Kalman Filter in the Real Time URBIS modelRichard KranenburgMaster scriptie June 11, 2010

Kalman Filter in the Real Time URBIS model Introduction Real Time URBIS model Problems and Goals Method Kalman filter equations Results Extensions on the Kalman Filter Conclusions

Introduction

Company: TNO Business Unit: ‘BenO’ Department: ‘ULO’ Accompanists:

Michiel Roemer (TNO) Jan Duyzer (TNO) Arjo Segers (TNO) Kees Vuik (TUDelft)

Real Time URBIS model

Real Time URBIS Model

URBIS Model, standard concentration fields 11 sources, 4 wind directions, 2 wind speeds

Real Time URBIS model Every hour interpolation between standard

concentration fields Correction for background and traffic fields

μ is the weight function dependent of wind direction (φ), wind speed (v), temperature (T), hour (h), day (d), month (m)

: standard concentration fields

88

1,

iiki

mk mc

im

Real Time URBIS model

Real Time URBIS model

Linear correction used by DCMR Average concentration of three stations

Schiedam Hoogvliet Maassluis

mmslkmslk

mhgvkhgvk

msdmksdmk

mk

DCMRk cycycycc ,,,,,,3

1

Real Time URBIS model

Problems and Goals

The model simulation can become negative No information about the uncertainty of the

simulation

Goal: Find an uncertainty interval for the concentration NO , which does not contain negative concentrations

x

Method

Kalman filter connects the model simulations with a series of measurements

Kalman filter corrects the model in two steps Forecast step Analysis step

Result is a (multivariate) Gaussian distribution of the unknown

Mean Covariance matrix

Kalman filter equations

Correction factor ( ) for each standard concentration field

Kalman filter calculates a Gaussian distribution for the unknown variable ( )

The concentration NO can be found in a log-normal distribution

88

1,

,

iiki

KFk

kiemc

x

Kalman filter equations

Second equation not linear in ( ), thus a linearization around

H: projection matrix, A: correlation matrix represents the uncertainty of the

measurements on time k

),0(~

ln)ln()ln(88

1,

1

,

kk

ki

ikikkk

kk

R

emHcHy

A

ki

Ν

0

kR

Kalman filter equations

The linearization results in:

with:kkk

kk

Hy

A

~~

1

88

1

,~

)ln()ln(~

j

mk

jkj

mkkk

c

mH

cyy

Kalman filter equations

Forecast step

represents the uncertainty of the model is the covariance matrix after the forecast step The temporal correlation matrix A is determined

with information from the measurements

),0(~T1

1

QNAAPP

A

kfk

kfk

Q

kP

Kalman filter equations

Analysis step

Minimum variance gain

TT)~

()~

(

)~~(

KKRHKIPHKIP

HyK

kfkk

fkk

fkk

1TT )~~

(~ k

fk

fk RHPHHPK

Kalman filter equations

Start of the iteration process:

Screening process: Before the analysis step is executed, the measurements are

screened. If difference between simulation and observation is too large,

that observation will be thrown away. In this application the criterion is twice the standard deviation

QP 00 ,0

Results

Results

For the whole domain on each hour an uncertainty interval for the concentration NO can be calculated

Annual mean of the widths of these uncertainty intervals

Population density on the whole domain Connection between population and

uncertainty

x

Results

Results

Results

Connection between uncertainty and population

Kalman filter reduced the uncertainty Absolute uncertainty: 14.5 % Relative uncertainty: 16.1 %

gp

gp

n

iii

n

iii

uU

uU

1,rel

rel

1 abs,

abs

pop

pop

Extensions of the Kalman filter

Goal: Minimize the uncertainty connected with the population

Methods: Add extra monitoring stations to the system Apply Kalman filter with different time scale and

add stations with other time scales Analyse the values of the correction factors