Combining HYCOM, AXBTs and Polynomial Chaos Methods to ... · Rely on Polynomial Chaos expansions to replace HYCOM by a polynomial series that could be either summed for MCMC or differentiated

The Problem The Tools The results Conclusions References

Combining HYCOM, AXBTs and PolynomialChaos Methods to Estimate Wind Drag

Parameters during Typhoon Fanapi

Mohamed Iskandarani Ashwanth Srinivasan Carlisle ThackerChia-Ying Lee Shuyi Chen,

University of MiamiOmar Knio Alen Alexandrian Justin Winokur Ihab Sraj,

Duke University

Funding: Office of Naval ResearchGulf of Mexico Research Initiative

May 20, 2013


Outline

The ProblemDrag ParameterizationBayesian formulation of inverse problem

The ToolsPolynomial Chaos

The resultsPC AnalysisThe inference posteriorsVariational Solution

Conclusions

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

3

V (m/s)

CD

×1

03

α = 1.1 α = 1 α = 0.4

~τ = ρaCDV ~VCD = CD0 + CD1(Ts − Ta)

CD0 = a0 + a1Ṽ + a2Ṽ 2

CD1 = b0 + b1Ṽ + b2Ṽ 2

Ṽ = max [ Vmin, min (Vmax,V ) ]

CD is drag coefficientV is wind speed at 10 m.CD saturates for V > Vmax

• Blue circles: aircraft observations (French et al., 2007),• red: wind tunnel (Donelan et al., 2004),• green: drop sondes (Powell et al., 2003),• magenta: HYCOM fit to COARE 2.5,• Problem: Vmax and CmaxD are not well-known and does CD

decrease for V > Vmax as drop sondes suggest?


Inverse Modeling Problem

• Perturb CD by introducing 3 control variables (α,Vmax,m)

CD′ = αCD for V < Vmax (1)

CD′ = α[CD + m(V − Vmax)] for V > Vmax (2)

• multiplicative factor 0.4 ≤ α ≤ 1.1• vary Vmax between 20 and 35 m/s• m is a linear slope modeling decrease for V > Vmax with−3.8× 10−5 ≤ m ≤ 0

• Use ITOP data to learn about likely distribution of α, Vmaxand m.


Bayes Theorem: p(θ |T ) ∝ p(T |θ) p(θ)• Likelihood: � = T −M is normally distributed

p(T |θ) =N∏

i=1

1√2πσ2

exp(−(Ti −Mi)2

2σ2

)(3)

• σ2 unknown, treated as hyper-parameter. Assume aJeffreys prior

p(σ2) =

{1σ2

for σ2 > 0,0 otherwise.

(4)

• Uninformed priors for α, Vmax and m:

p({α,Vmax,m}) =

{1

bi−ai for ai ≤ {α,Vmax,m} ≤ bi ,0 otherwise,

(5)

where [ai ,bi ] denote the parameter ranges.


Final Form of Bayes theorem

p({α,Vmax,m}, σ2|T ) ∝

[N∏

i=1

1√2πσ2

exp(−(Ti −Mi)2

2σ2

)]p(σ2)p(α)p(Vmax)p(m)

• Build full posterior with Markov Chain Monte Carlo (MCMC)MCMC requires O(105) estimates of Mi : prohibitive

• Solve for center and spread of posteriorminimization problem requiring access to cost functiongradient and Hessian: Needs an adjoint model

• Rely on Polynomial Chaos expansions to replace HYCOMby a polynomial series that could be either summed forMCMC or differentiated for the gradients.


What is Polynomial Chaos

• Series Representation of Model Output

M(x , t ,θ) =P∑

k=0

Mk (x , t) ψk (θ) (6)

• M(x , t ,θ): a model output (aka observable)• Mk (x , t): series coefficients• ψk (θ): orthogonal basis functions w.r.t. p(θ)• mean: E [M] = 〈M, ψ0〉 =

∑Pk=0 Mk (x , t) 〈ψk , ψ0〉 = M0(x , t)

• Variance: E[(M − E [M])2

]=∑P

k=1 M2k (x , t)

• Basic Questions• How to choose ψk ? Legendre polynomials• How to determine the coefficients Mk ? Projection• Where to truncate the series, P ? Monitor Variance


How do we determine PC coefficients• Series: M(x , t ,θ) =

∑Pk=0 Mk (x , t)ψk (θ)

• Projection:

Mk (x , t) = 〈M, ψk 〉 =∫

M(x , t ,θ)ψk (θ)ρ(θ)dθ

• Approximate integral with numerical Quadrature

Mk (x , t) ≈Q∑

q=1

M(x , t ,θq)ψk (θq)ωq

• θq/ωq quadrature points/weights• Quadrature requires an ensemble run at θq• Here we Used Adaptive quadrature requiring 6-iteration

levels for a total of 67 realizations

Figure: Fanapi’s JTWC track (black curve) and paths of C-130 flights.The yellow circles on the track represent the typhoon center at00:00 UTC. The circles on the flight paths mark the 119 AXBT drops.The 42× 42 km2 analysis box is also shown.

10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/14 20:35 UTC

Temperature (oC)

De

pth

(m

)

Simulated AXBT (29.54)Observed AXBT (29.43)

10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/15 22:58 UTC

Temperature (oC)

De

pth

(m

)


10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/17 21:87 UTC

Temperature (oC)

De

pth

(m

)


10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/14 22:44 UTC

Temperature (oC)

De

pth

(m

)


10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/15 23:67 UTC

Temperature (oC)

De

pth

(m

)


10 12 14 16 18 20 22 24 26 28 30 32−600

−500

−400

−300

−200−150−100

−500

09/17 23:96 UTC

Temperature (oC)

De

pth

(m

)


Figure: Comparison of HYCOM vertical temperature profiles withAXBT observations on Sep 14 (left), 15 (center) and 17 (right).Temperature averages over the first 50 m are shown in the legend.


PC Representation Errors

11 12 13 14 15 16 17 18 19 20 2124

25

26

27

28

29

30

Day of September

SS

T (

oC

)

11 12 13 14 15 16 17 18 19 20 2110−5

10−4

10−3

10−2

Day of September

Rela

tive e

rror

Evolution of the area-averaged SST realizations (blue) and ofthe corresponding PC estimates (red). The normalized rmserror (right panel) remains below 0.1% for the duration of thesimulation.

Longitude

La

titu

de

09/15 at 0 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Longitude

La

titu

de

09/15 at 50 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Longitude

La

titu

de

09/15 at 200 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Longitude

La

titu

de

09/18 at 0 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Longitude

La

titu

de

09/18 at 50 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Longitude

La

titu

de

09/18 at 200 m

120E 125E 130E 135E15N

20N

25N

30N

2

4

6

8

10x 10

−3

Figure: Normalized error between realizations and the correspondingPC surrogates at different depths; Top row: 00:00 UTC Sep 15;bottom row: 00:00 UTC Sep 18.


Depth Profile of Temperature Statistics

Day of September

Z in m

Mean Temperature

11 12 13 14 15 16 17 18 19 20 21−200

−150

−100

−50

0

18

20

22

24

26

28

30

Day of September

Z in m

Standard deviation

11 12 13 14 15 16 17 18 19 20 21−200

−150

−100

−50

0

0

0.2

0.4

0.6

0.8

1

50m-deep mixed layer2◦C cooling after Fanapi arrivesUncertainties confined to top 50 m.


SST Response Surface

Vmax

(m/s)

α

Response surface: 09/17

20 25 30 350.4

0.6

0.8

1

29.1

29.15

29.2

29.25

29.3

Vmax

(m/s)

α


20 25 30 350.4

0.6

0.8

1

28.6

28.8

29

29.2

Vmax

(m/s)

α


20 25 30 350.4

0.6

0.8

1

25

26

27

28

Figure: SST response surface as function of α and Vmax , with fixedm = 0. Plots are generated on different days, as indicated. SST’sdependence on Vmax decreases after 09/17.


Markov Chain Monte Carlo

0 5 10

x 104

20

25

30

35

Vm

ax

Iteration0 5 10

x 104

−4

−3

−2

−1

0x 10

−5

m

Iteration0 5 10

x 104

0.7

0.8

0.9

1

1.1

α

Iteration

0 5 10

x 104

0.4

0.5

0.6

0.7

0.8

0.9

σ2

Iteration

09/14 − 09/15

0 5 10

x 104

0

0.5

1

1.5σ

2

Iteration

09/15 − 09/16

0 5 10

x 104

0.5

1

1.5

2

2.5

σ2

Iteration

09/17 − 09/18

Figure: Top row: chain samples for Vmax , m and α. Bottom row: chainsamples for σ2 generated for different days, as indicated.

10 20 30 400

0.1

0.2

0.3

0.4

Vmax

pd

f

PosteriorPriorMAP

−6 −4 −2 0 2

x 10−5

0

1

2

3x 10

4

m

pd

f

PosteriorPriorMAP

0.9 0.95 1 1.05 1.10

20

40

60

80

α

pd

f

PosteriorPriorMAP

0.4 0.5 0.6 0.7 0.80

5

10

1509/14 − 09/15

σ2

pd

f

0.5 0.6 0.7 0.8 0.90

2

4

6

8

1009/15 − 09/16

σ2

pd

f

0.5 1 1.50

2

4

609/17 − 09/18

σ2

pd

f

1 0.087 3

Figure: Posterior distributions for the drag parameters (top) and thevariance between simulations and observations (bottom). Thenumbers show the Kullback-Liebler divergence quantifying thedistance between 2 prior and posterior pdfs, i.e. the information gain.


Remarks on posteriors

• Vmax exhibits a well-defined peak at 34 m/s.• Posterior of m resembles prior. Data added little to our

knowledge of m.• α shows a definite peak at 1.03 with a Gaussian

like-distribution.•√σ2 is a measure of the temperature error expected. This

error grows with time from about 0.75◦ to 1◦C.


Joint posterior PDFs

20 25 30 350.98

1

1.02

1.04

1.06

Vmax

α

5

10

15

20

25

0.98 1 1.02 1.04 1.060.6

0.8

1

1.2

1.4

1.6

σ2

α

100

200

300

400

500

Figure: Left: joint posterior distribution of α (left) and Vmax ; right: jointposterior of α and σ2, generated for Sep 17-Sep 18. Single peaklocated at Vmax = 34 m/s and α = 1.03. The posterior shows a tightestimate for α with little spread around it.


0 10 20 30 40 500.5

1

1.5

2

2.5

V (m/s)

CD

×10

3

09/12 − 09/1309/13 − 09/1409/14 − 09/1509/15 − 09/1609/17 − 09/18

Figure: Optimal wind drag coefficient CD using MAP estimate of thethree drag parameters. The symbols refer to AXBT data used in theBayesian inference.


Variational Form• maximize the posterior density, or equivalently, minimize

the negative of its logarithm

J (α,Vmax ,m, σ21, σ22, σ23, σ24, σ25) =5∑

d=1

[Jd +

(nd2

+ 1)

ln(σ2d)],

(7)where Jd is the misfit cost for day d , the ln(σ2d) terms comefrom the normalization factors of the Gaussian likelihoodfunctions and from the Jeffreys priors.

• The expression for Jd is:

Jd(α,Vmax ,m, σ2d) =1

2σ2d

∑i∈Id

[Mi − Ti ]2 , (8)

where Id is the set of nd indices of the observations fromday d .


Adjoint-Free GradientsMinimization requires cost function gradients

[∂J∂α

,∂J∂Vmax

,∂J∂m

]=

5∑d=1

1σ2d

∑i∈Id

(Mi − Ti)[∂Mi∂α

,∂Mi∂Vmax

,∂Mi∂m

]Compute them from PC expansion[

∂M∂α

,∂M∂Vmax

,∂M∂m

]=

P∑k=0

M̂k (x , t)[∂ψk∂α

,∂ψk∂Vmax

,∂ψk∂m

].

• ∂ψk∂α easy to compute

• No adjoint model needed• For Hessian just differentiate above again.


Figure: Posterior probability distributions for (top) drag parametersand (bottom) variances σ2d at selected days using variational methodand MCMC. The vertical lines correspond to the MAP values fromMCMC and optimal parameters found using the variational method.


Conclusions & Future Work

• Identified drag parameters from ITOP observations duringtyphoon Fanapi.

• PC instrumental to make calculations tractable eitherthrough MCMC or through adjoint-free minimization

• Vmax ≈ 34 m/s• Data uninformative regarding decrease in CD• CmaxD peaking around 2.3× 10

−3

• Surface temperature measurements more valuable thanones at depths > 75 m.

• Inference of Vmax and m hampered by lack of observationat wind speeds > 35 m/s.

• Future: Hurricane Model & other air-sea exchangecoefficients


Publications• I. Sraj, M. Iskandarani, A. Srinivasan, W. C. Thacker, and O.M. Knio, Computing

Model Gradients from a Polynomial Chaos based Surrogate for an InverseModeling Problem Monthly Weather Review, in revision.

• J. Winokur, P. Conrad, I. Sraj, M. Iskandarani, A. Srinivasan, W.C. Thacker, Y.Marzouk, O. M. Knio, A priori testing of sparse adaptive polynomial Chaosexpansions using an OGCM database, Computational Geosciences, in review.

• I. Sraj, M. Iskandarani, A. Srinivasan, W. C. Thacker, J. Winokur, A.Alexanderian, C-Y Lee S. S. Chen, O.M. Knio, Bayesian Inference of DragCoefficient Parameters using AXBT data from Typhoon Fanapi, Monthly WeatherReview, doi:10.1175/MWR-D-12-00228.1.

• A. Alexanderian, J. Winokur, I. Sraj, M. Iskandarani, A. Srinivasan,W. C. Thacker, and O. M. Knio, Global sensitivity analysis in an ocean generalcirculation model: a sparse spectral projection approach, ComputationalGeosciences, 16, Vol 3, pp 757–778, 2012.

• W. C. Thacker, A. Srinivasan, M. Iskandarani, O. M. Knio, and M. Le Henaff.Propagating oceanographic uncertainties using the method of polynomial chaosexpansion. Ocean Modelling, 43–44, pp 52–63, 2012.

• A. Srinivasan, J. Helgers, C. B. Paris, M. LeHenaff, H. Kang, V. Kourafalou,M. Iskandarani, W. C. Thacker, J. P. Zysman, N. F. Tsinoremas, and O. M. Knio.Many task computing for modeling the fate of oil discharged from the deep waterhorizon well blowout. In Many-Task Computing on Grids and Supercomputers(MTAGS), 2010 IEEE Workshop on, pages 1–7, November, 2010. IEEE.

http://dx.doi.org/10.1016/j.ocemod.2011.11.011http://dx.doi.org/10.1016/j.ocemod.2011.11.011http://dx.doi.org/10.1109/MTAGS.2010.5699424http://dx.doi.org/10.1109/MTAGS.2010.5699424


Bibliography

Donelan, M. A., B. K. Haus, N. Reul, W. J. Plant, M. Stiassnie,H. C. Graber, O. B. Brown, and E. S. Saltzman, 2004: On thelimiting aerodynamic roughness of the ocean in very strongwinds. Geophysical Research Letters, 31 (L18306), 1–5,doi:doi:10.1029/2004GL019460, URLhttp://dx.doi.org/10.1029/2004GL019460.

French, J. R., W. M. Drennan, J. A. Zhang, and P. G. Black,2007: Turbulent fluxes in the hurricane boundary layer. part i:Momentum flux. Journal of the Atmospheric Sciences, 64,1089–1102, doi:doi:10.1175/JAS3887.1.

Powell, M. D., P. J. Vickery, and T. A. Reinhold, 2003: Reduceddrag coefficient for high wind speeds in tropical cyclones.Nature, 422, 279–283, doi:doi:10.1038/nature01481, windmeasurement and wind stress calculation in (observed)hurricane conditions.

http://dx.doi.org/10.1029/2004GL019460

The ProblemDrag ParameterizationBayesian formulation of inverse problem

The ToolsPolynomial Chaos

The resultsPC AnalysisThe inference posteriorsVariational Solution

Conclusions

Documents

Combining HYCOM, AXBTs and Polynomial Chaos Methods to ... · Rely on Polynomial Chaos expansions to replace HYCOM by a polynomial series that could be either summed for MCMC or differentiated