  Weak gravitational lensing due to the inhomogeneous matter distribution in
the universe is an important systematic uncertainty in the use of standard
candles in cosmology. There are two different descriptions of weak lensing
amplification, one uses a local smoothness parameter $\tilde{\alpha}$, the
other uses reduced convergence $\eta= 1+ \kappa/|\kappa_{min}|$ (where $\kappa$
is convergence). The $\tilde{\alpha}$ description involves Dyer-Roeder distance
$D_A(\tilde{\alpha}|z)$ ($\tilde{\alpha}=1$ corresponds to a smooth universe);
it is simple and convenient, and has been used by the community to illustrate
the effect of weak lensing on point sources such as type Ia supernovae. Wang
(1999) has shown that the $\tilde{\alpha}$ description can be made realistic by
allowing $\tilde{\alpha}$ to be a local variable, the local smoothness
parameter. The $\eta$ description has been used by Wang, Holz, & Munshi (2002)
to derive a universal probability distribution (UPDF) for weak lensing
amplification. In this paper, we bridge the two different descriptions of weak
lensing amplification by relating the reduced convergence $\eta$ and the local
smoothness parameter $\tilde{\alpha}$. We give the variance of $\tilde{\alpha}$
in terms of the matter power spectrum, thus providing a quantitative guidance
to the use of Dyer-Roeder distances in illustrating the effect of weak lensing.
The by-products of this work include a corrected definition of the reduced
convergence, and simple and accurate analytical expressions for
$D_A(\tilde{\alpha}|z)$. Our results should be very useful in studying the weak
lensing of standard candles.
