A numerical integration-based Kalman filter for moderately nonlinear systems

Abstract

This paper introduces a computationally efficient data assimilation scheme based on Gaussian quadrature filtering that potentially outperforms current methods in data assimilation for moderately nonlinear systems. Moderately nonlinear systems, in this case, are systems with numerical models with small fourth and higher derivative terms. Gaussian quadrature filters are a family of filters that make simplifying Gaussian assumptions about filtering pdfs in order to numerically evaluate the integrals found in Bayesian data assimilation. These filters are differentiated by the varying quadrature rules to evaluate the arising integrals. The approach we present, denoted by Assumed Gaussian Reduced (AGR) filter, uses a reduced order version of the polynomial quadrature first proposed in Ito and Xiong [2000. Gaussian filters for nonlinear filtering problems. IEEE Trans. Automat. Control. 45, 910–927]. This quadrature uses the properties of Gaussian distributions to form an effectively higher order method increasing its efficiency. To construct the AGR filter, this quadrature is used to form a reduced order square-root filter, which will reduce computational costs and improve numerical robustness. For cases of sufficiently small fourth derivatives of the nonlinear model, we demonstrate that the AGR filter outperforms ensemble Kalman filters (EnKFs) for a Korteweg-de Vries model and a Boussinesq model.

Keywords:

• data assimilation
• Bayesian filtering
• Kalman filtering
• numerical integration

Acknowledgments

The authors would like to express their gratitude to Dr Nancy Baker from the U.S. Naval Research Laboratory for her insights and discussions that have greatly improved this manuscript. This research is supported by the Office of Naval Research (ONR) through the NRL Base Program PE 0601153N.

A. Formulas

A.1. Expectation formulas

Consider the expectation (A.1) $$E[x_t,Y_{T-1}]={\int }_{{\mathbb{R}}^n}{x}_{t}p(x_t|Y_{T-1})\mathrm{d}{x}_t$$(A.1) (A.2) $$={\int }_{{\mathbb{R}}^n}\left[{\int }_{{\mathbb{R}}^n}{x}_{t}p(x_t|x_{t-1})d{x}_t\right]p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}$$(A.2) (A.3) $$={\int }_{{\mathbb{R}}^n}f(x_{t-1})p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}$$(A.3)

where $E[·]$ is the expectation. Note that (A.2)(A.2) $$={\int }_{{\mathbb{R}}^n}\left[{\int }_{{\mathbb{R}}^n}{x}_{t}p(x_t|x_{t-1})d{x}_t\right]p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}$$(A.2) follows from (2.1)(2.1) $$p(x_t|Y_{T-1})=\int p(x_t|x_{t-1})p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}.$$(2.1) and Fubini’s theorem and (2.7)(2.7) $$={\int }_{{\mathbb{R}}^n}f(x_{t-1})p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}$$(2.7) follows from (2.3)(2.3) $$x_t=f(x_{t-1})+w_t$$(2.3) and w_t being Gaussian. Similarly, the predicted covariance is given by (A.4) $$E\left[x_t{x}_t^T\right]={\int }_{{\mathbb{R}}^n}{x}_t{x}_t^{T}p(x_t|Y_{T-1})\mathrm{d}{x}_t$$(A.4) (A.5) $$={\int }_{{\mathbb{R}}^n}\left[{\int }_{{\mathbb{R}}^n}{x}_t{x}_t^{T}p(x_t|x_{t-1})d{x}_t\right]p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}$$(A.5) (A.6) $$={\int }_{{\mathbb{R}}^n}f(x_{t-1})f{(x_{t-1})}^{T}p(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}+Q.$$(A.6)

A.2. Covariance

The cross covariance in (2.13)(2.13) $$=N\left(\left(\begin{array}{c}{x_t}^b\\ {{\hat{y}}_t}^b\end{array}\right),\left(\begin{array}{cc}{P_t}^b& {P_t}^b{H}^T\\ H{P_t}^b& H{P_t}^b{H}^T+R\end{array}\right)\right)$$(2.13) is computed via (A.7) $$P_t^{xy}=E[(x_t-x_t^b){({\widehat{y}}_t-{\widehat{y}}_t^b)}^{\mathrm{T}}]$$(A.7) (A.8) $$=\int (x_t-x_t^b){({\widehat{y}}_t-{\widehat{y}}_t^b)}^{T}p(x_t,{\widehat{y}}_t|Y_{t-1})\mathrm{d}{x}_t\mathrm{d}{\widehat{y}}_t$$(A.8) (A.9) $$={\int }_{{\mathbb{R}}^n}(x_t-x_t^b)\left[{\int }_{{\mathbb{R}}^d}{({\widehat{y}}_t-{\widehat{y}}_t^b)}^{T}p({\widehat{y}}_t|x_t)\mathrm{d}{\widehat{y}}_t\right]p(x_t|Y_{T-1})\mathrm{d}{x}_t$$(A.9) (A.10) $$={\int }_{{\mathbb{R}}^n}(x_t-x_t^b){(H{x}_t-H{x}_t^b)}^{\mathrm{T}}p(x_t|Y_{t-1})\mathrm{d}{x}_t$$(A.10) (A.11) $$={\int }_{{\mathbb{R}}^n}(x_t-x_t^b){(x_t-x_t^b)}^{\mathrm{T}}{H}^{\mathrm{T}}p(x_t|Y_{t-1})\mathrm{d}{x}_t$$(A.11) (A.12) $$=P_t^b{H}^T.$$(A.12)

B. Qquadrature error

B.1. Scalar quadrature error

The quadrature error in evaluating the integrals (2.18) and (2.19)(2.19) $$P_t^b={\int }_{{\mathbb{R}}^n}(f(x_{t-1})-x_t^b){(f(x_{t-1})-x_t^b)}^{T}N(x_{t-1}|Y_{T-1})\mathrm{d}{x}_{t-1}+Q.$$(2.19) comes from the low-order polynomial approximation (3.2)(3.2) $$\gamma (s)=\tilde{F}(0)+a_{1}s+\frac{1}{2}{a}_2{s}^2$$(3.2) . Consider the estimation error of (3.6)(3.6) $$=f(x_{t-1}^a)+\frac{1}{2}{a}_2.$$(3.6) given by (B.1) $$e_{\text{mean}}={\int }_{\mathbb{R}}(f(\sqrt{P}\eta +x_{t-1}^a)-\gamma (\eta ))N(\eta |0,1)\mathrm{d}\eta$$(B.1) (B.2) $$={\int }_{\mathbb{R}}(\frac{1}{6}{a}_3{\eta }^3+\frac{1}{24}{a}_4{\eta }^4+\cdots )\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{\eta }^2}\mathrm{d}\eta$$(B.2) (B.3) $$=\frac{3}{24}{a}_4+\cdots$$(B.3) where $a_3,a_4$ are the third and fourth derivatives, respectively, of f. Similarly, the quadrature error for (3.9)(3.9) $$=a_1^2+\frac{1}{2}{a}_2^2+Q.$$(3.9) is given by (B.4) $$e_{\text{covariance}}={\int }_{\mathbb{R}}(f(\sqrt{P}\eta +x_{t-1}^a)-x_{t-1}^a{)}^{2}N(\eta |0,1)\mathrm{d}\eta$$(B.4) (B.5) $$={\int }_{\mathbb{R}}{\left(\frac{1}{6}{a}_3{\eta }^3+\frac{1}{24}{a}_4{\eta }^4+\cdots -\left(\frac{3}{24}{a}_4+\cdots \right)\right)}^2\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{\eta }^2}\mathrm{d}\eta$$(B.5) (B.6) $$=\frac{15}{36}{a}_3^2+\frac{105}{576}{a}_4^2+\cdots .$$(B.6)

B.2. N-d quadrature error

The estimation error of (3.30) and (3.34)(3.34) $$=Q+\sum _{i=1}^n{a}_i{a}_i^T+\frac{1}{2}\sum _{i=1}^n{b}_i{b}_i^T.$$(3.34) is given by (B.7) $$e_{\text{mean}}=\frac{1}{{(2\pi )}^{n/2}}{\int }_{{\mathbb{R}}^n}\left(\frac{1}{2}\sum _{i\ne j}^n{b}_{i,j}{\eta }_i{\eta }_j+\frac{1}{6}\sum _{i,j,k=1}^n{c}_{i,j,k}{\eta }_i{\eta }_j{\eta }_k+\frac{1}{24}\sum _{i,j,k,\ell =1}^n{d}_{i,j,k,\ell }{\eta }_i{\eta }_j{\eta }_k{\eta }_{\ell }\cdots \right)e^{-1/2|\eta {|}^2}\mathrm{d}\eta$$(B.7) (B.8) $$=\frac{1}{8}\sum _{i=1}^n{d}_{i,i,i,i}+\frac{1}{24}\sum _{i=j,k=\ell ,i\ne k}{d}_{i,i,k,k}+\frac{1}{24}\sum _{i=k,j=\ell ,i\ne j}{d}_{i,j,i,j}+\frac{1}{24}\sum _{i=\ell ,j=k,i\ne k}{d}_{i,j,j,i}+\cdots$$(B.8) and (B.9) $$e_{\text{covariance}}=\frac{1}{{(2\pi )}^{n/2}}{\int }_{{\mathbb{R}}^n}\left(\frac{1}{2}\sum _{i\ne j}^n{b}_{i,j}{\eta }_i+\frac{1}{6}\sum _{i,j,k=1}^n{c}_{i,j,k}{\eta }_i{\eta }_j{\eta }_k+\cdots -\left(\frac{1}{8}\sum _{i=1}^n{d}_{i,i,i,i}+\cdots \right)\right)$$(B.9) (B.10) $$·{\left(\frac{1}{2}\sum _{i\ne j}^n{b}_{i,j}{\eta }_i+\frac{1}{6}\sum _{i,j,k=1}^n{c}_{i,j,k}{\eta }_i{\eta }_j{\eta }_k+\cdots -\left(\frac{1}{8}\sum _{i=1}^n{d}_{i,i,i,i}+\cdots \right)\right)}^{\mathrm{T}}{e}^{-1/2|\eta {|}^2}d\eta$$(B.10) (B.11) $$=\frac{1}{4}\sum _{i\ne j}^n{b}_{ij}{b}_{ij}^{\mathrm{T}}+\frac{15}{36}\sum _{i=1}^n{c}_{\mathit{iii}}{c}_{\mathit{iii}}^{\mathrm{T}}+\frac{1}{36}\sum _{i\ne j\ne k}^n{c}_{\mathit{ijk}}{c}_{\mathit{ijk}}^{\mathrm{T}}+\cdots$$(B.11) where $$c_{\mathit{ijk}}=\sum _{i,j,k=1}^n\frac{{\partial }^{3}f}{\partial x_i\partial x_j\partial x_k}\text{and}{d}_{\mathit{ijk}\ell }=\sum _{i,j,k,\ell =1}^n\frac{{\partial }^{4}f}{\partial x_i\partial x_j\partial x_k\partial x_{\ell }}.$$

Note that the error on the mean does not depend on the cross terms in the Jacobian.