Transcript Document

BICS, September 2007
Inversion imaging of the Sun-Earth System
Damien Allain, Cathryn Mitchell, Dimitriy Pokhotelov, Manuchehr Soleimani,
Paul Spencer, Jenna Tong, Ping Yin, Bettina Zapfe
Invert, Dept of E & E Engineering, University of Bath, UK
Plan
Tomography and the ionosphere
• Outline the basic problem
GPS imaging of electron density
• large-scale slow moving (mid/low latitude)
• medium-scale fast moving (high latitude)
• high-resolution imaging
• small-scale structure
System applications
Next steps
The ionosphere
Tenuous atmosphere above 100 km – ionised by EUV
Tomography applied to imaging the ionosphere
Along each continuous arc
measurements of timeevolving, biased TEC
Produce the time-evolving
3D distribution of electron
density
Tomography applied to imaging the ionosphere
Measure – integral of electron density
Solve for spatial field of electron density
Ground-receiver
tomography
Problems
• Incomplete data coverage
• Variability of the measurement biases
• Temporal changes in the ionosphere
Problem 1 - incomplete data coverage
?
?
10
?
?
10
10 10
If each of the measurements
(integrated quantities) are equal to 10,
find the density in each pixel …
Problem 1 - incomplete data coverage
?
?
10
?
?
10
If each of the measurements
(integrated quantities) are equal to 10,
find the density in each pixel …
10 10
Four equations, four unknowns … but there are many possible answers
because the equations are not all independent
5
5
8
2
7
3
5
5
2
8
3
7
… etc but if
vertical ratio is
known to be 4:1
8
2
2
8
… then the solution is unique
See for example Fremouw et al, 1992
Problem 1 - incomplete data coverage
Satellite-to-ground measurements are
biased in the vertical direction … this
means that the inversion is better
determined in the horizontal distribution
of electron density
Problem 1 - incomplete data coverage
High peak height large scale height
Low peak height small scale height
h
Example of basis set
constraints of MIDAS
EOF2
EOF1
EOF3
Problem 2 – variable measurement biases
5+c 15+c
?
?
?
?
Each set of satellite to receiver paths
is assumed to have a ‘constant’
measurement bias, c …
In terms of a mathematical solution, this just results in a slightly more
underdetermined problem,
because need to solve for c for each satellite-receiver pair
See for example Kunitsyn et al., 1994
Height (km)
Problem 2 – variable measurement biases
Large differences in the profile still
result in small TEC changes …
20 TECu
TECu
TECu
NmF2 from ground based data?
… so we need to use the
differential phase not the
calibrated code observations
Problem 3 – temporal changes
5 15
?
?
?
?
Now, we had a static solution, but what if
the ionosphere changes during the time we
collect the measurements?
time1 TEC =5 ; time2 TEC=15
Problem 3 – temporal changes
5 15
?
?
?
?
Now, we had a static solution, but what if
the ionosphere changes during the time we
collect the measurements?
time1 TEC =5 ; time2 TEC=15
This gives a time-evolving solution of electron density, where (applying
for example a linear time evolution) the solution is
4
1
1 4
Time 1
8
2
12
3
2
8
3 12
Time 2
Problem 4 – uneven data coverage
Some form of regularisation e.g. spherical harmonics
MIDAS – time-dependent inversion
A relatively short period is chosen for the time-dependent inversion, for
example one hour, and data collected at typically 30 second intervals
are considered. The change in the ray path geometry, defined in the D
matrix, multiplied by the unknown change in electron concentration (y) is
equal to the change in TEC, Tc.
Dy  Tc
The mapping matrix, X, is used to transform the problem to one for
which the unknowns are the linear (or other) changes in coefficients (G)
D(XG)  Tc
Spherical harmonics
and EOFs (X)
MIDAS – time-dependent inversion
The matrices can be re-written such that the ray path
geometry is multiplied directly by the mapping matrix to
create the basis set
DX(G)  Tc
.
and the change in the unknown contributions of each of
these line integrations of electron concentration is solved
for
Solve for G
G  (DX)1 Tc
The time-dependent solution to the inverse problem is then given by
y  XG
[electron density change] = [model electron density] [coefficients]
MIDAS – high latitude
Problems
• Grid geometry
• Limited ground-based data
• Severe gradients, localized features
• Fast moving structures
Solutions
• Rotated grid
• Convected background ionosphere
Convected ionosphere formulated in Kalman filter
H is the path-pixel
geometry defined by
the satellite orbits and
receivers
Hxr  z
measurements
x  Axt 1
State transition to
project prior into the
future
P  APt 1AT  Q
K  PHT (HPHT  R) 1
New density is formed
from projected previous
state and new
measurements
Pt  (I  KH)P
xt  x  K(z  Hx)
Variance in
observations (IFB)
MIDAS – high latitude
E-field from
Weimer
MIDAS – high latitude
Magnetic
field from
IGRF
MIDAS – high latitude
Velocity
used to
convect
‘background’
ionosphere
MIDAS – high latitude
MIDAS – high latitude
MIDAS – high latitude
MIDAS – high latitude
MIDAS – comparison to EISCAT
Electron density as a function of height and universal time 30th October 2003
EISCAT
radar
MIDAS
tomography
Acknowledgement: EISCAT Scientific Association, in particular Ian McCrea at CCLRC, UK
Extension of imaging to Antarctica
GPS data-sharing collaboration through International Polar Year 2007-2008
Arctic
Antarctic
Conjugate plasma controlled by electric field
High-resolution imaging
In collaboration with J-P Luntama, FMI
High-resolution imaging
In collaboration with J-P Luntama, FMI
Equatorial imaging and GPS Scintillation –
South America and Europe
In collaboration with Cornell University, USA
Ionosphere multi-scale problems – system effects
GPS
• Perturbs the signal propagation speed
proportional to total electron content –
tens of metres error at solar maximum
Credit: ESA
Ionosphere multi-scale problems – system effects
Space-based P-band radar (SAR)
• forest biomass estimation
• ice sheet thickness determination
Ionospheric impacts
• Faraday rotations from
several degrees to several
cycles in high sun-spot periods
• defocusing by ionospheric
irregularities
Summary and Further Work
Tomography and the ionosphere
GPS imaging of electron density
System applications
Next steps …
NASA movie - CME Sun-Earth connections
Next steps
Goal – to nowcast and forecast the Sun-Earth System
Models –
• Do we know all of the physics of the Sun-Earth System?
• Can we simplify it into a useful Sun-Earth model?
• Computational – how can we minimise the computational costs?
Multi-scale data assimilation (temporal and spatial) will be essential
Credit to ESA