  We present a numerical study of the transport and localization properties of
excitations in one-dimensional lattices with diagonal disordered mosaic
modulations. The model is characterized by the modulation period $\kappa$ and
the disorder strength $W$. We calculate the disorder averages $\langle
T\rangle$, $\langle \ln T\rangle$, and $\langle P\rangle$, where $T$ is the
transmittance and $P$ is the participation ratio, as a function of energy $E$
and system size $L$, for different values of $\kappa$ and $W$. For excitations
at quasiresonance energies determined by $\kappa$, we find power-law scaling
behaviors of the form $\langle T \rangle \propto L^{-\gamma_{a}}$, $\langle \ln
T \rangle \approx -\gamma_g \ln L$, and $\langle P \rangle \propto L^{\beta}$,
as $L$ increases to a large value. This behavior is in contrast to the
exponential localization behavior occurring at all other energies. The
appearance of sharp peaks in the participation ratio spectrum at quasiresonance
energies provides additional evidence for the existence of an anomalous
power-law localization phenomenon. The corresponding eigenstates demonstrate
multifractal behavior and exhibit unique node structures. In addition, we
investigate the time-dependent wave packet dynamics and calculate the mean
square displacement $\langle m^2(t) \rangle$, spatial probability distribution,
participation number, and return probability. When the wave packet's initial
momentum satisfies the quasiresonance condition, we observe a subdiffusive
spreading of the wave packet, characterized by $\langle m^2(t) \rangle\propto
t^{\eta}$ where $\eta$ is always less than 1. We also note the occurrence of
partial localization at quasiresonance energies, as indicated by the saturation
of the participation number and a nonzero value for the return probability at
long times.
