kalman filter in the real time urbis model richard kranenburg master scriptie june 11, 2010
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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