Data assimilation Derek Karssenberg, Faculty of Geosciences, Utrecht University

time Calibration Historical observations Compare with state variables z Adjust parameters p and inputs i

time Forecasting Historical observations calibration Forecast with calibrated model Historical observations

time Forecasting: data assimilation Current and future observations Compare with state variables z Adjust parameters p, inputs i, and state variables z 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 AcAc S: sample space, all possible outcomes (e.g., persons in a test) A: event (e.g., ill person) A c : complement of A (e.g., person not ill) Note that P(A c ) = 1 - P(A)

Understanding Bayes’ equation: Venn diagrams S BcBc S: sample space, all possible outcomes (e.g., persons in a test) B: event (e.g., person tested positive) B c : 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(B)P(B)

Bayes’ equation

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

Stochastic modelling: Monte Carlo simulation

Data assimilation: Filter

Particle filter

Apply Bayes’ equation at observation time steps 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 1 Step 2

Step 1: Apply Bayes’ equation to each realization (particle) i 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..

Combine

Proportionality proportional Same for all realizations i

Calculating weights observationsmodel state Measurement error variance of observations

Combine, resulting in ‘weights’ for each realization (particle) proportional

Step 2: resampling Copy the realizations a number of times proportional to Title text Step 1 Step 2

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) = t area (t)·L·h(s)for each t T(s,t) temperature field, each timestep t area (t) average temperature of study area (measured) L lapse rate, random variable h(s) elevation field (DEM)

Snow case study P(t) = p area (t) + Z(t) P(t)precipitation field p area (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,

Visualisations: 2. Statistics calculated over realizations Derek Karssenberg et al, Utrecht University, NL,

Number of Monte Carlo samples per snow cover Interval Probability density Cumulative probability density 0.1

Demo aguila Comparison of Techniques Monte Carlo Particle Filter Ens. Kalman Filter