data assimilation on a two-layer QG channel model
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data assimilation on a two-layer QG channel model MPO624 final project Ting-Chi Wu
description
data assimilation on a two-layer QG channel model. MPO624 final project Ting-Chi Wu. Data assimilation (1/2). The “Forecast” will be the “First Guess” of the next time step. Objective analysis. Data assimilation (2/2). DA cycle. Error = RMS of (DA run – Truth run) - PowerPoint PPT Presentation
Transcript of data assimilation on a two-layer QG channel model
data assimilation on a two-layer QG channel model
MPO624 final projectTing-Chi Wu
Data assimilation (1/2)
• The “Forecast” will be the “First Guess” of the next time step.
Objective analysis
DA cycle
Data assimilation (2/2)
Error = RMS of (DA run – Truth run)DA run starts from t=50day, but Truth run starts from t=100day
2-layer QG channel model
• Potential vorticity equation • One variable: streamfunction
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(∂
∂t+ U1
∂
∂x) ∇2ψ1 −
1
Rd12 (ψ1 −ψ2 )
⎡
⎣ ⎢
⎤
⎦ ⎥+
∂ψ1
∂x
dΠ1
dy= 0
(∂
∂t+ U2
∂
∂x) ∇2ψ2 −
1
Rd 22 (ψ2 −ψ1)
⎡
⎣ ⎢
⎤
⎦ ⎥+
∂ψ2
∂x
dΠ2
dy= 0
€
f0 = 0.83 ×10−4 s−1
€
β0 = 2 ×10−11m−1s−1
€
Rd12 =
gΔρ
ρf02 H1
Rd 22 =
gΔρ
ρf02 H 2
~29km
~50km
First guess (Background) : day 50 of true runObservation : day 100 of true run
Experiments
• DA run: only assimilate upper layer• Direct insertion (no objective analysis)
• Int=1day• Int=2day• Int=5day
• Optimum Interpolation• Div=2 gridpoints• Div=3 gridpoints• Div=4 gridpoints• Div=5 gridpoints
Direct insertion
With random error5 % of average value
€
ϕ i
ϕ g
˜ ϕ g
Observation
True value on gridpoint
Analyzed value on gridpoint
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ϕ i = ϕ i +ϕ i '; ϕ g = ϕ g +ϕ g '; ˜ ϕ g = ϕ g + ˜ ϕ g '
˜ ϕ g = ϕ g + α i(ϕ i −ϕ i)i=1
n
∑ ⇒ ˜ ϕ g ' = α iϕ i 'i=1
n
∑
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E = ( ˜ ϕ g −ϕ g )2 = (ϕ g + ˜ ϕ g ' −ϕ g −ϕ g ')2 = ( ˜ ϕ g ' −ϕ g ')2 ⇒∂E
∂α k
= 0,(k =1,2,......,n)
⇒ α jj =1
n
∑ ϕ i 'ϕ j ' = ϕ i 'ϕ g ' a ϕ i 'ϕ j '[ ] α j[ ] = ϕ i 'ϕ g '[ ]
Optimum Interpolation (1/2)
Use correlation instead of covariance
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μij =ϕ i 'ϕ j '
ϕ i '2 ϕ j '
2; 0 ≤ μ ij ≤1;
€
μ(r) = Aexp(−Br2) (GaussianForm) Schlatter(1975)
Optimum Interpolation (2/2)
observation Model gridpoint
observation Model gridpoint
For every gridpoint, pick 8 nearest observations
Optimum Interpolation (2/2)
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μij[ ]8x8
α i[ ]8x1
= μ ig[ ]8x1
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μij
€
μig
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˜ ϕ g ' = α iϕ i 'i=1
n
∑
With Objective Analysis
• Model gridpoints:• X=128; Y=65; Points: 8320
Div points
1 2 2112
2 3 946
3 4 544
4 5 338
After OA (1/3)First guess/background observation
Different spatial intervals
After OA (2/3)
After OA (3/3)
Future work
• Pick another time-period• Apply other assimilation scheme
