Observation impact in data assimilation: the effect of non-Gaussian observation error

Alison Fowler; Peter Jan Van Leeuwen

doi:10.3402/tellusa.v65i0.20035

Observation impact in data assimilation: the effect of non-Gaussian observation error

Tellus

Volume 65 (2013): Issue 1

By: Alison Fowler and Peter Jan Van Leeuwen

Open Access

|Dec 2013

Figures & Tables

Skewness (left) and kurtosis (right) of the GM2 distribution as a function of w and µ ₂–µ ₁ when σ ²=1.

Schematic of experimental setup in section 3. Left hand panel: Non-Gaussian prior and Gaussian likelihood as in Fowler and van Leeuwen (2012). Right hand panel: Non-Gaussian likelihood and Gaussian prior, which is the focus of this paper. In each case the non-Gaussian parameters are given by $w = 0.25$ , $σ^{2} = 1$ , $∣ μ_{1} - μ_{2} ∣ = 3$ . The variance of the Gaussian distributions are chosen such that in each case $σ_{y}^{2} / σ_{x}^{2} = \frac{32}{43}$ , for agreement with Fowler and van Leeuwen (2012), giving k=1849/512 and $κ = 2$ .

Comparison of $S = \frac{\partial μ_{a}}{\partial y}$ as a function of d when the likelihood is non-Gaussian (prior is Gaussian) (thin blue line) and when the prior is non-Gaussian (likelihood is Gaussian) (thin black line). In each case the non-Gaussian distribution is a two component Gaussian mixture with identical variances with parameter values as in Fig. 2. The variance of the Gaussian distributions, also given in Fig. 2, are chosen such that the Gaussian estimate to the sensitivity is the same in each case (bold, dashed line). Also marked on is $\frac{k}{k + 1}$ (bold blue line) and $\frac{1}{κ + 1}$ (bold black line), and d _p (black dashed line) and d _l (blue dashed line).

Comparison of S normalised by its Gaussian approximation when the likelihood is non-Gaussian (prior is Gaussian) (a, c) and when the prior is non-Gaussian (likelihood is Gaussian) (b, d). In the top two panels the effect of separating the Gaussian components (y-axis) is given when the weights are equal. In the bottom two panels the effect of varying the weights of the Gaussian components (y-axis) is given when $∣ μ_{1} - μ_{2} ∣ = 3$ . In all cases σ ²=1 and $σ_{y}^{2} / σ_{x}^{2}$ is kept constant at $\frac{32}{43}$ by varying the values of k and $κ$ .

Top: Comparison of MI normalised by its Gaussian approximation as a function of w (x axis) and µ ₂–µ ₁ (y axis) when (a) the likelihood is non-Gaussian (prior is Gaussian) and (b) when the prior is non-Gaussian (likelihood is Gaussian). Bottom: Comparison of $\int p (y) S d y$ normalised by its Gaussian approximation as a function of w (x axis) and $μ_{2} - μ_{1}$ (y axis) when (c) the likelihood is non-Gaussian (prior is Gaussian) and (d) when the prior is non-Gaussian (likelihood is Gaussian). In all cases σ ²=1 and $σ_{y}^{2} / σ_{b}^{2}$ is kept constant at $\frac{32}{43}$ by varying the values of k and $κ$ . Red contours indicate values above 1 and blue contours indicate values below one. The contours are separated by increments of 0.02.

Table 1. Comparison of a non-Gaussian likelihood's and non-Gaussian prior's effect on the observation impact

non-Gaussian likelihood/Gaussian priornon-Gaussian prior/Gaussian likelihood

\frac{\partial μ_{a}}{\partial μ_{y}} = 1 - σ_{a}^{2} / σ_{x}^{2}

in the scalar case.

\frac{\partial μ_{a}}{\partial μ_{y}} = σ_{a}^{2} / σ_{y}^{2}

in the scalar caseSensitivity is bounded by

- \infty

and 1.Sensitivity is bounded by 0 and ∞.As a function of innovation the peak in sensitivity coincides with a minimum in analysis error covariance.As a function of innovation the peak in sensitivity coincides with a maximum in analysis error covariance.Variability (as a function of d) in the error of the Gaussian approximation to sensitivity is smaller than the error in the relative entropy.Variability (as a function of d) in the error of the Gaussian approximation to sensitivity is larger than the error in the relative entropy.On average Gaussian approximation underestimates observation impact.On average Gaussian approximation overestimates observation impact.The error in the Gaussian approximation to the average sensitivity is larger than the error in the Gaussian approximation to mutual information (the average relative entropy).The error in the Gaussian approximation to the average sensitivity is smaller than the error in the Gaussian approximation to mutual information.

Left: Huber likelihood (black) $a = - 0.5$ , b=1 and σ ²=2, when $μ_{y} = - 19.4$ . Gaussian prior (blue), µ _x=0, variance of prior chosen such that for the variance of the full likelihood ( $σ_{y}^{2}$ not σ ²) $σ_{y}^{2} / σ_{x}^{2}$ is 32/43. Right: The sensitivity of the analysis to the observation.

Mutual information (red), sensitivity (black), relative entropy (blue) and the average sensitivity (black dashed) all normalised by their Gaussian approximations plotted as a function of d.

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DOI: https://doi.org/10.3402/tellusa.v65i0.20035 | Journal eISSN: 3035-9554

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Language: English

Page range: 20035 - 20035

Submitted on: Nov 6, 2012

Accepted on: May 1, 2013

Published on: Dec 1, 2013

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

Keywords:

mutual information,

relative entropy,

sensitivity

© 2013 Alison Fowler, Peter Jan Van Leeuwen, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 65 (2013): Issue 1

Observation impact in data assimilation: the effect of non-Gaussian observation error

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