error of each signal, weighted and added in quadrature [@Kahn1954; @Brennan1959]. The condition $\alpha_1 +\alpha_2 =1$ is required such that the merged signal is equal to the coherent environmental field observed by each sensor.

Because signal amplitudes are ideally equal in either sensor, optimizing the ratio $B_m/N_m$ leads to frequency dependent solutions for $\alpha_1(\omega)$ and $\alpha_2(\omega)$ which are independent of the environmental signal, and determined by the spectral composition of the noise floors: $$\begin{aligned}
\label{eq:alpha}
 \alpha_1(\omega)=\frac{\tilde{n_2}^2}{ \tilde{n_1}^2 +\tilde{n_2}^2}\\
\alpha_2(\omega)=\frac{\tilde{n_1}^2}{ \tilde{n_1}^2 +\tilde{n_2}^2}\end{aligned}$$ where $\tilde{n_1}^2$ and $\tilde{n_2}^2$ are computed as the spectral densities of the instrument noise. For FIELDS, the coefficients $\alpha_{MAG}$ and $\alpha_{SCM}$ correspond to an effective weighting in instrumental gain which preserves an optimized signal to noise ratio for the merged SCaM data product. The SCM sensor coordinate system is not initially aligned with the MAG sensor axes, accordingly a rotation matrix ${{\ensuremath{\bf{R}}}}$ is applied to bring the SCM measurements in sensor coordinates, ${{\ensuremath{\bf{B}}}}_{SCM}'$, into alignment with the MAG coordinate system, $$\begin{aligned}
  {{\ensuremath{\bf{R}}}} =\left(\begin{array}{ccc}
      0.8165 & -0.4082 &-0.4082\\
      0.0000 & -0.7071 & 0.7071\\
-0.577 & -0.577& -0.577
      \end{array}\right)
   \end{aligned}$$

$${{\ensuremath{\bf{B}}}}_{SCM}={{\ensuremath{\bf{R}}}}\cdot{{\ensuremath{\bf{B}}}}_{SCM}'.$$

Adhering to an optimal signal to noise merger, spectral composition of the noise of the rotated SCM vector time series 