circle within the projection radius $r_p$ $\leq$ 1.5 $\textrm{Mpc}$ is considered as the stacked galaxy sample, including group and field galaxies, while the outer annulus between $r_1$ (8.0 $\textrm{Mpc}$) and $r_2$ (9.5 $\textrm{Mpc}$) is as the field galaxy (or background) sample. The recovered sample is then the difference between the stacked galaxy and the field sample normalized to have the same area as the stacked galaxy sample. In addition, we also construct a background sample selected from a random position within the same redshift width as the stacked group galaxy sample, and we find that the difference between the corrected sample using the annulus background and that using the random background is negligible, and the results show good consistency.

Stacking Tests Using a Mock Catalog {#mt}
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To know the performance of the “background subtraction” method, we make use of a mock galaxy catalog, based on a semi-analytic galaxy formation model [@lag12], constructed for the Pan-STARRS1 Medium Deep Surveys [@mer13 also see Jian et al. 2014], with simulated photo-$z$ accuracy $\sigma_{\Delta z/(1+z_s)}$ $\sim$ 0.05 to test the method. We select clusters in the redshift range between $0.2 < z < 0.5$ with the host halo mass $M_h/M_{\odot} > 10^{14}$, and 