Data assimilation

Derek Karssenberg, Faculty of Geosciences, Utrecht University

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

Forecasting: data assimilation

Time steps with observations

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

Solve Bayes’ equation at each observation time

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)

Understanding Bayes’ equation: Venn diagrams

S

P(A B)

P A B( ) =

P A « B( )

P B( )

P(B)

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

Stochastic modelling: Monte Carlo simulation

Data assimilation: Filter

Particle filter

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

Title

text

Step 1Step 2

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

Demo

aguila

Title

text

Title

text

t = 61 t = 90 t = 140 Resampling

particles

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

Demo

aguila

Comparison ofTechniques

Monte Carlo

Particle Filter

Ens. Kalman Filter