in MAG sensor coordinates is then taken as the quadrature weighted error of the SCM sensor axis noise, assuming independence in each sensor channel: i.e. $$n_{SCMx}^2(\omega)= R^2_{xx'}n^2_{SCMx'} +R^2_{xy'}n^2_{SCMy'}+R^2_{xz'}n^2_{SCMz'}.$$ The MAG orthogonalization matrix and rotation from sensor to spacecraft coordinates is approximately equal to the identity matrix such that the measured noise floor for each sensor axis is used without contribution from the other axes.

![Empirically determined noise floors of PSP FIELDS MAGo $x,y,z$ axes (blue, green, black) and the SCM low frequency $x, y, z$ axes with the DFB sampling in a high gain state (red, orange, purple) rotated into MAGo sensor coordinates.[]{data-label="merging_noise"}](fig3)

Figure \[merging\_noise\] shows empirically determined noise-floors for both the SCM and MAGo associated with mean-zero stochastic fluctuations limiting each sensors sensitivity. Merging coefficients are obtained using Equation \[eq:alpha\]. Since the empirically determined noise spectrum is continuous, a smooth weighting of the merged signals is obtained by approximating the MAG merging coefficient using a real-valued rational function of the form $$\label{eq:ratfit}\hat{\alpha}_{MAG}(f)=\frac{N(f)}{D(f)} = \frac{ \sum_0^n A_n f^n }{1 +\sum_1^m B_mf^m}.$$ Below $f_0=2$ Hz the sensitivity of the SCM drops significantly and the full MAG signal is used, i.e. $\alpha_{MAG} =1$ for $f\leq 2$ Hz. Above 2 Hz Equation \[eq:ratfit\] 