993	0.9988	0.9985	0.9987	0.9982	0.9903	0.9906	0.9906	0.9911	1.0462
Spacing 9	0.9152	0.9220	0.9197	0.9184	0.9203	0.9184	0.9173	0.9031	0.8989	0.8989	0.8992	1.0461

Table 3

RMS error results for Scenario 2.

SCENARIO 2 OBSERVATION ERROR VARIANCE 0.1
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	0.8311	0.8296	0.8295	0.8295	0.8295	0.8295	0.8294	0.8068	0.8068	0.8068	0.8070	1.0469
Spacing 33	0.3701	0.3858	0.3763	0.3745	0.3856	0.3758	0.3739	0.5856	0.3495	0.3460	0.3459	1.0464
Spacing 9	0.1995	0.2175	152.6382	136.7548	0.2174	16.2293	109.381	3.4972	0.1947	0.1947	0.1940	1.0463
SCENARIO 2 OBSERVATION ERROR VARIANCE 1.0
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	0.9324	0.9329	0.9350	0.9326	0.9329	0.2349	0.9326	0.9161	0.9161	0.9161	0.9161	1.0469
Spacing 33	0.7279	0.7370	0.7352	0.7289	0.7370	0.7352	0.7288	0.7215	0.6971	0.6971	0.6971	1.0464
Spacing 9	0.5089	0.5280	0.5173	0.5119	0.5280	0.5170	0.5113	1.8282	0.4846	0.4846	0.4847	1.0463
SCENARIO 2 OBSERVATION ERROR VARIANCE 10.0
		OSTC			OVTC			HOVTC
	GAUSSIAN	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	(SG;G)	(C;C)	(SG;C)	HTDG	BACKGROUND ERROR
Single ob	1.0253	1.0252	0.0254	1.0252	1.0252	1.0254	1.0252	1.0207	1.0207	1.0207	1.0207	1.0469
Spacing 33	0.9897	0.9904	0.9907	0.9902	0.9904	0.9907	0.9901	0.9780	0.9782	0.9782	0.9782	1.0464
Spacing 9	0.8941	0.8998	0.8982	0.8969	0.8997	0.8982	0.8968	0.8882	0.8706	0.8706	0.8707	1.0463

**RMS errors resulting from DA with each kind of localisation in Gaussian Scenario 1**. Columns show results for a range of observation spacing and rows show a range of observation error variance.

**RMS errors resulting from DA with each kind of localisation in Gaussian Scenario 2**. Columns show results for a range of observation spacing and rows show a range of observation error variance.

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DOI: https://doi.org/10.16993/tellusa.35 | Journal eISSN: 3035-9554

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Language: English

Page range: 57 - 73

Submitted on: Feb 17, 2022

Accepted on: Mar 19, 2024

Published on: Apr 29, 2024

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

Ensemble methods,

Sampling error,

Optimal Localisation,

Hybrid Localisation,

Observation density

© 2024 Rebecca Susanne Atkinson, Jonathan Flowerdew, Sue Hughes, Ian Roulstone, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 76 (2024): Issue 1

Comparison of Traditional and Hybrid Forms of Optimal Localisation for Mitigation of Sampling Error in Ensemble Kalman Filters

Figures & Tables

Table 1

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Table 2

Table 3

Figure 6

Figure 7

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