1. Analysis of methods for estimating westward auroral electrojet current with meridian magnetometer chain data
М.А. Evdokimova, А.А. Petrukovich
Space Research Institute of the Russian Academy of Sciences, Мoscow
Introduction
The investigation goal is to choose optimal model for small number of stations.
Performance of several methods, available in literature, for various configurations of substorm currents was compared.
Optimal models (model modifications) may differ for sparse and dense networks.
Analysis of errors was carried out.

2. H
Auroral electrojet
Current is reconstructed with H- and Z-components of magnetic field. H is directed to the North, Z – vertical.
IMAGE network and magnetospheric substorm on 24 November, 1996 are used for the test

3. Solution algorithm OLS-method is used
−2𝑙𝑛𝐿= 1 𝜎 2 𝑘=1 𝑁 𝛿𝐻 𝑘 − 𝛿𝐻 m 𝑛 𝑘=1 𝑁 𝛿𝑍 𝑘 − 𝛿𝑍 m 𝑛 𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 (m𝑜𝑑𝑒𝑙 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡) ,
where 𝐿 – likelihood function, 𝑁 – number of stations, 𝛿𝐻 𝑘 and 𝛿𝑍 𝑘 - disturbances of the field, measured on the station k, 𝛿𝐻 m 𝑛 and 𝛿𝑍 m 𝑛 - calculated disturbances (model dependent).
Covariance matrix of model errors
where H – hessian of 𝑙𝑛𝐿 ,
Model 1, linear (current wires, evenly disturbed, A.L. Kotikov, Yu.O. Latov and O.A. Troshichev, 1987). Model parameters: currents 𝐼 𝑗
𝛿𝐻 m 1 = 𝜇 0 ℎ 2𝜋 𝑗=1 𝑀 𝐼 𝑗 ℎ 𝑥 𝑗 − 𝑥 𝑘 , 𝛿𝑍 m 1 = 𝜇 0 2𝜋 𝑗=1 𝑀 𝐼 𝑗 𝑥 𝑗 − 𝑥 𝑘 ℎ 𝑥 𝑗 − 𝑥 𝑘 2
where ℎ - height of the wires, 𝑥 𝑗 - coordinate of the wire j, 𝐼 𝑗 - currents on the previous time step.
𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛=

4. Model parameters: current densities 𝑗 𝑖 of the stripe 𝑖
Model 2, linear (current stripes, evenly disturbed V.A. Popov, V.O. Papitashvili, and J.F. Watermann, 2001).
Model parameters: current densities 𝑗 𝑖 of the stripe 𝑖
𝛿𝐻 m 2 = 𝜇 0 2𝜋 𝑖=1 𝑀 𝑗 𝑖 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 𝑖𝑘 +𝑑 ℎ −𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 𝑖𝑘 −𝑑 ℎ , 𝛿𝑍 m 1 = 𝜇 0 4𝜋 𝑖=1 𝑀 𝑗 𝑖 𝑙𝑛 ℎ 𝑥 𝑖𝑘 +𝑑 2 ℎ 𝑥 𝑖𝑘 −𝑑 2
where 𝑑 – half-width of a stripe, 𝑥 𝑖𝑘 - distance between station 𝑘 and projection of the center of the stripe 𝑖 to the ground.
𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛=
Model 3, non-linear. One current stripe.
Model parameters: current density, stripe center, stripe half-width.

5. Model 1. Dependence on the number of wires (𝜶=𝟎, 𝒒=𝟎), only H-component
For one wire result is far from measured on the tails of the plot. Greater 𝑀 leads to better precision.
When 𝑀>𝑁 (𝑀=16) with 𝛼=0, 𝑞=0 artifitial maxima between station appear, regularization is necessary.

6. Model 1. Dependence on the parameter 𝐪 (𝐌=𝟏𝟔, 𝛂=𝟎)
Inclusion of the regularization parameter leads to stabilization.
Model 1 with M=16 and with regularization works well

7. Model 2 Dependence on the parameter 𝐪 (𝐌=𝟓𝟎)
Current density ( ∗5.5∗10 5 A/ m)
Optimal model – with a large number of current stripes.
H, nT

8. Model 1. Reconstruction with different residual functions
Reconstruction of the H- (Z-) component using residual function with H- (Z-) component gives rather accurate result, reconstruction using residual function with Z- (H-) component deviates sharply from measured values.
Comparison of normalized standard deviations results that H- (Z-) field reconstructed with residual function included both components deviates insignificantly greater than H- (Z-) field reconstructed with H- (Z-) component.

9. Model 1. Influence of the boundary condition
Inclusion of the boundary condition (B=0 on the boundary latitudes) gives physically appropriate profiles of the field outside electrojet area
Current and H-field at 23:09
With boundary condition
Without boundary condition
( ∗5.5∗10 5 A)
Current
H, nT
latitude
latitude

10. Model 1. Correlation matrix
Part of correlation matrix for 16 wires of current:
Values of correlation matrix diagonal elements are greater than off-diagonal which become smaller moving off the diagonal. So, current wires, separated far from each other, have weak correlation.
However there is correlation between adjacent wires giving large errors.
Errors are large because number of current wires is large.

11. Model 3. Problem of initial condition for nonlinear model

12. Model 3. Z-component residual function
Residual function in non-linear model has local minima
Current density ( ∗5.5∗10 5 A/ m)
Half-width, degrees

13. Model 3. Correlation matrix
There is strong correlation between current density and half-width.

14. Separation of external and internal fields
This method are used in Model 1 and Model 2. It doesn’t work for small number of stations.

15. Summary
Small number of current wires doesn’t give physically appropriate profiles of current and field (peaks between stations, not enough details).
Greater number of current wires (stripes) gives instability, so regularization is necessary.
Using boundary condition helps to get physically appropriate profiles of current and field.
Non-linear model is optimal for small number of stations.
Finding a model with not correlated parameters is the further purpose of the research.
References
Kotikov A. L., Latov Yu. O., Troshichev O. A., Structure of auroral electrojets by the data from a meridional chain of magnetic stations, Geophysica, 1987, V. 23, P
Popov V. A., Papitashvili V. O., Watermann J. F., Modeling of equivalent ionospheric currents from meridian magnetometer chain data, Earth Planets Space, 2001, V. 53, P
Pudovkin M. I., Sources of bay-like disturbances, Izv. Acad. Sci., Ser. Geophys., 1960, V. 3, (Trans. Acad. Sci. USSR, Geophysics), P (in Russian)