Injecting a Grid of Sources

Figure 4.14shows the differences in RA and dec between injected and output catalogues. The mean position at all scales is accurate, confirming that STARFINDER, with its mod- ified beam, is not biased in any direction when finding sources. The errors in the position are given as the 1σ standard deviation; 66% of sources are found within one pixel of the true position of the source at fainter fluxes, falling to within 2 arcsec at brighter fluxes.

Figure 4.15 gives the log ratio of injected and output fluxes, showing the fractional effect of over- or underestimating the flux at that particular flux density. The first res-

(a) Right Ascension

(b) Declination

Figure 4.14: Accuracy in Right Ascension and declination within HeLMS using a grid of injected source catalogues. The dashed line marks the 1σ variation in the value, with the dotted line the error on the mean and is consistent with zero. As expected, accuracy increases as a function of flux.

Figure 4.15: Accuracy of flux within HeLMS using injected source catalogues. There is an overestimate of flux at every flux-level, corresponding to a value of 5-15 mJy difference in detection.

ult that stands out is the overestimation of fluxes at low fluxes. This ranges from a 10mJy overestimation for a source flux density of 20mJy (a 50% overestimation) to 5mJy at 100mJy source flux density (5% overestimation). The changing nature of this over- estimation suggests that its origin is not (entirely, at least) from a background offset. Further, at an injected source flux of 100mJy for 4560 sources, there has been a total of 1.41 × 10−6Jy/pix injected into the map at 250µm, translating to a 1.46 × 10−5 Jy/beam increase in flux across every pixel in the map. Therefore the injected sources themselves cannot be causing the flux overestimation.

At low fluxes, the overestimation could be greater due to a compounding number of reasons. Sources are detected in the map with STARFINDER if and only if the signal-to- noise ratio is greater than 3. HeLMS is constructed from fewer scans than other HerMES maps so the noise in each pixel is higher. Therefore a source detected with a flux of, for example, 25mJy in the HerMES map will be at a higher signal to noise value than in HeLMS and less prone to errors in the flux estimate. 25mJy is a low flux in the HeLMS map and it’s possible that sources detected at this flux are in truth lower, boosted due to Eddington biasing and thus this is an overestimate.

In section4.4fluxes above 100mJy in the catalogue were compared to fluxes in the HerS catalogue to check the calibration of source fluxes. There was an approximate 5%/4%/5% offset between the two sets of sources. When first calculated, this was thought to be due to a implementation difference or error in, for example, the beam used or the effect of the filter applied to the map. When performing estimations of accuracy using source injection, the sources have been added to the map before applying this upwards correction to the flux in the map. If instead the correction was needed due to a difference in map calibration, then injecting sources into the map before applying the correction will effectively change the flux of the injected source, i.e. from 100 to 105mJy. This is tested in the next section by injecting sources after the upwards flux correction has been applied to the map.

Figure 4.16shows the completeness, i.e. fraction of sources injected into the map that are recovered with source extraction. Unlike the completeness curves produced for the DR2 release, the injected fluxes probe the 20-100mJy range. This is where the transition from 0% to 100% detection is thought to occur with source extraction on these maps, estimated

from Viero et al. (2014). The fit was constructed using only results calculated from the

20-60mJy source injection maps. The fraction of sources recovered is the completeness as shown in figure 4.16. As the completeness is a measurement is the probability of recovering a source or not, the system can be modelled as a binomial distribution, with

10 20 30 40 50 60 70 80 90 100

Injected Flux [mJy]

0.0

0.2

0.4

0.6

0.8

1.0

Co

mp

let

en

ess

constrained logistics function

logistic function

HeLMS completeness

Figure 4.16: Completeness curve parameterised by equation 4.5. The blue dashed line represents the fit when the fits is forced between zero and one, the green solid line when the curve is allowed to freely vary. Fit parameters are given in table4.1

the variance given aspp(1 − p) where p probability of recovering a source or completeness. The completeness can be modelled as a logistics curve:

C(S) = B + A − B

(1 + e(−C(S−S0))₎ (4.5)

a smooth curve with a gradient represented by C and S0 the offset of the curve in relation

to the source flux S, B the minimum value and A the peak value. First we fix the peak to A = 1 and minimum value B = 0 i.e. 100% completeness at higher flux and zero completeness at very low fluxes. The fit is not particularly good, however if A and B are left to vary, the fit is better as seen in figure4.16, with a peak value of A = 0.901. This fit implies a maximum of only 90% of sources at any flux will be recovered in the map. The resultant completeness is alarming. To see whether this was a real trend, 80mJy and 100mJy source injected maps were made and the results overlaid. The results confirm the fit, that the maximum number of sources to be found is only ever ninety percent of the total number of sources.

gridded, full fit A B C S0

gridded, fixed fit 1.0 (fixed) 0.0 (fixed) 0.117 ± 0.010 39.25 ± 0.92 Full fit 0.025 ± 0.005 0.908 ± 0.002 0.139 ± 0.002 36.25 ± 0.12

Table 4.1: The fit parameters for the completeness curve for the grid of injected sources. Letting the minimum and maximum values of the fit vary is a far better fit to the results. Crucially, the completeness does not reach 1.0 at infinity, which would be expected. does not contribute substantially to reducing the χ2 fit and therefore those fluxes would be removed from the final catalogue. It is noted in Hurley et al. (submitted) that, for dense prior catalogues, this could lead to many sources with zero flux if a source is on a segment with many other sources or substantially brighter sources. However, in the case of HeLMS, these sources are instead not detected by STARFINDER, with very few sources in the final real catalogue attributed zero flux (< 0.1%). This runs contrary to expectation as at 100mJy sources should typically be at far greater than 3σ signal to noise and therefore detected by STARFINDER. Indeed, the pixels that the sources are on are at a signal to noise ratio greater than three and should pass the first stage of detection in STARFINDER. The hypothesis then is that sources are removed from the potential source candidate list when the correlation between the beam and source in the map is less than 0.5. This is implying that the beam, or the sources, are the wrong shape due to high instrumental noise effects or confusion.

The flux offset will now be explored through testing. To determine whether the injected grid itself is effecting the results (as grids are regular, large structures in the maps and thus could be effected by filtering) sources will be injected randomly to remove that possibility. Then, a comparison between the accuracy and completeness results will be shown, with the flux correction applied before and after the sources are injected into the maps again to see if the flux estimate will be improved.