# A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Ensemble Kalman Filters Based on Modified Cholesky Decomposition

#### 2.2. Gaussian Mixture Models Based Filters

## 3. Proposed Method

#### 3.1. Estimation of Hyper-Parameters—EM Method

#### 3.2. Sampling Method—Approaching the Posterior

- Step 1
- Let $k=1$, set $u=0$, go to step 2.
- Step 2
- Set ${\mathbf{x}}^{\left(u\right)}={\overline{\mathbf{x}}}_{k}^{b}$.
- Step 3
- Step 4
- Compute ${\mathbf{z}}^{\left(u\right)}$ via Equation (19).
- Step 5
- Set ${\mathbf{x}}^{(u+1)}$ according to Equation (20).
- Step 6
- If $u\le v$ set $u=u+1$ and go to step 3, set ${\overline{\mathbf{x}}}_{k}^{a}={\mathbf{x}}^{\left(v\right)}$ and go to step 7 otherwise.
- Step 7
- If $k\le K$ go to step 1, go to step 8 otherwise.
- Step 8
- The posterior mode approximations read ${\left\{{\overline{\mathbf{x}}}_{k}^{a}\right\}}_{k=1}^{K}$.

#### 3.3. Building the Posterior Ensemble

#### 3.4. Computational Complexity

- During the E-Step, the computations of weights (14a) depend on the calculation:$$\begin{array}{ccc}\hfill {\displaystyle -\frac{1}{2}\xb7{\u2225{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\u2225}_{{\left[{\widehat{\mathbf{B}}}_{k}^{\left(p\right)}\right]}^{-1}}^{2}}& =& -\frac{1}{2}\xb7{\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]}^{T}\xb7{\widehat{\mathbf{B}}}_{k}^{-1}\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]\hfill \\ & =& -\frac{1}{2}\xb7{\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]}^{T}\xb7{\mathbf{L}}_{k}^{T}\xb7{\mathbf{D}}_{k}\xb7{\mathbf{L}}_{k}\xb7\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]\hfill \\ & =& -\frac{1}{2}\xb7\underset{{\mathbf{s}}_{k}}{\overset{T}{\underbrace{\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]}}}\xb7{\mathbf{L}}_{k}^{T}\xb7{\mathbf{D}}_{k}^{T/2}\xb7{\mathbf{D}}_{k}^{1/2}\xb7{\mathbf{L}}_{k}\xb7\underset{{\mathbf{s}}_{k}}{\underbrace{\left[{\mathbf{x}}^{b\left[e\right]}-{\mathbf{x}}_{k}^{b\left(p\right)}\right]}}\hfill \\ & =& -\frac{1}{2}\xb7{\mathbf{s}}_{k}^{T}\xb7{\mathbf{L}}_{k}^{T}\xb7{\mathbf{D}}_{k}^{T/2}\xb7{\mathbf{D}}_{k}^{1/2}\xb7{\mathbf{L}}_{k}\xb7{\mathbf{s}}_{k}\hfill \\ & =& {\left[{\mathbf{D}}_{k}^{1/2}\xb7\underset{{\widehat{\mathbf{s}}}_{k}}{\underbrace{{\mathbf{L}}_{k}\xb7{\mathbf{s}}_{k}}}\right]}^{T}\xb7{\mathbf{D}}_{k}^{1/2}\xb7\underset{{\widehat{\mathbf{s}}}_{k}}{\underbrace{{\mathbf{L}}_{k}\xb7{\mathbf{s}}_{k}}}\hfill \\ & =& {\left[\underset{{\tilde{\mathbf{s}}}_{k}}{\underbrace{{\mathbf{D}}_{k}^{1/2}\xb7{\widehat{\mathbf{s}}}_{k}}}\right]}^{T}\xb7\underset{{\tilde{\mathbf{s}}}_{k}}{\underbrace{{\mathbf{D}}_{k}^{1/2}\xb7{\widehat{\mathbf{s}}}_{k}}}={\tilde{\mathbf{s}}}_{k}^{T}\xb7{\tilde{\mathbf{s}}}_{k}={\u2225{\tilde{\mathbf{s}}}_{k}\u2225}_{2}^{2}\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$From this step, given the special structure of ${\mathbf{L}}_{k}$, ${\tilde{\mathbf{s}}}_{k}\in {\mathbb{R}}^{n\times 1}$ can be computed with no more than $\mathcal{O}\left({\theta}^{2}\xb7n\right)$ long computations where $\theta $ denotes the maximum number of non-zero elements across all rows in ${\mathbf{L}}_{k}$ with $\theta \ll n$. Likewise, the number of long computations in order to obtain $\tilde{\mathbf{s}}\in {\mathbb{R}}^{n\times 1}$ is bounded by $\mathcal{O}\left(n\right)$ since ${\mathbf{D}}_{k}$ is diagonal. Thus, since there are K clusters, each E-step has the following operation counting:$$\mathcal{O}\left(K\xb7\left[n+{\theta}^{2}\xb7n\right]\right)\phantom{\rule{0.166667em}{0ex}}$$
- During the M-step, updating the centroids (15b) can be performed with no more than $\mathcal{O}\left({N}^{2}\xb7n\right)$ since ${\mathbf{D}}_{k}$ has only n components different from zero (the diagonal ones), the least square solution of (15c) is bounded by $\mathcal{O}\left({\theta}^{2}\xb7n\right)$ calculations since there are n model components, and the cost of (15e) is bounded by $\mathcal{O}\left(n\xb7{\theta}^{2}\right)$ since the multiplication of coefficients and model components is constrained to the neighbourhood of each model component. The computational effort of this method is then estimated as follows:$$\mathcal{O}\left(K\xb7\left[{\theta}^{2}\xb7n+{N}^{2}\xb7n\right]\right)\phantom{\rule{0.166667em}{0ex}}$$
- During the sampling procedure, the gradient (18b) can be efficiently computed as follows:$$\begin{array}{ccc}\hfill {\displaystyle \nabla {\widehat{\mathcal{J}}}_{k}\left(\mathbf{x}\right)}& =& {\widehat{\mathbf{B}}}_{k}^{-1}\xb7\left[\underset{{\mathbf{g}}_{k}}{\underbrace{\mathbf{x}-{\overline{\mathbf{x}}}_{k}^{b}}}\right]-{\mathbf{H}}_{{\mathbf{x}}^{\left(u\right)}}^{T}\xb7{\mathbf{R}}^{-1}\xb7\left[\underset{{\mathbf{f}}_{k}}{\underbrace{\mathbf{y}-{\mathcal{G}}_{u}\left(\mathbf{x}\right)}}\right]={\mathbf{L}}_{k}^{T}\xb7{\mathbf{D}}_{k}\xb7\underset{\widehat{{\mathbf{g}}_{k}}}{\underbrace{{\mathbf{L}}_{k}\xb7{\mathbf{g}}_{k}}}-{\mathbf{H}}_{{\mathbf{x}}^{\left(u\right)}}^{T}\xb7\underset{\widehat{{\mathbf{f}}_{k}}}{\underbrace{{\mathbf{R}}^{-1}\xb7{\mathbf{f}}_{k}}}\phantom{\rule{0.166667em}{0ex}}\hfill \\ & =& {\mathbf{L}}_{k}^{T}\xb7\underset{\tilde{{\mathbf{g}}_{k}}}{\underbrace{{\mathbf{D}}_{k}\xb7{\widehat{\mathbf{g}}}_{k}}}-\underset{\tilde{{\mathbf{f}}_{k}}}{\underbrace{{\mathbf{H}}_{{\mathbf{x}}^{\left(u\right)}}^{T}\xb7\widehat{{\mathbf{f}}_{k}}}}=\underset{\overline{{\mathbf{g}}_{k}}}{\underbrace{{\mathbf{L}}_{k}^{T}\xb7{\tilde{\mathbf{g}}}_{k}}}-\tilde{{\mathbf{f}}_{k}}=\overline{{\mathbf{g}}_{k}}-{\mathbf{f}}_{k}\phantom{\rule{0.166667em}{0ex}}\hfill \end{array}$$$$\mathcal{O}\left(v\xb7\left[n+{\theta}^{2}\xb7n\right]\right)\phantom{\rule{0.166667em}{0ex}}$$
- The posterior ensemble can be built (Section 3.3 [19]) with no more than$$\mathcal{O}\left(n\xb7{\theta}^{2}+m\xb7N\right)\phantom{\rule{0.166667em}{0ex}},$$

#### 3.5. Comparison of GM-EnKF-MCMC with Other Sampling Methods

## 4. Experimental Settings

- Starting with an initial random solution, a 4th order Runge Kutta method is employed in order to integrate it over a long time period from which initial condition ${\mathbf{x}}_{-2}^{*}\in {\mathbb{R}}^{n\times 1}$ is obtained.
- A perturbed background solution ${\tilde{\mathbf{x}}}_{-2}^{b}$ is formed at time ${t}_{-2}$ by drawing a sample from the Normal distribution,$$\begin{array}{c}\hfill {\displaystyle {\tilde{\mathbf{x}}}_{-2}^{b}\sim \mathcal{N}\left({\mathbf{x}}_{-2}^{*},\phantom{\rule{0.166667em}{0ex}}{0.05}^{2}\xb7\mathbf{I}\right)\phantom{\rule{0.166667em}{0ex}}}\end{array}$$
- An initial perturbed ensemble is built about the background state by taking samples from the distribution,$$\begin{array}{c}\hfill {\displaystyle {\tilde{\mathbf{x}}}_{-1}^{b\left[\widehat{e}\right]}\sim \mathcal{N}\left({\mathbf{x}}_{-1}^{b},\phantom{\rule{0.166667em}{0ex}}{0.05}^{2}\xb7\mathbf{I}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}1\le \widehat{e}\le \widehat{N}\phantom{\rule{0.166667em}{0ex}}}\end{array}$$
- Two assimilation windows are proposed for the tests, both of them consist of $M=15$ observations. In the first assimilation window, observations are taken every 16 h (time step of 0.1 time units) while in the last one, observations are available every 50 h (time step of 0.3 time units). We denote by $\delta t\in \{16,\phantom{\rule{0.166667em}{0ex}}50\}$ the elapsed time between two observations.
- The observational errors are described by the probability distribution,$$\begin{array}{c}\hfill {\displaystyle {\mathbf{y}}_{\ell}\sim \mathcal{N}\left({\mathcal{H}}_{k}\left({\mathbf{x}}_{\ell}^{*}\right)\phantom{\rule{0.166667em}{0ex}},{\left[{\u03f5}^{o}\right]}^{2}\xb7\mathbf{I}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4.pt}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}1\le \ell \le M\phantom{\rule{0.166667em}{0ex}}}\end{array}$$
- We consider the non-linear observation operator [32]:$$\left\{\mathcal{H}\left(\mathbf{x}\right)\right\}}_{j}=\frac{{\left\{\mathbf{x}\right\}}_{j}}{2}\xb7\left[{\left(\frac{\left|{\left\{\mathbf{x}\right\}}_{j}\right|}{2}\right)}^{\gamma -1}+1\right]\phantom{\rule{0.166667em}{0ex}$$
- We consider two percentages of observed components s from the model state $s\in \{70\%,\phantom{\rule{0.166667em}{0ex}}100\%\}$.
- The radius of influence is set to $r=1$ while the inflation factor is set to 1.02 (a typical value).
- We propose two ensemble sizes for the benchmark $N\in \{20,\phantom{\rule{0.166667em}{0ex}}80\}$. These members are randomly chosen from the pool ${\widehat{{\mathbf{X}}^{b}}}_{0}$ for different experiments in order to form the initial ensemble ${\mathbf{X}}_{0}^{b}$ for the assimilation window. Evidently, ${\mathbf{X}}_{0}^{b}\subset {\widehat{{\mathbf{X}}^{b}}}_{0}$.
- The $L-2$ norm of errors are utilized as a measure of accuracy at the assimilation step ℓ,$${\lambda}_{\ell}=\sqrt{{\left[{\mathbf{x}}_{\ell}^{*}-{\mathbf{x}}_{\ell}^{a}\right]}^{T}\xb7\left[{\mathbf{x}}_{\ell}^{*}-{\mathbf{x}}_{\ell}^{a}\right]}\phantom{\rule{0.166667em}{0ex}}$$
- The Root-Mean-Square-Error (RMSE) is used as a measure of performance. On average, on a given assimilation window,$$\lambda =\sqrt{\frac{1}{M}\xb7\sum _{\ell =1}^{M}{\lambda}_{\ell}^{2}}\phantom{\rule{0.166667em}{0ex}}$$

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Experimental results with the Lorenz-96 model (29). The time evolution of mean analysis errors and their standard deviation across the 10 different experimental configurations are reported for $\gamma \in \{1,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}5\}$, and $\delta t=16$.

**Figure 2.**Experimental results with the Lorenz-96 model (29). The time evolution of mean analysis errors and their standard deviation across the 10 different experimental configurations are reported for $\gamma \in \{1,\phantom{\rule{0.166667em}{0ex}}3,\phantom{\rule{0.166667em}{0ex}}5\}$, and $\delta t=50$.

**Figure 3.**Two dimensional projections of the steps performed by the sampling method. Blue dots denote prior ensemble members ${\mathbf{x}}^{b\left[e\right]}$, large blue dots stand for centroids ${\overline{\mathbf{x}}}_{k}^{b}$, dashed black lines together with black dots denote accepted states ${\mathbf{x}}^{\left(u\right)}$, and the red dot stands for the actual state of the system ${\mathbf{x}}^{*}$. Even for $K=N$, the method is able to obtain reasonable estimates of the actual state of the system. The variance explained in such plots is larger than $90\%$.

**Table 1.**Experimental results with the Lorenz-96 model. Mean of RMSE values are reported across the 10 different experimental configurations for different configuration of parameters $\delta t$, s, K, and $\gamma $. The number of ensemble members equals $N=20$.

$\mathit{s}$ | $\mathit{\delta}\mathit{t}$ | $\mathit{\gamma}=\mathbf{1}$ | $\mathit{\gamma}=\mathbf{3}$ | $\mathit{\gamma}=\mathbf{5}$ |
---|---|---|---|---|

70% | 16 h | |||

50 h | ||||

100% | 16 h | |||

50 h |

**Table 2.**Experimental results with the Lorenz-96 model. Mean of RMSE values are reported across the 10 different experimental configurations for different configuration of parameters $\delta t$, s, K, and $\gamma $. The number of ensemble members equals $N=80$.

s | $\mathit{\delta}\mathit{t}$ | $\mathit{\gamma}=\mathbf{1}$ | $\mathit{\gamma}=\mathbf{3}$ | $\mathbf{\gamma}=\mathbf{5}$ |
---|---|---|---|---|

70% | 16 h | |||

50 h | ||||

100% | 16 h | |||

50 h |

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**MDPI and ACS Style**

Nino-Ruiz, E.D.; Cheng, H.; Beltran, R. A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models. *Atmosphere* **2018**, *9*, 126.
https://doi.org/10.3390/atmos9040126

**AMA Style**

Nino-Ruiz ED, Cheng H, Beltran R. A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models. *Atmosphere*. 2018; 9(4):126.
https://doi.org/10.3390/atmos9040126

**Chicago/Turabian Style**

Nino-Ruiz, Elias D., Haiyan Cheng, and Rolando Beltran. 2018. "A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models" *Atmosphere* 9, no. 4: 126.
https://doi.org/10.3390/atmos9040126