in $B_y$, just as one might require a constant offset in modeling a scalar function with a Fourier series. While a constant offset is already built into the aforementioned basis, it was difficult for our optimization program to properly converge to a reasonable value without isolating it. We may isolate the necessary constant offset in $B_y$ in a simple manner: Impose that for the first term in the series $k_x = k_y = \phi_x = \phi_y = \phi_z = 0$. Since $k_x = 0$, we use Form 2 for this term. Furthermore our impositions force $k_y = 0$, so we have that $B_y = 1$, while $B_x = B_z = 0$. After this is done, given the symmetry of LGB fields, the phase shifts are no longer needed and can be set to zero for all other terms. The cumulation of our derivation and simplifications is that we obtain the follow series to model our magnetic field data:\
$$\mathbf{B}_{fit}(x,y,z) = 
\begin{bmatrix}
0\\b_y\\0
\end{bmatrix}
 + \displaystyle\sum_{i = 2}^{N} C_i \mathbf{B}_i (x,y,z;k_{xi},k_{zi}),$$

in which $\mathbf{B}_i$ is one of the basis functions in equation (3) and $b_y$, $C_i$, $k_{xi}$, and $k_{zi}$ are our fit parameters.

Fitting the Model to Data
=========================

In this 