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The High Level Tasks were designed to cover all steps of the aXe reduction without any restriction in functionality. Working with aXe therefore means to apply the four High Level Tasks to a set of data. All tasks are controlled through a set of configuration files which can be edited by the user and optimized for a given data set.
The core of the software package is written using ANSI C and Python (http://www.python.org) and is highly portable from one platform to another. aXe uses the third party libraries CFITSIO, GSL, and WCSLIB which have been successfully employed under Linux, Solaris, and MacOS X.
aXe is distributed as part of the STSDAS software package. Within STSDAS, aXe is located under the subpackages 'hst_calib/acs/axe/'. As a relatively young software project which is still under active development, the changes and improvements introduced with new releases are quite large. To give users the possibility to work always with the newest release, we also distribute aXe on the aXe webpages at http://www.stecf.org/software/slitless_software/axe/. On this webpage we always offer the latest aXe release for download.
In slitless spectroscopy there is no unique correspondence between pixel coordinates and wavelength. Consequently, the spectral reduction on the basis of the spectroscopic data alone is impossible. Additional information concerning the positions of the object must be added to facilitate the spectral reduction. In aXe this is done by providing Input Object Lists (IOL) at the beginning of the reduction process. In the Input Object List the object positions are given in the image-coordinate system or the world coordinate system. This allows the determination of the so called reference pixel for every object. The reference pixel is the undispersed object position in image coordinates on the grism data. For each individual object, it is then possible to assign a wavelength to each pixel.
In conventional spectroscopy the extraction of the 1D spectra from the 2D data is done along the direction of the slit or mask. In slitless spectroscopy, such a predefined extraction direction does not exist. It is in fact possible to define a different extraction direction for each object individually by adjusting the wavelength assignment to be constant along the chosen extraction direction (see Fig. 1.3). In aXe the default action is to set the extraction direction to be parallel to the object position angle as given in the Input Object List.
The absence of slits and masks also dramatically enhances the probability that spectra of different sources overlap each other. Even at large distances along the dispersion direction, the different orders of two objects still can overlap and create confusion problems. aXe can mark and extract several dispersion orders per object to properly record the source confusion or contamination for every order of each object.
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The beams follow the spectral trace of the spectrum, which is defined in the Configuration File. While the length of a beam is set by the length of the corresponding dispersion order, its width is adapted to the extraction width set by the user.
The left panel in Figure 1.2 shows an HRC/G800L grism exposure reduced in the HUDF HRC Parallels Program (cleaned from cosmic-ray hits). In the right panel, some of the beams marked and extracted in aXe are indicated. The numbers give the spectral order, and the letters denote the correspondence with the beam in the configuration file. The bright areas mark regions where beams overlap and contaminate their spectra mutually. The different extraction angles for the objects result in different shapes of the marked regions. For each beam the description of the spectral trace and the wavelength assignment is set up and the spectral reduction is done independently.
An important step in the aXe reduction process is the generation of the so called Pixel Extraction Table (PET). A PET is a multi-extension fits-table which stores in each extension the complete spectral description of all pixels of one beam. Figure 1.3 illustrates the geometry in a beam and shows various quantities stored in the PET. Important pixel information stored in the PET is:
The PETs are read and manipulated by many aXe tasks. For example, a flat-field correction is applied to the pixel values stored in the PETs. Since flat-fielding is a wavelength dependent operation, the assignment of a wavelength to each pixel is required before the correction values, derived from a 3D flatfield cube (see Chapt. 6.2), are applied.
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where
Integration over this function
to compute
, and
is trivial once
have
been computed, which are derived from simple trigonometry.
Once the one dimensional spectra have been generated, the final step of flux calibrating can be performed by applying a known sensitivity curve for the observing mode which was used. The output product of the aXe extraction process is a FITS binary table containing the set of extracted and calibrated spectra (Extracted Spectra File, see Chapt. 7.11).
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aXe has two different strategies for removal of the sky background from the spectra.
The first strategy is to perform a global subtraction of a scaled ``master-sky'' frame from each input spectrum image at the beginning of the reduction process. This removes the background signature from the images, so that the remaining signal can be assumed to originate from the sources only and is extracted without further background correction in the aXe reduction.
The second strategy is to make a local estimate of the sky background for each BEAM by interpolating between the adjacent pixels on either side of the BEAM. In this case, an individual sky estimate is made for every BEAM in each science image.
The homogeneous background of HST grism exposures makes the global background subtraction from the pipeline processed science images (i.e. _flt.fits files) feasible. Master sky images for both the ACS Wide Field Channel (WFC) and the High Resolution Channel (HRC) are available from the aXe webpages at http://www.stecf.org/instruments/ACSgrism/. These master sky images were created by combining several hundreds of WFC and HRC grism images from different science programs. The object signatures on the science images were removed using several techniques, including a two step median combination, to derive a high signal-to-noise image of the sky background. Figure 1.5a shows the HRC master sky image.
Scaling and subtraction of the master sky is done with the aXe task axeprep (see Fig. 1.1). Before scaling the master sky to the level of each science frame, the object spectra are masked out on both the science and the master sky image.
When reducing a dataset consisting of many individual exposures, it may be desirable to check the sky subtraction by co-adding all the sky-subtracted grism images (e.g. with the MultiDrizzle task). The co-added image also provides a way to quickly assess the quality of the background subtraction. Any deviations from zero in the mean background level of the combined image will also affect the spectra derived with the aXe reduction.
The second option for handling the sky background is to make a local estimate of the background for each object. In this case, aXe creates an individual background image for each object on the spectrum image. On the background image the pixel values at the positions of the object beams are derived by interpolating in each column between the pixel values on both sides of the beam. The number of pixels used in the interpolation as well as the degree of the interpolating polynomial can be chosen by the user. Figure 1.5b shows the background image corresponding to the grism image displayed in Fig. 1.2.
The background images are then processed in much the same way as the science images, resulting in a Background Pixel Extraction Table (BPET) for all BEAMs in a grism image. Thus, every PET has its corresponding BPET, derived from the background image, with the spectral information of the identical objects and beams in it. Finally, the BPET is subtracted from the PET and the background subtracted spectra are extracted.
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The information on the number of contaminating sources in Fig. 1.6 is stored in the object PET and fully propagated in the 1D extraction of the individual object spectra. As a final result each spectral element is accompanied by a flag which indicates whether its input pixels were also part of other object spectra. The regions of 1D spectra where the contamination flag is set must be used with care, since neighbouring sources also contribute to the extracted flux.
This contamination scheme is fast and very efficient in identifying problematic regions in the individual object spectra, but it does not assist the user to assess the severity of the contamination, and thereby to decide whether the contaminated spectrum might still be suitable for further scientific analysis.
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The basis of the quantitative contamination estimation is a model which estimates the dispersed contribution of every object to the grism image. The contributions of the individual objects are then coadded to a 2D contamination image, which is a quantitative model of the examined grism image. In the 1D extraction of the individual object spectra, the model contribution of the object itself is subtracted (to avoid self-contamination), and then the data from the modelled grism image is processed in parallel with the data from the real grism image.
As a result two spectra for every object are derived: one extracted from the real grism image; and a second one extracted from the modelled grism image. Since the model contribution of the object itself was excluded in the extraction of the latter spectrum, this spectrum is a quantitative estimate of the contamination from all other sources to the object spectrum in question. The accuracy of the contamination spectrum is set by the accuracy of the emission model which is needed as an input to compute the modelled grism image.
Two different emission models have been implemented, called the Gaussian Emission Model and the Fluxcube Model
Figure 1.7 displays, on the left side, the direct images of
the Gaussian emission
model in four filters for data taken in the HUDF. The right side of
Fig. 1.7 shows the modelled grism image computed from the
spectrum with the Gaussian emission model. The arrows point from the direct
image positions
of one object to the position of its first order spectrum in the modelled
grism image. The spectrum which was employed to model this object
is plotted in the lower part of Fig. 1.7. The interpolated
values in between
the data points derived from the AB-magnitudes (at
,
,
and
nm in this case)
are computed with a cubic spline; outside of the range of magnitudes
the spectrum is set to a constant extrapolation of the last data point.
The images in Fig. 1.7 cover the same area as the contamination image in Fig. 1.6. The direct images in Fig. 1.7 were only created for illustration purposes. In real aXe runs, each filter is just represented by a column in the Input Object List which gives the total AB-magnitude of the objects.
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The flux extensions in the fluxcube provide sufficient information to compute a model grism image. In the determination of the quantitative contamination however it is essential to derive the individual contribution of each object to the modelled grism image. This addition is necessary to be able to subtract the self contamination and to isolate the contamination from other sources for each individual object.
For this reason the first extension of a fluxcube image must contain the so called "Segmentation Image". In the segmentation image each pixel value is the (integer) number of the object to which the pixel is attributed. The SExtractor software provides the possibility to create a segmentation image (parameter setting: CHECKIMAGE_TYPE SEGMENTATION) as an additional output product of the source extraction.
The fluxcube files necessarily follow a rather complicated file format. To support the user in the creation of fluxcube files a new aXe task has been implemented. The new task works in a standard scenario with a multidrizzled grism image, one or several multidrizzled direct images and a segmentation image as input.
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As an illustration of the Fluxcube model, Figure 1.8 shows on
the left side the segmentation image and the filter images used to create
the fluxcube. The lower right part of Fig. 1.8 displays the
modelled grism image derived by the fluxcube emission model.
All images in Fig. 1.8 cover the identical area of
Figs. 1.6 and 1.7 in the HUDF.
The differences between the new quantitative contamination scheme and the
old, `geometrical' contamination is demonstrated in Figure 1.9.
In the lower panel the red line indicates spectral elements with the (old)
contamination flag set. All data longwards of 6200 AA is suspected to be
significantly influenced by contamination. The upper panel shows in black
the spectrum
extracted by aXe, and in red the the sum of the spectral contribution of
all contaminating sources according to the Gaussian emission model.
It is immediately clear that, within the accuracy of the emission model,
the contaminants contribution to the object spectrum is negligible
except at the longest wavelengths; there
are thus no restrictions for the scientific use of the object spectrum.
More details on quantitative comtamination are given in Kümmel et al. (2005).
To circumvent these drawbacks, a new reduction scheme is available in aXe 1.4, whereby all the individual 2D spectra of an object are coadded to a single deep 2D spectrum. The final, deep 1D spectrum is then extracted from this combined 2D spectral image. The combination of the individual 2D spectra is done with the ``Drizzle'' software, Fruchter & Hook (2002), which is available in the STSDAS package within IRAF.
The advantages of this technique as applied to slitless spectra can be summarised as follows:
These advantages come at the expense of a greater complexity of the reduction and significantly longer processing time. Also, the aXe drizzle reduction currently supports only first-order spectra.
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The drizzling within aXe is fully embedded in the aXe reduction flow and uses data products and tasks created and used in the non-drizzling part of aXe. The input for the drizzle combination consists of flatfielded and wavelength calibrated PETs extracted for each science image, which are converted to Drizzle PrePare files (DPP) using the drzprep task. Every first order beam in a PET is converted to a stamp image stored as an extension in a DPP. The drzprep task also computes the transformation coefficients which are required to drizzle the single stamp images of each object onto a single deep, combined 2D spectral image. These transformation coefficients are computed such that the combined drizzle image resembles an ideal long slit spectrum, with the dispersion direction parallel to the x-axis and cross-dispersion direction parallel to the y-axis. The wavelength scale and the pixel scale in the cross-dispersion direction can be set by the user with keyword settings in the aXe Configuration File.
To finally extract the 1D spectrum from the deep 2D spectral image,
aXe uses an (automatically created) adapted configuration file
that takes into account the modified spectrum of the drizzled images
(i.e. orthogonal wavelength and cross-dispersion and the Å
and
scales).
A detailed discussion of the drizzling used in aXe is given in Kümmel et al. (2004a).
Figure 1.10 illustrates the aXedrizzle process for one object. Panel a shows one individual grism image with an object marked. Panel b displays the stamp image for this object out of the grism image. Panel c shows the derived drizzled grism stamp image, and the final coadded 2D spectrum for this object is given in panel d. Panel d shows an image combined from 112 PETs with a total exposure time of 124 ksec. In both panels b and c, the `holes' resulting from the discarded cosmic ray-flagged pixels in this individual exposure are clearly visible.
The variables are:
In the original descriptions of optimal weighting, the extraction profile
is computed from the object spectrum itself by e.g.
averaging the pixel values in wavelength direction. In Horne (1986)
optimal weighting (or optimal extraction, as named there) is even
an iterative procedure which, starting from a normal extraction procedure using
equal weights, produces improved results for sky background, extraction
profile and, of course, the extracted spectrum.
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In ACS slitless spectroscopy such an approach is not viable since
To compute extraction profiles for all sources, the optimal
weighting as implemented in aXe uses the 2D models for the dispersed
objects, which were introduced in Chapt. 1.7.2 as the basis of
quantitative contamination. The source-specific models computed there
deliver a perfect basis to calculate the quantity
in
Eqn. 1.1.
The beam models are also used as an input to calculate the pixel
errors
according to the typical CCD noise model
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In all extraction modes (from individual grism images or from the combined 2D drizzled grism images) aXe delivers optimal weighted spectra as an optional addition to the usual, equally weighted ones. Figure 1.11 shows a comparison between two spectra extracted from the same data using equal and optimal weights. Results from both, observed as well as simulated data indicate that optimal weigthing in aXe improves the signal-to-noise ratio by a small, but significant amount as expected according to Horne (1986) and Robertson (1986).
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With variable extraction, the line of constant wavelength follows for every object a specific, marked direction. The major axis angle in the column THETA_IMAGE of the Input Object List is used in this mode to define the line of constant wavelength or extraction direction for every object individually. aXe mimics with the variable extraction direction individually oriented slits for all objects. This can help to maintain the instrumental resolution for small, extended objects. However for small angles between the trace and the extraction direction the finite instrumental resolution limits any improvements due to the variable extraction direction, and in addition the extraction becomes numerically unstable 1.1. aXe can switch (with the parameter SLITLESS_GEOM='YES', see below) the extraction direction from the major axis angle to a different angle which optimizes the resolution of the extracted spectra ([5], [2]).
The variable extraction width is determined for each object individually
to a scaled value
of the object extent in the extraction
direction.
The main parameters to specify extraction width and extraction direction
are extrfwhm (or
in Fig. 1.12),
orient and slitless_geom in the task
axecore. Figure 1.12 illustrates how those parameters
can be used to extract the flux of an object in various ways:
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Figure 1.13 shows the effect of the sensitivity adjustment for an extracted ACS/WFC spectrum. The lower panel shows a strong upturn at both wavelength ends due to the degraded resolution. Smoothing the sensitivity function using the appropriate Gaussian kernel suppresses this effect (upper panel).
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Since aXe2web requires specific python modules it
cannot be included in the STSDAS
software package. It is therefore distributed via the aXe
webpage at
http://www.stecf.org/software/slitless_software/axe/
in the aXe2html package.
aXe2web uses a standard aXe input catalogue and the aXe output files
to produce an html summary containing a variety of information
for each spectrum. This includes a reference number, magnitude
in the magnitude system of the direct object, the X and Y position
of the direct object, its Right Ascension and Declination,
a cut-out image showing the direct object, the spectrum stamp image
showing the 2D spectrum, a 1D extracted spectrum in counts and
the same in flux units.
The user can set various keywords to influence the html output. For example, it is possible to sort the objects with respect to an object property such as magnitude or Right Ascension.
In order to facilitate the navigation within a data set, an overview and an index page accompany the object pages. The overview page contains for each object the basic information sequence number, reference number, X,Y,RA,Dec and magnitude. The index page includes a table with the ordered reference number of all objects. Direct links, from both the overview page and the index page point to the corresponding locations of the objects in the object pages.
Figure 1.14 is a screenshot taken from Epoch 1 data of the HUDF HRC Parallels survey and shows the line covering the object whose coadded 2D spectrum is shown in Fig. 1.10d. The webpages created by aXe2web are located at the preview webpages: http://www.stecf.org/UDF/epoch1/hrc_udf.html.