In this paper Gatheral presents the following parametrization of the implied total variance $w(k,T) = \sigma_{BS}(k,T)^2T$ for each slice $k \mapsto w(k,T)$:
$$ w(k) = a + b\{\rho (k-m) + \sqrt{(k-m)^2 + \sigma^2} \}.$$
As far as I understand it, for each expiry $T$ one will have to calibrate a set of five parameters $\{a,b,m,\rho,\sigma\}$.
On the other hand I found the following article where in appendix A, a calibrated volatility surface is presented. But in their example there is an explicit dependence on $T$, so then they will have a "simple" specific expression for the whole volatility surface.
Another thing I've noticed when reading articles about various parametrizations that there seems to be some inconsistencies regarding implied total variance. Gatheral defines it as $\sigma_{BS}(k,T)^2T$ but I've seen in other articles people parametrizing on $\sigma_{BS}(k,T)^2$ or $\sigma_{BS}(k,T)$ instead.
Summarizing: My question is primarily if one has to calibrate SVI for each expiry slice or if it is possible to parametrizing the whole surface in a way such that the total number of parameter does not increase if more expiries are added.