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MEOPAR Winter School in Environmental Modeling Notes
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Participant intros
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Jenna Joyce: UoOttawa w/ Jackie Dawson
- Shipping in the Arctic (Corridors)
- Corridors & Environment/Culturally significant areas coexistence
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Ewelina Luczko: Baird & associates "where land meets water"
- Coastal engineering - wave modeling
- Wave energy generation in SWAN

Lindsay Chipman: UoColorado - Boulder
- Cycling of O2 and C in permeable sediment
- O2 flux from <w'O'>

Lei Ren: NUI Galway
- Surface flow fields from CODAR & Model
- Focus area: West Coast of Ireland

Russel Glazer: UoFlorida - Tallahassee
- Saturation over ice & water in models
- Cloud formation scheme

Blanche St-Béat: ULaval - Qc
- Food Webs: distinguish ecosystemséestablish stability
- Resilience vs Resistance: Wood house is better than hay or bricks?!

Feifei Sun: Andrea Scott student
- Predict ice thickness from data
- New assimilation methods

Dennis Monteban: Denmark university
- Study of Fjord west of Greenland / Wave-Ice damping
- Validated & Calibrated MIKE model

Ben Moore-Maley: UBC w/ Susan Allen
- 3D modeling of Salish Sea @ 500m resolution
- Wind driven circulation/interactions

Nancy Chen: DFO - St-John's
- Satellite SSH anomaly -> water transport
- Labrador & Scotian Shelf

Deborah Benkort: PhD Laval
- Krill aggregation & dispersion in GSL/GSE
- Effect of envt. on growth, distribution, reproduction

Becky Segal: MSc UVic
- Permafrost thaw sumps work before
- Ice prediction/obs and creating relevant products for communities

Charles Brunette: McGill with Bruno
- Predictability using Lagrangian methods
- Seasonal & Regional
- Later formation -> Thinner ice -> Minimum ice extent

Onur Bora: Coastal engineer/PhD student in Istanbul
- Hydrodynamics effects from shipping -> impact on sediment
- Water cooling/intake system design

Olivier Asselin:
- Figure out the atmosphere
- Energy constant vs \lambda for different processes: 2 slopes for everything?!
- CFD

Nathan Grivault: UoAlberta w/ Paul Myers
- Freshwater in the Arctic: Arctic -> Lower latitudes
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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