Variational Data Assimilation
in Coastal Ocean Problems with Instabilities
Alexander KurapovJ. S. Allen, G. D. Egbert, R. N. Miller
College of Oceanic and Atmospheric SciencesOregon State University
With support from ONR Physical Oceanography
Program
Grant N00014-98-1-0043
Model daily ave. surface T, velocities Oregon shelf, 7 Sept. 2005.
Using ROMS at 2 km resolution, forced with output from atmospheric ETA model forecast
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Variational data assimilation is computationally expensive. Why use it?
…Example using sequential Optimal Interpolation (OI):
HF radars (HF radars (Kosro))
Moorings (ADP, T, S: Moorings (ADP, T, S: Levine, Kosro, Boyd))
-Distant effect of assimilation of moored ADP currents (Kurapov et al., JGR, 2005a)
-Effects of velocity assimilation on other fields of interest (JGR, 2005b)
-Analysis of BML variability (JPO, 2005)
and limitations…
,a ft t t POMν ν
( )a f ft t t t ν ν G obs Hν
matrix matching observations to state vector
Time-invariant gain matrix
success…
COAST data, summer 2001
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OI corrects the ocean state, not forcing limited information on the source of model error
OI: Correction term is present in the momentum equations
Depth-ave, time-ave alongshore momentum term balance:
OI assumes time-invariant forecast error covariance (Pf):
Satisfactory performance on average over the season, but possibly difficulties predicting events (instabilities, relaxation from upwelling to downwelling, etc.), when state-dependent covariance is needed.
Snapshots of surf. velocity and density (<24.4 kg m-3), day 152, 2001: Less meandering in the data assimilation solution
no DA DA
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Variational Data Assimilation: Theory
Penalty Functional:
J(u)= || Ini. error ||2 + || OB Cond. error ||2 + || Dynamical error ||2 + || Obs. error ||2
Representer-based minimization algorithm (ref.: Chua and Bennett, OM, 2001):
Nontrivial covariances can be incorporated in J
Applicable both for strong-constraint (Dyn. error = 0) and weak-constraint cases
Involves search in a relatively small space spanned by representers
Provides error covariance in the prior and inverse solutions
Utilized in the emerging Inverse Ocean Modeling (IOM) System (Bennett et al.)
10 0
IC
ε C ε
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Details of the representer-based method:
0
Nonlinear model:
Initial condition:
Data: ( 1, , )
( ) (
) ( )
(0)
( )k k k k k K
N t tdt
t d e
uu f ε
u i ε
l u
var(J) = 0 Euler-Lagrange Eqns. (nonlinear and coupled):
1
ˆ ˆ( ) ( ) ( )
( ) 0
(1)
( 2 )
K
k k kk
b t tt
T
λA u λ u l
λ0
(3)
(4
ˆˆ( ) ( ) ( ; ) ( )
ˆ (0) (0) )
T
IC
N f t t dt
u
u C λ
u C λ
where
1@obs. loc. and timesˆ ˆ( ) { } ( )k db b u C d u
, , are assumed error covariances (dyn., IC, data)IC dC C C
ˆadjoint solution, inverse solution λ u
ˆ
ˆ( )N
u u
A uu
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Linearization and decoupling :
Try to find a sequence of linearized solutions {un}, n=1,…,N
{un} inverse solution
To obtain un, the original ELE are linearized with respect to un-1
An optimal linear combination forcing the adjoint equation is obtained iteratively, solving the matrix-vector equation: 1
( ) K
k k kk
b t t
l
@ obs. loc. and times( ) (5)F
d R C b d u
representer matrix (K K)
Direct representer method (for small K). To obtain R, compute and sample (at data locations and times) K representer functions. To obtain a representer solution:
-run the adjoint (AD) model, forced with -smooth the result with the model error covariance-run the tangent linear (TL) model
( ) k kt t l
Computational cost: (2 K + 1) N [equiv. fwd runs]
prior (“Fwd”) solution of the linearized system
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Use the conjugate gradient method to solve
To compute R where is any vector:
-run the AD model, forced with
-smooth the result with the model error covariance,
-run the TL model
Preconditioning. Compute directly and sample a number of representer functions (obtain selected columns of R):
11 21
121 21 11 21
pcond
R RR
R R R R
Indirect representer method (Egbert et al., 1994):
( ) k k kt t l
@ obs. loc. and times( ) F
d R C b d u
Computational cost: <<(2 K + 1) N [equiv. fwd runs], depends on the number of degrees of freedom in the problem (the size of the domain, length of the run, assumed error covariances)
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Implementation 1. Barotropic flow through the channel (Kuo jet)
(in collaboration with E. Di Lorenzo, A. Moore, H. Arango, et al.)
Use IROMS (in a version used originally for stability analyses):
-correct initial conditions-direct representer method
Kuo jet: 2D, 100 x 400 km
H=const=20 m, f=10-4 s-1
Periodic channel
No forcing
No dissipation
SSH mid-channel (NL ROMS): growth of instability is constrained
STEADY (SSH colored)
PROPAGATING TO THE RIGHT (PERIOD 6.7 days)
days
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Data assimilation experiment (with synthetic data):
Prior:
- symmetric jet, or
- no flow
“True” (equilibrated wave):
Data: SSH or velocities, sampled from the true solution
Goal: correct initial conditions
Assimilation window: T=3-12 days
Direct computation of representers.
Does {un} converge to the inv. solution?
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Dataless (Picard) iterations:
No data. True controls (Ini. Cond.). Wrong background u0 (e.g., prior solution)
Domain-averaged RMS error (surf. elevation) as a function of time:
Picard Iterations: Convergence to truth for a limited period (comparable to the period of nonlinear
oscillations)
Assimilate at 40 locations on days 3, 6, 9, 12)
Data assimilation: convergence for a longer period of time
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Choice of the IC error covariance affects convergence:
1( ) 0
(0) (0)
nn n
nIC
t
uA u u
u C λ
IC penalty term: 1
0 0 IC
ε C ε TL:
Choices:
1. CIC=2 I singular un (Bennett, 2002)
includes IC errors in , u, v
0 0
0 0
0 0IC uu
vv
C
C C
C
2. Cbell: bell-shaped, independently for , u, v
3. Cgeos: provide geostrophically balanced correction
u v
IC u uu uv
v vu vv
C C C
C C C C
C C C
Make an hypothesis about , , , ,u v uu uv vv C C C C C C
days
use Cgeos Cbell
RMS
u RMS
v RMS
(these cases: assim. at t=3 d, prior= no flow)
Lorenc (1981), Daley (1985)
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Implementation 2. Forced-dissipative nearshore flows
In response to steady forcing,
– steady flow,
– equilibrated shear waves,
– or “turbulent” regime,
depending on how large bottom friction is
DA: correct forcing, initial, cross-shore boundary conditions
Shown is Vorticity (NL ROMS, 2D, ADV_4C,biharmonic horiz. diff.)
Alongshore currents over variable beach topography (Slinn et al. 2000)
alongshore coord. (m)
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Inverse model of choice: AKM
TL and AD for shallow-water equations. Limited set of model options (straight periodic or OB channel, ADV_2C, harmonic horiz. diff., variable H)
Reasons to proceed with our TL&AD development:
1. Better understand how TL and AD are built in ROMS.2. Clarify details of (and possibly suggest effective solutions for):
- time-stepping in the TL and AD- inputs and outputs
3. Interface with IOM (AKM has been incorporated in IOM)4. Address the issue of instabilities vs. variational DA in a simpler, 2D set-up
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TL of time stepping momentum eqn.: Automatic Differentiation Tool / AKM
1 1
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11
1 2 1
__ _
( )In continuous form: rhs, where
1NL ROMS: rhs_ubar + rhs_ubar
1TL: + rhs _ tl_rhs_ub_ubar + + a
)r
(
n n n n n n nn
n n n nn n
nn n ntl D
tl u t
DuD H
t
D u D u u D uD
D u ul D tl uDD D
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1 1_ _ _ _ tl_rhs_uba1
TL AKM: r nn n n nn
nntl u tl D tl u t ulD DuD
No background rhs arrays in TL or AD
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Inputs and outputs in TL and AD (AKM):
Input: IC
Output: zeta, ubar, vbar every NHIS time steps
time
Inputs: forcing, boundary values (sms_time, bry_time)NL, TL:
interpolation
u = [TL] f, length (u) length (f) [AD] = [TL]T = [AD] v
Input in ADM is a vector of the same length as output in TLMOutput in ADM is a vector of the same length as input in TLM
Output: IC)
Input: a linear combination of data functionals (defined as a NetCDF file(s) of same format as a history file(s) in NL and TL, outside the code
time
Outputs: corrections to forcing, boundary values (at sms_time, bry_time)
AD to interpolation
AD:
- Avoid saving the AD solution every time step- Minimum (no) modifications to the code when using new data types
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Implementation of AKM in the forced-dissipative case:
True solution: 250 x 200 m, x=2 m, periodic channel, NHIS=1 min,
forcing is spatially variable, stationary in time (input: 2 identical fields at 0 and 60 min)
time-ave, alongshore-ave
u (m/s)
Instant vorticity: 5 min 10 min 30 min 60 min
time-ser. of v (m/s)
time, min
Prior: IC=0, forcing=0 (no flow)
u
v
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Dataless (Picard) Iterations: no data, correct forcing, start with incorrect background (no flow) – does the series of linearized problems converge to truth?
Time-series of u-RMS error: convergence for a limited time period
Iter 5
Data assimilation: , u, v every minute at 16 locations, T=30 min
-RMSE (m)
u-RMSE (m/s)
v-RMSE
Total of 1200 obs.
Computational requirements:Preconditioner: 24 rep.CGM: 20-30 iter. (for each linearized problem)
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Data assimilation, T=60 min (correcting time-invariant forcing):
Mean current is restored, but meandering is not predicted deterministically
no convergence in v
2: Assume forcing is time-dependent
Time-series of v-RMSE
Possible approach 1: DA in a series of short (30 min) time windows, correct both forcing and IC in each window
Look for additional controls!
3: Open boundary control 4: Modify linearization scheme, step control
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SUMMARY:
1. Clear advantages of using variational DA methods in problems of coastal and shelf circulation, to meet both scientific and operational needs
2. TL and AD ROMS (Kuo jet case): representer-based algorithm works for a period of time comparable to characteristic time scale of instabilities. Importance of using a dynamically based IC error covariance
3. AKM (nearshore, forced-dissipative case): experience building a TL&AD model. No background rhs arrays are needed. Clear definitions of inputs and outputs (no need to store the TL and AD solutions every time step; new data types can be incorporated without changes to TL&AD codes)
4. Possible approaches to overcome the problem of instabilities: look for additional controls (flow-forcing feedback; open boundary values)
