Datacube

Integral field spectroscopy (IFS Courtgs 1982) is a technique to produce a spectrum for each point in a contiguous two-dimensional field, resulting in a datacube with axes given by the two spatial coordinates in the field and the wavelength. The advantages of IFS are as follows. 

The position of the object in the field can be unambiguously determined post facto by summing the datacube in wavelength to produce a white-light image. In contrast, the position of the target with respect to the slit is difficult to determine in slit spectroscopy. 

From this, it can be seen that the optimum choice of 3D technique depends on the observing strategy appropriate to the particular scientific investigation. However there are other, second-order, considerations to be considered which violate this datacube theorem. These include whether the individual exposures are dominated by photon noise from the background or the detector, and the stability of the instrument and background over a long period of stepped exposures. 

Figure 17. Steps in the constmction of a datacube of the nucleus of NGC1068 from observations with the integral held unit of the Gemini Multiobject Spectrograph installed on the Gemini-north telescope. The datacube is illustrated by a few spectra distributed over the field and equivalently by a few slices at a given radial velocity in the light of the [OIII]5007 emission line. Only a few percent of the total data content is shown.

The adaptive spectrometer described in this paper produces either non-imaging, onedimensional or two-dimensional multispectral radiance datasets ( datacube in the case of two-dimensional spectral mapping) for gas or aerosol discrimination and classification. The spectral, temporal and spatial resolution of the data collected by the instrament are adjustable in real time, making it possible to keep the tradeoff between sensor parameters at optimum at all times. The instrument contains no macro-scale moving parts making it an excellent candidate for the development of a robust, compact, lightweight and low-power-consumption device suitable for field operation. 

The Sky Generator Module is the module that generates the datacube to be fed to the subsequent blocks. This data cube consists of series of 2-dimensional images of a Sky. These 2-dimensional images are defined by the field of view (FOVx and FOVy) of the instrument. Along the 3rd dimension of the Sky and where a source has been positioned, the spectrum of this source is stored. The number of samples N defines the number of the images and is determined such that there is no aliasing. 

Figure4.2 shows the structure of the sky datacube (left) and the spectrum for two pixels of the sky datacube (right) indicated as A (purple) and B (green) along the wavenumbers axis v. The axis 0 and By represent the position of the sources on the sky grid, and A9x and AOy are the pixel size or angular resolution.

Fig. 4.2 Sky datacube structure left) and spectra stored in two pixels of the sky datacube right)...

Fig. 4.3 Simulated sky datacube where 2 point sources (a, b) and 2 gaussian sources (c, d) with different spectra have been placed on a sky grid...

Figure4.3 shows an example of a simulated datacube, where 4 sources have been positioned on the sky grid (left) two gaussian sources, and two point sources. The spectra of these sources are different, corresponding to blackbodies of different temperatures which have been multiplied by Alters with different cut-on and cut-off wavenumbers (right). For the loading of more complex sky map data cubes such as science datacubes the simulator will interpolate or decimate to meet the instrument parameters.

The output of the instrument simulator FllnS is a set of FITS files where the interferograms corresponding to each baseline and to each FTS scan are stored for a given input sky datacube, Skyi . These FITS files also include fhe mefrology data for the FTS drive and pointing of the telescope, as well as all the information needed to calculate the baseline vector. 

Once the data has been loaded, if noise has been selected to be included in the simulation, the first step will be the noise reduction to increase the dynamic range (DR) and the signal to noise ratio (SNR). Next the reconstruction of the dirty datacube is achieved by extracting the spectral and spatial properties detailed below. To illustrate the data reduction process a master sky map is used. 

The next step is the extraction of the spatial features of Sp, this is the dirty datacube, Sky. To extract the spatial features one has to perform the 2-dimensional Fourier Transform of the m v-map. Combining each baseline position bj and each wavenumber Vk the dirty datacube is calculated as... 

Fig. 5.9 Synthesised datacube layer after 1 iteration of the blind deconvolution algorithm for the minimum wavenumber (25cm , / ), central wavenumber (118cm , centre) and maximum wavenumber (212cm, right) (top). Restored PSF at each of the previous wavenumbers (bottom)...

For a 2-iteration blind deconvolution, the results do not improve. Although the PSF beam size is still consistent with the one expected from theory, the ripples have vanished from the PSF and are still present in the recovered datacube. The spatial size of the gaussian source approximates to a point source for the maximum wavenumber image (right) and does not correspond to the expected spatial size. 

When 5 blind deconvolution iterations are applied to the dirty data cube, the beam size does not correspond to the one expected from theory anymore, from which one can infer that the recovered datacube is unrealistic. 

In conclusion, a blind deconvolution algorithm is not suitable for DFM dirty datacubes because for a proper restoration, more information regarding the dirty beam has to be applied. As the information of the dirty beam can be extracted from the known v-map, previous knowledge of the dirty beam can be used. One algorithm that makes use of a known dirty beam is the interferometric CLEAN algorithm. 

Figure 5.16 shows the obtained results for three layers of the datacube corresponding to the wavenumbers 25, 118 and 212cm (top), and the corresponding interferometric dirty beam (bottom). It can be observed that for the minimum wavenumber CLEAN can not separate the sources due to the size of the interferometric dirty... 

Fig. 5.17 Result of the wavenumber integration of the CLEAN datacube (left) and its logarithm (right)...

Finally, in Fig. 5.18 (left) the detected spectra for the central pixel of the gaussian source (blue), the point source (green) and the central pixel of the elliptical source (red) is shown. One can observe that the ripples due to the interferometric dirty beam have vanished and the detected spectra is in concordance with the input Master sky map. Again, for comparison. Fig. 5.18 (right) presents the detected spectra for three positions in the sky where no signal is expected. The presence of power in the low wavenumbers is due to the interferometric beam size. From 60cm the sources are resolved, the power in those pixels reduces and the modulation present in the dirty datacube (see Fig. 5.7) disappears because there are no ripples from the interferometric beam. 

Once the interferograms have been generated, one can proceed to extract the spectral and spatial features of the datacube using the algorithms described earUer in this chapter. 

The datacube reconstruction from detected interferograms has been performed. Initial steps consist of the noise reduction and time domain interpolation when instrument errors have been included in the simulation. Once the data has been reduced, the dirty datacube is calculated in two steps first through Fourier transforming in the wavenumber domain and second through two dimensional Fourier transforming in the spatial domain. 

It has been observed that instrumental artifacts appear at the dirty datacube. To reduce them, one can use interferometric data synthesis algorithms. Brute force... 

In the previous chapter a simulation of a Master sky datacube for FllnS was presented. However, this sky simulation was generated within the simulator itself. 

In the case concerning this chapter, a science sky datacube is the input to FllnS a simulation of a circumstellar disk around a Herbig Ae star kindly generated by Dr. Catherine Walsh at Leiden Observatory (Walsh et al. 2014). 

The science datacube selected for the next simulations corresponds to a proto-planetary disk surrounding a Herbig Ae star 10,000 K). As presented earlier in this chapter, Herbig Ae/Be stars are pre-main-sequence stars. The main difference with T Tauri stars is the mass, this being Af+ > Mq. Spectrally, their SED shows strong infrared radiation excess due to the presence of the drcumstellar accretion disk (Hillenbrand et al. 1992), this is, the thermal emission of circumstellar dust. 

Spatially, the science datacube is a disk model with a diameter of approximately 400 AU and with a gap of radius = 10 AU situated 130-140 parsecs from our Solar System, where the nearest regions of ongoing star formation are. At this distance, 1 AU subtends an angle of 7 milliarcseconds. 

Figure 6.2 shows the provided science datacube. Spatially, it consists of a face-on disk decreasing in intensity from its centre (left), while spectrally it contains the dust continuum only (right) which is the predominant at Far Infrared wavelengths. Three pixels (a, b and c in the picture) have been selected to show the intensity variation over wavenumber for comparison with the detected spectra later in this chapter. 

In this chapter the capabilities of FllnS for the simulation of an observation of an external science datacube have been presented. The science case selected and presented in Sect. 6.1 has been a circumstellar disc because of its importance in the understanding of how planetary systems and habitable planets are formed. Also, due to the nature of their structure and composition, cirumstellar disks can only be measured using interferometric systems at the Infrared wavelength range. 

In summary, this first version of FllnS allows the simulation and study of some instrumental effects during a simulated observation of a science datacube. Future versions of FllnS will include more instrumental errors and noise, such as pointing errors during an FTS scan which have not yet been included here for computational reasons. Future work on the software will be performed prior to its release to the scientific community to include additional instmmental effects and configurations. 

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