### Abstract

A statistically exactly solvable model for passive tracers is introduced as a test model for the authors' Nonlinear Extended Kalman Filter (NEKF) as well as other filtering algorithms. The model involves a Gaussian velocity field and a passive tracer governed by the advection-diffusion equation with an imposed mean gradient. The model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere. One of the attractive properties of the model is the existence of the exact statistical solution. In particular, this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real-time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales. Here, we extensively study the performance of the NEKF which uses the exact first and second order nonlinear statistics without any approximations due to linearization. The role of partial and sparse observations, the frequency of observations and the observation noise strength in recovering the true signal, its spectrum, and fat tail probability distribution are the central issues discussed here. The results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations.

Original language | English (US) |
---|---|

Pages (from-to) | 1602-1638 |

Number of pages | 37 |

Journal | Journal of Computational Physics |

Volume | 230 |

Issue number | 4 |

DOIs | |

State | Published - Feb 20 2011 |

### Fingerprint

### Keywords

- Data assimilation
- Exactly solvable model
- Nonlinear Kalman filter
- Turbulent advection-diffusion equation

### ASJC Scopus subject areas

- Computer Science Applications
- Physics and Astronomy (miscellaneous)

### Cite this

**Filtering a statistically exactly solvable test model for turbulent tracers from partial observations.** / Gershgorin, B.; Majda, A. J.

Research output: Contribution to journal › Article

*Journal of Computational Physics*, vol. 230, no. 4, pp. 1602-1638. https://doi.org/10.1016/j.jcp.2010.11.024

}

TY - JOUR

T1 - Filtering a statistically exactly solvable test model for turbulent tracers from partial observations

AU - Gershgorin, B.

AU - Majda, A. J.

PY - 2011/2/20

Y1 - 2011/2/20

N2 - A statistically exactly solvable model for passive tracers is introduced as a test model for the authors' Nonlinear Extended Kalman Filter (NEKF) as well as other filtering algorithms. The model involves a Gaussian velocity field and a passive tracer governed by the advection-diffusion equation with an imposed mean gradient. The model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere. One of the attractive properties of the model is the existence of the exact statistical solution. In particular, this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real-time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales. Here, we extensively study the performance of the NEKF which uses the exact first and second order nonlinear statistics without any approximations due to linearization. The role of partial and sparse observations, the frequency of observations and the observation noise strength in recovering the true signal, its spectrum, and fat tail probability distribution are the central issues discussed here. The results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations.

AB - A statistically exactly solvable model for passive tracers is introduced as a test model for the authors' Nonlinear Extended Kalman Filter (NEKF) as well as other filtering algorithms. The model involves a Gaussian velocity field and a passive tracer governed by the advection-diffusion equation with an imposed mean gradient. The model has direct relevance to engineering problems such as the spread of pollutants in the air or contaminants in the water as well as climate change problems concerning the transport of greenhouse gases such as carbon dioxide with strongly intermittent probability distributions consistent with the actual observations of the atmosphere. One of the attractive properties of the model is the existence of the exact statistical solution. In particular, this unique feature of the model provides an opportunity to design and test fast and efficient algorithms for real-time data assimilation based on rigorous mathematical theory for a turbulence model problem with many active spatiotemporal scales. Here, we extensively study the performance of the NEKF which uses the exact first and second order nonlinear statistics without any approximations due to linearization. The role of partial and sparse observations, the frequency of observations and the observation noise strength in recovering the true signal, its spectrum, and fat tail probability distribution are the central issues discussed here. The results of our study provide useful guidelines for filtering realistic turbulent systems with passive tracers through partial observations.

KW - Data assimilation

KW - Exactly solvable model

KW - Nonlinear Kalman filter

KW - Turbulent advection-diffusion equation

UR - http://www.scopus.com/inward/record.url?scp=78650533520&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78650533520&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2010.11.024

DO - 10.1016/j.jcp.2010.11.024

M3 - Article

AN - SCOPUS:78650533520

VL - 230

SP - 1602

EP - 1638

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 4

ER -