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 B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O B-Survey O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-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 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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: To compute the cross-covariance matrix more accurately, we compute the BFD moments for each image of empty sky. Using the BFD software enables us to properly account for the differing PSFs and world coordinate systems (i.e. registration) of each image. We then compute the empirical cross-covariance matrix of BFD moments, : where ( i, j ) are indexes of the moment element of the covariance matrix, and α and β index are the exposures of this sky patch. The summation is performed over all combinations of images where α ≠ β . We combine data from all 39 Balrog-variant tiles to compute a global cross-covariance matrix for the DES data from empty sky regions. Finally, we test how using the new cross-covariance matrix affects the noise tests described in Section 2.4 . Analogous to χ M , we now define χ MX , which is now calculated using the sum of the shot/read-noise covariance and : where σ M is from the shot/read-noise covariance matrix and σ X is from the cross-covariance matrix . Generalizing to the χ 2 test using arbitrary moments, we have For these tests, we combine moments from all 48 Balrog-injection tiles, and compute the μ and σ of the normalized histogram (as in Fig. 1 ) for each band. We have subtracted the local sky estimate for each postage stamp and included the variance due to that local sky in the flux moment variance term of . In Fig. 6 , we show the σ value for each band/moment combination using and using + . We find general improvement using our globally defined cross-covariance term, with σ lowering from ∼1.12 to ∼1.01 for the flux moment, and dropping to similar levels for the other moments ( σ = 1.026 in the worst case). We report the σ and μ values for all bands in Table 1 . Figure 6. σ of χ M distribution (blue) and χ MX distribution (red) for each BFD moment from all 48 Balrog-injection tiles. Including the cross-covariance matrix brings the σ of the distribution closer to the expected value of 1. The values of σ for each band/moment combination are given in Table 1 . Furthermore, in Fig. 7 , we show the full χ 2 test for M F only and for all four moments of a given band across all 48 Balrog tiles and we report the value in Table 2 . In all bands, we find that when including the cross-covariance matrix that characterizes the US noise, the CDF of the data more closely approaches the χ 2 distribution with the appropriate number of degrees of freedom. The values also reflect the improvement, as they approach the expected value of 1 for M F only and 4 when looking at the vector of four moments. Figure 7. CDF of (blue/light blue) and (red/pink) for the single BFD M F moment and for the four M F , M R , M 1 , and M 2 moments for each band over all 48 Balrog-injection tiles. The χ 2 CDFs with 1 (black dotted line) and 4 degrees of freedom (black dashed line) are plotted for reference. Again, we show that including the cross-covariance term brings the moment distribution closer to expectations. Mean χ 2 values for each