The idea is to feed the time series of time differences into an extended Kalman filter (EKF) using two models for the propagation time from the transmitter to the receiver:

1) Simple ray-geometry for oblique propagation in the ionosphere:

- based on arXiv:1104.2248 which uses the Breit and Tuve + Martyn theorems to convert vertical to oblique incidence
- using vertical electron density profiles from the IRI2016 model as an input
- group refractive index: 1/sqrt(1-f
_{N}^{2}/f^{2}), i.e., no magentoionic effects are taken into account - when integrating over the group refractive index in the vertical direction, analytic integration of 1/sqrt(f(h)) is performed in each height interval in order to avoid the spurious numerical singularity at the reflection point
- spherical coordinates are used (r
_{E}=6371km)

2) Alternatively, great-circle distances on the WGS84 ellipsoid are used:

- geographiclib adapted for octave

All receiver coordinates are assumed to be in WGS84, however in practice it is suspected that there are deviations from the true locations of up to 20 km, as the KiwiSDR positions are provided by the KiwiSDR owners and are not taken from the GPS position solutions used for time synchronization.

The plot below shows the output of the EKF

- implementation in octave following ADA285972
- state = [lat, lon] of the unknown position
- measurements = time differences between pairs of KiwiSDRs
- the measurements covariance matrix (constant) is build from the covariances of time differences (and is set to zero for time differences not sharing a common receiver)
- state covariance matrix = diag(0.01, 0.01)
^{2}deg^{2}(constant) - start covariance = diag(0.1, 0.1)
^{2}deg^{2} - there are about 60 measurements which are run 5 times sequentially through the EKF

Extended Kalman Filter (EKF) output; note that the range of the x(y) axis corresponds to approximately 17(67)km. |

The simple ionosphere propagation model improves the match to the known position but there is still a remaining bias of about 30km North, 4km West. Clearly, more studies are needed in order to find out if this improvement is by chance or real.

## 2 comments:

How about using baseline interferometry and looking purely for the Angle of Arrival as an alternative to TDOA?

This would certainly be interesting. But the angle of arrival can be skewed by magneto-ionic propagation effects and for large baselines coherence may be lost.

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