using small ensembles in high dimensions: hierarchical
TRANSCRIPT
![Page 1: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/1.jpg)
7/17/08
h Dimensions:roaches
Filters
tion (DAReS)
u, Kevin Raeder
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 1
Using Small Ensembles in HigHierarchical Bayesian App
to Adaptive Ensemble
Jeffrey AndersonIMAGe Data Assimilation Research Sec
Thanks to Nancy Collins, Tim Hoar, Hui Li
![Page 2: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/2.jpg)
7/17/08
ata Assimilation
re)ilable.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 2
How an Ensemble Filter Works for Geophysical D
Ensemble stateestimate after usingprevious observation(analysis).
Ensemble state attime of next obser-vation (prior).
tk tk+1
1. Use model to advanceensemble (3 members heto time at which next observation becomes ava
****
![Page 3: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/3.jpg)
7/17/08
ata Assimilation
=h(x), byember.
servationsments withed errors canequentially.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 3
How an Ensemble Filter Works for Geophysical D
2. Get prior ensemble sample of observation, yapplying forward operator h to each ensemble m
Theory: obfrom instruuncorrelatbe done s
y
****
h hh
![Page 4: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/4.jpg)
7/17/08
ata Assimilation
ibution
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 4
How an Ensemble Filter Works for Geophysical D
3. Getobserved valueandobservational error distrfrom observing system.
y
****
h hh
![Page 5: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/5.jpg)
7/17/08
ata Assimilation
mbleation errors).
y
ce betweenrs of ensem-imarily increment.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 5
How an Ensemble Filter Works for Geophysical D
4. Findincrement for each prior observation ense(this is a scalar problem for uncorrelated observ
y
****
h hh Note: Differen
different flavoble filters is probservation in
![Page 6: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/6.jpg)
7/17/08
ata Assimilation
ariable to linearlyble increments.
y
impact oftion increments onte variable can be independently!
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 6
How an Ensemble Filter Works for Geophysical D
5. Use ensemble samples of y and each state vregress observation increments onto state varia
y
****
h hh
Theory:observaeach stahandled
![Page 7: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/7.jpg)
7/17/08
ata Assimilation
variable are updated,bservation...
y
tk+2
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 7
How an Ensemble Filter Works for Geophysical D
6. When all ensemble members for each state have a new analysis. Integrate to time of next o
y
****
h hh
tk
![Page 8: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/8.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 8
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 91
State Variable
![Page 9: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/9.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 9
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 92
State Variable
![Page 10: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/10.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 10
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 93
State Variable
![Page 11: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/11.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 11
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 94
State Variable
![Page 12: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/12.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 12
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 95
State Variable
![Page 13: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/13.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 13
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 96
State Variable
![Page 14: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/14.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 14
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 97
State Variable
![Page 15: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/15.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 15
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 98
State Variable
![Page 16: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/16.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 16
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 99
State Variable
![Page 17: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/17.jpg)
7/17/08
le Model
latitude band.
35 40
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 17
Ensemble Filter for Lorenz-96 40-Variab
40 state variables: X1, X2,..., X40.
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.Acts ‘something’ like synoptic weather around a
5 10 15 20 25 30−10
−5
0
5
10
time 100
State Variable
![Page 18: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/18.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 18
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 102 truth
State Variable
time 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truthtime 102 truth
![Page 19: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/19.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 19
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 104 truth
State Variable
time 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truthtime 104 truth
![Page 20: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/20.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 20
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 106 truth
State Variable
time 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truthtime 106 truth
![Page 21: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/21.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 21
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 108 truth
State Variable
time 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truthtime 108 truth
![Page 22: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/22.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 22
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 110 truth
State Variable
time 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truthtime 110 truth
![Page 23: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/23.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 23
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 112 truth
State Variable
time 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truthtime 112 truth
![Page 24: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/24.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 24
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 114 truth
State Variable
time 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truthtime 114 truth
![Page 25: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/25.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 25
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 116 truth
State Variable
time 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truthtime 116 truth
![Page 26: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/26.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 26
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 118 truth
State Variable
time 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truthtime 118 truth
![Page 27: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/27.jpg)
7/17/08
bations
0.h’.
35 40ensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensembleensemble
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 27
Lorenz-96 is sensitive to small pertur
Introduce 20 ‘ensemble’ state estimates.Each is slightly perturbed for each Xi at time 10Refer to unperturbed control integration as ‘trut
5 10 15 20 25 30−10
−5
0
5
10
time 120 truth
State Variable
time 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truthtime 120 truth
![Page 28: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/28.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 28
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 201 truth ensemb
State Variable
time 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensemb
![Page 29: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/29.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 29
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 203 truth ensemb
State Variable
time 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensemb
![Page 30: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/30.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 30
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 205 truth ensemb
State Variable
time 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensemb
![Page 31: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/31.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 31
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 207 truth ensemb
State Variable
time 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensemb
![Page 32: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/32.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 32
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 209 truth ensemb
State Variable
time 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensemb
![Page 33: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/33.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 33
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 211 truth ensemb
State Variable
time 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensemb
![Page 34: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/34.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 34
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 213 truth ensemb
State Variable
time 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensemb
![Page 35: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/35.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 35
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 215 truth ensemb
State Variable
time 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensemb
![Page 36: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/36.jpg)
7/17/08
ed stations each step.
station location. from N(0, 1) to each..
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 36
Assimilate ‘observations’ from 40 randomly locat
Observations generated by interpolating truth toSimulate observational error: Add random drawStart from ‘climatological’ 20-member ensemble
5 10 15 20 25 30−10
−5
0
5
10
time 217 truth ensemb
State Variable
time 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensemb
![Page 37: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/37.jpg)
7/17/08
ed stations each step.
.
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
dent and right!
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 37
Assimilate ‘observations’ from 40 randomly locat
This isn’t working very well.Ensemble spread is reduced, but...,Ensemble is inconsistent with truth most places
5 10 15 20 25 30−10
−5
0
5
10
time 219 truth ensemb
State Variable
time 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensemb
ConfiConfident and WRONG.
![Page 38: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/38.jpg)
7/17/08
Filters
y
tk+2
pling Error;ian Assumption
Error;inearelation
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 38
Some Error Sources in Ensemble
y
****
h hh
tk
1. Model Error
2. h errors;Representativeness
4. SamGauss
5. SamplingAssuming LStatistical R
3. ‘Gross’ Obs. Errors
![Page 39: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/39.jpg)
7/17/08
gh sampling error.
hows expectedte value of sample
ation vs. trueation.
ted obs. reduced, increase error.
with localization.
let obs. impactted state.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 39
Observations impact unrelated state variables throu
Plot sabsolucorrelcorrel
Unrelasprea
Attack
Don’t unrela
0 0.5 10
0.2
0.4
0.6
0.8
1
True Correlation
Exp
ecte
d |S
ampl
e C
orre
latio
n|
10 Members20 Members40 Members80 Members
![Page 40: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/40.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 40
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 201 truth ensemb
Localization from Hierarchical Fi
time 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensemb
5 10 15 20 25 30−10
0
10
time 201 truth ensemb
State Variable
No Localization
time 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensembtime 201 truth ensemb
![Page 41: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/41.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 41
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 203 truth ensemb
Localization from Hierarchical Fi
time 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensemb
5 10 15 20 25 30−10
0
10
time 203 truth ensemb
State Variable
No Localization
time 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensembtime 203 truth ensemb
![Page 42: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/42.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 42
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 205 truth ensemb
Localization from Hierarchical Fi
time 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensemb
5 10 15 20 25 30−10
0
10
time 205 truth ensemb
State Variable
No Localization
time 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensembtime 205 truth ensemb
![Page 43: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/43.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 43
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 207 truth ensemb
Localization from Hierarchical Fi
time 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensemb
5 10 15 20 25 30−10
0
10
time 207 truth ensemb
State Variable
No Localization
time 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensembtime 207 truth ensemb
![Page 44: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/44.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 44
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 209 truth ensemb
Localization from Hierarchical Fi
time 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensemb
5 10 15 20 25 30−10
0
10
time 209 truth ensemb
State Variable
No Localization
time 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensembtime 209 truth ensemb
![Page 45: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/45.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 45
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 211 truth ensemb
Localization from Hierarchical Fi
time 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensemb
5 10 15 20 25 30−10
0
10
time 211 truth ensemb
State Variable
No Localization
time 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensembtime 211 truth ensemb
![Page 46: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/46.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 46
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 213 truth ensemb
Localization from Hierarchical Fi
time 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensemb
5 10 15 20 25 30−10
0
10
time 213 truth ensemb
State Variable
No Localization
time 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensembtime 213 truth ensemb
![Page 47: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/47.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 47
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 215 truth ensemb
Localization from Hierarchical Fi
time 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensemb
5 10 15 20 25 30−10
0
10
time 215 truth ensemb
State Variable
No Localization
time 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensembtime 215 truth ensemb
![Page 48: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/48.jpg)
7/17/08
bservation impact.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 48
Lorenz-96 Assimilation with localization of o
5 10 15 20 25 30−10
0
10
time 217 truth ensemb
Localization from Hierarchical Fi
time 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensemb
5 10 15 20 25 30−10
0
10
time 217 truth ensemb
State Variable
No Localization
time 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensembtime 217 truth ensemb
![Page 49: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/49.jpg)
7/17/08
bservation impact.with truth.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 49
Lorenz-96 Assimilation with localization of oEnsemble is much more consistent
5 10 15 20 25 30−10
0
10
time 219 truth ensemb
Localization from Hierarchical Fi
time 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensemb
5 10 15 20 25 30−10
0
10
time 219 truth ensemb
State Variable
No Localization
time 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensemb
![Page 50: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/50.jpg)
7/17/08
hical filter.maximizes signal.
35 40le obs
lter
le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40
ons
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 50
Localization computed by adaptive hierarcA tuning run of 4, 20-member ensembles
5 10 15 20 25 30−10
0
10
time 219 truth ensemb
Localization from Hierarchical Fi
time 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensembtime 219 truth ensemb
5 10 15 20 25 300
0.5
1
State Variable
3 Sample Observation Localizati
![Page 51: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/51.jpg)
7/17/08
ressure Obs. at 20N, 60E
XPERT USERS.
60 80 100
an factor level 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 60 80 100
s section at row 18
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 51
Localization in GCM can be very complex. Surface P
MUST HAVE ADAPTIVE HELP FOR NON-E
20 40 60 80 100−10
0
10
20
30
40
50
60u mean factor level 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100−10
0
10
20
30
40
50
60u mean factor level 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40−10
0
10
20
30
40
50
60u me
20 40 60 80 100−10
0
10
20
30
40
50
60u mean factor level 4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40 60 80 100−10
0
10
20
30
40
50
60u mean factor level 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
20 40
0.2
0.4
0.6
0.8
1cros
PS to U
![Page 52: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/52.jpg)
7/17/08
Filters
y
tk+2
pling Error;ian Assumption
Error;inearelation
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 52
Some Error Sources in Ensemble
y
****
h hh
tk
1. Model Error
2. h errors;Representativeness
4. SamGauss
5. SamplingAssuming LStatistical R
3. ‘Gross’ Obs. Errors
![Page 53: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/53.jpg)
7/17/08
d model error.
model.
10ys)
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 53
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
Time evolution for X1 shown.Assimilating model quickly diverges from ‘true’
0 5−5
0
5
10
model time (pseudo−da
F=8 F=6
![Page 54: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/54.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 54
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 301 truth ensemb
State Variable
time 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensemb
![Page 55: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/55.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 55
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 306 truth ensemb
State Variable
time 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensemb
![Page 56: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/56.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 56
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 311 truth ensemb
State Variable
time 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensemb
![Page 57: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/57.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 57
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 316 truth ensemb
State Variable
time 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensemb
![Page 58: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/58.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 58
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 321 truth ensemb
State Variable
time 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensemb
![Page 59: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/59.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 59
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 326 truth ensemb
State Variable
time 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensemb
![Page 60: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/60.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 60
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 331 truth ensemb
State Variable
time 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensemb
![Page 61: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/61.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 61
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 336 truth ensemb
State Variable
time 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensemb
![Page 62: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/62.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 62
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
5 10 15 20 25 30−10
−5
0
5
10
time 341 truth ensemb
State Variable
time 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensemb
![Page 63: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/63.jpg)
7/17/08
d model error
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 63
Assimilating in the presence of simulate
dXi / dt = (Xi+1 - Xi-2)Xi-1 - Xi + F.For truth, use F = 8.In assimilating model, use F = 6.
This isn’t working again!It will just keep getting worse.
5 10 15 20 25 30−10
−5
0
5
10
time 346 truth ensemb
State Variable
time 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensemb
![Page 64: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/64.jpg)
7/17/08
ance Inflation
=> ‘true’ distribution.ufficient prior variance.
ase spread of prior
.
−1 0
) x+
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 64
Model/Filter Error; Filter Divergence and Vari
1. History of observations and physical system 2. Sampling error, some model errors lead to ins
3. Naive solution is Variance inflation: just incre
4. For ensemble member i,
−4 −3 −20
0.5
1
Pro
babi
lity
"TRUE" Prior PDF
Variance Deficient PDF
inflate xi( ) λ xi x–(=
![Page 65: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/65.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 65
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 301 truth ensemb
Adaptive State Space Inflation
time 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensemb
5 10 15 20 25 30−10
0
10
time 301 truth ensemb
State Variable
No Inflation
time 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensembtime 301 truth ensemb
![Page 66: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/66.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 66
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 306 truth ensemb
Adaptive State Space Inflation
time 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensemb
5 10 15 20 25 30−10
0
10
time 306 truth ensemb
State Variable
No Inflation
time 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensembtime 306 truth ensemb
![Page 67: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/67.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 67
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 311 truth ensemb
Adaptive State Space Inflation
time 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensemb
5 10 15 20 25 30−10
0
10
time 311 truth ensemb
State Variable
No Inflation
time 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensembtime 311 truth ensemb
![Page 68: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/68.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 68
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 316 truth ensemb
Adaptive State Space Inflation
time 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensemb
5 10 15 20 25 30−10
0
10
time 316 truth ensemb
State Variable
No Inflation
time 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensembtime 316 truth ensemb
![Page 69: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/69.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 69
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 321 truth ensemb
Adaptive State Space Inflation
time 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensemb
5 10 15 20 25 30−10
0
10
time 321 truth ensemb
State Variable
No Inflation
time 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensembtime 321 truth ensemb
![Page 70: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/70.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 70
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 326 truth ensemb
Adaptive State Space Inflation
time 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensemb
5 10 15 20 25 30−10
0
10
time 326 truth ensemb
State Variable
No Inflation
time 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensembtime 326 truth ensemb
![Page 71: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/71.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 71
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 331 truth ensemb
Adaptive State Space Inflation
time 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensemb
5 10 15 20 25 30−10
0
10
time 331 truth ensemb
State Variable
No Inflation
time 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensembtime 331 truth ensemb
![Page 72: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/72.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 72
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 336 truth ensemb
Adaptive State Space Inflation
time 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensemb
5 10 15 20 25 30−10
0
10
time 336 truth ensemb
State Variable
No Inflation
time 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensembtime 336 truth ensemb
![Page 73: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/73.jpg)
7/17/08
f model error
rithm.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 73
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algo
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 341 truth ensemb
Adaptive State Space Inflation
time 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensemb
5 10 15 20 25 30−10
0
10
time 341 truth ensemb
State Variable
No Inflation
time 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensembtime 341 truth ensemb
![Page 74: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/74.jpg)
7/17/08
f model error
rithm.rror.
35 40
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
35 40le obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obsle obs
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 74
Assimilating with Inflation in presence oInflation is a function of state variable and time.Automatically selected by adaptive inflation algoIt can work, even in presence of severe model e
5 10 15 20 25 30
1.2
1.4
1.6inflation
5 10 15 20 25 30−10
0
10
time 346 truth ensemb
Adaptive State Space Inflation
time 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensemb
5 10 15 20 25 30−10
0
10
time 346 truth ensemb
State Variable
No Inflation
time 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensembtime 346 truth ensemb
![Page 75: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/75.jpg)
7/17/08
ltering
observed inconsistency.
.
posed to be unbiased.
2 4
hood
S.D.n
obs2
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 75
Adaptive Inflation for Ensemble Fi
1. For observed variable, have estimate of prior-
2. Expected(prior mean - observation) =
Assumes that prior and observation are supIs it model error or random chance?
−4 −2 00
0.2
0.4
0.6
0.8
Pro
babi
lity
Prior PDF
S.D.
Obs. Likeli
Expected Separatio
Actual 4.714 SDs
σprior2 σ+
![Page 76: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/76.jpg)
7/17/08
ltering
observed inconsistency.
.
rior and observation.
2 4
hood
S.D.tion
obs2
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 76
Adaptive Inflation for Ensemble Fi
1. For observed variable, have estimate of prior-
2. Expected(prior mean - observation) =
3. Inflating increases expected separation.Increases ‘apparent’ consistency between p
−4 −2 00
0.2
0.4
0.6
0.8
Pro
babi
lity
Prior PDF Obs. Likeli
Inflated S.D.Expected Separa
Actual 3.698 SDs
σprior2 σ+
![Page 77: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/77.jpg)
7/17/08
ltering
observed inconsistency.
2 4
hood
S.D.tion
yo.
or σobs2
+ N 0 θ,( )=
2⁄D2 2θ2⁄–( )exp
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 77
Adaptive Inflation for Ensemble Fi
1. For observed variable, have estimate of prior-
Distance, D, from prior mean y to obs. is
Prob. yo is observed givenλ:
−4 −2 00
0.2
0.4
0.6
0.8
Pro
babi
lity
Prior PDF Obs. Likeli
Inflated S.D.Expected Separa
Actual 3.698 SDsy
λ
N 0 λσpri2
,
p yo λ( ) 2Πθ2( )1–
=
![Page 78: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/78.jpg)
7/17/08
ltering
on factor,λ.
.p)
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 78
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
Assume prior is gaussian;
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. LikelihoodPrior PDF
p λ Yprev( ) N λp σλ,2,(=
![Page 79: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/79.jpg)
7/17/08
ltering
on factor,λ.
e've assumed aaussian for prior
.
ecall that
an be evaluatedrom normal PDF.
.
p λ Yprev( )
p yo λ( )
tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 79
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
Wg
R
cf
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. LikelihoodPrior PDF
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 80: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/80.jpg)
7/17/08
ltering
on factor,λ.
et
rom normal PDF.
ultiply by
o get
.
p yo λ 0.75=( )
λ 0.75= Yprev( )
λ 0.75= Yprev˙ yo,( )
tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 80
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
G
f
M
t
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. LikelihoodInflated Prior λ = 0.75
p
p
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 81: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/81.jpg)
7/17/08
ltering
on factor,λ.
et
rom normal PDF.
ultiply by
o get
.
p yo λ 1.5=( )
p λ 1.5= Yprev( )
p λ 1.5= Yprev˙ yo,( )
tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 81
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
G
f
M
t
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. LikelihoodInflated Prior λ = 1.5
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 82: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/82.jpg)
7/17/08
ltering
on factor,λ.
et
rom normal PDF.
ultiply by
o get
.
p yo λ 2.2=( )
p λ 2.2= Yprev( )
p λ 2.2= Yprev˙ yo,( )
tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 82
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
G
f
M
t
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. LikelihoodInflated Prior λ = 2.25
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 83: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/83.jpg)
7/17/08
ltering
on factor,λ.
epeat for a rangef values ofλ.
ow must get pos-erior in same forms prior (gaussian).
.tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 83
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
Ro
Nta
0 1 2 3 4 5 60
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
Likelihood y observed given λPosterior
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. Likelihood
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 84: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/84.jpg)
7/17/08
ltering
on factor,λ.
ery little informa-ion aboutλ in aingle observation.
osterior and priorre very similar.
ormalized poste-ior indistinguish-ble from prior.
.tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 84
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
Vts
Pa
Nra
1 20
1
2
Obs. Space Inflation Factor: λ
Prior λ PDF
Posterior
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. Likelihood
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 85: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/85.jpg)
7/17/08
ltering
on factor,λ.
ery little informa-ion aboutλ in aingle observation.
osterior and priorre very similar.
ifference showslight shift to largeralues ofλ.
.tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 85
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
Vts
Pa
Dsv
1 2
0.01
0
0.01
Obs. Space Inflation Factor: λ
λ: Posterior − Prior
Max density shifted to right
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. Likelihood
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 86: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/86.jpg)
7/17/08
ltering
on factor,λ.
ne option is to useaussian prior for.
elect max (mode)f posterior asean of updatedaussian.
o a fit for updatedtandard deviation.
.tion
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 86
Adaptive Inflation for Ensemble Fi
Use Bayesian statistics to get estimate of inflati
OGλ
SomG
Ds1 2
0
1
2
Obs. Space Inflation Factor: λ
Prior λ PDFFind Max by search
Max is new λ mean
−1 0 1 2 3 40
0.2
0.4
0.6
Observation: y
Obs. Likelihood
p λ Yprev yo,( ) p yo λ( ) p λ Yprev( ) normaliza⁄=
![Page 87: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/87.jpg)
7/17/08
ltering
ically!
6th order poly inθ
lved to give mode.
regressions.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 87
Adaptive Inflation for Ensemble Fi
A. Computing updated inflation mean, .
Mode of can be found analyt
Solving leads to
This can be reduced to a cubic equation and so
New is set to the mode.
This is relatively cheap compared to computing
λu
p yo λ( ) p λ Yprev( )
∂ p yo λ( ) p λ Yprev( ) ∂λ 0=⁄
λu
![Page 88: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/88.jpg)
7/17/08
ltering
d point, e.g. .
and .
.
λu σλ p,+
p λu σλ p,+( )
λu σλ p,+ ) p λu( )⁄
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 88
Adaptive Inflation for Ensemble Fi
A. Computing updated inflation variance,
1. Evaluate numerator at mean and secon
2. Find so goes through
3. Compute as where
σλ u,2
λu
σλ u,2 N λu σλ u,
2,( ) p λu( )
σλ u,2 σλ p,
2 2 rln⁄–= r p(=
![Page 89: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/89.jpg)
7/17/08
tion space.
space inflation?
int distribution.
ss changes inn onto state vari-flation.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 89
State Space Adaptive Inflation
Computations so far adapt inflation for observa
What is relation between observation and state
Have to use prior ensemble observation/state jo
Regreinflatioable in
y
****
h hh
y
tk+2
tk
![Page 90: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/90.jpg)
7/17/08
orithm:
te variable,λs,i.
e impact ofλs,i on
bservation then
ll 12th order:
.duct in this case!
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 90
Spatially varying adaptive inflation alg
Have a distribution forλ at each time for each sta
Use prior correlation from ensemble to determinprior variance for given observation.
If γ is correlation between state variable i and o
.
Equation for finding mode of posterior is now fuAnalytic solution appears unlikely.
Can do Taylor expansion ofθ aroundλs,i .
Retaining linear term is normally quite accurateThere is an analytic solution to find mode of pro
θ 1 γ λs i, 1–( )+[ ]2σprior
2σobs
2+=
![Page 91: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/91.jpg)
7/17/08
P
erature, moisture.
huge spread).
inds.
reanalysis.
day.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 91
Adaptive Inflation in Global NW
Model: CAM 3.1 T85L26.
Several million model variables: winds, temp
Initialized from a climatological distribution (
Observations: Radiosondes, ACARS, Satellite W
Subset of observations used in NCAR/NCEP
Several hundred thousand observations per
![Page 92: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/92.jpg)
7/17/08
ce Prior RMS, Spread
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 92
Adaptive Inflation in CAM; 500 hPa T Obs. Spa
![Page 93: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/93.jpg)
7/17/08
; 266 hPa U
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 93
Adaptive Inflation in CAM after 1-Month
![Page 94: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/94.jpg)
7/17/08
on in CAM
nsely observed regions.
embleto work well!
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 94
Adaptive Inflation for Numerical Weather Predicti
1. Largest inflation caused by model error in de
2. RMS reduced, spread increased.
3. Fewer observations rejected.
Combinedwith localization,allows80memberens
![Page 95: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/95.jpg)
7/17/08
lters: Summary
tio to minimize error.
ter system.e of estimate.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 95
Hierarchical Bayesian Methods for Adaptive Fi
1. Localization:Run an ensemble of ensembles.Use regression coefficient signal-to-noise ra
2. Inflation:Use each observation twice.
Once to adjust parameter (inflation) of filSecond time to adjust mean and varianc
![Page 96: Using Small Ensembles in High Dimensions: Hierarchical](https://reader031.vdocuments.us/reader031/viewer/2022012104/616a1b4c11a7b741a34edfd5/html5/thumbnails/96.jpg)
7/17/08
ticians.
Anderson: Joint Stats Meeting, Denver, 4 August, 2008 96
Conclusions
Lots of cool statistics being done poorly.
Work in small models can give insight.
Applications in large models are important.
Looking for collaborations with interested statis