Stokes I Simultaneous Image and Instrument Modeling

In this tutorial, we will create a preliminary reconstruction of the 2017 M87 data on April 6 by simultaneously creating an image and model for the instrument. By instrument model, we mean something akin to self-calibration in traditional VLBI imaging terminology. However, unlike traditional self-cal, we will solve for the gains each time we update the image self-consistently. This allows us to model the correlations between gains and the image.

To get started we load Comrade.

using Comrade


using Pyehtim
using LinearAlgebra

    CondaPkg Found dependencies: /home/runner/.julia/packages/CondaPkg/8GjrP/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/PythonCall/83z4q/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/Pyehtim/bQtHC/CondaPkg.toml
    CondaPkg Resolving changes
             + ehtim (pip)
             + libstdcxx
             + libstdcxx-ng
             + numpy
             + numpy (pip)
             + openssl
             + pandas
             + python
             + setuptools (pip)
             + uv
    CondaPkg Initialising pixi
             │ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
             │ init
             │ --format pixi
             └ /home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/StokesIImaging/.CondaPkg
✔ Created /home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/StokesIImaging/.CondaPkg/pixi.toml
    CondaPkg Wrote /home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/StokesIImaging/.CondaPkg/pixi.toml
             │ [dependencies]
             │ openssl = ">=3, <3.6"
             │ libstdcxx = ">=3.4,<15.0"
             │ uv = ">=0.4"
             │ libstdcxx-ng = ">=3.4,<15.0"
             │ pandas = "<2"
             │ numpy = ">=1.24, <2.0"
             │ 
             │     [dependencies.python]
             │     channel = "conda-forge"
             │     build = "*cp*"
             │     version = ">=3.10,!=3.14.0,!=3.14.1,<4, >=3.6,<=3.12"
             │ 
             │ [project]
             │ name = ".CondaPkg"
             │ platforms = ["linux-64"]
             │ channels = ["conda-forge"]
             │ channel-priority = "strict"
             │ description = "automatically generated by CondaPkg.jl"
             │ 
             │ [pypi-dependencies]
             │ ehtim = ">=1.2.10, <2.0"
             │ numpy = ">=1.24, <2.0"
             └ setuptools = "*"
    CondaPkg Installing packages
             │ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
             │ install
             └ --manifest-path /home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/StokesIImaging/.CondaPkg/pixi.toml
✔ The default environment has been installed.

For reproducibility we use a stable random number genreator

using StableRNGs
rng = StableRNG(42)

StableRNGs.LehmerRNG(state=0x00000000000000000000000000000055)

Load the Data

To download the data visit https://doi.org/10.25739/g85n-f134 First we will load our data:

obs = ehtim.obsdata.load_uvfits(joinpath(__DIR, "..", "..", "Data", "SR1_M87_2017_096_lo_hops_netcal_StokesI.uvfits"))

Python: <ehtim.obsdata.Obsdata object at 0x7effd0928710>

Now we do some minor preprocessing:

• Scan average the data since the data have been preprocessed so that the gain phases coherent.

• Add 1% systematic noise to deal with calibration issues that cause 1% non-closing errors.

obs = scan_average(obs).add_fractional_noise(0.02)

Python: <ehtim.obsdata.Obsdata object at 0x7effd0a61490>

Now we extract our complex visibilities.

dvis = extract_table(obs, Visibilities())

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

##Building the Model/Posterior

Now, we must build our intensity/visibility model. That is, the model that takes in a named tuple of parameters and perhaps some metadata required to construct the model. For our model, we will use a raster or ContinuousImage for our image model. The model is given below:

The model construction is very similar to Imaging a Black Hole using only Closure Quantities, except we include a large scale gaussian since we want to model the zero baselines. For more information about the image model please read the closure-only example.

function sky(θ, metadata)
    (; fg, c, σimg) = θ
    (; ftot, mimg) = metadata
    # Apply the GMRF fluctuations to the image
    rast = apply_fluctuations(CenteredLR(), mimg, σimg .* c.params)
    pimg = parent(rast)
    @. pimg = (ftot * (1 - fg)) * pimg
    m = ContinuousImage(rast, BSplinePulse{3}())
    # Add a large-scale gaussian to deal with the over-resolved mas flux
    g = modify(Gaussian(), Stretch(μas2rad(500.0), μas2rad(500.0)), Renormalize(ftot * fg))
    x, y = centroid(m)
    return shifted(m, -x, -y) + g
end

sky (generic function with 1 method)

Now, let's set up our image model. The EHT's nominal resolution is 20-25 μas. Additionally, the EHT is not very sensitive to a larger field of view. Typically 60-80 μas is enough to describe the compact flux of M87. Given this, we only need to use a small number of pixels to describe our image.

npix = 48
fovx = μas2rad(200.0)
fovy = μas2rad(200.0)

9.69627362219072e-10

Now let's form our cache's. First, we have our usual image cache which is needed to numerically compute the visibilities.

grid = imagepixels(fovx, fovy, npix, npix)

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 need to specify our image prior. For this work we will use a Gaussian Markov Random field prior Since we are using a Gaussian Markov random field prior we need to first specify our mean image. This behaves somewhat similary to a entropy regularizer in that it will start with an initial guess for the image structure. For this tutorial we will use a a symmetric Gaussian with a FWHM of 50 μas

using VLBIImagePriors
using Distributions
fwhmfac = 2 * sqrt(2 * log(2))
mpr = modify(Gaussian(), Stretch(μas2rad(60.0) ./ fwhmfac))
mimg = intensitymap(mpr, grid)

┌ 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
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 -4.54513e-10   3.03669e-9    5.61718e-9    1.01163e-8    1.77383e-8    3.02823e-8    5.03328e-8    8.14516e-8   1.28332e-7    1.96859e-7    2.9401e-7     4.2752e-7     6.05251e-7    8.3426e-7     1.11957e-6    1.46282e-6    1.86087e-6    2.30476e-6    2.77922e-6    3.26291e-6    3.72971e-6    4.15078e-6    4.4975e-6     4.74459e-6    4.87318e-6   4.87318e-6   4.74459e-6   4.4975e-6    4.15078e-6   3.72971e-6   3.26291e-6   2.77922e-6   2.30476e-6   1.86087e-6   1.46282e-6   1.11957e-6   8.3426e-7    6.05251e-7   4.2752e-7    2.9401e-7    1.96859e-7   1.28332e-7   8.14516e-8  5.03328e-8   3.02823e-8   1.77383e-8   1.01163e-8   5.61718e-9   3.03669e-9
 -4.34312e-10   5.46896e-9    1.01163e-8    1.82191e-8    3.1946e-8     5.45372e-8    9.06473e-8    1.46691e-7   2.3112e-7     3.54535e-7    5.29501e-7    7.69945e-7    1.09003e-6    1.50247e-6    2.01631e-6    2.63448e-6    3.35135e-6    4.15078e-6    5.00526e-6    5.87637e-6    6.71705e-6    7.47539e-6    8.09982e-6    8.54482e-6    8.77641e-6   8.77641e-6   8.54482e-6   8.09982e-6   7.47539e-6   6.71705e-6   5.87637e-6   5.00526e-6   4.15078e-6   3.35135e-6   2.63448e-6   2.01631e-6   1.50247e-6   1.09003e-6   7.69945e-7   5.29501e-7   3.54535e-7   2.3112e-7    1.46691e-7  9.06473e-8   5.45372e-8   3.1946e-8    1.82191e-8   1.01163e-8   5.46896e-9
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 -3.93911e-10   1.63708e-8    3.02823e-8    5.45372e-8    9.56274e-8    1.63252e-7    2.71345e-7    4.39106e-7   6.91838e-7    1.06127e-6    1.58501e-6    2.30476e-6    3.26291e-6    4.4975e-6     6.03564e-6    7.88608e-6    1.0032e-5     1.2425e-5     1.49828e-5    1.75904e-5    2.01069e-5    2.23769e-5    2.42461e-5    2.55782e-5    2.62714e-5   2.62714e-5   2.55782e-5   2.42461e-5   2.23769e-5   2.01069e-5   1.75904e-5   1.49828e-5   1.2425e-5    1.0032e-5    7.88608e-6   6.03564e-6   4.4975e-6    3.26291e-6   2.30476e-6   1.58501e-6   1.06127e-6   6.91838e-7   4.39106e-7  2.71345e-7   1.63252e-7   9.56274e-8   5.45372e-8   3.02823e-8   1.63708e-8
 -3.73711e-10   2.72103e-8    5.03328e-8    9.06473e-8    1.58944e-7    2.71345e-7    4.51007e-7    7.29848e-7   1.14992e-6    1.76396e-6    2.63448e-6    3.83079e-6    5.42336e-6    7.47539e-6    1.0032e-5     1.31076e-5    1.66743e-5    2.06518e-5    2.49032e-5    2.92374e-5    3.34201e-5    3.71931e-5    4.02999e-5    4.2514e-5     4.36662e-5   4.36662e-5   4.2514e-5    4.02999e-5   3.71931e-5   3.34201e-5   2.92374e-5   2.49032e-5   2.06518e-5   1.66743e-5   1.31076e-5   1.0032e-5    7.47539e-6   5.42336e-6   3.83079e-6   2.63448e-6   1.76396e-6   1.14992e-6   7.29848e-7  4.51007e-7   2.71345e-7   1.58944e-7   9.06473e-8   5.03328e-8   2.72103e-8
 -3.5351e-10    4.40334e-8    8.14516e-8    1.46691e-7    2.57213e-7    4.39106e-7    7.29848e-7    1.18108e-6   1.86087e-6    2.85454e-6    4.26328e-6    6.19922e-6    8.77641e-6    1.20971e-5    1.62343e-5    2.12116e-5    2.69834e-5    3.34201e-5    4.02999e-5    4.73137e-5    5.40824e-5    6.01882e-5    6.52158e-5    6.87987e-5    7.06633e-5   7.06633e-5   6.87987e-5   6.52158e-5   6.01882e-5   5.40824e-5   4.73137e-5   4.02999e-5   3.34201e-5   2.69834e-5   2.12116e-5   1.62343e-5   1.20971e-5   8.77641e-6   6.19922e-6   4.26328e-6   2.85454e-6   1.86087e-6   1.18108e-6  7.29848e-7   4.39106e-7   2.57213e-7   1.46691e-7   8.14516e-8   4.40334e-8
 -3.33309e-10   6.93772e-8    1.28332e-7    2.3112e-7     4.05255e-7    6.91838e-7    1.14992e-6    1.86087e-6   2.93191e-6    4.4975e-6     6.71705e-6    9.76724e-6    1.38277e-5    1.90598e-5    2.55782e-5    3.34201e-5    4.2514e-5     5.26553e-5    6.34949e-5    7.45455e-5    8.521e-5      9.483e-5      0.000102751   0.000108396   0.000111334  0.000111334  0.000108396  0.000102751  9.483e-5     8.521e-5     7.45455e-5   6.34949e-5   5.26553e-5   4.2514e-5    3.34201e-5   2.55782e-5   1.90598e-5   1.38277e-5   9.76724e-6   6.71705e-6   4.4975e-6    2.93191e-6   1.86087e-6  1.14992e-6   6.91838e-7   4.05255e-7   2.3112e-7    1.28332e-7   6.93772e-8
 -3.13109e-10   1.06424e-7    1.96859e-7    3.54535e-7    6.21655e-7    1.06127e-6    1.76396e-6    2.85454e-6   4.4975e-6     6.8991e-6     1.03039e-5    1.49828e-5    2.12116e-5    2.92374e-5    3.92365e-5    5.12659e-5    6.52158e-5    8.07724e-5    9.74002e-5    0.000114352   0.000130711   0.000145468   0.000157619   0.000166278   0.000170785  0.000170785  0.000166278  0.000157619  0.000145468  0.000130711  0.000114352  9.74002e-5   8.07724e-5   6.52158e-5   5.12659e-5   3.92365e-5   2.92374e-5   2.12116e-5   1.49828e-5   1.03039e-5   6.8991e-6    4.4975e-6    2.85454e-6  1.76396e-6   1.06127e-6   6.21655e-7   3.54535e-7   1.96859e-7   1.06424e-7
 -2.92908e-10   1.58944e-7    2.9401e-7     5.29501e-7    9.28446e-7    1.58501e-6    2.63448e-6    4.26328e-6   6.71705e-6    1.03039e-5    1.53889e-5    2.23769e-5    3.16796e-5    4.36662e-5    5.86e-5       7.65659e-5    9.74002e-5    0.000120634   0.000145468   0.000170785   0.000195218   0.000217257   0.000235405   0.000248338   0.000255069  0.000255069  0.000248338  0.000235405  0.000217257  0.000195218  0.000170785  0.000145468  0.000120634  9.74002e-5   7.65659e-5   5.86e-5      4.36662e-5   3.16796e-5   2.23769e-5   1.53889e-5   1.03039e-5   6.71705e-6   4.26328e-6  2.63448e-6   1.58501e-6   9.28446e-7   5.29501e-7   2.9401e-7    1.58944e-7
 -2.72708e-10   2.3112e-7     4.2752e-7     7.69945e-7    1.35005e-6    2.30476e-6    3.83079e-6    6.19922e-6   9.76724e-6    1.49828e-5    2.23769e-5    3.25382e-5    4.60652e-5    6.34949e-5    8.521e-5      0.000111334   0.000141629   0.000175414   0.000211524   0.000248338   0.000283865   0.000315913   0.000342302   0.000361107   0.000370894  0.000370894  0.000361107  0.000342302  0.000315913  0.000283865  0.000248338  0.000211524  0.000175414  0.000141629  0.000111334  8.521e-5     6.34949e-5   4.60652e-5   3.25382e-5   2.23769e-5   1.49828e-5   9.76724e-6   6.19922e-6  3.83079e-6   2.30476e-6   1.35005e-6   7.69945e-7   4.2752e-7    2.3112e-7
 -2.52507e-10   3.27203e-7    6.05251e-7    1.09003e-6    1.9113e-6     3.26291e-6    5.42336e-6    8.77641e-6   1.38277e-5    2.12116e-5    3.16796e-5    4.60652e-5    6.52158e-5    8.98914e-5    0.000120634   0.000157619   0.000200508   0.000248338   0.000299461   0.000351579   0.000401876   0.000447247   0.000484606   0.00051123    0.000525085  0.000525085  0.00051123   0.000484606  0.000447247  0.000401876  0.000351579  0.000299461  0.000248338  0.000200508  0.000157619  0.000120634  8.98914e-5   6.52158e-5   4.60652e-5   3.16796e-5   2.12116e-5   1.38277e-5   8.77641e-6  5.42336e-6   3.26291e-6   1.9113e-6    1.09003e-6   6.05251e-7   3.27203e-7
 -2.32307e-10   4.51007e-7    8.3426e-7     1.50247e-6    2.63448e-6    4.4975e-6     7.47539e-6    1.20971e-5   1.90598e-5    2.92374e-5    4.36662e-5    6.34949e-5    8.98914e-5    0.000123904   0.000166278   0.000217257   0.000276375   0.000342302   0.000412768   0.000484606   0.000553933   0.000616471   0.000667966   0.000704663   0.000723762  0.000723762  0.000704663  0.000667966  0.000616471  0.000553933  0.000484606  0.000412768  0.000342302  0.000276375  0.000217257  0.000166278  0.000123904  8.98914e-5   6.34949e-5   4.36662e-5   2.92374e-5   1.90598e-5   1.20971e-5  7.47539e-6   4.4975e-6    2.63448e-6   1.50247e-6   8.3426e-7    4.51007e-7
 -2.12106e-10   6.05251e-7    1.11957e-6    2.01631e-6    3.53547e-6    6.03564e-6    1.0032e-5     1.62343e-5   2.55782e-5    3.92365e-5    5.86e-5       8.521e-5      0.000120634   0.000166278   0.000223145   0.000291559   0.000370894   0.000459368   0.000553933   0.00065034    0.000743377   0.000827303   0.000896409   0.000945657   0.000971287  0.000971287  0.000945657  0.000896409  0.000827303  0.000743377  0.00065034   0.000553933  0.000459368  0.000370894  0.000291559  0.000223145  0.000166278  0.000120634  8.521e-5     5.86e-5      3.92365e-5   2.55782e-5   1.62343e-5  1.0032e-5    6.03564e-6   3.53547e-6   2.01631e-6   1.11957e-6   6.05251e-7
 -1.91905e-10   7.90813e-7    1.46282e-6    2.63448e-6    4.6194e-6     7.88608e-6    1.31076e-5    2.12116e-5   3.34201e-5    5.12659e-5    7.65659e-5    0.000111334   0.000157619   0.000217257   0.000291559   0.000380947   0.000484606   0.000600204   0.000723762   0.000849725   0.000971287   0.00108094    0.00117124    0.00123558    0.00126907   0.00126907   0.00123558   0.00117124   0.00108094   0.000971287  0.000849725  0.000723762  0.000600204  0.000484606  0.000380947  0.000291559  0.000217257  0.000157619  0.000111334  7.65659e-5   5.12659e-5   3.34201e-5   2.12116e-5  1.31076e-5   7.88608e-6   4.6194e-6    2.63448e-6   1.46282e-6   7.90813e-7
 -1.71705e-10   1.006e-6      1.86087e-6    3.35135e-6    5.87637e-6    1.0032e-5     1.66743e-5    2.69834e-5   4.2514e-5     6.52158e-5    9.74002e-5    0.000141629   0.000200508   0.000276375   0.000370894   0.000484606   0.000616471   0.000763525   0.000920704   0.00108094    0.00123558    0.00137508    0.00148994    0.0015718     0.0016144    0.0016144    0.0015718    0.00148994   0.00137508   0.00123558   0.00108094   0.000920704  0.000763525  0.000616471  0.000484606  0.000370894  0.000276375  0.000200508  0.000141629  9.74002e-5   6.52158e-5   4.2514e-5    2.69834e-5  1.66743e-5   1.0032e-5    5.87637e-6   3.35135e-6   1.86087e-6   1.006e-6
 -1.51504e-10   1.24597e-6    2.30476e-6    4.15078e-6    7.27813e-6    1.2425e-5     2.06518e-5    3.34201e-5   5.26553e-5    8.07724e-5    0.000120634   0.000175414   0.000248338   0.000342302   0.000459368   0.000600204   0.000763525   0.000945657   0.00114033    0.00133879    0.00153032    0.00170309    0.00184535    0.00194673    0.0019995    0.0019995    0.00194673   0.00184535   0.00170309   0.00153032   0.00133879   0.00114033   0.000945657  0.000763525  0.000600204  0.000459368  0.000342302  0.000248338  0.000175414  0.000120634  8.07724e-5   5.26553e-5   3.34201e-5  2.06518e-5   1.2425e-5    7.27813e-6   4.15078e-6   2.30476e-6   1.24597e-6
 -1.31304e-10   1.50247e-6    2.77922e-6    5.00526e-6    8.77641e-6    1.49828e-5    2.49032e-5    4.02999e-5   6.34949e-5    9.74002e-5    0.000145468   0.000211524   0.000299461   0.000412768   0.000553933   0.000723762   0.000920704   0.00114033    0.00137508    0.0016144     0.00184535    0.00205369    0.00222523    0.00234749    0.00241111   0.00241111   0.00234749   0.00222523   0.00205369   0.00184535   0.0016144    0.00137508   0.00114033   0.000920704  0.000723762  0.000553933  0.000412768  0.000299461  0.000211524  0.000145468  9.74002e-5   6.34949e-5   4.02999e-5  2.49032e-5   1.49828e-5   8.77641e-6   5.00526e-6   2.77922e-6   1.50247e-6
 -1.11103e-10   1.76396e-6    3.26291e-6    5.87637e-6    1.03039e-5    1.75904e-5    2.92374e-5    4.73137e-5   7.45455e-5    0.000114352   0.000170785   0.000248338   0.000351579   0.000484606   0.00065034    0.000849725   0.00108094    0.00133879    0.0016144     0.00189536    0.00216652    0.00241111    0.00261251    0.00275604    0.00283074   0.00283074   0.00275604   0.00261251   0.00241111   0.00216652   0.00189536   0.0016144    0.00133879   0.00108094   0.000849725  0.00065034   0.000484606  0.000351579  0.000248338  0.000170785  0.000114352  7.45455e-5   4.73137e-5  2.92374e-5   1.75904e-5   1.03039e-5   5.87637e-6   3.26291e-6   1.76396e-6
 -9.09026e-11   2.01631e-6    3.72971e-6    6.71705e-6    1.17779e-5    2.01069e-5    3.34201e-5    5.40824e-5   8.521e-5      0.000130711   0.000195218   0.000283865   0.000401876   0.000553933   0.000743377   0.000971287   0.00123558    0.00153032    0.00184535    0.00216652    0.00247646    0.00275604    0.00298626    0.00315032    0.00323571   0.00323571   0.00315032   0.00298626   0.00275604   0.00247646   0.00216652   0.00184535   0.00153032   0.00123558   0.000971287  0.000743377  0.000553933  0.000401876  0.000283865  0.000195218  0.000130711  8.521e-5     5.40824e-5  3.34201e-5   2.01069e-5   1.17779e-5   6.71705e-6   3.72971e-6   2.01631e-6
 -7.0702e-11    2.24395e-6    4.15078e-6    7.47539e-6    1.31076e-5    2.23769e-5    3.71931e-5    6.01882e-5   9.483e-5      0.000145468   0.000217257   0.000315913   0.000447247   0.000616471   0.000827303   0.00108094    0.00137508    0.00170309    0.00205369    0.00241111    0.00275604    0.0030672     0.0033234     0.00350599    0.00360101   0.00360101   0.00350599   0.0033234    0.0030672    0.00275604   0.00241111   0.00205369   0.00170309   0.00137508   0.00108094   0.000827303  0.000616471  0.000447247  0.000315913  0.000217257  0.000145468  9.483e-5     6.01882e-5  3.71931e-5   2.23769e-5   1.31076e-5   7.47539e-6   4.15078e-6   2.24395e-6
 -5.05014e-11   2.43138e-6    4.4975e-6     8.09982e-6    1.42025e-5    2.42461e-5    4.02999e-5    6.52158e-5   0.000102751   0.000157619   0.000235405   0.000342302   0.000484606   0.000667966   0.000896409   0.00117124    0.00148994    0.00184535    0.00222523    0.00261251    0.00298626    0.0033234     0.00360101    0.00379885    0.00390181   0.00390181   0.00379885   0.00360101   0.0033234    0.00298626   0.00261251   0.00222523   0.00184535   0.00148994   0.00117124   0.000896409  0.000667966  0.000484606  0.000342302  0.000235405  0.000157619  0.000102751  6.52158e-5  4.02999e-5   2.42461e-5   1.42025e-5   8.09982e-6   4.4975e-6    2.43138e-6
 -3.03009e-11   2.56496e-6    4.74459e-6    8.54482e-6    1.49828e-5    2.55782e-5    4.2514e-5     6.87987e-5   0.000108396   0.000166278   0.000248338   0.000361107   0.00051123    0.000704663   0.000945657   0.00123558    0.0015718     0.00194673    0.00234749    0.00275604    0.00315032    0.00350599    0.00379885    0.00400756    0.00411617   0.00411617   0.00400756   0.00379885   0.00350599   0.00315032   0.00275604   0.00234749   0.00194673   0.0015718    0.00123558   0.000945657  0.000704663  0.00051123   0.000361107  0.000248338  0.000166278  0.000108396  6.87987e-5  4.2514e-5    2.55782e-5   1.49828e-5   8.54482e-6   4.74459e-6   2.56496e-6
 -1.01003e-11   2.63448e-6    4.87318e-6    8.77641e-6    1.53889e-5    2.62714e-5    4.36662e-5    7.06633e-5   0.000111334   0.000170785   0.000255069   0.000370894   0.000525085   0.000723762   0.000971287   0.00126907    0.0016144     0.0019995     0.00241111    0.00283074    0.00323571    0.00360101    0.00390181    0.00411617    0.00422773   0.00422773   0.00411617   0.00390181   0.00360101   0.00323571   0.00283074   0.00241111   0.0019995    0.0016144    0.00126907   0.000971287  0.000723762  0.000525085  0.000370894  0.000255069  0.000170785  0.000111334  7.06633e-5  4.36662e-5   2.62714e-5   1.53889e-5   8.77641e-6   4.87318e-6   2.63448e-6
  1.01003e-11   2.63448e-6    4.87318e-6    8.77641e-6    1.53889e-5    2.62714e-5    4.36662e-5    7.06633e-5   0.000111334   0.000170785   0.000255069   0.000370894   0.000525085   0.000723762   0.000971287   0.00126907    0.0016144     0.0019995     0.00241111    0.00283074    0.00323571    0.00360101    0.00390181    0.00411617    0.00422773   0.00422773   0.00411617   0.00390181   0.00360101   0.00323571   0.00283074   0.00241111   0.0019995    0.0016144    0.00126907   0.000971287  0.000723762  0.000525085  0.000370894  0.000255069  0.000170785  0.000111334  7.06633e-5  4.36662e-5   2.62714e-5   1.53889e-5   8.77641e-6   4.87318e-6   2.63448e-6
  3.03009e-11   2.56496e-6    4.74459e-6    8.54482e-6    1.49828e-5    2.55782e-5    4.2514e-5     6.87987e-5   0.000108396   0.000166278   0.000248338   0.000361107   0.00051123    0.000704663   0.000945657   0.00123558    0.0015718     0.00194673    0.00234749    0.00275604    0.00315032    0.00350599    0.00379885    0.00400756    0.00411617   0.00411617   0.00400756   0.00379885   0.00350599   0.00315032   0.00275604   0.00234749   0.00194673   0.0015718    0.00123558   0.000945657  0.000704663  0.00051123   0.000361107  0.000248338  0.000166278  0.000108396  6.87987e-5  4.2514e-5    2.55782e-5   1.49828e-5   8.54482e-6   4.74459e-6   2.56496e-6
  5.05014e-11   2.43138e-6    4.4975e-6     8.09982e-6    1.42025e-5    2.42461e-5    4.02999e-5    6.52158e-5   0.000102751   0.000157619   0.000235405   0.000342302   0.000484606   0.000667966   0.000896409   0.00117124    0.00148994    0.00184535    0.00222523    0.00261251    0.00298626    0.0033234     0.00360101    0.00379885    0.00390181   0.00390181   0.00379885   0.00360101   0.0033234    0.00298626   0.00261251   0.00222523   0.00184535   0.00148994   0.00117124   0.000896409  0.000667966  0.000484606  0.000342302  0.000235405  0.000157619  0.000102751  6.52158e-5  4.02999e-5   2.42461e-5   1.42025e-5   8.09982e-6   4.4975e-6    2.43138e-6
  7.0702e-11    2.24395e-6    4.15078e-6    7.47539e-6    1.31076e-5    2.23769e-5    3.71931e-5    6.01882e-5   9.483e-5      0.000145468   0.000217257   0.000315913   0.000447247   0.000616471   0.000827303   0.00108094    0.00137508    0.00170309    0.00205369    0.00241111    0.00275604    0.0030672     0.0033234     0.00350599    0.00360101   0.00360101   0.00350599   0.0033234    0.0030672    0.00275604   0.00241111   0.00205369   0.00170309   0.00137508   0.00108094   0.000827303  0.000616471  0.000447247  0.000315913  0.000217257  0.000145468  9.483e-5     6.01882e-5  3.71931e-5   2.23769e-5   1.31076e-5   7.47539e-6   4.15078e-6   2.24395e-6
  9.09026e-11   2.01631e-6    3.72971e-6    6.71705e-6    1.17779e-5    2.01069e-5    3.34201e-5    5.40824e-5   8.521e-5      0.000130711   0.000195218   0.000283865   0.000401876   0.000553933   0.000743377   0.000971287   0.00123558    0.00153032    0.00184535    0.00216652    0.00247646    0.00275604    0.00298626    0.00315032    0.00323571   0.00323571   0.00315032   0.00298626   0.00275604   0.00247646   0.00216652   0.00184535   0.00153032   0.00123558   0.000971287  0.000743377  0.000553933  0.000401876  0.000283865  0.000195218  0.000130711  8.521e-5     5.40824e-5  3.34201e-5   2.01069e-5   1.17779e-5   6.71705e-6   3.72971e-6   2.01631e-6
  1.11103e-10   1.76396e-6    3.26291e-6    5.87637e-6    1.03039e-5    1.75904e-5    2.92374e-5    4.73137e-5   7.45455e-5    0.000114352   0.000170785   0.000248338   0.000351579   0.000484606   0.00065034    0.000849725   0.00108094    0.00133879    0.0016144     0.00189536    0.00216652    0.00241111    0.00261251    0.00275604    0.00283074   0.00283074   0.00275604   0.00261251   0.00241111   0.00216652   0.00189536   0.0016144    0.00133879   0.00108094   0.000849725  0.00065034   0.000484606  0.000351579  0.000248338  0.000170785  0.000114352  7.45455e-5   4.73137e-5  2.92374e-5   1.75904e-5   1.03039e-5   5.87637e-6   3.26291e-6   1.76396e-6
  1.31304e-10   1.50247e-6    2.77922e-6    5.00526e-6    8.77641e-6    1.49828e-5    2.49032e-5    4.02999e-5   6.34949e-5    9.74002e-5    0.000145468   0.000211524   0.000299461   0.000412768   0.000553933   0.000723762   0.000920704   0.00114033    0.00137508    0.0016144     0.00184535    0.00205369    0.00222523    0.00234749    0.00241111   0.00241111   0.00234749   0.00222523   0.00205369   0.00184535   0.0016144    0.00137508   0.00114033   0.000920704  0.000723762  0.000553933  0.000412768  0.000299461  0.000211524  0.000145468  9.74002e-5   6.34949e-5   4.02999e-5  2.49032e-5   1.49828e-5   8.77641e-6   5.00526e-6   2.77922e-6   1.50247e-6
  1.51504e-10   1.24597e-6    2.30476e-6    4.15078e-6    7.27813e-6    1.2425e-5     2.06518e-5    3.34201e-5   5.26553e-5    8.07724e-5    0.000120634   0.000175414   0.000248338   0.000342302   0.000459368   0.000600204   0.000763525   0.000945657   0.00114033    0.00133879    0.00153032    0.00170309    0.00184535    0.00194673    0.0019995    0.0019995    0.00194673   0.00184535   0.00170309   0.00153032   0.00133879   0.00114033   0.000945657  0.000763525  0.000600204  0.000459368  0.000342302  0.000248338  0.000175414  0.000120634  8.07724e-5   5.26553e-5   3.34201e-5  2.06518e-5   1.2425e-5    7.27813e-6   4.15078e-6   2.30476e-6   1.24597e-6
  1.71705e-10   1.006e-6      1.86087e-6    3.35135e-6    5.87637e-6    1.0032e-5     1.66743e-5    2.69834e-5   4.2514e-5     6.52158e-5    9.74002e-5    0.000141629   0.000200508   0.000276375   0.000370894   0.000484606   0.000616471   0.000763525   0.000920704   0.00108094    0.00123558    0.00137508    0.00148994    0.0015718     0.0016144    0.0016144    0.0015718    0.00148994   0.00137508   0.00123558   0.00108094   0.000920704  0.000763525  0.000616471  0.000484606  0.000370894  0.000276375  0.000200508  0.000141629  9.74002e-5   6.52158e-5   4.2514e-5    2.69834e-5  1.66743e-5   1.0032e-5    5.87637e-6   3.35135e-6   1.86087e-6   1.006e-6
  1.91905e-10   7.90813e-7    1.46282e-6    2.63448e-6    4.6194e-6     7.88608e-6    1.31076e-5    2.12116e-5   3.34201e-5    5.12659e-5    7.65659e-5    0.000111334   0.000157619   0.000217257   0.000291559   0.000380947   0.000484606   0.000600204   0.000723762   0.000849725   0.000971287   0.00108094    0.00117124    0.00123558    0.00126907   0.00126907   0.00123558   0.00117124   0.00108094   0.000971287  0.000849725  0.000723762  0.000600204  0.000484606  0.000380947  0.000291559  0.000217257  0.000157619  0.000111334  7.65659e-5   5.12659e-5   3.34201e-5   2.12116e-5  1.31076e-5   7.88608e-6   4.6194e-6    2.63448e-6   1.46282e-6   7.90813e-7
  2.12106e-10   6.05251e-7    1.11957e-6    2.01631e-6    3.53547e-6    6.03564e-6    1.0032e-5     1.62343e-5   2.55782e-5    3.92365e-5    5.86e-5       8.521e-5      0.000120634   0.000166278   0.000223145   0.000291559   0.000370894   0.000459368   0.000553933   0.00065034    0.000743377   0.000827303   0.000896409   0.000945657   0.000971287  0.000971287  0.000945657  0.000896409  0.000827303  0.000743377  0.00065034   0.000553933  0.000459368  0.000370894  0.000291559  0.000223145  0.000166278  0.000120634  8.521e-5     5.86e-5      3.92365e-5   2.55782e-5   1.62343e-5  1.0032e-5    6.03564e-6   3.53547e-6   2.01631e-6   1.11957e-6   6.05251e-7
  2.32307e-10   4.51007e-7    8.3426e-7     1.50247e-6    2.63448e-6    4.4975e-6     7.47539e-6    1.20971e-5   1.90598e-5    2.92374e-5    4.36662e-5    6.34949e-5    8.98914e-5    0.000123904   0.000166278   0.000217257   0.000276375   0.000342302   0.000412768   0.000484606   0.000553933   0.000616471   0.000667966   0.000704663   0.000723762  0.000723762  0.000704663  0.000667966  0.000616471  0.000553933  0.000484606  0.000412768  0.000342302  0.000276375  0.000217257  0.000166278  0.000123904  8.98914e-5   6.34949e-5   4.36662e-5   2.92374e-5   1.90598e-5   1.20971e-5  7.47539e-6   4.4975e-6    2.63448e-6   1.50247e-6   8.3426e-7    4.51007e-7
  2.52507e-10   3.27203e-7    6.05251e-7    1.09003e-6    1.9113e-6     3.26291e-6    5.42336e-6    8.77641e-6   1.38277e-5    2.12116e-5    3.16796e-5    4.60652e-5    6.52158e-5    8.98914e-5    0.000120634   0.000157619   0.000200508   0.000248338   0.000299461   0.000351579   0.000401876   0.000447247   0.000484606   0.00051123    0.000525085  0.000525085  0.00051123   0.000484606  0.000447247  0.000401876  0.000351579  0.000299461  0.000248338  0.000200508  0.000157619  0.000120634  8.98914e-5   6.52158e-5   4.60652e-5   3.16796e-5   2.12116e-5   1.38277e-5   8.77641e-6  5.42336e-6   3.26291e-6   1.9113e-6    1.09003e-6   6.05251e-7   3.27203e-7
  2.72708e-10   2.3112e-7     4.2752e-7     7.69945e-7    1.35005e-6    2.30476e-6    3.83079e-6    6.19922e-6   9.76724e-6    1.49828e-5    2.23769e-5    3.25382e-5    4.60652e-5    6.34949e-5    8.521e-5      0.000111334   0.000141629   0.000175414   0.000211524   0.000248338   0.000283865   0.000315913   0.000342302   0.000361107   0.000370894  0.000370894  0.000361107  0.000342302  0.000315913  0.000283865  0.000248338  0.000211524  0.000175414  0.000141629  0.000111334  8.521e-5     6.34949e-5   4.60652e-5   3.25382e-5   2.23769e-5   1.49828e-5   9.76724e-6   6.19922e-6  3.83079e-6   2.30476e-6   1.35005e-6   7.69945e-7   4.2752e-7    2.3112e-7
  2.92908e-10   1.58944e-7    2.9401e-7     5.29501e-7    9.28446e-7    1.58501e-6    2.63448e-6    4.26328e-6   6.71705e-6    1.03039e-5    1.53889e-5    2.23769e-5    3.16796e-5    4.36662e-5    5.86e-5       7.65659e-5    9.74002e-5    0.000120634   0.000145468   0.000170785   0.000195218   0.000217257   0.000235405   0.000248338   0.000255069  0.000255069  0.000248338  0.000235405  0.000217257  0.000195218  0.000170785  0.000145468  0.000120634  9.74002e-5   7.65659e-5   5.86e-5      4.36662e-5   3.16796e-5   2.23769e-5   1.53889e-5   1.03039e-5   6.71705e-6   4.26328e-6  2.63448e-6   1.58501e-6   9.28446e-7   5.29501e-7   2.9401e-7    1.58944e-7
  3.13109e-10   1.06424e-7    1.96859e-7    3.54535e-7    6.21655e-7    1.06127e-6    1.76396e-6    2.85454e-6   4.4975e-6     6.8991e-6     1.03039e-5    1.49828e-5    2.12116e-5    2.92374e-5    3.92365e-5    5.12659e-5    6.52158e-5    8.07724e-5    9.74002e-5    0.000114352   0.000130711   0.000145468   0.000157619   0.000166278   0.000170785  0.000170785  0.000166278  0.000157619  0.000145468  0.000130711  0.000114352  9.74002e-5   8.07724e-5   6.52158e-5   5.12659e-5   3.92365e-5   2.92374e-5   2.12116e-5   1.49828e-5   1.03039e-5   6.8991e-6    4.4975e-6    2.85454e-6  1.76396e-6   1.06127e-6   6.21655e-7   3.54535e-7   1.96859e-7   1.06424e-7
  3.33309e-10   6.93772e-8    1.28332e-7    2.3112e-7     4.05255e-7    6.91838e-7    1.14992e-6    1.86087e-6   2.93191e-6    4.4975e-6     6.71705e-6    9.76724e-6    1.38277e-5    1.90598e-5    2.55782e-5    3.34201e-5    4.2514e-5     5.26553e-5    6.34949e-5    7.45455e-5    8.521e-5      9.483e-5      0.000102751   0.000108396   0.000111334  0.000111334  0.000108396  0.000102751  9.483e-5     8.521e-5     7.45455e-5   6.34949e-5   5.26553e-5   4.2514e-5    3.34201e-5   2.55782e-5   1.90598e-5   1.38277e-5   9.76724e-6   6.71705e-6   4.4975e-6    2.93191e-6   1.86087e-6  1.14992e-6   6.91838e-7   4.05255e-7   2.3112e-7    1.28332e-7   6.93772e-8
  3.5351e-10    4.40334e-8    8.14516e-8    1.46691e-7    2.57213e-7    4.39106e-7    7.29848e-7    1.18108e-6   1.86087e-6    2.85454e-6    4.26328e-6    6.19922e-6    8.77641e-6    1.20971e-5    1.62343e-5    2.12116e-5    2.69834e-5    3.34201e-5    4.02999e-5    4.73137e-5    5.40824e-5    6.01882e-5    6.52158e-5    6.87987e-5    7.06633e-5   7.06633e-5   6.87987e-5   6.52158e-5   6.01882e-5   5.40824e-5   4.73137e-5   4.02999e-5   3.34201e-5   2.69834e-5   2.12116e-5   1.62343e-5   1.20971e-5   8.77641e-6   6.19922e-6   4.26328e-6   2.85454e-6   1.86087e-6   1.18108e-6  7.29848e-7   4.39106e-7   2.57213e-7   1.46691e-7   8.14516e-8   4.40334e-8
  3.73711e-10   2.72103e-8    5.03328e-8    9.06473e-8    1.58944e-7    2.71345e-7    4.51007e-7    7.29848e-7   1.14992e-6    1.76396e-6    2.63448e-6    3.83079e-6    5.42336e-6    7.47539e-6    1.0032e-5     1.31076e-5    1.66743e-5    2.06518e-5    2.49032e-5    2.92374e-5    3.34201e-5    3.71931e-5    4.02999e-5    4.2514e-5     4.36662e-5   4.36662e-5   4.2514e-5    4.02999e-5   3.71931e-5   3.34201e-5   2.92374e-5   2.49032e-5   2.06518e-5   1.66743e-5   1.31076e-5   1.0032e-5    7.47539e-6   5.42336e-6   3.83079e-6   2.63448e-6   1.76396e-6   1.14992e-6   7.29848e-7  4.51007e-7   2.71345e-7   1.58944e-7   9.06473e-8   5.03328e-8   2.72103e-8
  3.93911e-10   1.63708e-8    3.02823e-8    5.45372e-8    9.56274e-8    1.63252e-7    2.71345e-7    4.39106e-7   6.91838e-7    1.06127e-6    1.58501e-6    2.30476e-6    3.26291e-6    4.4975e-6     6.03564e-6    7.88608e-6    1.0032e-5     1.2425e-5     1.49828e-5    1.75904e-5    2.01069e-5    2.23769e-5    2.42461e-5    2.55782e-5    2.62714e-5   2.62714e-5   2.55782e-5   2.42461e-5   2.23769e-5   2.01069e-5   1.75904e-5   1.49828e-5   1.2425e-5    1.0032e-5    7.88608e-6   6.03564e-6   4.4975e-6    3.26291e-6   2.30476e-6   1.58501e-6   1.06127e-6   6.91838e-7   4.39106e-7  2.71345e-7   1.63252e-7   9.56274e-8   5.45372e-8   3.02823e-8   1.63708e-8
  4.14112e-10   9.58947e-9    1.77383e-8    3.1946e-8     5.60153e-8    9.56274e-8    1.58944e-7    2.57213e-7   4.05255e-7    6.21655e-7    9.28446e-7    1.35005e-6    1.9113e-6     2.63448e-6    3.53547e-6    4.6194e-6     5.87637e-6    7.27813e-6    8.77641e-6    1.03039e-5    1.17779e-5    1.31076e-5    1.42025e-5    1.49828e-5    1.53889e-5   1.53889e-5   1.49828e-5   1.42025e-5   1.31076e-5   1.17779e-5   1.03039e-5   8.77641e-6   7.27813e-6   5.87637e-6   4.6194e-6    3.53547e-6   2.63448e-6   1.9113e-6    1.35005e-6   9.28446e-7   6.21655e-7   4.05255e-7   2.57213e-7  1.58944e-7   9.56274e-8   5.60153e-8   3.1946e-8    1.77383e-8   9.58947e-9
  4.34312e-10   5.46896e-9    1.01163e-8    1.82191e-8    3.1946e-8     5.45372e-8    9.06473e-8    1.46691e-7   2.3112e-7     3.54535e-7    5.29501e-7    7.69945e-7    1.09003e-6    1.50247e-6    2.01631e-6    2.63448e-6    3.35135e-6    4.15078e-6    5.00526e-6    5.87637e-6    6.71705e-6    7.47539e-6    8.09982e-6    8.54482e-6    8.77641e-6   8.77641e-6   8.54482e-6   8.09982e-6   7.47539e-6   6.71705e-6   5.87637e-6   5.00526e-6   4.15078e-6   3.35135e-6   2.63448e-6   2.01631e-6   1.50247e-6   1.09003e-6   7.69945e-7   5.29501e-7   3.54535e-7   2.3112e-7    1.46691e-7  9.06473e-8   5.45372e-8   3.1946e-8    1.82191e-8   1.01163e-8   5.46896e-9
  4.54513e-10   3.03669e-9    5.61718e-9    1.01163e-8    1.77383e-8    3.02823e-8    5.03328e-8    8.14516e-8   1.28332e-7    1.96859e-7    2.9401e-7     4.2752e-7     6.05251e-7    8.3426e-7     1.11957e-6    1.46282e-6    1.86087e-6    2.30476e-6    2.77922e-6    3.26291e-6    3.72971e-6    4.15078e-6    4.4975e-6     4.74459e-6    4.87318e-6   4.87318e-6   4.74459e-6   4.4975e-6    4.15078e-6   3.72971e-6   3.26291e-6   2.77922e-6   2.30476e-6   1.86087e-6   1.46282e-6   1.11957e-6   8.3426e-7    6.05251e-7   4.2752e-7    2.9401e-7    1.96859e-7   1.28332e-7   8.14516e-8  5.03328e-8   3.02823e-8   1.77383e-8   1.01163e-8   5.61718e-9   3.03669e-9
  4.74713e-10   1.64166e-9    3.03669e-9    5.46896e-9    9.58947e-9    1.63708e-8    2.72103e-8    4.40334e-8   6.93772e-8    1.06424e-7    1.58944e-7    2.3112e-7     3.27203e-7    4.51007e-7    6.05251e-7    7.90813e-7    1.006e-6      1.24597e-6    1.50247e-6    1.76396e-6    2.01631e-6    2.24395e-6    2.43138e-6    2.56496e-6    2.63448e-6   2.63448e-6   2.56496e-6   2.43138e-6   2.24395e-6   2.01631e-6   1.76396e-6   1.50247e-6   1.24597e-6   1.006e-6     7.90813e-7   6.05251e-7   4.51007e-7   3.27203e-7   2.3112e-7    1.58944e-7   1.06424e-7   6.93772e-8   4.40334e-8  2.72103e-8   1.63708e-8   9.58947e-9   5.46896e-9   3.03669e-9   1.64166e-9

Now we can form our metadata we need to fully define our model. We will also fix the total flux to be the observed value 1.1. This is because total flux is degenerate with a global shift in the gain amplitudes making the problem degenerate. To fix this we use the observed total flux as our value.

skymeta = (; ftot = 1.1, mimg = mimg ./ flux(mimg))

(ftot = 1.1, mimg = [1.6419398126916194e-9 3.0372105473444726e-9 5.4698920279304e-9 9.591108334634163e-9 1.6373632090899278e-8 2.7214944149646553e-8 4.404088682310447e-8 6.938903568369913e-8 1.0644170854928012e-7 1.5897140421510779e-7 2.3115983313045458e-7 3.272592438419348e-7 4.510842476340416e-7 6.053542398020764e-7 7.909475845251705e-7 1.0061714917927168e-6 1.2461849459797968e-6 1.502723858077153e-6 1.7642578193002676e-6 2.0166526840829715e-6 2.2443281409323587e-6 2.431799639052888e-6 2.5654019043369063e-6 2.6349308626683512e-6 2.6349308626683512e-6 2.5654019043369063e-6 2.431799639052888e-6 2.244328140932357e-6 2.0166526840829715e-6 1.7642578193002676e-6 1.502723858077153e-6 1.2461849459797968e-6 1.0061714917927168e-6 7.909475845251705e-7 6.053542398020754e-7 4.5108424763404237e-7 3.272592438419348e-7 2.3115983313045418e-7 1.5897140421510802e-7 1.0644170854928012e-7 6.938903568369913e-8 4.404088682310447e-8 2.72149441496466e-8 1.6373632090899278e-8 9.591108334634163e-9 5.4698920279304e-9 3.0372105473444726e-9 1.6419398126916194e-9; 3.0372105473444726e-9 5.618140103307805e-9 1.0118040644152322e-8 1.7741341777273437e-8 3.0287448845822705e-8 5.034137961561296e-8 8.146549888102353e-8 1.2835373709777678e-7 1.9689264940429196e-7 2.940604898402379e-7 4.275924597718894e-7 6.053542398020764e-7 8.344019823778814e-7 1.1197659425728733e-6 1.4630708918486186e-6 1.8611855585014145e-6 2.3051551784149337e-6 2.7796930899776574e-6 3.263470692162403e-6 3.7303430704848267e-6 4.151490236519409e-6 4.498269337078975e-6 4.745402761905639e-6 4.8740153236803395e-6 4.8740153236803395e-6 4.745402761905639e-6 4.498269337078975e-6 4.151490236519405e-6 3.7303430704848267e-6 3.263470692162403e-6 2.7796930899776574e-6 2.3051551784149337e-6 1.8611855585014145e-6 1.4630708918486186e-6 1.1197659425728733e-6 8.344019823778828e-7 6.053542398020764e-7 4.275924597718886e-7 2.940604898402384e-7 1.9689264940429196e-7 1.2835373709777678e-7 8.146549888102353e-8 5.034137961561296e-8 3.0287448845822705e-8 1.7741341777273437e-8 1.0118040644152322e-8 5.618140103307805e-9 3.0372105473444726e-9; 5.4698920279304e-9 1.0118040644152322e-8 1.822217755239726e-8 3.195143123585701e-8 5.4546457153905775e-8 9.066276662156884e-8 1.467160329962283e-7 2.3115983313045458e-7 3.5459561217323976e-7 5.295909203636775e-7 7.700765391304108e-7 1.0902182376728397e-6 1.502723858077153e-6 2.0166526840829736e-6 2.6349308626683538e-6 3.3519191015081937e-6 4.151490236519409e-6 5.00611361509195e-6 5.8773772987359084e-6 6.718195365326499e-6 7.476664193933783e-6 8.101199176950392e-6 8.546276371706097e-6 8.777902337498369e-6 8.777902337498369e-6 8.546276371706097e-6 8.101199176950392e-6 7.476664193933778e-6 6.718195365326499e-6 5.8773772987359084e-6 5.00611361509195e-6 4.151490236519409e-6 3.3519191015081937e-6 2.6349308626683538e-6 2.0166526840829715e-6 1.5027238580771545e-6 1.0902182376728397e-6 7.700765391304094e-7 5.295909203636784e-7 3.5459561217323976e-7 2.3115983313045458e-7 1.467160329962283e-7 9.066276662156917e-8 5.4546457153905775e-8 3.195143123585701e-8 1.822217755239726e-8 1.0118040644152322e-8 5.4698920279304e-9; 9.591108334634163e-9 1.7741341777273437e-8 3.195143123585701e-8 5.6024805766717734e-8 9.564374893731387e-8 1.5897140421510802e-7 2.572572474391242e-7 4.053240888209999e-7 6.217608892412182e-7 9.286040500087317e-7 1.350280311758629e-6 1.911628451260389e-6 2.6349308626683487e-6 3.53607242475841e-6 4.620183950447215e-6 5.8773772987359084e-6 7.279374511474409e-6 8.77790233749836e-6 1.0305607881811155e-5 1.1779928969907525e-5 1.3109855898349644e-5 1.420493833330359e-5 1.4985352932052472e-5 1.5391494354896698e-5 1.5391494354896698e-5 1.4985352932052472e-5 1.420493833330359e-5 1.3109855898349632e-5 1.1779928969907525e-5 1.0305607881811155e-5 8.77790233749836e-6 7.279374511474415e-6 5.8773772987359084e-6 4.620183950447215e-6 3.5360724247584035e-6 2.6349308626683512e-6 1.911628451260389e-6 1.3502803117586267e-6 9.286040500087334e-7 6.217608892412182e-7 4.053240888209999e-7 2.572572474391242e-7 1.5897140421510834e-7 9.564374893731387e-8 5.6024805766717734e-8 3.195143123585701e-8 1.7741341777273437e-8 9.591108334634163e-9; 1.6373632090899278e-8 3.0287448845822705e-8 5.4546457153905775e-8 9.564374893731387e-8 1.6327993619244742e-7 2.713908752539494e-7 4.391813099509043e-7 6.919562657809534e-7 1.0614502196981392e-6 1.5852830082272205e-6 2.3051551784149337e-6 3.263470692162409e-6 4.498269337078971e-6 6.0366692680024615e-6 7.887429643946967e-6 1.0033669748197654e-5 1.2427114358863923e-5 1.4985352932052472e-5 1.759340276874101e-5 2.011031637643879e-5 2.2380724912588598e-5 2.425021447245457e-5 2.5582513209206184e-5 2.627586480137954e-5 2.627586480137954e-5 2.5582513209206184e-5 2.425021447245457e-5 2.2380724912588574e-5 2.011031637643879e-5 1.759340276874101e-5 1.4985352932052472e-5 1.2427114358863933e-5 1.0033669748197654e-5 7.887429643946967e-6 6.0366692680024505e-6 4.498269337078975e-6 3.263470692162409e-6 2.305155178414932e-6 1.5852830082272232e-6 1.0614502196981392e-6 6.919562657809534e-7 4.391813099509043e-7 2.713908752539499e-7 1.6327993619244742e-7 9.564374893731387e-8 5.4546457153905775e-8 3.0287448845822705e-8 1.6373632090899278e-8; 2.7214944149646553e-8 5.034137961561296e-8 9.066276662156884e-8 1.5897140421510802e-7 2.713908752539494e-7 4.510842476340432e-7 7.299721134277668e-7 1.1501144659097052e-6 1.7642578193002719e-6 2.6349308626683538e-6 3.8314449163481485e-6 5.424280460691016e-6 7.476664193933783e-6 1.0033669748197674e-5 1.3109855898349667e-5 1.6677164864658932e-5 2.0655357427124005e-5 2.4907457358543318e-5 2.9242349595704554e-5 3.3425762468469194e-5 3.7199454289878896e-5 4.0306770594388894e-5 4.2521211196819014e-5 4.367364486286884e-5 4.367364486286884e-5 4.2521211196819014e-5 4.0306770594388894e-5 3.719945428987886e-5 3.3425762468469194e-5 2.9242349595704554e-5 2.4907457358543318e-5 2.065535742712402e-5 1.6677164864658932e-5 1.3109855898349667e-5 1.0033669748197654e-5 7.476664193933791e-6 5.424280460691016e-6 3.831444916348146e-6 2.634930862668358e-6 1.7642578193002719e-6 1.1501144659097052e-6 7.299721134277668e-7 4.510842476340439e-7 2.713908752539494e-7 1.5897140421510802e-7 9.066276662156884e-8 5.034137961561296e-8 2.7214944149646553e-8; 4.404088682310447e-8 8.146549888102353e-8 1.467160329962283e-7 2.572572474391242e-7 4.391813099509043e-7 7.299721134277668e-7 1.1812855119128452e-6 1.8611855585014124e-6 2.8550298879665212e-6 4.264006248603274e-6 6.200278457379807e-6 8.777902337498369e-6 1.2099195198373104e-5 1.6237097947765598e-5 2.1215170485438063e-5 2.6988007996490334e-5 3.342576246846914e-5 4.0306770594388826e-5 4.73217582922288e-5 5.409161281960394e-5 6.019843168713919e-5 6.522688094420994e-5 6.881042414061877e-5 7.067536277060657e-5 7.067536277060657e-5 6.881042414061877e-5 6.522688094420994e-5 6.0198431687139135e-5 5.409161281960394e-5 4.73217582922288e-5 4.0306770594388826e-5 3.342576246846914e-5 2.6988007996490334e-5 2.1215170485438063e-5 1.623709794776558e-5 1.2099195198373111e-5 8.777902337498369e-6 6.200278457379796e-6 4.264006248603281e-6 2.8550298879665212e-6 1.8611855585014124e-6 1.1812855119128452e-6 7.299721134277695e-7 4.391813099509043e-7 2.572572474391242e-7 1.467160329962283e-7 8.146549888102353e-8 4.404088682310447e-8; 6.938903568369913e-8 1.2835373709777678e-7 2.3115983313045458e-7 4.053240888209999e-7 6.919562657809534e-7 1.1501144659097052e-6 1.8611855585014124e-6 2.932408506022366e-6 4.498269337078975e-6 6.718195365326483e-6 9.76890735320735e-6 1.3830107031477007e-5 1.9063001404492862e-5 2.5582513209206164e-5 3.342576246846911e-5 4.252121119681889e-5 5.2664276130395184e-5 6.350571354983377e-5 7.455824375071895e-5 8.52245520669049e-5 9.484620827958101e-5 0.00010276882905545298 0.00010841491442445243 0.00011135323611774409 0.00011135323611774409 0.00010841491442445243 0.00010276882905545298 9.484620827958101e-5 8.52245520669049e-5 7.455824375071895e-5 6.350571354983377e-5 5.2664276130395184e-5 4.252121119681889e-5 3.342576246846911e-5 2.558251320920614e-5 1.906300140449289e-5 1.3830107031477007e-5 9.768907353207345e-6 6.7181953653264934e-6 4.498269337078975e-6 2.932408506022366e-6 1.8611855585014124e-6 1.150114465909707e-6 6.919562657809534e-7 4.053240888209999e-7 2.3115983313045458e-7 1.2835373709777678e-7 6.938903568369913e-8; 1.0644170854928012e-7 1.9689264940429196e-7 3.5459561217323976e-7 6.217608892412182e-7 1.0614502196981392e-6 1.7642578193002719e-6 2.8550298879665212e-6 4.498269337078975e-6 6.90027565646087e-6 1.0305607881811166e-5 1.49853529320525e-5 2.12151704854381e-5 2.924234959570453e-5 3.9243179965565206e-5 5.12746031365021e-5 6.522688094421001e-5 8.07861858231187e-5 9.741678330418785e-5 0.0001143711938502203 0.0001307331459929699 0.0001454926179507512 0.00015764579580172894 0.00016630680351526565 0.000170814143590246 0.000170814143590246 0.00016630680351526565 0.00015764579580172894 0.0001454926179507512 0.00013073314599296995 0.0001143711938502203 9.741678330418779e-5 8.07861858231187e-5 6.522688094421001e-5 5.12746031365021e-5 3.924317996556517e-5 2.924234959570458e-5 2.12151704854381e-5 1.4985352932052484e-5 1.0305607881811182e-5 6.90027565646087e-6 4.498269337078975e-6 2.8550298879665212e-6 1.7642578193002753e-6 1.0614502196981392e-6 6.217608892412182e-7 3.5459561217323976e-7 1.9689264940429196e-7 1.0644170854928012e-7; 1.5897140421510779e-7 2.940604898402379e-7 5.295909203636775e-7 9.286040500087317e-7 1.5852830082272205e-6 2.6349308626683538e-6 4.264006248603274e-6 6.718195365326483e-6 1.0305607881811166e-5 1.5391494354896675e-5 2.2380724912588574e-5 3.168499912956171e-5 4.3673644862868684e-5 5.860995196355548e-5 7.65789630049792e-5 9.741678330418758e-5 0.00012065470929131058 0.00014549261795075095 0.0001708141435902456 0.00019525082863865447 0.00021729419880417281 0.00023544505127523585 0.0002483803242702325 0.00025511206684333235 0.00025511206684333235 0.0002483803242702325 0.00023544505127523585 0.00021729419880417281 0.00019525082863865455 0.0001708141435902456 0.0001454926179507509 0.00012065470929131058 9.741678330418758e-5 7.65789630049792e-5 5.860995196355542e-5 4.367364486286876e-5 3.168499912956171e-5 2.2380724912588537e-5 1.5391494354896698e-5 1.0305607881811166e-5 6.718195365326483e-6 4.264006248603274e-6 2.634930862668358e-6 1.5852830082272205e-6 9.286040500087317e-7 5.295909203636775e-7 2.940604898402379e-7 1.5897140421510779e-7; 2.3115983313045458e-7 4.275924597718894e-7 7.700765391304108e-7 1.350280311758629e-6 2.3051551784149337e-6 3.8314449163481485e-6 6.200278457379807e-6 9.76890735320735e-6 1.49853529320525e-5 2.2380724912588574e-5 3.254374371086372e-5 4.607306042046038e-5 6.350571354983377e-5 8.5224552066905e-5 0.00011135323611774424 0.0001416534469446525 0.0001754436441188102 0.00021156037120802129 0.00024838032427023277 0.0002839136333325858 0.00031596683053652545 0.00034235992965458496 0.00036116907059275224 0.00037095767689938896 0.00037095767689938896 0.00036116907059275224 0.00034235992965458496 0.00031596683053652545 0.0002839136333325859 0.00024838032427023277 0.00021156037120802118 0.0001754436441188102 0.0001416534469446525 0.00011135323611774424 8.522455206690495e-5 6.350571354983388e-5 4.607306042046038e-5 3.2543743710863696e-5 2.238072491258862e-5 1.49853529320525e-5 9.76890735320735e-6 6.200278457379807e-6 3.831444916348156e-6 2.3051551784149337e-6 1.350280311758629e-6 7.700765391304108e-7 4.275924597718894e-7 2.3115983313045458e-7; 3.272592438419348e-7 6.053542398020764e-7 1.0902182376728397e-6 1.911628451260389e-6 3.263470692162409e-6 5.424280460691016e-6 8.777902337498369e-6 1.3830107031477007e-5 2.12151704854381e-5 3.168499912956171e-5 4.607306042046038e-5 6.522688094421008e-5 8.990676067944877e-5 0.0001206547092913108 0.00015764579580172894 0.0002005426258832297 0.0002483803242702331 0.0002995117541436506 0.00035163876009556546 0.0004019442292485252 0.00044732280967758225 0.00048468823576848475 0.0005113168466191734 0.0005251748419931728 0.0005251748419931728 0.0005113168466191734 0.00048468823576848475 0.00044732280967758225 0.0004019442292485254 0.00035163876009556546 0.0002995117541436505 0.0002483803242702331 0.0002005426258832297 0.00015764579580172894 0.00012065470929131066 8.990676067944894e-5 6.522688094421008e-5 4.60730604204603e-5 3.168499912956177e-5 2.12151704854381e-5 1.3830107031477007e-5 8.777902337498369e-6 5.424280460691025e-6 3.263470692162409e-6 1.911628451260389e-6 1.0902182376728397e-6 6.053542398020764e-7 3.272592438419348e-7; 4.510842476340416e-7 8.344019823778814e-7 1.502723858077153e-6 2.6349308626683487e-6 4.498269337078971e-6 7.476664193933783e-6 1.2099195198373104e-5 1.9063001404492862e-5 2.924234959570453e-5 4.3673644862868684e-5 6.350571354983377e-5 8.990676067944877e-5 0.0001239247607560054 0.00016630680351526535 0.0002172941988041729 0.00027642189248223184 0.0003423599296545847 0.0004128379467279341 0.00048468823576848383 0.0005540277735561485 0.0006165762368821491 0.0006680796105313319 0.0007047836826611451 0.0007238851245138184 0.0007238851245138184 0.0007047836826611451 0.0006680796105313319 0.0006165762368821491 0.0005540277735561485 0.00048468823576848383 0.0004128379467279341 0.0003423599296545847 0.00027642189248223184 0.0002172941988041729 0.00016630680351526516 0.00012392476075600558 8.990676067944877e-5 6.35057135498337e-5 4.367364486286876e-5 2.924234959570453e-5 1.9063001404492862e-5 1.2099195198373104e-5 7.476664193933798e-6 4.498269337078971e-6 2.6349308626683487e-6 1.502723858077153e-6 8.344019823778814e-7 4.510842476340416e-7; 6.053542398020764e-7 1.1197659425728733e-6 2.0166526840829736e-6 3.53607242475841e-6 6.0366692680024615e-6 1.0033669748197674e-5 1.6237097947765598e-5 2.5582513209206164e-5 3.9243179965565206e-5 5.860995196355548e-5 8.5224552066905e-5 0.0001206547092913108 0.00016630680351526535 0.00022318342780520376 0.00029160841953678227 0.00037095767689938896 0.00045944640284331504 0.0005540277735561492 0.0006504507307528019 0.0007435042643351424 0.0008274441883429026 0.0008965616220954052 0.0009458184201288619 0.0009714525203496029 0.0009714525203496029 0.0009458184201288619 0.0008965616220954052 0.0008274441883429026 0.0007435042643351424 0.0006504507307528019 0.0005540277735561492 0.00045944640284331504 0.00037095767689938896 0.00029160841953678227 0.0002231834278052034 0.0001663068035152656 0.0001206547092913108 8.522455206690495e-5 5.860995196355557e-5 3.9243179965565206e-5 2.5582513209206164e-5 1.6237097947765598e-5 1.003366974819769e-5 6.0366692680024615e-6 3.53607242475841e-6 2.0166526840829736e-6 1.1197659425728733e-6 6.053542398020764e-7; 7.909475845251705e-7 1.4630708918486186e-6 2.6349308626683538e-6 4.620183950447215e-6 7.887429643946967e-6 1.3109855898349667e-5 2.1215170485438063e-5 3.342576246846911e-5 5.12746031365021e-5 7.65789630049792e-5 0.00011135323611774424 0.00015764579580172894 0.0002172941988041729 0.00029160841953678227 0.0003810115794930779 0.0004846882357684846 0.0006003063969065673 0.0007238851245138195 0.0008498700438767397 0.0009714525203496029 0.0010811272789849299 0.001171435174231512 0.0012357934340146508 0.0012692865993680492 0.0012692865993680492 0.0012357934340146508 0.001171435174231512 0.0010811272789849299 0.0009714525203496029 0.0008498700438767397 0.0007238851245138195 0.0006003063969065675 0.0004846882357684846 0.0003810115794930779 0.00029160841953678184 0.0002172941988041732 0.00015764579580172894 0.00011135323611774417 7.657896300497933e-5 5.12746031365021e-5 3.342576246846911e-5 2.1215170485438063e-5 1.3109855898349693e-5 7.887429643946967e-6 4.620183950447215e-6 2.6349308626683538e-6 1.4630708918486186e-6 7.909475845251705e-7; 1.0061714917927168e-6 1.8611855585014145e-6 3.3519191015081937e-6 5.8773772987359084e-6 1.0033669748197654e-5 1.6677164864658932e-5 2.6988007996490334e-5 4.252121119681889e-5 6.522688094421001e-5 9.741678330418758e-5 0.0001416534469446525 0.0002005426258832297 0.00027642189248223184 0.00037095767689938896 0.0004846882357684846 0.0006165762368821498 0.0007636551330652295 0.0009208607370055721 0.0010811272789849292 0.0012357934340146508 0.0013753116747516273 0.0014901931554700534 0.001572063872976649 0.0016146708279858348 0.001614670827985835 0.001572063872976649 0.0014901931554700534 0.0013753116747516271 0.0012357934340146508 0.0010811272789849292 0.0009208607370055721 0.0007636551330652295 0.0006165762368821498 0.0004846882357684846 0.00037095767689938853 0.00027642189248223216 0.0002005426258832297 0.00014165344694465232 9.741678330418775e-5 6.522688094421001e-5 4.252121119681889e-5 2.6988007996490334e-5 1.667716486465896e-5 1.0033669748197654e-5 5.8773772987359084e-6 3.3519191015081937e-6 1.8611855585014145e-6 1.0061714917927168e-6; 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1.0061714917927168e-6 1.8611855585014145e-6 3.3519191015081937e-6 5.8773772987359084e-6 1.0033669748197654e-5 1.6677164864658932e-5 2.6988007996490334e-5 4.252121119681889e-5 6.522688094421001e-5 9.741678330418758e-5 0.0001416534469446525 0.0002005426258832297 0.00027642189248223184 0.00037095767689938896 0.0004846882357684846 0.0006165762368821498 0.0007636551330652295 0.0009208607370055721 0.0010811272789849292 0.0012357934340146508 0.0013753116747516273 0.0014901931554700534 0.001572063872976649 0.0016146708279858348 0.001614670827985835 0.001572063872976649 0.0014901931554700534 0.0013753116747516271 0.0012357934340146508 0.0010811272789849292 0.0009208607370055721 0.0007636551330652295 0.0006165762368821498 0.0004846882357684846 0.00037095767689938853 0.00027642189248223216 0.0002005426258832297 0.00014165344694465232 9.741678330418775e-5 6.522688094421001e-5 4.252121119681889e-5 2.6988007996490334e-5 1.667716486465896e-5 1.0033669748197654e-5 5.8773772987359084e-6 3.3519191015081937e-6 1.8611855585014145e-6 1.0061714917927168e-6; 7.909475845251705e-7 1.4630708918486186e-6 2.6349308626683538e-6 4.620183950447215e-6 7.887429643946967e-6 1.3109855898349667e-5 2.1215170485438063e-5 3.342576246846911e-5 5.12746031365021e-5 7.65789630049792e-5 0.00011135323611774424 0.00015764579580172894 0.0002172941988041729 0.00029160841953678227 0.0003810115794930779 0.0004846882357684846 0.0006003063969065673 0.0007238851245138195 0.0008498700438767397 0.0009714525203496029 0.0010811272789849299 0.001171435174231512 0.0012357934340146508 0.0012692865993680492 0.0012692865993680492 0.0012357934340146508 0.001171435174231512 0.0010811272789849299 0.0009714525203496029 0.0008498700438767397 0.0007238851245138195 0.0006003063969065675 0.0004846882357684846 0.0003810115794930779 0.00029160841953678184 0.0002172941988041732 0.00015764579580172894 0.00011135323611774417 7.657896300497933e-5 5.12746031365021e-5 3.342576246846911e-5 2.1215170485438063e-5 1.3109855898349693e-5 7.887429643946967e-6 4.620183950447215e-6 2.6349308626683538e-6 1.4630708918486186e-6 7.909475845251705e-7; 6.053542398020754e-7 1.1197659425728733e-6 2.0166526840829715e-6 3.5360724247584035e-6 6.0366692680024505e-6 1.0033669748197654e-5 1.623709794776558e-5 2.558251320920614e-5 3.924317996556517e-5 5.860995196355542e-5 8.522455206690495e-5 0.00012065470929131066 0.00016630680351526516 0.0002231834278052034 0.00029160841953678184 0.00037095767689938853 0.00045944640284331455 0.0005540277735561487 0.0006504507307528012 0.0007435042643351416 0.0008274441883429017 0.0008965616220954041 0.0009458184201288609 0.0009714525203496018 0.0009714525203496018 0.0009458184201288609 0.0008965616220954041 0.0008274441883429017 0.0007435042643351416 0.0006504507307528012 0.0005540277735561485 0.00045944640284331455 0.00037095767689938853 0.00029160841953678184 0.00022318342780520322 0.00016630680351526544 0.00012065470929131066 8.522455206690484e-5 5.8609951963555525e-5 3.924317996556517e-5 2.558251320920614e-5 1.623709794776558e-5 1.0033669748197674e-5 6.0366692680024505e-6 3.5360724247584035e-6 2.0166526840829715e-6 1.1197659425728733e-6 6.053542398020754e-7; 4.5108424763404237e-7 8.344019823778828e-7 1.5027238580771545e-6 2.6349308626683512e-6 4.498269337078975e-6 7.476664193933791e-6 1.2099195198373111e-5 1.906300140449289e-5 2.924234959570458e-5 4.367364486286876e-5 6.350571354983388e-5 8.990676067944894e-5 0.00012392476075600558 0.0001663068035152656 0.0002172941988041732 0.00027642189248223216 0.0003423599296545851 0.0004128379467279347 0.0004846882357684846 0.0005540277735561492 0.0006165762368821499 0.0006680796105313329 0.0007047836826611461 0.0007238851245138195 0.0007238851245138195 0.0007047836826611461 0.0006680796105313329 0.0006165762368821499 0.0005540277735561492 0.0004846882357684846 0.0004128379467279347 0.0003423599296545853 0.00027642189248223216 0.0002172941988041732 0.00016630680351526544 0.00012392476075600575 8.990676067944894e-5 6.350571354983377e-5 4.367364486286884e-5 2.924234959570458e-5 1.906300140449289e-5 1.2099195198373111e-5 7.476664193933804e-6 4.498269337078975e-6 2.6349308626683512e-6 1.5027238580771545e-6 8.344019823778828e-7 4.5108424763404237e-7; 3.272592438419348e-7 6.053542398020764e-7 1.0902182376728397e-6 1.911628451260389e-6 3.263470692162409e-6 5.424280460691016e-6 8.777902337498369e-6 1.3830107031477007e-5 2.12151704854381e-5 3.168499912956171e-5 4.607306042046038e-5 6.522688094421008e-5 8.990676067944877e-5 0.0001206547092913108 0.00015764579580172894 0.0002005426258832297 0.0002483803242702331 0.0002995117541436506 0.00035163876009556546 0.0004019442292485252 0.00044732280967758225 0.00048468823576848475 0.0005113168466191734 0.0005251748419931728 0.0005251748419931728 0.0005113168466191734 0.00048468823576848475 0.00044732280967758225 0.0004019442292485254 0.00035163876009556546 0.0002995117541436505 0.0002483803242702331 0.0002005426258832297 0.00015764579580172894 0.00012065470929131066 8.990676067944894e-5 6.522688094421008e-5 4.60730604204603e-5 3.168499912956177e-5 2.12151704854381e-5 1.3830107031477007e-5 8.777902337498369e-6 5.424280460691025e-6 3.263470692162409e-6 1.911628451260389e-6 1.0902182376728397e-6 6.053542398020764e-7 3.272592438419348e-7; 2.3115983313045418e-7 4.275924597718886e-7 7.700765391304094e-7 1.3502803117586267e-6 2.305155178414932e-6 3.831444916348146e-6 6.200278457379796e-6 9.768907353207345e-6 1.4985352932052484e-5 2.2380724912588537e-5 3.2543743710863696e-5 4.60730604204603e-5 6.35057135498337e-5 8.522455206690495e-5 0.00011135323611774417 0.00014165344694465232 0.00017544364411880986 0.000211560371208021 0.00024838032427023244 0.0002839136333325856 0.00031596683053652507 0.0003423599296545846 0.0003611690705927517 0.00037095767689938853 0.00037095767689938853 0.0003611690705927517 0.0003423599296545846 0.00031596683053652507 0.0002839136333325856 0.00024838032427023244 0.0002115603712080209 0.00017544364411880995 0.00014165344694465232 0.00011135323611774417 8.522455206690484e-5 6.350571354983377e-5 4.60730604204603e-5 3.2543743710863635e-5 2.2380724912588574e-5 1.4985352932052484e-5 9.768907353207345e-6 6.200278457379796e-6 3.831444916348152e-6 2.305155178414932e-6 1.3502803117586267e-6 7.700765391304094e-7 4.275924597718886e-7 2.3115983313045418e-7; 1.5897140421510802e-7 2.940604898402384e-7 5.295909203636784e-7 9.286040500087334e-7 1.5852830082272232e-6 2.634930862668358e-6 4.264006248603281e-6 6.7181953653264934e-6 1.0305607881811182e-5 1.5391494354896698e-5 2.238072491258862e-5 3.168499912956177e-5 4.367364486286876e-5 5.860995196355557e-5 7.657896300497933e-5 9.741678330418775e-5 0.0001206547092913108 0.0001454926179507512 0.0001708141435902459 0.00019525082863865482 0.0002172941988041732 0.0002354450512752363 0.00024838032427023293 0.00025511206684333273 0.00025511206684333273 0.00024838032427023293 0.0002354450512752363 0.0002172941988041732 0.00019525082863865493 0.0001708141435902459 0.00014549261795075114 0.0001206547092913108 9.741678330418775e-5 7.657896300497933e-5 5.8609951963555525e-5 4.367364486286884e-5 3.168499912956177e-5 2.2380724912588574e-5 1.539149435489673e-5 1.0305607881811182e-5 6.7181953653264934e-6 4.264006248603281e-6 2.634930862668363e-6 1.5852830082272232e-6 9.286040500087334e-7 5.295909203636784e-7 2.940604898402384e-7 1.5897140421510802e-7; 1.0644170854928012e-7 1.9689264940429196e-7 3.5459561217323976e-7 6.217608892412182e-7 1.0614502196981392e-6 1.7642578193002719e-6 2.8550298879665212e-6 4.498269337078975e-6 6.90027565646087e-6 1.0305607881811166e-5 1.49853529320525e-5 2.12151704854381e-5 2.924234959570453e-5 3.9243179965565206e-5 5.12746031365021e-5 6.522688094421001e-5 8.07861858231187e-5 9.741678330418785e-5 0.0001143711938502203 0.0001307331459929699 0.0001454926179507512 0.00015764579580172894 0.00016630680351526565 0.000170814143590246 0.000170814143590246 0.00016630680351526565 0.00015764579580172894 0.0001454926179507512 0.00013073314599296995 0.0001143711938502203 9.741678330418779e-5 8.07861858231187e-5 6.522688094421001e-5 5.12746031365021e-5 3.924317996556517e-5 2.924234959570458e-5 2.12151704854381e-5 1.4985352932052484e-5 1.0305607881811182e-5 6.90027565646087e-6 4.498269337078975e-6 2.8550298879665212e-6 1.7642578193002753e-6 1.0614502196981392e-6 6.217608892412182e-7 3.5459561217323976e-7 1.9689264940429196e-7 1.0644170854928012e-7; 6.938903568369913e-8 1.2835373709777678e-7 2.3115983313045458e-7 4.053240888209999e-7 6.919562657809534e-7 1.1501144659097052e-6 1.8611855585014124e-6 2.932408506022366e-6 4.498269337078975e-6 6.718195365326483e-6 9.76890735320735e-6 1.3830107031477007e-5 1.9063001404492862e-5 2.5582513209206164e-5 3.342576246846911e-5 4.252121119681889e-5 5.2664276130395184e-5 6.350571354983377e-5 7.455824375071895e-5 8.52245520669049e-5 9.484620827958101e-5 0.00010276882905545298 0.00010841491442445243 0.00011135323611774409 0.00011135323611774409 0.00010841491442445243 0.00010276882905545298 9.484620827958101e-5 8.52245520669049e-5 7.455824375071895e-5 6.350571354983377e-5 5.2664276130395184e-5 4.252121119681889e-5 3.342576246846911e-5 2.558251320920614e-5 1.906300140449289e-5 1.3830107031477007e-5 9.768907353207345e-6 6.7181953653264934e-6 4.498269337078975e-6 2.932408506022366e-6 1.8611855585014124e-6 1.150114465909707e-6 6.919562657809534e-7 4.053240888209999e-7 2.3115983313045458e-7 1.2835373709777678e-7 6.938903568369913e-8; 4.404088682310447e-8 8.146549888102353e-8 1.467160329962283e-7 2.572572474391242e-7 4.391813099509043e-7 7.299721134277668e-7 1.1812855119128452e-6 1.8611855585014124e-6 2.8550298879665212e-6 4.264006248603274e-6 6.200278457379807e-6 8.777902337498369e-6 1.2099195198373104e-5 1.6237097947765598e-5 2.1215170485438063e-5 2.6988007996490334e-5 3.342576246846914e-5 4.0306770594388826e-5 4.73217582922288e-5 5.409161281960394e-5 6.019843168713919e-5 6.522688094420994e-5 6.881042414061877e-5 7.067536277060657e-5 7.067536277060657e-5 6.881042414061877e-5 6.522688094420994e-5 6.0198431687139135e-5 5.409161281960394e-5 4.73217582922288e-5 4.0306770594388826e-5 3.342576246846914e-5 2.6988007996490334e-5 2.1215170485438063e-5 1.623709794776558e-5 1.2099195198373111e-5 8.777902337498369e-6 6.200278457379796e-6 4.264006248603281e-6 2.8550298879665212e-6 1.8611855585014124e-6 1.1812855119128452e-6 7.299721134277695e-7 4.391813099509043e-7 2.572572474391242e-7 1.467160329962283e-7 8.146549888102353e-8 4.404088682310447e-8; 2.72149441496466e-8 5.034137961561296e-8 9.066276662156917e-8 1.5897140421510834e-7 2.713908752539499e-7 4.510842476340439e-7 7.299721134277695e-7 1.150114465909707e-6 1.7642578193002753e-6 2.634930862668358e-6 3.831444916348156e-6 5.424280460691025e-6 7.476664193933798e-6 1.003366974819769e-5 1.3109855898349693e-5 1.667716486465896e-5 2.065535742712404e-5 2.4907457358543362e-5 2.9242349595704604e-5 3.3425762468469255e-5 3.719945428987896e-5 4.0306770594388975e-5 4.252121119681908e-5 4.367364486286892e-5 4.367364486286892e-5 4.252121119681908e-5 4.0306770594388975e-5 3.7199454289878936e-5 3.3425762468469255e-5 2.9242349595704604e-5 2.4907457358543362e-5 2.065535742712406e-5 1.667716486465896e-5 1.3109855898349693e-5 1.0033669748197674e-5 7.476664193933804e-6 5.424280460691025e-6 3.831444916348152e-6 2.634930862668363e-6 1.7642578193002753e-6 1.150114465909707e-6 7.299721134277695e-7 4.5108424763404475e-7 2.713908752539499e-7 1.5897140421510834e-7 9.066276662156917e-8 5.034137961561296e-8 2.72149441496466e-8; 1.6373632090899278e-8 3.0287448845822705e-8 5.4546457153905775e-8 9.564374893731387e-8 1.6327993619244742e-7 2.713908752539494e-7 4.391813099509043e-7 6.919562657809534e-7 1.0614502196981392e-6 1.5852830082272205e-6 2.3051551784149337e-6 3.263470692162409e-6 4.498269337078971e-6 6.0366692680024615e-6 7.887429643946967e-6 1.0033669748197654e-5 1.2427114358863923e-5 1.4985352932052472e-5 1.759340276874101e-5 2.011031637643879e-5 2.2380724912588598e-5 2.425021447245457e-5 2.5582513209206184e-5 2.627586480137954e-5 2.627586480137954e-5 2.5582513209206184e-5 2.425021447245457e-5 2.2380724912588574e-5 2.011031637643879e-5 1.759340276874101e-5 1.4985352932052472e-5 1.2427114358863933e-5 1.0033669748197654e-5 7.887429643946967e-6 6.0366692680024505e-6 4.498269337078975e-6 3.263470692162409e-6 2.305155178414932e-6 1.5852830082272232e-6 1.0614502196981392e-6 6.919562657809534e-7 4.391813099509043e-7 2.713908752539499e-7 1.6327993619244742e-7 9.564374893731387e-8 5.4546457153905775e-8 3.0287448845822705e-8 1.6373632090899278e-8; 9.591108334634163e-9 1.7741341777273437e-8 3.195143123585701e-8 5.6024805766717734e-8 9.564374893731387e-8 1.5897140421510802e-7 2.572572474391242e-7 4.053240888209999e-7 6.217608892412182e-7 9.286040500087317e-7 1.350280311758629e-6 1.911628451260389e-6 2.6349308626683487e-6 3.53607242475841e-6 4.620183950447215e-6 5.8773772987359084e-6 7.279374511474409e-6 8.77790233749836e-6 1.0305607881811155e-5 1.1779928969907525e-5 1.3109855898349644e-5 1.420493833330359e-5 1.4985352932052472e-5 1.5391494354896698e-5 1.5391494354896698e-5 1.4985352932052472e-5 1.420493833330359e-5 1.3109855898349632e-5 1.1779928969907525e-5 1.0305607881811155e-5 8.77790233749836e-6 7.279374511474415e-6 5.8773772987359084e-6 4.620183950447215e-6 3.5360724247584035e-6 2.6349308626683512e-6 1.911628451260389e-6 1.3502803117586267e-6 9.286040500087334e-7 6.217608892412182e-7 4.053240888209999e-7 2.572572474391242e-7 1.5897140421510834e-7 9.564374893731387e-8 5.6024805766717734e-8 3.195143123585701e-8 1.7741341777273437e-8 9.591108334634163e-9; 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3.0372105473444726e-9 5.618140103307805e-9 1.0118040644152322e-8 1.7741341777273437e-8 3.0287448845822705e-8 5.034137961561296e-8 8.146549888102353e-8 1.2835373709777678e-7 1.9689264940429196e-7 2.940604898402379e-7 4.275924597718894e-7 6.053542398020764e-7 8.344019823778814e-7 1.1197659425728733e-6 1.4630708918486186e-6 1.8611855585014145e-6 2.3051551784149337e-6 2.7796930899776574e-6 3.263470692162403e-6 3.7303430704848267e-6 4.151490236519409e-6 4.498269337078975e-6 4.745402761905639e-6 4.8740153236803395e-6 4.8740153236803395e-6 4.745402761905639e-6 4.498269337078975e-6 4.151490236519405e-6 3.7303430704848267e-6 3.263470692162403e-6 2.7796930899776574e-6 2.3051551784149337e-6 1.8611855585014145e-6 1.4630708918486186e-6 1.1197659425728733e-6 8.344019823778828e-7 6.053542398020764e-7 4.275924597718886e-7 2.940604898402384e-7 1.9689264940429196e-7 1.2835373709777678e-7 8.146549888102353e-8 5.034137961561296e-8 3.0287448845822705e-8 1.7741341777273437e-8 1.0118040644152322e-8 5.618140103307805e-9 3.0372105473444726e-9; 1.6419398126916194e-9 3.0372105473444726e-9 5.4698920279304e-9 9.591108334634163e-9 1.6373632090899278e-8 2.7214944149646553e-8 4.404088682310447e-8 6.938903568369913e-8 1.0644170854928012e-7 1.5897140421510779e-7 2.3115983313045458e-7 3.272592438419348e-7 4.510842476340416e-7 6.053542398020764e-7 7.909475845251705e-7 1.0061714917927168e-6 1.2461849459797968e-6 1.502723858077153e-6 1.7642578193002676e-6 2.0166526840829715e-6 2.2443281409323587e-6 2.431799639052888e-6 2.5654019043369063e-6 2.6349308626683512e-6 2.6349308626683512e-6 2.5654019043369063e-6 2.431799639052888e-6 2.244328140932357e-6 2.0166526840829715e-6 1.7642578193002676e-6 1.502723858077153e-6 1.2461849459797968e-6 1.0061714917927168e-6 7.909475845251705e-7 6.053542398020754e-7 4.5108424763404237e-7 3.272592438419348e-7 2.3115983313045418e-7 1.5897140421510802e-7 1.0644170854928012e-7 6.938903568369913e-8 4.404088682310447e-8 2.72149441496466e-8 1.6373632090899278e-8 9.591108334634163e-9 5.4698920279304e-9 3.0372105473444726e-9 1.6419398126916194e-9])

To make the Gaussian Markov random field efficient we first precompute a bunch of quantities that allow us to scale things linearly with the number of image pixels. The returns a functional that accepts a single argument related to the correlation length of the field. The second argument defines the underlying random field of the Markov process. Here we are using a zero mean and unit variance Gaussian Markov random field. For this tutorial we will use the first order random field

cprior = corr_image_prior(grid, dvis)

HierarchicalPrior(
	map: 
	ConditionalMarkov(
Random Field: VLBIImagePriors.GaussMarkovRandomField
Graph: MarkovRandomFieldGraph{1}(
dims: (48, 48)
)
)	hyper prior: 
	Truncated(Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.03623874772901475)
θ: 27.594772520225487
)
; lower=1.0, upper=96.0)

)

Putting everything together the total prior is then our image prior, a prior on the standard deviation of the MRF, and a prior on the fractional flux of the Gaussian component.

prior = (
    c = cprior,
    σimg = truncated(Normal(0.0, 0.5); lower = 0.0),
    fg = Uniform(0.0, 1.0),
)

(c = HierarchicalPrior(
	map: 
	ConditionalMarkov(
Random Field: VLBIImagePriors.GaussMarkovRandomField
Graph: MarkovRandomFieldGraph{1}(
dims: (48, 48)
)
)	hyper prior: 
	Truncated(Distributions.InverseGamma{Float64}(
invd: Distributions.Gamma{Float64}(α=1.0, θ=0.03623874772901475)
θ: 27.594772520225487
)
; lower=1.0, upper=96.0)

)
, σimg = Truncated(Distributions.Normal{Float64}(μ=0.0, σ=0.5); lower=0.0), fg = Distributions.Uniform{Float64}(a=0.0, b=1.0))

Now we can construct our sky model.

skym = SkyModel(sky, prior, grid; metadata = skymeta)

SkyModel
  with map: 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)
)
   )

Unlike other imaging examples (e.g., Imaging a Black Hole using only Closure Quantities) we also need to include a model for the instrument, i.e., gains. The gains will be broken into two components

• Gain amplitudes which are typically known to 10-20%, except for LMT, which has amplitudes closer to 50-100%.

• Gain phases which are more difficult to constrain and can shift rapidly.

gain(x) = exp(x.lg + 1im * x.gp)
G = SingleStokesGain(gain)

intpr = (
    lg = ArrayPrior(IIDSitePrior(ScanSeg(), Normal(0.0, 0.2)); LM = IIDSitePrior(ScanSeg(), Normal(0.0, 1.0))),
    gp = ArrayPrior(IIDSitePrior(ScanSeg(), DiagonalVonMises(0.0, inv(π^2))); refant = SEFDReference(0.0), phase = true),
)
intmodel = InstrumentModel(G, intpr)

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

To form the posterior we just combine the skymodel, instrument model and the data. To utilize gradients of the posterior we also need to load Enzyme.jl. Under the hood, Comrade will use Enzyme to compute the gradients of the posterior.

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

VLBIPosterior
ObservedSkyModel
  with map: sky
   on grid: 
FourierDualDomain(
Algorithm: VLBISkyModels.NFFTAlg{Float64, AbstractNFFTs.PrecomputeFlags, UInt32}(1, 1.0e-9, AbstractNFFTs.TENSOR, 0x00000000)
Image Domain: 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)
)
Visibility Domain: UnstructuredDomain(
executor: ComradeBase.Serial()
Dimensions: 
274-element StructArray(::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}) with eltype @NamedTuple{U::Float64, V::Float64, Ti::Float64, Fr::Float64}:
 (U = -4.405690154666661e9, V = -4.523017159111106e9, Ti = 0.9166666567325592, Fr = 2.27070703125e11)
 (U = 787577.6145833326, V = -1.6838098888888871e6, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = -4.444299918222218e9, V = -4.597825294222218e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 1.337045162666665e9, V = -3.765300401777774e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = -1.336260540444443e9, V = 3.763616127999996e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 4.445088654222218e9, V = 4.596145080888884e9, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 5.781345607111105e9, V = 8.325259893333325e8, Ti = 1.2166666388511658, Fr = 2.27070703125e11)
 (U = 757554.6649305547, V = -1.6707483020833314e6, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 1.4806382151111097e9, V = -3.741479615999996e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = -4.455366328888884e9, V = -4.673060451555551e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = -1.4798758791111097e9, V = 3.739809735111107e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 4.456123861333328e9, V = 4.671391715555551e9, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 5.936013027555549e9, V = 9.315912497777768e8, Ti = 1.516666665673256, Fr = 2.27070703125e11)
 (U = 722830.7065972214, V = -1.6582321493055536e6, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = -4.438811278222218e9, V = -4.748261176888884e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 1.615060401777776e9, V = -3.715306943999996e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = -1.6143345528888872e9, V = 3.713649343999996e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 4.439536184888884e9, V = 4.746598001777773e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 6.053865329777771e9, V = 1.0329465084444433e9, Ti = 1.816666603088379, Fr = 2.27070703125e11)
 (U = 683620.5937499993, V = -1.6463396631944429e6, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = -4.394744988444439e9, V = -4.822934485333328e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = 1.7394626631111093e9, V = -3.686947740444441e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = -1.7387777884444425e9, V = 3.6853017173333297e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = 4.395427669333329e9, V = 4.821289443555551e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = 6.134203007999993e9, V = 1.1359907128888876e9, Ti = 2.1166666746139526, Fr = 2.27070703125e11)
 (U = -1.8643712995555537e9, V = 3.651452543999996e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 6.178850375111104e9, V = 1.2516714417777765e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 4.314477112888885e9, V = 4.903116785777773e9, Ti = 2.449999988079071, Fr = 2.27070703125e11)
 (U = 587276.6180555549, V = -1.6236182569444429e6, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 1.9658264106666646e9, V = -3.6206990791111073e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = -4.212778346666662e9, V = -4.976839239111106e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = -1.965238759111109e9, V = 3.6190756053333297e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 6.178607487999993e9, V = 1.356139594666665e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 4.213364039111107e9, V = 4.975216568888884e9, Ti = 2.7500000596046448, Fr = 2.27070703125e11)
 (U = 535806.2881944438, V = -1.6141240694444429e6, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 2.0544588266666646e9, V = -3.586710200888885e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = -4.085601863111107e9, V = -5.046995640888884e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = -2.0539194026666646e9, V = 3.5850976213333297e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 4.0861423288888845e9, V = 5.045379299555551e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 6.140063729777771e9, V = 1.4602817422222207e9, Ti = 3.0500001311302185, Fr = 2.27070703125e11)
 (U = 481010.2230902773, V = -1.6055275520833316e6, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 2.130349845333331e9, V = -3.5513328995555515e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = -3.933104519111107e9, V = -5.114784767999995e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = -2.1298682808888867e9, V = 3.5497276302222185e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
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 (U = 416648.93055555515, V = -1.5970948472222206e6, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 2.578938929777775e9, V = -4.734788238222218e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
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 (U = 2.1991734328888865e9, V = -3.5106569031111073e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -2.5785191964444413e9, V = 4.733192191999995e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -2.198756124444442e9, V = 3.509060273777774e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
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 (U = 3.735526620444441e9, V = 5.185227192888883e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 6.314048611555549e9, V = 4.520333351111106e8, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
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 (U = 6.120220771555549e9, V = 6.626024835555549e8, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 5.588211185777772e9, V = 1.8712975182222202e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 238138.85156249977, V = -1.5812476874999984e6, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.092881095111105e9, V = -4.199598421333329e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.903268209777775e9, V = -4.595170062222218e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.299867178666664e9, V = -3.396077226666663e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -3.062767004444441e9, V = -5.359950862222217e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.050181503999994e9, V = -4.210810559999995e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -2.8907034879999967e9, V = 4.600886271999995e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.189606492444442e9, V = 3.955690115555551e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = -2.2976930559999976e9, V = 3.400280575999996e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 2.79301371733333e9, V = -8.035218719999992e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 6.034046399999994e8, V = -1.199091349333332e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 3.1008046079999967e9, V = 5.350639359999994e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 8.155639423999991e9, V = 1.1603480675555544e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.966032455111105e9, V = 7.64783736888888e8, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 5.36262916266666e9, V = 1.963875015111109e9, Ti = 4.583333194255829, Fr = 2.27070703125e11)
 (U = 163232.87499999983, V = -1.5773965972222206e6, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.388079544888883e9, V = -4.101135303111107e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.3048084479999976e9, V = -3.3528182257777743e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.768253631999997e9, V = -5.414730879999994e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.388055679999994e9, V = -4.101249678222218e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.9831417315555525e9, V = -4.539868501333328e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.356877311999994e9, V = -4.110922695111107e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.975202047999997e9, V = 4.544589937777773e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.4049356302222195e9, V = 4.387324826666662e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.4049011413333306e9, V = 4.3861474933333284e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -2.305032220444442e9, V = 3.356104334222219e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 3.083277411555552e9, V = -7.48315953777777e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 3.0832264319999967e9, V = -7.484370453333325e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 6.783317208888881e8, V = -1.187050453333332e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 2.8027259164444413e9, V = 5.407289457777772e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 8.156328149333324e9, V = 1.3135985066666653e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 8.156297400888881e9, V = 1.3134812622222207e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.751382712888883e9, V = 8.74867279999999e8, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 5.073054606222218e9, V = 2.0619144959999979e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = -5.356795278222217e9, V = 4.1110577777777734e9, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 30750.899088541635, V = 116905.95138888876, Ti = 4.916666686534882, Fr = 2.27070703125e11)
 (U = 99881.10751488084, V = -1.575297547619046e6, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.59465437866666e9, V = -4.018611114666662e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.030624810666663e9, V = -4.494683207111106e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.2960602239999976e9, V = -3.3182473599999967e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.594626830222217e9, V = -4.0187286115555515e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -2.5172720142222195e9, V = -5.45444669155555e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.579493814857137e9, V = -4.023605065142853e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -3.0274068479999967e9, V = 4.496662674285709e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.5640353279999976e9, V = 4.7607165511111057e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.564000689777775e9, V = 4.7595243111111057e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -2.2970926933333306e9, V = 3.3193651078095202e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.2985927679999967e9, V = -7.003643235555549e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 7.345665155555549e8, V = -1.1764336924444432e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 3.2985603128888855e9, V = -7.004829137777771e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 2.5374800944761877e9, V = 5.44992148723809e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 8.111924807111103e9, V = 1.4358354488888874e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 4.813340984888884e9, V = 2.136196949333331e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 5.547893447111105e9, V = 9.597677279999989e8, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 8.111898709333324e9, V = 1.4357162488888874e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = -5.579455049142851e9, V = 4.0237272990476146e9, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 27098.63357204858, V = 117340.62348090265, Ti = 5.183333337306976, Fr = 2.27070703125e11)
 (U = 45690.53906249995, V = -1.5743099999999984e6, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.773825464888883e9, V = -3.933187214222218e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.2760630257777753e9, V = -3.283890659555552e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.0632664319999967e9, V = -4.448890581333328e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.773791857777772e9, V = -3.9333102648888845e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -2.253957283555553e9, V = -5.490298367999994e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.744240127999994e9, V = -3.946954495999996e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -3.0586252799999967e9, V = 4.455449599999995e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.710558015999997e9, V = 5.1570311111111057e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.7105363982222195e9, V = 5.1558591999999946e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -2.2804378879999976e9, V = 3.2883723519999967e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.4977656675555515e9, V = -6.492943324444438e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 7.872002471111102e8, V = -1.164999438222221e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 3.497740145777774e9, V = -6.494127164444437e8, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 2.3015257599999976e9, V = 5.482690303999994e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 8.027784391111103e9, V = 1.5571086684444427e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.317234005333327e9, V = 1.0414041991111101e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 8.027758634666658e9, V = 1.556993443555554e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 4.530021788444439e9, V = 2.206406378666664e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = -5.744238079999993e9, V = 3.9470612479999957e9, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 23313.64084201386, V = 117719.41384548598, Ti = 5.449999988079071, Fr = 2.27070703125e11)
 (U = 5.924709319111105e9, V = -3.8452848639999957e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 5.924688497777772e9, V = -3.845403854222218e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.08089890133333e9, V = -4.402725802666661e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.244918698666664e9, V = -3.2499217208888855e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.8438056746666636e9, V = 5.57437258666666e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 2.8437909475555525e9, V = 5.573208195555549e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.6797841706666627e9, V = -5.953657653333327e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 3.679770524444441e9, V = -5.954820355555549e8, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 8.35980113777777e8, V = -1.1528026026666653e9, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = 19414.168077256923, V = 118040.4856770832, Ti = 5.716666638851166, Fr = 2.27070703125e11)
 (U = -92093.64322916657, V = -1.5751088611111094e6, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 3.081715143111108e9, V = -4.344833365333329e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 6.07242564266666e9, V = -3.732578396444441e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 2.190544647111109e9, V = -3.208251676444441e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 6.05836305066666e9, V = -3.7438256071111073e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = -3.0829770239999967e9, V = 4.34976796444444e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 2.99071138133333e9, V = 6.122509119999994e8, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = -2.1976609848888865e9, V = 3.211308942222219e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 3.881882190222218e9, V = -5.2432489066666615e8, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = 8.911678026666657e8, V = -1.1365769386666653e9, Ti = 6.049999952316284, Fr = 2.27070703125e11)
 (U = -150875.45052083317, V = -1.5769051249999984e6, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.157184056888882e9, V = -3.6406887111111073e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.157173617777771e9, V = -3.6408075306666627e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 2.1349524657777758e9, V = -3.175750087111108e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.065380067555552e9, V = -4.298637105777774e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.14621098666666e9, V = -3.6535959466666627e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -3.0691708159999967e9, V = 4.304277162666662e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.091809799111108e9, V = 6.579570684444437e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 3.091795150222219e9, V = 6.578363697777771e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -2.1445705386666644e9, V = 3.1792032426666636e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 4.0222341617777734e9, V = -4.6493393599999946e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 4.0222294044444404e9, V = -4.650499119999995e8, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 9.304277457777768e8, V = -1.1228893119999988e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = -6.146197247999993e9, V = 3.653717759999996e9, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 10322.971652560753, V = 118544.06944444432, Ti = 6.316666603088379, Fr = 2.27070703125e11)
 (U = 6.211784632888882e9, V = -3.5477483875555515e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 6.211780195555549e9, V = -3.547863480888885e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 2.068902503111109e9, V = -3.1441626808888855e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.034031331555552e9, V = -4.2528075377777734e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.177752227555552e9, V = 7.050634364444437e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 3.177752810666663e9, V = 7.049488924444437e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 4.1428866915555515e9, V = -4.035797448888885e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 9.651305439999989e8, V = -1.108645550222221e9, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 4.1428811519999957e9, V = -4.036979662222218e8, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = 6186.6973334418335, V = 118668.12304687487, Ti = 6.583333253860474, Fr = 2.27070703125e11)
 (U = -266936.3645833331, V = -1.583141166666665e6, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.235954645333326e9, V = -3.4543274026666627e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 2.987818652444441e9, V = -4.207558826666662e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.235956423111104e9, V = -3.454211199999996e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 1.9927185599999979e9, V = -3.1136442239999967e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 6.23440708266666e9, V = -3.468812543999996e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -2.9972223146666636e9, V = 4.213749674666662e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 3.2481352177777743e9, V = 7.533482702222215e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 3.2481338097777743e9, V = 7.532299875555547e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -2.006915669333331e9, V = 3.1172445866666636e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 4.243231630222218e9, V = -3.4069073333333296e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 4.243239466666662e9, V = -3.4056780622222185e8, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 9.951025973333323e8, V = -1.0939171839999988e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -6.23440452266666e9, V = 3.4689294506666627e9, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = 2020.1427137586786, V = 118729.78884548598, Ti = 6.849999904632568, Fr = 2.27070703125e11)
 (U = -343950.03624999966, V = -1.589413219999998e6, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.223212003555549e9, V = -3.337163527111108e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.223215416888882e9, V = -3.337282766222219e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 2.909506687999997e9, V = -4.152156949333329e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 1.8838133048888867e9, V = -3.07722722133333e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 6.225276682239994e9, V = -3.340931624959996e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -2.913978071039997e9, V = 4.1530716876799955e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 3.313704135111108e9, V = 8.14994561777777e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 3.3137085368888855e9, V = 8.148773351111102e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -1.889491799039998e9, V = 3.077254901759997e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 4.339403889777773e9, V = -2.5993809599999973e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 4.339408696888885e9, V = -2.6005522088888863e8, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = 1.0256997831111101e9, V = -1.0749325617777767e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -6.225278402559994e9, V = 3.3410462617599964e9, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -3199.665127224389, V = 118718.70724826376, Ti = 7.183333396911621, Fr = 2.27070703125e11)
 (U = -401459.5972222218, V = -1.5953378194444429e6, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 6.178709461333326e9, V = -3.243972359111108e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 2.8307861475555525e9, V = -4.1090220657777734e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 6.178720483555549e9, V = -3.244096682666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 1.786251743999998e9, V = -3.049647466666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 6.179113016888882e9, V = -3.242380138666663e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 3.347918890666663e9, V = 8.650494186666657e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -2.8311915662222195e9, V = 4.107428693333329e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 3.347926620444441e9, V = 8.649315555555546e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -1.7866596017777758e9, V = 3.048053795555552e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 4.392453788444439e9, V = -1.9432878977777755e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 4.392462890666661e9, V = -1.944448759999998e8, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = 1.0445360586666656e9, V = -1.0593764586666656e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -6.179119544888882e9, V = 3.2424975217777743e9, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -7359.556450737839, V = 118639.36892361098, Ti = 7.450000047683716, Fr = 2.27070703125e11)
 (U = -453840.1640624995, V = -1.6017645034722206e6, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 2.7382116835555525e9, V = -4.0671804444444404e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.103944163555549e9, V = -3.151683946666663e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 1.6799544319999983e9, V = -3.023603527111108e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.103956750222216e9, V = -3.1518036479999967e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 6.10439825066666e9, V = -3.150082410666663e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 3.3657344497777743e9, V = 9.154982524444435e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -2.7386646755555525e9, V = 4.065578346666662e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 3.365744974222219e9, V = 9.153749564444435e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -1.6804040142222204e9, V = 3.02200078933333e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 4.423988807111106e9, V = -1.2808080399999987e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 4.424001663999995e9, V = -1.2819521311111099e8, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = 1.0582553617777767e9, V = -1.0435755999999989e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -6.104410581333326e9, V = 3.150201891555552e9, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -11483.574490017349, V = 118497.78168402765, Ti = 7.7166666984558105, Fr = 2.27070703125e11)
 (U = -504000.3645833328, V = -1.6089620833333316e6, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 5.999277468444439e9, V = -3.060741980444441e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 2.6322108302222195e9, V = -4.0268253795555515e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 5.999288348444438e9, V = -3.060856782222219e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 1.565423818666665e9, V = -2.999217870222219e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 5.999780252444438e9, V = -3.0591319537777743e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -2.6327225742222195e9, V = 4.025219150222218e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 3.3670615039999967e9, V = 9.660856764444433e8, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 3.367077162666663e9, V = 9.659668408888879e8, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 4.433867007999995e9, V = -6.164012344444438e7, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -1.5659301759999983e9, V = 2.99760936533333e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 4.433851406222218e9, V = -6.1523151111111045e7, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = 1.0667900266666656e9, V = -1.0276084337777767e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -5.999801315555549e9, V = 3.059254300444441e9, Ti = 7.983333349227905, Fr = 2.27070703125e11)
 (U = -15551.297851562484, V = 118294.64453124987, Ti = 7.983333349227905, Fr = 2.27070703125e11)
)
   )
ObservedInstrumentModel
  with Jones: SingleStokesGain
  with reference basis: PolarizedTypes.CirBasis()Data Products: Comrade.EHTVisibilityDatum

Optimization and Sampling

Now we need to actually compute our image. For this we will first follow standard approaches in VLBI and find the maximum a posteriori (MAP) estimate of the image and instrument model. This is done using the comrade_opt function which accepts the posterior, an optimization algorithm, and some keyword arguments. For this tutorial we will use the L-BFGS algorithm. For more information about the optimization algorithms available see the Optimization.jl docs.

using Optimization, OptimizationLBFGSB
xopt, sol = comrade_opt(
    post, LBFGSB(); initial_params = prior_sample(rng, post),
    maxiters = 2000, g_tol = 1.0e-1
);

Warning

Fitting gains tends to be very difficult, meaning that optimization can take a lot longer. The upside is that we usually get nicer images.

First we will evaluate our fit by plotting the residuals

using CairoMakie
using DisplayAs
res = residuals(post, xopt)
plotfields(res[1], :uvdist, :res) |> DisplayAs.PNG |> DisplayAs.Text

These look reasonable, although there may be some minor overfitting. This could be improved in a few ways, but that is beyond the goal of this quick tutorial. Plotting the image, we see that we have a much cleaner version of the closure-only image from Imaging a Black Hole using only Closure Quantities.

g = imagepixels(fovx, fovy, 128, 128)
img = intensitymap(skymodel(post, xopt), g)
imageviz(img, size = (500, 400)) |> DisplayAs.PNG |> DisplayAs.Text

Because we also fit the instrument model, we can inspect their parameters. First, let's query the posterior object with the optimal parameters to get the instrument model.

intopt = instrumentmodel(post, xopt)

126-element Comrade.SiteArray{ComplexF64, 1, Vector{ComplexF64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}:
    1.0040293761152883 + 0.0im
    -0.542912011322623 + 0.8447488716846258im
    1.0094561521911194 + 0.0im
   -0.5977814497784117 - 0.8354183223192608im
   -0.4066784032871718 + 0.7281011420274763im
  -0.43084990828842185 + 1.0013895179059313im
    1.0045608555934253 + 0.0im
   -0.6401643662134519 - 0.8072407013061348im
   -0.5502912288737307 + 0.8175365728993372im
   -0.2073983222714196 + 1.0238609132639303im
    1.0191646593423844 + 0.0im
   -0.6601444396818366 - 0.7767154041688832im
   -0.5513655297326726 + 0.7360667152900291im
  -0.01890397637825507 + 1.015837548796599im
    0.9957598120744683 + 0.0im
   -0.7130695073446806 - 0.7539577092923735im
   -0.6099189870401683 + 0.7119774718182891im
    0.1627338597780665 + 0.9457052870567974im
    1.0365662919353826 + 0.0im
   -0.6405230363177328 + 0.24068894792199216im
    -0.793202467468735 - 0.39056210270046027im
    1.0204855811836735 + 0.0im
   -0.7537565340530135 - 0.6872619334660244im
   -0.4148225054761631 + 0.4754022689786969im
    0.5084407158372986 + 0.8396173597826596im
    1.0089074861953797 + 0.0im
    -0.775362041963762 - 0.6624901423244121im
  -0.38715760724931625 + 0.4469262106804389im
    0.6770806578071197 + 0.7452674717937005im
     1.020635491678828 + 0.0im
   -0.7948262888318026 - 0.6379197834782946im
  -0.37598912510094973 + 0.46556446596710543im
    0.8166845617393569 + 0.5886747209155306im
    1.0040028173589564 + 0.0im
   -0.8043234700961073 - 0.6406938378802872im
  -0.24792667991697856 + 0.9833023379054495im
   -0.3144768342182342 + 0.4518304521536984im
    0.9629528609044776 + 0.3847773944860618im
    0.9880801987293939 + 0.0im
   -0.7990510607744304 - 0.677873044445302im
  -0.08720658466238143 + 0.9621559231781845im
  -0.29983530476739206 + 0.5394566741766882im
     0.991352846106808 + 0.13345175452149158im
    1.0518384838164683 + 0.0im
   -0.7703977447124638 - 0.6294644803825625im
   0.10441928997931493 + 0.9795105364052304im
  -0.26360889370684915 + 0.7122895197284221im
    1.0543683098693273 - 0.11974365587197719im
    0.9863127141577835 + 0.0im
   -0.7542431678673122 - 0.7302566341658618im
   0.30652133504507584 + 0.9386865773090219im
    0.7325864487030177 - 0.7142512552041325im
  -0.16478066176487316 + 0.6740126870711126im
    1.0176458808879227 - 0.4201523279471729im
     1.037608486328644 + 0.0im
   -0.7521161690345942 - 0.6552773751159284im
    0.5257471795489718 + 0.8451324699062479im
    0.8350273742134242 - 0.5844006768502329im
  -0.03859907646502712 + 0.6900262245673658im
    0.8686767367517294 - 0.712315627086379im
  -0.28697912186142116 - 0.9695605230919648im
    0.9679369282827742 + 0.0im
   -0.7706859901758379 - 0.7272414892205886im
    0.6854619403954885 + 0.725912410906822im
    0.8756063747540546 - 0.4932373496842482im
   0.09769770939229963 + 0.6465591135652219im
     0.804025672311504 - 1.0053154732060348im
    -0.511683923964811 - 0.8756581498939432im
    0.8887541024857569 + 0.0im
   -0.7769450469469426 - 0.8632719881372393im
    0.7927357745626514 + 0.5905319543912638im
    0.9330663813088711 - 0.4124002051861197im
    0.2453929591808163 + 0.61867338912498im
    0.4529112041002772 - 0.8963657715625163im
   -0.7407420189829292 - 0.7137296779083568im
    1.0314324899298726 + 0.0im
    0.9214780620099418 + 0.3902469371224779im
    0.9525184194638568 - 0.3526264884127928im
    0.4462921186951578 + 0.49489071945342283im
   -0.8958573676245667 - 0.48685785511433577im
    1.0722074859786608 + 0.0im
   -0.5922541066319646 - 0.7725295549981953im
    1.0112978796796246 + 0.10481145214174312im
    0.9930112634387458 - 0.3045486795264113im
    0.7024486754816703 + 0.059717589555136286im
    1.0219773276805102 + 0.0im
   -0.5663109430696038 - 0.8419354982672179im
    1.0061759949193463 - 0.09676871092183695im
     0.974062925253343 - 0.2867099962535364im
    0.5003941086133642 - 0.3275369696150911im
   -1.0009607826455786 - 0.14056178177984793im
     1.086569749900304 + 0.0im
    0.9730751738493548 - 0.3618543710213341im
    0.9805426065679806 - 0.270882414637963im
    0.3466622436913784 - 0.5354877672636293im
    -1.015690316335403 + 0.0230996455931306im
     1.108579479818797 + 0.0im
   -0.4792720488073152 - 0.8035158738294329im
    0.8230891326771097 - 0.6035275767124129im
    0.9783286175241427 - 0.2721908039277571im
   0.14463411234155554 - 0.5206488237187736im
   -0.9953781507422211 + 0.18589901770196351im
     1.028678680147231 + 0.0im
  -0.42776352906218595 - 0.9126196653081987im
    0.6758203574713723 - 0.7764091507018676im
    0.9741296183994905 - 0.2897702659267935im
 -0.008640456623592684 - 0.5862032213356426im
   -0.9852913676092595 + 0.2659061913180039im
    1.0040740790571774 + 0.0im
   -0.4015368624873917 - 0.9549081053727548im
   0.48245578948815565 - 0.8940971313209183im
    0.9670386108906946 - 0.3101757287993789im
  -0.14180843504474588 - 0.7283086126556813im
   -0.9661020706154363 + 0.3294667630356128im
    0.9809321260892585 + 0.0im
   -0.3290458130115615 - 0.9924210096383523im
   0.28766933984848114 - 0.950643909992987im
    0.9544351139191172 - 0.3439689390115571im
   -0.2546522927117744 - 0.6807332339319366im
   -0.9588755021911265 + 0.32421610370282417im
    1.0108095360436242 + 0.0im
  -0.26376142807913305 - 0.9938167451646918im
   0.18653656302120278 - 0.8990677790314365im
    0.9614418422819101 - 0.3469530087500489im
  -0.32799052823232056 - 0.6148416480526001im
   -0.9568558497408134 + 0.3271646891730872im

This returns a SiteArray object which contains the gains as a flat vector with metadata about the sites, time, and frequency. To visualize the gains we can use the plotcaltable function which automatically plots the gains. Since the gains are complex we will first plot the phases.

plotcaltable(angle.(intopt)) |> DisplayAs.PNG |> DisplayAs.Text

Due to the a priori calibration of the data, the gain phases are quite stable and just drift over time. Note that the one outlier around 2.5 UT is when ALMA leaves the array. As a result we shift the reference antenna to APEX on that scan causing the gain phases to jump.

We can also plot the gain amplitudes

plotcaltable(abs.(intopt)) |> DisplayAs.PNG |> DisplayAs.Text

Here we find relatively stable gains for most stations. The exception is LMT which has a large offset after 2.5 UT. This is a known issue with the LMT in 2017 and is due to pointing issues. However, we can see the power of simultaneous imaging and instrument modeling as we are able to solve for the gain amplitudes and get a reasonable image.

One problem with the MAP estimate is that it does not provide uncertainties on the image. That is we are unable to statistically assess which components of the image are certain. Comrade is really a Bayesian imaging and calibration package for VLBI. Therefore, our goal is to sample from the posterior distribution of the image and instrument model. This is a very high-dimensional distribution with typically 1,000 - 100,000 parameters. To sample from this very high dimensional distribution, Comrade has an array of samplers that can be used. However, note that Comrade also satisfies the LogDensityProblems.jl interface. Therefore, you can use any package that supports LogDensityProblems.jl if you have your own fancy sampler.

For this example, we will use HMC, specifically the NUTS algorithm. For However, due to the need to sample a large number of gain parameters, constructing the posterior can take a few minutes. Therefore, for this tutorial, we will only do a quick preliminary run.

using AdvancedHMC
chain = sample(rng, post, NUTS(0.8), 700; n_adapts = 500, initial_params = xopt)

PosteriorSamples
  Samples size: (700,)
  sampler used: AHMC
Mean
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                                               │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                         │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, fg::Float64}                                                                                                                                                            │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                          │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [-0.0203117 0.0125146 … 0.0442767 0.0280257; -0.0179332 -0.00781317 … 0.0483923 0.0381917; … ; -0.00472493 -0.0379754 … -0.0760904 -0.0159854; 0.0363968 0.0691855 … -0.0707402 0.0377206], hyperparams = 36.4491), σimg = 1.86604, fg = 0.359772) │ (lg = [0.0286319, 0.0303722, 0.00457908, 0.0269108, -0.310516, 0.145881, 0.00199939, 0.0297768, -0.126014, 0.100187  …  -0.0278908, 0.0150418, -0.775605, 0.0130881, 0.00553084, 0.031299, -0.0935263, 0.0228266, -0.824023, 0.0108655], gp = [0.0, 0.568357, 0.0, -2.19206, 1.5031, 0.419998, 0.0, -2.24204, 1.55682, 0.222756  …  -1.86402, -0.275258, -2.4615, 1.67485, 0.0, -1.83054, -1.98322, -0.255207, -1.63569, 1.61059]) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
Std. Dev.
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                       │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                   │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, fg::Float64}                                                                                                                                    │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                    │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [0.527661 0.578031 … 0.590051 0.566547; 0.589631 0.662874 … 0.668907 0.568095; … ; 0.570926 0.622272 … 0.599102 0.602008; 0.544088 0.611502 … 0.565635 0.545255], hyperparams = 22.6278), σimg = 0.220805, fg = 0.0746884) │ (lg = [0.144792, 0.149124, 0.0207412, 0.0212394, 0.0840549, 0.0632296, 0.0200318, 0.0206659, 0.0770866, 0.05972  …  0.0602391, 0.0186713, 0.0853227, 0.0185099, 0.017082, 0.01986, 0.0617581, 0.0174532, 0.0878219, 0.0182886], gp = [0.0, 0.745322, 0.0, 0.0176934, 0.311663, 0.749835, 0.0, 0.0185135, 0.310486, 0.756302  …  0.287415, 0.282797, 1.14668, 2.30703, 0.0, 0.0169989, 0.291868, 0.281945, 2.30294, 2.36608]) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

Note

The above sampler will store the samples in memory, i.e. RAM. For large models this can lead to out-of-memory issues. To avoid this we recommend using the saveto = DiskStore() kwargs which periodically saves the samples to disk limiting memory useage. You can load the chain using load_samples(diskout) where diskout is the object returned from sample.

Now we prune the adaptation phase

chain = chain[(begin + 500):end]

PosteriorSamples
  Samples size: (200,)
  sampler used: AHMC
Mean
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                                           │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                        │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, fg::Float64}                                                                                                                                                        │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                         │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [-0.0672346 -0.038531 … -0.00749518 0.0342577; -0.103355 -0.0421284 … 0.0690295 0.0422826; … ; 0.00760585 0.025193 … 0.00132555 0.0179066; 0.0279111 0.0817919 … 0.0137633 0.00815569], hyperparams = 37.7893), σimg = 1.92943, fg = 0.402455) │ (lg = [0.037117, -6.41088e-5, 0.00362527, 0.029268, -0.3498, 0.131922, 0.00314347, 0.0291703, -0.161038, 0.0903737  …  -0.0526113, 0.0147699, -0.825818, 0.011841, 0.00677289, 0.0280795, -0.114784, 0.0232885, -0.876686, 0.0089633], gp = [0.0, -0.18481, 0.0, -2.19144, 1.16001, -0.342488, 0.0, -2.2421, 1.20961, -0.546826  …  -2.21434, -0.29247, -2.17276, 2.11625, 0.0, -1.82858, -2.33625, -0.267357, 1.09774, 2.07552]) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
Std. Dev.
┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┬─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky                                                                                                                                                                                                                                     │ instrument                                                                                                                                                                                                                                                                                                                                                                                                                      │
│ @NamedTuple{c::@NamedTuple{params::Matrix{Float64}, hyperparams::Float64}, σimg::Float64, fg::Float64}                                                                                                                                  │ @NamedTuple{lg::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}, gp::Comrade.SiteArray{Float64, 1, Vector{Float64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}}                                                                       │
├─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = (params = [0.46806 0.579064 … 0.533598 0.514154; 0.603091 0.665278 … 0.614683 0.583536; … ; 0.545579 0.627571 … 0.58647 0.573685; 0.489476 0.562241 … 0.563933 0.508339], hyperparams = 24.0407), σimg = 0.215505, fg = 0.0528678) │ (lg = [0.152447, 0.160753, 0.019661, 0.0197162, 0.0663424, 0.0507968, 0.0184361, 0.0207215, 0.0668174, 0.0494759  …  0.0426215, 0.018159, 0.0591762, 0.0187076, 0.0164268, 0.0204455, 0.0443826, 0.0182911, 0.0602373, 0.0163622], gp = [0.0, 0.607635, 0.0, 0.0195781, 0.250624, 0.615163, 0.0, 0.0172711, 0.242475, 0.622193  …  0.162521, 0.232234, 2.09497, 1.93392, 0.0, 0.0181901, 0.171183, 0.232095, 2.82885, 2.00262]) │
└─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

Warning

This should be run for likely an order of magnitude more steps to properly estimate expectations of the posterior

Now that we have our posterior, we can put error bars on all of our plots above. Let's start by finding the mean and standard deviation of the gain phases

mchain = Comrade.rmap(mean, chain);
schain = Comrade.rmap(std, chain);

Now we can use the measurements package to automatically plot everything with error bars. First we create a caltable the same way but making sure all of our variables have errors attached to them.

using Measurements
gmeas = instrumentmodel(post, (; instrument = map((x, y) -> Measurements.measurement.(x, y), mchain.instrument, schain.instrument)))
ctable_am = caltable(abs.(gmeas))
ctable_ph = caltable(angle.(gmeas))

─────────┬───────────────────┬────────────────────────────────────────────────────────────────────────────────────
      Ti │                Fr │      AA            AP          AZ          JC           LM          PV          SM
─────────┼───────────────────┼────────────────────────────────────────────────────────────────────────────────────
 0.92 hr │ 227.070703125 GHz │ 0.0±0.0       missing     missing     missing      missing  -0.18±0.61     missing
 1.22 hr │ 227.070703125 GHz │ 0.0±0.0   -2.191±0.02     missing     missing    1.16±0.25  -0.34±0.62     missing
 1.52 hr │ 227.070703125 GHz │ 0.0±0.0  -2.242±0.017     missing     missing    1.21±0.24  -0.55±0.62     missing
 1.82 hr │ 227.070703125 GHz │ 0.0±0.0  -2.273±0.018     missing     missing    1.23±0.23  -0.73±0.63     missing
 2.12 hr │ 227.070703125 GHz │ 0.0±0.0   -2.33±0.016     missing     missing    1.28±0.23  -0.92±0.63     missing
 2.45 hr │ 227.070703125 GHz │ missing       0.0±0.0     missing     missing    1.77±0.22   1.27±0.63     missing
 2.75 hr │ 227.070703125 GHz │ 0.0±0.0  -2.399±0.017     missing     missing     1.27±0.2  -1.29±0.63     missing
 3.05 hr │ 227.070703125 GHz │ 0.0±0.0   -2.433±0.02     missing     missing    1.27±0.19  -1.48±0.62     missing
 3.35 hr │ 227.070703125 GHz │ 0.0±0.0  -2.465±0.019     missing     missing    1.24±0.17  -1.68±0.61     missing
 3.68 hr │ 227.070703125 GHz │ 0.0±0.0  -2.467±0.017    0.91±0.2     missing    1.19±0.16   -1.91±0.6     missing
 3.98 hr │ 227.070703125 GHz │ 0.0±0.0  -2.437±0.019    0.77±0.2     missing    1.11±0.15  -2.13±0.58     missing
 4.28 hr │ 227.070703125 GHz │ 0.0±0.0  -2.458±0.023   0.59±0.19     missing    0.98±0.14  -2.32±0.68     missing
 4.58 hr │ 227.070703125 GHz │ 0.0±0.0  -2.374±0.017   0.41±0.18   -1.1±0.12    0.89±0.13    -1.7±2.0     missing
 4.92 hr │ 227.070703125 GHz │ 0.0±0.0   -2.425±0.02    0.2±0.17  -0.97±0.11    0.73±0.12   -0.89±2.6  -2.22±0.11
 5.18 hr │ 227.070703125 GHz │ 0.0±0.0  -2.385±0.017  0.015±0.17  -0.88±0.11    0.54±0.12    0.39±2.8  -2.46±0.11
 5.45 hr │ 227.070703125 GHz │ 0.0±0.0  -2.304±0.018  -0.15±0.16  -0.76±0.12    0.31±0.11     1.4±2.4  -2.72±0.12
 5.72 hr │ 227.070703125 GHz │ 0.0±0.0       missing  -0.38±0.15  -0.66±0.14  -0.044±0.11     missing    -2.5±1.5
 6.05 hr │ 227.070703125 GHz │ 0.0±0.0  -2.226±0.021  -0.68±0.15  -0.54±0.16   -0.81±0.11     missing     missing
 6.32 hr │ 227.070703125 GHz │ 0.0±0.0  -2.163±0.019  -0.89±0.15  -0.48±0.18    -1.5±0.11     missing    0.64±2.9
 6.58 hr │ 227.070703125 GHz │ 0.0±0.0       missing  -1.16±0.15  -0.41±0.19   -1.94±0.11     missing     1.7±2.4
 6.85 hr │ 227.070703125 GHz │ 0.0±0.0   -2.107±0.02  -1.46±0.15  -0.36±0.21   -2.28±0.11     missing     2.3±1.7
 7.18 hr │ 227.070703125 GHz │ 0.0±0.0   -2.01±0.018  -1.72±0.15  -0.31±0.22   -2.62±0.12     missing     2.4±1.5
 7.45 hr │ 227.070703125 GHz │ 0.0±0.0  -1.968±0.018  -1.98±0.16  -0.29±0.23   -2.83±0.12     missing     2.4±1.5
 7.72 hr │ 227.070703125 GHz │ 0.0±0.0   -1.89±0.018  -2.21±0.16  -0.29±0.23     -2.2±2.1     missing     2.1±1.9
 7.98 hr │ 227.070703125 GHz │ 0.0±0.0  -1.829±0.018  -2.34±0.17  -0.27±0.23      1.1±2.8     missing     2.1±2.0
─────────┴───────────────────┴────────────────────────────────────────────────────────────────────────────────────

Now let's plot the phase curves

plotcaltable(ctable_ph) |> DisplayAs.PNG |> DisplayAs.Text

and now the amplitude curves

plotcaltable(ctable_am) |> DisplayAs.PNG |> DisplayAs.Text

Finally let's construct some representative image reconstructions.

samples = skymodel.(Ref(post), chain[begin:5:end])
imgs = intensitymap.(samples, Ref(g))

mimg = mean(imgs)
simg = std(imgs)
fig = Figure(; size = (700, 700));
axs = [Axis(fig[i, j], xreversed = true, aspect = 1) for i in 1:2, j in 1:2]
image!(axs[1, 1], mimg, colormap = :afmhot); axs[1, 1].title = "Mean"
image!(axs[1, 2], simg ./ (max.(mimg, 1.0e-8)), colorrange = (0.0, 2.0), colormap = :afmhot);axs[1, 2].title = "Frac. Uncer."
image!(axs[2, 1], imgs[1], colormap = :afmhot);
image!(axs[2, 2], imgs[end], colormap = :afmhot);
hidedecorations!.(axs)
fig |> DisplayAs.PNG |> DisplayAs.Text

And viola, you have just finished making a preliminary image and instrument model reconstruction. In reality, you should run the sample step for many more MCMC steps to get a reliable estimate for the reconstructed image and instrument model parameters.

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