  The observation of large azimuthal anisotropy or $v_2$ for hadrons above
$p_T>5$ GeV/$c$ in Au+Au collisions at $\sqrt{s_{\rm nn}}=200$ GeV has been a
longstanding challenge for jet quenching models based on perturbative QCD
(pQCD). Using a simple jet absorption model, we seek to clarify the situation
by exploring in detail how the calculated $v_2$ varies with choices of the
collision geometry as well as choices of the path length dependence and
thermalization time $\tau_0$ in the energy loss formula. Besides the change of
eccentricity due to distortion from gluon saturation or event-by-event
fluctuation, we find that the $v_2$ is also sensitive to the centrality
dependence of multiplicity and the relative size between the matter profile and
the jet profile. We find that the $v_2$ calculated for the naive quadratic path
length dependence of energy loss, even including eccentricity fluctuation and
the gluon saturation, is not enough to describe the experimental data at high
$p_T$ ($\sim$ 6 GeV/$c$) in Au+Au collisions. However, it can match the full
centrality dependence of $v_2$ data if higher power path length dependence of
energy loss is allowed. We also find that the calculated $v_2$ is sensitive to
the assumption of the early time dynamics but generally increases with
$\tau_0$, opposite to what one expects for elliptic flow. This study attests to
the importance of confining the initial geometry, possibly by combining jet
quenching $v_2$ with elliptic flow and other jet quenching observables, for
proper interpretation of the experimental data.
