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2020.01.30 Data Assimilation Workshop
Application of machine learning methods to model bias correction:
Lorenz-96 model experiments
Data Assimilation Research Team, RIKEN Center for Computational Science
Arata Amemiya*, Takemasa Miyoshi
Shlok Mohta
Graduate School of Science, the University of Tokyo
Motivation: bias in weather and climate models
Weather and climate models have model biases from various sources
โข Truncation errorโข Approximation of unresolved physical processes
- Convection- Small-scale topography- Turbulence- Cloud microphysics
(From JMA website)
Treatment of forecast error in data assimilation
๐ท๐ก+1๐
= ๐ด๐ท๐ก๐๐ด๐ป
Kalman filter
๐๐ก+1๐
=๐(๐๐ก๐)
๐ด = ๐๐/๐๐
http://www.data-assimilation.riken.jp/jp/research/index.html
๐ท๐ = ๐ฐ โ ๐ฒ๐ฏ ๐ท๐
๐ฒ = ๐ท๐๐ฏ๐ ๐ฏ๐ท๐๐ฏ๐ + ๐นโ1
๐๐ = ๐๐ +๐ฒ ๐ โ ๐ป(๐๐)
Model bias leads tothe underestimation of forecast(background) error
Previous analysis
Analysis mean
Observation
Forecast mean
Update state and forecast error covariance
Calculate analysis state and error covariance
Calculate Kalman gain
Treatment of imperfect model
Insufficient model error degrades the performance of Kalman filter
1. Covariance inflation
2. Correction of systematic bias component
๐ท๐ โ ๐ผ ๐ท๐
๐๐ก+1๐
= ๐๐ก+1๐
+ ๐
๐๐ก+1๐ = ๐๐ก+1
๐+๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐
Relaxation-to-prior
multiplicative inflation
๐ท๐ โ ๐ท๐ +๐ธadditive inflation
๐ท๐ โ (1 โ ๐ผ) ๐ท๐ + ๐ผ๐ท๐
Bias correction with simple functional form
Dimensionality reduction can be applied using Singular Value Decomposition (SVD) (Danforth et al. 2007)
Set of training data {๐ฟ๐, ๐๐}โ bias correction term ๐ซ(๐) estimation
๐ซ0 = ฮค๐ฟ๐ ฮ๐ก
๐ณ๐โฒ = ๐ช๐ฟ๐,๐ ๐ช๐,๐โ1๐โฒ โ ฮ๐ก
๐ซ ๐ = ๐ซ0 + ๐ณ๐โฒ ๐โฒ = ๐๐ โ ๐๐
Steady component
๐ช : correlation matrix
Linearly-dependent component (Leith , 1978)
โ โOfflineโ bias correction
๐
๐๐ก๐ = ๐true(๐)
๐
๐๐ก๐ = ๐model ๐
๐ฟ๐ = ๐๐ก ๐ก + ฮ๐ก โ ๐๐ ๐ก + ฮ๐ก
๐๐ก ๐ก , ๐๐ก ๐ก + ฮ๐ก โฆ
๐๐ ๐ก + ฮ๐ก
๐
๐๐ก๐ = ๐model ๐ + ๐ซ(๐)
Simplest form: linear dependency
Bias correction with nonlinear basis functions
โข Coupled Lorenz96 system (Wilks et al. 2005, Arnold et al. 2013)
โข Real case: All-sky satellite infrared brightness temperature (Otkin et al. 2018)
Probability of (obs โ fcst) vs obs
Higher order polynomials :
Neural networks :
โข Coupled Lorenz96 system (Watson et al. 2019)
(Fig.2 of Otkin et al. 2018)
Online bias correction
โข Kalman filter
= Simultaneous estimation of state variables and bias correction terms
- Polynomials (Pulido et al. 2018)
โ โOnlineโ bias correction
- Legendre polynomials(Cameron and Bell, 2016; for Satellite sounding in UK Met Office operational model)
- Steady component (Dee and Da Sliva 1998, Baek et al 2006)
โข Variational data assimilation (โVarBCโ)
sequential treatment / augmented state
Penalty function forbias-corrected obs error
Penalty function forregularization
Localization
Localization is built-in in LETKF
(Pathak et al. 2018, Watson et al. 2019)
Also in ML-based data driven modelling
โ Reduced matrix size -> low cost
In high dimensional spatiotemporal system,geographically (and temporally) local interaction is usually dominant
โ Highly effective parallelization(Miyoshi and Yamane, 2007)
Also used in simultaneous parameter estimation(Aksoy et al. 2006)
(Pathak et al. 2018)
The goal of this study
โ ML-based online bias correction using Long-Short Term Memory (LSTM)
โ Combined with LETKF with similar localization
Test experiments with coupled Lorenz96 model
โข Experimental Setup
โข Online bias correction with simple linear regression as a reference
โข (Online bias correction with LSTM)
bias correction in LETKF system
Model
๐๐ก+1๐ = ๐๐ก+1
๐+๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐
observationforecast
๐๐ก+1๐
Bias correction
Analysis
Bias correction function ๐(๐๐) update
๐๐ก+1 = ๐ด ๐๐ก
๐๐ก+1๐
= ๐(๐๐ก+1๐
)
LETKF
๐๐ก+1
Lorenz96 model
๐
๐๐ก๐ฅ๐ = ๐ฅ๐โ1 ๐ฅ๐+1 โ ๐ฅ๐โ2 โ ๐ฅ๐ + ๐น
๐ฅ๐ ๐ฅ๐+1
โข 1-D cyclic domainโข Chaotic behavior for sufficiently large ๐น
Wind @ 250hPaNCEP GFS
https://earth.nullschool.net/
(Lorenz 1996)
๐ = 1,2,โฆ , ๐พ
๐ฅ = ๐น
๐พ = 40, ๐น = 8
Largest Lyapunov exp.
Mean ๐ฅ ๐ก
Coupled Lorenz96 model
๐
๐๐ก๐ฅ๐ = ๐ฅ๐โ1 ๐ฅ๐+1 โ ๐ฅ๐โ2 โ ๐ฅ๐ + ๐น โ
โ๐
๐
๐=๐ฝ ๐โ1 +1
๐๐ฝ
๐ฆ๐
๐
๐๐ก๐ฆ๐ = โ๐๐ ๐ฆ๐+1 ๐ฆ๐+2 โ ๐ฆ๐โ1 โ ๐๐ฆ๐ +
โ๐
๐๐ฅint[(๐โ1)โ๐ฝ]+1
๐ = 1,2,โฆ , ๐พ
๐ = 1,2,โฆ , ๐พ๐ฝ
โNature runโ
๐
๐๐ก๐ฅ๐ = ๐ฅ๐โ1 ๐ฅ๐+1 โ ๐ฅ๐โ2 โ ๐ฅ๐ + ๐น + ๐ด sin 2๐
๐
๐พ
Forecast model (Danforth and Kalnay, 2008)
Coupling terms
Multi-scale interaction
๐ฆ1 ๐ฆ๐ฝ
Parameters used in this study๏ผ
(Wilks, 2005)
โฆ โฆ
(๐พ = 8)
Large scale (Slow) variables
Small scale (fast) variables
๐พ = 16, ๐ฝ = 16
โ = 1, ๐ = 20, ๐ = 50
๐น = 16, A = 1
Data assimilation experiment
Observation operator๏ผidentical๏ผobs = model grid๏ผ
Error standard deviation๏ผ 0.1
Interval: 0.025 / 0.05 / 0.1 (cf: doubling time โ 0.2)
Member๏ผ 20Localization๏ผ Gaussian weighting (length scale = 3 grids)
Covariance inflation: multiplicative (factor: ๐ผ)
Obs interval Minimum RMSE Inflation ๐ผ
0.025 0.0760 2.9
0.05 0.0851 5.8
0.1 0.0902 15.0
โObservationโ
LETKF configuration
Large inflation factor ๐ผ is needed to stabilize DA cycle
โNature runโ + random error
โControl runโ: Data assimilation without bias correction
Example: simple linear regression
Model
๐๐ก+1๐ = ๐๐ก+1
๐+๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐
๐๐ก+1 = ๐ป ๐๐ก+1true + ๐obs
observationforecast
๐๐ก+1๐
Bias correction
AnalysisLETKF
๐๐ก+1๐
= ๐พ๐+๐๐
๐๐ก+1๐
+ ๐๐ก+1๐
Online bias correction by linear regression
๐พ๐ก+1๐ = ๐พ๐ก+1
๐+ ๐ฝ๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐โ (๐๐ก+1
๐)๐ โ ๐๐ก+1
๐ ๐
๐๐ก+1 = ๐ด ๐๐ก
๐๐ก+1 = (1 โ ๐พ)๐๐ก
๐พ๐ก+1 = 1 โ ๐พ ๐พ๐ก
๐๐ก+1๐ = ๐๐ก+1
๐+ ๐ฝ๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐
Online bias correction by linear regression
๐ฝ = 0.02, ๐พ = 0.999 (half-life: ~37) ๐๐
๐๐ โ ๐๐
๐๐ โ ๐๐ :correction
๐๐ โ ๐๐ has large negative correlation with ๐๐
Interval (day) Min. infl. RMSE
0.025 2.9 0.0760
0.1 15.0 0.0902
Interval (day) Min. infl. RMSE
0.025 2.0 0.0625
0.1 8.2 0.0849
and mostly offset by bias correction term
Stable with smaller inflation factor ๐ผ
Correction term
Time 0-30 120-150
240-270 360-390
Bias: Forecast(before correction)-analysisCorrection
Nonlinear bias correction with ML
Model
๐๐ก+1๐ = ๐๐ก+1
๐+๐ฒ ๐๐ก+1 โ ๐ป ๐๐ก+1
๐
observationforecast
๐๐ก+1๐
Bias correction
Analysis
Train the network ๐ with ๐๐ก+1๐
, ๐๐ก+1๐
๐๐ก+1 = ๐ด ๐๐ก
๐๐ก+1๐
= ๐(๐๐ก+1๐
, โฆ )
LETKF
๐๐ก+1
RNN
Plan: LSTM implementation
โข Activation:tanh / sigmoid(recurrent)
LSTM
โข No regularization / dropout
๐๐ก๐, ๐๐ก
๐
Input
๐๐กโฮ๐ก๐
โฆ๐๐ก๐
nx=9, nt=5
Output๐๐ก๐
nx=1
LSTMnx=15
Densenx=10
Densenx=5
Lorenz96LETKF ๐๐ก
๐
Python TensorFlowTensorflow LSTM is implemented and integrated with LETKF codes
Network architecture
โข 1 LSTM + 3 Dense layers
โข LETKF-like Localization
Input : localized area
Output : one grid point
Summary
โข LSTM-based bias correction is implemented and to be tested
โข The efficiency of localization ?
โข Systematic model bias degrades forecasts and analysis
โข โOnlineโ bias correction with data assimilation has been studied using a fixed basis function set
โข โOfflineโ bias correction can be performed by ML as nonlinear regression
โข Online learning ?