Interferometry

Alongside CCD imaging data, PyAutoLens supports the modeling of interferometer data from submillimeter and radio observatories. The dataset is fitted directly in the uv-plane, circumventing issues that arise when fitting a ‘dirty image’ such as correlated noise.

Real Space Mask

To begin, we define a real-space mask. Although interferometer lens modeling is performed in the uv-plane and therefore Fourier space, we still need to define the grid of coordinates in real-space from which the lensed source’s images are computed. It is this image that is mapped to Fourier space to compare to the uv-plane data.

real_space_mask = al.Mask2D.circular(
    shape_native=imaging.shape_native, pixel_scales=imaging.pixel_scales, sub_size=1, radius=3.0
)
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Interferometer Data

We next load an Interferometer dataset from fits files, which follows the same API that we have seen for an Imaging object.

dataset_path = "/path/to/dataset/folder"

interferometer = al.Interferometer.from_fits(
    visibilities_path=path.join(dataset_path, "visibilities.fits"),
    noise_map_path=path.join(dataset_path, "noise_map.fits"),
    uv_wavelengths_path=path.join(dataset_path, "uv_wavelengths.fits"),
    real_space_mask=real_space_mask
)

interferometer_plotter = aplt.InterferometerPlotter(interferometer=interferometer)
interferometer_plotter.figures_2d(visibilities=True, uv_wavelengths=True)

Here is what the interferometer visibilities and uv wavelength (which represent the interferometer’s baselines) looks like (these are representative of an ALMA dataset with ~ 1 million visibilities):

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UV-Plane FFT

To perform uv-plane modeling, PyAutoLens Fourier transforms the lensed image (computed via a Tracer) from real-space to the uv-plane. This operation uses a Transformer object, of which there are multiple available in PyAutoLens. This includes a direct Fourier transform which performs the exact Fourier transform without approximation.

transformer_class = al.TransformerDFT

However, the direct Fourier transform is inefficient. For ~10 million visibilities, it requires thousands of seconds to perform a single transform. To model a lens, we’ll perform tens of thousands of transforms, making this approach unfeasible for high quality ALMA and radio datasets.

For this reason, PyAutoLens supports the non-uniform fast fourier transform algorithm PyNUFFT (https://github.com/jyhmiinlin/pynufft), which is significantly faster, being able too perform a Fourier transform of ~10 million in less than a second!

transformer_class = al.TransformerNUFFT

To perform a fit, we follow the same process we did for imaging. We do not need to mask an interferometer dataset, but we will apply the settings above:

interferometer = interferometer.apply_settings(
    settings=al.SettingsInterferometer(transformer_class=transformer_class)
)

Fitting

The interferometer can now be passed to a FitInterferometer object to fit it to a data-set:

fit = al.FitInterferometer(
    interferometer=interferometer, tracer=tracer
)

Here is what the image of the tracer looks like before it is Fourier transformed to the uv-plane:

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And here is what the Fourier transformed model visibilities look like:

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To show the fit to the real and imaginary visibilities, we plot the residuals and chi-squared values as a function uv-distance:

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Pixelized Sources

Interferometer data can also be modeled using pixelized source’s, which again perform the source reconstruction by directly fitting the visibilities in the uv-plane. The source reconstruction is visualized in real space:

Computing this source recontruction would be extremely inefficient if PyAutoLens used a traditional approach to linear algebra which explicitly stored in memory the values required to solve for the source fluxes. In fact, for an interferometer dataset of ~10 million visibilities this would require hundreds of GB of memory!

PyAutoLens uses the library PyLops (https://pylops.readthedocs.io/en/latest/) to represent this calculation as a sequence of memory-light linear operators.

The combination of PyNUFFT and PyLops makes the analysis of ~10 million visibilities from observatories such as ALMA and JVLA feasible in PyAutoLens. However, the largest datasets may still require a degree of augmentation, averaging or tapering. Rest assured, we are actively working on new solution that will make the analysis of hundreds of millions of visibilities feasible.

Lens Modeling

It is straight forward to fit a lens model to an interferometer dataset, using the same API that we saw for imaging data in the modeling overview example.

Whereas we previously used an AnalysisImaging object, we instead use an AnalysisInterferometer object which fits the lens model in the correct way for an interferometer dataset. This includes mapping the lens model from real-space to the uv-plane via the Fourier transform discussed above:

lens_galaxy_model = af.Model(al.Galaxy, redshift=0.5, mass=al.mp.EllIsothermal)
source_galaxy_model = af.Model(al.Galaxy, redshift=1.0, disk=al.lp.EllExponential)

model = af.Collection(lens=lens_galaxy_model, source=source_galaxy_model)

search = af.DynestyStatic(name="overview_interferometer")

analysis = al.AnalysisInterferometer(dataset=interferometer)

result = search.fit(model=model, analysis=analysis)

Simulations

Simulated interferometer datasets can be generated using the SimulatorInterferometer object, which includes adding Gaussian noise to the visibilities:

real_space_grid = al.Grid2D.uniform(shape_native=real_space_mask.shape_native, pixel_scales=real_space_mask.pixel_scales)

simulator = al.SimulatorInterferometer(
    uv_wavelengths=uv_wavelengths,
    exposure_time=300.0,
    background_sky_level=1.0,
    noise_sigma=0.01,
)

interferometer = simulator.from_tracer_and_grid(tracer=tracer, grid=real_space_grid)

Wrap-Up

The interferometer package of the autolens_workspace contains numerous example scripts for performing interferometer modeling and simulating strong lens interferometer datasets.