Documentation on imrm

Task: imrm
Purpose: Compute rotation measure image from position angle images
Categories: image analysis

        IMRM computes  rotation measure and zero wavelength position
        angle images from at least 2 position angle images at 
        different frequencies.   This is done via a linear least
        squares fit to:

                       PA = PA_0 + RM*LAMBDA**2

        where RM is the rotation measure (rad/m**2) and PA_0 is the 
        position angle at zero wavelength.  The output rotation
        measure image is in rad/m**2, and the output position 
        angle image is in degrees.  Optionally, plots of the fits
 	can be made.

        The more frequencies you have the better.  It is very important
        to try and get at least two sufficiently close that there is
        no ambiguity between them.

        By default, IMRM attempts to remove n*pi ambiguities from the
        data.  Its algorithm is (pixel by pixel)

         0) First remove angle according to the amount given by the
           user (keyword "rmi") and the equation PA = RM*LAMBDA**2

         1) Put the position angles of the first two frequencies 
           in the range +/- 90 degrees.

         2) Remove 180 degree ambiguity from the position angles given
           by the FIRST TWO IMAGES (keyword in).  Thus, it modifies
           the position angle of the second frequency by 180 degrees 
           so that the absolute value of the angle between the 
           two position angles is less than 90 degrees.

         3) Compute the initial RM and PA_0 from these FIRST
           TWO position angles.
        
         4) This RM and PA_0 is used to predict the expected position
           angle at the other frequencies according to the expression
           PA = PA_0 + RM*LAMBDA**2.  Integer amounts of 180 degrees
           are then added or subtracted to the position angles at the
           remaining frequencies in order to make the position angle
           as close as possible to the expected value.

         5) Then a least squares fit is used to solve for the RM and PA_0
        
         6) Finally, the procedure is repeated from step 0) where the
          initial guess is now the value just determined above in
          step 5).   

        The order in which the images are given is thus very important.
        You should generally give your images in order of decreasing 
        frequency, with the assumption being that the smallest angle
        between the first two represents a rough guess for the RM
        with no ambiguities.  However, if you are very certain abou
        the lack of ambiguity between certain frequencies, or there 
        are some of particularly high S/N and likely lack of ambiguity,
        you may like to try these.  Its a nasty business and it is VERY
        important that you look at the results carefully.

        The attempt to remove ambiguities can be turned off with
        keyword "options=ambiguous".  In this case, its algorithm is

         0) First remove angle according to the intial guess given
           by the user (keyword "rmi").

        1) Put all position angles in the range +/- 90 degrees

         2) Then a least squares fit is used to solve for the RM and PA_0
        
        In principle, you should never need to use this option.
        If there are no ambiguities, the first algorithm shouldn't
        find any !

        There are also a variety of methods offered with which to blank the 
        output images.  Most of these require error images associated with
        the input position angle images. Use the program IMPOL to make 
        the position angle images and position angle error images.

Key: in
        Up to 5 input position angle (positive N -  E) images 
        (in degrees) at different frequencies.  Generally, you should
        give the images in order of decreasing frequency.
        Wild card expansion is supported, no default.

Key: inerr
        Up to 5 position angle error images (in degrees) used for 
        weighting the data during the least squares fit.  They are 
        assumed to be in one-to-one association with the position 
        angle images. If no error images are given, each position 
        angle image is given equal weight and we must assume a goodness
        of fit of unity in order to find the output image errors.
        Wild card expansion is supported, default is no error images.

Key: rmi
        An amount of rotation measure to remove from the data before fitting.
        If you have a good idea of this, it helps enormously in removing 
        ambiguities. See the detailed use in the discussion of the algorithm
        above.  See also options=guess where it is used slightly differently.
        Default is 0

Key: rm
        Two values. The output fitted rotation measure image in 
        rad/m**2, and optionally, its associated error image.
        The default is no output RM images.

Key: pa0
        The output fitted (at zero wavelength) position angle image 
        in degrees, and optionally, its associated error image.
        The default is no output PA images.

Key: qcut
        Blank the output image (RM or PA) pixels if the goodness of fit
        (Q) is less than this value.  If Q is larger than about 0.1 say,
        the fit is believable.  If it is greater than 0.001, the fit 
        may be acceptable if the errors are non-normal or too small. If
        Q is less than 0.001 the model can be called into question.  The
        probability distribution for position angle images approximates
        a Gaussian at high S/N ratios.  At low S/N ratios (roughly, when
        P/sigma   2) it is non-Gaussian.  If you don't specify error 
        images, Q cannot be determined and is assumed to be one.  This is 
        also true if you give IMRM position angle images at two 
        frequencies only.
        Default is 0.001

Key: errcut
        Blank the output image (RM or PA) pixels if ANY of the input PA 
        image pixels has an error greater than this value (degrees).
        Default is no input error based blanking.

Key: rmcut
        Blank pixels in BOTH the output RM and PA_0 images when the error 
        in the fitted RM is greater than this value (rad/m**2).
        Errors can be worked out if you give input error images,
        or if you input images at more than two frequencies AND we
        assume the goodness of fit is unity.
        Default is no fitted RM error based blanking.

Key: pacut
        Blank pixels in BOTH the output RM and PA_0 images when the
        error in the fitted PA_0 is greater than this value (degrees).
        Errors can be worked out if you give input error images,
        or if you input images at more than two frequencies AND we
        assume the goodness of fit is unity.
        Default is no fitted PA_0 error based blanking.

Key: device
        PGPLOT plotting device to see the fits to the data.  The absolute 
        pixel numbers in x and y are also written into the corner of the 
        plot (unless options=accumulate).
        
        No default.

Key: nxy
        Number of subplots per page in the x and y directions, to put 
        on the plotting device.  See options=accumulate
        The default is 10x10

Key: csize
        PGPLOT character height.
        Default is 1.0

Key: options
        Task enrichment options.  Minimum match is active,

        "relax"      issue warnings instead of a fatal error when image
                     axis descriptors are inconsistent with each other,
        	     and when the input image headers do not indicate that
        	     they are position angle images (btype=position_angle)
        "guess"      when removing ambiguities, this option causes IMRM to
        	     use the rotation measure input through the keyword 
        	     "rmi" in step 3 above (on the first pass only), rather
        	     than working it out from the first two frequencies. By 
        	     default, angle is removed from the data according to 
        	     the value of "rmi" and then the first guess made from 
        	     the first two frequencies.  The angle is not removed 
        	     in this way with this option.  This may prove useful if 
        	     you have two close but perhaps noisy frequencies which 
        	     is causing the initial guess of the RM to be wrong 
        	     (because of noise) and driving the subsequent turn 
        	     removal off.
        "ambiguous"  Do not try to remove ambiguites. 
        "accumulate" means put all the plots on one sub-plot, rather than
         	     the default, which is to put the plot for each 
        	     spatial pixel on a spearate subplot
        "yindependent"
        	     By default, the sub-plots are all drawn with the same
        	     Y-axis scale, that embraces all sub-plots.  This option
        	     forces each sub-plot to be scaled independently.