  Satellites in core level spectra of photoelectron spectroscopy (PES) can
provide crucial information on the electronic structure and chemical bonding in
materials, particular in transition metal oxides. This paper explores
satellites of the Ti 1$s$ and 2$p$ core level spectra of SrTiO$_3$ and TiO$_2$.
Conventionally, soft x-ray PES (SXPS) probes the Ti 2$p$ core level; however,
it is not ideal to fully capture satellite features due to its inherent
spin-orbit-splitting (SOS). Here, hard x-ray PES(HAXPES) provides access to the
Ti 1$s$ spectrum instead, which allows us to study intrinsic charge responses
upon core-hole creation without the complication from SOS and with favorable
intrinsic linewidths. The experimental spectra are theoretically analyzed by
two impurity models, including an Anderson impurity model (AIM) built on local
density approximation (LDA) and dynamical mean-field theory (DMFT), and a
conventional TiO$_6$ cluster model. The theoretical results emphasize the
importance of explicit inclusion of higher-order Ti-O charge-transfer processes
beyond the nearest-neighboring Ti-O bond to simulate the core level spectra of
SrTiO$_3$ and TiO$_2$. The AIM approach with continuous bath orbitals provided
by LDA+DMFT represents the experimental spectra well. Crucially, with the aid
of the LDA+DMFT method, this paper provides a robust prescription of how to use
the computationally cheap cluster model in fitting analyses of core level
spectra.
