Commit 8fbed8ef authored by Arthanor's avatar Arthanor

Data Assimilation Part 1

parent 92ce9f0f
......@@ -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
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- 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 = (I-KH)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[(I-KH)BH^t + KR]dK = 0 for any dK
=> (I-KH)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 = \sigma^2_b I and R = \sigma^2_o I
xa = xb + BH^t (HBH^t+R)^-1 (y - Hx^b)
xa = xb + (sigma^2_b)(y-xb)/(sigma^2_o + sigma^2_b)***
IF \sigma^2_o >> \sigma^2_b => xa -> xb because sig/(sig+sig) -> 0
IF \sigma^2_b >> \sigma^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, Space-time distributions) to diagnose DA
Can take a while for assimilation to kick in
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