ner_tags: O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-CelestialObject B-CelestialObject B-CelestialObject B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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-CelestialObject O B-CelestialObject O O O O O O O O O O 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-CelestialObject B-CelestialObject O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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: pixels and 1's to the valid ones. This form of the 2PT function is the same as the unbiased measure described in Kleiner Dickman ( 1984 ), though we used the column density of each pixel without subtracting the mean column density. In this way, we are more sensitive to the correlation of the density field, rather than density fluctuations. We adopted a flat weighting, where all pixel pairs were assigned with an identical weight. This otherwise unbiased estimator may be biased when the scales of the lags (i.e., the spatial separation of two pixels in a pair) are comparable with the observational FOV (Kleiner Dickman 1984 ). Figure 14 shows the obtained 2PT correlation functions with the directional information averaged out. Figure 14. Normalized 2PT functions for all the sources, showing the correlation strengths as a function of spatial separation. For separations with pixel pairs of less than total pixel number, the correlation strengths are plotted with dashed lines. The gray filled region indicates approximately where the rapidly decreasing (RD) components end. The averaged 2PT correlation functions of W49A, G10.6-0.4, W33, W43-Main, and G10.2-0.3, have a common feature: a rapidly decreasing (RD) correlation strength at short lags, and a sudden transition to a shallower decrease at larger lags. The RD component corresponds to localized clumps/cores, while the shallower decreasing (SD) component comes from the broader molecular cloud. We note that all the derived 2PT correlation functions show a steep decrease in correlation strength at large lags because of the limited FOV. To avoid this bias in our analysis, we exclude the range of lags that have pixel pairs less than ∼90% of the number of total pixels. For sources like W43-south and G10.2-0.3, where the structures of column density are composed of widely scattered overdensities, the correlation strengths at shorter lags do not decrease as significantly as the correlation strengths of the more centrally condensed sources, like W49A, W33, and G10.6-0.4. They also do not show an obvious RD component at short lags. There are some bumps seen in the 2PT functions for several sources at larger lags, which may present the characteristic scales of some localized overdensities. We do not quantify these features in this paper. The average azimuthal profiles of our obtained 2PT functions are intended to capture the major similarities and differences between the target sources. We hypothesize that the steeply decreasing component and the shallower one represent structures that were created or supported with different physical mechanisms, and thereby have different characteristic spatial scales and spatial distributions. More discussion about the physical implications is deferred to Section 4.1 . For the sake of quantifying and archiving the observed 2PT functions for comparison with other observations, theories, and numerical simulations, we perform segmented linear fits of these 2PT functions in log