Electric field calculations for real-time space weather alerting systems

Space weather alerting systems for power systems require real-time calculations of the electric fields that drive geomagnetically induced currents. In this paper, we present a new method for calculating the Earth impulse response that can be convolved with real-time magnetic field data to give the required electric fields. We start with the Earth transfer function which can be expressed in two ways: as a relationship between the electric field, E, and the geomagnetic field, B, which has characteristics equivalent to that of a high-pass filter, and between E and the time derivative of the magnetic field, dB/dt, which has characteristics of a low-pass filter. An inverse Fourier transform of these transfer functions should then give the corresponding Earth impulse responses in the time domain. This works well for a uniform conductivity model for which the inverse Fourier transform of the transfer function has an analytic solution. However, the inverse Fourier transform of the transfer function for a non-uniform conductivity model requires numerical calculations and produces an acausal impulse response with oscillations because of the Gibbs phenomenon. To investigate the origin of the Gibbs oscillations, the real and imaginary parts of the transfer functions are transformed separately. This shows that the Gibbs oscillations arise from the inverse transform of the real and imaginary parts of the high-pass transfer function and the imaginary part of the low-pass transfer function. A new method is introduced that just transforms the real part of the low-pass transfer function and uses the requirement of causality to construct the full low-pass impulse response. From this, the derivative theorem of convolution is used to obtain the high-pass impulse response. Electric fields can then be calculated by convolution of the low-pass impulse response with the rate of change of the magnetic field, dB/dt, or by convolution of the high-pass impulse response with the magnetic field, B. Tests of the new method by comparison with analytic solutions for specified earth models and synthetic magnetic field data gave very high correlation coefficients, slopes near 1.0 and intercepts near zero, showing the accuracy of the new method. The method can be used with any Earth transfer function whether obtained from magnetotelluric measurements or from 1-D, 2-D or 3-D conductivity models. Thus, it provides a versatile technique that avoids the Gibbs phenomenon and produces a causal impulse response suitable for time domain calculations of electric fields.