(Marquardt 1963) to find the coefficients in the following equation:

$$f(C,X)=\sum\limits_{i,j=1}^N C_{ij}X_iX_j+\sum\limits_{i=1}^N C^{'}_iX_i+C^{''}.
\label{eq-basis-function}$$

In the above equation, $f(x)$= log   and X is:

$$X= \left( \begin{array}{c}
m_g\\
m_r\\
m_i\\
R_{50i}\\
\end{array} \right)$$

Here, N=4 and X$_i$ (i=1 to 4) in which $m_{g}$ , $m_{r}$, $m_i$ are apparent magnitudes. These are the individual variables that comprise $g-r$ and $\mu_i$ in the parametrization of Zhang et al. (2009) in Eqs. \[eq-zhnag\] and \[eq-zhnag\_new\]. We use AC1 as the training set and determine the 15 coefficients required for the above parametrization.

Fig. \[fig-alfa-gass-4-par\] shows the estimated vs. observed gas fractions derived from Eqn. \[eq-basis-function\], which is a representation of the matrix-based solution. The residual tilt in Fig. \[fig-alfa-gass-two-par\] in the AC1 data (blue points) is now completely removed and scatter is reduced from 0.31 dex in the linear calibration (Eq. \[eq-zhnag\_new\]) to 0.22 dex. This demonstrates that although $g-r$ and $\mu_i$ *can* be used for  calibration, a more complex representation, such as the matrix form, can provide a better match (due to more complex connections between the parameters). Moreover, this matrix form can also remove the deviation and skew seen in Fig. \[fig-alfa-gass-two-par\] for AC1, which is obtained using only three (different) coefficients.

