Estimation of error covariance matricesin data assimilation

Pierre Tandeo

Associate professor at IMT-Atlantique, Brest, France

20th of February 2018

Before starting

My organization:

I New engineering school:Telecom Bretagne + MinesNantes

I Top 10 best school

I Topics: numeric, energy,environment

Works in collaboration with:

State-space model

Nonlinear formulation with additive and Gaussian noises:{x(t) =M (x(t − dt) + η(t) (1)

y(t) = Hx(t) + ε(t) (2)

with:

I M the dynamical model (physical or statistical)

I H the transformation matrix from state x to observations y

I η(t) ∼ N (0,Q(t)) the model error

I ε(t) ∼ N (0,R(t)) the observation error

⇒ Estimate Q and R is a key point in data assimilation

State of the art

I 4 main families of methods to jointly estimate Q and R

I More than 50 papers in data assimilation from the 90’s

I No comparison between all these methods

I Poor links with signal processing and statistical communities

⇒ Review on estimation of Q and R is the goal of this presentation

Importance of errors

Simple univariate, linear and Gaussian state-space model:{x(k) = 0.95x(k − 1) + η(k)

y(k) = x(k) + ε(k)

with η(k) ∼ N (0,Qt = 1) and ε(k) ∼ N (0,Rt = 1)

Importance of errors

I Bad Q/R ratio:I Q too low (top)I R too low (bottom)

I Impact on state reconstruction (RMSE)

Importance of errors

I Good Q/R ratio:I Q and R too low (top)I Q and R too high (bottom)

I Impact on uncertainty quantification (% in CI)

OutlineFiltering/smoothing methods

Data Assimilation algorithmsKalman equations

Innovation-based methodsInnovations in the observation spaceLag-innovation statistics

Likelihood-based methodsBayesian approachesMaximization of the total likelihood

ApplicationsMeteorologySpatial oceanographyEcology

Conclusions and perspectivesConclusionsPerspectives

Data Assimilation algorithms

I Variational approaches:I filtering/smoothing covariances not estimatedI adjoint model needed

I Particle filter/smoother:I optimal in theory but...I ... not relevant for high-dimensional problems

I Kalman-based methods:I robust and widely usedI additive Gaussian assumptions as in Eq. (1-2)

Kalman equations

Filtering step:

K(k) = Pf (k)H>(

HPf (k)H> + R(k))−1

xa(k) = xf (k) + K(k)(

y(k)−Hxf (k))

Innovation-based methods

Innovation statistics (mean and covariance):

d(k) = y(k)−Hxf (k) (3)

Σ(k) = HPf (k)H> + R(k) (4)

Historical references:

I [Daley, 1992]

I [Dee, 1995]

Issues:

I difficult to estimate Q and R...

I ... using only the current innovation

I pointed out by [Blanchet et al., 1997]

Innovations in the observation space

Main references:

I [Desroziers et al., 2005] (use of various innovations amongdo−f (k) = y(k)−Hxf (k) and do−a(k) = y(k)−Hxa(k))

I [Li et al., 2009] and [Miyoshi, 2011] (estimation of covarianceinflation for Pf and covariance R)

Solve the following system: E[do−f (k)do−f (k)>

]= HPf (k)H> + R(k) = Σ(k) (5)

E[do−a(k)do−f (k)>

]= R(k) (6)

Lag-innovation statistics

Main references:

I [Mehra, 1970] (signal processing community)

I [Berry and Sauer, 2013] (only lag-1 innovation)

I [Harlim et al., 2014] (various lag-L innovations)

I [Zhen and Harlim, 2015] (comparison lag-1 VS lag-L)

Solve the following system (example of lag-0 VS lag-1):E[d(k)d(k)>

]= HPf (k)H> + R(k) = Σ(k) (7)

E[d(k)d(k − 1)>

]= HF(k)Pf (k − 1)H>

−HF(k)K(k − 1)Σ(k − 1) (8)

Bayesian approaches

Main references:

I classic in the statistical community

I [Stroud and Bengtsson, 2007] (scalar parameters for Q and R)

I [Ueno and Nakamura, 2016] (parameterization of R)

I [Stroud et al., 2017] (spatial parameterization of R)

Write the joint distribution of x, Q and R:

p (x(k),Q,R|y(0 : k)) =

p (x(k)|Q,R, y(0 : k)) p (Q,R|y(0 : k)) (9)

withp (Q,R|y(0 : k)) ∝p (y(k)|Q,R, y(0 : k − 1)) p (Q,R|y(0 : k − 1)) (10)

Maximization of the total likelihood

Main references:

I [Shumway and Stoffer, 1982] (linear and Gaussian case)

I [Ueno et al., 2010] (using grid-based algorithm)

I [Ueno and Nakamura, 2014] (using EM, only for R)

I [Dreano et al., 2017] (using EM, for both Q and R)

Maximize the likelihood function:

L(Q,R) =

p (x(0))K∏

k=1

p (x(k)|x(k − 1),Q)K∏

k=0

p (y(k)|x(k),R) (11)

⇒ estimate iteratively Q and R using the ExpectationMaximization (EM) algorithm

Meteorology (1D example)1

I In situ wind speed data (Brest, France)I Statistical AR(1) model for the temporal dynamics

05/1 10/1 15/1 20/1 25/1 30/1time(t)

0

2

4

6

8

10

12

14

16Timeseries of hourly­recorded wind intensity (January 2012) in Brest (France)

0 20 40 60 80 100iteration

−1800

−1600

−1400

−1200

Log­likelihood

0 20 40 60 80 100iterations­ r)

0.60

0.65

0.70

0.75

0.80

0.85

Q­estimates

0 20 40 60 80 100iterations­ r)

0.2

0.4

0.6

0.8

1.0

1.2

R­estimates

1from T.T.T. Chau, PhD candidate at Univ. Rennes II, France

Spatial oceanography (2D example)2

I Interpolation of daily SST (Sea Surface Temperature)

I 40 years of noisy satellite images (AVHRR-Pathfinder)

2from Autret and Tandeo 2017, ”Atlantic European North West Shelf Seas - High Resolution L4 Sea Surface

Temperature Reprocessed”, http://marine.copernicus.eu

Spatial oceanography (2D example)

I Statistical AR(1) model for the spatio-temporal dynamics

I Estimate Q (statistical model error) and R (satellite error)

Ecology (1D multivariate ODE example)3

I Biogeochemical model with80 variables (various oceandepths)

I Satellite observations ofChl-a (8 days, 4 km)

I Monitoring blooms in theRed Sea

3from Dreano 2017, ”Data and Dynamics Driven Approaches for Modelling and Forecasting the Red Sea

Chlorophyll”, PhD dissertation

Conlusions

I Q and R are crucial in data assimilation:I for prediction/filtering/smoothing modesI for uncertainty quantification

I Review paper in preparation:I exhaustive bibliography in the data assimilation communityI comparison between the 4 main methods

I Free Python code (https://github.com/ptandeo/CEDA):

Perspectives

I Combine different approaches:I first offline for calibration of Q and R...I ... and then online for adaptive estimation of Q(k) and R(k)

I Possible extensions:I deal with the initial condition (background xb and B)I parameterize Q and R for high dimensional problems

I Ongoing works:I using conditional particle filter/smootherI using fully data-driven approaches (analog data assimilation)

Thank you! Any questions?

Berry, T. and Sauer, T. (2013).

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Blanchet, I., Frankignoul, C., and Cane, M. A. (1997).

A comparison of adaptive kalman filters for a tropical pacific ocean model.Monthly Weather Review, 125(1):40–58.

Daley, R. (1992).

Estimating Model-Error Covariances for Application to Atmospheric Data Assimilation.Monthly Weather Review, 120(8):1735–1746.

Dee, D. P. (1995).

On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation.Monthly Weather Review, 123(4):1128–1145.

Desroziers, G., Berre, L., Chapnik, B., and Poli, P. (2005).

Diagnosis of observation, background and analysis-error statistics in observation space.Quarterly Journal of the Royal Meteorological Society, 131(613):3385–3396.

Dreano, D., Tandeo, P., Pulido, M., Chonavel, T., AIt-El-Fquih, B., and Hoteit, I. (2017).

Estimating model error covariances in nonlinear state-space models using Kalman smoothing and theexpectation-maximisation algorithm.Quarterly Journal of the Royal Meteorological Society, 143(705):1877–1885.

Harlim, J., Mahdi, A., and Majda, A. J. (2014).

An ensemble Kalman filter for statistical estimation of physics constrained nonlinear regression models.Journal of Computational Physics, 257:782–812.

Li, H., Kalnay, E., and Miyoshi, T. (2009).

Simultaneous estimation of covariance inflation and observation errors within an ensemble Kalman filte.Quarterly Journal of the Royal Meteorological Society, 135(2):523–533.

Mehra, R. K. (1970).

On the Identification of Variances and Adaptive Kalman Filtering.IEEE Transactions on Automatic Control, 15(2):175–184.

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The Gaussian Approach to Adaptive Covariance Inflation and Its Implementation with the Local EnsembleTransform Kalman Filter.Monthly Weather Review, 139(5):1519–1535.

Shumway, R. H. and Stoffer, D. S. (1982).

An Approach to Time Series Smoothing and Forecasting using the EM Algorithm.Journal of Time Series Analysis, 3(4):253–264.

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Sequential State and Variance Estimation within the Ensemble Kalman Filter.Monthly Weather Review, 135(9):3194–3208.

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A Bayesian adaptive ensemble Kalman filter for sequential state and parameter estimation.Monthly Weather Review.

Ueno, G., Higuchi, T., Kagimoto, T., and Hirose, N. (2010).

Maximum likelihood estimation of error covariances in ensemble-based filters and its application to acoupled atmosphere-ocean model.Quarterly Journal of the Royal Meteorological Society, 136(650):1316–1343.

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Iterative algorithm for maximum-likelihood estimation of the observation-error covariance matrix forensemble-based filters.Quarterly Journal of the Royal Meteorological Society, 140(678):295–315.

Ueno, G. and Nakamura, N. (2016).

Bayesian estimation of the observation-error covariance matrix in ensemble-based filters.Quarterly Journal of the Royal Meteorological Society, 142(698):2055–2080.

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Adaptive error covariances estimation methods for ensemble Kalman filters.Journal of Computational Physics, 294:619–638.