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JeanPierre Auclair
MWSNotes
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Mar 09, 2017
by
Arthanor
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Data Assimilation Part 1
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@@ 68,4 +68,104 @@ Olivier Asselin:
Nathan Grivault: UoAlberta w/ Paul Myers

Freshwater in the Arctic: Arctic > Lower latitudes

Impact on circulation: Export vs Forcing
\ No newline at end of file

Impact on circulation: Export vs Forcing
## Data Assimilation
### Development of data assimilation

developped for weather forecasting (fast time scale, highly nonlinear)

applied for operational oceanography (slower time scales, poorly observed)

starting to be used in sea ice
### Why?

Obtain an improved state that can be used to initialize a forecast

Obtain consistent states that can be used in process studies or hindcasting

To challenge models with data and vice versa, leading to improvements in models and/or observational methods (OSSEs)
### Assimilation cycle
1 Observation
2 Assimilation system
3 Analysis
4 Forecast model
5 Background
6 Repeat
Note: Need really good observations otherwise they can contaminate runs
### Components of a DA system

DA is a state estimation problem

Need a prognostic model to solve time evolution of the state

the state is all you need to characterize the system

prevent model drift
### DA problem
given a mapping from a state to observations, try to get from observation to state
for y = H(x), try to get x = H^1(y) but size(x) and size(y) don't match so it's not invertible
### Best Linear Unbiased Estimator (BLUE)
x_a = L x_b + K y
a: state estimate at a given time
b: background (state of the model for a given time)
y: obs for a given time
Best estimate minimizes analysis error.
E_a = x_a  x_t
E_b = x_b  x_t
E_o = y  x_t
x_t + E_a = L(x_t + E_b) + K(E_o + H(x_t))
assume errors are unbiased:
<E_
?
>
= 0
<x_t>
= L
<x_t>
+ KH
<x_t>
<x_t>
= (L+KH)
<x_t>
> L+KH = I > L = I  KH
Plug back in:
x_a = (IKH)x_b + Ky
= x_b + K (y  H x_b) <== Kalman filter equation
**
Need to get K
'best' K is the one that minimizes trace of A where A =
<E
^
a
E
^
a*
>
1 subtract x^t from both sides of
**
to get an eqn for E^a
2 A =
<E
^
a
E
^
a*
>
3 A(K+dK)  A(K) = A(dK)
4 Trace A(dK) = 0
2[(IKH)BH^t + KR]dK = 0 for any dK
=> (IKH)BH^t + KR = 0
==> BH^t  KHBH^t  KR =0
===> K(HBH^t + R) = BH^t
====> K = BH^t (HBH^t+R)^1
Try H = I, ie you have 1 observation for every state variable
Try B =
\s
igma^2_b I and R =
\s
igma^2_o I
xa = xb + BH^t (HBH^t+R)^1 (y  Hx^b)
xa = xb + (sigma^2_b)(yxb)/(sigma^2_o + sigma^2_b)
**
*
IF
\s
igma^2_o >>
\s
igma^2_b => xa > xb because sig/(sig+sig) > 0
IF
\s
igma^2_b >>
\s
igma^2_o => xa > y because sig/(sig+sig) > 1
If model is biased, need to include bias in
**
*
and have a coupled error/bias data assimilation
Generally treat things as a minimization problem.
## Getting B and R:
Error covariance matrices:
B: Model error covariance matrix for the model: how state variables vary with each others
R: Obs error covariance matrix
Innovations: y  Hxb
Correlated B&R act as low&high pass filters of innovations.
Using ensembles to create error correlation function: Not bad but spurious noise at longer range.
=> Use localization.
## Case studies
T&S from buoy in Labrador sea using EnOI > Improved profiles
Sea ice concentration from passive microwave
Look at innovations (PDF, Spacetime distributions) to diagnose DA
Can take a while for assimilation to kick in
\ No newline at end of file
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