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