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Inversion imaging of the Sun-Earth System

Discussing ionosphere tomography applied to imaging the Sun-Earth system, addressing problems and solutions, with focus on incomplete data coverage, variable measurement biases, temporal changes, and uneven data coverage.

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Inversion imaging of the Sun-Earth System

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  1. 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

  2. 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

  3. The ionosphere Tenuous atmosphere above 100 km – ionised by EUV

  4. Tomography applied to imaging the ionosphere Along each continuous arc measurements of time-evolving, biased TEC Produce the time-evolving 3D distribution of electron density

  5. Ground-receiver tomography Tomography applied to imaging the ionosphere • Measure – integral of electron density • Solve for spatial field of electron density • Problems • Incomplete data coverage • Variability of the measurement biases • Temporal changes in the ionosphere

  6. Problem 1 - incomplete data coverage ? ? ? ? If each of the measurements (integrated quantities) are equal to 10, find the density in each pixel … 10 10 10 10

  7. Problem 1 - incomplete data coverage ? 5 8 7 ? 5 3 2 ? 3 5 2 ? 5 8 7 If each of the measurements (integrated quantities) are equal to 10, find the density in each pixel … 10 10 10 10 Four equations, four unknowns … but there are many possible answers because the equations are not all independent … 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

  8. 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

  9. Problem 1 - incomplete data coverage h EOF2 EOF1 EOF3 High peak height large scale height Low peak height small scale height Example of basis set constraints of MIDAS

  10. 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

  11. Problem 2 – variable measurement biases Large differences in the profile still result in small TEC changes … Height (km) 20 TECu TECu TECu … so we need to use the differential phase not the calibrated code observations NmF2 from ground based data?

  12. 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

  13. Problem 3 – temporal changes 12 8 4 ? 1 3 ? 2 2 ? 1 3 4 12 ? 8 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 Time 2 Time 1

  14. Problem 4 – uneven data coverage Some form of regularisation e.g. spherical harmonics

  15. 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. 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) Spherical harmonics and EOFs (X)

  16. 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 and the change in the unknown contributions of each of these line integrations of electron concentration is solved for . Solve for G The time-dependent solution to the inverse problem is then given by [electron density change] = [model electron density] [coefficients]

  17. MIDAS – high latitude • Problems • Grid geometry • Limited ground-based data • Severe gradients, localized features • Fast moving structures • Solutions • Rotated grid • Convected background ionosphere

  18. Convected ionosphere formulated in Kalman filter H is the path-pixel geometry defined by the satellite orbits and receivers measurements State transition to project prior into the future Variance in observations (IFB) New density is formed from projected previous state and new measurements

  19. MIDAS – high latitude E-field from Weimer

  20. MIDAS – high latitude Magnetic field from IGRF

  21. MIDAS – high latitude Velocity used to convect ‘background’ ionosphere

  22. MIDAS – high latitude

  23. MIDAS – high latitude

  24. MIDAS – high latitude

  25. MIDAS – high latitude

  26. 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

  27. Extension of imaging to Antarctica GPS data-sharing collaboration through International Polar Year 2007-2008 Arctic Antarctic Conjugate plasma controlled by electric field

  28. High-resolution imaging In collaboration with J-P Luntama, FMI

  29. High-resolution imaging In collaboration with J-P Luntama, FMI

  30. Equatorial imaging and GPS Scintillation – South America and Europe In collaboration with Cornell University, USA

  31. 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

  32. 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

  33. Summary and Further Work • Tomography and the ionosphere • GPS imaging of electron density • System applications • Next steps …

  34. NASA movie - CME Sun-Earth connections

  35. 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

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