Module MeasurementEquations

Description (classes)

Prerequisite

• MeasurementSets

Etymology

Synopsis

J = G D C E P T F K

The measured visibility from a radio telescope is then given by:

V_i,j=X_i,j (M_i,j integral directProduct(J_i, Conjugate(J_j)) S I + A_i,j)

where the elements in the equation are

• M is the 4 by 4 complex closure error matrix
• K is the Fourier phase Factor Jones matrix
• G is the antenna gain Jones matrix
• D is the polarization leakage Jones matrix
• C is the configuration Jones matrix
• E is the electric field pattern Jones matrix
• P is the receptor position angle Jones matrix
• T is the atmospheric gain Jones matrix
• F is the Faraday rotation Jones matrix
• A is the complex additive component
• X is the non-linear correlator function
• S is the Stokes conversion matrix
• I is the (real) sky brightness 4-vector

and the integral is over time, frequency, sky position. The direct product of two 2 by 2 matrices gives a 4 by 4 matrix in which every possible product of the 2 by 2 matrix elements appears.

Manipulation of the equation in this form is possible but is much too expensive for most uses so we break it down into two parts and also use a special machine for the Fourier summation. This loses some generality but makes the use of the HBS measurement equation feasible.

The split is such that the class VisEquation expresses the visibility-plane part of the ME:

J = G D C P

where the visibility is V_i,j=X_i,j (M_i,j integral directProduct(J_i, Conjugate(J_j)) Vsky_i,j + A_i,j) and the integral is over time, frequency.

and the class SkyEquation expresses the sky-plane part of the ME:

J = E T F K

Vsky_i,j=integral directProduct(J_i, Conjugate(J_j)) S I and the integral is over time, frequency, sky position.

The last integral (over K) amounts to a Fourier transform and so the SkyEquation is given FT machines to use for this purpose. Note that we have chosen to move the position of the parallactic angle term. This has been done for expediency but will lead to some difficulties in difficult cases.

The terms G, D, C, etc, are represented by the classes in the module MeasurementComponents. This classes can typically do two basic things: apply a correction to a VisBuffer (which is a holder of a chunk of visibility data), and solve for its own internal parameters. Solution is accomplished using gradients of chi-squared obtained via standard services of the MeasurementEquation.

The SkyBrightness is modelled by a special type of MeasurementComponent called a SkyModel. This has an interface to the SkyEquation via a PagedImage.

Another type of MeasurementComponent is the Fourier transform machine FTMachine which is used for performing forward and inverse Fourier transforms. The class GridFT implements a straightforward grid and degrid FFT-based Fourier transform. We anticipate that other FTMachines will be needed for e.g. wide-field imaging.

Visibility Data is held in a MeasurementSet. To expedite processing, we use a VisibilityIterator (found in this module) to iterate through the MeasurementSet as needed. Setting up the iterator is relatively expensive so we store the iterator in a VisSet (also found in this module). Thus one should construct a VisSet and then use the iterator method to retrieve the iterator. Once one has a VisibilityIterator, it can be used to access the actual visibility data in chunk by using the VisBuffer (also in this module). This scheme may seem baroque but it is needed to cut down on superfluous otherhead of various types.

Example

      // Create a VisSet from a MeasurementSet on disk
      VisSet vs("3c84.MS");

      // Now make an FTMachine with a 2048 by 2048
      // complex pixel cache of 16 by 16 tiles,
      // using Spheriodal Function gridding
      GridFT ft(2048*2048, 16, "SF")

      // Create an ImageSkyModel from an image on disk
      PagedImage<Float> im("3c84.modelImage"));
      ImageSkyModel ism(im);

      // For the imaging, we need a SkyEquation and
      // an FTMachine
      SkyEquation se(vs, ft);

      // Predict the visibility set for the model
      se.predict();

      // Make a VisEquation
      VisEquation ve(vs);

      // Solve for calibration of G matrix every 5 minutes
      GJones gj(vs, 5*60);
      ve.solve(gj);

      // Solve for calibration of D matrix every 12 hours
      DJones dj(vs, 12*60*60);
      ve.solve(dj);

      // Now use the SkyEquaton to make a Clean Image
      HogbomCleanImageSkyModel csm(ism);
      if (se.solve(csm)) {
        Image<StokesVector> cleanImage=csm.getImage();
        cleanImage.setName("3c84.cleanImage");
      }

Motivation