# Real-time Monitoring of Pollutant Diffusion States and Source Using Fuzzy Adaptive Kalman Filter

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## Abstract

An inverse analysis method for the real-time monitoring of pollutant diffusion is developed based on fuzzy adaptive Kalman filter (FAKF) coupled with weighted recursive least squares algorithm (WRLSA). In the monitoring process, the discrete diffusion states equation is established first. Then, the FAKF is adopted to realize the precise monitoring of the pollution diffusion states while the WRLSA is used to monitor the pollutant source in real time. Finally, the simulations are presented to validate the effectiveness of the technique, which shows that this technique has wide applications in situations with several different kinds of sources and measurement noises. Besides, the results demonstrate the strong robustness of this method to have great monitoring performance.

## Keywords

Diffusion state Fuzzy adaptive Kalman filter Inverse analysis Pollutant source Real-time monitoring## Nomenclature

*e, e*_{c}inputs of fuzzy inference unit

*f*number of measurement points

*C*pollutant concentration

*D*_{x},*D*_{y}diffusion coefficients

*EC*concentration monitoring error

*ES*_{o}pollutant source monitoring error

*L*_{x},*L*_{y}length and width

*S*_{o}pollutant source strength

**B**_{b}sensitive matrix of WRLSA

**H**measurement matrix

**I**unit matrix

**K**gain matrix of Kalman filter

**K**_{b}gain matrix of WRLSA

**M**sensitive matrix of WRLSA

**P**covariance of state estimation errors

**P**_{b}error covariance of source estimation

**Q**process noise covariance

**R**measurement noise covariance

**S**residual variance

**Z**output vector

- \( \overline{\boldsymbol{Z}} \)
sequence of measurement residual

## Greek Symbols

*γ*weighting factor

*σ*_{q}standard deviation of process noises

*σ*_{r}standard deviation of measurement noises

*τ*time

**Ф**state transition matrix

**Ψ**input matrix

## Superscripts

- ^
monitored result in equations

^{T}transpose of a vector or a matrix

*k*the

*k*th time step

## Subscripts

- exa
exact result

- mon
monitored result

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