  Recently, the interest on theoretical and experimental studies of dynamic
properties of the magnetic domain wall (MDW) of ferromagnetic thin films with
disorder placed in an external magnetic field has increased. In order to study
global and local measurable observables, we consider the $(1+1)$-dimensional
model introduced by Buceta and Muraca [Physica A 390 (2011) 4192], based on
rules of evolution that describe the MDW avalanches. From the values of the
roughness exponents, global $\zeta$, local $\zeta_\text{loc}$, and spectral
$\zeta_s$, obtained from the global interface width, hight-difference
correlation function and structure function, respectively, recent works have
concluded that the universality classes should be analyzed in the context of
the anomalous scaling theory. We show that the model is included in the group
of systems with intrinsic anomalous scaling ($\zeta\simeq 1.5 $,
$\zeta_\text{loc}=\zeta_s\simeq 0.5 $), and that the surface of the MDW is
multi-affine. With these results, we hope to establish in short term the
scaling relations that verify the critical exponents of the model, including
the dynamic exponent $z$, the exponents of the distributions of avalanche-size
$\tau$ and -duration $\alpha$, among others.
