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.

julia

using Comrade


using Pyehtim
using LinearAlgebra

    CondaPkg Found dependencies: /home/runner/.julia/packages/DimensionalData/hv9KC/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/CondaPkg/0UqYV/CondaPkg.toml
    CondaPkg Found dependencies: /home/runner/.julia/packages/PythonCall/wkBj7/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
             + xarray
    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, >=3, <3.6"
             │ libstdcxx = ">=3.4,<15.0"
             │ uv = ">=0.4"
             │ libstdcxx-ng = ">=3.4,<15.0"
             │ pandas = "<2"
             │ xarray = "*"
             │ numpy = ">=1.24, <2.0"
             │ 
             │     [dependencies.python]
             │     channel = "conda-forge"
             │     build = "*cp*"
             │     version = ">=3.10,<3.14, >=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.
/home/runner/work/Comrade.jl/Comrade.jl/examples/intermediate/StokesIImaging/.CondaPkg/.pixi/envs/default/lib/python3.11/site-packages/ehtim/__init__.py:58: UserWarning: pkg_resources is deprecated as an API. See https://setuptools.pypa.io/en/latest/pkg_resources.html. The pkg_resources package is slated for removal as early as 2025-11-30. Refrain from using this package or pin to Setuptools<81.
  import pkg_resources

For reproducibility we use a stable random number genreator

julia

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:

julia

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

Python: <ehtim.obsdata.Obsdata object at 0x7f3620546090>

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.

julia

obs = scan_average(obs).add_fractional_noise(0.02)

Python: <ehtim.obsdata.Obsdata object at 0x7f3635fe93d0>

Now we extract our complex visibilities.

julia

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.

julia

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}())
    x0, y0 = centroid(m)
    # 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))
    return shifted(m, -x0, -y0) + 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.

julia

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.

julia

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

julia

using VLBIImagePriors
using Distributions
fwhmfac = 2 * sqrt(2 * log(2))
mpr = modify(Gaussian(), Stretch(μas2rad(50.0) ./ 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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 -4.34312e-10   2.01456e-11   4.88461e-11   1.13961e-10   2.55833e-10   5.52631e-10   1.14865e-9    2.29731e-9    4.42105e-9    8.18667e-9    1.4587e-8     2.50092e-8    4.12582e-8    6.54933e-8    1.00037e-7    1.47028e-7    2.07928e-7    2.82947e-7    3.70486e-7    4.66784e-7    5.65894e-7    6.60132e-7    7.40973e-7    8.00295e-7    8.31714e-7   8.31714e-7   8.00295e-7   7.40973e-7   6.60132e-7   5.65894e-7   4.66784e-7   3.70486e-7   2.82947e-7   2.07928e-7   1.47028e-7   1.00037e-7   6.54933e-8   4.12582e-8   2.50092e-8   1.4587e-8    8.18667e-9   4.42105e-9   2.29731e-9   1.14865e-9   5.52631e-10  2.55833e-10  1.13961e-10  4.88461e-11  2.01456e-11
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 -3.93911e-10   9.76922e-11   2.3687e-10    5.52631e-10   1.24061e-9    2.67987e-9    5.57017e-9    1.11403e-8    2.1439e-8     3.96996e-8    7.07367e-8    1.21277e-7    2.00074e-7    3.17597e-7    4.85109e-7    7.12982e-7    1.00831e-6    1.3721e-6     1.7966e-6     2.26358e-6    2.74419e-6    3.20118e-6    3.5932e-6     3.88087e-6    4.03323e-6   4.03323e-6   3.88087e-6   3.5932e-6    3.20118e-6   2.74419e-6   2.26358e-6   1.7966e-6    1.3721e-6    1.00831e-6   7.12982e-7   4.85109e-7   3.17597e-7   2.00074e-7   1.21277e-7   7.07367e-8   3.96996e-8   2.1439e-8    1.11403e-8   5.57017e-9   2.67987e-9   1.24061e-9   5.52631e-10  2.3687e-10   9.76922e-11
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 -3.5351e-10    4.0611e-10    9.84676e-10   2.29731e-9    5.15728e-9    1.11403e-8    2.31554e-8    4.63108e-8    8.91227e-8    1.65033e-7    2.94055e-7    5.04154e-7    8.31714e-7    1.32026e-6    2.01662e-6    2.96389e-6    4.19157e-6    5.70385e-6    7.46854e-6    9.40977e-6    1.14077e-5    1.33074e-5    1.49371e-5    1.61329e-5    1.67663e-5   1.67663e-5   1.61329e-5   1.49371e-5   1.33074e-5   1.14077e-5   9.40977e-6   7.46854e-6   5.70385e-6   4.19157e-6   2.96389e-6   2.01662e-6   1.32026e-6   8.31714e-7   5.04154e-7   2.94055e-7   1.65033e-7   8.91227e-8   4.63108e-8   2.31554e-8   1.11403e-8   5.15728e-9   2.29731e-9   9.84676e-10  4.0611e-10
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 -2.72708e-10   4.42105e-9    1.07195e-8    2.50092e-8    5.61438e-8    1.21277e-7    2.52077e-7    5.04154e-7    9.70218e-7    1.7966e-6     3.20118e-6    5.48838e-6    9.0543e-6     1.43728e-5    2.19535e-5    3.22659e-5    4.56308e-5    6.2094e-5     8.13049e-5    0.000102438   0.000124188   0.000144869   0.00016261    0.000175628   0.000182523  0.000182523  0.000175628  0.00016261   0.000144869  0.000124188  0.000102438  8.13049e-5   6.2094e-5    4.56308e-5   3.22659e-5   2.19535e-5   1.43728e-5   9.0543e-6    5.48838e-6   3.20118e-6   1.7966e-6    9.70218e-7   5.04154e-7   2.52077e-7   1.21277e-7   5.61438e-8   2.50092e-8   1.07195e-8   4.42105e-9
 -2.52507e-10   7.29349e-9    1.76842e-8    4.12582e-8    9.26216e-8    2.00074e-7    4.15857e-7    8.31714e-7    1.60059e-6    2.96389e-6    5.28105e-6    9.0543e-6     1.49371e-5    2.37111e-5    3.62172e-5    5.32297e-5    7.52781e-5    0.000102438   0.00013413    0.000168994   0.000204875   0.000238993   0.000268261   0.000289738   0.000301113  0.000301113  0.000289738  0.000268261  0.000238993  0.000204875  0.000168994  0.00013413   0.000102438  7.52781e-5   5.32297e-5   3.62172e-5   2.37111e-5   1.49371e-5   9.0543e-6    5.28105e-6   2.96389e-6   1.60059e-6   8.31714e-7   4.15857e-7   2.00074e-7   9.26216e-8   4.12582e-8   1.76842e-8   7.29349e-9
 -2.32307e-10   1.15777e-8    2.80719e-8    6.54933e-8    1.47028e-7    3.17597e-7    6.60132e-7    1.32026e-6    2.54078e-6    4.70488e-6    8.38315e-6    1.43728e-5    2.37111e-5    3.76391e-5    5.74912e-5    8.44968e-5    0.000119497   0.00016261    0.000212919   0.000268261   0.00032522    0.000379378   0.000425837   0.00045993    0.000477986  0.000477986  0.00045993   0.000425837  0.000379378  0.00032522   0.000268261  0.000212919  0.00016261   0.000119497  8.44968e-5   5.74912e-5   3.76391e-5   2.37111e-5   1.43728e-5   8.38315e-6   4.70488e-6   2.54078e-6   1.32026e-6   6.60132e-7   3.17597e-7   1.47028e-7   6.54933e-8   2.80719e-8   1.15777e-8
 -2.12106e-10   1.76842e-8    4.2878e-8     1.00037e-7    2.24575e-7    4.85109e-7    1.00831e-6    2.01662e-6    3.88087e-6    7.1864e-6     1.28047e-5    2.19535e-5    3.62172e-5    5.74912e-5    8.78141e-5    0.000129063   0.000182523   0.000248376   0.00032522    0.000409751   0.000496752   0.000579475   0.000650439   0.000702513   0.000730093  0.000730093  0.000702513  0.000650439  0.000579475  0.000496752  0.000409751  0.00032522   0.000248376  0.000182523  0.000129063  8.78141e-5   5.74912e-5   3.62172e-5   2.19535e-5   1.28047e-5   7.1864e-6    3.88087e-6   2.01662e-6   1.00831e-6   4.85109e-7   2.24575e-7   1.00037e-7   4.2878e-8    1.76842e-8
 -1.91905e-10   2.59911e-8    6.30193e-8    1.47028e-7    3.30066e-7    7.12982e-7    1.48195e-6    2.96389e-6    5.70385e-6    1.05621e-5    1.88195e-5    3.22659e-5    5.32297e-5    8.44968e-5    0.000129063   0.000189689   0.000268261   0.000365047   0.000477986   0.000602225   0.000730093   0.000851675   0.000955973   0.00103251    0.00107304   0.00107304   0.00103251   0.000955973  0.000851675  0.000730093  0.000602225  0.000477986  0.000365047  0.000268261  0.000189689  0.000129063  8.44968e-5   5.32297e-5   3.22659e-5   1.88195e-5   1.05621e-5   5.70385e-6   2.96389e-6   1.48195e-6   7.12982e-7   3.30066e-7   1.47028e-7   6.30193e-8   2.59911e-8
 -1.71705e-10   3.67569e-8    8.91227e-8    2.07928e-7    4.66784e-7    1.00831e-6    2.09579e-6    4.19157e-6    8.06647e-6    1.49371e-5    2.66148e-5    4.56308e-5    7.52781e-5    0.000119497   0.000182523   0.000268261   0.000379378   0.000516254   0.000675975   0.000851675   0.00103251    0.00120445    0.00135195    0.00146019    0.00151751   0.00151751   0.00146019   0.00135195   0.00120445   0.00103251   0.000851675  0.000675975  0.000516254  0.000379378  0.000268261  0.000182523  0.000119497  7.52781e-5   4.56308e-5   2.66148e-5   1.49371e-5   8.06647e-6   4.19157e-6   2.09579e-6   1.00831e-6   4.66784e-7   2.07928e-7   8.91227e-8   3.67569e-8
 -1.51504e-10   5.00184e-8    1.21277e-7    2.82947e-7    6.35194e-7    1.3721e-6     2.85193e-6    5.70385e-6    1.09768e-5    2.03262e-5    3.62172e-5    6.2094e-5     0.000102438   0.00016261    0.000248376   0.000365047   0.000516254   0.000702513   0.00091986    0.00115895    0.00140503    0.001639      0.00183972    0.00198701    0.00206502   0.00206502   0.00198701   0.00183972   0.001639     0.00140503   0.00115895   0.00091986   0.000702513  0.000516254  0.000365047  0.000248376  0.00016261   0.000102438  6.2094e-5    3.62172e-5   2.03262e-5   1.09768e-5   5.70385e-6   2.85193e-6   1.3721e-6    6.35194e-7   2.82947e-7   1.21277e-7   5.00184e-8
 -1.31304e-10   6.54933e-8    1.58799e-7    3.70486e-7    8.31714e-7    1.7966e-6     3.73427e-6    7.46854e-6    1.43728e-5    2.66148e-5    4.74222e-5    8.13049e-5    0.00013413    0.000212919   0.00032522    0.000477986   0.000675975   0.00091986    0.00120445    0.00151751    0.00183972    0.00214609    0.0024089     0.00260176    0.0027039    0.0027039    0.00260176   0.0024089    0.00214609   0.00183972   0.00151751   0.00120445   0.00091986   0.000675975  0.000477986  0.00032522   0.000212919  0.00013413   8.13049e-5   4.74222e-5   2.66148e-5   1.43728e-5   7.46854e-6   3.73427e-6   1.7966e-6    8.31714e-7   3.70486e-7   1.58799e-7   6.54933e-8
 -1.11103e-10   8.25164e-8    2.00074e-7    4.66784e-7    1.04789e-6    2.26358e-6    4.70488e-6    9.40977e-6    1.81086e-5    3.35326e-5    5.97483e-5    0.000102438   0.000168994   0.000268261   0.000409751   0.000602225   0.000851675   0.00115895    0.00151751    0.00191195    0.0023179     0.0027039     0.00303502    0.00327801    0.0034067    0.0034067    0.00327801   0.00303502   0.0027039    0.0023179    0.00191195   0.00151751   0.00115895   0.000851675  0.000602225  0.000409751  0.000268261  0.000168994  0.000102438  5.97483e-5   3.35326e-5   1.81086e-5   9.40977e-6   4.70488e-6   2.26358e-6   1.04789e-6   4.66784e-7   2.00074e-7   8.25164e-8
 -9.09026e-11   1.00037e-7    2.42555e-7    5.65894e-7    1.27039e-6    2.74419e-6    5.70385e-6    1.14077e-5    2.19535e-5    4.06524e-5    7.24344e-5    0.000124188   0.000204875   0.00032522    0.000496752   0.000730093   0.00103251    0.00140503    0.00183972    0.0023179     0.00281005    0.00327801    0.00367944    0.00397401    0.00413003   0.00413003   0.00397401   0.00367944   0.00327801   0.00281005   0.0023179    0.00183972   0.00140503   0.00103251   0.000730093  0.000496752  0.00032522   0.000204875  0.000124188  7.24344e-5   4.06524e-5   2.19535e-5   1.14077e-5   5.70385e-6   2.74419e-6   1.27039e-6   5.65894e-7   2.42555e-7   1.00037e-7
 -7.0702e-11    1.16696e-7    2.82947e-7    6.60132e-7    1.48195e-6    3.20118e-6    6.65371e-6    1.33074e-5    2.56094e-5    4.74222e-5    8.44968e-5    0.000144869   0.000238993   0.000379378   0.000579475   0.000851675   0.00120445    0.001639      0.00214609    0.0027039     0.00327801    0.00382389    0.00429217    0.0046358     0.0048178    0.0048178    0.0046358    0.00429217   0.00382389   0.00327801   0.0027039    0.00214609   0.001639     0.00120445   0.000851675  0.000579475  0.000379378  0.000238993  0.000144869  8.44968e-5   4.74222e-5   2.56094e-5   1.33074e-5   6.65371e-6   3.20118e-6   1.48195e-6   6.60132e-7   2.82947e-7   1.16696e-7
 -5.05014e-11   1.30987e-7    3.17597e-7    7.40973e-7    1.66343e-6    3.5932e-6     7.46854e-6    1.49371e-5    2.87456e-5    5.32297e-5    9.48445e-5    0.00016261    0.000268261   0.000425837   0.000650439   0.000955973   0.00135195    0.00183972    0.0024089     0.00303502    0.00367944    0.00429217    0.0048178     0.00520351    0.0054078    0.0054078    0.00520351   0.0048178    0.00429217   0.00367944   0.00303502   0.0024089    0.00183972   0.00135195   0.000955973  0.000650439  0.000425837  0.000268261  0.00016261   9.48445e-5   5.32297e-5   2.87456e-5   1.49371e-5   7.46854e-6   3.5932e-6    1.66343e-6   7.40973e-7   3.17597e-7   1.30987e-7
 -3.03009e-11   1.41473e-7    3.43024e-7    8.00295e-7    1.7966e-6     3.88087e-6    8.06647e-6    1.61329e-5    3.1047e-5     5.74912e-5    0.000102438   0.000175628   0.000289738   0.00045993    0.000702513   0.00103251    0.00146019    0.00198701    0.00260176    0.00327801    0.00397401    0.0046358     0.00520351    0.0056201     0.00584074   0.00584074   0.0056201    0.00520351   0.0046358    0.00397401   0.00327801   0.00260176   0.00198701   0.00146019   0.00103251   0.000702513  0.00045993   0.000289738  0.000175628  0.000102438  5.74912e-5   3.1047e-5    1.61329e-5   8.06647e-6   3.88087e-6   1.7966e-6    8.00295e-7   3.43024e-7   1.41473e-7
 -1.01003e-11   1.47028e-7    3.56491e-7    8.31714e-7    1.86713e-6    4.03323e-6    8.38315e-6    1.67663e-5    3.22659e-5    5.97483e-5    0.000106459   0.000182523   0.000301113   0.000477986   0.000730093   0.00107304    0.00151751    0.00206502    0.0027039     0.0034067     0.00413003    0.0048178     0.0054078     0.00584074    0.00607005   0.00607005   0.00584074   0.0054078    0.0048178    0.00413003   0.0034067    0.0027039    0.00206502   0.00151751   0.00107304   0.000730093  0.000477986  0.000301113  0.000182523  0.000106459  5.97483e-5   3.22659e-5   1.67663e-5   8.38315e-6   4.03323e-6   1.86713e-6   8.31714e-7   3.56491e-7   1.47028e-7
  1.01003e-11   1.47028e-7    3.56491e-7    8.31714e-7    1.86713e-6    4.03323e-6    8.38315e-6    1.67663e-5    3.22659e-5    5.97483e-5    0.000106459   0.000182523   0.000301113   0.000477986   0.000730093   0.00107304    0.00151751    0.00206502    0.0027039     0.0034067     0.00413003    0.0048178     0.0054078     0.00584074    0.00607005   0.00607005   0.00584074   0.0054078    0.0048178    0.00413003   0.0034067    0.0027039    0.00206502   0.00151751   0.00107304   0.000730093  0.000477986  0.000301113  0.000182523  0.000106459  5.97483e-5   3.22659e-5   1.67663e-5   8.38315e-6   4.03323e-6   1.86713e-6   8.31714e-7   3.56491e-7   1.47028e-7
  3.03009e-11   1.41473e-7    3.43024e-7    8.00295e-7    1.7966e-6     3.88087e-6    8.06647e-6    1.61329e-5    3.1047e-5     5.74912e-5    0.000102438   0.000175628   0.000289738   0.00045993    0.000702513   0.00103251    0.00146019    0.00198701    0.00260176    0.00327801    0.00397401    0.0046358     0.00520351    0.0056201     0.00584074   0.00584074   0.0056201    0.00520351   0.0046358    0.00397401   0.00327801   0.00260176   0.00198701   0.00146019   0.00103251   0.000702513  0.00045993   0.000289738  0.000175628  0.000102438  5.74912e-5   3.1047e-5    1.61329e-5   8.06647e-6   3.88087e-6   1.7966e-6    8.00295e-7   3.43024e-7   1.41473e-7
  5.05014e-11   1.30987e-7    3.17597e-7    7.40973e-7    1.66343e-6    3.5932e-6     7.46854e-6    1.49371e-5    2.87456e-5    5.32297e-5    9.48445e-5    0.00016261    0.000268261   0.000425837   0.000650439   0.000955973   0.00135195    0.00183972    0.0024089     0.00303502    0.00367944    0.00429217    0.0048178     0.00520351    0.0054078    0.0054078    0.00520351   0.0048178    0.00429217   0.00367944   0.00303502   0.0024089    0.00183972   0.00135195   0.000955973  0.000650439  0.000425837  0.000268261  0.00016261   9.48445e-5   5.32297e-5   2.87456e-5   1.49371e-5   7.46854e-6   3.5932e-6    1.66343e-6   7.40973e-7   3.17597e-7   1.30987e-7
  7.0702e-11    1.16696e-7    2.82947e-7    6.60132e-7    1.48195e-6    3.20118e-6    6.65371e-6    1.33074e-5    2.56094e-5    4.74222e-5    8.44968e-5    0.000144869   0.000238993   0.000379378   0.000579475   0.000851675   0.00120445    0.001639      0.00214609    0.0027039     0.00327801    0.00382389    0.00429217    0.0046358     0.0048178    0.0048178    0.0046358    0.00429217   0.00382389   0.00327801   0.0027039    0.00214609   0.001639     0.00120445   0.000851675  0.000579475  0.000379378  0.000238993  0.000144869  8.44968e-5   4.74222e-5   2.56094e-5   1.33074e-5   6.65371e-6   3.20118e-6   1.48195e-6   6.60132e-7   2.82947e-7   1.16696e-7
  9.09026e-11   1.00037e-7    2.42555e-7    5.65894e-7    1.27039e-6    2.74419e-6    5.70385e-6    1.14077e-5    2.19535e-5    4.06524e-5    7.24344e-5    0.000124188   0.000204875   0.00032522    0.000496752   0.000730093   0.00103251    0.00140503    0.00183972    0.0023179     0.00281005    0.00327801    0.00367944    0.00397401    0.00413003   0.00413003   0.00397401   0.00367944   0.00327801   0.00281005   0.0023179    0.00183972   0.00140503   0.00103251   0.000730093  0.000496752  0.00032522   0.000204875  0.000124188  7.24344e-5   4.06524e-5   2.19535e-5   1.14077e-5   5.70385e-6   2.74419e-6   1.27039e-6   5.65894e-7   2.42555e-7   1.00037e-7
  1.11103e-10   8.25164e-8    2.00074e-7    4.66784e-7    1.04789e-6    2.26358e-6    4.70488e-6    9.40977e-6    1.81086e-5    3.35326e-5    5.97483e-5    0.000102438   0.000168994   0.000268261   0.000409751   0.000602225   0.000851675   0.00115895    0.00151751    0.00191195    0.0023179     0.0027039     0.00303502    0.00327801    0.0034067    0.0034067    0.00327801   0.00303502   0.0027039    0.0023179    0.00191195   0.00151751   0.00115895   0.000851675  0.000602225  0.000409751  0.000268261  0.000168994  0.000102438  5.97483e-5   3.35326e-5   1.81086e-5   9.40977e-6   4.70488e-6   2.26358e-6   1.04789e-6   4.66784e-7   2.00074e-7   8.25164e-8
  1.31304e-10   6.54933e-8    1.58799e-7    3.70486e-7    8.31714e-7    1.7966e-6     3.73427e-6    7.46854e-6    1.43728e-5    2.66148e-5    4.74222e-5    8.13049e-5    0.00013413    0.000212919   0.00032522    0.000477986   0.000675975   0.00091986    0.00120445    0.00151751    0.00183972    0.00214609    0.0024089     0.00260176    0.0027039    0.0027039    0.00260176   0.0024089    0.00214609   0.00183972   0.00151751   0.00120445   0.00091986   0.000675975  0.000477986  0.00032522   0.000212919  0.00013413   8.13049e-5   4.74222e-5   2.66148e-5   1.43728e-5   7.46854e-6   3.73427e-6   1.7966e-6    8.31714e-7   3.70486e-7   1.58799e-7   6.54933e-8
  1.51504e-10   5.00184e-8    1.21277e-7    2.82947e-7    6.35194e-7    1.3721e-6     2.85193e-6    5.70385e-6    1.09768e-5    2.03262e-5    3.62172e-5    6.2094e-5     0.000102438   0.00016261    0.000248376   0.000365047   0.000516254   0.000702513   0.00091986    0.00115895    0.00140503    0.001639      0.00183972    0.00198701    0.00206502   0.00206502   0.00198701   0.00183972   0.001639     0.00140503   0.00115895   0.00091986   0.000702513  0.000516254  0.000365047  0.000248376  0.00016261   0.000102438  6.2094e-5    3.62172e-5   2.03262e-5   1.09768e-5   5.70385e-6   2.85193e-6   1.3721e-6    6.35194e-7   2.82947e-7   1.21277e-7   5.00184e-8
  1.71705e-10   3.67569e-8    8.91227e-8    2.07928e-7    4.66784e-7    1.00831e-6    2.09579e-6    4.19157e-6    8.06647e-6    1.49371e-5    2.66148e-5    4.56308e-5    7.52781e-5    0.000119497   0.000182523   0.000268261   0.000379378   0.000516254   0.000675975   0.000851675   0.00103251    0.00120445    0.00135195    0.00146019    0.00151751   0.00151751   0.00146019   0.00135195   0.00120445   0.00103251   0.000851675  0.000675975  0.000516254  0.000379378  0.000268261  0.000182523  0.000119497  7.52781e-5   4.56308e-5   2.66148e-5   1.49371e-5   8.06647e-6   4.19157e-6   2.09579e-6   1.00831e-6   4.66784e-7   2.07928e-7   8.91227e-8   3.67569e-8
  1.91905e-10   2.59911e-8    6.30193e-8    1.47028e-7    3.30066e-7    7.12982e-7    1.48195e-6    2.96389e-6    5.70385e-6    1.05621e-5    1.88195e-5    3.22659e-5    5.32297e-5    8.44968e-5    0.000129063   0.000189689   0.000268261   0.000365047   0.000477986   0.000602225   0.000730093   0.000851675   0.000955973   0.00103251    0.00107304   0.00107304   0.00103251   0.000955973  0.000851675  0.000730093  0.000602225  0.000477986  0.000365047  0.000268261  0.000189689  0.000129063  8.44968e-5   5.32297e-5   3.22659e-5   1.88195e-5   1.05621e-5   5.70385e-6   2.96389e-6   1.48195e-6   7.12982e-7   3.30066e-7   1.47028e-7   6.30193e-8   2.59911e-8
  2.12106e-10   1.76842e-8    4.2878e-8     1.00037e-7    2.24575e-7    4.85109e-7    1.00831e-6    2.01662e-6    3.88087e-6    7.1864e-6     1.28047e-5    2.19535e-5    3.62172e-5    5.74912e-5    8.78141e-5    0.000129063   0.000182523   0.000248376   0.00032522    0.000409751   0.000496752   0.000579475   0.000650439   0.000702513   0.000730093  0.000730093  0.000702513  0.000650439  0.000579475  0.000496752  0.000409751  0.00032522   0.000248376  0.000182523  0.000129063  8.78141e-5   5.74912e-5   3.62172e-5   2.19535e-5   1.28047e-5   7.1864e-6    3.88087e-6   2.01662e-6   1.00831e-6   4.85109e-7   2.24575e-7   1.00037e-7   4.2878e-8    1.76842e-8
  2.32307e-10   1.15777e-8    2.80719e-8    6.54933e-8    1.47028e-7    3.17597e-7    6.60132e-7    1.32026e-6    2.54078e-6    4.70488e-6    8.38315e-6    1.43728e-5    2.37111e-5    3.76391e-5    5.74912e-5    8.44968e-5    0.000119497   0.00016261    0.000212919   0.000268261   0.00032522    0.000379378   0.000425837   0.00045993    0.000477986  0.000477986  0.00045993   0.000425837  0.000379378  0.00032522   0.000268261  0.000212919  0.00016261   0.000119497  8.44968e-5   5.74912e-5   3.76391e-5   2.37111e-5   1.43728e-5   8.38315e-6   4.70488e-6   2.54078e-6   1.32026e-6   6.60132e-7   3.17597e-7   1.47028e-7   6.54933e-8   2.80719e-8   1.15777e-8
  2.52507e-10   7.29349e-9    1.76842e-8    4.12582e-8    9.26216e-8    2.00074e-7    4.15857e-7    8.31714e-7    1.60059e-6    2.96389e-6    5.28105e-6    9.0543e-6     1.49371e-5    2.37111e-5    3.62172e-5    5.32297e-5    7.52781e-5    0.000102438   0.00013413    0.000168994   0.000204875   0.000238993   0.000268261   0.000289738   0.000301113  0.000301113  0.000289738  0.000268261  0.000238993  0.000204875  0.000168994  0.00013413   0.000102438  7.52781e-5   5.32297e-5   3.62172e-5   2.37111e-5   1.49371e-5   9.0543e-6    5.28105e-6   2.96389e-6   1.60059e-6   8.31714e-7   4.15857e-7   2.00074e-7   9.26216e-8   4.12582e-8   1.76842e-8   7.29349e-9
  2.72708e-10   4.42105e-9    1.07195e-8    2.50092e-8    5.61438e-8    1.21277e-7    2.52077e-7    5.04154e-7    9.70218e-7    1.7966e-6     3.20118e-6    5.48838e-6    9.0543e-6     1.43728e-5    2.19535e-5    3.22659e-5    4.56308e-5    6.2094e-5     8.13049e-5    0.000102438   0.000124188   0.000144869   0.00016261    0.000175628   0.000182523  0.000182523  0.000175628  0.00016261   0.000144869  0.000124188  0.000102438  8.13049e-5   6.2094e-5    4.56308e-5   3.22659e-5   2.19535e-5   1.43728e-5   9.0543e-6    5.48838e-6   3.20118e-6   1.7966e-6    9.70218e-7   5.04154e-7   2.52077e-7   1.21277e-7   5.61438e-8   2.50092e-8   1.07195e-8   4.42105e-9
  2.92908e-10   2.57864e-9    6.2523e-9     1.4587e-8     3.27467e-8    7.07367e-8    1.47028e-7    2.94055e-7    5.65894e-7    1.04789e-6    1.86713e-6    3.20118e-6    5.28105e-6    8.38315e-6    1.28047e-5    1.88195e-5    2.66148e-5    3.62172e-5    4.74222e-5    5.97483e-5    7.24344e-5    8.44968e-5    9.48445e-5    0.000102438   0.000106459  0.000106459  0.000102438  9.48445e-5   8.44968e-5   7.24344e-5   5.97483e-5   4.74222e-5   3.62172e-5   2.66148e-5   1.88195e-5   1.28047e-5   8.38315e-6   5.28105e-6   3.20118e-6   1.86713e-6   1.04789e-6   5.65894e-7   2.94055e-7   1.47028e-7   7.07367e-8   3.27467e-8   1.4587e-8    6.2523e-9    2.57864e-9
  3.13109e-10   1.44721e-9    3.50899e-9    8.18667e-9    1.83784e-8    3.96996e-8    8.25164e-8    1.65033e-7    3.17597e-7    5.8811e-7     1.04789e-6    1.7966e-6     2.96389e-6    4.70488e-6    7.1864e-6     1.05621e-5    1.49371e-5    2.03262e-5    2.66148e-5    3.35326e-5    4.06524e-5    4.74222e-5    5.32297e-5    5.74912e-5    5.97483e-5   5.97483e-5   5.74912e-5   5.32297e-5   4.74222e-5   4.06524e-5   3.35326e-5   2.66148e-5   2.03262e-5   1.49371e-5   1.05621e-5   7.1864e-6    4.70488e-6   2.96389e-6   1.7966e-6    1.04789e-6   5.8811e-7    3.17597e-7   1.65033e-7   8.25164e-8   3.96996e-8   1.83784e-8   8.18667e-9   3.50899e-9   1.44721e-9
  3.33309e-10   7.81538e-10   1.89496e-9    4.42105e-9    9.92491e-9    2.1439e-8     4.45613e-8    8.91227e-8    1.71512e-7    3.17597e-7    5.65894e-7    9.70218e-7    1.60059e-6    2.54078e-6    3.88087e-6    5.70385e-6    8.06647e-6    1.09768e-5    1.43728e-5    1.81086e-5    2.19535e-5    2.56094e-5    2.87456e-5    3.1047e-5     3.22659e-5   3.22659e-5   3.1047e-5    2.87456e-5   2.56094e-5   2.19535e-5   1.81086e-5   1.43728e-5   1.09768e-5   8.06647e-6   5.70385e-6   3.88087e-6   2.54078e-6   1.60059e-6   9.70218e-7   5.65894e-7   3.17597e-7   1.71512e-7   8.91227e-8   4.45613e-8   2.1439e-8    9.92491e-9   4.42105e-9   1.89496e-9   7.81538e-10
  3.5351e-10    4.0611e-10    9.84676e-10   2.29731e-9    5.15728e-9    1.11403e-8    2.31554e-8    4.63108e-8    8.91227e-8    1.65033e-7    2.94055e-7    5.04154e-7    8.31714e-7    1.32026e-6    2.01662e-6    2.96389e-6    4.19157e-6    5.70385e-6    7.46854e-6    9.40977e-6    1.14077e-5    1.33074e-5    1.49371e-5    1.61329e-5    1.67663e-5   1.67663e-5   1.61329e-5   1.49371e-5   1.33074e-5   1.14077e-5   9.40977e-6   7.46854e-6   5.70385e-6   4.19157e-6   2.96389e-6   2.01662e-6   1.32026e-6   8.31714e-7   5.04154e-7   2.94055e-7   1.65033e-7   8.91227e-8   4.63108e-8   2.31554e-8   1.11403e-8   5.15728e-9   2.29731e-9   9.84676e-10  4.0611e-10
  3.73711e-10   2.03055e-10   4.92338e-10   1.14865e-9    2.57864e-9    5.57017e-9    1.15777e-8    2.31554e-8    4.45613e-8    8.25164e-8    1.47028e-7    2.52077e-7    4.15857e-7    6.60132e-7    1.00831e-6    1.48195e-6    2.09579e-6    2.85193e-6    3.73427e-6    4.70488e-6    5.70385e-6    6.65371e-6    7.46854e-6    8.06647e-6    8.38315e-6   8.38315e-6   8.06647e-6   7.46854e-6   6.65371e-6   5.70385e-6   4.70488e-6   3.73427e-6   2.85193e-6   2.09579e-6   1.48195e-6   1.00831e-6   6.60132e-7   4.15857e-7   2.52077e-7   1.47028e-7   8.25164e-8   4.45613e-8   2.31554e-8   1.15777e-8   5.57017e-9   2.57864e-9   1.14865e-9   4.92338e-10  2.03055e-10
  3.93911e-10   9.76922e-11   2.3687e-10    5.52631e-10   1.24061e-9    2.67987e-9    5.57017e-9    1.11403e-8    2.1439e-8     3.96996e-8    7.07367e-8    1.21277e-7    2.00074e-7    3.17597e-7    4.85109e-7    7.12982e-7    1.00831e-6    1.3721e-6     1.7966e-6     2.26358e-6    2.74419e-6    3.20118e-6    3.5932e-6     3.88087e-6    4.03323e-6   4.03323e-6   3.88087e-6   3.5932e-6    3.20118e-6   2.74419e-6   2.26358e-6   1.7966e-6    1.3721e-6    1.00831e-6   7.12982e-7   4.85109e-7   3.17597e-7   2.00074e-7   1.21277e-7   7.07367e-8   3.96996e-8   2.1439e-8    1.11403e-8   5.57017e-9   2.67987e-9   1.24061e-9   5.52631e-10  2.3687e-10   9.76922e-11
  4.14112e-10   4.52254e-11   1.09656e-10   2.55833e-10   5.74327e-10   1.24061e-9    2.57864e-9    5.15728e-9    9.92491e-9    1.83784e-8    3.27467e-8    5.61438e-8    9.26216e-8    1.47028e-7    2.24575e-7    3.30066e-7    4.66784e-7    6.35194e-7    8.31714e-7    1.04789e-6    1.27039e-6    1.48195e-6    1.66343e-6    1.7966e-6     1.86713e-6   1.86713e-6   1.7966e-6    1.66343e-6   1.48195e-6   1.27039e-6   1.04789e-6   8.31714e-7   6.35194e-7   4.66784e-7   3.30066e-7   2.24575e-7   1.47028e-7   9.26216e-8   5.61438e-8   3.27467e-8   1.83784e-8   9.92491e-9   5.15728e-9   2.57864e-9   1.24061e-9   5.74327e-10  2.55833e-10  1.09656e-10  4.52254e-11
  4.34312e-10   2.01456e-11   4.88461e-11   1.13961e-10   2.55833e-10   5.52631e-10   1.14865e-9    2.29731e-9    4.42105e-9    8.18667e-9    1.4587e-8     2.50092e-8    4.12582e-8    6.54933e-8    1.00037e-7    1.47028e-7    2.07928e-7    2.82947e-7    3.70486e-7    4.66784e-7    5.65894e-7    6.60132e-7    7.40973e-7    8.00295e-7    8.31714e-7   8.31714e-7   8.00295e-7   7.40973e-7   6.60132e-7   5.65894e-7   4.66784e-7   3.70486e-7   2.82947e-7   2.07928e-7   1.47028e-7   1.00037e-7   6.54933e-8   4.12582e-8   2.50092e-8   1.4587e-8    8.18667e-9   4.42105e-9   2.29731e-9   1.14865e-9   5.52631e-10  2.55833e-10  1.13961e-10  4.88461e-11  2.01456e-11
  4.54513e-10   8.63485e-12   2.09365e-11   4.88461e-11   1.09656e-10   2.3687e-10    4.92338e-10   9.84676e-10   1.89496e-9    3.50899e-9    6.2523e-9     1.07195e-8    1.76842e-8    2.80719e-8    4.2878e-8     6.30193e-8    8.91227e-8    1.21277e-7    1.58799e-7    2.00074e-7    2.42555e-7    2.82947e-7    3.17597e-7    3.43024e-7    3.56491e-7   3.56491e-7   3.43024e-7   3.17597e-7   2.82947e-7   2.42555e-7   2.00074e-7   1.58799e-7   1.21277e-7   8.91227e-8   6.30193e-8   4.2878e-8    2.80719e-8   1.76842e-8   1.07195e-8   6.2523e-9    3.50899e-9   1.89496e-9   9.84676e-10  4.92338e-10  2.3687e-10   1.09656e-10  4.88461e-11  2.09365e-11  8.63485e-12
  4.74713e-10   3.56128e-12   8.63485e-12   2.01456e-11   4.52254e-11   9.76922e-11   2.03055e-10   4.0611e-10    7.81538e-10   1.44721e-9    2.57864e-9    4.42105e-9    7.29349e-9    1.15777e-8    1.76842e-8    2.59911e-8    3.67569e-8    5.00184e-8    6.54933e-8    8.25164e-8    1.00037e-7    1.16696e-7    1.30987e-7    1.41473e-7    1.47028e-7   1.47028e-7   1.41473e-7   1.30987e-7   1.16696e-7   1.00037e-7   8.25164e-8   6.54933e-8   5.00184e-8   3.67569e-8   2.59911e-8   1.76842e-8   1.15777e-8   7.29349e-9   4.42105e-9   2.57864e-9   1.44721e-9   7.81538e-10  4.0611e-10   2.03055e-10  9.76922e-11  4.52254e-11  2.01456e-11  8.63485e-12  3.56128e-12

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.

julia

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

(ftot = 1.1, mimg = [3.5612926136987482e-12 8.634895734275082e-12 2.0145713255487537e-11 4.522559713081575e-11 9.769269325671449e-11 2.0305606556587994e-10 4.061121311317591e-10 7.815415460537175e-10 1.447219108186102e-9 2.5786512967024023e-9 4.4210666159488215e-9 7.293527272855018e-9 1.15777528654888e-8 1.7684266463795293e-8 2.599117639243255e-8 3.67570741562095e-8 5.001865894743796e-8 6.549366049671296e-8 8.25168414944769e-8 1.0003731789487592e-7 1.166964363656801e-7 1.3098732099342573e-7 1.4147413171036213e-7 1.4702829662483802e-7 1.4702829662483802e-7 1.4147413171036213e-7 1.3098732099342573e-7 1.166964363656801e-7 1.0003731789487592e-7 8.25168414944769e-8 6.549366049671285e-8 5.001865894743796e-8 3.67570741562095e-8 2.599117639243255e-8 1.768426646379523e-8 1.15777528654888e-8 7.293527272855018e-9 4.421066615948815e-9 2.5786512967024106e-9 1.447219108186102e-9 7.815415460537175e-10 4.061121311317591e-10 2.0305606556587994e-10 9.769269325671449e-11 4.522559713081575e-11 2.0145713255487537e-11 8.634895734275082e-12 3.5612926136987482e-12; 8.634895734275082e-12 2.0936618365757505e-11 4.884634662835752e-11 1.0965634057779147e-10 2.368707971446701e-10 4.923403226202322e-10 9.846806452404627e-10 1.8949663771573584e-9 3.509002898489335e-9 6.252332368429731e-9 1.0719548603267813e-8 1.7684266463795293e-8 2.8072023187914533e-8 4.287819441307118e-8 6.301956129538964e-8 8.912311827874256e-8 1.212778481380712e-7 1.587993436624031e-7 2.0007463578975186e-7 2.4255569627614246e-7 2.8294826342072426e-7 3.175986873248062e-7 3.430255553045695e-7 3.564924731149702e-7 3.564924731149702e-7 3.430255553045695e-7 3.175986873248062e-7 2.8294826342072426e-7 2.4255569627614246e-7 2.0007463578975186e-7 1.587993436624031e-7 1.212778481380712e-7 8.912311827874256e-8 6.301956129538964e-8 4.287819441307118e-8 2.807202318791463e-8 1.7684266463795293e-8 1.0719548603267791e-8 6.252332368429754e-9 3.509002898489335e-9 1.8949663771573584e-9 9.846806452404627e-10 4.923403226202322e-10 2.368707971446701e-10 1.0965634057779147e-10 4.884634662835752e-11 2.0936618365757505e-11 8.634895734275082e-12; 2.0145713255487537e-11 4.884634662835752e-11 1.1396136363835936e-10 2.5583461131528484e-10 5.526333269935996e-10 1.1486585673815487e-9 2.297317134763089e-9 4.421066615948807e-9 8.186707562089107e-9 1.4587054545709983e-8 2.5009329473718933e-8 4.125842074723837e-8 6.549366049671273e-8 1.0003731789487575e-7 1.4702829662483776e-7 2.079294111394597e-7 2.8294826342072373e-7 3.704880916956411e-7 4.667857454627196e-7 5.658965268414476e-7 6.60134731955813e-7 7.40976183391281e-7 8.002985431590045e-7 8.317176445578389e-7 8.317176445578389e-7 8.002985431590045e-7 7.40976183391281e-7 6.60134731955813e-7 5.658965268414476e-7 4.667857454627196e-7 3.704880916956404e-7 2.8294826342072373e-7 2.079294111394597e-7 1.4702829662483776e-7 1.0003731789487539e-7 6.549366049671273e-8 4.125842074723837e-8 2.5009329473718887e-8 1.4587054545710036e-8 8.186707562089107e-9 4.421066615948807e-9 2.297317134763089e-9 1.1486585673815487e-9 5.526333269935996e-10 2.5583461131528484e-10 1.1396136363835936e-10 4.884634662835752e-11 2.0145713255487537e-11; 4.522559713081575e-11 1.0965634057779147e-10 2.5583461131528484e-10 5.743292836907731e-10 1.2406198723625213e-9 2.5786512967024023e-9 5.1573025934047955e-9 9.924958978900158e-9 1.837853707810478e-8 3.274683024835631e-8 5.6144046375829066e-8 9.262202292391026e-8 1.4702829662483752e-7 2.2457618550331592e-7 3.3006736597790707e-7 4.667857454627196e-7 6.351973746496115e-7 8.317176445578389e-7 1.0478985679474042e-6 1.2703947492992233e-6 1.481952366782562e-6 1.6634352891156783e-6 1.7966094840265274e-6 1.8671429818508787e-6 1.8671429818508787e-6 1.7966094840265274e-6 1.6634352891156783e-6 1.481952366782562e-6 1.2703947492992233e-6 1.0478985679474042e-6 8.317176445578389e-7 6.351973746496115e-7 4.667857454627196e-7 3.3006736597790707e-7 2.2457618550331592e-7 1.4702829662483802e-7 9.262202292391026e-8 5.614404637582897e-8 3.274683024835643e-8 1.837853707810478e-8 9.924958978900158e-9 5.1573025934047955e-9 2.578651296702407e-9 1.2406198723625213e-9 5.743292836907731e-10 2.5583461131528484e-10 1.0965634057779147e-10 4.522559713081575e-11; 9.769269325671449e-11 2.368707971446701e-10 5.526333269935996e-10 1.2406198723625213e-9 2.6798871508169387e-9 5.5701948924213965e-9 1.1140389784842775e-8 2.143909720653547e-8 3.9699835915600776e-8 7.073706585518068e-8 1.2127784813807075e-7 2.0007463578975112e-7 3.1759868732480456e-7 4.851113925522823e-7 7.129849462299381e-7 1.0083129807262293e-6 1.3721022212182733e-6 1.7966094840265248e-6 2.263586107365787e-6 2.7442044424365453e-6 3.2011941726360163e-6 3.5932189680530496e-6 3.88089114041826e-6 4.033251922904918e-6 4.033251922904918e-6 3.88089114041826e-6 3.5932189680530496e-6 3.2011941726360163e-6 2.7442044424365453e-6 2.263586107365787e-6 1.7966094840265248e-6 1.3721022212182733e-6 1.0083129807262293e-6 7.129849462299381e-7 4.851113925522823e-7 3.1759868732480573e-7 2.0007463578975112e-7 1.2127784813807057e-7 7.073706585518093e-8 3.9699835915600776e-8 2.143909720653547e-8 1.1140389784842775e-8 5.570194892421407e-9 2.6798871508169387e-9 1.2406198723625213e-9 5.526333269935996e-10 2.368707971446701e-10 9.769269325671449e-11; 2.0305606556587994e-10 4.923403226202322e-10 1.1486585673815487e-9 2.5786512967024023e-9 5.5701948924213965e-9 1.157775286548882e-8 2.315550573097756e-8 4.456155913937119e-8 8.251684149447703e-8 1.4702829662483776e-7 2.5207824518155823e-7 4.158588222789203e-7 6.601347319558142e-7 1.0083129807262331e-6 1.4819523667825646e-6 2.095797135894812e-6 2.8519397849197576e-6 3.734285963701764e-6 4.704905491994815e-6 5.703879569839515e-6 6.653741156462725e-6 7.468571927403523e-6 8.066503845809857e-6 8.383188543579245e-6 8.383188543579245e-6 8.066503845809857e-6 7.468571927403523e-6 6.653741156462725e-6 5.703879569839515e-6 4.704905491994815e-6 3.734285963701764e-6 2.85193978491976e-6 2.095797135894812e-6 1.4819523667825646e-6 1.0083129807262312e-6 6.601347319558153e-7 4.158588222789203e-7 2.5207824518155776e-7 1.4702829662483828e-7 8.251684149447703e-8 4.456155913937119e-8 2.315550573097756e-8 1.1577752865488842e-8 5.5701948924213965e-9 2.5786512967024023e-9 1.1486585673815487e-9 4.923403226202322e-10 2.0305606556587994e-10; 4.061121311317591e-10 9.846806452404627e-10 2.297317134763089e-9 5.1573025934047955e-9 1.1140389784842775e-8 2.315550573097756e-8 4.631101146195504e-8 8.912311827874205e-8 1.650336829889538e-7 2.9405659324967445e-7 5.041564903631155e-7 8.317176445578389e-7 1.320269463911624e-6 2.016625961452459e-6 2.963904733565124e-6 4.191594271789614e-6 5.703879569839505e-6 7.46857192740351e-6 9.409810983989606e-6 1.1407759139679001e-5 1.330748231292542e-5 1.4937143854807011e-5 1.6133007691619673e-5 1.6766377087158443e-5 1.6766377087158443e-5 1.6133007691619673e-5 1.4937143854807011e-5 1.330748231292542e-5 1.1407759139679001e-5 9.409810983989606e-6 7.46857192740351e-6 5.703879569839505e-6 4.191594271789614e-6 2.963904733565124e-6 2.016625961452459e-6 1.3202694639116285e-6 8.317176445578389e-7 5.041564903631147e-7 2.9405659324967556e-7 1.650336829889538e-7 8.912311827874205e-8 4.631101146195504e-8 2.31555057309776e-8 1.1140389784842775e-8 5.1573025934047955e-9 2.297317134763089e-9 9.846806452404627e-10 4.061121311317591e-10; 7.815415460537175e-10 1.8949663771573584e-9 4.421066615948807e-9 9.924958978900158e-9 2.143909720653547e-8 4.456155913937119e-8 8.912311827874205e-8 1.7151277765228381e-7 3.1759868732480573e-7 5.658965268414456e-7 9.702227851045645e-7 1.6005970863180064e-6 2.5407894985984374e-6 3.88089114041826e-6 5.7038795698395e-6 8.066503845809836e-6 1.0976817769746173e-5 1.43728758722122e-5 1.810868885892628e-5 2.1953635539492345e-5 2.5609553381088117e-5 2.8745751744424377e-5 3.104712912334606e-5 3.2266015383239325e-5 3.2266015383239325e-5 3.104712912334606e-5 2.8745751744424377e-5 2.5609553381088117e-5 2.1953635539492345e-5 1.810868885892628e-5 1.43728758722122e-5 1.0976817769746181e-5 8.066503845809836e-6 5.7038795698395e-6 3.880891140418253e-6 2.5407894985984416e-6 1.6005970863180064e-6 9.702227851045628e-7 5.658965268414476e-7 3.1759868732480573e-7 1.7151277765228381e-7 8.912311827874205e-8 4.4561559139371265e-8 2.143909720653547e-8 9.924958978900158e-9 4.421066615948807e-9 1.8949663771573584e-9 7.815415460537175e-10; 1.447219108186102e-9 3.509002898489335e-9 8.186707562089107e-9 1.837853707810478e-8 3.9699835915600776e-8 8.251684149447703e-8 1.650336829889538e-7 3.1759868732480573e-7 5.881131864993532e-7 1.0478985679474042e-6 1.796609484026531e-6 2.963904733565132e-6 4.70490549199481e-6 7.186437936106125e-6 1.056215571129304e-5 1.493714385480706e-5 2.0326315988787573e-5 2.6614964625850903e-5 3.353275417431698e-5 4.065263197757515e-5 4.742247573704209e-5 5.322992925170177e-5 5.749150348884896e-5 5.97485754192282e-5 5.97485754192282e-5 5.749150348884896e-5 5.322992925170177e-5 4.742247573704209e-5 4.065263197757515e-5 3.353275417431698e-5 2.6614964625850903e-5 2.032631598878759e-5 1.493714385480706e-5 1.056215571129304e-5 7.186437936106112e-6 4.704905491994819e-6 2.963904733565132e-6 1.796609484026531e-6 1.047898567947408e-6 5.881131864993532e-7 3.1759868732480573e-7 1.650336829889538e-7 8.251684149447703e-8 3.9699835915600776e-8 1.837853707810478e-8 8.186707562089107e-9 3.509002898489335e-9 1.447219108186102e-9; 2.5786512967024023e-9 6.252332368429731e-9 1.4587054545709983e-8 3.274683024835631e-8 7.073706585518068e-8 1.4702829662483776e-7 2.9405659324967445e-7 5.658965268414456e-7 1.0478985679474042e-6 1.8671429818508722e-6 3.201194172636014e-6 5.2810778556465e-6 8.383188543579206e-6 1.2804776690544057e-5 1.8819621967979195e-5 2.661496462585082e-5 3.6217377717852536e-5 4.742247573704192e-5 5.9748575419228004e-5 7.243475543570508e-5 8.449724569034398e-5 9.484495147408377e-5 0.0001024382135243524 0.00010645985850340317 0.00010645985850340317 0.0001024382135243524 9.484495147408377e-5 8.449724569034398e-5 7.243475543570508e-5 5.9748575419228004e-5 4.742247573704192e-5 3.621737771785257e-5 2.661496462585082e-5 1.8819621967979195e-5 1.2804776690544035e-5 8.383188543579222e-6 5.2810778556465e-6 3.2011941726360103e-6 1.8671429818508787e-6 1.0478985679474042e-6 5.658965268414456e-7 2.9405659324967445e-7 1.4702829662483802e-7 7.073706585518068e-8 3.274683024835631e-8 1.4587054545709983e-8 6.252332368429731e-9 2.5786512967024023e-9; 4.4210666159488215e-9 1.0719548603267813e-8 2.5009329473718933e-8 5.6144046375829066e-8 1.2127784813807075e-7 2.5207824518155823e-7 5.041564903631155e-7 9.702227851045645e-7 1.796609484026531e-6 3.201194172636014e-6 5.488408884873099e-6 9.054344429463164e-6 1.43728758722122e-5 2.1953635539492382e-5 3.2266015383239386e-5 4.563103655871605e-5 6.209425824669222e-5 8.130526395515017e-5 0.00010243821352435257 0.0001241885164933844 0.0001448695108714104 0.0001626105279103003 0.00017562908431593906 0.00018252414623486416 0.00018252414623486416 0.00017562908431593906 0.0001626105279103003 0.0001448695108714104 0.00012418851649338445 0.00010243821352435257 8.130526395515017e-5 6.209425824669228e-5 4.563103655871605e-5 3.2266015383239386e-5 2.1953635539492345e-5 1.4372875872212226e-5 9.054344429463164e-6 5.488408884873096e-6 3.201194172636025e-6 1.796609484026531e-6 9.702227851045645e-7 5.041564903631155e-7 2.5207824518155866e-7 1.2127784813807075e-7 5.6144046375829066e-8 2.5009329473718933e-8 1.0719548603267813e-8 4.4210666159488215e-9; 7.293527272855018e-9 1.7684266463795293e-8 4.125842074723837e-8 9.262202292391026e-8 2.0007463578975112e-7 4.158588222789203e-7 8.317176445578389e-7 1.6005970863180064e-6 2.963904733565132e-6 5.2810778556465e-6 9.054344429463164e-6 1.4937143854807035e-5 2.371123786852098e-5 3.621737771785263e-5 5.322992925170173e-5 7.527848787191693e-5 0.00010243821352435266 0.00013413101669726773 0.00016899449138068818 0.00020487642704870515 0.00023899430167691242 0.0002682620333945352 0.0002897390217428208 0.0003011139514876675 0.0003011139514876675 0.0002897390217428208 0.0002682620333945352 0.00023899430167691231 0.00020487642704870529 0.00016899449138068818 0.00013413101669726773 0.00010243821352435266 7.527848787191693e-5 5.322992925170173e-5 3.621737771785257e-5 2.371123786852102e-5 1.4937143854807035e-5 9.054344429463147e-6 5.28107785564652e-6 2.963904733565132e-6 1.6005970863180064e-6 8.317176445578389e-7 4.1585882227892094e-7 2.0007463578975112e-7 9.262202292391026e-8 4.125842074723837e-8 1.7684266463795293e-8 7.293527272855018e-9; 1.15777528654888e-8 2.8072023187914533e-8 6.549366049671273e-8 1.4702829662483752e-7 3.1759868732480456e-7 6.601347319558142e-7 1.320269463911624e-6 2.5407894985984374e-6 4.70490549199481e-6 8.383188543579206e-6 1.43728758722122e-5 2.371123786852098e-5 3.763924393595837e-5 5.749150348884881e-5 8.449724569034405e-5 0.00011949715083845609 0.00016261052791030013 0.00021291971700680657 0.0002682620333945348 0.0003252210558206002 0.0003793798058963355 0.0004258394340136128 0.00045993202791079024 0.00047798860335382403 0.00047798860335382403 0.00045993202791079024 0.0004258394340136128 0.0003793798058963353 0.00032522105582060036 0.0002682620333945348 0.00021291971700680657 0.00016261052791030018 0.00011949715083845609 8.449724569034405e-5 5.749150348884871e-5 3.7639243935958425e-5 2.371123786852098e-5 1.4372875872212187e-5 8.383188543579237e-6 4.70490549199481e-6 2.5407894985984374e-6 1.320269463911624e-6 6.601347319558153e-7 3.1759868732480456e-7 1.4702829662483752e-7 6.549366049671273e-8 2.8072023187914533e-8 1.15777528654888e-8; 1.7684266463795293e-8 4.287819441307118e-8 1.0003731789487575e-7 2.2457618550331592e-7 4.851113925522823e-7 1.0083129807262331e-6 2.016625961452459e-6 3.88089114041826e-6 7.186437936106125e-6 1.2804776690544057e-5 2.1953635539492382e-5 3.621737771785263e-5 5.749150348884881e-5 8.781454215796957e-5 0.00012906406153295754 0.00018252414623486427 0.0002483770329867688 0.00032522105582060074 0.00040975285409741024 0.0004967540659735375 0.0005794780434856418 0.000650442111641201 0.0007025163372637559 0.0007300965849394566 0.0007300965849394566 0.0007025163372637559 0.000650442111641201 0.0005794780434856414 0.0004967540659735377 0.00040975285409741024 0.00032522105582060074 0.00024837703298676895 0.00018252414623486427 0.00012906406153295754 8.78145421579694e-5 5.7491503488848916e-5 3.621737771785263e-5 2.1953635539492362e-5 1.2804776690544102e-5 7.186437936106125e-6 3.88089114041826e-6 2.016625961452459e-6 1.0083129807262346e-6 4.851113925522823e-7 2.2457618550331592e-7 1.0003731789487575e-7 4.287819441307118e-8 1.7684266463795293e-8; 2.599117639243255e-8 6.301956129538964e-8 1.4702829662483776e-7 3.3006736597790707e-7 7.129849462299381e-7 1.4819523667825646e-6 2.963904733565124e-6 5.7038795698395e-6 1.056215571129304e-5 1.8819621967979195e-5 3.2266015383239386e-5 5.322992925170173e-5 8.449724569034405e-5 0.00012906406153295754 0.00018968990294816794 0.0002682620333945353 0.0003650482924697282 0.00047798860335382485 0.0006022279029753348 0.0007300965849394566 0.0008516788680272268 0.0009559772067076492 0.0010325124922636595 0.0010730481335781406 0.0010730481335781406 0.0010325124922636595 0.0009559772067076492 0.0008516788680272265 0.0007300965849394566 0.0006022279029753348 0.00047798860335382463 0.00036504829246972854 0.0002682620333945353 0.00018968990294816794 0.0001290640615329573 8.449724569034421e-5 5.322992925170173e-5 3.226601538323935e-5 1.8819621967979263e-5 1.056215571129304e-5 5.7038795698395e-6 2.963904733565124e-6 1.4819523667825673e-6 7.129849462299381e-7 3.3006736597790707e-7 1.4702829662483776e-7 6.301956129538964e-8 2.599117639243255e-8; 3.67570741562095e-8 8.912311827874256e-8 2.079294111394597e-7 4.667857454627196e-7 1.0083129807262293e-6 2.095797135894812e-6 4.191594271789614e-6 8.066503845809836e-6 1.493714385480706e-5 2.661496462585082e-5 4.563103655871605e-5 7.527848787191693e-5 0.00011949715083845609 0.00018252414623486427 0.0002682620333945353 0.00037937980589633605 0.0005162562461318299 0.000675977965522753 0.0008516788680272268 0.0010325124922636595 0.0012044558059506696 0.0013519559310455054 0.0014601931698789128 0.0015175192235853429 0.0015175192235853429 0.0014601931698789128 0.0013519559310455054 0.0012044558059506696 0.0010325124922636597 0.0008516788680272268 0.0006759779655227528 0.0005162562461318303 0.00037937980589633605 0.0002682620333945353 0.00018252414623486392 0.0001194971508384563 7.527848787191693e-5 4.563103655871602e-5 2.6614964625850903e-5 1.493714385480706e-5 8.066503845809836e-6 4.191594271789614e-6 2.0957971358948143e-6 1.0083129807262293e-6 4.667857454627196e-7 2.079294111394597e-7 8.912311827874256e-8 3.67570741562095e-8; 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1.3098732099342573e-7 3.175986873248062e-7 7.40976183391281e-7 1.6634352891156783e-6 3.5932189680530496e-6 7.468571927403523e-6 1.4937143854807011e-5 2.8745751744424377e-5 5.322992925170177e-5 9.484495147408377e-5 0.0001626105279103003 0.0002682620333945352 0.0004258394340136128 0.000650442111641201 0.0009559772067076492 0.0013519559310455054 0.0018397281116431623 0.002408911611901339 0.0030350384471706845 0.0036794562232863232 0.004292192534312561 0.004817823223802676 0.005203536893129606 0.005407823724182016 0.005407823724182018 0.005203536893129606 0.004817823223802676 0.00429219253431256 0.0036794562232863245 0.0030350384471706845 0.0024089116119013388 0.0018397281116431631 0.0013519559310455054 0.0009559772067076492 0.0006504421116412 0.00042583943401361353 0.0002682620333945352 0.00016261052791030005 9.484495147408411e-5 5.322992925170177e-5 2.8745751744424377e-5 1.4937143854807011e-5 7.46857192740353e-6 3.5932189680530496e-6 1.6634352891156783e-6 7.40976183391281e-7 3.175986873248062e-7 1.3098732099342573e-7; 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8.25168414944769e-8 2.0007463578975186e-7 4.667857454627196e-7 1.0478985679474042e-6 2.263586107365787e-6 4.704905491994815e-6 9.409810983989606e-6 1.810868885892628e-5 3.353275417431698e-5 5.9748575419228004e-5 0.00010243821352435257 0.00016899449138068818 0.0002682620333945348 0.00040975285409741024 0.0006022279029753348 0.0008516788680272268 0.0011589560869712824 0.0015175192235853427 0.001911954413415297 0.002317912173942565 0.002703911862091009 0.0030350384471706845 0.0032780228327792802 0.003406715472108905 0.003406715472108905 0.0032780228327792802 0.0030350384471706845 0.002703911862091008 0.0023179121739425657 0.001911954413415297 0.0015175192235853427 0.0011589560869712833 0.0008516788680272268 0.0006022279029753348 0.00040975285409740943 0.0002682620333945353 0.00016899449138068818 0.00010243821352435248 5.97485754192282e-5 3.353275417431698e-5 1.810868885892628e-5 9.409810983989606e-6 4.704905491994819e-6 2.263586107365787e-6 1.0478985679474042e-6 4.667857454627196e-7 2.0007463578975186e-7 8.25168414944769e-8; 6.549366049671285e-8 1.587993436624031e-7 3.704880916956404e-7 8.317176445578389e-7 1.7966094840265248e-6 3.734285963701764e-6 7.46857192740351e-6 1.43728758722122e-5 2.6614964625850903e-5 4.742247573704192e-5 8.130526395515017e-5 0.00013413101669726773 0.00021291971700680657 0.00032522105582060074 0.00047798860335382463 0.0006759779655227528 0.0009198640558215815 0.00120445580595067 0.0015175192235853427 0.0018397281116431623 0.002146096267156281 0.0024089116119013388 0.002601768446564804 0.00270391186209101 0.00270391186209101 0.002601768446564804 0.0024089116119013388 0.0021460962671562807 0.001839728111643163 0.0015175192235853427 0.0012044558059506696 0.000919864055821582 0.0006759779655227528 0.00047798860335382463 0.0003252210558206002 0.00021291971700680693 0.00013413101669726773 8.130526395515009e-5 4.742247573704209e-5 2.6614964625850903e-5 1.43728758722122e-5 7.46857192740351e-6 3.734285963701767e-6 1.7966094840265248e-6 8.317176445578389e-7 3.704880916956404e-7 1.587993436624031e-7 6.549366049671285e-8; 5.001865894743796e-8 1.212778481380712e-7 2.8294826342072373e-7 6.351973746496115e-7 1.3721022212182733e-6 2.85193978491976e-6 5.703879569839505e-6 1.0976817769746181e-5 2.032631598878759e-5 3.621737771785257e-5 6.209425824669228e-5 0.00010243821352435266 0.00016261052791030018 0.00024837703298676895 0.00036504829246972854 0.0005162562461318303 0.0007025163372637562 0.000919864055821582 0.0011589560869712833 0.0014050326745275118 0.0016390114163896412 0.0018397281116431631 0.0019870162638941503 0.00206502498452732 0.00206502498452732 0.0019870162638941503 0.0018397281116431631 0.0016390114163896412 0.0014050326745275127 0.0011589560869712833 0.000919864055821582 0.0007025163372637566 0.0005162562461318303 0.00036504829246972854 0.0002483770329867685 0.0001626105279103005 0.00010243821352435266 6.209425824669217e-5 3.621737771785269e-5 2.032631598878759e-5 1.0976817769746181e-5 5.703879569839505e-6 2.851939784919763e-6 1.3721022212182733e-6 6.351973746496115e-7 2.8294826342072373e-7 1.212778481380712e-7 5.001865894743796e-8; 3.67570741562095e-8 8.912311827874256e-8 2.079294111394597e-7 4.667857454627196e-7 1.0083129807262293e-6 2.095797135894812e-6 4.191594271789614e-6 8.066503845809836e-6 1.493714385480706e-5 2.661496462585082e-5 4.563103655871605e-5 7.527848787191693e-5 0.00011949715083845609 0.00018252414623486427 0.0002682620333945353 0.00037937980589633605 0.0005162562461318299 0.000675977965522753 0.0008516788680272268 0.0010325124922636595 0.0012044558059506696 0.0013519559310455054 0.0014601931698789128 0.0015175192235853429 0.0015175192235853429 0.0014601931698789128 0.0013519559310455054 0.0012044558059506696 0.0010325124922636597 0.0008516788680272268 0.0006759779655227528 0.0005162562461318303 0.00037937980589633605 0.0002682620333945353 0.00018252414623486392 0.0001194971508384563 7.527848787191693e-5 4.563103655871602e-5 2.6614964625850903e-5 1.493714385480706e-5 8.066503845809836e-6 4.191594271789614e-6 2.0957971358948143e-6 1.0083129807262293e-6 4.667857454627196e-7 2.079294111394597e-7 8.912311827874256e-8 3.67570741562095e-8; 2.599117639243255e-8 6.301956129538964e-8 1.4702829662483776e-7 3.3006736597790707e-7 7.129849462299381e-7 1.4819523667825646e-6 2.963904733565124e-6 5.7038795698395e-6 1.056215571129304e-5 1.8819621967979195e-5 3.2266015383239386e-5 5.322992925170173e-5 8.449724569034405e-5 0.00012906406153295754 0.00018968990294816794 0.0002682620333945353 0.0003650482924697282 0.00047798860335382485 0.0006022279029753348 0.0007300965849394566 0.0008516788680272268 0.0009559772067076492 0.0010325124922636595 0.0010730481335781406 0.0010730481335781406 0.0010325124922636595 0.0009559772067076492 0.0008516788680272265 0.0007300965849394566 0.0006022279029753348 0.00047798860335382463 0.00036504829246972854 0.0002682620333945353 0.00018968990294816794 0.0001290640615329573 8.449724569034421e-5 5.322992925170173e-5 3.226601538323935e-5 1.8819621967979263e-5 1.056215571129304e-5 5.7038795698395e-6 2.963904733565124e-6 1.4819523667825673e-6 7.129849462299381e-7 3.3006736597790707e-7 1.4702829662483776e-7 6.301956129538964e-8 2.599117639243255e-8; 1.768426646379523e-8 4.287819441307118e-8 1.0003731789487539e-7 2.2457618550331592e-7 4.851113925522823e-7 1.0083129807262312e-6 2.016625961452459e-6 3.880891140418253e-6 7.186437936106112e-6 1.2804776690544035e-5 2.1953635539492345e-5 3.621737771785257e-5 5.749150348884871e-5 8.78145421579694e-5 0.0001290640615329573 0.00018252414623486392 0.00024837703298676836 0.0003252210558206002 0.00040975285409740943 0.0004967540659735367 0.0005794780434856407 0.0006504421116412 0.0007025163372637547 0.0007300965849394552 0.0007300965849394552 0.0007025163372637547 0.0006504421116412 0.0005794780434856405 0.0004967540659735367 0.00040975285409740943 0.0003252210558206002 0.0002483770329867685 0.00018252414623486392 0.0001290640615329573 8.781454215796923e-5 5.749150348884881e-5 3.621737771785257e-5 2.195363553949232e-5 1.280477669054408e-5 7.186437936106112e-6 3.880891140418253e-6 2.016625961452459e-6 1.0083129807262312e-6 4.851113925522823e-7 2.2457618550331592e-7 1.0003731789487539e-7 4.287819441307118e-8 1.768426646379523e-8; 1.15777528654888e-8 2.807202318791463e-8 6.549366049671273e-8 1.4702829662483802e-7 3.1759868732480573e-7 6.601347319558153e-7 1.3202694639116285e-6 2.5407894985984416e-6 4.704905491994819e-6 8.383188543579222e-6 1.4372875872212226e-5 2.371123786852102e-5 3.7639243935958425e-5 5.7491503488848916e-5 8.449724569034421e-5 0.0001194971508384563 0.00016261052791030042 0.00021291971700680693 0.0002682620333945353 0.00032522105582060074 0.0003793798058963361 0.00042583943401361353 0.000459932027910791 0.00047798860335382485 0.00047798860335382485 0.000459932027910791 0.00042583943401361353 0.00037937980589633605 0.00032522105582060085 0.0002682620333945353 0.00021291971700680693 0.0001626105279103005 0.0001194971508384563 8.449724569034421e-5 5.749150348884881e-5 3.763924393595849e-5 2.371123786852102e-5 1.4372875872212212e-5 8.38318854357925e-6 4.704905491994819e-6 2.5407894985984416e-6 1.3202694639116285e-6 6.601347319558153e-7 3.1759868732480573e-7 1.4702829662483802e-7 6.549366049671273e-8 2.807202318791463e-8 1.15777528654888e-8; 7.293527272855018e-9 1.7684266463795293e-8 4.125842074723837e-8 9.262202292391026e-8 2.0007463578975112e-7 4.158588222789203e-7 8.317176445578389e-7 1.6005970863180064e-6 2.963904733565132e-6 5.2810778556465e-6 9.054344429463164e-6 1.4937143854807035e-5 2.371123786852098e-5 3.621737771785263e-5 5.322992925170173e-5 7.527848787191693e-5 0.00010243821352435266 0.00013413101669726773 0.00016899449138068818 0.00020487642704870515 0.00023899430167691242 0.0002682620333945352 0.0002897390217428208 0.0003011139514876675 0.0003011139514876675 0.0002897390217428208 0.0002682620333945352 0.00023899430167691231 0.00020487642704870529 0.00016899449138068818 0.00013413101669726773 0.00010243821352435266 7.527848787191693e-5 5.322992925170173e-5 3.621737771785257e-5 2.371123786852102e-5 1.4937143854807035e-5 9.054344429463147e-6 5.28107785564652e-6 2.963904733565132e-6 1.6005970863180064e-6 8.317176445578389e-7 4.1585882227892094e-7 2.0007463578975112e-7 9.262202292391026e-8 4.125842074723837e-8 1.7684266463795293e-8 7.293527272855018e-9; 4.421066615948815e-9 1.0719548603267791e-8 2.5009329473718887e-8 5.614404637582897e-8 1.2127784813807057e-7 2.5207824518155776e-7 5.041564903631147e-7 9.702227851045628e-7 1.796609484026531e-6 3.2011941726360103e-6 5.488408884873096e-6 9.054344429463147e-6 1.4372875872212187e-5 2.1953635539492362e-5 3.226601538323935e-5 4.563103655871602e-5 6.209425824669217e-5 8.130526395515009e-5 0.00010243821352435248 0.00012418851649338426 0.0001448695108714103 0.00016261052791030005 0.0001756290843159388 0.00018252414623486392 0.00018252414623486392 0.0001756290843159388 0.00016261052791030005 0.00014486951087141022 0.00012418851649338431 0.00010243821352435248 8.130526395515009e-5 6.209425824669217e-5 4.563103655871602e-5 3.226601538323935e-5 2.195363553949232e-5 1.4372875872212212e-5 9.054344429463147e-6 5.488408884873085e-6 3.2011941726360218e-6 1.796609484026531e-6 9.702227851045628e-7 5.041564903631147e-7 2.5207824518155823e-7 1.2127784813807057e-7 5.614404637582897e-8 2.5009329473718887e-8 1.0719548603267791e-8 4.421066615948815e-9; 2.5786512967024106e-9 6.252332368429754e-9 1.4587054545710036e-8 3.274683024835643e-8 7.073706585518093e-8 1.4702829662483828e-7 2.9405659324967556e-7 5.658965268414476e-7 1.047898567947408e-6 1.8671429818508787e-6 3.201194172636025e-6 5.28107785564652e-6 8.383188543579237e-6 1.2804776690544102e-5 1.8819621967979263e-5 2.6614964625850903e-5 3.621737771785266e-5 4.742247573704209e-5 5.97485754192282e-5 7.243475543570533e-5 8.449724569034428e-5 9.484495147408411e-5 0.00010243821352435276 0.00010645985850340356 0.00010645985850340356 0.00010243821352435276 9.484495147408411e-5 8.449724569034428e-5 7.243475543570533e-5 5.97485754192282e-5 4.742247573704209e-5 3.621737771785269e-5 2.6614964625850903e-5 1.8819621967979263e-5 1.280477669054408e-5 8.38318854357925e-6 5.28107785564652e-6 3.2011941726360218e-6 1.8671429818508857e-6 1.047898567947408e-6 5.658965268414476e-7 2.9405659324967556e-7 1.4702829662483855e-7 7.073706585518093e-8 3.274683024835643e-8 1.4587054545710036e-8 6.252332368429754e-9 2.5786512967024106e-9; 1.447219108186102e-9 3.509002898489335e-9 8.186707562089107e-9 1.837853707810478e-8 3.9699835915600776e-8 8.251684149447703e-8 1.650336829889538e-7 3.1759868732480573e-7 5.881131864993532e-7 1.0478985679474042e-6 1.796609484026531e-6 2.963904733565132e-6 4.70490549199481e-6 7.186437936106125e-6 1.056215571129304e-5 1.493714385480706e-5 2.0326315988787573e-5 2.6614964625850903e-5 3.353275417431698e-5 4.065263197757515e-5 4.742247573704209e-5 5.322992925170177e-5 5.749150348884896e-5 5.97485754192282e-5 5.97485754192282e-5 5.749150348884896e-5 5.322992925170177e-5 4.742247573704209e-5 4.065263197757515e-5 3.353275417431698e-5 2.6614964625850903e-5 2.032631598878759e-5 1.493714385480706e-5 1.056215571129304e-5 7.186437936106112e-6 4.704905491994819e-6 2.963904733565132e-6 1.796609484026531e-6 1.047898567947408e-6 5.881131864993532e-7 3.1759868732480573e-7 1.650336829889538e-7 8.251684149447703e-8 3.9699835915600776e-8 1.837853707810478e-8 8.186707562089107e-9 3.509002898489335e-9 1.447219108186102e-9; 7.815415460537175e-10 1.8949663771573584e-9 4.421066615948807e-9 9.924958978900158e-9 2.143909720653547e-8 4.456155913937119e-8 8.912311827874205e-8 1.7151277765228381e-7 3.1759868732480573e-7 5.658965268414456e-7 9.702227851045645e-7 1.6005970863180064e-6 2.5407894985984374e-6 3.88089114041826e-6 5.7038795698395e-6 8.066503845809836e-6 1.0976817769746173e-5 1.43728758722122e-5 1.810868885892628e-5 2.1953635539492345e-5 2.5609553381088117e-5 2.8745751744424377e-5 3.104712912334606e-5 3.2266015383239325e-5 3.2266015383239325e-5 3.104712912334606e-5 2.8745751744424377e-5 2.5609553381088117e-5 2.1953635539492345e-5 1.810868885892628e-5 1.43728758722122e-5 1.0976817769746181e-5 8.066503845809836e-6 5.7038795698395e-6 3.880891140418253e-6 2.5407894985984416e-6 1.6005970863180064e-6 9.702227851045628e-7 5.658965268414476e-7 3.1759868732480573e-7 1.7151277765228381e-7 8.912311827874205e-8 4.4561559139371265e-8 2.143909720653547e-8 9.924958978900158e-9 4.421066615948807e-9 1.8949663771573584e-9 7.815415460537175e-10; 4.061121311317591e-10 9.846806452404627e-10 2.297317134763089e-9 5.1573025934047955e-9 1.1140389784842775e-8 2.315550573097756e-8 4.631101146195504e-8 8.912311827874205e-8 1.650336829889538e-7 2.9405659324967445e-7 5.041564903631155e-7 8.317176445578389e-7 1.320269463911624e-6 2.016625961452459e-6 2.963904733565124e-6 4.191594271789614e-6 5.703879569839505e-6 7.46857192740351e-6 9.409810983989606e-6 1.1407759139679001e-5 1.330748231292542e-5 1.4937143854807011e-5 1.6133007691619673e-5 1.6766377087158443e-5 1.6766377087158443e-5 1.6133007691619673e-5 1.4937143854807011e-5 1.330748231292542e-5 1.1407759139679001e-5 9.409810983989606e-6 7.46857192740351e-6 5.703879569839505e-6 4.191594271789614e-6 2.963904733565124e-6 2.016625961452459e-6 1.3202694639116285e-6 8.317176445578389e-7 5.041564903631147e-7 2.9405659324967556e-7 1.650336829889538e-7 8.912311827874205e-8 4.631101146195504e-8 2.31555057309776e-8 1.1140389784842775e-8 5.1573025934047955e-9 2.297317134763089e-9 9.846806452404627e-10 4.061121311317591e-10; 2.0305606556587994e-10 4.923403226202322e-10 1.1486585673815487e-9 2.578651296702407e-9 5.570194892421407e-9 1.1577752865488842e-8 2.31555057309776e-8 4.4561559139371265e-8 8.251684149447703e-8 1.4702829662483802e-7 2.5207824518155866e-7 4.1585882227892094e-7 6.601347319558153e-7 1.0083129807262346e-6 1.4819523667825673e-6 2.0957971358948143e-6 2.851939784919763e-6 3.734285963701767e-6 4.704905491994819e-6 5.703879569839521e-6 6.653741156462732e-6 7.46857192740353e-6 8.066503845809865e-6 8.38318854357925e-6 8.38318854357925e-6 8.066503845809865e-6 7.46857192740353e-6 6.653741156462732e-6 5.703879569839521e-6 4.704905491994819e-6 3.734285963701767e-6 2.851939784919763e-6 2.0957971358948143e-6 1.4819523667825673e-6 1.0083129807262312e-6 6.601347319558153e-7 4.1585882227892094e-7 2.5207824518155823e-7 1.4702829662483855e-7 8.251684149447703e-8 4.4561559139371265e-8 2.31555057309776e-8 1.1577752865488842e-8 5.570194892421407e-9 2.578651296702407e-9 1.1486585673815487e-9 4.923403226202322e-10 2.0305606556587994e-10; 9.769269325671449e-11 2.368707971446701e-10 5.526333269935996e-10 1.2406198723625213e-9 2.6798871508169387e-9 5.5701948924213965e-9 1.1140389784842775e-8 2.143909720653547e-8 3.9699835915600776e-8 7.073706585518068e-8 1.2127784813807075e-7 2.0007463578975112e-7 3.1759868732480456e-7 4.851113925522823e-7 7.129849462299381e-7 1.0083129807262293e-6 1.3721022212182733e-6 1.7966094840265248e-6 2.263586107365787e-6 2.7442044424365453e-6 3.2011941726360163e-6 3.5932189680530496e-6 3.88089114041826e-6 4.033251922904918e-6 4.033251922904918e-6 3.88089114041826e-6 3.5932189680530496e-6 3.2011941726360163e-6 2.7442044424365453e-6 2.263586107365787e-6 1.7966094840265248e-6 1.3721022212182733e-6 1.0083129807262293e-6 7.129849462299381e-7 4.851113925522823e-7 3.1759868732480573e-7 2.0007463578975112e-7 1.2127784813807057e-7 7.073706585518093e-8 3.9699835915600776e-8 2.143909720653547e-8 1.1140389784842775e-8 5.570194892421407e-9 2.6798871508169387e-9 1.2406198723625213e-9 5.526333269935996e-10 2.368707971446701e-10 9.769269325671449e-11; 4.522559713081575e-11 1.0965634057779147e-10 2.5583461131528484e-10 5.743292836907731e-10 1.2406198723625213e-9 2.5786512967024023e-9 5.1573025934047955e-9 9.924958978900158e-9 1.837853707810478e-8 3.274683024835631e-8 5.6144046375829066e-8 9.262202292391026e-8 1.4702829662483752e-7 2.2457618550331592e-7 3.3006736597790707e-7 4.667857454627196e-7 6.351973746496115e-7 8.317176445578389e-7 1.0478985679474042e-6 1.2703947492992233e-6 1.481952366782562e-6 1.6634352891156783e-6 1.7966094840265274e-6 1.8671429818508787e-6 1.8671429818508787e-6 1.7966094840265274e-6 1.6634352891156783e-6 1.481952366782562e-6 1.2703947492992233e-6 1.0478985679474042e-6 8.317176445578389e-7 6.351973746496115e-7 4.667857454627196e-7 3.3006736597790707e-7 2.2457618550331592e-7 1.4702829662483802e-7 9.262202292391026e-8 5.614404637582897e-8 3.274683024835643e-8 1.837853707810478e-8 9.924958978900158e-9 5.1573025934047955e-9 2.578651296702407e-9 1.2406198723625213e-9 5.743292836907731e-10 2.5583461131528484e-10 1.0965634057779147e-10 4.522559713081575e-11; 2.0145713255487537e-11 4.884634662835752e-11 1.1396136363835936e-10 2.5583461131528484e-10 5.526333269935996e-10 1.1486585673815487e-9 2.297317134763089e-9 4.421066615948807e-9 8.186707562089107e-9 1.4587054545709983e-8 2.5009329473718933e-8 4.125842074723837e-8 6.549366049671273e-8 1.0003731789487575e-7 1.4702829662483776e-7 2.079294111394597e-7 2.8294826342072373e-7 3.704880916956411e-7 4.667857454627196e-7 5.658965268414476e-7 6.60134731955813e-7 7.40976183391281e-7 8.002985431590045e-7 8.317176445578389e-7 8.317176445578389e-7 8.002985431590045e-7 7.40976183391281e-7 6.60134731955813e-7 5.658965268414476e-7 4.667857454627196e-7 3.704880916956404e-7 2.8294826342072373e-7 2.079294111394597e-7 1.4702829662483776e-7 1.0003731789487539e-7 6.549366049671273e-8 4.125842074723837e-8 2.5009329473718887e-8 1.4587054545710036e-8 8.186707562089107e-9 4.421066615948807e-9 2.297317134763089e-9 1.1486585673815487e-9 5.526333269935996e-10 2.5583461131528484e-10 1.1396136363835936e-10 4.884634662835752e-11 2.0145713255487537e-11; 8.634895734275082e-12 2.0936618365757505e-11 4.884634662835752e-11 1.0965634057779147e-10 2.368707971446701e-10 4.923403226202322e-10 9.846806452404627e-10 1.8949663771573584e-9 3.509002898489335e-9 6.252332368429731e-9 1.0719548603267813e-8 1.7684266463795293e-8 2.8072023187914533e-8 4.287819441307118e-8 6.301956129538964e-8 8.912311827874256e-8 1.212778481380712e-7 1.587993436624031e-7 2.0007463578975186e-7 2.4255569627614246e-7 2.8294826342072426e-7 3.175986873248062e-7 3.430255553045695e-7 3.564924731149702e-7 3.564924731149702e-7 3.430255553045695e-7 3.175986873248062e-7 2.8294826342072426e-7 2.4255569627614246e-7 2.0007463578975186e-7 1.587993436624031e-7 1.212778481380712e-7 8.912311827874256e-8 6.301956129538964e-8 4.287819441307118e-8 2.807202318791463e-8 1.7684266463795293e-8 1.0719548603267791e-8 6.252332368429754e-9 3.509002898489335e-9 1.8949663771573584e-9 9.846806452404627e-10 4.923403226202322e-10 2.368707971446701e-10 1.0965634057779147e-10 4.884634662835752e-11 2.0936618365757505e-11 8.634895734275082e-12; 3.5612926136987482e-12 8.634895734275082e-12 2.0145713255487537e-11 4.522559713081575e-11 9.769269325671449e-11 2.0305606556587994e-10 4.061121311317591e-10 7.815415460537175e-10 1.447219108186102e-9 2.5786512967024023e-9 4.4210666159488215e-9 7.293527272855018e-9 1.15777528654888e-8 1.7684266463795293e-8 2.599117639243255e-8 3.67570741562095e-8 5.001865894743796e-8 6.549366049671296e-8 8.25168414944769e-8 1.0003731789487592e-7 1.166964363656801e-7 1.3098732099342573e-7 1.4147413171036213e-7 1.4702829662483802e-7 1.4702829662483802e-7 1.4147413171036213e-7 1.3098732099342573e-7 1.166964363656801e-7 1.0003731789487592e-7 8.25168414944769e-8 6.549366049671285e-8 5.001865894743796e-8 3.67570741562095e-8 2.599117639243255e-8 1.768426646379523e-8 1.15777528654888e-8 7.293527272855018e-9 4.421066615948815e-9 2.5786512967024106e-9 1.447219108186102e-9 7.815415460537175e-10 4.061121311317591e-10 2.0305606556587994e-10 9.769269325671449e-11 4.522559713081575e-11 2.0145713255487537e-11 8.634895734275082e-12 3.5612926136987482e-12])

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

julia

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.

julia

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.

julia

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.

julia

G = SingleStokesGain() do x
    lg = x.lg
    gp = x.gp
    return exp(lg + 1im * gp)
end

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.

julia

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: UnstructredDomain(
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)
 (U = 6.063457521777771e9, V = 1.5634511359999983e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 3.9335861404444404e9, V = 5.113178993777773e9, Ti = 3.3499998450279236, Fr = 2.27070703125e11)
 (U = 416648.93055555515, V = -1.5970948472222206e6, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 2.578938929777775e9, V = -4.734788238222218e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = -3.735105663999996e9, V = -5.186825713777772e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (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)
 (U = 3.797641537777774e8, V = -1.2241308906666653e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 3.735526620444441e9, V = 5.185227192888883e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 6.314048611555549e9, V = 4.520333351111106e8, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 5.934280931555549e9, V = 1.6761668835555537e9, Ti = 3.6833333373069763, Fr = 2.27070703125e11)
 (U = 364284.04464285675, V = -1.591376535714284e6, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = -3.5324296035555515e9, V = -5.248264732444439e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 2.246733795555553e9, V = -3.4730721351111073e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 2.7040352142222195e9, V = -4.690125752888884e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = -2.687901513142854e9, V = 4.694663167999995e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = -2.2408562468571405e9, V = 3.476569380571425e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 4.573011991111106e8, V = -1.2170540408888876e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 3.561408036571425e9, V = 5.238641298285708e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 5.779150734222217e9, V = 1.7751998862222204e9, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 6.236464440888882e9, V = 5.581377528888882e8, Ti = 3.98333340883255, Fr = 2.27070703125e11)
 (U = 292810.0842803027, V = -1.5850529772727257e6, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 2.81236712533333e9, V = -4.643490716444439e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 2.280368668444442e9, V = -3.43479938133333e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -3.307854179555552e9, V = -5.306092316444439e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -2.812399111529409e9, V = 4.641748931764702e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = -2.280159088941174e9, V = 3.43308967905882e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 5.3200019644444394e8, V = -1.2086904924444432e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (U = 3.307383679999996e9, V = 5.304687194352936e9, Ti = 4.283333480358124, Fr = 2.27070703125e11)
 (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.

julia

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

julia

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.

julia

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.

julia

intopt = instrumentmodel(post, xopt)

126-element Comrade.SiteArray{ComplexF64, 1, Vector{ComplexF64}, Vector{Comrade.IntegrationTime{Int64, Float64}}, Vector{Comrade.FrequencyChannel{Float64, Int64}}, Vector{Symbol}}:
   1.0105689540450347 + 0.0im
 -0.14457050870382424 - 1.0063044637273695im
   1.0076239550859565 + 0.0im
   -0.598161238472178 - 0.8357427050939629im
   0.6337864829468699 + 0.46394989296291944im
 -0.35005000805474873 - 1.0535798509126475im
   1.0041902597000445 + 0.0im
  -0.6401814807574002 - 0.8071113644369282im
   0.6966429026830845 + 0.6102216672538459im
  -0.5440754768126357 - 0.9095494319218614im
   1.0198356987640462 + 0.0im
  -0.6600511085979225 - 0.7765235656836619im
   0.6114377250138789 + 0.5936965909592231im
  -0.6875754585792183 - 0.765444325328169im
   0.9976618122362416 + 0.0im
  -0.7125718001496244 - 0.7535365590725749im
   0.5698773094265052 + 0.6292655578986104im
  -0.7807952951743828 - 0.5798122866034535im
   1.0428320301569498 + 0.0im
  0.13773763089593813 + 0.5824003984111575im
   0.8497072230254785 - 0.2775562967673232im
   1.0224910900370412 + 0.0im
  -0.7527830935773998 - 0.6864275154869833im
  0.35588710745758007 + 0.3973037163853282im
  -0.9717217015947958 - 0.24459642388424022im
   1.0110543703522055 + 0.0im
  -0.7743419428015395 - 0.6615022136219695im
  0.32151423936375023 + 0.35792023943365636im
  -1.0299908206064807 - 0.05065042988392803im
    1.023032961043191 + 0.0im
  -0.7934104463980683 - 0.6371451203238113im
   0.3223829992843362 + 0.33894212714188476im
   -1.018377151864932 + 0.16229298352133423im
   1.0083003006025024 + 0.0im
  -0.8023414630643779 - 0.6392085134122582im
   0.9047463891857779 + 0.3409835404792646im
   0.2987429554022417 + 0.2811781921562723im
  -0.9774701586837249 + 0.40111051601892783im
   0.9856052039411173 + 0.0im
  -0.8008956462468615 - 0.6796952279081022im
   0.9073107350789896 + 0.19633165942199232im
  0.35044385266049655 + 0.2759507819946495im
  -0.8304264676951753 + 0.5854235015156253im
   1.0420276896115699 + 0.0im
  -0.7767595283940503 - 0.6347209190981523im
    0.954812629376856 + 0.03750543712681113im
  0.45952907798417836 + 0.2707102790390364im
  -0.7258187241479234 + 0.7900896992932007im
   0.9737866089419885 + 0.0im
  -0.7631976872206655 - 0.7390936050387897im
   0.9568329555086622 - 0.13184528231445453im
   0.4005095993297889 - 0.9588293331070786im
  0.42794719698606926 + 0.20170842968218983im
  -0.5330951957677562 + 0.9659945844695722im
    1.029357569029594 + 0.0im
  -0.7572196867605288 - 0.6598933546913142im
   0.9190818474373834 - 0.3268227858773617im
   0.5851907699389712 - 0.8370682176534688im
  0.43864948288576916 + 0.12563581339245863im
 -0.27621385768349943 + 1.0657788804320536im
   -0.603675956152816 - 0.8148097305929687im
   0.9585100371324182 + 0.0im
  -0.7758256125317947 - 0.7325485967864123im
   0.8511242872096741 - 0.482701738262474im
   0.6962071319698152 - 0.7244038069933002im
   0.4230655863738308 + 0.03617252445853528im
 -0.09392216772566424 + 1.2237630652707434im
  -0.7416583271826658 - 0.6910168529574343im
   0.8750354187387177 + 0.0im
  -0.7886594160592241 - 0.8763447360514823im
   0.7633283287226331 - 0.5998207987003922im
   0.8158148893512529 - 0.6108049130534648im
  0.42423502026299226 - 0.057923045088310365im
  0.09210390832307583 + 0.909829608774863im
  -0.8810081820973308 - 0.5290761139438159im
     1.01186285884117 + 0.0im
   0.6239261638529487 - 0.7603009891626784im
   0.8839839419468818 - 0.49864954739769046im
   0.3772847132882303 - 0.19940146125293376im
  -0.9608745710345306 - 0.33876338043364823im
   1.0628051175958397 + 0.0im
  -0.5969603445830516 - 0.7788913794607079im
   0.3868755349771389 - 0.9094667579126873im
   0.9567614770894226 - 0.38001787792259045im
   0.1431381287055951 - 0.42437438502022273im
   1.0134750753609545 + 0.0im
  -0.5707211581867707 - 0.8486921721848754im
  0.19658496824689453 - 0.9564374350946024im
    0.966499055968478 - 0.30988880441168265im
  -0.1325906223240408 - 0.3582226974646178im
  -1.0031697618390405 - 0.11622860357697629im
    1.084944804307568 + 0.0im
 -0.06761637931658664 - 0.9892729004625296im
   0.9872028041843197 - 0.24492844199192085im
  -0.2923709936744424 - 0.2873823169689105im
  -1.0159901437173635 - 0.0033798977330092273im
   1.1129751565043406 + 0.0im
 -0.47749781152552906 - 0.8005586839344486im
 -0.34356320877197555 - 0.9008305786132667im
   0.9952211026567963 - 0.20142568328282298im
  -0.3174402580154428 - 0.1507138648870082im
  -1.0068571894784182 + 0.1142579937523862im
    1.035167630928549 + 0.0im
 -0.42572104843329595 - 0.9086594794138803im
  -0.5591357072856167 - 0.782438088042471im
   1.0024880610818105 - 0.1699649456695587im
 -0.38652951955841514 - 0.04467473319463661im
  -1.0116570553579385 + 0.14496308273962355im
   1.0097328931581713 + 0.0im
  -0.4002765029243772 - 0.9519101158179379im
  -0.7228423683579105 - 0.6044230932194378im
   1.0036894580583515 - 0.15718251894589141im
 -0.49895023319205506 + 0.04927893622713054im
   -1.007071910221982 + 0.17660358251543146im
   0.9835839040327009 + 0.0im
   -0.328718032678165 - 0.991078568978706im
   -0.820819847948041 - 0.4097943043533259im
   1.0007969546408992 - 0.16611424306872777im
  -0.4772778400465269 + 0.1467467671289736im
  -1.0020113862861302 + 0.14588011215834232im
   1.0075878613428388 + 0.0im
 -0.26433482251598567 - 0.9954975539426171im
  -0.7991341071114537 - 0.28651995269159397im
   1.0103146366696825 - 0.1499309893280345im
   -0.434455282355566 + 0.21563473080044518im
  -1.0007612444938105 + 0.13139929449276155im

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.

julia

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

julia

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.

julia

using AdvancedHMC
chain = sample(rng, post, NUTS(0.8), 1_000; n_adapts = 500, initial_params = xopt)

PosteriorSamples
  Samples size: (1000,)
  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 = [-0.00320475 0.027415 … 0.0137095 0.0305493; 0.0249811 0.0775529 … 0.0358077 0.00517739; … ; 0.0130793 0.0272808 … 0.00111261 0.01094; 0.00665214 0.00737358 … -0.0297941 -0.0135941], hyperparams = 34.8438), σimg = 1.97018, fg = 0.411961) │ (lg = [0.0287117, 0.00930652, 0.00565042, 0.0282176, -0.35023, 0.133668, 0.00378885, 0.0290853, -0.169728, 0.0939602  …  -0.050907, 0.0144701, -0.846221, 0.0129815, 0.00605535, 0.0301058, -0.115477, 0.021683, -0.894874, 0.0117271], gp = [0.0, -0.836533, 0.0, -2.19194, 0.908799, -1.00233, 0.0, -2.24138, 0.968794, -1.21547  …  -2.40306, -0.130955, 0.471738, 0.868588, 0.0, -1.83138, -2.52837, -0.102505, 1.99917, 0.661261]) │
└──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
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 = [0.578748 0.590178 … 0.578015 0.569989; 0.610492 0.652271 … 0.64756 0.616641; … ; 0.597905 0.620046 … 0.616718 0.604602; 0.541334 0.577731 … 0.567595 0.566789], hyperparams = 23.0362), σimg = 0.218083, fg = 0.0520185) │ (lg = [0.144574, 0.148522, 0.0203718, 0.0209889, 0.077473, 0.0607939, 0.0186947, 0.0204404, 0.0701576, 0.059969  …  0.0469703, 0.0189855, 0.0595455, 0.0183912, 0.0169146, 0.0201348, 0.0492678, 0.0177603, 0.0622357, 0.017804], gp = [0.0, 0.411236, 0.0, 0.0167036, 0.193324, 0.413993, 0.0, 0.0167328, 0.189528, 0.418665  …  0.192142, 0.23615, 2.96832, 2.80852, 0.0, 0.0190207, 0.189866, 0.235292, 2.14799, 2.87113]) │
└──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

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

julia

chain = chain[501:end]

PosteriorSamples
  Samples size: (500,)
  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 = [-0.00994655 0.0111828 … 0.0371309 0.0335698; 0.0151948 0.0565936 … 0.0515289 0.016179; … ; 0.0286185 0.0667015 … 0.0412181 0.00880748; 0.0366191 0.0431943 … -0.00700657 0.00559568], hyperparams = 36.2503), σimg = 1.96233, fg = 0.417284) │ (lg = [0.0221051, 0.0180831, 0.0075834, 0.0266355, -0.344337, 0.136293, 0.0038359, 0.0295135, -0.167692, 0.0992928  …  -0.0562578, 0.0148357, -0.857104, 0.0124816, 0.00606694, 0.0296154, -0.12008, 0.0210091, -0.903442, 0.0118535], gp = [0.0, -1.01715, 0.0, -2.19152, 0.853047, -1.18408, 0.0, -2.24126, 0.915342, -1.40053  …  -2.44956, -0.0829902, 1.05548, 0.425921, 0.0, -1.83075, -2.57566, -0.0533579, 2.70602, 0.150474]) │
└──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
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 = [0.606134 0.602088 … 0.569351 0.577725; 0.650066 0.65063 … 0.670033 0.616654; … ; 0.581942 0.623896 … 0.607798 0.574299; 0.518496 0.589882 … 0.552723 0.571374], hyperparams = 22.8446), σimg = 0.217592, fg = 0.0571483) │ (lg = [0.137851, 0.138195, 0.0203663, 0.0215301, 0.0821456, 0.0616225, 0.0190938, 0.0205183, 0.0738983, 0.0623049  …  0.0477396, 0.0187713, 0.057439, 0.0188053, 0.0168885, 0.0204477, 0.04933, 0.0176551, 0.0613634, 0.0181667], gp = [0.0, 0.375708, 0.0, 0.0163525, 0.166584, 0.37866, 0.0, 0.0172682, 0.162651, 0.382598  …  0.162818, 0.232455, 2.83267, 2.93136, 0.0, 0.0186529, 0.159728, 0.231389, 1.06744, 2.96137]) │
└──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘

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

julia

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.

julia

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  -1.02±0.38     missing
 1.22 hr │ 227.070703125 GHz │ 0.0±0.0  -2.192±0.016      missing      missing   0.85±0.17  -1.18±0.38     missing
 1.52 hr │ 227.070703125 GHz │ 0.0±0.0  -2.241±0.017      missing      missing   0.92±0.16   -1.4±0.38     missing
 1.82 hr │ 227.070703125 GHz │ 0.0±0.0  -2.275±0.018      missing      missing   0.95±0.16  -1.59±0.38     missing
 2.12 hr │ 227.070703125 GHz │ 0.0±0.0  -2.328±0.017      missing      missing   1.01±0.16  -1.79±0.39     missing
 2.45 hr │ 227.070703125 GHz │ missing       0.0±0.0      missing      missing   1.52±0.16    0.4±0.39     missing
 2.75 hr │ 227.070703125 GHz │ 0.0±0.0  -2.403±0.018      missing      missing   1.01±0.16  -2.12±0.56     missing
 3.05 hr │ 227.070703125 GHz │ 0.0±0.0  -2.434±0.017      missing      missing   1.02±0.16   -2.24±0.8     missing
 3.35 hr │ 227.070703125 GHz │ 0.0±0.0  -2.466±0.018      missing      missing    1.0±0.16    -2.1±1.4     missing
 3.68 hr │ 227.070703125 GHz │ 0.0±0.0  -2.468±0.018    0.63±0.17      missing   0.97±0.16    -1.9±1.9     missing
 3.98 hr │ 227.070703125 GHz │ 0.0±0.0  -2.439±0.019     0.5±0.17      missing   0.89±0.16   -0.82±2.7     missing
 4.28 hr │ 227.070703125 GHz │ 0.0±0.0  -2.457±0.019    0.33±0.17      missing   0.77±0.16     0.3±2.8     missing
 4.58 hr │ 227.070703125 GHz │ 0.0±0.0  -2.373±0.021    0.16±0.17    -1.14±0.2   0.69±0.16     1.5±2.4     missing
 4.92 hr │ 227.070703125 GHz │ 0.0±0.0   -2.424±0.02  -0.043±0.17   -0.96±0.21   0.54±0.16     2.0±1.7  -2.21±0.21
 5.18 hr │ 227.070703125 GHz │ 0.0±0.0  -2.385±0.017   -0.22±0.17   -0.84±0.21   0.35±0.16   2.27±0.97  -2.42±0.21
 5.45 hr │ 227.070703125 GHz │ 0.0±0.0   -2.302±0.02   -0.38±0.17   -0.69±0.22   0.13±0.16    2.2±0.57  -2.52±0.83
 5.72 hr │ 227.070703125 GHz │ 0.0±0.0       missing   -0.61±0.17   -0.56±0.22  -0.22±0.16     missing    -2.2±1.7
 6.05 hr │ 227.070703125 GHz │ 0.0±0.0  -2.225±0.019    -0.9±0.17   -0.41±0.23  -0.99±0.16     missing     missing
 6.32 hr │ 227.070703125 GHz │ 0.0±0.0  -2.162±0.019   -1.11±0.17   -0.33±0.23  -1.68±0.16     missing    -1.2±2.7
 6.58 hr │ 227.070703125 GHz │ 0.0±0.0       missing   -1.38±0.17   -0.24±0.23  -2.13±0.16     missing   -0.21±3.0
 6.85 hr │ 227.070703125 GHz │ 0.0±0.0  -2.108±0.018   -1.68±0.17   -0.18±0.23  -2.47±0.15     missing    0.82±2.8
 7.18 hr │ 227.070703125 GHz │ 0.0±0.0    -2.01±0.02   -1.95±0.17   -0.12±0.23    -2.6±1.1     missing    0.86±2.8
 7.45 hr │ 227.070703125 GHz │ 0.0±0.0  -1.968±0.018   -2.21±0.16  -0.089±0.23    -2.0±2.2     missing    0.97±2.8
 7.72 hr │ 227.070703125 GHz │ 0.0±0.0  -1.891±0.019   -2.45±0.16  -0.083±0.23     1.1±2.8     missing    0.43±2.9
 7.98 hr │ 227.070703125 GHz │ 0.0±0.0  -1.831±0.019   -2.58±0.16  -0.053±0.23     2.7±1.1     missing    0.15±3.0
─────────┴───────────────────┴─────────────────────────────────────────────────────────────────────────────────────

Now let's plot the phase curves

julia

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

and now the amplitude curves

julia

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

Finally let's construct some representative image reconstructions.

julia

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

mimg = mean(imgs)
simg = std(imgs)
fig = Figure(; resolution = (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.

This page was generated using Literate.jl.

Stokes I Simultaneous Image and Instrument Modeling ​

Load the Data ​

Optimization and Sampling ​

Stokes I Simultaneous Image and Instrument Modeling

Load the Data

Optimization and Sampling