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# Description

Invert and deconvolve images with selected algorithm

NOTE: clean is no longer being actively developed; it has been refactored, and the new version of the task is called tclean. Much more information on the context for clean/tclean can be found in the Sythesis Imaging chapter.

## Basics of clean

The CLEAN algorithm (more details available here) is the most popular and widely-studied method for reconstructing a model image based on interferometer data. It iteratively removes at each step a fraction of the flux in the brightest pixel in a defined region of the current “dirty” image, and places this in the model image. The clean task implements the CLEAN algorithm for single-field data. The user can choose from a number of options for the particular flavor of CLEAN to use. Often, the first step in imaging is to make a simple gridded Fourier inversion of the calibrated data to make a “dirty” image. This can then be examined to look for the presence of noticeable emission above the noise, and to assess the quality of the calibration by searching for artifacts in the image. This is done using clean with niter=0.

ALERT: For large fractional bandwidths, the psf in clean may vary considerably with frequency in data cubes. To accommodate this fact we have introduced a per-plane psf (dirty beam) when the change is larger than half the size of a pixel. Analysis tasks in CASA can deal with such beam variation. If a single beam size is requested, imsmooth can be invoked on the clean products to smooth to a common, uniform beam for all channels.

The clean task has many options:

• Make 'dirty' image and 'dirty' beam (PSF)
• Multi-frequency-continuum images or spectral channel imaging
• Full Stokes imaging
• Mosaicking of several pointings
• Multi-scale cleaning
• Widefield cleaning
• Interactive clean boxing
• Use starting model (e.g. from single-dish)

## imagermode parameter

CASA supports several methods for data deconvolution and imaging. These methods can be set using the parameter imagermode, which chooses the mode of operation of clean, either as single-field deconvolution using image-plane major and minor cycles only (imagermode=''), single-field deconvolution using Cotton-Schwab (CS) residual visibilities for major cycles (imagermode='csclean'), or multi-field mosaics using CS major cycles (imagermode='mosaic').

The default imagermode='csclean' choice specifies the Cotton-Schwab algorithm (more details available here). This opens up the following sub-parameters:

imagermode          =  'csclean'   #  Options: 'csclean' or 'mosaic'; '', uses psfmode
cyclefactor    =        1.5   #  Controls how often major cycles are done. (e.g. 5 for frequently)
cyclespeedup   =         -1   #  Cycle threshold doubles in this number of iterations

In the CS mode, cleaning is split into minor and major cycles. For each field, a minor cycle is performed using the PSF algorithm specified by the psfmode parameter. At major-cycle breakpoints, the points thus found are subtracted from the original visibilities. A fast variant does a convolution using a FFT (Fast Fourier transform). This will be faster for large numbers of visibilities. If you want to be extra careful, double the image size from that used for the Clark clean and set a mask to clean only the inner quarter or less (this is not done by default). This is probably the best choice for high-fidelity deconvolution of images without lots of large-scale structure.

NOTE: When using the Cotton-Schwab algorithm with a threshold, there may be strange behavior when you hit the threshold with a major cycle. In particular, it may be above threshold again at the start of the next major cycle. This is particularly noticeable when cleaning a cube, where different channels will hit the threshold at different times.

In the empty mode (imagermode=''), the major and minor clean cycles work off of the gridded FFT dirty image, with residuals updated using the PSF calculation algorithm set by the psfmode parameter. This method is not recommended for high dynamic range or high fidelity imaging applications, but can be significantly faster than CS clean (the default).

NOTE: For this option only, if mask='' (no mask or box set) then it will clean the inner quarter of the image by default.

ALERT: You will see a warning message in the logger, similar to this:

Zero Pixels selected with a Flux limit of 0.000551377 and a maximum Residual of 0.00751239

whenever it finds 0 pixels above the threshold. This is normal, and not a problem if you’ve specified a non-zero threshold. On the other hand, if you get this warning with the threshold set to the default of '0Jy', then you should look carefully at your inputs or your data, since this usually means that the masking is bad.

The option imagermode='mosaic' is for multi-field mosaics. This choice opens up the following sub-parameters:

imagermode          =   'mosaic'   #  Use csclean or mosaic.  If ’’, use psfmode
mosweight      =      False   #  Individually weight the fields of the mosaic
ftmachine      =   'mosaic'   #  Gridding method for the image
scaletype      =    'SAULT'   #  Controls scaling of pixels in the image plane.
cyclefactor    =        1.5   #  change depth in between of  csclean cycle
cyclespeedup   =         -1   #  Cycle threshold doubles in this number of iterations

#### psfmode parameter

The psfmode parameter chooses the “algorithm” that will be used to calculate the synthesized beam for use during the minor cycles in the image plane. There are 3 choices: 'clark' (default), 'hogbom', and 'clarkstokes'.

In the 'clark' algorithm, the cleaning is split into minor and major cycles. In the minor cycles only the brightest points are cleaned, using a subset of the point spread function. In the major cycle, the points thus found are subtracted correctly by using an FFT-based convolution. This algorithm is reasonably fast. Also, for polarization imaging, Clark searches for the peak in

$I^2 + Q^2 + U^2 + V^2$.

The 'hogbom' algorithm is the “Classic” image-plane CLEAN, where model pixels are found iteratively by searching for the peak. Each point is subtracted from the full residual image using the shifted and scaled point spread function. In general, this is not a good choice for most imaging problems (clark or csclean are preferred) as it does not calculate the residuals accurately. But in some cases, with poor uv-coverage and/or a PSF with bad sidelobes, the Hogbom algorithm will do better as it uses a smaller beam patch. For polarization cleaning, Hogbom searches for clean peak in I, Q, U, and V independently.

In the 'clarkstokes' algorithm, the Clark psf is used, but for polarization imaging the Stokes planes are cleaned sequentially for components instead of jointly as in 'clark'. This means that this is the same as 'clark' for Stokes I imaging only. This option can also be combined with imagermode='csclean'.

## Data weighting

Data weighting during imaging allows for the improvement of the dynamic range and the ability to adjust the synthesized beam associated with the produced image. The weight given to each visibility sample can be adjusted to fit the desired output. There are several reasons to adjust the weighting, including improving sensitivity to extended sources or accounting for noise variation between samples.The user can adjust the weighting using clean and changing the weighting parameter with six options: 'natural', 'uniform', 'briggs',  'superuniform', 'briggsabs', and 'radial'.

#### Natural weighting

For weighting='natural', visibilities are weighted only by the data weights, which are calculated during filling and calibration and should be equal to the inverse noise variance on that visibility. Imaging weight $w_i$ of sample $\dot\imath$ is given by:

$w_i = \omega_i = \frac{1}{{\sigma_i}^2}$

where the data weight $\omega_i$ is determined from $\sigma_i$, the rms noise on visibility $\dot\imath$. When data is gridded into the same uv-cell for imaging, the weights are summed, and thus a higher uv density results in higher imaging weights. No sub-parameters are linked to this mode choice. It is the default imaging weight mode, and it should produce “optimum” image with with the lowest noise (highest signal-to-noise ratio).

NOTE: This generally produces images with the poorest angular resolution, since the density of visibilities falls radially in the uv-plane.

#### Uniform weighting

For weighting='uniform', the data weights are calculated as in 'natural' weighting. The data is then gridded to a number of cells in the uv-plane, and after all data is gridded the uv-cells are re-weighted to have “uniform” imaging weights. This pumps up the influence on the image of data with low weights (they are multiplied up to be the same as for the highest weighted data), which sharpens resolution and reduces the sidelobe level in the field-of-view, but increases the rms image noise. No sub-parameters are linked to this mode choice.

For uniform weighting, we first grid the inverse variance $\omega_i$ for all selected data onto a grid with uv cell-size given by 2 ∕ FOV, where FOV is the specified field of view (defaults to the image field of view). This forms the gridded weights $W_k$. The weight of the $\dot\imath$-th sample is then:

$w_i = \frac{w_i}{W_k}$

#### Briggs weighting

The weighting='briggs' mode is an implementation of the flexible weighting scheme developed by Dan Briggs in his PhD thesis, which can be viewed here.

This choice brings up the sub-parameters:

weighting      =   'briggs'   #   Weighting to apply to visibilities
robust    =        0.0   #   Briggs robustness parameter
npixels   =          0   #   number of pixels to determine uv-cell size 0=> field of view

The actual weighting scheme used is:

$w_i = \frac{\omega_i}{1 + W_k f^2}$

where $W_k$ is defined as in 'uniform' and 'superuniform' weighting, and

$f^2 = \frac{(5 \times 10^{-\text{R}})^2}{\frac{\Sigma_k W_k^2}{\Sigma_i \omega_i}}$

and R is the robust sub-parameter.

The key parameter is the robust sub-parameter, which sets R in the Briggs equations. The scaling of R is such that robust=0 gives a good trade-off between resolution and sensitivity. The robust R takes value between -2.0 (close to uniform weighting) to 2.0 (close to natural).

Superuniform weighting can be combined with Briggs weighting using the npixels sub-parameter. This works as in ’superuniform’ weighting.

#### Superuniform weighting

The weighting='superuniform' mode is similar to the 'uniform' weighting mode but there is now an additional npixels sub-parameter that specifies a change to the number of cells on a side (with respect to uniform weighting) to define a uv-plane patch for the weighting renormalization. If npixels=0, you get uniform weighting.

#### Briggsabs weighting

For weighting='briggsabs', a slightly different Briggs weighting is used, with:

$w_i = \frac{\omega_i}{W_k \text{R}^2 + 2\sigma_\text{R}^2}$

where R is the robust parameter and $\sigma_\text{R}$ is the noise parameter.

This choice brings up the sub-parameters:

weighting      = 'briggsabs'  #   Weighting to apply to visibilities
robust    =      0.0     #   Briggs robustness parameter
noise     =  '0.0Jy'     #   noise parameter for briggs weighting when rmode='abs'
npixels   =        0     #   number of pixels to determine uv-cell size 0=> field of view

Otherwise, this works as weighting='briggs' above.

The weighting='radial' mode is a seldom-used option that increases the weight by the radius in the uv-plane, i.e.:

$w_i = \omega_i \times \sqrt{u_i^2 + v_i^2}$

Technically, this would be called an inverse uv-taper, since it depends on uv-coordinates and not on the data per-se. Its effect is to reduce the rms sidelobes for an east-west synthesis array. This option has limited utility.

## Output images with parameter imagename

The value of the imagename parameter is used as the root name of the output image. Depending on the particular task and the options chosen, one or more images with names built from that root will be created. For example, the clean task run with imagename='ngc5921' a series of output images will be created with the names ngc5921.clean, ngc5921.residual, ngc5921.model, etc. If an image with that name already exists, it will in general be overwritten. Beware using names of existing images however. If the clean is run using an imagename where <imagename>.residual and <imagename>.model already exist  then clean will continue starting from these (effectively restarting from the end of the previous clean). Thus, if multiple runs of clean are run consecutively with the same imagename, then the cleaning is incremental (as in the difmap package).

The output image may also have a different beam per plane. For datasets with very large fractional bandwidth, clean will use a different PSF for each channel when the PSF changes by more than half a pixel as a function of frequency. To smooth to a common resolution, one can either use the parameter resmooth to smooth to the smallest common possible beam, restoringbeam for an arbitrary, larger beam, or the task imsmooth after cleaning. Data analysis tasks such as immoments in CASA support changing beams per plane.

There is some differences between the output images based on the algorithm used during a clean. The following is a list of differences between MS-MFS (nterms>1) and standard imaging, in the current CASA release:

1. Iterations always proceed as cs-clean major/minor cycles, and uses the full psf during minor cycle iterations. There are currently no user-controls on the cyclespeedup, and the flux-limit per major cycle is chosen as 10% of the peak residual. In future releases, this will be made more adaptive/controllable.
2. Currently, the following options are not supported for nterms>1: psfmode, pbcorr, minpb, imagermode='mosaic', gridmode='aprojection', cyclespeedup, and allowed are one of Stokes I, Q, U, V, RR, LL, XX, YY at a time. More options and combinations are currently under development and testing. Under 'Using CASA'→'Other Documentation'→'Imaging Algorithms in CASA' you can find the latest implementations.

## Mosaic imaging

The clean task contains the capability to image multiple pointing centers together into a single “mosaic” image. This ability is controlled by setting imagermode='mosaic'. The key parameter that controls how clean produces the mosaic is the ftmachine sub-parameter. For ftmachine='ft', clean will perform a weighted combination of the images produced by transforming each mosaic pointing separately. This can be slow, as the individual sub-images must be recombined in the image plane.

NOTE: This option is preferred for data taken with sub-optimal mosaic sampling (e.g. fields too far apart, on a sparse irregular pattern, etc.)

If ftmachine='mosaic', then the data are gridded onto a single uv-plane which is then transformed to produce the single output image. This is accomplished by using a gridding kernel that approximates the  transform of the primary beam pattern. Note that for this mode the <imagename>.flux image includes this convolution kernel in its effective weighted response pattern (needed to “primary-beam correct” the output image). For this mode only, an additional image <imagename>.flux.pbcoverage is produced that is the primary-beam coverage only used to compute the minpb cutoff.

The flatnoise parameter determines whether the minor cycle performs on the the residual with or without a primary beam correction. Whereas the former has the correct fluxes, the latter has a uniform noise, which allows for a simpler deconvolution in particular at the the edges of the mosaic where the primary beam correction is largest.

ALERT: In order to avoid aliasing artifacts for ftmachine='mosaic' in the mosaic image, due to the discrete sampling of the mosaic pattern on the sky, you should make an image in which the desired unmasked part of the image (above minpb) lies within the inner quarter. In other words, make an image twice as big as necessary to encompass the mosaic.

It is also important to choose an appropriate phasecenter for your output mosaic image. The phase center should not be at the edge of an image with pointings around it. In that case, FFT aliasing may creep into the image.

#### Mosaic threshold parameter

For mosaics, the specification of the threshold is not straightforward, as it is in the single field case. This is because the different fields can be observed to different depths, and get different weights in the mosaic. We now provide internal rescaling (based on scaletype) so clean does its component search on a properly weighted and scaled version of the sky. For ftmachine='ft', the minor cycles of the deconvolution are performed on an image that has been weighted to have constant noise, as in 'SAULT' weighting. This is equivalent to making a dirty mosaic by coadding dirty images made from the individual pointings with a sum of the mosaic contributions to a given pixel weighted by so as to give constant noise across the image. This means that the flux scale can vary across the mosaic depending on the effective noise (higher weighted regions have lower noise, and thus will have higher “fluxes” in the 'SAULT' map). Effectively, the flux scale that threshold applies to is that at the center of the highest-weighted mosaic field, with higher-noise regions down-scaled accordingly. Compared to the true sky, this image has a factor of the PB, plus a scaling map (returned in the .flux image). You will preferentially find components in the low-noise regions near mosaic centers. When ftmachine='mosaic' and scaletype='SAULT', the deconvolution is also performed on a “constant noise image”, as detailed above for 'ft'.

ALERT: The intrinsic image made using ftmachine='mosaic' is equivalent to a dirty mosaic that is formed by coadding dirty images made from the individual fields after apodizing each by the PB function. Thus compared to the true sky, this has a factor of the PB 2 in it. You would thus preferentially find components in the centers of the mosaic fields (even more so than in the 'ft' mosaics). We now rescale this image internally at major-cycle (and interactive) boundaries based on scaletype, and do not have a way to clean on the raw unscaled dirty image (as was done in previous released versions).

## Multi-scale cleaning

The CASA multi-scale algorithm uses “Multi-scale CLEAN” to deconvolve using delta-functions and circular Gaussians as the basis functions for the model, instead of just delta-functions or pixels as in the other clean algorithms. This algorithm is still in the experimental stage, mostly because we are working on better algorithms for setting the scales for the Gaussians. The sizes of the Gaussians are set using the scales sub-parameter.

Multi-scale cleaning is also not as sensitive to the loop gain as regular cleaning algorithms. A loop gain of 0.3 may still work fine and will considerably speed up the processing time. Increasing the cyclefactor by a few may provide better stability in the solution, in particular when the data exhibit a severely non-Gaussian dirty beam.

Inside the Toolkit: The im.setscales method sets the multi-scale Gaussian widths. In addition to choosing a list of sizes in pixels, you can just pick a number of scales and get a geometric series of sizes.

To activate multi-scale mode, specify a non-blank list of scales in the multiscale parameter. A good rule of thumb for starters is [ 0, 2xbeam, 5xbeam ], and maybe adding larger scales up to the maximum scale the interferometer can image. E.g. for a 2 arcsecond beam:

multiscale = [0,6,10,30] # Four scales including point sources

These are given in numbers of pixels, and specify FWHM of the Gaussians used to compute the filtered images. Setting the multiscale parameter to a non-empty list opens up the sub-parameter:

multiscale = [0, 6, 10, 30]  # set deconvolution scales (pixels)
negcomponent = -1       # Stop cleaning if the
# largest scale finds this number of neg
# components
smallscalebias = 0.6    # a bias to give more weight
# toward smaller scales

The negcomponent sub-parameter is here to set the point at which the clean terminates because of negative components. For negcomponent > 0, component search will cease when this number of negative  components are found at the largest scale. If negcomponent = -1, then component search will continue even if the largest component is negative. Increasing smallscalebias gives more weight to small scales. A value of 1.0 weighs the largest scale to zero and a value < 0.2 weighs all scales nearly equally. The default of 0.6 is usually a good number as it corresponds to a weighting that approximates the normalization of each component by its area. Depending on the image, however, it may be necessary to tweak the smallscalebias for a better convergence of the algorithm.

NOTE: Currently smallscalebias is ignored by the MS-MFS algorithm. It will be available in a future release.

#### MS-MFS Algorithm

The MS-MFS (multiscale-multifrequency synthesis) algorithm combines the concepts of multi-scale and multi-frequency synthesis cleaning for wideband synthesis imaging. Setting the mode='mfs' sub-parameter nterms>1 runs the MS-MFS algorithm, and the choice of nterms should depend on the expected shape and SNR of the spectral structure, across the chosen bandwidth. The MS-MFS algorithm requires the multiscale parameter to be set. For point-source deconvolution, set multiscale=[0] (also the default). Output images represent Taylor-coefficients of the sky spectrum (images with file-name extensions of tt0,tt1,etc). A spectral index map is also computed as the ratio of the first two terms, following this convention:

$I(\nu) = I(ref_\nu) \times (\nu/\nu_0)^\alpha$

NOTE: Unlike standard multi-scale cleaning (multiscale= [0,6,10,....] with nterms=1), with higher nterms the largest specified scale size must lie within the sampled range of the interferometer. If not, there can be an ambiguity in the spectral reconstruction at very large spatial scales.

Additionally, a spectral-index error image is made by treating Taylor-coefficient residuals as errors, and propagating them through the division used to compute spectral-index. It is meant to be a guide to which parts of the spectral-index image to trust, and the values may not always represent a statistically-correct error. For more details about this algorithm, please refer to the paper titled "A multi-scale multi-frequency deconvolution algorithm for synthesis imaging in radio interferometry" [1] .

NOTE: The software implementation of the MS-MFS algorithm for nterms>1 currently does not allow combination with mosaics and pbcor.

## Polarization Imaging

The stokes parameter specifies the Stokes parameters for the resulting images, with standard imaging only using the stokes='I' for the total intensity measurement.

NOTE: Forming Stokes Q and U images requires the presence of cross-hand polarizations (e.g. RL and LR for circularly polarized systems such as the VLA) in the data. Stokes V requires both parallel hands (RR and :LL) for circularly polarized systems or the cross-hands (XY and YX) for linearly polarized systems such as ALMA and ATCA.

This parameter is specified as a string of up to four letters and can indicate stokes parameters themselves, Right/Left hand polarization products, or linear polarization products (X/Y). For example,

stokes = 'I' # Intensity only
stokes = 'IQU' # Intensity and linear polarization
stokes = 'IV' # Intensity and circular polarization
stokes = 'IQUV' # All Stokes imaging
stokes = 'RR' # Right hand polarization only
stokes = 'XXYY' # Both linear polarizations

are common choices (see the inline help of clean for a full range of possible options). The output image will have planes (along the “polarization axis”) corresponding to the chosen Stokes parameters. If as input to deconvolution tasks such as clean, the stokes parameter includes polarization planes other than I, then choosing psfmode='hogbom' or psfmode='clarkstokes' will clean (search for components) each plane sequentially, while psfmode='clark' will deconvolve jointly.

ALERT: As of Release 3.2, clean expects that all input polarizations are present. E.g. if you have RR and LL dual polarization data and you flagged parts of RR but not LL, clean will ignore both polarizations in slice. It is possible to split out a polarization product with split and image separately. But you will not be able to combine these part-flagged data in the uv-domain. We will remove that restriction in a future CASA release.

## Hints on clean with flanking fields

There are two ways of specifying multi-field images for clean: (a) the task parameters are used to define the first (main) field and a text file containing definitions of all additional fields is supplied to the outlierfile task parameter, or (b) all fields are specified as lists for each task parameter.

For the first example, the outlier file must contain the following parameters per field: imagename, imsize, and phasecenter. Optional parameters include mask and modelimage. The parameter set for each field must begin with imagename. Parameters can be listed in a single line or span multiple lines. The task inputs are:

imagename = 'M1_0'
outlierfile='outlier.txt'
imsize = [1024,1024]
phasecenter = 'J2000 13h27m20.98 43d26m28.0'

The contents of outlier file 'outlier.txt' are:

imagename = 'M1_1'imsize = [128,128]phasecenter = 'J2000 13h30m52.159 43d23m08.02'mask = ['out1.mask', 'circle[[40pix,40pix],5pix]' ]modelimage = 'out1.model'imagename = 'M1_2'imsize = [128,128]phasecenter = 'J2000 13h24m08.16 43d09m48.0'

In this example, the first field 'M1_0' is defined using main task parameters. The next two 'M1_1' and 'M1_2' are listed in the file 'outlier.txt'.  A mask and modelimage has been supplied only for the second field (M1_1). Fields with unspecified masks will use the full field for cleaning.

For the second example, the inputs are instead included in the main parameters, using brackets to signify multiple inputs. Parameters that support lists for multi-field specification are imagename, imsize, phasecenter, mask, and modelimage. The task inputs are:

imagename = ['M1_0','M1_1','M1_2]
imsize = [[1024,1024],[128,128],[128,128]]
phasecenter = ['J2000 13h27m20.98 43d26m28.0',
'J2000 13h30m52.159 43d23m08.02',
'J2000 13h24m08.16 43d09m48.0']
modelimage=[[''],['out1.model'],['']]

NOTE: All lists must have the same length.

In both examples, the following images will be made:

• M1_0.image, M1_1.image, M1_2.image (cleaned images)
• M1.0.model, M1_1.model, M1_2.model (model images)
• M1.0.residual, M1_1.residual, M1_2.residual (residual images)

NOTE: The old AIPS-style outlier-file and boxfile formats have been deprecated. However, due to user-requests, they will continue be supported in CASA 3.4. Note that the old outlier file format does not support the specification of modelimage and mask for each field. The new format is more complete, and less ambiguous, so please consider updating your scripts.

# Parameters

#### vis

Name(s) of input visibility file(s). default: none; example: vis='ngc5921.ms'; vis=['ngc5921a.ms','ngc5921b.ms']; multiple MSes

#### imagename

Pre-name of output images.

default: none; example: imagename='m2'

Output images are:

• m2.image; cleaned and restored image with or without primary beam correction
• m2.psf; point-spread function (dirty beam)
• m2.flux;  relative sky sensitivity over field
• m2.flux.pbcoverage;  relative pb coverage over field (gets created only for ft='mosaic')
• m2.model; image of clean components
• m2.residual; image of residuals
• m2.interactive.mask; image containing clean regions

To include outlier fields: imagename=['n5921','outlier1','outlier2']

#### outlierfile

Text file name which contains image names, sizes, field centers (See 'HINTS ON CLEAN WITH FLANKING FIELDS' above for the format of this outlier file.)

#### field

Select fields to image or mosaic.  Use field ID(s) or name(s). ['go listobs' to obtain the list id's or names]

default: '' all fields; If field string is a non-negative integer, it is assumed to be a field index otherwise, it is assumed to be a field name
examples: field='0~2'; field IDs 0,1,2
field='0,4,5~7'; field IDs 0,4,5,6,7
field='3C286,3C295'; field named 3C286 and 3C295
field = '3,4C*'; field id 3, all names starting with 4C
For multiple MS input, a list of field strings can be used:
field = ['0~2','0~4']; field IDs 0-2 for the first MS and 0-4 for the second
field = '0~2'; field IDs 0-2 for all input MSes

#### spw

Select spectral window/channels

NOTE:  Channels de-selected here will contain all zeros if selected by the parameter mode subparameters.

default: '' all spectral windows and channels
examples: spw='0~2,4'; spws 0,1,2,4 (all channels)
spw='0:5~61'; spw 0, channels 5 to 61
spw='<2';   spws less than 2 (i.e. 0,1)
spw='0,10,3:3~45'; spw 0,10 all channels, spw 3, channels 3 to 45.
spw='0~2:2~6'; spw 0,1,2 with channels 2 through 6 in each.
For multiple MS input, a list of spw strings can be used:
spw=['0','0~3']; spw ids 0 for the first MS and 0-3 for the second
spw='0~3' spw ids 0-3 for all input MS
spw='3:10~20;50~60' for multiple channel ranges within spw id 3
spw='3:10~20;50~60,4:0~30' for different channel ranges for spw ids 3 and 4
spw='0:0~10,1:20~30,2:1;2;3'; spw 0, channels 0-10, spw 1, channels 20-30, and spw 2, channels, 1,2 and 3
spw='1~4;6:15~48' for channels 15 through 48 for spw ids 1,2,3,4 and 6

#### selectdata

Other data selection parameters
default: True

### selectdata=True expandable parameters (See help par.selectdata for more on these)

#### timerange

Select data based on time range:
default: '' (all)
examples: timerange = 'YYYY/MM/DD/hh:mm:ss~YYYY/MM/DD/hh:mm:ss'

NOTE: If YYYY/MM/DD is missing, date defaults to first day in data set.

timerange='09:14:0~09:54:0' picks 40 min on first day
timerange='25:00:00~27:30:00' picks 1 hr to 3 hr 30min on NEXT day
timerange='09:44:00' pick data within one integration of time
timerange='>10:24:00' data after this time
For multiple MS input, a list of timerange strings can be used:
timerange=['09:14:0~09:54:0','>10:24:00']
timerange='09:14:0~09:54:0'; apply the same timerange for all input MSes

#### uvrange

Select data within uvrange (default units meters)
default: '' (all)
example: uvrange='0~1000klambda'; uvrange from 0-1000 kilo-lambda
uvrange='>4klambda';uvranges greater than 4 kilo lambda
For multiple MS input, a list of uvrange strings can be used:
uvrange=['0~1000klambda','100~1000klamda']
uvrange='0~1000klambda'; apply 0-1000 kilo-lambda for all input MSes

#### antenna

Select data based on antenna/baseline
default: '' (all)
If antenna string is a non-negative integer, it is assumed to be an antenna index, otherwise, it is considered an antenna name.
antenna='5&6'; baseline between antenna index 5 and index 6.
antenna='VA05&VA06'; baseline between VLA antenna 5 and 6.
antenna='5&6;7&8'; baselines 5-6 and 7-8
antenna='5'; all baselines with antenna index 5
antenna='05'; all baselines with antenna number 05 (VLA old name)
antenna='5,6,9'; all baselines with antennas 5,6,9 index number
For multiple MS input, a list of antenna strings can be used:
antenna=['5','5&6'];
antenna='5'; antenna index 5 for all input MSes

#### scan

Scan number range. [Check 'go listobs' to insure the scan numbers are in order.]
default: '' (all)
examples: scan='1~5'
For multiple MS input, a list of scan strings can be used:
scan=['0~100','10~200']
scan='0~100; scan ids 0-100 for all input MSes

#### observation

Observation ID range.
default: '' (all); example: observation='1~5'

#### intent

Scan intent (case sensitive)
default: '' (all); examples: intent='TARGET_SOURCE', intent='TARGET_SOURCE1,TARGET_SOURCE2', intent='TARGET_POINTING*'

#### mode: Frequency Specification

NOTE: Channels deselected with spw parameter will contain all zeros.

default: 'mfs'; examples: mode = 'mfs' means produce one image from all specified data, mode = 'channel' use with nchan, start, width to specify output image cube, mode = 'velocity' channels are specified in velocity, mode = 'frequency', channels are specified in frequency.

### mode='mfs' expandable parameters

Make a continuum image from the selected frequency channels/range using Multi-frequency synthesis algorithm for wide-band narrow field imaging.
examples: spw = '0,1'; mode = 'mfs' will produce one image made from all channels in spw 0 and 1
spw='0:5~28^2'; mode = 'mfs' will produce one image made with channels (5,7,9,...,25,27)

#### nterms

Number of Taylor terms to be used to model the frequency dependence of the sky emission. nterms=1 is equivalent to assuming no frequency dependence. nterms>1 runs the MS-MFS algorithm, and the choice of nterms should depend on the expected shape and SNR of the spectral structure, across the chosen bandwidth. Output images represent taylor-coefficients of the sky spectrum (images with file-name extensions of tt0,tt1,etc). A spectral index map is also computed as the ratio of the first two terms (following the convention of $I(nu) = I(ref_nu) x (nu/nu_0)^\alpha$). Additionally, a spectral-index error image is made by treating taylor-coefficient residuals as errors, and propagating them through the division used to compute spectral-index. It is meant to be a guide to which parts of the spectral-index image to trust, and the values may not always represent a statistically-correct error.

NOTE: The software implementation of the MS-MFS algorithm for nterms>1 currently does not allow combination with mosaics, and pbcor.

#### reffreq

The reference frequency (for nterms>1) about which the Taylor expansion if done.
reffreq='' defaults to the middle frequency of the selected range.

### mode='channel', 'velocity', and 'frequency' expandable parameters

#### nchan

Total number of channels in the output image.
default: -1; Automatically selects enough channels to cover data selected by 'spw' consistent with 'start' and 'width'. It is often easiest to leave nchan at the default value. example: nchan=100.

#### start

First channel, velocity, or frequency.
For mode='channel'; This selects the channel index number from the MS (0 based) that you want to correspond to the first channel of the output cube. The output cube will be in frequency space with the first channel having the frequency of the MS channel selected by startstart=0 refers to the first channel in the first selected spw, even if that channel is de-selected in the spw parameter. Channels de-selected by the spw parameter will be filled with zeros if included by the start parameter. For example, spw=3~8:3~100 and start=2 will produce a cube that starts on the third channel (recall 0 based) of spw index 3, and the first channel will be blank. example: start=5
For mode='velocity' or 'frequency': default=''; starts at first input channel of first input spw; examples: start='5.0km/s' or start='22.3GHz'

#### width

Output channel width
For mode='channel', default=1; >1 indicates channel averaging; example: width=4
For mode= 'velocity' or 'frequency', default=''; width of first input channel, or more precisely, the difference in frequencies between the first two selected channels. For example, if channels 1 and 3 are selected with spw, then the default width will be the difference between their frequencies, and not the width of channel 1. Similarly, if the selected data has uneven channel-spacing, the default width will be picked from the first two selected channels. In this case, please specify the desired width. When specifying the width, one must give units. examples: width='1.0km/s', or width='24.2kHz'. Setting width>0 gives channels of increasing frequency for mode='frequency', and increasing velocity for mode='velocity'.

#### interpolation

Interpolation type for spectral gridding onto the uv-plane. Options: 'nearest', 'linear', or 'cubic'.
default = 'linear'

NOTE: 'linear' and 'cubic' interpolation requires data points on both sides of each image frequency. Errors are therefore possible at edge channels, or near flagged data channels. When image channel width is much larger than the data channel width there is nothing much to be gained using linear or cubic thus not worth the extra computation involved.

#### resmooth

If the cube has a different restoring beam/channel. Restore image to a common beam or leave as is; (default) options: True or False
default = False

#### chaniter

Specify how spectral CLEAN is performed,
default: chaniter=False; example: chaniter=True; step through channels

#### outframe

For mode='velocity', 'frequency', or 'channel': default spectral reference frame of output image; Options: '','LSRK','LSRD','BARY','GEO','TOPO','GALACTO', ''LGROUP','CMB'
default: ''; same as input data; example: frame='bary' for Barycentric frame

#### veltype

For mode='velocity' gives the velocity definition;  Options: 'radio','optical'

NOTE: The viewer always defaults to displaying the 'radio' frame, but that can be changed in the position tracking pull down.

mode='channel' examples:
spw = '0'; mode = 'channel': nchan=3; start=5; width=4 will produce an image with 3 output planes: plane 1 contains data from channels (5+6+7+8), plane 2 contains data from channels (9+10+11+12), plane 3 contains data from channels (13+14+15+16)
spw = '0:0~63^3'; mode='channel'; nchan=21; start = 0; width = 1 will produce an image with 20 output planes: plane 1 contains data from channel 0, plane 2 contains date from channel 2, plane 21 contains data from channel 61
spw = '0:0~40^2'; mode = 'channel'; nchan = 3; start = 5; width = 4 will produce an image with three output planes: plane 1 contains channels (5,7), plane 2 contains channels (13,15), plane 3 contains channels (21,23)

#### psfmode

method of PSF calculation to use during minor cycles:
default: 'clark': Options: 'clark','clarkstokes', 'hogbom'
'clark'  use smaller beam (faster, usually good enough); for stokes images clean components peaks are searched in the I^2+Q^2+U^2+V^2 domain
'clarkstokes' locate clean components independently in each stokes image
'hogbom' full-width of image (slower, better for poor uv-coverage)

NOTEpsfmode will also be used to clean if imagermode = ''.

#### imagermode

Advanced imaging e.g. mosaic or Cotton-Schwab clean
default: imagermode='csclean': Options: '', 'csclean', 'mosaic'
''  => psfmode cleaning algorithm used

NOTE: imagermode 'mosaic' (and/or) any gridmode not blank (and/or) nterms>1 : will always use CS style clean.

### imagermode='mosaic' expandable parameter(s)

Make a mosaic of the different pointings (uses csclean style too)

#### mosweight

Individually weight the fields of the mosaic. Default: mosweight = False; Example mosweight = True, this performs the weight density calculation for each field indepedently when using Briggs (including uniform) weighting. This can be useful if some of your fields are more sensitive than others (i.e. due to time spent on-source) or if you have relatively poor uv-coverage (e.g., snap-shot). If False, the weight density is calculated from the average uv distribution of all the fields.

#### ftmachine

Gridding method for the mosaic; Options: 'mosaic' , 'ft' or 'wproject'. default: 'mosaic'; 'ft' or 'wproject' implies standard interferometric 2D or widefield gridding. The residual visibilities are imaged for each pointing and combined in the image plane with the appropriate PB to make the mosaic. 'mosaic' (grid using the Fourier transform of PB as convolution function and mosaic combination is done in visibilities). ONLY if imagermode='mosaic' is chosen and ftmachine='mosaic', is heterogeneous imaging (CARMA, ALMA) or wideband beam accounting possible using the right convolution derived from primary beams for each baseline and for different frequencies

NOTE: ftmachine='mosaic' uses Fourier transforms of the primary beams/pointing for mosaicing. Making an image which is too small for the pointing coverages will cause aliasing due to standard Fourier transform wrap around.

#### scaletype

Controls scaling of pixels in the image plane. (controls what is seen if interactive=True) It does *not* affect the scaling of the *final* image that is done by pbcor. default='SAULT'; example: scaletype='PBCOR'; Options: 'PBCOR','SAULT'. 'SAULT' when interactive=True shows the residual with constant noise across the mosaic. Can also be achieved by setting pbcor=False. 'PBCOR' uses the SAULT scaling scheme for deconvolution, but if interactive=True shows the primary beam corrected image during interactive.

#### cyclefactor

Controls the threshhold at which the deconvolution cycle will pause to degrid and subtract the model from the visibilities. With poor PSFs, reconcile often (cyclefactor=4 or 5) for reliability. With good PSFs, use cyclefactor = 1.5 to 2.0 for speed.

NOTE: threshold = cyclefactor * max sidelobe * max residual

default: 1.5; example: cyclefactor=4

#### cyclespeedup

The major cycle threshold doubles in this number of iterations.
default: -1 (no doubling); example: cyclespeedup=3; Try cyclespeedup = 50 to speed up cleaning.

#### flatnoise

Controls whether searching for clean components is done in a constant noise residual image (True) or in an optimal signal-to-noise residual image (False) when ftmosaic='mosaic' is chosen. default=True

### imagermode='csclean' expandable parameter(s)

Image using the Cotton-Schwab algorithm in between major cycles.

#### cyclefactor

See above, under imagermode='mosaic'.

#### cyclespeedup

See above, under imagermode='mosaic'.

#### gridmode

This parameter is now provided to access more advanced deconvolution capabilities.

### gridmode='' expandable parameters

The default value of '' has no effect.

### gridmode='widefield' expandable parameters

Apply corrections for non-coplanar effects during imaging using the W-Projection algorithm [2] or faceting or a combination of the two.

#### wprojplanes

The number of pre-computed w-planes used for the W-Projection algorithm. wprojplanes=1 disables correction for non-coplanar effects. default value wprojpanes=-1 means clean will determine the number to use.

#### facets

The number of facets on each side of the image (i.e. the total number of facets is 'facets x facets'). If wprojplanes>1, W-Projection is done for each facet. Usually when many wprojection convolution functions sizes are  above ~400 pixels, it might be faster to use a few facets with wprojection.

### gridmode='aprojection' expandable parameters

Corrects for the (E)VLA time-varying PB effects including polarization squint using the A-Projection algorithm [3]. This can optinally include w-projection also.

#### wprojplanes

The number of pre-computed w-planes used for W-Projection algorithm. wprojplanes=1 disables correction for non-coplanar effects.

#### cfcache

The name of the directory to store the convolution functions and weighted sensitivty pattern function. These functions can be reused again if the image parameters are unchanged. If the image parameters change, a new cache must be created (or the existing one removed).

#### rotpainc

The Parallactic Angle increment (in degrees) used for OTF rotation of the convolution function.

#### painc

The Parallactic Angle increment (in degrees) used to compute the convolution functions.

#### multiscale

set of scales to use in deconvolution. If set, cleans with several resolutions using Hogbom clean. The scale sizes are in units of cellsize. So if cell='2arcsec', a multiscale scale=10 => 20arcsec. The first scale is recommended to  be 0 (point), we suggest the second be on the order of synthesized beam, the third 3-5 times the synthesized beam, etc.. Avoid making the largest scale too large relative to the image width or the scale of the lowest measured spatial frequency.  For example, if the synthesized beam is 10" FWHM and cell='2', try multiscale = [0,5,15]. default: multiscale=[] (standard clean with psfmode algorithm, no multi-scale). Example: multiscale = [0,5,15]

### multiscale expandable parameter(s)

#### negcomponent

Stop component search when the largest scale has found this number of negative components; -1 means continue component search even if the largest component is negative. default: -1; example: negcomponent=50

#### smallscalebias

A bias toward smaller scales. The peak flux found at each scale is weighted by a factor = 1 - smallscalebias*scale/max_scale, so that Fw = F*factor. Typically the values range from 0.2 to 1.0. default: 0.6

#### imsize

Image size in pixels (x, y). DOES NOT HAVE TO BE A POWER OF 2 (but has to be even and factorizable to 2,3,5,7 only). default = [256,256]; examples: imsize=[350,350], imsize = 500 is equivalent to [500,500]. If include outlier fields, e.g., [[400,400],[100,100]] or use outlierfile. Avoid odd-numbered imsize.

#### cell

Cell size (x,y). default= '1.0arcsec'; examples: cell=['0.5arcsec,'0.5arcsec'], cell=['1arcmin', '1arcmin'], cell = '1arcsec' is equivalent to ['1arcsec','1arcsec'], cell = 2.0 is equivalent to ['2arcsec', '2arcsec']

#### phasecenter

Direction measure or fieldid for the mosaic center. default: '' = first field selected; examples: phasecenter=6, phasecenter='J2000 19h30m00 -40d00m00', phasecenter='J2000 292.5deg  -40.0deg', phasecenter='J2000 5.105rad  -0.698rad'. If include outlier fields, e.g. ['J2000 19h30m00 -40d00m00',J2000 19h25m00 -38d40m00'] or use outlierfile.

#### restfreq

Specify rest frequency to use for output image. default='' Occasionally it is necessary to set this (for example some VLA spectral line data). For example, for NH_3 (1,1) put restfreq='23.694496GHz'

#### stokes

Stokes parameters to image. default='I'; example: stokes='IQUV'; Options: 'I','Q','U','V','IV','QU','IQ','UV','IQU','IUV','IQUV','RR','LL','XX','YY','RRLL','XXYY'

#### niter

Maximum number iterations. If niter=0, then no cleaning is done ("invert" only). (niter=0 can be used instead of the 'ft' task to predict/save a model) For cube or multi field images, niter is the maximum number of iteration clean will use for each image plane. The number of iterations used may be less that niter if threshold value is reached. default: 500; example: niter=5000

#### gain

Loop gain for CLEANing. default: 0.1; example: gain=0.5

#### threshold

Flux level at which to stop CLEANing. default: '0.0mJy'; examples: threshold='2.3mJy'  (always include units), threshold = '0.0023Jy', threshold = '0.0023Jy/beam' (okay also)

#### interactive

Use interactive clean (with GUI viewer). Interactive clean allows the user to build the cleaning mask interactively using the viewer. The viewer will appear every npercycle interation, but modify as needed. The final interactive mask is saved in the file imagename_interactive.mask. The initial masks use the union of mask and cleanbox (see below). default: interactive=False; example: interactive=True

### interactive=True expandable parameters

#### npercycle

This is the number of iterations between each interactive update of the mask. It is important to modify this number interactively during the cleaning, starting with a low number like 20, but then increasing as more extended emission is encountered.

Specification of cleanbox(es), mask image(s), primary beam coverage level, and/or region(s) to be used for cleaning. clean tends to perform better, and is less likely to diverge, if the clean component placement is limited by a mask to where real emission is expected to be. As long as the image has the same shape (size), mask images (e.g. from a previous interactive session) can be used for a new execution.

NOTE: The initial clean mask actually used is the union of what is specified in mask and <imagename>.mask.

default: [] or '' : no masking; Possible specification types:
(a) Cleanboxes, specified using the CASA region format (http://casaguides.nrao.edu/index.php?title=CASA_Region_Format)
examples: mask='box [ [ 100pix , 130pix] , [120pix, 150pix ] ]', mask='circle [ [ 120pix , 40pix] ,6pix ]', mask='circle[[19h58m52.7s,+40d42m06.04s ], 30.0arcsec]'
If used with a spectral cube, it will apply to all channels.
Multiple regions may be specified as a list of pixel ranges.
examples: mask= ['circle [ [ 120pix , 40pix] ,6pix ]', 'box [ [ 100pix , 130pix] , [120pix, 150pix ] ]' ]
(b) Filename with cleanbox shapes defined using the CASA region format.
example: mask='mycleanbox.txt'; The file 'mycleanbox.txt' contains:

box [ [ 100pix , 130pix ] , [ 120pix, 150pix ] ]circle [ [ 150pix , 150pix] ,10pix ]rotbox [ [ 60pix , 50pix ] , [ 30pix , 30pix ] , 30deg ]

Multiple mask files may be specified.
(d) Filename for region specification (e.g. from viewer).
(e) Combinations of the above options.
example: mask=['mycleanbox.txt', 'myimage.mask', 'myregion.rgn','circle [ [ 120pix , 40pix] ,6pix ]']
(f) Threshold on primary-beam.
A number between 0 and 1, used as a threshhold of primary beam coverage. The primary beam coverage map (imagename + '.flux(.pbcoverage)') will be made and the clean component placement will be limited to where it is > the number.
(g) True or False.
True: like (f), but use minpb as the number.
False: go maskless (and expect trouble).
(For masks for multiple fields, please see 'HINTS ON CLEAN WITH FLANKING FIELDS')

### Apply additional uv tapering of the visibilities. default: uvtaper=False; example: uvtaper=Trueuvtaper=True expandable parameters

#### outertaper

uv-taper on outer baselines in uv-plane, [bmaj, bmin, bpa] taper Gaussian scale in uv or angular units.

NOTE: The on-sky FWHM in arcsec is roughly the uvtaper / 200 (klambda).

default: outertaper=[]; no outer taper applied; examples: outertaper=['5klambda'] circular taper FWHM=5 kilo-lambda, outertaper=['5klambda','3klambda','45.0deg'], outertaper=['10arcsec'] on-sky FWHM 10 arcseconds, outertaper=['300.0'] default units are lambda in aperture plane

#### modelimage

Name of model image(s) to initialize cleaning. If multiple images, then these will be added together to form initial staring model.

NOTE: these are in addition to any initial model in the <imagename>.model image file.

default: '' (none); examples: modelimage='orion.model', modelimage=['orion.model','sdorion.image']

NOTE: If the units in the image are Jy/beam as in a single-dish image, then it will be converted to Jy/pixel as in a model image, using the restoring beam in the image header and zeroing negatives. If the image is in Jy/pixel then it is taken as is.

When nterms>1, a one-to-one mapping is done between images in this list and Taylor-coefficients. If more than nterms images are specified, only the first nterms are used. It is valid to supply fewer than nterms model images. Example: Supply an estimate of the continuum flux from a previous imaging run.

#### weighting

Weighting to apply to visibilities. default='natural'; example: weighting='uniform'; Options: 'natural','uniform','briggs', 'superuniform','briggsabs','radial'

### weighting expandable parameters

For details on weighting please see Chapter3 of late Dr. Brigg's thesis (http://www.aoc.nrao.edu/dissertations/dbriggs)

For weighting='briggs' and 'briggsabs':

#### robust

Brigg's robustness parameter. default=0.0; example: robust=0.5; Options: -2.0 to 2.0; -2 (uniform)/+2 (natural)

#### npixels

uv-box used for weight calculation a box going from -npixel/2 to +npixel/2 on each side around a point is used to calculate weight density. 0 means box is pixel size. default = 0; example: npixels=2

EXEMPTION: When choosing superuniform, it does not make sense to use npixels=0 as it is uniform thus if npixels is 0, it will be forced to 6 or a box from -3pixels to 3pixels.

For weighting='briggsabs'

#### noise

noise parameter to use for Briggs "abs" weighting. example: noise='1.0mJy'

#### restoringbeam

Output Gaussian restoring beam for clean image, [bmaj, bmin, bpa] elliptical Gaussian restoring beam. Default units are in arc-seconds for bmaj,bmin, degrees for bpa. default: restoringbeam=[]; Use PSF calculated from dirty beam. examples: restoringbeam=['10arcsec'] circular Gaussian FWHM 10 arcseconds, restoringbeam=['10.0','5.0','45.0deg'] 10"x5" at 45 degrees

#### pbcor

Output primary beam-corrected image. If pbcor=False, the final output image is NOT corrected for the PB pattern (particularly important for mosaics), and therefore is not "flux correct". Correction can also be done after the fact using immath to divide <imagename>.image by the <imagename>.flux image. default: pbcor=False, output un-corrected image; example: pbcor=True, output pb-corrected image (masked outside minpb)

#### minpb

Minimum PB level to use for pb-correction and pb-based masking. default=0.2; example: minpb=0.01
When imagermode is *not* 'mosaic': minpb is applied to the flux image (sensitivity-weighted pb). minpb is used to create a mask, only when pbcor=True
When imagermode='mosaic': minpb is applied to the flux.pbcoverage image (mosaic pb with equal weight per pointing). minpb is always used to create a mask (regardless of pbcor=True/False).

#### usescratch

If True will create scratch columns if they are not there. And after clean completes the predicted model visibility is from the clean components are written to the MS. This increases the MS size by the data volume. if False then the model is saved in the MS header and the calculation of the visibilities is done on the fly when using calibration or plotms. Use True if you want to access the model visibilities in python, say.

#### allowchunk

Partition the image cube by channel-chunks. default=False;
False: Major cycle grids all channels. Minor cycle steps through all channels before the next major cycle.
True: Major and minor cycles are performed one chunk at a time, and output images cubes are concatenated.

#### async

Run asynchronously. default = False; do not run asychronously

Citation Number 1 Rau and Cornwell, AA, Volume 532, 2011 (ADS)
Citation Number 2 Cornwell et al. IEEE JSTSP, 2008 (IEEE)
Citation Number 3 Bhatnagar et al., AandA, 487, 419, 2008 (A&A)


text:
In: Chemistry

# A 2.67 g sample of plant tissue was analyzed for its zinccontent. 23.25 ml of...

A 2.67 g sample of plant tissue was analyzed for its zinc content. 23.25 ml of a standardized 0.010 Molar solution of EDTA was used to titrate the sample (by complexing with the zinc ions), determine the mass of zinc in the sample and express the value as;

a. ppm zinc?

b. ppb zinc?

c. What mass of zinc product would form if the zinc (2+) ions were instead allowed to precipitate as zinc phosphate?

3Zn2+(aq) + 2PO43-(aq)Zn3(PO4)2(s)

## Solutions

##### Expert Solution

The mass of zinc present is calculated:

m Zn = M * V * MM = 0.01 M * 0.02325 L * 65.38 g / mol = 0.015 g

a) The ppm is calculated:

ppm = mg / Kg = 0.015 g Zn / 2.67 g * (1000 mg / 1 g) * (1000 g / 1 Kg) = 5618 ppm

b) The ppb is calculated:

ppb = ppm * 1000 = 5618 * 1000 = 5618000

c) The mass of Zn3 (PO4) 2 formed is calculated:

m Zn3 (PO4) 2 = 0.015 g Zn * (1 mol / 65.38 g) * (1 mol Zn3 (PO4) 2/3 mol Zn) * (386.1 g Zn3 (PO4) 2/1 mol) = 0.030 g

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