  In this paper, we study a random field constructed from the two-dimensional
Gaussian free field (GFF) by modifying the variance along the scales in the
neighborhood of each point. The construction can be seen as a local martingale
transform and is akin to the time-inhomogeneous branching random walk. In the
case where the variance takes finitely many values, we compute the first order
of the maximum and the log-number of high points. These quantities were
obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when
the variance is constant on all scales. The proof relies on a truncated second
moment method proposed by Kistler (2015), which streamlines the proof of the
previous results. We also discuss possible extensions of the construction to
the continuous GFF.
