Gradient of the Perturbed Field

The exact analytical gradient of the perturbed field $B^{(1)}$ can be calculated by simply replacing the convolution kernel, $H$, with its gradient and neglecting the first term, since it is proportional to $\chi_1 B^{(0)}$ and has zero gradient at the centre of the voxel. This can be also be seen from equation 24 which gives

$$\frac{\partial B^{(1)}}{\partial q} = \sum_{\mathbf{x}'} \chi_1(\... ..., \left. \frac{\partial H}{\partial q} \right\vert _{(\mathbf{x}- \mathbf{x}')}$$ (26)

where $q$ stands for $x, y$ or $z$.