Kernels for the test case with different metallicity. Averaging kernels (left) and cross-term kernels (right) for the age indicator inversion (top panels), the acoustic radius inversion (middle panels), and the mean density (lower panels). The SOLA method is in blue, the ⟨Δν⟩ estimate in red and when implemented, the KBCD approach is in green. The target function in all panels is plotted in black. (Colour online.)