View
215
Download
0
Tags:
Embed Size (px)
Citation preview
time
Calibration
Historicalobservations
Compare with state variables z
Adjust parameters pand inputs i
time
Forecasting
Historicalobservations
calibration
Forecast with calibrated modelHistoricalobservations
time
Forecasting: data assimilation
Current and futureobservations
Compare with state variables z
Adjust parameters p, inputs i, and state variablesz
Forecast with calibrated model
Data assimilation techniques
• Direct insertion of observations– Replace model state variables with observed variables
• Probabilistic techniques using Bayes’ equation– Adjust model state variables
– Adjustment is related to uncertainty in model state variables relative to the uncertainty in observations
Understanding Bayes’ equation: Venn diagrams
A
S
Ac
S: sample space, all possible outcomes (e.g., persons in a test)
A: event (e.g., ill person)
Ac: complement of A (e.g., person not ill)
Note that P(Ac) = 1 - P(A)
Understanding Bayes’ equation: Venn diagrams
S
Bc
S: sample space, all possible outcomes (e.g., persons in a test)
B: event (e.g., person tested positive)
Bc: complement of A (e.g., person not tested positive)B
Understanding Bayes’ equation: Venn diagrams
S
A B
A B: intersection, e.g. persons that are ill and are tested positive
Understanding Bayes’ equation: Venn diagrams
P(A|B)
P(A|B): conditional probability, the probability of A given that B occurs (e.g. the probability that the person is ill given it is tested positive)
Bayes’ equation
P A B( ) =
P A « B( )
P B( )
P B A( ) =
P A « B( )
P A( )€ P A « B( ) = P B A( ) P A( )
Data assimilation techniques that use Bayes’ equation
• (Ensemble) Kalman filters– Adjust model state by changing state variables of model realizations
• Particle filters– Adjust model state by duplicating (cloning) model realizations
Probabilistic data assimilation
for each t
set of stochastic variables
Solve by using Monte Carlo simulation:
n realizations of variables representing state of the model
( ) Nnf nt
nt 1,..., for ,)(
1)( == −xx
( )1−= tt f XX
tX
)(ntx
Apply Bayes’ equation at observation time steps
p xt yt( )=p yt xt( ) p xt( )
p yt( )
Prior: PDF modelPrior: PDF of observations
Prior: PDF of observations given the model
Posterior: probability distribution function (PDF) of model given the observations
Apply Bayes’ equation at observation times
Step 1:
Apply Bayes’ equation to realizations of the model
Results in a ‘weight’ assigned to each realization
Step 2:
Clone each realization a number of times proportional to the
weight of the realization
Step 1: Apply Bayes’ equation to each realization (particle) i
p xt(i) yt( )=
p yt xt(i)
( ) p xt(i)
( )
p yt( )
Prior: PDF of model realization i
Prior: PDF of observations
Prior: PDF of observations given the model realization i
Posterior: probability distribution function (PDF) of realization i given the observations
Combine..
p xt(i) yt( )=
p yt xt(i)
( ) p xt(i)
( )
p yt( )
p yt( )= p yt xt
( j)( ) p xt
( j)( )
j=1
N
Â
p xt(i) yt( )=
p yt xt(i)
( ) p xt(i)
( )
p yt xt( j)
( ) p xt( j)
( )j=1
N
Â
Combine
p xt
(i)( )=1/ N
p xt(i) yt( )=
p yt xt(i)
( )
p yt xt( j)
( )j=1
N
Â
p xt(i) yt( )=
p yt xt(i)
( ) p xt(i)
( )
p yt xt( j)
( ) p xt( j)
( )j=1
N
Â
Proportionality
p xt(i) yt( )=
p yt xt(i)
( )
p yt xt( j)
( )j=1
N
Âfi
proportionalSame forall realizations i
Calculating weights
p yt xt
(i )( ) = exp -
1
2yt - Ht xt
(i )( )ÈÎÍ
˘˚̇TRt
- 1 yt - Ht xt(i )( )È
Î͢˚̇
Ê
ËÁÁÁ
ˆ
¯˜̃˜
p yt xt(i )
( ) = exp -xt , j
(i ) - yt , j( )2
2s j2
j= 1
n
ÂÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃̃
observationsmodel state
Measurement error variance of observations
Combine, resulting in ‘weights’ for each realization (particle)
p xt(i) yt( )=
p yt xt(i)
( )
p yt xt( j)
( )j=1
N
Â
p yt xt(i )
( ) = exp -xt , j
(i ) - yt , j( )2
2s j2
j= 1
n
ÂÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃̃
p xt(i ) yt( ) : exp -
xt , j(i ) - yt , j( )
2
2s j2
j= 1
n
ÂÊ
Ë
ÁÁÁÁÁÁ
ˆ
¯
˜̃˜̃˜̃̃
proportional
Step 2: resampling
Copy the realizations a number of times proportional to p xt
(i ) yt( )
Title
text
Step 1Step 2
p xt
(i ) yt( )
Catchment model: snowfall, melt, discharge
Catchment in Alpes, one winter, time step 1 day
Simplified model for illustrative purposes only
Stochastic inputs: temperature lapse rate, precipitation
Filter data: snow thickness fields
Snow case study
Catchment in Alpes, one winter, time step 1 day
T(s,t) = tarea(t)·L·h(s) for each t
T(s,t) temperature field, each timesteptarea(t) average temperature of study area (measured)L lapse rate, random variableh(s) elevation field (DEM)
Snow case study
P(t) = parea(t) + Z(t)
P(t) precipitation fieldparea(t) average precipitation of study area (measured)Z(t) random variable with zero mean
Snowmelt linear function of temperature
Filter data: snow thickness fields at day 61, 90, 140
Visualisations: 1. Realizations
Derek Karssenberg et al, Utrecht University, NL, http://pcraster.geo.uu.nl
Visualisations: 2. Statistics calculated over realizations
Derek Karssenberg et al, Utrecht University, NL, http://pcraster.geo.uu.nl
Number of Monte Carlo
samples per snow cover
Interval
Probability density
Cumulative probability density
0.1