Absolute offset estimation of SPIRE maps via

cross-calibration with Planck-HFI

Luca Conversi & Bernhard Schulz

June 10, 2013

1 Introduction

Herschel-SPIRE detectors are only sensitive to relative variations, as a consequence theabsolute brightness of the observed region is unknown and maps are constructed such thatthey have zero median.

Planck-HFI detectors are similar to the SPIRE ones, however its observing strategyallows it to (almost) observe a sky’s great circle every minute (having a 1 rpm spinningrate). By comparing the sky brightness as measured by COBE-FIRAS at the galacticpoles (where the dust emission is lower), HFI is capable of setting an absolute offset to itsmaps.

SPIRE and HFI share two channels with overlapping wavebands (see Fig. 1): SPIRE-PMW and HFI-857 have a similar filter profile, while SPIRE-PLW and HFI-545 are shiftedby ∼ 10%.

As of HCSS 10, a new task named zeroPointCorrection is available: this task willcalculate the absolute offset for a SPIRE map based on cross-calibration with HFI-545 andHFI-857 maps, colour-correcting HFI to SPIRE wavebands assuming a grey body functionwith fixed β.

Sec. 2 summarises the steps needed to colour-correct SPIRE data for extended emission,while Sec. 3 explains how the colour correction from HFI to SPIRE wavebands is computed.Finally, the zeroPointCorrection task algorithm is outlined in Sec. 4.

2 SPIRE extended source calibration

The SPIRE instrument is based on point source calibration with Neptune as prime cali-brator [1, 5, 4, 3]: in order to compare SPIRE results to Planck, the first step needed itto correct the SPIRE data to extended source calibration.

The relation between the flux of a point source having a spectrum of the type SSky(ν) =SSky(νi)(ν/νi)

α and the in-band measured one SMeas(νi) is given by:

SMeas(νi) =

∫νSSky(νi)

(ν

νi

)αF (ν) η(ν) dν∫

νF (ν) η(ν) dν

=SSky(νi)

KmonP(α)(1)

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where η(ν) is the aperture efficiency, F (ν) is the filter’s response function for a pointsource, νi the nominal central frequency of each band and KmonP the pipeline conversionfactor :

KmonP(α) =

∫νF (ν) η(ν) dν∫

ν

(ν

νi

)αF (ν) η(ν) dν

(2)

For an extended source, the frequency dependence of the beam solid angle Ων(ν) mustbe taken into account too:

Ων(ν) = Ω(νi) ·(ν

νi

)−2γ(3)

being γ = 0.85 (a previous recommendation was that γ = 1, corresponding to a λ2 trend).Thus, eq. 1 becomes:

SMeas(νi) =

∫νISky(νi)

(ν

νi

)αF (ν) η(ν) Ων(ν) dν∫

νF (ν) η(ν) dν

=ISky(νi)

KmonE(α)(4)

where ISky is the extended source surface brightness. Note that KmonE converts mapsfrom Jy/beam to MJy/sr, hence its values are not close to unity:

KmonE(α) =

∫νF (ν) η(ν) dν∫

ν

(ν

νi

)αF (ν) η(ν) Ων(ν) dν

(5)

=

∫νF (ν) η(ν) dν

Ων(νi)

∫ν

(ν

νi

)α−2γF (ν) η(ν) dν

(6)

In the specific case of the cross-calibration with Planck, SPIRE maps have been cor-rected to extended emission assuming a source having spectral index α = −1, hencemultiplying them by the factor KPtoE [1, 2]:

KPtoE(α = −1) =KmonE(−1)

KmonP(−1)=

∫νν−1 F (ν) η(ν) dν

Ων(νi)

∫νν−1 F (ν) η(ν)

(ν

νi

)−2γdν

(7)

3 Colour correction of HFI data

As mentioned in Sec. 1, SPIRE and HFI have 2 overlapping wavebands. However, asshown in Fig. 1, the filter profiles are not identical: colour-correction from HFI to SPIREwaveband is thus needed.

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The HFI pipeline assumes a source’s spectrum of the type νI(ν) = const, i.e.:

IPipe(ν) = IPipe(νi) ·(ν

νi

)−1(8)

where νi is the nominal central frequency of each pass-band: 545 or 857 GHz.However, the actual spectrum of the sky is closer to that of a grey body with a spectral

index β:

IGB(ν, T, β) =2hν3

c2· 1

ehν/KT − 1· νβ (9)

Hence, a proper colour correction calculation implies to go from the HFI pipeline skyspectrum νHFIIPipe(νHFI) = const, to the grey body spectrum in the HFI wavebandIGB(νHFI, T, β), followed by the grey body spectrum in the SPIRE waveband IGB(νSPIRE,T, β) and finally the SPIRE pipeline spectrum νSPIREIPipe(νSPIRE) = const.

Moreover, the measured quantity is different because of the filters’ chain. For a generalextended source having spectrum ISky(ν), the measured brightness will be:

IMeas,i =

∫νISky(ν)F ′i (ν) dν∫

νF ′i (ν) dν

(10)

where F ′i (ν) is the extended-emission-adjusted filter pass-band function for a given channel:F ′i (ν) = F (ν) for HFI, while for SPIRE aperture correction and beam area must be takeninto account, i.e. F ′i (ν) = F (ν) η(ν) Ων(ν). Fig. 1 shows the relative response function ofboth HFI and SPIRE wavebands.

From eqs. 8 and 10, it follows that the correction factor K to go from the HFI pipelinespectrum to the grey body spectrum is:

Ki =IGB(νi, T, β)

IPipe(νi)=

IGB(νi, T, β)

∫ν

F ′i (ν)

νdν

1

νi

∫νIGB(ν, T, β)F ′i (ν) dν

(11)

A similar expression can be obtained to go from the grey body spectrum to the SPIREpipeline one, although inverted.

Instead, the correction of go from HFI to SPIRE wavebands will be simply the ratioIGB(νSPIRE, T, β)/IGB(νHFI, T, β). Thus, the total colour correction to go from e.g. HFI-545 to SPIRE-PLW will be:

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Figure 1: Filters’ response function F ′i (ν) for the five wavebands: HFI-545, HFI-857,SPIRE-PLW, SPIRE-PMW and SPIRE-PSW.

K545 → PLW(T, β) =

IGB(ν545, T, β)

∫ν

F ′545(ν)

νdν

1

ν545

∫νIGB(ν, T, β)F ′545(ν) dν

· IGB(νPLW, T, β)

IGB(ν545, T, β)·

·

1

νPLW

∫νIGB(ν, T, β)F ′PLW(ν) dν

IGB(νPLW, T, β)

∫ν

F ′PLW(ν)

νdν

=

ν545

∫νIGB(ν, T, β)F ′PLW(ν) dν ·

∫ν

F ′545(ν)

νdν

νPLW

∫νIGB(ν, T, β)F ′545(ν) dν ·

∫ν

F ′PLW(ν)

νdν

(12)

Similarly, it is possible to calculate K857 → PMW and K857 → PSW.It is evident that the colour correction depends on the assumed grey body function.

For our purposes, we assumed a grey body with varying temperature and fixedspectral index β = 1.8. The colour correction factors computed under this assumptionare reported in Fig. 2.

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Furthermore, it is convenient to calculate the ratio RHFI(T, β) of the HFI flux densitiesfor a template of temperatures to eliminate this dependence from the colour correctionfactors:

RHFI(T ) =IGB(ν545, T, β = 1.8)

IGB(ν857, T, β = 1.8)(13)

Fig. 3 shows the same colour correction factors shown in Fig. 2, this time as function ofthe ratio RHFI.

4 zeroPointCorrection task description

The zeroPointCorrection task estimates the absolute offset of SPIRE maps, in MJy/sr,compering them to the equivalent sky area as observed by HFI. To run properly, the taskneeds the following inputs:

• The two Planck HFI-545 and HFI-857 full-sky maps, provided in HealPIX pixelisa-tion scheme, galactic coordinate system, in units of MJy/sr and under the assump-tion νS(ν) = const.

• A colour correction table, computed as described in Sec. 3. The table is given in theform of a FITS file and must contain at least the following:

1. three columns titled k545toPLW, k857toPMW and k857toPSW, containing respec-tively the colour correction factors to go from HFI-545 to PLW (K545 → PLW),from HFI-857 to PMW (K857 → PMW) and from HFI-857 to PSW (K857 → PSW),computed as described in eq. 12;

2. a column titled ratio545 857, containing the ratio of the HFI flux densities(RHFI), as described in eq. 13;

3. three the metadata parameters, named k4P PLW, k4P PMW and k4P PSW, i.e. thepipeline conversion factors KmonP(α = −1) as outlined in eq. 2 for the threeSPIRE wavebands;

4. three the metadata parameters, named k4E PLW, k4E PMW and k4E PSW, i.e. theextended emission colour correction factors KPtoE(α = −1) as outlined in eq. 7for the three SPIRE wavebands.

• A SPIRE level-2 context: this must contain three SPIRE maps named extdPLW,extdPMW and extdPSW, each created with the relative gain correction applied.

As an option, the user may also change the full width to half-maximum (FWHM ) of the2D Gaussian function used in the convolution of SPIRE maps to HFI beam (default value:8 arcmin) and the gain correction factors (default value: 1, i.e. no correction applied).

Then, the zeroPointCorrection runs the following steps, also illustrated in Fig. 4:

1. it load the the three SPIRE maps extdPLW, extdPMW and extdPSW and converts themfrom Jy/beam to MJy/sr using the KmonP and KmonE factors;

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Figure 2: The three colour correction factor (K545 → PLW, K857 → PMW and K857 → PSW)and the ratio RHFI as a function of dust temperature, assuming a fixed spectral indexβ = 1.8.

Figure 3: The three colour correction factor (K545 → PLW, K857 → PMW and K857 → PSW)as a function of the ratio RHFI, assuming a fixed spectral index β = 1.8.

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SPIRE&

PMW&or&PLW&Expand&&Map&

Convolve&Kernel&is&convolu:on&of&&

8’&Gaussian&&and&&HFI&Beam&model&

(Smaller&~3.5’&Gaussian)&&&

HFII857&

HFII545&

Ra:o&HFII857/HFII545&

HFII857&to&PMW&

HFII545&to&PLW&

colour&Correc:on

&Convolved&&SPIRE&map&

Colour&corrected&HFI&map&

All&sky&HealPIX&maps&

Ra:o&map&

Mask&where&&edge&effects&<0.1%&

Embedding&reduces&edge&effects&in&the&subsequent&SPIRE&map&convolu:on&

Embedded&SPIRE&map& Convolved&&

SPIRE&map&

Colour&corrected&HFI&map&

Convolve&

Flux&HFI&Map&

Flux&SPIRE

&Map&

2nd&Scale&2nd&Offset&2nd&Median&Offset&

Compare&within&mask&

SPIRE&map&with&corrected&zeroI

point&

Difference&map&

Fit&Residual&Offset&and&

Scale&

Output&

Colour&correct&HFI&map&to&SPIRE&filter&bands&PSW&or&PLW&based&on&fit&of&GreyIBody&dust&model&with&β=1.8&to&HFI&fluxes.&

HFI&

Flux&HFI&Map&

Flux&SPIRE

&Map&

1st&Scale&1st&Offset&1st&Median&Offset&

Fit&Offset&and&Scale&

Embed&&median&offset&

adjusted&SPIRE&map&into&HFI&map&

The&total&offset&is&the&sum&of&the&first&and&the&second&median&differences.&&The&scale&parameter&provided&the&comparison&of&photometric&gains&shown&above&and&is&not&part&of&the&standard&processing.&

Subtract&second&median&offset&from&SPIRE&map.&

Figure 4: Step-by-step description of the calculations performed internally by thezeroPointCorrection task to estimate the absolute offset to add to the SPIRE maps.

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2. in order to avoid edge effects, it enlarges the SPIRE maps by 0.5 degrees in eachdirection (i.e. it adds a ring of 1 deg2);

3. it reads the HFI maps, extracts and re-projects HFI data based on SPIRE astrom-etry;

4. it applies the colour correction factors K545 → PLW, K857 → PMW and K857 → PSW toHFI data. The colour correction factors are computed, pixel by pixel, interpolatingthe ratio RHFI obtained from the colour correction table and comparing it to theratio of maps HFI-545/HFI-857.

5. The task convolves SPIRE maps with a 2D Gaussian beam function, which by defaulthas a FWHM of 8 arcmin;

6. then, the difference maps (HFI-545 − PLW), (HFI-857 − PMW) and (HFI-857− PSW) are computed: a first estimation of the PLW, PMW and PSW offsets,respectively, is obtained as the median of the these difference maps.

7. The offsets are applied full-resolution SPIRE maps obtained in point 1. above;

8. these SPIRE maps are then embedded in the HFI colour corrected maps obtainedin step 4.

9. The task runs the convolution on SPIRE maps embedded into HFI ones;

10. SPIRE maps are de-embedded: the result is the original SPIRE maps, convolvedwith the defined Gaussian beam and with edge effects reduced, especially for mapssmaller then ∼1 deg2.

11. The difference maps (HFI-545 − PLW), (HFI-857 − PMW) and (HFI-857 − PSW)are computed: a second estimation of the PLW, PMW and PSW offsets, respectively,is obtained as the median of the these difference maps. The statistical error on theoffset estimation is competed as the difference map standard deviation.

12. Finally, the total offsets are applied to the full resolution SPIRE maps extdPLW,extdPMW and extdPSW. Their values, as well as their estimated error, are reported inthe maps metadata and the level-2 context is updated.

As described above, the zeroPointCorrection task computes the offset as the medianof the difference map between an HFI colour corrected map and the corresponding SPIREone. However, this method works under the assumption that the relative gain betweenSPIRE and HFI is 1. Nevertheless, extended testing have shown that a relative gain ispresent: with colour correction table version 2.3, it is estimated to be HFI-545/PLW ∼0.98 and HFI-857/PMW ∼ 0.93. As a consequence, it is suggested to use these valueswhen launching the zeroPointCorrection task.

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References

[1] Herschel Science Centre & SPIRE ICC, SPIRE Observers’ Manual.

[2] Herschel Science Centre & SPIRE ICC, SPIRE Data Reduction Guide,http://herschel.esac.esa.int/hcss-doc-10.0/index.jsp#spire_drg:

spire-photometer

[3] G. J. Bendo et al., Flux calibration of the Herschel-SPIRE photometer, MNRAS, 2013.

[4] B. M. Swinyard et al., In-flight calibration of the Herschel-SPIRE instrument, A&A,518, 2010.

[5] M. J. Griffin et al., The Herschel-SPIRE instrument and its in-flight performance,A&A, 518, 2010.

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