Stochastic magnetic field models of the observatory era, in the

Stochastic magnetic eld models of the
observatory era, in the perspective of core
dynamics reconstructions
Nicolas Gillet1 , Dominique Jault1,2 , Chris Finlay2,3 , Nils Olsen3
1 ISTerre,
CNRS, University of Grenoble, 2 ETH Zurich, 3 DTU Space Copenhagen
with support from ISSI and the French space research agency CNES
ateliers `inversion', Grenoble, 17 janvier 2012
1 / 22
composante Y (vers l'est) à Chambon-la-Forêt
Y (t )
dY /dt
0
65
60
−500
55
−1000
50
dYg/dt (nT/y)
45
Y (nT)
−1500
−2000
40
35
30
−2500
25
−3000
20
−3500
1930
1940
1950
1960
1970
time (y)
1980
1990
2000
2010
15
1930
1940
1950
1960
1970
time (y)
1980
1990
2000
2010
⇒ la dérivée seconde ∂ 2 B/∂ t 2 ne semble pas dénie aux périodes
décennales (`jerks' ou secousses)
2 / 22
virtual axial dipole moment
past 8 000 y
past 2 million y
Gennevey et al, 2008
Ziegler et al, 2011
les propriétés statistiques de B(t ) aux périodes T > 1000 ans sont
semblables à celles de ∂ B/∂ t pour T < 1000 ans,
i.e. ∂ B/∂ t non dénie vers les basses fréquences
(`archeomagnetic jerks')
3 / 22
moyenne, covariance, corrélation
d'une série temporelle
moyenne :
variance :
S (t )
µS = E [S (t )]
h
i
σS 2 = E (S (t ) − µ)2
covariance :
CS (τ ) = E ([S (t ) − µ] [S (t + τ ) − µ]) = σS 2 ρS (τ )
corrélation :
ρS (τ ) = CS (τ )/CS (0)
4 / 22
a exible familly of correlation function
Matérn functions,
Rasmussen & Williams 2006
√
ρν,τc (τ ) ∝
1
nu=1/2
nu=3/2
nu=5/2
2ντ
τc
correlation fn.
0.6
•
0.4
0.2
0
0.5
1
1.5
2
tau/tauc
2.5
Kν
2ντ
!
τc
modied Bessel function of the
second kind and of order ν
k -times dierentiable if k < ν
∗ ν → ∞ : squared-exponential (SE)
∗ ν = 1/2 : Laplace or AR(1)
∗ ν = 3/2 : AR(2)
• Kν
0.8
0
√
!ν
3
5 / 22
case of C∞ SE correlation
Hulot & Le Mouël (1994), Hongre et al (1998)
dY/dt (nT/y)
1940
1960
1980
1940
1960
1980
1940
1960
HUA (12 S, 75 W)
1980
10
5
0
-5
-10
-15
-20
-25
-30
60
40
20
0
-20
-40
2000
80
60
40
20
0
-20
-40
-60
1920
50
40
30
20
10
0
-10
-20
-30
-40
2000
-20
-30
-40
-50
-60
-70
-80
-90
-100
1920
dX/dt (nT/y)
20
0
-20
-40
-60
-80
1920
dZ/dt (nT/y)
2
dY/dt (nT/y)
dX/dt (nT/y)
ρ
1
2
(τ ) = exp − (τ /τc )
dZ/dt (nT/y)
SE ,τc
2000
1920
1940
1960
1980
2000
1920
1940
1960
1980
2000
1920
1940
1960
1980
2000
KAK (36 N, 140 E)
6 / 22
correlation function and power spectrum f (ω)
•
•
correlation function uniquely
determined from the knowledge of
the (positive denite) spectral
density (and vice versa)
for Matérn functions :
f (ω) ∝ ω −(2ν+1) as ω → ∞
∗ ν = 1/2 : f (ω) ∝ ω −2
∗ ν = 3/2 : f (ω) ∝ ω −4
Constable & Johnson, 2005
7 / 22
AR(2) Matérn series (ν = 3/2)
synthetic series
-0.4
4
nu=3/2
nu=5/2
nu=3/2
nu=5/2
3.5
-0.45
3
-0.5
S(t)
dS/dt
2.5
-0.55
-0.6
1.5
-0.65
-0.7
2
1
0
0.02
0.04
0.06
0.08
time (tauc units)
0.1
0.5
0
0.02
0.04
0.06
0.08
time (tauc units)
0.1
8 / 22
AR process and stochastic dierential equation (SDE)
•
autoregressive process of order p :
X (ti ) =
p
X
αk X (ti −k ) + (ti )
1
k=
•
with (t ) = white noise process
ρ3/2,τc : continuous AR(2) process that obeys the SDE
d 2X
3
− 2 X = (t )
2
dt
τc
discontinuous second time derivative (`secular acceleration')
9 / 22
AR process and stochastic dierential equation (SDE)
•
•
∂ Br
is also a stochastic process, dened as
∂t
Z t
∂ Br
∂ Br
(t ) =
(t0 ) +
φ(Br ) + (t 0 ) dt 0 ,
∂t
∂t
t0
which time derivative is not dened.
∂ Br
is an AR(1) process that obeys a SDE of the form
∂t
dX
1
+ X = (t )
dt
τc
10 / 22
stochastic component in the ow evolution
in absence of diusion (η = 0), the radial induction equation at the
core-mantle boundary
η
∂ Br
= −∇H · (uBr ) + ∇2 (rBr ) ,
∂t
r
suggest to write the ow u(t ) as
u(t ) = u(t0 ) +
Z
t
F(u) + e(t 0 ) dt 0
t0
where the relative strength of the deterministic F and stochastic e
components remain to be determined.
11 / 22
need for stochastic magnetic eld models (1)
•
existing models are regularized in an ad hoc fashion :
• time derivatives penalized
• covariances ignored
⇒ issue 1 : rapid changes possibly ltered out (loss of
information)
⇒ issue 2 : biased posterior covariances
if model = data for a second inverse problem (core ows, data
assimilation algorithms) : unrealistic data error statistics
12 / 22
need for stochastic magnetic eld models (2)
•
in the perspective of reconstructing the core ow, there is a
need for eld models Br (t ) that satisfy our prior knowledge
• Thèse de F. Labbé :
utiliser les données satellitaires pour
- donner une ébauche de la partie déterministe F des `forces'
qui gouvernent la dynamique, et de l'amplitude de la
composante stochastique e
- valider un modèle dynamique à partir de ces quantités
• Thèse de G. Héllio :
- utiliser cette représentation stochastique pour l'analyse des
données anciennes (historiques, archéomagnétiques)
- tester la validité des hypothèses (stochastiques + physiques)
vers les basses fréquences
13 / 22
regularized satellite eld models
•
•
•
(Olsen et al, Lesur et al, Finlay et al)
spherical harmonic coecients
gnm (t ) of degree n and order m
temporal parameterization with
B-splines of order 6
∂t3 Br penalized at the CMB to
image the secular acceleration
τmf
2 (n )
=
τsv 2 (n) =
E gnm 2
h
i
E (ġnm )2
h
i
E (ġnm )2
h
i
E (g̈nm )2
CHAOS-3
GRIMM-2
GUFMsat-E3
103
tau MF (up) and SV (down) / yr
properties in 2005.0
102
101
1
2
3
4
5
SH degree
6
7 8 9 10
14
14 / 22
synthetic exercise with AR(2) coecients
•
c
Cġν,τ
(0)
nm
ν
(ν − 1)τc2
Cgnm (0)
√
ν = 3/2 ⇒ τc (n) = 3τmf (n)
ν,τc
•
• satellite eld model
synthetic satellite data
generated assuming coecients
gnm (t ) follow an AR(2) process
Matérn formulation imposes
=
same regularization as used for
pre-existing eld models
gufm-sat-e3
• spline t to synthetic AR(2) process
data, minimizing 3rd-time derivatives
tau-MF-AR2-REG
tau-SV-AR2-REG
tau-MF-GUFM-SAT-E3
tau-SV-GUFM-SAT-E3
103
time / yr
•
102
101
1
2
3
4
SH degree
5
6
7
8 9 10
⇒ the common feature,
`∀n, τsv (n) ' 10 y',
is an eect of the regularization
15 / 22
ensemble of eld models
•
forward problem : co-estimation of internal eld + external
dipole (magnetospheric ring current)
y = A(m) +e
, Ce = E eeT
m=0
+δ m , Cm = E δ mδ mT
•
expectation model m∗ and posterior covariance matrix
∗
−1
1
C∗ = ∇A(m∗ )T C−
e ∇A(m ) + Cm
h
•
•
i−1
from a Newthon algorithm ; no damping parameter to tune !
ensemble of eld models generated as follow :
1- Choleski decomposition C∗ = UUT + random unit vectors v
2- all models m = m∗ + Uv t the data y (if any), and satisfy the
required covariances (if no data is available).
updated version of the gufm1 dataset (Jackson et al 2000), plus
subselection of Oersted and Champ data (Finlay et al 2012)
16 / 22
predictions at the Kakioka observatory
dZ/dt (nT/y)
dY/dt (nT/y)
dX/dt (nT/y)
internal + external + induced
50
40
30
20
10
0
-10
-20
-30
-40
1900
10
5
0
-5
-10
-15
-20
-25
-30
1900
60
40
20
0
-20
-40
1900
1920
1940
1960
1980
2000
1920
1940
1960
1980
2000
1920
1940
1960
1980
2000
17 / 22
external eld models q10 (dipole coord.)
40
ensemble
OHM: annual means
OHM: 1997-2010 average
q10 geomag. coord. (nT)
30
20
10
0
-10
-20
1840 1860 1880 1900 1920 1940 1960 1980 2000
time (y)
•
•
•
good comparison with recent model (Nils Olsen, pers. comm.)
way to reconstruct low-frequency external changes that are
badly captured by (high-frequency) indices such as Dst
(despite the larger dispersion at earlier epochs)
weaker activity at the begining of the XXth century
18 / 22
dipole decay models ∂t g10
35
ensemble
gufm1
quad
grimm2
30
dg10/dt (nT/y)
25
20
15
10
5
0
-5
1840
•
•
•
1860
1880
1900
1920 1940
time (y)
1960
1980
2000
larger dispersion at earlier epochs
mis-match with gufm1 (outside the ensemble range for
1920-1970) dues to dierent data sets
mis-match pre-1900 due to methodology
19 / 22
resolution of SV coefs ∂ gnm /∂ t
104
realizations
perturbations
var(dglm/dt)
102
10
0
10-2
10-4
10-6
•
•
0
2
4
6
8
SH degree l
10
12
14
stationary properties of the ∂ gnm /∂ t
larger SV errors at earlier epochs
20 / 22
implications, perspectives (1)
•
stochastic approach with realistic prior information :
• joint inversion of strongly inhomogeneous data without
ad hoc
changes in statistical model properties
• potential to produce realistic posterior covariances at older
epochs (past centuries and millenia)
• expectation + posterior covariances : an entry for data
assimilation algorithms (Fournier et al 2010)
•
triggerring of torsional Alfvén waves (Gillet et al 2010) :
induction in the core from decadal modulation of interannual
q10 changes (G. Legaut, PhD Thesis)
21 / 22
implications, perspectives (2)
•
AR(2) processes and rapid core dynamics :
• suggested over 11000 y time-scales (Constable & Johnson 2005)
• secular acceleration ∂t2 Br not dened over such time-scales
• observed SV changes and underlying core physics driven by ∂t
inside the core, to which ∂t u is enslaved (Canet et al 2009)
•
•
B
most relevent strategy for dynamical reconstruction ?
(Kalman Filter, 4D-var...) depending on the relative strength
of the stochastic and deterministic `forces'
AR(1) processes at millenial frequencies :
archeomagnetic jerks (Gallet et al 2003) versus millenial
oscillations (Nilsson et al 2011)
22 / 22