1. Scale-dependent localization: test with quasi-geostrophic models
Michael Ying
Group meeting, Dec 2016

2. Motivation Localization distance:
too large: suffer from sampling error
too small: lose useful covariance information
Higher model and observation resolution:
Smaller optimal localization distance.
shorter correlation length for small-scale dynamics; and
more sampling error due to rank deficiency.
Multi-scale localization (Zhang et al SCL; Li et al. 2015; Miyoshi Kondo 2015; Buehner et al. 2015)
In this study, I attempt to use simplified model to demonstrate the necessity of scale awareness in ensemble filtering.

3. Surface quasi-geostrophic model (SQG)
2D turbulence maintained by Markovian forcing from k=1~4
domain size: 256*256, maximum wavenumber k=128, cyclic boundary condition
full animation
(Held et al. 1995, Eq 2)

4. Kinetic energy spectra and predictability
Reference power
-5/3
Error power

5. Covariance structure at different scales
All scales k=1~4 k=4~10 k=10~50
N=2000
N=200
N=20

6. Sampling error at each scale
Sampling error = RMS Difference of using 20 and 2000 members

7. Scale dependency in analysis error
Fixed localization distance l for all observation, 80 members, results from 1 cycle,
observation resolution: Δobs=4 grid points; optimal l ?
obs error
reference power
prior error
analysis error with
l =

8. Remedy of larger sampling error at small scale
reference power
prior error
analysis error with obs thin:
Δobs =
l = 2 Δobs
Thinning observation
Super observation
Reduce impact for correlated obs using AOEI (Huber norm)
SCL (Zhang et al. 2009)
Cons:
Losing information when throwing away observation / impact
Better solution:
treat analysis increment at each scale separately

9. One of the bred vectors (fast-growing error modes)
local bred vector

10. Bred vector dimension (local effective dimension)
Ψ members can span the subspace formed by the k bred vectors.
Indicator of necessary ensemble size
1 of 20 bred vectors
bred vector dimension

11. Next… Test the scale-dependent localization: How to estimate ?
Run cycling experiment …