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 BICS, September 2007

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 Plan

The ionosphere Tenuous atmosphere above 100 km – ionised by EUV

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

Ground-receiver tomography 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 Tomography applied to imaging the ionosphere

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

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 Problem 1 - incomplete data coverage … then the solution is unique If each of the measurements (integrated quantities) are equal to 10, find the density in each pixel … See for example Fremouw et al, 1992

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

Low peak height small scale height High peak height large scale height Example of basis set constraints of MIDAS h EOF 1 EOF 2 EOF 3 Problem 1 - incomplete data coverage

? ? ? ? Problem 2 – variable measurement biases 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 5+c15+c See for example Kunitsyn et al., 1994

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

? ? ? ? Problem 3 – temporal changes Now, we had a static solution, but what if the ionosphere changes during the time we collect the measurements? time1 TEC =5 ; time2 TEC=

? ? ? ? Now, we had a static solution, but what if the ionosphere changes during the time we collect the measurements? time1 TEC =5 ; time2 TEC= This gives a time-evolving solution of electron density, where (applying for example a linear time evolution) the solution is Time 1 Time 2 Problem 3 – temporal changes

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

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) MIDAS – time-dependent inversion Spherical harmonics and EOFs (X)

The time-dependent solution to the inverse problem is then given by 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. MIDAS – time-dependent inversion [electron density change] = [model electron density] [coefficients] Solve for G

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

State transition to project prior into the future Convected ionosphere formulated in Kalman filter New density is formed from projected previous state and new measurements H is the path-pixel geometry defined by the satellite orbits and receivers measurements 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 – comparison to EISCAT Acknowledgement: EISCAT Scientific Association, in particular Ian McCrea at CCLRC, UK Electron density as a function of height and universal time 30 th October 2003 EISCAT radar MIDAS tomography

Conjugate plasma controlled by electric field ArcticAntarctic GPS data-sharing collaboration through International Polar Year Extension of imaging to Antarctica

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

High-resolution imaging

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

GPS Ionosphere multi-scale problems – system effects Credit: ESA Perturbs the signal propagation speed proportional to total electron content – tens of metres error at solar maximum

Space-based P-band radar (SAR) forest biomass estimation ice sheet thickness determination Ionosphere multi-scale problems – system effects Ionospheric impacts Faraday rotations from several degrees to several cycles in high sun-spot periods defocusing by ionospheric irregularities

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

NASA movie - CME Sun-Earth connections

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 Next steps Credit to ESA