Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
Data Assimilation and Driver Estimation forSpace Weather Models using Ensemble Filters
Alexey V. Morozov
UofM Committee:Dennis S. BernsteinAaron J. RidleyPierre T. KabambaIlya V. Kolmanovsky
NCAR Collaborators:Nancy CollinsTimothy J. HoarJeffrey L. Anderson
March 29, 2013
AV Morozov, UM Data Assimilation and Driver Estimation 1/56
Overview
1 Summary
2 Problem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 2/56
1 SummaryMain ContributionsPublications
2 Problem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 3/56
Main Contributions
Contributions relating to data assimilation are:
modified the Data Assimilation Research Testbed (DART) tointerface it with the Global Ionosphere-Thermosphere Model(GITM),
developed a novel inflation technique for the Ensemble AdjustmentKalman Filter (EAKF) as applied to GITM for purposes of dataassimilation and driver estimation, and
introduced an ability to assimilate Total Electron Content (TEC)measurements into the DART-GITM interface.
Contributions relating to adaptive control are:
described the Retrospective Cost Adaptive Control (RCAC) stabilitymargins for plants with uncertain nonminimum-phase zeros,
introduced a convex constraint on the controller pole locations toimprove RCAC transient and steady-state performance, and
modeled nonlinear system and described the achievable amplitudeand frequency control ranges.
AV Morozov, UM Data Assimilation and Driver Estimation 4/56
Main Contributions
Contributions relating to data assimilation are:
modified the Data Assimilation Research Testbed (DART) tointerface it with the Global Ionosphere-Thermosphere Model(GITM),
developed a novel inflation technique for the Ensemble AdjustmentKalman Filter (EAKF) as applied to GITM for purposes of dataassimilation and driver estimation, and
introduced an ability to assimilate Total Electron Content (TEC)measurements into the DART-GITM interface.
Contributions relating to adaptive control are:
described the Retrospective Cost Adaptive Control (RCAC) stabilitymargins for plants with uncertain nonminimum-phase zeros,
introduced a convex constraint on the controller pole locations toimprove RCAC transient and steady-state performance, and
modeled nonlinear system and described the achievable amplitudeand frequency control ranges.
AV Morozov, UM Data Assimilation and Driver Estimation 4/56
Publications
1 M. S. Fledderjohn, M. S. Holzel, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “On the Accuracy ofLeast Squares Algorithms for Estimating Zeros,” Proc. Amer. Contr. Conf., Baltimore, MD, June 2010.
2 A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “A Computational Study of the Performance andRobustness Properties of Retrospective Cost Adaptive Control,” AIAA Guid. Nav. Contr. Conf., Toronto,August 2010.
3 A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “Retrospective Adaptive Control of a Planar MultilinkArm with Nonminimum-Phase Zeros,” Proc. Conf. Dec. Contr., pp. 3706–3711, Atlanta, GA, December2010.
4 A. V. Morozov, A. M. D’Amato, J. B. Hoagg, and D. S. Bernstein, “Retrospective Cost Adaptive Controlfor Nonminimum-Phase Systems with Uncertain Nonminimum-Phase Zeros Using Convex Optimization,”Proc. Amer. Contr. Conf., pp. 1188–2293, San Francisco, CA, June 2011.
5 A. M. D’Amato, E. D. Sumer, K. S. Mitchell, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “AdaptiveOutput Feedback Control of the NASA GTM Model with Unknown Nonminimum-Phase Zeros,” AIAAGuid. Nav. Contr. Conf., Portland, OR, August 2011.
6 A. V. Morozov, A. A. Ali, A. M. D’Amato, A. J. Ridley, S. L. Kukreja, and D. S. Bernstein,“Retrospective-Cost-Based Model Refinement for System Emulation and Subsystem Identification,” Proc.Conf. Dec. Contr., pp. 2142–2147, Orlando, FL, December 2011.
7 E. D. Sumer, A. M. D’Amato, A. V. Morozov, J. B. Hoagg, and D. S. Bernstein, “Robustness ofRetrospective Cost Adaptive Control to Markov-Parameter Uncertainty,” Proc. Conf. Dec. Contr., pp.6085–6090, Orlando, FL, December 2011.
8 M. W. Isaacs, J. B. Hoagg, A. V. Morozov, and D. S. Bernstein, “A Numerical Study on Controlling aNonlinear Multilink Arm Using a Retrospective Cost Model Reference Adaptive Controller,” Proc. Conf.Dec. Contr., pp. 8008–8013, Orlando, FL, December 2011.
9 A. V. Morozov, A. J. Ridley, D. S. Bernstein, N. Collins, T. J. Hoar, and J. L. Anderson, “DataAssimilation and Driver Estimation for the Global Ionosphere-Thermosphere Model Using the EnsembleAdjustment Kalman Filter”, Journal of Atmospheric and Solar-Terrestrial Physics, 2013, Submitted.
10 A. V. Morozov, A. G. Burrell, A. J. Ridley, D. S. Bernstein, “Assimilation of the Total Electron ContentMeasurements into the Global Ionosphere-Thermosphere Model”, Journal of Atmospheric andSolar-Terrestrial Physics, 2013, To be submitted.
AV Morozov, UM Data Assimilation and Driver Estimation 5/56
1 Summary
2 Problem StatementBig PictureProblem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 6/56
AV Morozov, UM Data Assimilation and Driver Estimation 7/56
laneposition
steeringwheel angle
Car
AV Morozov, UM Data Assimilation and Driver Estimation 8/56
outputinputPlant
AV Morozov, UM Data Assimilation and Driver Estimation 9/56
+
- output
desired
input
correction
+
error
Controller
Plant-output
AV Morozov, UM Data Assimilation and Driver Estimation 10/56
Where does model inversion come in?
+
-
yy d
u
+
eyC
P-
The goal is to make y = y, that is yy
= 1 or e = 0, that is ey
= 0
y = Pd
y = P (y − Cy)
y = Py − PCy(1 + PC)y = Py
y
y=
P
1 + PC
P
1 + PC
want= 1
Pw= 1 + PC
Cw= 1− P−1
e = y − y
e = y −P
1 + PCy
e =1 + PC − P
1 + PCy
e
y=
1 + PC − P1 + PC
1 + PC − P1 + PC
w= 0
1 + PC − P w= 0
Cw= 1− P−1
AV Morozov, UM Data Assimilation and Driver Estimation 11/56
Example of inversion in frequency domain
Consider a linear plant P = 1z+0.5
, which can be described in time domain asyk = −0.5yk−1 + dk−1. Suppose the goal is for the plant output y to follow desiredcommand y (command following). RCAC converges to controller C = 0.7104
z−0.5797
0 50 100 150 200
−1
0
1
(a) Time step [samp]
y, desired
positio
n [m
]
0 50 100 150 200−2
0
2
4
(b) Time step [samp]
e, positio
n e
rror
[m]
0 50 100 150 200−2
0
2
(c) Time step [samp]
u, c
ontr
ol fo
rce [N
]
0 50 100 150 200−2
−1
0
1
(d) Time step [samp]
θ, c
ontr
ol gain
s
0 1 2 3−10
−8
−6
−4
−2
0
2
4
6
(e) Frequency [rad\samp]
Magnitude [d
B]
0 1 2 3−200
−150
−100
−50
0
50
(f) Frequency [rad\samp]
Phase [d
eg]
Plant Inverse (1−P−1
)
Controller (C)
Closed Loop (P/(1+PC))
Command frequency (w)
AV Morozov, UM Data Assimilation and Driver Estimation 12/56
How is data assimilation on GITM similar to driving a car?
+
- output
desired
input
correction
+
error
Controller
Plant-output
states+ states-
densitydesired
F10.7+
error
GITM
DART
-density
AV Morozov, UM Data Assimilation and Driver Estimation 13/56
Problem Statement
We want to drive GITM output (black) to match the satellite data (red).
AV Morozov, UM Data Assimilation and Driver Estimation 14/56
1 Summary
2 Problem Statement
3 GITMInputs and OutputsEquationsImplementation
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 15/56
Inputs and Outputs
Inputs Outputs
Solar flux index F10.7 or I∞ → → Ns Neutral number densities
Cooling rates Le(X) →
G
→ ρ Neutral mass density
Heating efficiency ε → I → p Neutral pressure
Thermal conductivity κc →
T
→ T Neutral temperature normalized
Earth magnetic field simplified →
M
→ u Neutral velocity
or APEX → Nj Ion number densities
Interplanetary simplified → → Tj Ion temperature normalized
magnetic field or ACE data → v Ion velocity
Hemispheric simplified →power index or POES data
Initialize using simplified →or MSIS/IRI
Table 21 compiled from 1
1Ridley, A. J., Y. Deng, and G. Toth. “The Global IonosphereThermosphereModel,” Journal of Atmospheric and Solar-Terrestrial Physics 68, no. 8 (2006)
AV Morozov, UM Data Assimilation and Driver Estimation 16/56
GITM Vertical Equations
∂ Ns
∂t+∂ur,s∂r
+2ur,sr
+ ur,s∂ Ns
∂r=
1
NsSs, (continuity) (1)
∂ur,s∂t
+ ur,s∂ur,s∂r
+uθr
∂ur,s∂θ
+uφ
r cos(θ)
∂ur,s∂φ
+k
Ms
∂T
∂r+k T
Ms
∂ Ns
∂r=
g + Fs +u2θ + u2φ
r+ cos2(θ)Ω2r + 2 cos(θ)Ωuφ, (momentum) (2)
∂ T
∂t+ ur
∂ T
∂r+ (γ − 1) T
(2urr
+∂ur∂r
)=
k
cvρmnQ , (energy) (3)
AV Morozov, UM Data Assimilation and Driver Estimation 17/56
GITM Vertical Equations and source terms
(red - outputs, green - inputs)
∂ Ns
∂t+∂ur,s∂r
+2ur,sr
+ ur,s∂ Ns
∂r=
1
NsSs, (continuity) (4)
∂ur,s∂t
+ ur,s∂ur,s∂r
+uθr
∂ur,s∂θ
+uφ
r cos(θ)
∂ur,s∂φ
+k
Ms
∂T
∂r+k T
Ms
∂ Ns
∂r=
g + Fs +u2θ + u2φ
r+ cos2(θ)Ω2r + 2 cos(θ)Ωuφ, (momentum) (5)
∂ T
∂t+ ur
∂ T
∂r+ (γ − 1) T
(2urr
+∂ur∂r
)=
k
cvρmnQ , (energy) (6)
Ss =∂
∂r
[NsKe
(∂Ns∂r− ∂N
∂r
)]+ Cs, (7)
Fs =ρiρsνin(vr − ur,s) +
kT
Ms
∑q 6=s
NqNDqs
(ur,q − ur,s), (8)
Q = QEUV +QNO +QO +∂
∂r
((κc + κeddy)
∂T
∂r
)+Ne
mimn
mi + mnνin(v − u)2.
(9)AV Morozov, UM Data Assimilation and Driver Estimation 18/56
Implementation
In the CHAMP simulations, 5 resolution in longitude and latitude is used.
Variable resolution in altitude (from about 2km to about 18km) is used to spanthe range between about 100km and 660km.
The atmosphere is broken up into 32 (8 in longitude, 4 in latitude) blocks toallow for parallel computation.
A typical EAKF run (20 ensemble members) requires 640 CPUs (cores) andabout 8 wall hours per 24 simulated hours on NASA Pleiades supercomputer.
(a) Longitude [deg]
Latitu
de [deg]
0 60 120 180 240 300 360−90
−60
−30
0
30
60
90
Mass D
ensity [kg m
−3]
1
2
3
4
5
6
7x 10
−12
Horizontal resolution.
0 60 120 180 240 300 360100
128
153
191
240
297
361
428
499
570
642
(b) Longitude [deg]
Altitude (
km
)
Log
10 (
Mass D
ensity [kg m
−3])
−14
−13
−12
−11
−10
−9
−8
−7
−6
Vertical resolution.
AV Morozov, UM Data Assimilation and Driver Estimation 19/56
1 Summary
2 Problem Statement
3 GITM
4 EAKFBig PictureKalman FilterExtended Kalman Filter (EKF)Unscented Kalman Filter (UKF)Ensemble Kalman Filter (EnKF)Ensemble Adjustment Kalman Filter (EAKF)
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future WorkAV Morozov, UM Data Assimilation and Driver Estimation 20/56
Pros and Cons of These Filters
KF → EKF → UKF→→
EnKFEAKF
KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.EKF → UKF The advantage of UKF over EKF is that it is more accurate
in propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not
require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but
utilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.
AV Morozov, UM Data Assimilation and Driver Estimation 21/56
Pros and Cons of These Filters
KF → EKF → UKF→→
EnKFEAKF
KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.
EKF → UKF The advantage of UKF over EKF is that it is more accuratein propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not
require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but
utilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.
AV Morozov, UM Data Assimilation and Driver Estimation 21/56
Pros and Cons of These Filters
KF → EKF → UKF→→
EnKFEAKF
KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.EKF → UKF The advantage of UKF over EKF is that it is more accurate
in propagating means and covariances through the nonlinear system anddoes not require linearization.
UKF → EnKF The advantage of EnKF over UKF is that it does notrequire 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but
utilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.
AV Morozov, UM Data Assimilation and Driver Estimation 21/56
Pros and Cons of These Filters
KF → EKF → UKF→→
EnKFEAKF
KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.EKF → UKF The advantage of UKF over EKF is that it is more accurate
in propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not
require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.
UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, bututilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.
AV Morozov, UM Data Assimilation and Driver Estimation 21/56
Pros and Cons of These Filters
KF → EKF → UKF→→
EnKFEAKF
KF → EKF The advantage of EKF over KF is that it allows the system tobe nonlinear.EKF → UKF The advantage of UKF over EKF is that it is more accurate
in propagating means and covariances through the nonlinear system anddoes not require linearization.UKF → EnKF The advantage of EnKF over UKF is that it does not
require 2n ensemble members (sigma points, particles), and allows the userto pick N , the number of ensemble members. The newest versions ofEnKF allow for localization, which improves computational performanceeven more than just reduction in number of ensemble members.UKF → EAKF EAKF is similar to EnKF in being an ensemble filter, but
utilizes different update equations. Some advantages of EAKF over UKFare addition of localization functionality and decreased computational load.
AV Morozov, UM Data Assimilation and Driver Estimation 21/56
Kalman Filter
Consider the linear discrete-time system given by
xk = Fk−1xk−1 +Gk−1uk−1 + wk−1, wk ∼ N(0, Qk), (10)
yk = Hkxk + vk, vk ∼ N(0, Rk), (11)
where xk ∈ Rn, yk ∈ Rm, uk ∈ Rp, wk is the process noise, vk is the measurementnoise, and wk and vk are uncorrelated (i.e. E(wkv
Tj ) = 0).
If Fk, Gk, Hk, Qk, Rk, are known, the discrete time Kalman Filter2 is given by
x−k = Fk−1x+k−1 +Gk−1uk−1, (prior estimate) (12)
P−k = Fk−1P+k−1F
Tk−1 +Qk−1, (prior error covariance) (13)
Kk = P−k HTk (HkP
−k H
Tk +Rk)−1, (estimator gain) (14)
P+k = (I −KkHk)P−k , (posterior error covariance) (15)
x+k = x−k +Kk(yk −Hkx−k ). (posterior estimate) (16)
2Simon, Dan. Optimal state estimation: Kalman, H infinity, and nonlinearapproaches. Wiley-Interscience, 2006.
AV Morozov, UM Data Assimilation and Driver Estimation 22/56
Kalman Filter and Discrete Algebraic Riccati Equation
As an aside, we demonstrate that Discrete Algebraic Riccati Equation (DARE) can bederived from the update equations derived so far.Substituting (16) into (12), we realize that prior state estimate can be updated directlywithout computation of the posterior estimate as in
x−k+1 = Fk(I −KkHk)x+k−1 + FkKkykGkuk. (17)
Similarly, by substituting (15) and (14) into (13), it can be shown that prior errorcovariance matrix can also be updated in one step, as given by
P−k+1 = Fk(P−k −[P−k H
Tk (HkP
−k H
Tk +Rk)−1
]HkP
−k )FTk +Qk (18)
= FkP−k F
Tk − FkP
−k H
Tk (HkP
−k H
Tk +Rk)−1HkP
−k F
Tk +Qk, (19)
which is the Discrete Algebraic Riccati Equation.
AV Morozov, UM Data Assimilation and Driver Estimation 23/56
Extended Kalman Filter (EKF)
Now consider the nonlinear discrete-time system given by
xk = fk−1(xk−1, uk−1, wk−1), wk ∼ N(0, Qk), (20)
yk = hk(xk, vk), vk ∼ N(0, Rk), (21)
where functions fk(·) and hk(·) are known explicitly and hence can be linearized via
Fk−1 =∂fk−1
∂x
∣∣∣x+k−1
, Lk−1 =∂fk−1
∂w
∣∣∣x+k−1
, (22)
Hk =∂hk
∂x
∣∣∣x−k
, Mk =∂hk
∂v
∣∣∣x−k
. (23)
Next, EKF3 can be updated via
x−k = fk−1 (x+k−1, uk−1, 0), (24)
P−k = Fk−1P+k−1F
Tk−1 + Lk−1Qk−1L
Tk−1, (25)
Kk = P−k HTk (HkP
−k H
Tk +MkRkM
Tk )−1, (26)
P+k = (I −KkHk)P−k , (27)
x+k = x−k +Kk
[yk − hk (x−k , 0)
]. (28)
3Simon, Dan. Optimal state estimation: Kalman, H infinity, and nonlinearapproaches. Wiley-Interscience, 2006.
AV Morozov, UM Data Assimilation and Driver Estimation 24/56
Unscented Kalman Filter (UKF)
Consider the nonlinear discrete-time system with additive noise given by
xk = fk−1(xk−1, uk−1) + wk−1, (29)
yk = hk(xk) + vk. (30)
UKF propagates 2n realizations (sigma points, x(i)) of the state’s probability densityfunction, as defined and propagated by
x(i)k−1 = x+
k−1 + (−1)bi−1nc(√
nP+k−1
)Ti, (31)
x(i)k = fk−1(x
(i)k−1, uk−1) , y
(i)k = hk(x
(i)k ) . (32)
Accordingly, UKF can be updated via
x−k =1
2n
∑2n
i=1x
(i)k , y−k =
1
2n
∑2n
i=1y
(i)k , (33)
P−k =∑2ni=1(x
(i)k − x
−k )(x
(i)k − x
−k )T /(2n) +Qk−1 , (34)
Kk = PxyP−1y , Pxy =
∑2n
i=1(x
(i)k − x
−k )(y
(i)k − y
−k )T /(2n) (35)
P+k = P−k −KkPyK
Tk , Py =
∑2n
i=1(y
(i)k − y
−k )(y
(i)k − y
−k )T /(2n) +Rk (36)
x+k = x−k +Kk(yk − yk). (37)
AV Morozov, UM Data Assimilation and Driver Estimation 25/56
Ensemble Kalman Filter (EnKF)
EnKF generates N vectors of initial conditions (x+1 ) and feeds these into the model
given by (21). Model started from different initial conditions is referred to as differentensemble members.
Mean states estimate is calculated via µx−k =∑Ni=1
x−k,i
N.
hk(·) still needs to be known explicitly and its linearization Hk needs to be computedabout ensemble mean.Accordingly, N EnKF ensemble members can be updated via4
x−k,i = fk−1(x+k−1,i, uk−1, 0), (38)
P−k =∑Ni=1(x−k,i − µx
−k )(x−k,i − µx
−k )T /(N − 1) , (39)
Kk = P−k HTk (HkP
−k H
Tk +MkRkM
Tk )−1, (40)
P+k = (I −KkHk)P−k , (41)
x+k,i = x−k,i +Kk
[yk − hk(x−k,i, 0)
]. (42)
4Evensen, Geir. “Sequential data assimilation with a nonlinear quasi-geostrophicmodel using Monte Carlo methods to forecast error statistics.” J. Geophys. Res., - 99(1994): 10-10.
AV Morozov, UM Data Assimilation and Driver Estimation 26/56
Ensemble Adjustment Kalman Filter (EAKF)
First, define joint5 state-observation vector as
zk =
[xkyk
]. (43)
Recall that xk ∈ Rn, yk ∈ Rm. We then define H ∈ Rm×(n+m) such that yk = Hzk,i.e. H = [ 0m×n Im×m ].Accordingly, N EAKF ensembles can be updated via6
z−k,i = [fk−1(x+k−1,i, uk−1, 0); hk(x+
k−1,i, 0)], (44)
P−k =∑N
i=1(z−k,i − µz
−k )(z−k,i − µz
−k )T /(N − 1), (45)
Ak = (FTk )−1GTk (UTk )−1BTk (GTk )−1FTk , (46)
P+k = [(P−k )−1 +HTR−1
k H]−1 , µz+k = P+
k [(P−k )−1µz−k +HTR−1k yk] , (47)
z+k,i = ATk (z−k,i − µz
−k ) + µz+
k . (48)
5Tarantola, Albert. Inverse problem theory and methods for model parameterestimation. Society for Industrial Mathematics, 2005.
6Anderson, Jeffrey L. “An ensemble adjustment Kalman filter for dataassimilation.” Monthly Weather Review 129.12 (2001): 2884-2903.
AV Morozov, UM Data Assimilation and Driver Estimation 27/56
EAKF: intuition
0 1 2 3 4 5 6 7−1.5
−1
−0.5
0
0.5
1
1.5
k
x[k
]
y [1]
x−[1]
x+[1]
x[k]y[k]
x−i [k ]
x+i [k ]
AV Morozov, UM Data Assimilation and Driver Estimation 28/56
Filter Divergence
Consider linear system
xk = 0.5xk−1 + uk−1, uk = 1.0 + sin(0.5k), (49)
yk = xk + vk, vk ∼ N(0, 0.2). (50)
If the driver varies with time but we assume it is constant, ensemble mightcollapse.
1 5 10 15 20 25 30 35 40 45−6
−4
−2
0
2
4
6
(a) Time k
State
sk
True state s k
Measurement yk
EAKF mean µ [ s k]EAKF spread µ [ s k] ± σ [ s k]EAKF ini t i al p df ( s+
1 )
1 5 10 15 20 25 30 35 40 45−3
−2
−1
0
1
2
3
(b) Time k
Inputuk
True input uk
EAKF mean µ [ uk]EAKF spread µ [ uk] ± σ [ uk]EAKF ini t i al p df ( u+
1 )
AV Morozov, UM Data Assimilation and Driver Estimation 29/56
Ensemble Inflation
Ensemble inflation given by
x−k =√λ(x−k − µ[x−k ]) + µ[x−k ], (51)
with λ = 2.0 alleviates filter divergence.
1 5 10 15 20 25 30 35 40 45−6
−4
−2
0
2
4
6
(a) Time k
Sta
tesk
True state s k
Measurement yk
EAKF mean µ [ s k]EAKF spread µ [ s k] ± σ [ s k]EAKF ini t i al p df ( s+
1 )
1 5 10 15 20 25 30 35 40 45−3
−2
−1
0
1
2
3
(b) Time k
Inputuk
True input uk
EAKF mean µ [ uk]EAKF spread µ [ uk] ± σ [ uk]EAKF ini t i al p df ( u+
1 )
AV Morozov, UM Data Assimilation and Driver Estimation 30/56
Differential Inflation
Inflating driver by
u−k =
√σ2i√
σ2[u−k ](u−k − µ[u−k ]) + µ[u−k ], (52)
with σ2i = 0.12 removes the lag in the driver estimate.
1 5 10 15 20 25 30 35 40 45−6
−4
−2
0
2
4
6
Sta
tesk
True state s k
Measu rem ent y k
EAKF mean µ [ s k]EAKF sp re ad µ [ s k] ± σ [ s k]EAKF in i t ial p df ( s+
1 )
1 5 10 15 20 25 30 35 40 45−3
−2
−1
0
1
2
3
T im e k
Inputu
k
True inpu t u k
EAKF mean µ [ u k]EAKF sp re ad µ [ u k] ± σ [ u k]EAKF in i t ial p df ( u+
1 )
AV Morozov, UM Data Assimilation and Driver Estimation 31/56
1 Summary
2 Problem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral DensitySimulated CHAMP dataReal CHAMP data
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 32/56
CHAMP and GRACE orbits
AV Morozov, UM Data Assimilation and Driver Estimation 33/56
Sim: Does actual density track the desired density?
Measurement source: ρ from GITM truth simulation (F10.7 = 148) isinterpolated to the CHAMP location
Ensemble members: 20 GITM instances with F10.7 = N(130, 25)
AV Morozov, UM Data Assimilation and Driver Estimation 34/56
Same plot, but orbit averages
AV Morozov, UM Data Assimilation and Driver Estimation 35/56
Sim: Does error at the measurement location go to zero?
Define RMS percentage error RMSPE4=
√(ρ−ρ)2√ρ2
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(a) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 2%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 36/56
Sim: Does error at the validation location go to zero?
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(b) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 4%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 37/56
Sim: Does F10.7 estimate converge to the true value?
AV Morozov, UM Data Assimilation and Driver Estimation 38/56
Real: Does actual density track the desired density?
Measurement source: real CHAMP ρ with real CHAMP uncertainty
Ensemble members: 20 GITM instances with F10.7 = N(130, 25)
AV Morozov, UM Data Assimilation and Driver Estimation 39/56
Real: Does error at the measurement location go to zero?
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450
10
20
30
40
50
60
70
80
90
100
(c) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 7%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 40/56
Real: Does error at the validation location go to zero?
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450
10
20
30
40
50
60
70
80
90
100
(d) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 52%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 41/56
Real: Does F10.7 estimate converge to the NOAA value?
AV Morozov, UM Data Assimilation and Driver Estimation 42/56
1 Summary
2 Problem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron ContentSimulated TEC dataReal TEC data
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 43/56
Real TEC data
AV Morozov, UM Data Assimilation and Driver Estimation 44/56
Simulated TEC data
0 60 120 180 240 300 360Longitude [deg]
−90
−60
−30
0
30
60
90La
titu
de [
deg]
2002-12-01 00:30:00 0
2
4
6
8
10
12
14
16
18
20
TEC
[T
EC
U]
AV Morozov, UM Data Assimilation and Driver Estimation 45/56
Sim: True and Estimated TEC
AV Morozov, UM Data Assimilation and Driver Estimation 46/56
Sim: TEC error
AV Morozov, UM Data Assimilation and Driver Estimation 47/56
Sim: average error and driver
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
0
2
4
6
8
10Average VTEC error [TECU]
Average VTEC error
Average VTEC spread (± SD)
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
100
150
200
250
300
F10
.7 [SFU
]
F10.7 measured
F10.7 estimated
F10.7 estimated ± SD
AV Morozov, UM Data Assimilation and Driver Estimation 48/56
Sim: average error without driver estimation
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
0
2
4
6
8
10Average VTEC error [TECU]
Average VTEC error
Average VTEC spread (± SD)
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
100
150
200
250
300
F10
.7 [SFU
]
F10.7 measured
F10.7 estimated
F10.7 estimated ± SD
AV Morozov, UM Data Assimilation and Driver Estimation 49/56
Real: True and Estimated TEC
AV Morozov, UM Data Assimilation and Driver Estimation 50/56
Real: TEC error
AV Morozov, UM Data Assimilation and Driver Estimation 51/56
Real: average error and driver
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
0
2
4
6
8
10
12
14
16Average VTEC error [TECU]
Average VTEC error
Average VTEC spread (± SD)
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
100
150
200
250
300
F10
.7 [SFU
]
F10.7 estimated
F10.7 estimated ± SD
AV Morozov, UM Data Assimilation and Driver Estimation 52/56
Real: average error without driver estimation
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
0
2
4
6
8
10
12
14
16Average VTEC error [TECU]
Average VTEC error
Average VTEC spread (± SD)
0 3 6 9 12 15 18 21 24Hours since 00UT 12/1/2002
100
150
200
250
300
F10
.7 [SFU
]
F10.7 estimated
F10.7 estimated ± SD
AV Morozov, UM Data Assimilation and Driver Estimation 53/56
1 Summary
2 Problem Statement
3 GITM
4 EAKF
5 Assimilating CHAMP Neutral Density
6 Assimilating GPS Total Electron Content
7 Conclusions and Future Work
AV Morozov, UM Data Assimilation and Driver Estimation 54/56
Conclusions
1 DART-GITM interface: This dissertation developed the frameworkfor data assimilation of satellite data into a space weather model. Inparticular, it introduced driver estimation and demonstrated thatdrivers need to be inflated differently from the model states.
2 Differential inflation: The driver estimate was inflated differentlyfrom the rest of the variables, that is its spread was set to beconstant to allow for continuous updating.
3 CHAMP density assimilation: The data assimilation techniquewas first demonstrated by using sparse thermospheric measurements.
4 GPS TEC assimilation: The interface was then modified to handlemore global (ionospheric) measurements coming from the GPSsatellites. It was found that the driver estimation in this more globalcase was not needed.
AV Morozov, UM Data Assimilation and Driver Estimation 55/56
Future Work
1 F10.7 Localization: F10.7 estimate currently affects both theday-side and the night-side. It would be interesting to see the effectof localizing F10.7.
2 Slant TEC: Only vertical total electron content measurements wereconsidered so far. Implementing slant TEC would allow us to solve abroader class of problems.
3 Heating efficiency: Heating efficiency is a stronger driver for TEC,so estimating it using TEC data should be easier than estimatingF10.7.
4 Geomagnetic storms: This study only considered geomagneticquiet times, so performing data assimilation during geomagneticstorms is subject of future work.
AV Morozov, UM Data Assimilation and Driver Estimation 56/56
Questions?
AV Morozov, UM Data Assimilation and Driver Estimation 57/56
RCAC Review
Consider the multi-input, multi-output discrete-time system
x(k + 1) = Ax(k) + Bu(k) +D1w(k), (53)
y(k) = Cx(k) +Du(k) +D2w(k), (54)
z(k) = E1x(k) + E2u(k) + E0w(k). (55)
Our goal is to develop an adaptive controller that generates a control signal u that minimizes the performance z inthe presence of the exogenous signal w. For this presentation we consider SISO plants with no direct feedthroughand z(k) = y(k) in command following context (i.e. E1 = C, E2 = D = 0, E0 = D2 6= 0, D1 = 0). We
define Markov parameters as Hi4= CAi−1B for i > 0.
z(k) =n∑
i=1
−αiz(k − i) +n∑
i=d
βiu(k − i) +n∑
i=0
γiw(k − i), (56)
u(k)4=
nc∑i=1
Mi(k)u(k−i) +
nc∑i=1
Ni(k)y(k−i), u(k) = θ(k)φ(k), u(k) = θ(k)φ(k),
(57)
Z(k)4=
z(k − 1):
z(k − pc)
, U(k)4=
u(k − 1):
u(k − pc)
, U(k)4=
u(k − 1):
u(k − pc)
, (58)
Bzu4= [ 0d Hd × poly(NMPz) ], or Bzu = [ 0d Hd Hd+1 ], (59)
Z(θ(k), k)4= Z(k) − Bzu
(U(k) − U(k)
), (60)
where green highlighting represents known variables, yellow - measured variables, and red - variables to be found.
AV Morozov, UM Data Assimilation and Driver Estimation 58/56
RCAC Review continued
We now consider the cost function
J(θ, k)4= Z
T(θ, k)Z(θ, k) + ζ(k)tr
[(θ−θ)T(θ−θ)
], (61)
where the positive scalar ζ(k) is the learning rate. Substituting (60) into (61), the cost function can be written asthe quadratic form
J(θ, k) =(vec θ
)TA(k) vec θ + B(k)T vec θ + C(k) , (62)
whereD(k)
4=
pc∑i=1
φT
(k − i)⊗ (BzuLi),
f(k)4= Z(k)− BzuU(k),
A(k)4= D
T(k)D(k) + ζ(k)Inclu(lu+ly),
B(k)4= 2D
T(k)f(k)− 2ζ(k)vec θ(k),
C(k)4= f(k)
Tf(k) + ζ(k)tr
[θT
(k)θ(k)]. (63)
Since A(k) is positive definite, J(θ, k) has the strict global minimizer
θ(k) = − 12
vec−1(A(k)−1B(k)). (64)
The controller gain update law is θ(k + 1) = θ(k).
AV Morozov, UM Data Assimilation and Driver Estimation 59/56
Computational Study of RCAC Robustness
Consider the discrete-time system
G(z) =z − 1.4
(z − .5)(z − .6)(z − .7). (65)
With NMP zero location known exactlyRCAC achieves transient performance ofabout 1.6. Transient and steady stateperformances are defined as
ztr = maxk|z(k)|, (66)
zss = maxk=900:1000
|z(k)|. (67)
200 400 600 800 1000−2
0
2
Perform
ance
z(k)
200 400 600 800 1000
−0.5
0
0.5
Controlu(k)
200 400 600 800 1000
−1
−0.5
0
0.5
Controller
coeff
-sθ(k)
k [steps]
AV Morozov, UM Data Assimilation and Driver Estimation 60/56
Computational Study of RCAC Robustness continued
When location of the nonminimum phase zero is uncertain, transient performance canbecome unbounded
(a) Plant NMP−zero locations
NM
P z
ero
estim
ate
locations
2 4 6 8
2
3
4
5
6
7
8
1
1.2
1.4
1.6
1.8
2
(b) Plant NMP−zero locations
NM
P z
ero
estim
ate
locations
2 4 6 8
2
3
4
5
6
7
8
−5
−4
−3
−2
−1
0
1
2
In this plot, the true nonminimum phase zero (x-axis) and zero estimate (y-axis) arevaried from 1.1 to 8.1. We conclude that RCAC is more robust to overestimating thelocation of the nonminimum phase zero than to underestimating it. Additionally, thecases with NMP zeros further out on the real axis result in greater stability marginsthan those with NMP zeros closer to 1.
AV Morozov, UM Data Assimilation and Driver Estimation 61/56
Computational Study of RCAC Robustness continued
We now generate 50 stable second order plants with random poles and NMP zero at 2and test RCAC with NMP zero estimates varying from 1.4 to 10.
2 3 4 5 6 7 8 9 100
25
50
75
100
Estimate of the nonminimum phase zero
Perc
ent o
f pla
nts
with
uns
tabl
e cl
osed
loop
resp
onse
For these 50 random plants, the stability of the closed-loop system is less sensitive tooverestimating the location of the NMP zero than it is to underestimating the locationof the NMP zero. However, none of the 50 plants have an upward margin in excess of10, which corresponds to 5 times the true value of the NMP zero.
AV Morozov, UM Data Assimilation and Driver Estimation 62/56
Computational Study of RCAC Robustness continued
We can extend these findings to plants with higher order and bigger relative degree.
23456
1
2
3
4
5
Plant order
Pla
nt re
lative d
egre
e
1.5 2 2.51.5 2 2.51.5 2 2.5
Estimated nonminimum−phase zero
1.5 2 2.51.5 2 2.5 0
50
100
0
50
100
0
50
100
Perc
ent of pla
nts
with u
nsta
ble
clo
sed−
loop r
esponse
0
50
100
0
50
100
We find that robustness to the estimate of the NMP zero increases in rows from leftto right (with decreasing order), and in columns from bottom to top (with increasingrelative degree).
AV Morozov, UM Data Assimilation and Driver Estimation 63/56
Convex Constraint on Pole Locations
The denominator coefficients of the controller (60) are given by
den(θ(k))4= [1 −M1 −M2 · · · −Mnc ]. (68)
We modify the problem of minimizing (62) by imposing a maximum singular valueconstraint on the companion-form matrix
K4=
M1 M2 . . . Mnc−1 Mnc
1 0 . . . 0 00 1 . . . 0 0...
.... . .
......
0 0 . . . 1 0
, (69)
σmax(K) ≤ γ, γ > 1. (70)
−1 −0.5 0 0.5 1
−1
−0.5
0
0.5
1
Real Axis
Imagin
ary
Axis
AV Morozov, UM Data Assimilation and Driver Estimation 64/56
Convex Constraint on Pole Locations continued
Convex constraint improves transient performance
200 400 600 800 1000−5
0
5
10
Perform
ance
z(k)
200 400 600 800 1000−10
0
10
Controlu(k)
200 400 600 800 1000−2
−1
0
1
Controller
coeff
-sθ(k)
k [steps]
RCAC
200 400 600 800 1000−2
0
2
Perform
ance
z(k)
200 400 600 800 1000
−0.5
0
0.5
Controlu(k)
200 400 600 800 1000
−1
−0.5
0
0.5
Controller
coeff
-sθ(k)
k [steps]
CC-RCAC
AV Morozov, UM Data Assimilation and Driver Estimation 65/56
Convex Constraint on Pole Locations continued
Evolution of the controller poles is shown as a function of time in terms of color.CC-RCAC poles settle twice as fast.
−1 −0.5 0 0.5 1 1.5−1
−0.5
0
0.5
1
Imaginary
Axis
Real Axis
1
1000
k[steps]
RCAC
−1 −0.5 0 0.5 1 1.5−1
−0.5
0
0.5
1
Imaginary
Axis
Real Axis
1
1000
k[steps]
CC-RCAC
AV Morozov, UM Data Assimilation and Driver Estimation 66/56
Nonlinear Multilink Arm
Consider planar multilink arm
-ıA
6A
θ1•p1
•q1
θ2•p2
•q2•p3
7ıB1Z
ZB1
1ıB2
BBM
B2
···
θN•pN
•qN
m1l21
3+m2l21
m2l1l22
cos(θ1 − θ2)m2l1l2
2cos(θ1 − θ2)
m2l22
3
[ θ1θ2
]+
[m2l1l2
2sin(θ1 − θ2)θ2
2
−m2l1l22
sin(θ1 − θ2)θ21
]
+
[c1 + c2 −c2−c2 c2
] [θ1θ2
]+
[k1 + k2 −k2
−k2 k2
] [θ1θ2
]=
[u(t)
0
].
(71)
AV Morozov, UM Data Assimilation and Driver Estimation 67/56
Nonlinear Multilink Arm continued
Nonlinear system can be controlled for small command magnitudes and frequencies.
0 10 20 30 40 50−0.5
0
0.5
Perform
ance
z(k)
0 10 20 30 40 50
−20
0
20
Controlu(k)
Time (sec)
RCAC performance at empiricallyfound maximum commandamplitude. Region plot for RC-MRAC, taken from [1]
AV Morozov, UM Data Assimilation and Driver Estimation 68/56
[1] M. W. Isaacs, J. B. Hoagg, A. V. Morozov, and D. S. Bernstein, “A Numerical Study on Controlling a
Nonlinear Multilink Arm Using a Retrospective Cost Model Reference Adaptive Controller,” Proc. Conf. Dec.
Contr., pp. 8008–8013, Orlando, FL, December 2011.
Vertical Energy Equation
∂T
∂t+ ur
∂T
∂r+ (γ − 1)T
(2urr
+∂ur∂r
)=
k
cv ρ mnQ , (72)
Q = QEUV +QNO+QO+∂
∂r
((κc+κeddy)
∂T
∂r
)+Ne
mimn
mi + mnνin(v − u)2,
QEUV =∑s
∑λ
[Ns(z) I∞(λ) e−sec(χ)
∑sNs(z)σ
as (λ)Hs(z)
], (73)
I∞(λ) = f(λ)
1 + a(λ)
F10.7 +⟨F10.7
⟩81d
2+ 80
, (74)
where equation (73) is a combination of equation (9.17)7 and notes from AOSS495. Equation (74) is an empirical model.8
7Schunk, R. W., and A. F. Nagy. “Ionospheres: Physics, Plasma Physics, andChemistry.” Cambridge University Press, 2004.
8Richards, P. G., J. A. Fennelly, and D. G. Torr, “EUVAC: A Solar EUV FluxModel for Aeronomic Calculations”, J. Geophys. Res., 99(A5), (1994)
AV Morozov, UM Data Assimilation and Driver Estimation 69/56
EAKF Implementation notes
First, some notes on matrices that were not defined above.1 Fk comes from SVD of P−k = FkDkFTk .2 Gk is a square root of Dk, as in Gk = D
1/2k .
3 Uk comes from SVD of GTk FTk HTR−1HFkGk = UkJkUTk .
4 Bk is a square root of I + Jk, as in Bk = (In+m + Jk)−1/2.
Second, here are some notes
The EAKF procedure presented so far is not exactly what is implemented inDART.The procedure presented here is the closest EAKF representation to otherfilters, but does not incorporate localization and is not optimized forparallel implementation.The version that is implemented in DART is described in 9 and 10.
9Anderson, Jeffrey L. “A local least squares framework for ensemble filtering.”Monthly Weather Review 131.4 (2003): 634-642.
10Anderson, Jeffrey L., and Nancy Collins. “Scalable implementations of ensemblefilter algorithms for data assimilation.” Journal of Atmospheric and OceanicTechnology 24.8 (2007): 1452-1463
AV Morozov, UM Data Assimilation and Driver Estimation 70/56
8 ResultsSimulated measurement above Ann ArborSimulated measurement at Subsolar PointSimulated measurement at CHAMP locationReal measurement from CHAMPReal measurement from CHAMP with advanced IMF and HPImodels
AV Morozov, UM Data Assimilation and Driver Estimation 71/56
Simulated measurement above Ann Arbor
Measurement source: GITM truth simulation with an input ofF10.7 fixed at about 150. ρ at (82.5W, 42.5N, 394km) is
recorded with associated uncertainty (R) of 2.6× 10−12kg/m3.
Ensemble members: 20 GITM instances prespun for 2 days prior toDec 01 with F10.7 values coming from normal distribution∼ N(130, 25). F−10.7 is inflated using Pi = 49.
ρ
EAKF
F+10.7F−10.7
F10.7 GITM Truth
GITM Ensemble
x−k x+k+1
√Pi
P−k
AV Morozov, UM Data Assimilation and Driver Estimation 72/56
ρ above Ann Arbor
EAKF estimates of ρ are brought within the uncertainty bounds of GITMtruth simulation.
AV Morozov, UM Data Assimilation and Driver Estimation 73/56
Localization
The effect of measurement assimilation can be restricted to a region to avoid updatinguncorrelated states.11
Correlation function with horizontal cutoffof 30 is shown to the right and below,and vertical cutoff of 100km is shownbottom right.
11Gaspari, G., and S. E. Cohn. “Construction of correlation functions in two andthree dimensions.” Quarterly Journal of the Royal Meteorological Society 125.554(2006): 723-757.
AV Morozov, UM Data Assimilation and Driver Estimation 74/56
ρ at subsolar point
Note, ρ at subsolar point does not vary too much since F10.7 is relativelyconstant. The localized AA measurement is tripled in intensity.
AV Morozov, UM Data Assimilation and Driver Estimation 75/56
ρ at CHAMP location
The localized AA measurement is tripled in intensity.
AV Morozov, UM Data Assimilation and Driver Estimation 76/56
ρ at CHAMP location RMSPE
We define RMS percentage error RMSPE4=
√(ρ−ρ)2√ρ2
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
Time (hrs) since 2002−12−01 00:00:00 UTC
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 24%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 77/56
Simulated measurement above Ann Arbor Summary
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
Time (hrs) since 2002−12−01 00:00:00 UTC
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 24%
GITM without EAKFEAKF posterior
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
Time (hrs) since 2002−12−01 00:00:00 UTC
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 41%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 78/56
F10.7 estimate AA
So what we learned is that it is hard to get a good estimate of F10.7
based on measurement that is fixed in longitude.
AV Morozov, UM Data Assimilation and Driver Estimation 79/56
Simulated measurement at Subsolar Point
Measurement source: ρ at subsolar point is recorded with associated
uncertainty (R) of 2.6× 10−12kg/m3.
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(c) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 3%
GITM without EAKFEAKF posterior
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(d) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 4%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 80/56
F10.7 estimate SP
Conclusion here is that ρ at subsolar point is more closely related to F10.7
than ρ at any point fixed in longitude (for example, Ann Arbor).
AV Morozov, UM Data Assimilation and Driver Estimation 81/56
Simulated measurement at CHAMP location
Measurement source: ρ at CHAMP location is recorded with associated
uncertainty (R) of 2.6× 10−12kg/m3.
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(a) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 2%
GITM without EAKFEAKF posterior
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
(b) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 4%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 82/56
F10.7 estimate CL
We conclude that this case is better conditioned than AA or SP ones.
AV Morozov, UM Data Assimilation and Driver Estimation 83/56
Real measurement from CHAMP
Measurement source: ρ from real CHAMP has associated uncertainty (Rk)
that varies, but has a mean of 2.6× 10−12kg/m3.
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450
10
20
30
40
50
60
70
80
90
100
(c) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 7%
GITM without EAKFEAKF posterior
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 450
10
20
30
40
50
60
70
80
90
100
(d) Hours since 00UT 01/12/2002
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 52%
GITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 84/56
F10.7 estimate CR
We conclude that EAKF F10.7 estimate did not converge to thecommonly accepted value in order to compensate for model bias.
AV Morozov, UM Data Assimilation and Driver Estimation 85/56
Real measurement from CHAMP with IMF and HPI
Measurement source: ρ from real CHAMP has associated uncertainty (Rk)
that varies, but has a mean of 2.6× 10−12kg/m3.
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
Time (hrs) since 2002−12−01 00:00:00 UTC
Abs
olut
e pe
rcen
tage
err
or a
long
CH
AM
P p
ath
RMSPE (2nd day) along CHAMP path = 9%
GITM with F10.7 from NOAAGITM without EAKFEAKF posteriorPercent of observations used
0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 480
10
20
30
40
50
60
70
80
90
100
Time (hrs) since 2002−12−01 00:00:00 UTC
Abs
olut
e pe
rcen
tage
err
or a
long
GR
AC
E p
ath
RMSPE (2nd day) along GRACE path = 46%
GITM with F10.7 from NOAAGITM without EAKFEAKF posterior
AV Morozov, UM Data Assimilation and Driver Estimation 86/56
F10.7 estimate CR
We conclude that EAKF F10.7 estimate did not converge to thecommonly accepted value in order to compensate for model bias, evenmore advanced IMF and HPI models are used.
AV Morozov, UM Data Assimilation and Driver Estimation 87/56