The residual covariance $\mathbf{N}$, as defined in Eisel & Egbert (2001), is one of the two building block matrices of the impedance covariance. It can be derived from first principles by setting up the estimation of the impedance as a linear regression problem. Consider the output channels $\mathbf{E}$ as the response variable, $\mathbf{H}$ as the input variable, and the impedance $\mathbf{Z}$ as the unknown parameter matrix, resulting in
where $\mathbf{E}$ and $\mathbf{H}$ are defined, for simplicity of notation, as horizontal, rather than vertical, vectors. If we were to transpose both sides of the equation, we would arrive at the more traditional definition of the impedance. To further generalize the analysis, we note that the same expression may be used for a variety of output channels, so that the definition of $\mathbf{E}$ may include the vertical magnetic field component, as well as, or instead of the electric field output channels, and the same analysis would apply.
Then, for the single-station magnetotelluric processing, the residual covariance estimate is
an expression that is easy to obtain from the cross-power spectra. The scalar variance $\hat\sigma^2_\nu$ is the inverse of AVGT value from the EDI SPECTRA file, the number of independent averages in time, at each frequency.
Similarly, for the remote-reference magnetotelluric processing,
where $\hat{\mathbf{Z}}$ is an impedance estimate.
See also impedance, covariance and inverse signal covariance.
Eisel, M. and Egbert, G.D., 2001. On the stability of magnetotelluric transfer function estimates and the reliability of their variances. Geophysical Journal International, 144(1), pp.65-82.
Chave, A.D. and Jones, A.G. eds., 2012. The magnetotelluric method: Theory and practice. Cambridge University Press.
Kelbert, A., 2019. EMTF XML: New Data Interchange Format and Conversion Tools for Electromagnetic Transfer Functions. Geophysics, 85(1), pp.1-69.