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Kalman Filter in Real Time URBIS
Richard Kranenburg
06-01-2010
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Introduction
TNO – Technisch Natuurwetenschappelijk Onderzoek
Kerngebied – Bouw en Ondergrond Business Unit – Milieu en Leefomgeving
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Introduction
AccompanistsMichiel Roemer (TNO)Jan Duyzer (TNO)Arjo Segers (TNO)Kees Vuik (TUDelft)
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Problem
Current situation Real Time URBIS model gives one value for
the concentration NOx in the DCMR area Wanted situation
Uncertainty interval for the concentration NOx
Uncertainty interval dependent of the place in the domain
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DCMR-area
Introduction - Uncertainty analysis - Kalman filter - Application on population
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URBIS model
11 emission sources Traffic
CAR Zone Cards Road near Road far
Background Abroad Rest of the Netherlands DCMR-area
Shipping Ship sea Ship inland
Industry Industry
Rest
Winddirections North East South West
Wind speeds 1.5 m/s 5.5 m/s
Total 88 standard concentration fields for the concentration NOx
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Real Time URBIS
Gives for every hour an expected concentration NOx for the whole DCMR-area, based on input parameters Wind direction (φ) Wind speed (v) Temperature (T) Month (m) Weekday (d) Hour (h)
State equation:
88
1iii
RTUk mC
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Measurement locations
DCMR-Stations Schiedam Hoogvliet Maassluis Overschie Ridderkerk Rotterdam Noord
RIVM-Stations Schiedamsevest Vlaardingen Bentinckplein
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Uncertainty Real Time URBIS
Compare Real Time URBIS simulations with the observations on the nine measurement locations
Both observations and model simulations have a log-normal distribution
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Log-normal distributions
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Correction of the model
Differences between model and measurements plotted with respect to 6 input parameters
h: hour of the day φ: wind direction
eCC RTUcorr
)()( hc
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Uncertainty of the model
Standard deviation of the differences between the corrected model and the observations
v: wind speed
)(v
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Result of uncertainty analysis
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Kalman filter
Smooth random errors in the model of a dynamical system
In a real time application, measurements on time k are directly available to filter the state on time k.
Two results after application New expected concentration NOx Uncertainty interval for the concentration NOx
New state equation:
88
1iii
KF iemC
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Kalman filter equations
Forecast
Analysis
k
f
kA
1
kT
kfk QAAPP 1
f
kk
m
kkf
k
a
kcHyK
111111lnln
111T
1111 kkkkfkk
ak KRKHKIPHKIP
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Kalman filter on emission source ‘Background’ In the vector all values are equal to zero,
except the entries corresponding with the source ‘background’
Linearization of the Kalman filter equations Matrix A estimated with measurements in
Schipluiden and Westmaas Matrix R estimated with measurements at
Bentinckplein
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Kalman filter on emission source ‘Background’ Screening criterion:
Pfabs,k : Model uncertainty after forecast step
Rabs,k : Uncertainty of the measurements
kabsfkabs
i
sd
iikRPemy
fki
fki
,,2
288
1
2
,21
,
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Kalman filter on emission source ‘Background’
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Kalman filter on all emission sources State equation:
Screening Criterion
88
1iii
KF iemC
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Kalman filter on all emission sources Matrix A estimated with measurements in
Schipluiden, Westmaas, Overschie, Ridderkerk, Maassluis, Vlaardingen
Matrix R estimated with measurements at Bentinckplein
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Kalman filter on all emission sources
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Connection with population
Each grid cell, every hour a certainty interval Annual mean of the width of these intervals per grid
cell Amount of large widths of these intervals per grid cell
Number of postal zipcodes per grid cell Population density 1.99 people per grid cell Number of people per grid cell
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Connection with population
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Connection with population
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Connection with population
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Connection with population
Introduction - Uncertainty analysis - Kalman filter - Application on population
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Conclusions
The differences between the observations and the model simulation are not only caused by inaccuracies in the background
Uncertainty interval has large width in the industrial region around Pernis and on the main roads
The application of the Kalman filter makes it possible to correct the model values every hour, with help of the observations
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Further investigation
Add measurement stations to reduce uncertainty of the Kalman filter results
State optimal setting of measurement stations
Increase time scale (Day, Week or Month)