Polarized Image and Instrumental Modeling

In this tutorial, we will analyze a simulated simple polarized dataset to demonstrate Comrade's polarized imaging capabilities.

Introduction to Polarized Imaging

The EHT is a polarized interferometer. However, like all VLBI interferometers, it does not directly measure the Stokes parameters (I, Q, U, V). Instead, it measures components related to the electric field at the telescope along two directions using feeds. There are two types of feeds at telescopes: circular, which measure $R / L$ components of the electric field, and linear feeds, which measure $X / Y$ components of the electric field. Most sites in the EHT use circular feeds, meaning they measure the right (R) and left electric field (L) at each telescope. Although note that ALMA actually uses linear feeds. Currently Comrade has the ability to fit natively mixed polarization data however, the publically released EHT data has been converted to circular polarization. For a VLBI array whose feeds are purely circluar the coherency matrices are given by,

C_{i j} = (\begin{matrix} R R^{*} & R L^{*} \\ L R^{*} & L L^{*} \end{matrix}) .

These coherency matrices are the fundamental object in interferometry and what the telescope observes. For a perfect interferometer, these circular coherency matrices are related to the usual Fourier transform of the stokes parameters by

(\begin{matrix} \tilde{I} \\ \tilde{Q} \\ \tilde{U} \\ \tilde{V} \end{matrix}) = \frac{1}{2} (\begin{matrix} R R^{*} + L L^{*} \\ R L^{*} + L R^{*} \\ i (L R^{*} - R L^{*}) \\ R R^{*} - L L^{*} \end{matrix}) .

Note

In this tutorial, we stick to circular feeds but Comrade has the capabilities to model linear (XX,XY, ...) and mixed basis coherencies (e.g., RX, RY, ...).

In reality, the measure coherencies are corrupted by both the atmosphere and the telescope itself. In Comrade we use the RIME formalism ^[1] to represent these corruptions, namely our measured coherency matrices $V_{i j}$ are given by

V_{i j} = J_{i} C_{i j} J_{j}^{†}

where $J$ is known as a Jones matrix and $i j$ denotes the baseline $i j$ with sites $i$ and $j$ .

Comrade is highly flexible with how the Jones matrices are formed and provides several convenience functions that parameterize standard Jones matrices. These matrices include:

JonesG which builds the set of complex gain Jones matrices

G = (\begin{matrix} g_{a} & 0 \\ 0 & g_{b} \end{matrix})

JonesD which builds the set of complex d-terms Jones matrices

D = (\begin{matrix} 1 & d_{a} \\ d_{b} & 1 \end{matrix})

JonesR is the ideal telescope response $R$ . This transformation is special and combines two things using the decomposition $R = F B$ . The first, $B$ , is the transformation from some reference basis to the observed coherency basis (this allows for mixed basis measurements). The second is the feed rotation, $F$ , that transforms from some reference axis to the axis of the telescope as the source moves in the sky. The feed rotation matrix F for circular feeds in terms of the per station feed rotation angle $φ$ is

F = (\begin{matrix} e^{- i φ} & 0 \\ 0 & e^{i φ} \end{matrix})

In the rest of the tutorial, we are going to solve for all of these instrument model terms in while re-creating the polarized image from the first EHT results on M87.

Load the Data

To get started we will load Comrade

julia

using Comrade

Load the Data

julia

using VLBIFiles

For reproducibility we use a stable random number genreator

julia

using StableRNGs
rng = StableRNG(48)

StableRNGs.LehmerRNG(state=0x00000000000000000000000000000061)

Now we will load some synthetic polarized data.

julia

fname = Base.download(
    "https://github.com/ptiede/ComradeTestData/releases/download/Data/eht_2017_data.uvfits",
    joinpath(__DIR, "m87polarized.uvfits")
)
uvd = VLBIFiles.load(VLBIFiles.UVData, fname)

VLBIFiles.UVData("/home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/PolarizedImaging/m87polarized.uvfits", VLBIFiles.UVHeader(SIMPLE  =                    T / conforms to FITS standard
BITPIX  =                  -32 / array data type
NAXIS   =                    7 / number of array dimensions
NAXIS1  =                    0
NAXIS2  =                    3
NAXIS3  =                    4
NAXIS4  =                    1
NAXIS5  =                    1
NAXIS6  =                    1
NAXIS7  =                    1
EXTEND  =                    T
GROUPS  =                    T / has groups
PCOUNT  =                    9 / number of parameters
GCOUNT  =                 8646 / number of groups
OBSRA   =    187.7059307575226
OBSDEC  =    12.39112323919932
OBJECT  = 'M87     '          
MJD     =              57849.0
DATE-OBS= '2017-04-06'        
BSCALE  =                  1.0
BZERO   =                  0.0
BUNIT   = 'JY      '          
VELREF  =                    3
EQUINOX = 'J2000   '          
ALTRPIX =                  1.0
ALTRVAL =                  0.0
TELESCOP= 'VLBA    '          
INSTRUME= 'VLBA    '          
OBSERVER= 'EHT     '          
CTYPE2  = 'COMPLEX '          
CRVAL2  =                  1.0
CDELT2  =                  1.0
CRPIX2  =                  1.0
CROTA2  =                  0.0
CTYPE3  = 'STOKES  '          
CRVAL3  =                 -1.0
CDELT3  =                 -1.0
CRPIX3  =                  1.0
CROTA3  =                  0.0
CTYPE4  = 'FREQ    '          
CRVAL4  =     2.27070703125e11
CDELT4  =              1.856e9
CRPIX4  =                  1.0
CROTA4  =                  0.0
CTYPE6  = 'RA      '          
CRVAL6  =    187.7059307575226
CDELT6  =                  1.0
CRPIX6  =                  1.0
CROTA6  =                  0.0
CTYPE7  = 'DEC     '          
CRVAL7  =    12.39112323919932
CDELT7  =                  1.0
CRPIX7  =                  1.0
CROTA7  =                  0.0
PTYPE1  = 'UU---SIN'          
PSCAL1  =  4.4039146672722e-12
PZERO1  =                  0.0
PTYPE2  = 'VV---SIN'          
PSCAL2  =  4.4039146672722e-12
PZERO2  =                  0.0
PTYPE3  = 'WW---SIN'          
PSCAL3  =  4.4039146672722e-12
PZERO3  =                  0.0
PTYPE4  = 'BASELINE'          
PSCAL4  =                  1.0
PZERO4  =                  0.0
PTYPE5  = 'DATE    '          
PSCAL5  =                  1.0
PZERO5  =                  0.0
PTYPE6  = 'DATE    '          
PSCAL6  =                  1.0
PZERO6  =                  0.0
PTYPE7  = 'INTTIM  '          
PSCAL7  =                  1.0
PZERO7  =                  0.0
PTYPE8  = 'TAU1    '          
PSCAL8  =                  1.0
PZERO8  =                  0.0
PTYPE9  = 'TAU2    '          
PSCAL9  =                  1.0
PZERO9  =                  0.0
HISTORY AIPS SORT ORDER='TB', "M87", Dates.Date("2017-04-06"), [:RR, :LL, :RL, :LR], 2.27070703125e11 Hz), VLBIFiles.FrequencyWindow[VLBIFiles.FrequencyWindow(1, 1, 2.270707f11 Hz, 1.856f9 Hz, 1, 1, 1.0f0)], VLBIFiles.AntArray[VLBIFiles.AntArray("VLBA", 2.270707f11 Hz, {1 = Antenna AA, 2 = Antenna AP, 3 = Antenna AZ, 4 = Antenna JC, 5 = Antenna LM, 6 = Antenna PV, 7 = Antenna SM}, [0.0, 0.0, 0.0])])

Unlike non-polarized tutorials, we also need an array file describing the antenna feed rotation parameters. We pass it via the arrayfile keyword on the data product.

julia

dvis = extract_table(
    uvd, Coherencies(;
        time_average = VLBI.GapBasedScans(),
    )
)

EHTObservationTable{Comrade.EHTCoherencyDatum{Float64}}
  source:      M87
  mjd:         57849
  bandwidth:   1.856e9
  sites:       [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples:    274

Often the EHT does not take into acount feed offsets. To load those we read in the standard ehtim style array.txt files.

julia

Comrade.reset_mounts!(dvis, Comrade.load_array_txt(joinpath(__DIR, "../../Data/array.txt")))

EHTObservationTable{Comrade.EHTCoherencyDatum{Float64}}
  source:      M87
  mjd:         57849
  bandwidth:   1.856e9
  sites:       [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples:    274

Inflate noise by 1% and drop short (uvdist < 0.1 Gλ) baselines.

julia

add_fractional_noise!(dvis, 0.01)
dvis = flag(d -> hypot(d.baseline.U, d.baseline.V) < 0.1e9, dvis)

EHTObservationTable{Comrade.EHTCoherencyDatum{Float64}}
  source:      M87
  mjd:         57849
  bandwidth:   1.856e9
  sites:       [:AA, :AP, :AZ, :JC, :LM, :PV, :SM]
  nsamples:    242

##Building the Model/Posterior

To build the model, we first break it down into two parts:

The image or sky model. In Comrade, all polarized image models are written in terms of the Stokes parameters. In this tutorial, we will use a polarized image model based on Pesce (2021)^[2], and parameterizes each pixel in terms of the Poincare sphere. This parameterization ensures that we have all the physical properties of stokes parameters. Note that we also have a parameterization in terms of hyperbolic trig functions VLBISkyModels.PolExp2Map
The instrument model. The instrument model specifies the model that describes the impact of instrumental and atmospheric effects. We will be using the $J = G D R$ decomposition we described above. However, to parameterize the R/L complex gains, we will be using a gain product and ratio decomposition. The reason for this decomposition is that in realistic measurements, the gain ratios and products have different temporal characteristics. Namely, many of the EHT observations tend to demonstrate constant R/L gain ratios across an nights observations, compared to the gain products, which vary every scan. Additionally, the gain ratios tend to be smaller (i.e., closer to unity) than the gain products. Using this apriori knowledge, we can build this into our model and reduce the total number of parameters we need to model.

First we specify our sky model with the @sky macro. Each name ~ dist line contributes an entry to the prior; everything else is the model body. The metadata fields (mimg, ftot, cprior) are passed as keyword arguments and are in scope inside both the prior expressions and the body.

julia

using Distributions
using VLBIImagePriors
@sky function sky(grid; mimg, ftot, cprior)
    σs ~ ntuple(Returns(VLBITruncated(VLBIGaussian(0.0, 0.5); lower = 0.0)), 4)
    as ~ ntuple(Returns(cprior), 4)
    # Build the stokes I model
    δs = ntuple(Val(4)) do i
        σs[i] .* as[i].params
    end

    # Convert hyperbolic polarization parameterization to Stokes parameters.
    pmap = VLBISkyModels.PolExp2Map!(δs..., axisdims(mimg))

    # We now add a mean image. Namely, we assume that `pmap` are multiplicative fluctuations
    # about some mean image `mimg`. We also compute the total flux of the Stokes I image
    # for normalization purposes below.
    bpmap = baseimage(pmap)
    bmimg = baseimage(mimg)
    ft = zero(ftot)
    bpmapI = stokes(bpmap, :I)
    bpmapQ = stokes(bpmap, :Q)
    bpmapU = stokes(bpmap, :U)
    bpmapV = stokes(bpmap, :V)

    @inbounds for i in eachindex(bpmap, bmimg)
        bpmapI[i] = bmimg[i] * bpmapI[i]
        bpmapQ[i] = bmimg[i] * bpmapQ[i]
        bpmapU[i] = bmimg[i] * bpmapU[i]
        bpmapV[i] = bmimg[i] * bpmapV[i]
        ft += bpmapI[i]
    end

    # Renormlize to get the correct total flux.
    @inbounds for i in eachindex(bpmap)
        bpmap[i] *= ftot / ft
    end

    m = ContinuousImage(pmap, BSplinePulse{3}())
    x, y = centroid(pmap)
    return shifted(m, -x, -y)
end

sky (generic function with 1 method)

Now, we define the metadata needed to build the model. We specify our image grid and cache model needed to define the polarimetric image model. Our image will be a 10x10 raster with a 60μas FOV.

julia

fovx = μas2rad(200.0)
fovy = μas2rad(200.0)
nx = ny = 48
grid = imagepixels(fovx, fovy, nx, ny)

fwhmfac = 2 * sqrt(2 * log(2))
mpr = modify(Gaussian(), Stretch(μas2rad(50.0) ./ fwhmfac))
mimg_raw = intensitymap(mpr, grid)
mimg = mimg_raw ./ Comrade.flux(mimg_raw)

┌ 48×48 IntensityMap{Float64, 2} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
  ↓ X Sampled{Float64} LinRange{Float64}(-4.747133960864206e-10, 4.747133960864206e-10, 48) ForwardOrdered Regular Points,
  → Y Sampled{Float64} LinRange{Float64}(-4.747133960864206e-10, 4.747133960864206e-10, 48) ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
  ↓ →          -4.74713e-10  -4.54513e-10  -4.34312e-10  -4.14112e-10  -3.93911e-10  -3.73711e-10  -3.5351e-10   -3.33309e-10  -3.13109e-10  -2.92908e-10  -2.72708e-10  -2.52507e-10  -2.32307e-10  -2.12106e-10  -1.91905e-10  -1.71705e-10  -1.51504e-10  -1.31304e-10  -1.11103e-10  -9.09026e-11  -7.0702e-11   -5.05014e-11  -3.03009e-11  -1.01003e-11  1.01003e-11  3.03009e-11  5.05014e-11  7.0702e-11   9.09026e-11  1.11103e-10  1.31304e-10  1.51504e-10  1.71705e-10  1.91905e-10  2.12106e-10  2.32307e-10  2.52507e-10  2.72708e-10  2.92908e-10  3.13109e-10  3.33309e-10  3.5351e-10   3.73711e-10  3.93911e-10  4.14112e-10  4.34312e-10  4.54513e-10  4.74713e-10
 -4.74713e-10   3.56129e-12   8.6349e-12    2.01457e-11   4.52256e-11   9.76927e-11   2.03056e-10   4.06112e-10   7.81542e-10   1.44722e-9    2.57865e-9    4.42107e-9    7.29353e-9    1.15778e-8    1.76843e-8    2.59912e-8    3.67571e-8    5.00187e-8    6.54937e-8    8.25168e-8    1.00037e-7    1.16696e-7    1.30987e-7    1.41474e-7    1.47028e-7   1.47028e-7   1.41474e-7   1.30987e-7   1.16696e-7   1.00037e-7   8.25168e-8   6.54937e-8   5.00187e-8   3.67571e-8   2.59912e-8   1.76843e-8   1.15778e-8   7.29353e-9   4.42107e-9   2.57865e-9   1.44722e-9   7.81542e-10  4.06112e-10  2.03056e-10  9.76927e-11  4.52256e-11  2.01457e-11  8.6349e-12   3.56129e-12
 -4.54513e-10   8.6349e-12    2.09366e-11   4.88463e-11   1.09656e-10   2.36871e-10   4.9234e-10    9.84681e-10   1.89497e-9    3.509e-9      6.25233e-9    1.07195e-8    1.76843e-8    2.8072e-8     4.28782e-8    6.30196e-8    8.91231e-8    1.21278e-7    1.58799e-7    2.00075e-7    2.42556e-7    2.82948e-7    3.17599e-7    3.43026e-7    3.56492e-7   3.56492e-7   3.43026e-7   3.17599e-7   2.82948e-7   2.42556e-7   2.00075e-7   1.58799e-7   1.21278e-7   8.91231e-8   6.30196e-8   4.28782e-8   2.8072e-8    1.76843e-8   1.07195e-8   6.25233e-9   3.509e-9     1.89497e-9   9.84681e-10  4.9234e-10   2.36871e-10  1.09656e-10  4.88463e-11  2.09366e-11  8.6349e-12
 -4.34312e-10   2.01457e-11   4.88463e-11   1.13961e-10   2.55835e-10   5.52633e-10   1.14866e-9    2.29732e-9    4.42107e-9    8.18671e-9    1.45871e-8    2.50093e-8    4.12584e-8    6.54937e-8    1.00037e-7    1.47028e-7    2.07929e-7    2.82948e-7    3.70488e-7    4.66786e-7    5.65897e-7    6.60135e-7    7.40976e-7    8.00299e-7    8.31718e-7   8.31718e-7   8.00299e-7   7.40976e-7   6.60135e-7   5.65897e-7   4.66786e-7   3.70488e-7   2.82948e-7   2.07929e-7   1.47028e-7   1.00037e-7   6.54937e-8   4.12584e-8   2.50093e-8   1.45871e-8   8.18671e-9   4.42107e-9   2.29732e-9   1.14866e-9   5.52633e-10  2.55835e-10  1.13961e-10  4.88463e-11  2.01457e-11
 -4.14112e-10   4.52256e-11   1.09656e-10   2.55835e-10   5.74329e-10   1.24062e-9    2.57865e-9    5.1573e-9     9.92496e-9    1.83785e-8    3.27468e-8    5.6144e-8     9.2622e-8     1.47028e-7    2.24576e-7    3.30067e-7    4.66786e-7    6.35197e-7    8.31718e-7    1.0479e-6     1.27039e-6    1.48195e-6    1.66344e-6    1.79661e-6    1.86714e-6   1.86714e-6   1.79661e-6   1.66344e-6   1.48195e-6   1.27039e-6   1.0479e-6    8.31718e-7   6.35197e-7   4.66786e-7   3.30067e-7   2.24576e-7   1.47028e-7   9.2622e-8    5.6144e-8    3.27468e-8   1.83785e-8   9.92496e-9   5.1573e-9    2.57865e-9   1.24062e-9   5.74329e-10  2.55835e-10  1.09656e-10  4.52256e-11
 -3.93911e-10   9.76927e-11   2.36871e-10   5.52633e-10   1.24062e-9    2.67989e-9    5.57019e-9    1.11404e-8    2.14391e-8    3.96998e-8    7.07371e-8    1.21278e-7    2.00075e-7    3.17599e-7    4.85111e-7    7.12985e-7    1.00831e-6    1.3721e-6     1.79661e-6    2.26359e-6    2.7442e-6     3.20119e-6    3.59322e-6    3.88089e-6    4.03325e-6   4.03325e-6   3.88089e-6   3.59322e-6   3.20119e-6   2.7442e-6    2.26359e-6   1.79661e-6   1.3721e-6    1.00831e-6   7.12985e-7   4.85111e-7   3.17599e-7   2.00075e-7   1.21278e-7   7.07371e-8   3.96998e-8   2.14391e-8   1.11404e-8   5.57019e-9   2.67989e-9   1.24062e-9   5.52633e-10  2.36871e-10  9.76927e-11
 -3.73711e-10   2.03056e-10   4.9234e-10    1.14866e-9    2.57865e-9    5.57019e-9    1.15778e-8    2.31555e-8    4.45616e-8    8.25168e-8    1.47028e-7    2.52078e-7    4.15859e-7    6.60135e-7    1.00831e-6    1.48195e-6    2.0958e-6     2.85194e-6    3.73429e-6    4.70491e-6    5.70388e-6    6.65374e-6    7.46857e-6    8.0665e-6     8.38319e-6   8.38319e-6   8.0665e-6    7.46857e-6   6.65374e-6   5.70388e-6   4.70491e-6   3.73429e-6   2.85194e-6   2.0958e-6    1.48195e-6   1.00831e-6   6.60135e-7   4.15859e-7   2.52078e-7   1.47028e-7   8.25168e-8   4.45616e-8   2.31555e-8   1.15778e-8   5.57019e-9   2.57865e-9   1.14866e-9   4.9234e-10   2.03056e-10
 -3.5351e-10    4.06112e-10   9.84681e-10   2.29732e-9    5.1573e-9     1.11404e-8    2.31555e-8    4.6311e-8     8.91231e-8    1.65034e-7    2.94057e-7    5.04156e-7    8.31718e-7    1.32027e-6    2.01663e-6    2.9639e-6     4.19159e-6    5.70388e-6    7.46857e-6    9.40981e-6    1.14078e-5    1.33075e-5    1.49371e-5    1.6133e-5     1.67664e-5   1.67664e-5   1.6133e-5    1.49371e-5   1.33075e-5   1.14078e-5   9.40981e-6   7.46857e-6   5.70388e-6   4.19159e-6   2.9639e-6    2.01663e-6   1.32027e-6   8.31718e-7   5.04156e-7   2.94057e-7   1.65034e-7   8.91231e-8   4.6311e-8    2.31555e-8   1.11404e-8   5.1573e-9    2.29732e-9   9.84681e-10  4.06112e-10
 -3.33309e-10   7.81542e-10   1.89497e-9    4.42107e-9    9.92496e-9    2.14391e-8    4.45616e-8    8.91231e-8    1.71513e-7    3.17599e-7    5.65897e-7    9.70223e-7    1.6006e-6     2.54079e-6    3.88089e-6    5.70388e-6    8.0665e-6     1.09768e-5    1.43729e-5    1.81087e-5    2.19536e-5    2.56096e-5    2.87458e-5    3.10471e-5    3.2266e-5    3.2266e-5    3.10471e-5   2.87458e-5   2.56096e-5   2.19536e-5   1.81087e-5   1.43729e-5   1.09768e-5   8.0665e-6    5.70388e-6   3.88089e-6   2.54079e-6   1.6006e-6    9.70223e-7   5.65897e-7   3.17599e-7   1.71513e-7   8.91231e-8   4.45616e-8   2.14391e-8   9.92496e-9   4.42107e-9   1.89497e-9   7.81542e-10
 -3.13109e-10   1.44722e-9    3.509e-9      8.18671e-9    1.83785e-8    3.96998e-8    8.25168e-8    1.65034e-7    3.17599e-7    5.88113e-7    1.0479e-6     1.79661e-6    2.9639e-6     4.70491e-6    7.18644e-6    1.05622e-5    1.49371e-5    2.03263e-5    2.6615e-5     3.35328e-5    4.06526e-5    4.74225e-5    5.32299e-5    5.74915e-5    5.97486e-5   5.97486e-5   5.74915e-5   5.32299e-5   4.74225e-5   4.06526e-5   3.35328e-5   2.6615e-5    2.03263e-5   1.49371e-5   1.05622e-5   7.18644e-6   4.70491e-6   2.9639e-6    1.79661e-6   1.0479e-6    5.88113e-7   3.17599e-7   1.65034e-7   8.25168e-8   3.96998e-8   1.83785e-8   8.18671e-9   3.509e-9     1.44722e-9
 -2.92908e-10   2.57865e-9    6.25233e-9    1.45871e-8    3.27468e-8    7.07371e-8    1.47028e-7    2.94057e-7    5.65897e-7    1.0479e-6     1.86714e-6    3.20119e-6    5.28108e-6    8.38319e-6    1.28048e-5    1.88196e-5    2.6615e-5     3.62174e-5    4.74225e-5    5.97486e-5    7.24348e-5    8.44972e-5    9.4845e-5     0.000102438   0.00010646   0.00010646   0.000102438  9.4845e-5    8.44972e-5   7.24348e-5   5.97486e-5   4.74225e-5   3.62174e-5   2.6615e-5    1.88196e-5   1.28048e-5   8.38319e-6   5.28108e-6   3.20119e-6   1.86714e-6   1.0479e-6    5.65897e-7   2.94057e-7   1.47028e-7   7.07371e-8   3.27468e-8   1.45871e-8   6.25233e-9   2.57865e-9
 -2.72708e-10   4.42107e-9    1.07195e-8    2.50093e-8    5.6144e-8     1.21278e-7    2.52078e-7    5.04156e-7    9.70223e-7    1.79661e-6    3.20119e-6    5.48841e-6    9.05434e-6    1.43729e-5    2.19536e-5    3.2266e-5     4.5631e-5     6.20943e-5    8.13053e-5    0.000102438   0.000124189   0.00014487    0.000162611   0.000175629   0.000182524  0.000182524  0.000175629  0.000162611  0.00014487   0.000124189  0.000102438  8.13053e-5   6.20943e-5   4.5631e-5    3.2266e-5    2.19536e-5   1.43729e-5   9.05434e-6   5.48841e-6   3.20119e-6   1.79661e-6   9.70223e-7   5.04156e-7   2.52078e-7   1.21278e-7   5.6144e-8    2.50093e-8   1.07195e-8   4.42107e-9
 -2.52507e-10   7.29353e-9    1.76843e-8    4.12584e-8    9.2622e-8     2.00075e-7    4.15859e-7    8.31718e-7    1.6006e-6     2.9639e-6     5.28108e-6    9.05434e-6    1.49371e-5    2.37112e-5    3.62174e-5    5.32299e-5    7.52785e-5    0.000102438   0.000134131   0.000168994   0.000204876   0.000238994   0.000268262   0.000289739   0.000301114  0.000301114  0.000289739  0.000268262  0.000238994  0.000204876  0.000168994  0.000134131  0.000102438  7.52785e-5   5.32299e-5   3.62174e-5   2.37112e-5   1.49371e-5   9.05434e-6   5.28108e-6   2.9639e-6    1.6006e-6    8.31718e-7   4.15859e-7   2.00075e-7   9.2622e-8    4.12584e-8   1.76843e-8   7.29353e-9
 -2.32307e-10   1.15778e-8    2.8072e-8     6.54937e-8    1.47028e-7    3.17599e-7    6.60135e-7    1.32027e-6    2.54079e-6    4.70491e-6    8.38319e-6    1.43729e-5    2.37112e-5    3.76392e-5    5.74915e-5    8.44972e-5    0.000119497   0.000162611   0.00021292    0.000268262   0.000325221   0.00037938    0.000425839   0.000459932   0.000477989  0.000477989  0.000459932  0.000425839  0.00037938   0.000325221  0.000268262  0.00021292   0.000162611  0.000119497  8.44972e-5   5.74915e-5   3.76392e-5   2.37112e-5   1.43729e-5   8.38319e-6   4.70491e-6   2.54079e-6   1.32027e-6   6.60135e-7   3.17599e-7   1.47028e-7   6.54937e-8   2.8072e-8    1.15778e-8
 -2.12106e-10   1.76843e-8    4.28782e-8    1.00037e-7    2.24576e-7    4.85111e-7    1.00831e-6    2.01663e-6    3.88089e-6    7.18644e-6    1.28048e-5    2.19536e-5    3.62174e-5    5.74915e-5    8.78145e-5    0.000129064   0.000182524   0.000248377   0.000325221   0.000409753   0.000496754   0.000579478   0.000650442   0.000702516   0.000730097  0.000730097  0.000702516  0.000650442  0.000579478  0.000496754  0.000409753  0.000325221  0.000248377  0.000182524  0.000129064  8.78145e-5   5.74915e-5   3.62174e-5   2.19536e-5   1.28048e-5   7.18644e-6   3.88089e-6   2.01663e-6   1.00831e-6   4.85111e-7   2.24576e-7   1.00037e-7   4.28782e-8   1.76843e-8
 -1.91905e-10   2.59912e-8    6.30196e-8    1.47028e-7    3.30067e-7    7.12985e-7    1.48195e-6    2.9639e-6     5.70388e-6    1.05622e-5    1.88196e-5    3.2266e-5     5.32299e-5    8.44972e-5    0.000129064   0.00018969    0.000268262   0.000365048   0.000477989   0.000602228   0.000730097   0.000851679   0.000955977   0.00103251    0.00107305   0.00107305   0.00103251   0.000955977  0.000851679  0.000730097  0.000602228  0.000477989  0.000365048  0.000268262  0.00018969   0.000129064  8.44972e-5   5.32299e-5   3.2266e-5    1.88196e-5   1.05622e-5   5.70388e-6   2.9639e-6    1.48195e-6   7.12985e-7   3.30067e-7   1.47028e-7   6.30196e-8   2.59912e-8
 -1.71705e-10   3.67571e-8    8.91231e-8    2.07929e-7    4.66786e-7    1.00831e-6    2.0958e-6     4.19159e-6    8.0665e-6     1.49371e-5    2.6615e-5     4.5631e-5     7.52785e-5    0.000119497   0.000182524   0.000268262   0.00037938    0.000516256   0.000675978   0.000851679   0.00103251    0.00120446    0.00135196    0.00146019    0.00151752   0.00151752   0.00146019   0.00135196   0.00120446   0.00103251   0.000851679  0.000675978  0.000516256  0.00037938   0.000268262  0.000182524  0.000119497  7.52785e-5   4.5631e-5    2.6615e-5    1.49371e-5   8.0665e-6    4.19159e-6   2.0958e-6    1.00831e-6   4.66786e-7   2.07929e-7   8.91231e-8   3.67571e-8
 -1.51504e-10   5.00187e-8    1.21278e-7    2.82948e-7    6.35197e-7    1.3721e-6     2.85194e-6    5.70388e-6    1.09768e-5    2.03263e-5    3.62174e-5    6.20943e-5    0.000102438   0.000162611   0.000248377   0.000365048   0.000516256   0.000702516   0.000919864   0.00115896    0.00140503    0.00163901    0.00183973    0.00198702    0.00206502   0.00206502   0.00198702   0.00183973   0.00163901   0.00140503   0.00115896   0.000919864  0.000702516  0.000516256  0.000365048  0.000248377  0.000162611  0.000102438  6.20943e-5   3.62174e-5   2.03263e-5   1.09768e-5   5.70388e-6   2.85194e-6   1.3721e-6    6.35197e-7   2.82948e-7   1.21278e-7   5.00187e-8
 -1.31304e-10   6.54937e-8    1.58799e-7    3.70488e-7    8.31718e-7    1.79661e-6    3.73429e-6    7.46857e-6    1.43729e-5    2.6615e-5     4.74225e-5    8.13053e-5    0.000134131   0.00021292    0.000325221   0.000477989   0.000675978   0.000919864   0.00120446    0.00151752    0.00183973    0.0021461     0.00240891    0.00260177    0.00270391   0.00270391   0.00260177   0.00240891   0.0021461    0.00183973   0.00151752   0.00120446   0.000919864  0.000675978  0.000477989  0.000325221  0.00021292   0.000134131  8.13053e-5   4.74225e-5   2.6615e-5    1.43729e-5   7.46857e-6   3.73429e-6   1.79661e-6   8.31718e-7   3.70488e-7   1.58799e-7   6.54937e-8
 -1.11103e-10   8.25168e-8    2.00075e-7    4.66786e-7    1.0479e-6     2.26359e-6    4.70491e-6    9.40981e-6    1.81087e-5    3.35328e-5    5.97486e-5    0.000102438   0.000168994   0.000268262   0.000409753   0.000602228   0.000851679   0.00115896    0.00151752    0.00191195    0.00231791    0.00270391    0.00303504    0.00327802    0.00340672   0.00340672   0.00327802   0.00303504   0.00270391   0.00231791   0.00191195   0.00151752   0.00115896   0.000851679  0.000602228  0.000409753  0.000268262  0.000168994  0.000102438  5.97486e-5   3.35328e-5   1.81087e-5   9.40981e-6   4.70491e-6   2.26359e-6   1.0479e-6    4.66786e-7   2.00075e-7   8.25168e-8
 -9.09026e-11   1.00037e-7    2.42556e-7    5.65897e-7    1.27039e-6    2.7442e-6     5.70388e-6    1.14078e-5    2.19536e-5    4.06526e-5    7.24348e-5    0.000124189   0.000204876   0.000325221   0.000496754   0.000730097   0.00103251    0.00140503    0.00183973    0.00231791    0.00281007    0.00327802    0.00367946    0.00397403    0.00413005   0.00413005   0.00397403   0.00367946   0.00327802   0.00281007   0.00231791   0.00183973   0.00140503   0.00103251   0.000730097  0.000496754  0.000325221  0.000204876  0.000124189  7.24348e-5   4.06526e-5   2.19536e-5   1.14078e-5   5.70388e-6   2.7442e-6    1.27039e-6   5.65897e-7   2.42556e-7   1.00037e-7
 -7.0702e-11    1.16696e-7    2.82948e-7    6.60135e-7    1.48195e-6    3.20119e-6    6.65374e-6    1.33075e-5    2.56096e-5    4.74225e-5    8.44972e-5    0.00014487    0.000238994   0.00037938    0.000579478   0.000851679   0.00120446    0.00163901    0.0021461     0.00270391    0.00327802    0.00382391    0.00429219    0.00463582    0.00481782   0.00481782   0.00463582   0.00429219   0.00382391   0.00327802   0.00270391   0.0021461    0.00163901   0.00120446   0.000851679  0.000579478  0.00037938   0.000238994  0.00014487   8.44972e-5   4.74225e-5   2.56096e-5   1.33075e-5   6.65374e-6   3.20119e-6   1.48195e-6   6.60135e-7   2.82948e-7   1.16696e-7
 -5.05014e-11   1.30987e-7    3.17599e-7    7.40976e-7    1.66344e-6    3.59322e-6    7.46857e-6    1.49371e-5    2.87458e-5    5.32299e-5    9.4845e-5     0.000162611   0.000268262   0.000425839   0.000650442   0.000955977   0.00135196    0.00183973    0.00240891    0.00303504    0.00367946    0.00429219    0.00481782    0.00520354    0.00540782   0.00540782   0.00520354   0.00481782   0.00429219   0.00367946   0.00303504   0.00240891   0.00183973   0.00135196   0.000955977  0.000650442  0.000425839  0.000268262  0.000162611  9.4845e-5    5.32299e-5   2.87458e-5   1.49371e-5   7.46857e-6   3.59322e-6   1.66344e-6   7.40976e-7   3.17599e-7   1.30987e-7
 -3.03009e-11   1.41474e-7    3.43026e-7    8.00299e-7    1.79661e-6    3.88089e-6    8.0665e-6     1.6133e-5     3.10471e-5    5.74915e-5    0.000102438   0.000175629   0.000289739   0.000459932   0.000702516   0.00103251    0.00146019    0.00198702    0.00260177    0.00327802    0.00397403    0.00463582    0.00520354    0.00562013    0.00584077   0.00584077   0.00562013   0.00520354   0.00463582   0.00397403   0.00327802   0.00260177   0.00198702   0.00146019   0.00103251   0.000702516  0.000459932  0.000289739  0.000175629  0.000102438  5.74915e-5   3.10471e-5   1.6133e-5    8.0665e-6    3.88089e-6   1.79661e-6   8.00299e-7   3.43026e-7   1.41474e-7
 -1.01003e-11   1.47028e-7    3.56492e-7    8.31718e-7    1.86714e-6    4.03325e-6    8.38319e-6    1.67664e-5    3.2266e-5     5.97486e-5    0.00010646    0.000182524   0.000301114   0.000477989   0.000730097   0.00107305    0.00151752    0.00206502    0.00270391    0.00340672    0.00413005    0.00481782    0.00540782    0.00584077    0.00607008   0.00607008   0.00584077   0.00540782   0.00481782   0.00413005   0.00340672   0.00270391   0.00206502   0.00151752   0.00107305   0.000730097  0.000477989  0.000301114  0.000182524  0.00010646   5.97486e-5   3.2266e-5    1.67664e-5   8.38319e-6   4.03325e-6   1.86714e-6   8.31718e-7   3.56492e-7   1.47028e-7
  1.01003e-11   1.47028e-7    3.56492e-7    8.31718e-7    1.86714e-6    4.03325e-6    8.38319e-6    1.67664e-5    3.2266e-5     5.97486e-5    0.00010646    0.000182524   0.000301114   0.000477989   0.000730097   0.00107305    0.00151752    0.00206502    0.00270391    0.00340672    0.00413005    0.00481782    0.00540782    0.00584077    0.00607008   0.00607008   0.00584077   0.00540782   0.00481782   0.00413005   0.00340672   0.00270391   0.00206502   0.00151752   0.00107305   0.000730097  0.000477989  0.000301114  0.000182524  0.00010646   5.97486e-5   3.2266e-5    1.67664e-5   8.38319e-6   4.03325e-6   1.86714e-6   8.31718e-7   3.56492e-7   1.47028e-7
  3.03009e-11   1.41474e-7    3.43026e-7    8.00299e-7    1.79661e-6    3.88089e-6    8.0665e-6     1.6133e-5     3.10471e-5    5.74915e-5    0.000102438   0.000175629   0.000289739   0.000459932   0.000702516   0.00103251    0.00146019    0.00198702    0.00260177    0.00327802    0.00397403    0.00463582    0.00520354    0.00562013    0.00584077   0.00584077   0.00562013   0.00520354   0.00463582   0.00397403   0.00327802   0.00260177   0.00198702   0.00146019   0.00103251   0.000702516  0.000459932  0.000289739  0.000175629  0.000102438  5.74915e-5   3.10471e-5   1.6133e-5    8.0665e-6    3.88089e-6   1.79661e-6   8.00299e-7   3.43026e-7   1.41474e-7
  5.05014e-11   1.30987e-7    3.17599e-7    7.40976e-7    1.66344e-6    3.59322e-6    7.46857e-6    1.49371e-5    2.87458e-5    5.32299e-5    9.4845e-5     0.000162611   0.000268262   0.000425839   0.000650442   0.000955977   0.00135196    0.00183973    0.00240891    0.00303504    0.00367946    0.00429219    0.00481782    0.00520354    0.00540782   0.00540782   0.00520354   0.00481782   0.00429219   0.00367946   0.00303504   0.00240891   0.00183973   0.00135196   0.000955977  0.000650442  0.000425839  0.000268262  0.000162611  9.4845e-5    5.32299e-5   2.87458e-5   1.49371e-5   7.46857e-6   3.59322e-6   1.66344e-6   7.40976e-7   3.17599e-7   1.30987e-7
  7.0702e-11    1.16696e-7    2.82948e-7    6.60135e-7    1.48195e-6    3.20119e-6    6.65374e-6    1.33075e-5    2.56096e-5    4.74225e-5    8.44972e-5    0.00014487    0.000238994   0.00037938    0.000579478   0.000851679   0.00120446    0.00163901    0.0021461     0.00270391    0.00327802    0.00382391    0.00429219    0.00463582    0.00481782   0.00481782   0.00463582   0.00429219   0.00382391   0.00327802   0.00270391   0.0021461    0.00163901   0.00120446   0.000851679  0.000579478  0.00037938   0.000238994  0.00014487   8.44972e-5   4.74225e-5   2.56096e-5   1.33075e-5   6.65374e-6   3.20119e-6   1.48195e-6   6.60135e-7   2.82948e-7   1.16696e-7
  9.09026e-11   1.00037e-7    2.42556e-7    5.65897e-7    1.27039e-6    2.7442e-6     5.70388e-6    1.14078e-5    2.19536e-5    4.06526e-5    7.24348e-5    0.000124189   0.000204876   0.000325221   0.000496754   0.000730097   0.00103251    0.00140503    0.00183973    0.00231791    0.00281007    0.00327802    0.00367946    0.00397403    0.00413005   0.00413005   0.00397403   0.00367946   0.00327802   0.00281007   0.00231791   0.00183973   0.00140503   0.00103251   0.000730097  0.000496754  0.000325221  0.000204876  0.000124189  7.24348e-5   4.06526e-5   2.19536e-5   1.14078e-5   5.70388e-6   2.7442e-6    1.27039e-6   5.65897e-7   2.42556e-7   1.00037e-7
  1.11103e-10   8.25168e-8    2.00075e-7    4.66786e-7    1.0479e-6     2.26359e-6    4.70491e-6    9.40981e-6    1.81087e-5    3.35328e-5    5.97486e-5    0.000102438   0.000168994   0.000268262   0.000409753   0.000602228   0.000851679   0.00115896    0.00151752    0.00191195    0.00231791    0.00270391    0.00303504    0.00327802    0.00340672   0.00340672   0.00327802   0.00303504   0.00270391   0.00231791   0.00191195   0.00151752   0.00115896   0.000851679  0.000602228  0.000409753  0.000268262  0.000168994  0.000102438  5.97486e-5   3.35328e-5   1.81087e-5   9.40981e-6   4.70491e-6   2.26359e-6   1.0479e-6    4.66786e-7   2.00075e-7   8.25168e-8
  1.31304e-10   6.54937e-8    1.58799e-7    3.70488e-7    8.31718e-7    1.79661e-6    3.73429e-6    7.46857e-6    1.43729e-5    2.6615e-5     4.74225e-5    8.13053e-5    0.000134131   0.00021292    0.000325221   0.000477989   0.000675978   0.000919864   0.00120446    0.00151752    0.00183973    0.0021461     0.00240891    0.00260177    0.00270391   0.00270391   0.00260177   0.00240891   0.0021461    0.00183973   0.00151752   0.00120446   0.000919864  0.000675978  0.000477989  0.000325221  0.00021292   0.000134131  8.13053e-5   4.74225e-5   2.6615e-5    1.43729e-5   7.46857e-6   3.73429e-6   1.79661e-6   8.31718e-7   3.70488e-7   1.58799e-7   6.54937e-8
  1.51504e-10   5.00187e-8    1.21278e-7    2.82948e-7    6.35197e-7    1.3721e-6     2.85194e-6    5.70388e-6    1.09768e-5    2.03263e-5    3.62174e-5    6.20943e-5    0.000102438   0.000162611   0.000248377   0.000365048   0.000516256   0.000702516   0.000919864   0.00115896    0.00140503    0.00163901    0.00183973    0.00198702    0.00206502   0.00206502   0.00198702   0.00183973   0.00163901   0.00140503   0.00115896   0.000919864  0.000702516  0.000516256  0.000365048  0.000248377  0.000162611  0.000102438  6.20943e-5   3.62174e-5   2.03263e-5   1.09768e-5   5.70388e-6   2.85194e-6   1.3721e-6    6.35197e-7   2.82948e-7   1.21278e-7   5.00187e-8
  1.71705e-10   3.67571e-8    8.91231e-8    2.07929e-7    4.66786e-7    1.00831e-6    2.0958e-6     4.19159e-6    8.0665e-6     1.49371e-5    2.6615e-5     4.5631e-5     7.52785e-5    0.000119497   0.000182524   0.000268262   0.00037938    0.000516256   0.000675978   0.000851679   0.00103251    0.00120446    0.00135196    0.00146019    0.00151752   0.00151752   0.00146019   0.00135196   0.00120446   0.00103251   0.000851679  0.000675978  0.000516256  0.00037938   0.000268262  0.000182524  0.000119497  7.52785e-5   4.5631e-5    2.6615e-5    1.49371e-5   8.0665e-6    4.19159e-6   2.0958e-6    1.00831e-6   4.66786e-7   2.07929e-7   8.91231e-8   3.67571e-8
  1.91905e-10   2.59912e-8    6.30196e-8    1.47028e-7    3.30067e-7    7.12985e-7    1.48195e-6    2.9639e-6     5.70388e-6    1.05622e-5    1.88196e-5    3.2266e-5     5.32299e-5    8.44972e-5    0.000129064   0.00018969    0.000268262   0.000365048   0.000477989   0.000602228   0.000730097   0.000851679   0.000955977   0.00103251    0.00107305   0.00107305   0.00103251   0.000955977  0.000851679  0.000730097  0.000602228  0.000477989  0.000365048  0.000268262  0.00018969   0.000129064  8.44972e-5   5.32299e-5   3.2266e-5    1.88196e-5   1.05622e-5   5.70388e-6   2.9639e-6    1.48195e-6   7.12985e-7   3.30067e-7   1.47028e-7   6.30196e-8   2.59912e-8
  2.12106e-10   1.76843e-8    4.28782e-8    1.00037e-7    2.24576e-7    4.85111e-7    1.00831e-6    2.01663e-6    3.88089e-6    7.18644e-6    1.28048e-5    2.19536e-5    3.62174e-5    5.74915e-5    8.78145e-5    0.000129064   0.000182524   0.000248377   0.000325221   0.000409753   0.000496754   0.000579478   0.000650442   0.000702516   0.000730097  0.000730097  0.000702516  0.000650442  0.000579478  0.000496754  0.000409753  0.000325221  0.000248377  0.000182524  0.000129064  8.78145e-5   5.74915e-5   3.62174e-5   2.19536e-5   1.28048e-5   7.18644e-6   3.88089e-6   2.01663e-6   1.00831e-6   4.85111e-7   2.24576e-7   1.00037e-7   4.28782e-8   1.76843e-8
  2.32307e-10   1.15778e-8    2.8072e-8     6.54937e-8    1.47028e-7    3.17599e-7    6.60135e-7    1.32027e-6    2.54079e-6    4.70491e-6    8.38319e-6    1.43729e-5    2.37112e-5    3.76392e-5    5.74915e-5    8.44972e-5    0.000119497   0.000162611   0.00021292    0.000268262   0.000325221   0.00037938    0.000425839   0.000459932   0.000477989  0.000477989  0.000459932  0.000425839  0.00037938   0.000325221  0.000268262  0.00021292   0.000162611  0.000119497  8.44972e-5   5.74915e-5   3.76392e-5   2.37112e-5   1.43729e-5   8.38319e-6   4.70491e-6   2.54079e-6   1.32027e-6   6.60135e-7   3.17599e-7   1.47028e-7   6.54937e-8   2.8072e-8    1.15778e-8
  2.52507e-10   7.29353e-9    1.76843e-8    4.12584e-8    9.2622e-8     2.00075e-7    4.15859e-7    8.31718e-7    1.6006e-6     2.9639e-6     5.28108e-6    9.05434e-6    1.49371e-5    2.37112e-5    3.62174e-5    5.32299e-5    7.52785e-5    0.000102438   0.000134131   0.000168994   0.000204876   0.000238994   0.000268262   0.000289739   0.000301114  0.000301114  0.000289739  0.000268262  0.000238994  0.000204876  0.000168994  0.000134131  0.000102438  7.52785e-5   5.32299e-5   3.62174e-5   2.37112e-5   1.49371e-5   9.05434e-6   5.28108e-6   2.9639e-6    1.6006e-6    8.31718e-7   4.15859e-7   2.00075e-7   9.2622e-8    4.12584e-8   1.76843e-8   7.29353e-9
  2.72708e-10   4.42107e-9    1.07195e-8    2.50093e-8    5.6144e-8     1.21278e-7    2.52078e-7    5.04156e-7    9.70223e-7    1.79661e-6    3.20119e-6    5.48841e-6    9.05434e-6    1.43729e-5    2.19536e-5    3.2266e-5     4.5631e-5     6.20943e-5    8.13053e-5    0.000102438   0.000124189   0.00014487    0.000162611   0.000175629   0.000182524  0.000182524  0.000175629  0.000162611  0.00014487   0.000124189  0.000102438  8.13053e-5   6.20943e-5   4.5631e-5    3.2266e-5    2.19536e-5   1.43729e-5   9.05434e-6   5.48841e-6   3.20119e-6   1.79661e-6   9.70223e-7   5.04156e-7   2.52078e-7   1.21278e-7   5.6144e-8    2.50093e-8   1.07195e-8   4.42107e-9
  2.92908e-10   2.57865e-9    6.25233e-9    1.45871e-8    3.27468e-8    7.07371e-8    1.47028e-7    2.94057e-7    5.65897e-7    1.0479e-6     1.86714e-6    3.20119e-6    5.28108e-6    8.38319e-6    1.28048e-5    1.88196e-5    2.6615e-5     3.62174e-5    4.74225e-5    5.97486e-5    7.24348e-5    8.44972e-5    9.4845e-5     0.000102438   0.00010646   0.00010646   0.000102438  9.4845e-5    8.44972e-5   7.24348e-5   5.97486e-5   4.74225e-5   3.62174e-5   2.6615e-5    1.88196e-5   1.28048e-5   8.38319e-6   5.28108e-6   3.20119e-6   1.86714e-6   1.0479e-6    5.65897e-7   2.94057e-7   1.47028e-7   7.07371e-8   3.27468e-8   1.45871e-8   6.25233e-9   2.57865e-9
  3.13109e-10   1.44722e-9    3.509e-9      8.18671e-9    1.83785e-8    3.96998e-8    8.25168e-8    1.65034e-7    3.17599e-7    5.88113e-7    1.0479e-6     1.79661e-6    2.9639e-6     4.70491e-6    7.18644e-6    1.05622e-5    1.49371e-5    2.03263e-5    2.6615e-5     3.35328e-5    4.06526e-5    4.74225e-5    5.32299e-5    5.74915e-5    5.97486e-5   5.97486e-5   5.74915e-5   5.32299e-5   4.74225e-5   4.06526e-5   3.35328e-5   2.6615e-5    2.03263e-5   1.49371e-5   1.05622e-5   7.18644e-6   4.70491e-6   2.9639e-6    1.79661e-6   1.0479e-6    5.88113e-7   3.17599e-7   1.65034e-7   8.25168e-8   3.96998e-8   1.83785e-8   8.18671e-9   3.509e-9     1.44722e-9
  3.33309e-10   7.81542e-10   1.89497e-9    4.42107e-9    9.92496e-9    2.14391e-8    4.45616e-8    8.91231e-8    1.71513e-7    3.17599e-7    5.65897e-7    9.70223e-7    1.6006e-6     2.54079e-6    3.88089e-6    5.70388e-6    8.0665e-6     1.09768e-5    1.43729e-5    1.81087e-5    2.19536e-5    2.56096e-5    2.87458e-5    3.10471e-5    3.2266e-5    3.2266e-5    3.10471e-5   2.87458e-5   2.56096e-5   2.19536e-5   1.81087e-5   1.43729e-5   1.09768e-5   8.0665e-6    5.70388e-6   3.88089e-6   2.54079e-6   1.6006e-6    9.70223e-7   5.65897e-7   3.17599e-7   1.71513e-7   8.91231e-8   4.45616e-8   2.14391e-8   9.92496e-9   4.42107e-9   1.89497e-9   7.81542e-10
  3.5351e-10    4.06112e-10   9.84681e-10   2.29732e-9    5.1573e-9     1.11404e-8    2.31555e-8    4.6311e-8     8.91231e-8    1.65034e-7    2.94057e-7    5.04156e-7    8.31718e-7    1.32027e-6    2.01663e-6    2.9639e-6     4.19159e-6    5.70388e-6    7.46857e-6    9.40981e-6    1.14078e-5    1.33075e-5    1.49371e-5    1.6133e-5     1.67664e-5   1.67664e-5   1.6133e-5    1.49371e-5   1.33075e-5   1.14078e-5   9.40981e-6   7.46857e-6   5.70388e-6   4.19159e-6   2.9639e-6    2.01663e-6   1.32027e-6   8.31718e-7   5.04156e-7   2.94057e-7   1.65034e-7   8.91231e-8   4.6311e-8    2.31555e-8   1.11404e-8   5.1573e-9    2.29732e-9   9.84681e-10  4.06112e-10
  3.73711e-10   2.03056e-10   4.9234e-10    1.14866e-9    2.57865e-9    5.57019e-9    1.15778e-8    2.31555e-8    4.45616e-8    8.25168e-8    1.47028e-7    2.52078e-7    4.15859e-7    6.60135e-7    1.00831e-6    1.48195e-6    2.0958e-6     2.85194e-6    3.73429e-6    4.70491e-6    5.70388e-6    6.65374e-6    7.46857e-6    8.0665e-6     8.38319e-6   8.38319e-6   8.0665e-6    7.46857e-6   6.65374e-6   5.70388e-6   4.70491e-6   3.73429e-6   2.85194e-6   2.0958e-6    1.48195e-6   1.00831e-6   6.60135e-7   4.15859e-7   2.52078e-7   1.47028e-7   8.25168e-8   4.45616e-8   2.31555e-8   1.15778e-8   5.57019e-9   2.57865e-9   1.14866e-9   4.9234e-10   2.03056e-10
  3.93911e-10   9.76927e-11   2.36871e-10   5.52633e-10   1.24062e-9    2.67989e-9    5.57019e-9    1.11404e-8    2.14391e-8    3.96998e-8    7.07371e-8    1.21278e-7    2.00075e-7    3.17599e-7    4.85111e-7    7.12985e-7    1.00831e-6    1.3721e-6     1.79661e-6    2.26359e-6    2.7442e-6     3.20119e-6    3.59322e-6    3.88089e-6    4.03325e-6   4.03325e-6   3.88089e-6   3.59322e-6   3.20119e-6   2.7442e-6    2.26359e-6   1.79661e-6   1.3721e-6    1.00831e-6   7.12985e-7   4.85111e-7   3.17599e-7   2.00075e-7   1.21278e-7   7.07371e-8   3.96998e-8   2.14391e-8   1.11404e-8   5.57019e-9   2.67989e-9   1.24062e-9   5.52633e-10  2.36871e-10  9.76927e-11
  4.14112e-10   4.52256e-11   1.09656e-10   2.55835e-10   5.74329e-10   1.24062e-9    2.57865e-9    5.1573e-9     9.92496e-9    1.83785e-8    3.27468e-8    5.6144e-8     9.2622e-8     1.47028e-7    2.24576e-7    3.30067e-7    4.66786e-7    6.35197e-7    8.31718e-7    1.0479e-6     1.27039e-6    1.48195e-6    1.66344e-6    1.79661e-6    1.86714e-6   1.86714e-6   1.79661e-6   1.66344e-6   1.48195e-6   1.27039e-6   1.0479e-6    8.31718e-7   6.35197e-7   4.66786e-7   3.30067e-7   2.24576e-7   1.47028e-7   9.2622e-8    5.6144e-8    3.27468e-8   1.83785e-8   9.92496e-9   5.1573e-9    2.57865e-9   1.24062e-9   5.74329e-10  2.55835e-10  1.09656e-10  4.52256e-11
  4.34312e-10   2.01457e-11   4.88463e-11   1.13961e-10   2.55835e-10   5.52633e-10   1.14866e-9    2.29732e-9    4.42107e-9    8.18671e-9    1.45871e-8    2.50093e-8    4.12584e-8    6.54937e-8    1.00037e-7    1.47028e-7    2.07929e-7    2.82948e-7    3.70488e-7    4.66786e-7    5.65897e-7    6.60135e-7    7.40976e-7    8.00299e-7    8.31718e-7   8.31718e-7   8.00299e-7   7.40976e-7   6.60135e-7   5.65897e-7   4.66786e-7   3.70488e-7   2.82948e-7   2.07929e-7   1.47028e-7   1.00037e-7   6.54937e-8   4.12584e-8   2.50093e-8   1.45871e-8   8.18671e-9   4.42107e-9   2.29732e-9   1.14866e-9   5.52633e-10  2.55835e-10  1.13961e-10  4.88463e-11  2.01457e-11
  4.54513e-10   8.6349e-12    2.09366e-11   4.88463e-11   1.09656e-10   2.36871e-10   4.9234e-10    9.84681e-10   1.89497e-9    3.509e-9      6.25233e-9    1.07195e-8    1.76843e-8    2.8072e-8     4.28782e-8    6.30196e-8    8.91231e-8    1.21278e-7    1.58799e-7    2.00075e-7    2.42556e-7    2.82948e-7    3.17599e-7    3.43026e-7    3.56492e-7   3.56492e-7   3.43026e-7   3.17599e-7   2.82948e-7   2.42556e-7   2.00075e-7   1.58799e-7   1.21278e-7   8.91231e-8   6.30196e-8   4.28782e-8   2.8072e-8    1.76843e-8   1.07195e-8   6.25233e-9   3.509e-9     1.89497e-9   9.84681e-10  4.9234e-10   2.36871e-10  1.09656e-10  4.88463e-11  2.09366e-11  8.6349e-12
  4.74713e-10   3.56129e-12   8.6349e-12    2.01457e-11   4.52256e-11   9.76927e-11   2.03056e-10   4.06112e-10   7.81542e-10   1.44722e-9    2.57865e-9    4.42107e-9    7.29353e-9    1.15778e-8    1.76843e-8    2.59912e-8    3.67571e-8    5.00187e-8    6.54937e-8    8.25168e-8    1.00037e-7    1.16696e-7    1.30987e-7    1.41474e-7    1.47028e-7   1.47028e-7   1.41474e-7   1.30987e-7   1.16696e-7   1.00037e-7   8.25168e-8   6.54937e-8   5.00187e-8   3.67571e-8   2.59912e-8   1.76843e-8   1.15778e-8   7.29353e-9   4.42107e-9   2.57865e-9   1.44722e-9   7.81542e-10  4.06112e-10  2.03056e-10  9.76927e-11  4.52256e-11  2.01457e-11  8.6349e-12   3.56129e-12

We use again use a GMRF prior similar to the Imaging a Black Hole using only Closure Quantities tutorial for the log-ratio transformed image. We use the same correlated image prior for the inverse-logit transformed total polarization. The mean total polarization fraction p0 is centered at -2.0 with a standard deviation of 2.0 which logit transformed puts most of the prior mass < 0.8 fractional polarization. The standard deviation of the total polarization fraction pσ again uses a Half-normal process. The angular parameters of the polarizaton are given by a uniform prior on the sphere.

julia

cprior = corr_image_prior(grid, dvis; order = 2)

HierarchicalPrior(
	map: 
	ConditionalMarkov(
Random Field: VLBIImagePriors.GaussMarkovRandomField
Graph: MarkovRandomFieldGraph{2}(
dims: (48, 48)
)
)	hyper prior: 
	ProbabilityTransports.Truncated{ProbabilityTransports.PushforwardDistribution{ProbabilityTransports.ScaleShift{Float64, Float64}, ProbabilityTransports.StdInverseGamma{Float64, Float64, 0, Float64}, 0, Float64}, Float64, Int64, Float64}(
untruncated: PushforwardDistribution(f=ScaleShift, base=StdInverseGamma)
lower: 1.0
upper: 96
logtp: -0.28744561520404394
lcdf: 1.0369068061208359e-12
)

)

The total flux is fixed to 0.6 since it is degenerate with an overall shift in the gain amplitudes.

julia

skym = sky(grid; mimg, ftot = 0.6, cprior)

SkyModel
  with map: _sky_sky
   on grid: 
RectiGrid(
executor: ComradeBase.Serial()
Dimensions: 
(↓ X Sampled{Float64} LinRange{Float64}(-4.747133960864206e-10, 4.747133960864206e-10, 48) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-4.747133960864206e-10, 4.747133960864206e-10, 48) ForwardOrdered Regular Points)
)
   )

Now we build the instrument model. Due to the complexity of VLBI the instrument model is critical to the success of imaging and getting reliable results. For this example we will use the standard instrument model used in polarized EHT analyses expressed in the RIME formalism. Our Jones decomposition will be given by GDR, where G are the complex gains, D are the d-terms, and R is what we call the ideal instrument response, which is how an ideal interferometer using the feed basis we observe relative to some reference basis.

Given the possible flexibility in different parameterizations of the individual Jones matrices each Jones matrix requires the user to specify a function that converts from parameters to specific parameterization f the jones matrices.

We bundle the whole instrument model — the Jones matrices and their priors — in a single block with the @instrument macro. Each Jones term is its own @jones block: the name ~ ArrayPrior(...) lines inside are that term's priors, and the block builds and returns the Jones matrix. The macro lifts each @jones body — and the combination function — into a named function for us (named functions, unlike closures, serialize reliably).

The individual Jones matrices are:

JonesG — the complex gains, in an amplitude/phase decomposition. The gain matrix is diagonal, so it returns a 2-tuple (R feed, L feed).
JonesD — the leakage (d-)terms in the small-leakage limit [1 d1; d2 1], using a re-im parameterization, returning a 2-tuple.
JonesR — the response matrix (basis transform plus feed rotation). It has no free parameters, so its @jones block has no ~ lines.

We combine them with JonesSandwich using the standard J = adjoint(R)*G*D*R.

Note

For arrays with non-zero leakage, feed rotation calibration does not remove the impact of feed rotations on the instrument model: the R and D matrices do not commute, so feed rotation must be modelled. This is why we keep add_fr = true and apply adjoint(R) in the composition. This is especially important for interferometers using a mixture of linear and circular feeds.

For the instrument prior we use IID priors for the gains and d-terms. The IIDSitePrior segments are ScanSeg() (independent per scan), TrackSeg() (constant over the track), and IntegSeg() (changes each integration time). For released EHT data the gains are stable over a scan, while the d-terms are stable over the track.

julia

@instrument function instrument()
    # Complex gains: amplitude/phase decomposition, returning (gR, gL).
    G = @jones begin
        lgR ~ ArrayPrior(IIDSitePrior(ScanSeg(), VLBIGaussian(0.0, 0.2)); LM = IIDSitePrior(ScanSeg(), VLBIGaussian(0.0, 1.0)))
        lgrat ~ ArrayPrior(IIDSitePrior(ScanSeg(), VLBIGaussian(0.0, 0.1)))
        gpR ~ ArrayPrior(IIDSitePrior(ScanSeg(), DiagonalVonMises(0.0, inv(π^2))); refant = SEFDReference(0.0), phase = true)
        gprat ~ ArrayPrior(IIDSitePrior(ScanSeg(), DiagonalVonMises(0.0, inv(0.1^2))); refant = SingleReference(:AA, 0.0), phase = true)
        gR = exp(complex(lgR, gpR))
        gL = gR * exp(complex(lgrat, gprat))
        return JonesG((gR, gL))
    end
    # Leakage/d-terms (re-im parameterization), returning (dR, dL).
    D = @jones begin
        dRx ~ ArrayPrior(IIDSitePrior(TrackSeg(), VLBIGaussian(0.0, 0.2)))
        dRy ~ ArrayPrior(IIDSitePrior(TrackSeg(), VLBIGaussian(0.0, 0.2)))
        dLx ~ ArrayPrior(IIDSitePrior(TrackSeg(), VLBIGaussian(0.0, 0.2)))
        dLy ~ ArrayPrior(IIDSitePrior(TrackSeg(), VLBIGaussian(0.0, 0.2)))
        return JonesD((complex(dRx, dRy), complex(dLx, dLy)))
    end
    # The ideal response (basis transform + feed rotation); no free parameters.
    R = @jones begin
        return JonesR(; add_fr = true)
    end
    # Combine into the full Jones matrix J = adjoint(R)*G*D*R.
    return JonesSandwich(G, D, R) do g, d, r
        return adjoint(r) * g * d * r
    end
end

instrument (generic function with 1 method)

Building the instrument model is now a single call.

julia

intmodel = instrument()

InstrumentModel
  with Jones: JonesSandwich
  with reference basis: PolarizedTypes.CirBasis()

intmodel = InstrumentModel(JonesR(;add_fr=true)) Putting it all together, we form our likelihood and posterior objects for optimization and sampling, and specifying to use Enzyme.Reverse with runtime activity for AD.

julia

using Enzyme
post = VLBIPosterior(skym, intmodel, dvis);

Reconstructing the Image and Instrument Effects

To sample from this posterior, it is convenient to move from our constrained parameter space to an unconstrained one (i.e., the support of the transported posterior is (-∞, ∞)). We do this with transport_to(post, TVFlat()), which maps the posterior into the flat (TVFlat) latent space.

julia

tpost = transport_to(post, TVFlat());

We can also query the dimension of our posterior or the number of parameters we will sample.

Warning

This can often be different from what you would expect. This difference is especially true when using angular variables, where we often artificially increase the dimension of the parameter space to make sampling easier.

Now we optimize. Unlike other imaging examples, we move straight to gradient optimizers due to the higher dimension of the space. In addition the only AD package that can currently work with the polarized Comrade posterior is Enzyme.

julia

using Optimization, OptimizationLBFGSB
xopt, sol = comrade_opt(
    post, LBFGSB();
    initial_params = prior_sample(rng, post), maxiters = 2_000
)

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Final objective value:     -17139.898090465696
)

Now let's evaluate our fits by plotting the residuals

julia

using CairoMakie
res = residuals(post, xopt)
fig = Figure(; size = (800, 600))
plotfields!(fig[1, 1], res[1], uvdist, x -> Comrade.measurement(x)[1, 1] / noise(x)[1, 1], axis_kwargs = (; ylabel = "RR Residual"))
plotfields!(fig[2, 1], res[1], uvdist, x -> Comrade.measurement(x)[2, 1] / noise(x)[2, 1], axis_kwargs = (; ylabel = "LR Residual"))
plotfields!(fig[1, 2], res[1], uvdist, x -> Comrade.measurement(x)[1, 2] / noise(x)[1, 2], axis_kwargs = (; ylabel = "RL Residual"))
plotfields!(fig[2, 2], res[1], uvdist, x -> Comrade.measurement(x)[2, 2] / noise(x)[2, 2], axis_kwargs = (; ylabel = "LL Residual"))
fig |> DisplayAs.PNG |> DisplayAs.Text

These look reasonable, although there may be some minor overfitting. Let's plot the results

julia

gpl = refinespatial(grid, 2)
img = intensitymap(skymodel(post, xopt), gpl)
fig = imageviz(
    img, adjust_length = true, colormap = :cmr_gothic, pcolormap = :rainbow1,
    pcolorrange = (0.0, 0.3), plot_total = false
);
fig |> DisplayAs.PNG |> DisplayAs.Text

Note

The image looks a little noisy. This is an artifact of the MAP image. To get a publication quality image we recommend sampling from the posterior and averaging the samples. The results will be essentially identical to the results from EHTC VII.

We can also analyze the instrument model. For example, we can look at the gain ratios and products. To grab the ratios and products we can use the caltable function which will return analyze the gprat array and convert it to a uniform table. We can then plot the gain phases and amplitudes.

julia

gphase_ratio = caltable(xopt.instrument.gprat)
gamp_ratio = caltable(exp.(xopt.instrument.lgrat))

┌─────────┬───────────────────┬─────────┬─────────┬─────────┬─────────┬─────────
│      Ti │                Fr │      AA │      AP │      AZ │      JC │      L ⋯
│     Any │               Any │     Any │     Any │     Any │     Any │     An ⋯
├─────────┼───────────────────┼─────────┼─────────┼─────────┼─────────┼─────────
│ 0.92 hr │ 227.070705664 GHz │   1.137 │ missing │ missing │ missing │ missin ⋯
│ 1.22 hr │ 227.070705664 GHz │   1.132 │   1.098 │ missing │ missing │   0.96 ⋯
│ 1.52 hr │ 227.070705664 GHz │   1.112 │    1.13 │ missing │ missing │   0.98 ⋯
│ 1.82 hr │ 227.070705664 GHz │   1.165 │   1.146 │ missing │ missing │   0.96 ⋯
│ 2.12 hr │ 227.070705664 GHz │   0.996 │   0.923 │ missing │ missing │   1.08 ⋯
│ 2.45 hr │ 227.070705664 GHz │ missing │   1.086 │ missing │ missing │   1.16 ⋯
│ 2.75 hr │ 227.070705664 GHz │   1.095 │   1.006 │ missing │ missing │   1.05 ⋯
│ 3.05 hr │ 227.070705664 GHz │    1.05 │   1.027 │ missing │ missing │   1.12 ⋯
│ 3.35 hr │ 227.070705664 GHz │   1.055 │   1.073 │ missing │ missing │   1.11 ⋯
│ 3.68 hr │ 227.070705664 GHz │   1.051 │   0.951 │   1.048 │ missing │   1.14 ⋯
│ 3.98 hr │ 227.070705664 GHz │   1.057 │   0.941 │   1.033 │ missing │   1.14 ⋯
│ 4.28 hr │ 227.070705664 GHz │    1.04 │    1.02 │   1.062 │ missing │    1.0 ⋯
│ 4.58 hr │ 227.070705664 GHz │   1.014 │   1.001 │   1.094 │     1.0 │   1.08 ⋯
│ 4.92 hr │ 227.070705664 GHz │   1.031 │   0.904 │   1.085 │     1.0 │   1.05 ⋯
│ 5.18 hr │ 227.070705664 GHz │   1.034 │   1.038 │   1.081 │     1.0 │   1.08 ⋯
│       ⋮ │                 ⋮ │       ⋮ │       ⋮ │       ⋮ │       ⋮ │        ⋱
└─────────┴───────────────────┴─────────┴─────────┴─────────┴─────────┴─────────
                                                   3 columns and 10 rows omitted

Plotting the phases first, we see large trends in the righ circular polarization phase. This is expected due to a lack of image centroid and the absense of absolute phase information in VLBI. However, the gain phase difference between the left and right circular polarization is stable and close to zero. This is expected since gain ratios are typically stable over the course of an observation and the constant offset was removed in the EHT calibration process.

julia

gphaseR = caltable(xopt.instrument.gpR)
fig = plotcaltable(gphaseR, gphase_ratio, labels = ["R Phase", "L/R Phase"]);
fig |> DisplayAs.PNG |> DisplayAs.Text

Moving to the amplitudes we see largely stable gain amplitudes on the right circular polarization except for LMT which is known and due to pointing issues during the 2017 observation. Again the gain ratios are stable and close to unity. Typically we expect that apriori calibration should make the gain ratios close to unity.

julia

gampr = caltable(exp.(xopt.instrument.lgR))
fig = plotcaltable(gampr, gamp_ratio, labels = ["R Amp", "L/R Amp"], axis_kwargs = (; limits = (nothing, (0.6, 1.3))));
fig |> DisplayAs.PNG |> DisplayAs.Text

To sample from the posterior, you can then just use the sample function from AdvancedHMC like in the other imaging examples. For example

julia

using AdvancedHMC
chain = sample(rng, post, NUTS(0.8), 2_000, n_adapts = 1000, progress = true, initial_params = xopt)

This page was generated using Literate.jl.

Hamaker J.P, Bregman J.D., Sault R.J. (1996) [https://articles.adsabs.harvard.edu/pdf/1996A%26AS..117..137H] ↩︎
Pesce D. (2021) [https://ui.adsabs.harvard.edu/abs/2021AJ....161..178P/abstract] ↩︎

Polarized Image and Instrumental Modeling ​

Introduction to Polarized Imaging ​

Load the Data ​

Load the Data ​

Reconstructing the Image and Instrument Effects ​

Polarized Image and Instrumental Modeling

Introduction to Polarized Imaging

Load the Data

Load the Data

Reconstructing the Image and Instrument Effects