  Optical absorption in amorphous tungsten oxide ($\textit{a}\mathrm{WO}_{3}$),
for photon energies below that of the band gap, can be rationalized in terms of
electronic transitions between localized states. For the study of this
phenomenon, we employed the differential coloration efficiency concept, defined
as the derivative of the optical density with respect to the inserted charge.
We also made use of its extension to a complex quantity in the context of
frequency-resolved studies. Combined $\textit{in situ}$ electrochemical and
optical experiments were performed on electrochromic
$\textit{a}\mathrm{WO}_{3}$ thin films for a wide lithium intercalation range
using an optical wavelength of $810~\mathrm{nm}$ ($1.53~\mathrm{eV}$).
Quasi-equilibrium measurements were made by chronopotentiometry (CP). Dynamic
frequency-dependent measurements were carried out by simultaneous
electrochemical and color impedance spectroscopy (SECIS). The differential
coloration efficiency obtained from CP changes sign at a critical intercalation
level. Its response exhibits an excellent agreement with a theoretical model
that considers electronic transitions between $\mathrm{W}^{4+}$,
$\mathrm{W}^{5+}$, and $\mathrm{W}^{6+}$ sites. For the SECIS experiment, the
low-frequency limit of the differential coloration efficiency shows a general
trend similar to that from CP. However, it does not change sign at a critical
ion insertion level. This discrepancy could be due to degradation effects
occurring in the films at high $\mathrm{Li}^+$ insertion levels. The
methodology and results presented in this work can be of great interest both
for the study of optical absorption in disordered materials and for
applications in electrochromism.
