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Relation between Five Data Assimilation Methods for a Simplistic - - PowerPoint PPT Presentation
Relation between Five Data Assimilation Methods for a Simplistic - - PowerPoint PPT Presentation
Relation between Five Data Assimilation Methods for a Simplistic Weakly Non-linear Problem Trond Mannseth Uni Research CIPR Methods Methods Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates) Methods Randomized
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Methods
Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates)
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Methods
Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates) Ensemble Smoother (ES) (Sim. est., A single update)
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Methods
Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates) Ensemble Smoother (ES) (Sim. est., A single update) Ensemble Smoother with Multiple Data Assimilations (ESMDA) (Sim. est., Multiple upd. with inflated data covariance)
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Methods
Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates) Ensemble Smoother (ES) (Sim. est., A single update) Ensemble Smoother with Multiple Data Assimilations (ESMDA) (Sim. est., Multiple upd. with inflated data covariance) Half-iteration Ensemble Kalman Filter (HIEnKF) (Sequential est., A single upd.)
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Methods
Randomized Maximum Likelihood (RML) (Simultaneous estimation, Iterative updates) Ensemble Smoother (ES) (Sim. est., A single update) Ensemble Smoother with Multiple Data Assimilations (ESMDA) (Sim. est., Multiple upd. with inflated data covariance) Half-iteration Ensemble Kalman Filter (HIEnKF) (Sequential est., A single upd.) Half-iteration Ensemble Kalman Filter with MDA (HIEnKFMDA) (Seq. est., Multiple upd. with inflated data covariance)
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Motivation
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Motivation
Methods are equivalent for gauss-linear problems
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Motivation
Methods are equivalent for gauss-linear problems Methods behave differently for non-linear problems
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Motivation
Methods are equivalent for gauss-linear problems Methods behave differently for non-linear problems RML is iterative, ES is purely non-iterative, while ESMDA, HIEnKF and HIEnKFMDA ‘lie somewhere in between’
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Motivation
Methods are equivalent for gauss-linear problems Methods behave differently for non-linear problems RML is iterative, ES is purely non-iterative, while ESMDA, HIEnKF and HIEnKFMDA ‘lie somewhere in between’ Systematic differences for weakly non-linear problems?
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Investigation
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Investigation
Compare methods on simplistic, weakly non-linear parameter estimation problem
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Investigation
Compare methods on simplistic, weakly non-linear parameter estimation problem Focus on differences in data handling – remove other differences
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Investigation
Compare methods on simplistic, weakly non-linear parameter estimation problem Focus on differences in data handling – remove other differences Asymptotic calculations (additional assumptions) to first order in non-linearity strength
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Investigation
Compare methods on simplistic, weakly non-linear parameter estimation problem Focus on differences in data handling – remove other differences Asymptotic calculations (additional assumptions) to first order in non-linearity strength Numerical calculations with full methods and relaxed assumptions
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Simplistic Weakly Non-linear Parameter Estimation Problem
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Simplistic Weakly Non-linear Parameter Estimation Problem
Estimate x from d, where
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Simplistic Weakly Non-linear Parameter Estimation Problem
Estimate x from d, where d = (d1 . . . dD)T,
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Simplistic Weakly Non-linear Parameter Estimation Problem
Estimate x from d, where d = (d1 . . . dD)T, di = yi (xref ) + ǫi; ǫi ∼ N(0, σ2
i ),
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Simplistic Weakly Non-linear Parameter Estimation Problem
Estimate x from d, where d = (d1 . . . dD)T, di = yi (xref ) + ǫi; ǫi ∼ N(0, σ2
i ),
yi(x) = M
m=1 cimx1+nim m
; |nim| ‘not too big’
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Focus on Diff. in Data Handling – Remove Other Diff.
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Focus on Diff. in Data Handling – Remove Other Diff.
Equip ES (and ESMDA, HIEnKF, HIEnKFMDA) with local gains
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Focus on Diff. in Data Handling – Remove Other Diff.
Equip ES (and ESMDA, HIEnKF, HIEnKFMDA) with local gains Kalman gain (global) K = Cxy (Cyy + Cd)−1
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Focus on Diff. in Data Handling – Remove Other Diff.
Equip ES (and ESMDA, HIEnKF, HIEnKFMDA) with local gains Kalman gain (global) K = Cxy (Cyy + Cd)−1 Replace Cxy by CxG T
e and Cyy by GeCxG T e
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Focus on Diff. in Data Handling – Remove Other Diff.
Equip ES (and ESMDA, HIEnKF, HIEnKFMDA) with local gains Kalman gain (global) K = Cxy (Cyy + Cd)−1 Replace Cxy by CxG T
e and Cyy by GeCxG T e
→ Local gains Ke = CxG T
e
- GeCxG T
e + Cd
−1
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Additional Assumptions facilitating Asymptotic Calculations
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Additional Assumptions facilitating Asymptotic Calculations
I consider the updates of a single (arbitrary) ensemble member
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Additional Assumptions facilitating Asymptotic Calculations
I consider the updates of a single (arbitrary) ensemble member Assumptions Univariate x → yi(x) = x1+ni
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Additional Assumptions facilitating Asymptotic Calculations
I consider the updates of a single (arbitrary) ensemble member Assumptions Univariate x → yi(x) = x1+ni Negligible data error → di = yi (xref ),
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Additional Assumptions . . . (continued)
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Additional Assumptions . . . (continued)
So far
|n1| ∧ |n2| ‘not too big’
xref d1 d2 y1 (x) y2 (x)
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Additional Assumptions . . . (continued)
So far I assume
|n1| ∧ |n2| ‘not too big’ n1 = n2 = n, |n| ≪ 1
xref d1 d2 y1 (x) y2 (x) xref d, d y (x) , y (x)
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Results from Asymptotic Calculations to O(n)
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2)
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML|
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML| Q = |d(ln d − ln xprior) − (d − xprior)|
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML| Q = |d(ln d − ln xprior) − (d − xprior)| ∆ES = Qn + O(n2)
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML| Q = |d(ln d − ln xprior) − (d − xprior)| ∆ES = Qn + O(n2) ∆ESMDA = A−1Qn + O(n2)
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML| Q = |d(ln d − ln xprior) − (d − xprior)| ∆ES = Qn + O(n2) ∆ESMDA = A−1Qn + O(n2) ∆HIEnKF = D−1Qn + O(n2)
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Results from Asymptotic Calculations to O(n)
xRML = d − (d ln d)n + O(n2) Define ∆method = |xmethod − xRML| Q = |d(ln d − ln xprior) − (d − xprior)| ∆ES = Qn + O(n2) ∆ESMDA = A−1Qn + O(n2) ∆HIEnKF = D−1Qn + O(n2) ∆HIEnKFMDA = (AD)−1Qn + O(n2)
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Results from Asymptotic Calculations to O(n)
∆ES ≈ Qn ∆ESMDA ≈ A−1Qn ∆HIEnKF ≈ D−1Qn ∆HIEnKFMDA ≈ (AD)−1Qn
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D
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Numerical Results with Full Methods
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D
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Numerical Results with Full Methods
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D
‘Ranking’ stable for n ∈ [−0.5, 5]
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- Num. Calc. with Full Methods and Relaxed Assumptions
d = (d1 . . . dD)T, di = yi (xref ) + ǫi; ǫi ∼ N(0, σ2
i ),
yi(x) = M
m=1 cimx1+nim m
; ¯ nim = 0.4
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- Num. Calc. with Full Methods and Relaxed Assumptions
d = (d1 . . . dD)T, di = yi (xref ) + ǫi; ǫi ∼ N(0, σ2
i ),
yi(x) = M
m=1 cimx1+nim m
; ¯ nim = 0.4 M = 1, 2, 5.
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- Num. Calc. with Full Methods and Relaxed Assumptions
d = (d1 . . . dD)T, di = yi (xref ) + ǫi; ǫi ∼ N(0, σ2
i ),
yi(x) = M
m=1 cimx1+nim m
; ¯ nim = 0.4 M = 1, 2, 5. Draw 300 realizations of xref , xprior, nim, cim
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- Num. Res. with Full Methods and Relaxed Assumptions
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 1, Arbitrary realization
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 1, Mean
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- Num. Res. with Full Methods and Relaxed Assumptions
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 2, Arbitrary realization
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 2, Mean
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- Num. Res. with Full Methods and Relaxed Assumptions
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 5, Arbitrary realization
2 4 6 8 10 12 14 16 18 20 dES dHIEnKF dESMDA2 dESMDA4 dHIEnKFMDA2 dHIEnKFMDA4
D M = 5, Mean
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Summary
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Summary
Compared five different ways to assimilate data on simplistic, weakly non-linear parameter estimation problem
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Summary
Compared five different ways to assimilate data on simplistic, weakly non-linear parameter estimation problem Asymptotic calculations to first order in non-linearity strength (relying on further simplifications) reveals nature of similarity with iterative methods for (local-gain) ESMDA, HIEnKF and HIEnKFMDA methods
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