ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: The overall null results are just due to cancellation globally. Figure 4. Additive bias c 1 for different boundary sizes ( l x , l y ). All c 1 values are units of 10 −3 . The variances are about 4 × 10 −5 . The parameter 2 cs defined in Equation ( 2 ) has a best-fit value of 408 ± 12 in the current simulation. To further check the validity of the relation in the equation, one can quantitatively check the change of the bias c 1 with the ratio of the axis sizes l x / l y for a fixed image area l x l y , or the other way around. The results are shown in Figure 5 , which are quite consistent with Equation ( 2 ). Figure 5. Change of the additive bias c 1 as a function of l x . The upper panel is under the condition of fixed axis ratio ( l x / l y ), and the lower one is for fixed total area (). The solid curves are fittings from Equation ( 2 ) using the data shown in Figure 4 . The variances of the data points are about 4 × 10 −5 . Another prediction of the model is that the boundary effect should be more significant for galaxies of larger average sizes. To observe this phenomenon, we generate another set of simulations with RW galaxies that are larger by 50% than those in the previous set of simulations. The results are shown in Figure 6 , from which we can clearly see that c 1 is larger than those reported in Figure 4 under the same condition of ( l x , l y ). Figure 6. Similar to Figure 4 . The radii of the RW galaxies used here are larger than those used in Figure 4 by 50%. All c 1 values are in units of 10 −3 . The variances are about 4 × 10 −5 . We also expect the boundary effect to be more significant for more elliptical galaxies. To observe this, we repeat the simulations twice with the Galsim galaxies. Instead of using the ellipticity probability distribution function, this time, we set the galaxy ellipticities at two specific values: e = 0.2 and e = 0.8. The results are shown in Figures 7 and 8 accordingly. One can see that the c 1 of the more elliptical cases are 10 times larger than their counterparts with smaller ellipticities. Figure 7. Similar to Figure 4 , but with Galsim galaxies. The ellipticity of these galaxies are set at 0.2. All c 1 values are in units of 10 −3 . The variances are about 1 × 10 −5 . Figure 8. Same as Figure 7 , but with more elliptical Galsim galaxies. The ellipticity of the galaxies is set to 0.8. All c 1 values are in units of 10 −3 . The variances are about 1 × 10 −5 . Overall, we can draw the conclusion that the additive bias due to the boundary is also influenced by the galaxy morphologies, such as their radii, ellipticities, etc. The results in this section only represent the bias levels for several special cases. 2.3. The Solution and Its Application to Real Data As we have shown, source selection is a statistically anisotropic process near the image boundaries. The resulting shear bias cannot be treated as a universal additive bias, as it is significant only in the neighborhood of the boundaries. To