Radio Interferometer Measurement Equation#

Functions used to compute the terms of the Radio Interferometer Measurement Equation (RIME). It describes the response of an interferometer to a sky model.

\[V_{pq} = G_{p} \left( \sum_{s} E_{ps} L_{p} K_{ps} B_{s} K_{qs}^H L_{q}^H E_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

\(G_{p}\) represents direction-independent effects.
\(E_{ps}\) represents direction-dependent effects.
\(L_{p}\) represents the feed rotation.
\(K_{ps}\) represents the phase delay term.
\(B_{s}\) represents the brightness matrix.

The RIME is more formally described in the following four papers:

Numpy#

`predict_vis`(time_index, antenna1, antenna2)	Multiply Jones terms together to form model visibilities according to the following formula:
`phase_delay`(lm, uvw, frequency[, convention])	Computes the phase delay (K) term:
`parallactic_angles`(times, antenna_positions, ...)	Computes parallactic angles per timestep for the given reference antenna position and field centre.
`feed_rotation`(parallactic_angles[, feed_type])	Computes the 2x2 feed rotation (L) matrix from the `parallactic_angles`.
`transform_sources`(lm, parallactic_angles, ...)	Creates beam sampling coordinates suitable for use in `beam_cube_dde()` by:
`beam_cube_dde`(beam, beam_lm_extents, ...)	Evaluates Direction Dependent Effects along a source's path by interpolating the values of a complex beam cube at the source location.
`zernike_dde`(coords, coeffs, noll_index, ...)	Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients `coeffs` and noll indexes `noll_index` at the specified coordinates `coords`.
`wsclean_predict`(uvw, lm, source_type, flux, ...)	Predict visibilities from a WSClean sky model.

africanus.rime.predict_vis(time_index, antenna1, antenna2, dde1_jones=None, source_coh=None, dde2_jones=None, die1_jones=None, base_vis=None, die2_jones=None)[source]#

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} E_{ps} X_{pqs} E_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

\(B_{{pq}}\) represent base coherencies.
\(E_{{ps}}\) represents Direction-Dependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (per-source).
\(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:

time_indexnumpy.ndarray: Time index used to look up the antenna Jones index for a particular baseline with shape (row,). Obtainable via np.unique(time, return_inverse=True)[1].
antenna1numpy.ndarray: Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).
antenna2numpy.ndarray: Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).
dde1_jonesnumpy.ndarray, optional: \(E_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)
source_cohnumpy.ndarray, optional: \(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)
dde2_jonesnumpy.ndarray, optional: \(E_{qs}\) Direction-Dependent Jones terms for the second antenna. This is usually the same array as dde1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (source,time,ant,chan,corr_1,corr_2)
die1_jonesnumpy.ndarray, optional: \(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
base_visnumpy.ndarray, optional: \(B_{pq}\) base coherencies, added to source coherency summation before multiplication with die1_jones and die2_jones. shape (row,chan,corr_1,corr_2).
die2_jonesnumpy.ndarray, optional: \(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. This is usually the same array as die1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (time,ant,chan,corr_1,corr_2)

Returns:

visibilitiesnumpy.ndarray: Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.
The inputs to this function involve row, time and ant (antenna) dimensions.
Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.
The row dimension must be an increasing partial order in time.

africanus.rime.phase_delay(lm, uvw, frequency, convention='fourier')[source]#

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]

Parameters:

lmnumpy.ndarray: LM coordinates of shape (source, 2) with L and M components in the last dimension.
uvwnumpy.ndarray: UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.
frequencynumpy.ndarray: frequencies of shape (chan,)
convention{‘fourier’, ‘casa’}: Uses the \(e^{-2 \pi \mathit{i}}\) sign convention if fourier and \(e^{2 \pi \mathit{i}}\) if casa.

Returns:

complex_phasenumpy.ndarray: complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses the CASA sign convention.

africanus.rime.parallactic_angles(times, antenna_positions, field_centre, backend='casa')[source]#

Computes parallactic angles per timestep for the given reference antenna position and field centre.

Parameters:

timesnumpy.ndarray

Array of Mean Julian Date times in seconds with shape (time,),

antenna_positionsnumpy.ndarray

Antenna positions of shape (ant, 3) in metres in the ITRF frame.

field_centrenumpy.ndarray

Field centre of shape (2,) in radians

backend{‘casa’, ‘test’}, optional

Backend to use for calculating the parallactic angles.

casa defers to an implementation depending on python-casacore. This backend should be used by default.
test creates parallactic angles by multiplying the times and antenna_position arrays. It exist solely for testing.

Returns:

parallactic_anglesnumpy.ndarray: Parallactic angles of shape (time,ant)

africanus.rime.feed_rotation(parallactic_angles, feed_type='linear')[source]#

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]

Parameters:

parallactic_anglesnumpy.ndarray: floating point parallactic angles. Of shape (pa0, pa1, ..., pan).
feed_type{‘linear’, ‘circular’}: The type of feed

Returns:

feed_matrixnumpy.ndarray: Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)

africanus.rime.transform_sources(lm, parallactic_angles, pointing_errors, antenna_scaling, frequency, dtype=None)[source]#

Creates beam sampling coordinates suitable for use in beam_cube_dde() by:

Rotating lm coordinates by the parallactic_angles
Adding pointing_errors
Scaling by antenna_scaling

Parameters:

lmnumpy.ndarray: LM coordinates of shape (src,2) in radians offset from the phase centre.
parallactic_anglesnumpy.ndarray: parallactic angles of shape (time, antenna) in radians.
pointing_errorsnumpy.ndarray: LM pointing errors for each antenna at each timestep in radians. Has shape (time, antenna, 2)
antenna_scalingnumpy.ndarray: antenna scaling factor for each channel and each antenna. Has shape (antenna, chan)
frequencynumpy.ndarray: frequencies for each channel. Has shape (chan,)
dtypenumpy.dtype, optional: Numpy dtype of result array. Should be float32 or float64. Defaults to float64

Returns:

coordsnumpy.ndarray: coordinates of shape (3, src, time, antenna, chan) where each coordinate component represents l, m and frequency, respectively.

africanus.rime.beam_cube_dde(beam, beam_lm_extents, beam_freq_map, lm, parallactic_angles, point_errors, antenna_scaling, frequency)[source]#

Evaluates Direction Dependent Effects along a source’s path by interpolating the values of a complex beam cube at the source location.

Parameters:

beamnumpy.ndarray

Complex beam cube of shape (beam_lw, beam_mh, beam_nud, corr, corr). beam_lw, beam_mh and beam_nud define the size of the cube in the l, m and frequency dimensions, respectively.

beam_lm_extentsnumpy.ndarray

lm extents of the beam cube of shape (2, 2). [[lower_l, upper_l], [lower_m, upper_m]].

beam_freq_mapnumpy.ndarray

Beam frequency map of shape (beam_nud,). This array is used to define interpolation along the (chan,) dimension.

lmnumpy.ndarray

Source lm coordinates of shape (source, 2). These coordinates are:

Scaled if the associated frequency lies outside the beam cube.

Offset by pointing errors: point_errors

Rotated by parallactic angles: parallactic_angles.

Scaled by antenna scaling factors: antenna_scaling.

parallactic_anglesnumpy.ndarray

Parallactic angles of shape (time, ant).

point_errorsnumpy.ndarray

Pointing errors of shape (time, ant, chan, 2).

antenna_scalingnumpy.ndarray

Antenna scaling factors of shape (ant, chan, 2)

frequencynumpy.ndarray

Frequencies of shape (chan,).

Returns:

ddesnumpy.ndarray: Direction Dependent Effects of shape (source, time, ant, chan, corr, corr)

Notes

Sources are clamped to the provided beam_lm_extents.
Frequencies outside the cube (i.e. outside beam_freq_map) introduce linear scaling to the lm coordinates of a source.

africanus.rime.zernike_dde(coords, coeffs, noll_index, parallactic_angles, frequency_scaling, antenna_scaling, pointing_errors)[source]#

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.

Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.

Parameters:

coordsnumpy.ndarray: Float coordinates at which to evaluate the zernike polynomials. Has shape (3, source, time, ant, chan). The three components in the first dimension represent l, m and frequency coordinates, respectively.
coeffsnumpy.ndarray: complex Zernicke polynomial coefficients. Has shape (ant, chan, corr_1, ..., corr_n, poly) where poly is the number of polynomial coefficients and corr_1, ..., corr_n are a variable number of correlation dimensions.
noll_indexnumpy.ndarray: Noll index associated with each polynomial coefficient. Has shape (ant, chan, corr_1, ..., corr_n, poly). correlation dimensions.
parallactic_anglesnumpy.ndarray: Parallactic angle rotation. Has shape (time, ant).
frequency_scalingnumpy.ndarray: The scaling of frequency of the beam. Has shape (chan,).
antenna_scalingnumpy.ndarray: The antenna scaling. Has shape (ant, chan, 2).
pointing_errorsnumpy.ndarray: The pointing error. Has shape (time, ant, chan, 2).

Returns:

ddenumpy.ndarray: complex values with shape (source, time, ant, chan, corr_1, ..., corr_n)

africanus.rime.wsclean_predict(uvw, lm, source_type, flux, coeffs, log_poly, ref_freq, gauss_shape, frequency)[source]#

Predict visibilities from a WSClean sky model.

Parameters:

uvwnumpy.ndarray: UVW coordinates of shape (row, 3)
lmnumpy.ndarray: Source LM coordinates of shape (source, 2), in radians. Derived from the Ra and Dec fields.
source_typenumpy.ndarray: Strings defining the source type of shape (source,). Should be either "POINT" or "GAUSSIAN". Contains the Type field.
fluxnumpy.ndarray: Source flux of shape (source,). Contains the I field.
coeffsnumpy.ndarray: Source Polynomial coefficients of shape (source, coeffs). Contains the SpectralIndex field.
log_polynumpy.ndarray: Source polynomial type of shape (source,). If True, logarithmic polynomials are used. If False, standard polynomials are used. Contains the LogarithmicSI field.
ref_freqnumpy.ndarray: Source Reference frequency of shape (source,). Contains the ReferenceFrequency field.
gauss_shapenumpy.ndarray: Gaussian shape parameters of shape (source, 3) used when the corresponding source_type is "GAUSSIAN". The 3 components should contain the MajorAxis, MinorAxis and Orientation fields in radians, respectively.
frequencynumpy.ndarray: Frequency of shape (chan,).

Returns:

visibilitiesnumpy.ndarray: Complex visibilities of shape (row, chan, 1)

Cuda#

`predict_vis`(time_index, antenna1, antenna2)	Multiply Jones terms together to form model visibilities according to the following formula:
`phase_delay`(lm, uvw, frequency)	Computes the phase delay (K) term:
`feed_rotation`(parallactic_angles[, feed_type])	Computes the 2x2 feed rotation (L) matrix from the `parallactic_angles`.
`beam_cube_dde`(beam, beam_lm_ext, ...)	Evaluates Direction Dependent Effects along a source's path by interpolating the values of a complex beam cube at the source location.

africanus.rime.cuda.predict_vis(time_index, antenna1, antenna2, dde1_jones=None, source_coh=None, dde2_jones=None, die1_jones=None, base_vis=None, die2_jones=None)[source]#

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} E_{ps} X_{pqs} E_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

\(B_{{pq}}\) represent base coherencies.
\(E_{{ps}}\) represents Direction-Dependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (per-source).
\(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:

time_indexcupy.ndarray: Time index used to look up the antenna Jones index for a particular baseline with shape (row,). Obtainable via cp.unique(time, return_inverse=True)[1].
antenna1cupy.ndarray: Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).
antenna2cupy.ndarray: Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).
dde1_jonescupy.ndarray, optional: \(E_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)
source_cohcupy.ndarray, optional: \(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)
dde2_jonescupy.ndarray, optional: \(E_{qs}\) Direction-Dependent Jones terms for the second antenna. This is usually the same array as dde1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (source,time,ant,chan,corr_1,corr_2)
die1_jonescupy.ndarray, optional: \(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
base_viscupy.ndarray, optional: \(B_{pq}\) base coherencies, added to source coherency summation before multiplication with die1_jones and die2_jones. shape (row,chan,corr_1,corr_2).
die2_jonescupy.ndarray, optional: \(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. This is usually the same array as die1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (time,ant,chan,corr_1,corr_2)

Returns:

visibilitiescupy.ndarray: Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.
The inputs to this function involve row, time and ant (antenna) dimensions.
Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.
The row dimension must be an increasing partial order in time.

africanus.rime.cuda.phase_delay(lm, uvw, frequency)[source]#

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]

Parameters:

lmcupy.ndarray: LM coordinates of shape (source, 2) with L and M components in the last dimension.
uvwcupy.ndarray: UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.
frequencycupy.ndarray: frequencies of shape (chan,)
convention{‘fourier’, ‘casa’}: Uses the \(e^{-2 \pi \mathit{i}}\) sign convention if fourier and \(e^{2 \pi \mathit{i}}\) if casa.

Returns:

complex_phasecupy.ndarray: complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses the CASA sign convention.

africanus.rime.cuda.feed_rotation(parallactic_angles, feed_type='linear')[source]#

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]

Parameters:

parallactic_anglescupy.ndarray: floating point parallactic angles. Of shape (pa0, pa1, ..., pan).
feed_type{‘linear’, ‘circular’}: The type of feed

Returns:

feed_matrixcupy.ndarray: Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)

africanus.rime.cuda.beam_cube_dde(beam, beam_lm_ext, beam_freq_map, lm, parangles, pointing_errors, antenna_scaling, frequencies)[source]#

Evaluates Direction Dependent Effects along a source’s path by interpolating the values of a complex beam cube at the source location.

Parameters:

beamcupy.ndarray

Complex beam cube of shape (beam_lw, beam_mh, beam_nud, corr, corr). beam_lw, beam_mh and beam_nud define the size of the cube in the l, m and frequency dimensions, respectively.

beam_lm_extentscupy.ndarray

lm extents of the beam cube of shape (2, 2). [[lower_l, upper_l], [lower_m, upper_m]].

beam_freq_mapcupy.ndarray

Beam frequency map of shape (beam_nud,). This array is used to define interpolation along the (chan,) dimension.

lmcupy.ndarray

Source lm coordinates of shape (source, 2). These coordinates are:

Scaled if the associated frequency lies outside the beam cube.

Offset by pointing errors: point_errors

Rotated by parallactic angles: parallactic_angles.

Scaled by antenna scaling factors: antenna_scaling.

parallactic_anglescupy.ndarray

Parallactic angles of shape (time, ant).

point_errorscupy.ndarray

Pointing errors of shape (time, ant, chan, 2).

antenna_scalingcupy.ndarray

Antenna scaling factors of shape (ant, chan, 2)

frequencycupy.ndarray

Frequencies of shape (chan,).

Returns:

ddescupy.ndarray: Direction Dependent Effects of shape (source, time, ant, chan, corr, corr)

Notes

Sources are clamped to the provided beam_lm_extents.
Frequencies outside the cube (i.e. outside beam_freq_map) introduce linear scaling to the lm coordinates of a source.

Dask#

`predict_vis`(time_index, antenna1, antenna2)	Multiply Jones terms together to form model visibilities according to the following formula:
`phase_delay`(lm, uvw, frequency[, convention])	Computes the phase delay (K) term:
`parallactic_angles`(times, antenna_positions, ...)	Computes parallactic angles per timestep for the given reference antenna position and field centre.
`feed_rotation`(parallactic_angles, feed_type)	Computes the 2x2 feed rotation (L) matrix from the `parallactic_angles`.
`transform_sources`(lm, parallactic_angles, ...)	Creates beam sampling coordinates suitable for use in `beam_cube_dde()` by:
`beam_cube_dde`(beam, beam_lm_extents, ...)	Evaluates Direction Dependent Effects along a source's path by interpolating the values of a complex beam cube at the source location.
`zernike_dde`(coords, coeffs, noll_index, ...)	Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients `coeffs` and noll indexes `noll_index` at the specified coordinates `coords`.
`wsclean_predict`(uvw, lm, source_type, flux, ...)	Predict visibilities from a WSClean sky model.

africanus.rime.dask.predict_vis(time_index, antenna1, antenna2, dde1_jones=None, source_coh=None, dde2_jones=None, die1_jones=None, base_vis=None, die2_jones=None, streams=None)[source]#

Multiply Jones terms together to form model visibilities according to the following formula:

\[V_{pq} = G_{p} \left( B_{pq} + \sum_{s} E_{ps} X_{pqs} E_{qs}^H \right) G_{q}^H\]

where for antenna \(p\) and \(q\), and source \(s\):

\(B_{{pq}}\) represent base coherencies.
\(E_{{ps}}\) represents Direction-Dependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (per-source).
\(G_{{p}}\) represents Direction-Independent Jones terms.

Generally, \(E_{ps}\), \(G_{p}\), \(X_{pqs}\) should be formed by using the RIME API functions and combining them together with einsum().

Please read the Notes

Parameters:

time_indexdask.array.Array: Time index used to look up the antenna Jones index for a particular baseline with shape (row,). Obtainable via time.map_blocks(lambda a: np.unique(a, return_inverse=True)[1]).
antenna1dask.array.Array: Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape (row,).
antenna2dask.array.Array: Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape (row,).
dde1_jonesdask.array.Array, optional: \(E_{ps}\) Direction-Dependent Jones terms for the first antenna. shape (source,time,ant,chan,corr_1,corr_2)
source_cohdask.array.Array, optional: \(X_{pqs}\) Direction-Dependent Coherency matrix for the baseline. with shape (source,row,chan,corr_1,corr_2)
dde2_jonesdask.array.Array, optional: \(E_{qs}\) Direction-Dependent Jones terms for the second antenna. This is usually the same array as dde1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (source,time,ant,chan,corr_1,corr_2)
die1_jonesdask.array.Array, optional: \(G_{ps}\) Direction-Independent Jones terms for the first antenna of the baseline. with shape (time,ant,chan,corr_1,corr_2)
base_visdask.array.Array, optional: \(B_{pq}\) base coherencies, added to source coherency summation before multiplication with die1_jones and die2_jones. shape (row,chan,corr_1,corr_2).
die2_jonesdask.array.Array, optional: \(G_{ps}\) Direction-Independent Jones terms for the second antenna of the baseline. This is usually the same array as die1_jones as this preserves the symmetry of the RIME. predict_vis will perform the conjugate transpose internally. shape (time,ant,chan,corr_1,corr_2)
streams{False, True}: If True the coherencies are serially summed in a linear chain. If False, dask uses a tree style reduction algorithm.

Returns:

visibilitiesdask.array.Array: Model visibilities of shape (row,chan,corr_1,corr_2)

Notes

Direction-Dependent terms (dde{1,2}_jones) and Independent (die{1,2}_jones) are optional, but if one is present, the other must be present.
The inputs to this function involve row, time and ant (antenna) dimensions.
Each row is associated with a pair of antenna Jones matrices at a particular timestep via the time_index, antenna1 and antenna2 inputs.
The row dimension must be an increasing partial order in time.
The ant dimension should only contain a single chunk equal to the number of antenna. Since each row can contain any antenna, random access must be preserved along this dimension.
The chunks in the row and time dimension must align. This subtle point must be understood otherwise invalid results will be produced by the chunking scheme. In the example below we have four unique time indices [0,1,2,3], and four unique antenna [0,1,2,3] indexing 10 rows.
```
#  Row indices into the time/antenna indexed arrays
time_idx = np.asarray([0,0,1,1,2,2,2,2,3,3])
ant1 = np.asarray(    [0,0,0,0,1,1,1,2,2,3]
ant2 = np.asarray(    [0,1,2,3,1,2,3,2,3,3])
```
A reasonable chunking scheme for the row and time dimension would be (4,4,2) and (2,1,1) respectively. Another way of explaining this is that the first four rows contain two unique timesteps, the second four rows contain one unique timestep and the last two rows contain one unique timestep.

Some rules of thumb:
1. The number chunks in row and time must match although the individual chunk sizes need not.
2. Unique timesteps should not be split across row chunks.
3. For a Measurement Set whose rows are ordered on the TIME column, the following is a good way of obtaining the row chunking strategy:
```
import numpy as np
import pyrap.tables as pt

ms = pt.table("data.ms")
times = ms.getcol("TIME")
unique_times, chunks = np.unique(times, return_counts=True)
```
4. Use aggregate_chunks() to aggregate multiple row and time chunks into chunks large enough such that functions operating on the resulting data can drop the GIL and spend time processing the data. Expanding the previous example:
```
# Aggregate row
utimes = unique_times.size
# Single chunk for each unique time
time_chunks = (1,)*utimes
# Aggregate row chunks into chunks <= 10000
aggregate_chunks((chunks, time_chunks), (10000, utimes))
```

africanus.rime.dask.phase_delay(lm, uvw, frequency, convention='fourier')[source]#

Computes the phase delay (K) term:

\[ \begin{align}\begin{aligned}& {\Large e^{-2 \pi i (u l + v m + w (n - 1))} }\\& \textrm{where } n = \sqrt{1 - l^2 - m^2}\end{aligned}\end{align} \]

Parameters:

lmdask.array.Array: LM coordinates of shape (source, 2) with L and M components in the last dimension.
uvwdask.array.Array: UVW coordinates of shape (row, 3) with U, V and W components in the last dimension.
frequencydask.array.Array: frequencies of shape (chan,)
convention{‘fourier’, ‘casa’}: Uses the \(e^{-2 \pi \mathit{i}}\) sign convention if fourier and \(e^{2 \pi \mathit{i}}\) if casa.

Returns:

complex_phasedask.array.Array: complex of shape (source, row, chan)

Notes

Corresponds to the complex exponential of the Van Cittert-Zernike Theorem.

MeqTrees uses the CASA sign convention.

africanus.rime.dask.parallactic_angles(times, antenna_positions, field_centre, **kwargs)[source]#

Computes parallactic angles per timestep for the given reference antenna position and field centre.

Parameters:

timesdask.array.Array

Array of Mean Julian Date times in seconds with shape (time,),

antenna_positionsdask.array.Array

Antenna positions of shape (ant, 3) in metres in the ITRF frame.

field_centredask.array.Array

Field centre of shape (2,) in radians

backend{‘casa’, ‘test’}, optional

Backend to use for calculating the parallactic angles.

casa defers to an implementation depending on python-casacore. This backend should be used by default.
test creates parallactic angles by multiplying the times and antenna_position arrays. It exist solely for testing.

Returns:

parallactic_anglesdask.array.Array: Parallactic angles of shape (time,ant)

africanus.rime.dask.feed_rotation(parallactic_angles, feed_type)[source]#

Computes the 2x2 feed rotation (L) matrix from the parallactic_angles.

\[\begin{split}\textrm{linear} \begin{bmatrix} cos(pa) & sin(pa) \\ -sin(pa) & cos(pa) \end{bmatrix} \qquad \textrm{circular} \begin{bmatrix} e^{-i pa} & 0 \\ 0 & e^{i pa} \end{bmatrix}\end{split}\]

Parameters:

parallactic_anglesnumpy.ndarray: floating point parallactic angles. Of shape (pa0, pa1, ..., pan).
feed_type{‘linear’, ‘circular’}: The type of feed

Returns:

feed_matrixnumpy.ndarray: Feed rotation matrix of shape (pa0, pa1,...,pan,2,2)

africanus.rime.dask.transform_sources(lm, parallactic_angles, pointing_errors, antenna_scaling, frequency, dtype=None)[source]#

Creates beam sampling coordinates suitable for use in beam_cube_dde() by:

Rotating lm coordinates by the parallactic_angles
Adding pointing_errors
Scaling by antenna_scaling

Parameters:

lmdask.array.Array: LM coordinates of shape (src,2) in radians offset from the phase centre.
parallactic_anglesdask.array.Array: parallactic angles of shape (time, antenna) in radians.
pointing_errorsdask.array.Array: LM pointing errors for each antenna at each timestep in radians. Has shape (time, antenna, 2)
antenna_scalingdask.array.Array: antenna scaling factor for each channel and each antenna. Has shape (antenna, chan)
frequencydask.array.Array: frequencies for each channel. Has shape (chan,)
dtypenumpy.dtype, optional: Numpy dtype of result array. Should be float32 or float64. Defaults to float64

Returns:

coordsdask.array.Array: coordinates of shape (3, src, time, antenna, chan) where each coordinate component represents l, m and frequency, respectively.

africanus.rime.dask.beam_cube_dde(beam, beam_lm_extents, beam_freq_map, lm, parallactic_angles, point_errors, antenna_scaling, frequencies)[source]#

Evaluates Direction Dependent Effects along a source’s path by interpolating the values of a complex beam cube at the source location.

Parameters:

beamdask.array.Array

Complex beam cube of shape (beam_lw, beam_mh, beam_nud, corr, corr). beam_lw, beam_mh and beam_nud define the size of the cube in the l, m and frequency dimensions, respectively.

beam_lm_extentsdask.array.Array

lm extents of the beam cube of shape (2, 2). [[lower_l, upper_l], [lower_m, upper_m]].

beam_freq_mapdask.array.Array

Beam frequency map of shape (beam_nud,). This array is used to define interpolation along the (chan,) dimension.

lmdask.array.Array

Source lm coordinates of shape (source, 2). These coordinates are:

Scaled if the associated frequency lies outside the beam cube.

Offset by pointing errors: point_errors

Rotated by parallactic angles: parallactic_angles.

Scaled by antenna scaling factors: antenna_scaling.

parallactic_anglesdask.array.Array

Parallactic angles of shape (time, ant).

point_errorsdask.array.Array

Pointing errors of shape (time, ant, chan, 2).

antenna_scalingdask.array.Array

Antenna scaling factors of shape (ant, chan, 2)

frequencydask.array.Array

Frequencies of shape (chan,).

Returns:

ddesdask.array.Array: Direction Dependent Effects of shape (source, time, ant, chan, corr, corr)

Notes

Sources are clamped to the provided beam_lm_extents.
Frequencies outside the cube (i.e. outside beam_freq_map) introduce linear scaling to the lm coordinates of a source.

africanus.rime.dask.zernike_dde(coords, coeffs, noll_index, parallactic_angle, frequency_scaling, antenna_scaling, pointing_errors)[source]#

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients coeffs and noll indexes noll_index at the specified coordinates coords.

Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.

Parameters:

coordsdask.array.Array: Float coordinates at which to evaluate the zernike polynomials. Has shape (3, source, time, ant, chan). The three components in the first dimension represent l, m and frequency coordinates, respectively.
coeffsdask.array.Array: complex Zernicke polynomial coefficients. Has shape (ant, chan, corr_1, ..., corr_n, poly) where poly is the number of polynomial coefficients and corr_1, ..., corr_n are a variable number of correlation dimensions.
noll_indexdask.array.Array: Noll index associated with each polynomial coefficient. Has shape (ant, chan, corr_1, ..., corr_n, poly). correlation dimensions.
parallactic_anglesdask.array.Array: Parallactic angle rotation. Has shape (time, ant).
frequency_scalingdask.array.Array: The scaling of frequency of the beam. Has shape (chan,).
antenna_scalingdask.array.Array: The antenna scaling. Has shape (ant, chan, 2).
pointing_errorsdask.array.Array: The pointing error. Has shape (time, ant, chan, 2).

Returns:

ddedask.array.Array: complex values with shape (source, time, ant, chan, corr_1, ..., corr_n)

africanus.rime.dask.wsclean_predict(uvw, lm, source_type, flux, coeffs, log_poly, ref_freq, gauss_shape, frequency)[source]#

Predict visibilities from a WSClean sky model.

Parameters:

uvwdask.array.Array: UVW coordinates of shape (row, 3)
lmdask.array.Array: Source LM coordinates of shape (source, 2), in radians. Derived from the Ra and Dec fields.
source_typedask.array.Array: Strings defining the source type of shape (source,). Should be either "POINT" or "GAUSSIAN". Contains the Type field.
fluxdask.array.Array: Source flux of shape (source,). Contains the I field.
coeffsdask.array.Array: Source Polynomial coefficients of shape (source, coeffs). Contains the SpectralIndex field.
log_polydask.array.Array: Source polynomial type of shape (source,). If True, logarithmic polynomials are used. If False, standard polynomials are used. Contains the LogarithmicSI field.
ref_freqdask.array.Array: Source Reference frequency of shape (source,). Contains the ReferenceFrequency field.
gauss_shapedask.array.Array: Gaussian shape parameters of shape (source, 3) used when the corresponding source_type is "GAUSSIAN". The 3 components should contain the MajorAxis, MinorAxis and Orientation fields in radians, respectively.
frequencydask.array.Array: Frequency of shape (chan,).

Returns:

visibilitiesdask.array.Array: Complex visibilities of shape (row, chan, 1)