fact that gravitational magnification and image multiplicity are not equally prominent features of strong lensing at different depths. The relatively poor performances of the magnification method for deep surveys ($X_{\rm lim}>23$) or the image multiplicity method for shallower surveys ($X_{\rm lim}<23$) cannot be appreciably improved by modifying the criteria for selecting the candidates. We demonstrate these intrinsic limitations in Fig. \[sn\_rate\_effects\], where we show supernova yields for different choices of the free parameters defining the detection criteria in the two methods. Reducing the magnitude gap $\Delta m$ in the magnification method results naturally in a higher detection rate. However, the improvement is limited solely to magnitudes $\gtrsim 22$ where the method is clearly outperformed by the image multiplicity technique. Furthermore, this inevitably leads to a higher false positive detection rates due to intrinsically bright type Ia supernovae.

Relaxing the selection criteria for the image multiplicity method by including all systems with at least two detectable images (for all images configurations) improves the supernova yields only at small limiting magnitudes, e.g.by a factor of 4 at $z_{\rm lim}\approx 21$ (see the green dashed curve in Fig. \[sn\_rate\_effects\]). The resulting detection rates, however, are still smaller than those of the magnification method. 