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.
where for antenna \(p\) and \(q\), and source \(s\):
\(G_{p}\) represents directionindependent effects.
\(E_{ps}\) represents directiondependent 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¶

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

Computes the phase delay (K) term: 

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

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

Creates beam sampling coordinates suitable for use in 

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

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients 

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 DirectionDependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (persource).
\(G_{{p}}\) represents DirectionIndependent 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_index
numpy.ndarray
Time index used to look up the antenna Jones index for a particular baseline with shape
(row,)
. Obtainable vianp.unique(time, return_inverse=True)[1]
. antenna1
numpy.ndarray
Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. antenna2
numpy.ndarray
Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. dde1_jones
numpy.ndarray
, optional \(E_{ps}\) DirectionDependent Jones terms for the first antenna. shape
(source,time,ant,chan,corr_1,corr_2)
 source_coh
numpy.ndarray
, optional \(X_{pqs}\) DirectionDependent Coherency matrix for the baseline. with shape
(source,row,chan,corr_1,corr_2)
 dde2_jones
numpy.ndarray
, optional \(E_{qs}\) DirectionDependent 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_jones
numpy.ndarray
, optional \(G_{ps}\) DirectionIndependent Jones terms for the first antenna of the baseline. with shape
(time,ant,chan,corr_1,corr_2)
 base_vis
numpy.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_jones
numpy.ndarray
, optional \(G_{ps}\) DirectionIndependent 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)
 time_index
 Returns
 visibilities
numpy.ndarray
Model visibilities of shape
(row,chan,corr_1,corr_2)
 visibilities
Notes
DirectionDependent 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
andant
(antenna) dimensions.Each
row
is associated with a pair of antenna Jones matrices at a particular timestep via thetime_index
,antenna1
andantenna2
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
 lm
numpy.ndarray
LM coordinates of shape
(source, 2)
with L and M components in the last dimension. uvw
numpy.ndarray
UVW coordinates of shape
(row, 3)
with U, V and W components in the last dimension. frequency
numpy.ndarray
frequencies of shape
(chan,)
 convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
.
 lm
 Returns
 complex_phase
numpy.ndarray
complex of shape
(source, row, chan)
 complex_phase
Notes
Corresponds to the complex exponential of the Van CittertZernike 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
 times
numpy.ndarray
Array of Mean Julian Date times in seconds with shape
(time,)
, antenna_positions
numpy.ndarray
Antenna positions of shape
(ant, 3)
in metres in the ITRF frame. field_centre
numpy.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 onpythoncasacore
. This backend should be used by default.test
creates parallactic angles by multiplying thetimes
andantenna_position
arrays. It exist solely for testing.
 times
 Returns
 parallactic_angles
numpy.ndarray
Parallactic angles of shape
(time,ant)
 parallactic_angles
 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_angles
numpy.ndarray
floating point parallactic angles. Of shape
(pa0, pa1, ..., pan)
. feed_type{‘linear’, ‘circular’}
The type of feed
 parallactic_angles
 Returns
 feed_matrix
numpy.ndarray
Feed rotation matrix of shape
(pa0, pa1,...,pan,2,2)
 feed_matrix
 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 theparallactic_angles
Adding
pointing_errors
Scaling by
antenna_scaling
 Parameters
 lm
numpy.ndarray
LM coordinates of shape
(src,2)
in radians offset from the phase centre. parallactic_angles
numpy.ndarray
parallactic angles of shape
(time, antenna)
in radians. pointing_errors
numpy.ndarray
LM pointing errors for each antenna at each timestep in radians. Has shape
(time, antenna, 2)
 antenna_scaling
numpy.ndarray
antenna scaling factor for each channel and each antenna. Has shape
(antenna, chan)
 frequency
numpy.ndarray
frequencies for each channel. Has shape
(chan,)
 dtype
numpy.dtype
, optional Numpy dtype of result array. Should be float32 or float64. Defaults to float64
 lm
 Returns
 coords
numpy.ndarray
coordinates of shape
(3, src, time, antenna, chan)
where each coordinate component represents l, m and frequency, respectively.
 coords
 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
 beam
numpy.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_extents
numpy.ndarray
lm extents of the beam cube of shape
(2, 2)
.[[lower_l, upper_l], [lower_m, upper_m]]
. beam_freq_map
numpy.ndarray
Beam frequency map of shape
(beam_nud,)
. This array is used to define interpolation along the(chan,)
dimension. lm
numpy.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_angles
numpy.ndarray
Parallactic angles of shape
(time, ant)
. point_errors
numpy.ndarray
Pointing errors of shape
(time, ant, chan, 2)
. antenna_scaling
numpy.ndarray
Antenna scaling factors of shape
(ant, chan, 2)
 frequency
numpy.ndarray
Frequencies of shape
(chan,)
.
 beam
 Returns
 ddes
numpy.ndarray
Direction Dependent Effects of shape
(source, time, ant, chan, corr, corr)
 ddes
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 indexesnoll_index
at the specified coordinatescoords
.Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.
 Parameters
 coords
numpy.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. coeffs
numpy.ndarray
complex Zernicke polynomial coefficients. Has shape
(ant, chan, corr_1, ..., corr_n, poly)
wherepoly
is the number of polynomial coefficients andcorr_1, ..., corr_n
are a variable number of correlation dimensions. noll_index
numpy.ndarray
Noll index associated with each polynomial coefficient. Has shape
(ant, chan, corr_1, ..., corr_n, poly)
. correlation dimensions. parallactic_angles
numpy.ndarray
Parallactic angle rotation. Has shape
(time, ant)
. frequency_scaling
numpy.ndarray
The scaling of frequency of the beam. Has shape
(chan,)
. antenna_scaling
numpy.ndarray
The antenna scaling. Has shape
(ant, chan, 2)
. pointing_errors
numpy.ndarray
The pointing error. Has shape
(time, ant, chan, 2)
.
 coords
 Returns
 dde
numpy.ndarray
complex values with shape
(source, time, ant, chan, corr_1, ..., corr_n)
 dde
 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
 uvw
numpy.ndarray
UVW coordinates of shape
(row, 3)
 lm
numpy.ndarray
Source LM coordinates of shape
(source, 2)
, in radians. Derived from theRa
andDec
fields. source_type
numpy.ndarray
Strings defining the source type of shape
(source,)
. Should be either"POINT"
or"GAUSSIAN"
. Contains theType
field. flux
numpy.ndarray
Source flux of shape
(source,)
. Contains theI
field. coeffs
numpy.ndarray
Source Polynomial coefficients of shape
(source, coeffs)
. Contains theSpectralIndex
field. log_poly
numpy.ndarray
Source polynomial type of shape
(source,)
. If True, logarithmic polynomials are used. If False, standard polynomials are used. Contains theLogarithmicSI
field. ref_freq
numpy.ndarray
Source Reference frequency of shape
(source,)
. Contains theReferenceFrequency
field. gauss_shape
numpy.ndarray
Gaussian shape parameters of shape
(source, 3)
used when the correspondingsource_type
is"GAUSSIAN"
. The 3 components should contain theMajorAxis
,MinorAxis
andOrientation
fields in radians, respectively. frequency
numpy.ndarray
Frequency of shape
(chan,)
.
 uvw
 Returns
 visibilities
numpy.ndarray
Complex visibilities of shape
(row, chan, 1)
 visibilities
Cuda¶

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

Computes the phase delay (K) term: 

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

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 DirectionDependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (persource).
\(G_{{p}}\) represents DirectionIndependent 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_index
cupy.ndarray
Time index used to look up the antenna Jones index for a particular baseline with shape
(row,)
. Obtainable viacp.unique(time, return_inverse=True)[1]
. antenna1
cupy.ndarray
Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. antenna2
cupy.ndarray
Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. dde1_jones
cupy.ndarray
, optional \(E_{ps}\) DirectionDependent Jones terms for the first antenna. shape
(source,time,ant,chan,corr_1,corr_2)
 source_coh
cupy.ndarray
, optional \(X_{pqs}\) DirectionDependent Coherency matrix for the baseline. with shape
(source,row,chan,corr_1,corr_2)
 dde2_jones
cupy.ndarray
, optional \(E_{qs}\) DirectionDependent 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_jones
cupy.ndarray
, optional \(G_{ps}\) DirectionIndependent Jones terms for the first antenna of the baseline. with shape
(time,ant,chan,corr_1,corr_2)
 base_vis
cupy.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_jones
cupy.ndarray
, optional \(G_{ps}\) DirectionIndependent 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)
 time_index
 Returns
 visibilities
cupy.ndarray
Model visibilities of shape
(row,chan,corr_1,corr_2)
 visibilities
Notes
DirectionDependent 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
andant
(antenna) dimensions.Each
row
is associated with a pair of antenna Jones matrices at a particular timestep via thetime_index
,antenna1
andantenna2
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
 lm
cupy.ndarray
LM coordinates of shape
(source, 2)
with L and M components in the last dimension. uvw
cupy.ndarray
UVW coordinates of shape
(row, 3)
with U, V and W components in the last dimension. frequency
cupy.ndarray
frequencies of shape
(chan,)
 convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
.
 lm
 Returns
 complex_phase
cupy.ndarray
complex of shape
(source, row, chan)
 complex_phase
Notes
Corresponds to the complex exponential of the Van CittertZernike 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_angles
cupy.ndarray
floating point parallactic angles. Of shape
(pa0, pa1, ..., pan)
. feed_type{‘linear’, ‘circular’}
The type of feed
 parallactic_angles
 Returns
 feed_matrix
cupy.ndarray
Feed rotation matrix of shape
(pa0, pa1,...,pan,2,2)
 feed_matrix
 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
 beam
cupy.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_extents
cupy.ndarray
lm extents of the beam cube of shape
(2, 2)
.[[lower_l, upper_l], [lower_m, upper_m]]
. beam_freq_map
cupy.ndarray
Beam frequency map of shape
(beam_nud,)
. This array is used to define interpolation along the(chan,)
dimension. lm
cupy.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_angles
cupy.ndarray
Parallactic angles of shape
(time, ant)
. point_errors
cupy.ndarray
Pointing errors of shape
(time, ant, chan, 2)
. antenna_scaling
cupy.ndarray
Antenna scaling factors of shape
(ant, chan, 2)
 frequency
cupy.ndarray
Frequencies of shape
(chan,)
.
 beam
 Returns
 ddes
cupy.ndarray
Direction Dependent Effects of shape
(source, time, ant, chan, corr, corr)
 ddes
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¶

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

Computes the phase delay (K) term: 

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

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

Creates beam sampling coordinates suitable for use in 

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

Computes Direction Dependent Effects by evaluating Zernicke Polynomials defined by coefficients 

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 DirectionDependent Jones terms.
\(X_{{pqs}}\) represents a coherency matrix (persource).
\(G_{{p}}\) represents DirectionIndependent 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_index
dask.array.Array
Time index used to look up the antenna Jones index for a particular baseline with shape
(row,)
. Obtainable viatime.map_blocks(lambda a: np.unique(a, return_inverse=True)[1])
. antenna1
dask.array.Array
Antenna 1 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. antenna2
dask.array.Array
Antenna 2 index used to look up the antenna Jones for a particular baseline. with shape
(row,)
. dde1_jones
dask.array.Array
, optional \(E_{ps}\) DirectionDependent Jones terms for the first antenna. shape
(source,time,ant,chan,corr_1,corr_2)
 source_coh
dask.array.Array
, optional \(X_{pqs}\) DirectionDependent Coherency matrix for the baseline. with shape
(source,row,chan,corr_1,corr_2)
 dde2_jones
dask.array.Array
, optional \(E_{qs}\) DirectionDependent 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_jones
dask.array.Array
, optional \(G_{ps}\) DirectionIndependent Jones terms for the first antenna of the baseline. with shape
(time,ant,chan,corr_1,corr_2)
 base_vis
dask.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_jones
dask.array.Array
, optional \(G_{ps}\) DirectionIndependent 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. IfFalse
, dask uses a tree style reduction algorithm.
 time_index
 Returns
 visibilities
dask.array.Array
Model visibilities of shape
(row,chan,corr_1,corr_2)
 visibilities
Notes
DirectionDependent 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
andant
(antenna) dimensions.Each
row
is associated with a pair of antenna Jones matrices at a particular timestep via thetime_index
,antenna1
andantenna2
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 eachrow
can contain any antenna, random access must be preserved along this dimension.The chunks in the
row
andtime
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]
indexing10
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
andtime
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:
The number chunks in
row
andtime
must match although the individual chunk sizes need not.Unique timesteps should not be split across row chunks.
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)
Use
aggregate_chunks()
to aggregate multiplerow
andtime
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
 lm
dask.array.Array
LM coordinates of shape
(source, 2)
with L and M components in the last dimension. uvw
dask.array.Array
UVW coordinates of shape
(row, 3)
with U, V and W components in the last dimension. frequency
dask.array.Array
frequencies of shape
(chan,)
 convention{‘fourier’, ‘casa’}
Uses the \(e^{2 \pi \mathit{i}}\) sign convention if
fourier
and \(e^{2 \pi \mathit{i}}\) ifcasa
.
 lm
 Returns
 complex_phase
dask.array.Array
complex of shape
(source, row, chan)
 complex_phase
Notes
Corresponds to the complex exponential of the Van CittertZernike 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
 times
dask.array.Array
Array of Mean Julian Date times in seconds with shape
(time,)
, antenna_positions
dask.array.Array
Antenna positions of shape
(ant, 3)
in metres in the ITRF frame. field_centre
dask.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 onpythoncasacore
. This backend should be used by default.test
creates parallactic angles by multiplying thetimes
andantenna_position
arrays. It exist solely for testing.
 times
 Returns
 parallactic_angles
dask.array.Array
Parallactic angles of shape
(time,ant)
 parallactic_angles
 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_angles
numpy.ndarray
floating point parallactic angles. Of shape
(pa0, pa1, ..., pan)
. feed_type{‘linear’, ‘circular’}
The type of feed
 parallactic_angles
 Returns
 feed_matrix
numpy.ndarray
Feed rotation matrix of shape
(pa0, pa1,...,pan,2,2)
 feed_matrix
 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 theparallactic_angles
Adding
pointing_errors
Scaling by
antenna_scaling
 Parameters
 lm
dask.array.Array
LM coordinates of shape
(src,2)
in radians offset from the phase centre. parallactic_angles
dask.array.Array
parallactic angles of shape
(time, antenna)
in radians. pointing_errors
dask.array.Array
LM pointing errors for each antenna at each timestep in radians. Has shape
(time, antenna, 2)
 antenna_scaling
dask.array.Array
antenna scaling factor for each channel and each antenna. Has shape
(antenna, chan)
 frequency
dask.array.Array
frequencies for each channel. Has shape
(chan,)
 dtype
numpy.dtype
, optional Numpy dtype of result array. Should be float32 or float64. Defaults to float64
 lm
 Returns
 coords
dask.array.Array
coordinates of shape
(3, src, time, antenna, chan)
where each coordinate component represents l, m and frequency, respectively.
 coords
 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
 beam
dask.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_extents
dask.array.Array
lm extents of the beam cube of shape
(2, 2)
.[[lower_l, upper_l], [lower_m, upper_m]]
. beam_freq_map
dask.array.Array
Beam frequency map of shape
(beam_nud,)
. This array is used to define interpolation along the(chan,)
dimension. lm
dask.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_angles
dask.array.Array
Parallactic angles of shape
(time, ant)
. point_errors
dask.array.Array
Pointing errors of shape
(time, ant, chan, 2)
. antenna_scaling
dask.array.Array
Antenna scaling factors of shape
(ant, chan, 2)
 frequency
dask.array.Array
Frequencies of shape
(chan,)
.
 beam
 Returns
 ddes
dask.array.Array
Direction Dependent Effects of shape
(source, time, ant, chan, corr, corr)
 ddes
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 indexesnoll_index
at the specified coordinatescoords
.Decomposition of a voxel beam cube into Zernicke polynomial coefficients can be achieved through the use of the eidos package.
 Parameters
 coords
dask.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. coeffs
dask.array.Array
complex Zernicke polynomial coefficients. Has shape
(ant, chan, corr_1, ..., corr_n, poly)
wherepoly
is the number of polynomial coefficients andcorr_1, ..., corr_n
are a variable number of correlation dimensions. noll_index
dask.array.Array
Noll index associated with each polynomial coefficient. Has shape
(ant, chan, corr_1, ..., corr_n, poly)
. correlation dimensions. parallactic_angles
dask.array.Array
Parallactic angle rotation. Has shape
(time, ant)
. frequency_scaling
dask.array.Array
The scaling of frequency of the beam. Has shape
(chan,)
. antenna_scaling
dask.array.Array
The antenna scaling. Has shape
(ant, chan, 2)
. pointing_errors
dask.array.Array
The pointing error. Has shape
(time, ant, chan, 2)
.
 coords
 Returns
 dde
dask.array.Array
complex values with shape
(source, time, ant, chan, corr_1, ..., corr_n)
 dde
 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
 uvw
dask.array.Array
UVW coordinates of shape
(row, 3)
 lm
dask.array.Array
Source LM coordinates of shape
(source, 2)
, in radians. Derived from theRa
andDec
fields. source_type
dask.array.Array
Strings defining the source type of shape
(source,)
. Should be either"POINT"
or"GAUSSIAN"
. Contains theType
field. flux
dask.array.Array
Source flux of shape
(source,)
. Contains theI
field. coeffs
dask.array.Array
Source Polynomial coefficients of shape
(source, coeffs)
. Contains theSpectralIndex
field. log_poly
dask.array.Array
Source polynomial type of shape
(source,)
. If True, logarithmic polynomials are used. If False, standard polynomials are used. Contains theLogarithmicSI
field. ref_freq
dask.array.Array
Source Reference frequency of shape
(source,)
. Contains theReferenceFrequency
field. gauss_shape
dask.array.Array
Gaussian shape parameters of shape
(source, 3)
used when the correspondingsource_type
is"GAUSSIAN"
. The 3 components should contain theMajorAxis
,MinorAxis
andOrientation
fields in radians, respectively. frequency
dask.array.Array
Frequency of shape
(chan,)
.
 uvw
 Returns
 visibilities
dask.array.Array
Complex visibilities of shape
(row, chan, 1)
 visibilities