Modeling the Power Spectrum of an AGN with Markov Random Field Expansion
In this tutorial, we will reconstruct VLBA data observations of the AGN XYZ using closures. However we will also model the power spectrum of the AGN using a Markov Random Field expansion Where we can fit multiple scales of the power spectrum simultaneously up to some order.
In this tutorial, we will do closure-only modeling of the AGN DA 193 observed with the VLBA at 15 GHz with the Mojave AGN project. Unlike the previous tutorials, we will constrain the power spectrum slope and using a Markov random field expansion. This will allow us to model more complex and multi-scale processes in AGN, which is expected to be common in black hole jets.
To get started, we will load Comrade
julia
using Comrade
using LinearAlgebra
LinearAlgebra.BLAS.set_num_threads(1)Pyehtim loads eht-imaging using PythonCall this is necessary to load uvfits files currently.
julia
using Pyehtim
using NonuniformFFTs CondaPkg Found dependencies: /home/runner/.julia/packages/CondaPkg/8GjrP/CondaPkg.toml
CondaPkg Found dependencies: /home/runner/.julia/packages/PythonCall/83z4q/CondaPkg.toml
CondaPkg Found dependencies: /home/runner/.julia/packages/Pyehtim/bQtHC/CondaPkg.toml
CondaPkg Resolving changes
+ ehtim (pip)
+ libstdcxx
+ libstdcxx-ng
+ numpy
+ numpy (pip)
+ openssl
+ pandas
+ python
+ setuptools (pip)
+ uv
CondaPkg Initialising pixi
│ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
│ init
│ --format pixi
└ /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/FitPS/.CondaPkg
✔ Created /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/FitPS/.CondaPkg/pixi.toml
CondaPkg Wrote /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/FitPS/.CondaPkg/pixi.toml
│ [dependencies]
│ openssl = ">=3, <3.6"
│ libstdcxx = ">=3.4,<15.0"
│ uv = ">=0.4"
│ libstdcxx-ng = ">=3.4,<15.0"
│ pandas = "<2"
│ numpy = ">=1.24, <2.0"
│
│ [dependencies.python]
│ channel = "conda-forge"
│ build = "*cp*"
│ version = ">=3.10,!=3.14.0,!=3.14.1,<4, >=3.6,<=3.12"
│
│ [project]
│ name = ".CondaPkg"
│ platforms = ["linux-64"]
│ channels = ["conda-forge"]
│ channel-priority = "strict"
│ description = "automatically generated by CondaPkg.jl"
│
│ [pypi-dependencies]
│ ehtim = ">=1.2.10, <2.0"
│ numpy = ">=1.24, <2.0"
└ setuptools = "*"
CondaPkg Installing packages
│ /home/runner/.julia/artifacts/cefba4912c2b400756d043a2563ef77a0088866b/bin/pixi
│ install
└ --manifest-path /home/runner/work/Comrade.jl/Comrade.jl/examples/advanced/FitPS/.CondaPkg/pixi.toml
✔ The default environment has been installed.For reproducibility we use a stable random number genreator
julia
using StableRNGs
rng = StableRNG(11)StableRNGs.LehmerRNG(state=0x00000000000000000000000000000017)Load the Data
For this tutorial we will image publicly available VLBA data of the AGN 1308+326 observed on 2021/03/19 at 43 GHz as part of the Boston University blazar monitoring program.
julia
file = Base.download("https://www.bu.edu/blazars/VLBA_GLAST/1308/1308+326Q.2021-03-19.UVP.gz")
obs0 = ehtim.obsdata.load_uvfits(file)Python: <ehtim.obsdata.Obsdata object at 0x7efe824115d0>Now we do some minor preprocessing:
Scan average the data since the data have been preprocessed so that the gain phases are coherent.
Add 0.5% systematic noise to deal with calibration issues such as leakage.
julia
obs = scan_average(obs0).add_fractional_noise(0.005)Python: <ehtim.obsdata.Obsdata object at 0x7efe925be410>For this tutorial we will only use closure quantities to reconstruct the image however, polarized or complex visibilities can also be used with instrumental models following the other tutorials.
julia
dlcamp, dcphase = extract_table(obs, LogClosureAmplitudes(; snrcut = 3), ClosurePhases(; snrcut = 3))(EHTObservationTable{Comrade.EHTLogClosureAmplitudeDatum{:I}}
source: 1308+326
mjd: 59293
bandwidth: 4.86e8
sites: [:BR, :FD, :HN, :KP, :LA, :MK, :NL, :OV, :PT, :SC]
nsamples: 291, EHTObservationTable{Comrade.EHTClosurePhaseDatum{:I}}
source: 1308+326
mjd: 59293
bandwidth: 4.86e8
sites: [:BR, :FD, :HN, :KP, :LA, :MK, :NL, :OV, :PT, :SC]
nsamples: 300)Build the Model/Posterior
Most of the model building here will look very similar to the previous Imaging a Black Hole using only Closure Quantities tutorial. However, we will be utilizing a more complex image prior. Specifically, VLBIImagePriors provides a basic framework for building stationary Gaussian random fields with cyclic boundary conditions. To define the random field we just need to define a spectral model. For this work we will use a Markovian spectral model. Namely, our power spectrum will be modeled as
where σ is the marginal variance of the image, ρs are the coefficients of the Markovian expansion, and k is the norm of the spatial wavenumber.
julia
using VLBIImagePriors ## Defines the `MarkovPS` power spectrum model and `StationaryRandomField`
using DistributionsWe use the @sky macro to define the model and prior in a single block. Each name ~ dist line contributes an entry to the prior; everything else is the model body. Metadata fields (mimg, pl, cprior, ρmax) are passed as keyword arguments and are in scope inside both the prior expressions and the body.
julia
@sky function sky(grid; mimg, pl, cprior, ρmax)
c ~ cprior
ρs ~ ntuple(Returns(Uniform(0.01, ρmax)), 3)
σimg ~ Exponential(2.0)
fb ~ Uniform(0.0, 1.0)
# Apply the GMRF fluctuations to the image
x = genfield(StationaryRandomField(MarkovPS(ρs), pl), c)
x .= σimg .* x
fbn = fb / length(mimg)
mb = mimg .* (1 - fb) .+ fbn
rast = apply_fluctuations(CenteredLR(), mb, x)
m = ContinuousImage(rast, BSplinePulse{3}())
return m
endsky (generic function with 1 method)For this tutorial we decided to image a very compact AGN. Thus, we will use a small FOV for a 15 GHz observation. Namely, we will use a 5000 μas FOV with 64x64 pixels.
julia
nx = 64
ny = 64
fovx = μas2rad(1_000)
fovy = fovx * ny / nx
grid = imagepixels(fovx, fovy, nx, ny, μas2rad(150.0), -μas2rad(150.0))RectiGrid(
executor: ComradeBase.Serial()
Dimensions:
(↓ X Sampled{Float64} LinRange{Float64}(-3.1134128583753017e-9, 1.6589718150466935e-9, 64) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-1.6589718150466935e-9, 3.1134128583753017e-9, 64) ForwardOrdered Regular Points)
)Since we are using a Gaussian Markov random field prior we need to first specify our mean image. For this work we will use a symmetric Gaussian with a FWHM equal to the approximate beamsize of the array. This models the fact that we expect the AGN core to be compact.
julia
fwhmfac = 2 * sqrt(2 * log(2))
mpr = modify(TBlob(3.0), Stretch(beamsize(dlcamp) / 4 / fwhmfac))
imgpr = intensitymap(mpr, grid)
mimg = imgpr ./ sum(imgpr)┌ 64×64 IntensityMap{Float64, 2} ┐
├────────────────────────────────┴─────────────────────────────────────── dims ┐
↓ X Sampled{Float64} LinRange{Float64}(-3.1134128583753017e-9, 1.6589718150466935e-9, 64) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-1.6589718150466935e-9, 3.1134128583753017e-9, 64) ForwardOrdered Regular Points
└──────────────────────────────────────────────────────────────────────────────┘
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1.12871e-9 2.78813e-7 3.26032e-7 3.82163e-7 4.48941e-7 5.28403e-7 6.22911e-7 7.35153e-7 8.6812e-7 1.02502e-6 1.20914e-6 1.42356e-6 1.67075e-6 1.952e-6 2.26672e-6 2.61147e-6 2.97919e-6 3.35842e-6 3.73308e-6 4.08305e-6 4.38575e-6 4.61866e-6 4.76258e-6 4.80485e-6 4.74164e-6 4.57865e-6 4.3301e-6 4.0161e-6 3.65942e-6 3.28236e-6 2.9043e-6 2.5404e-6 2.20121e-6 1.89302e-6 1.6186e-6 1.37812e-6 1.16999e-6 9.9157e-7 8.39716e-7 7.11146e-7 6.02681e-7 5.11388e-7 4.3464e-7 3.70145e-7 3.15926e-7 2.70305e-7 2.31867e-7 1.99425e-7 1.71991e-7 1.48739e-7 1.28988e-7 1.12168e-7 9.78078e-8 8.55166e-8 7.49686e-8 6.5893e-8 5.80639e-8 5.12925e-8 4.5421e-8 4.03169e-8 3.58689e-8 3.19832e-8 2.85806e-8 2.55942e-8 2.29669e-8
1.20446e-9 2.50565e-7 2.9103e-7 3.38663e-7 3.94735e-7 4.60701e-7 5.38201e-7 6.29042e-7 7.35153e-7 8.58505e-7 1.00097e-6 1.16412e-6 1.34895e-6 1.55547e-6 1.78232e-6 2.02624e-6 2.28164e-6 2.5404e-6 2.79183e-6 3.02321e-6 3.22081e-6 3.37133e-6 3.46372e-6 3.49077e-6 3.45031e-6 3.34557e-6 3.18465e-6 2.97919e-6 2.74272e-6 2.48886e-6 2.22999e-6 1.97632e-6 1.73545e-6 1.51247e-6 1.31023e-6 1.12978e-6 9.70866e-7 8.32362e-7 7.12614e-7 6.09715e-7 5.21693e-7 4.4664e-7 3.82778e-7 3.28505e-7 2.82402e-7 2.43232e-7 2.09932e-7 1.81593e-7 1.57442e-7 1.36828e-7 1.19199e-7 1.04095e-7 9.11253e-8 7.99654e-8 7.03408e-8 6.20216e-8 5.4814e-8 4.85553e-8 4.31081e-8 3.83562e-8 3.42016e-8 3.05611e-8 2.7364e-8 2.45504e-8 2.20688e-8
1.28021e-9 2.24699e-7 2.59278e-7 2.99596e-7 3.46573e-7 4.01238e-7 4.64716e-7 5.38201e-7 6.22911e-7 7.20018e-7 8.30538e-7 9.55179e-7 1.09415e-6 1.24693e-6 1.412e-6 1.5866e-6 1.7665e-6 1.94599e-6 2.11793e-6 2.27416e-6 2.40616e-6 2.50588e-6 2.56673e-6 2.5845e-6 2.55791e-6 2.48886e-6 2.3821e-6 2.24458e-6 2.08453e-6 1.91045e-6 1.73035e-6 1.5511e-6 1.37812e-6 1.21533e-6 1.06522e-6 9.29102e-7 8.07321e-7 6.99553e-7 6.05013e-7 5.22645e-7 4.51259e-7 3.89637e-7 3.36597e-7 2.9103e-7 2.51931e-7 2.18398e-7 1.8964e-7 1.64964e-7 1.43776e-7 1.25561e-7 1.09881e-7 9.6364e-8 8.46911e-8 7.45929e-8 6.58406e-8 5.824e-8 5.16264e-8 4.58601e-8 4.08222e-8 3.64118e-8 3.25429e-8 2.9142e-8 2.61466e-8 2.35031e-8 2.11656e-8
1.35596e-9 2.01166e-7 2.3065e-7 2.64711e-7 3.04011e-7 3.49267e-7 4.01238e-7 4.60701e-7 5.28403e-7 6.05013e-7 6.91031e-7 7.86689e-7 8.9182e-7 1.00572e-6 1.12698e-6 1.25339e-6 1.38183e-6 1.50827e-6 1.62791e-6 1.73545e-6 1.82549e-6 1.89302e-6 1.93404e-6 1.94599e-6 1.92811e-6 1.88153e-6 1.80913e-6 1.71517e-6 1.60478e-6 1.48336e-6 1.35616e-6 1.22783e-6 1.10223e-6 9.82292e-7 8.70061e-7 7.66787e-7 6.73059e-7 5.88952e-7 5.1417e-7 4.48172e-7 3.9027e-7 3.39703e-7 2.95698e-7 2.57502e-7 2.24407e-7 1.95765e-7 1.7099e-7 1.49563e-7 1.31025e-7 1.14977e-7 1.01073e-7 8.90132e-8 7.85392e-8 6.94296e-8 6.14947e-8 5.45716e-8 4.85211e-8 4.32241e-8 3.85784e-8 3.44965e-8 3.09036e-8 2.77353e-8 2.49364e-8 2.24592e-8 2.02629e-8
1.43172e-9 1.7987e-7 2.04969e-7 2.33709e-7 2.66558e-7 3.04011e-7 3.46573e-7 3.94735e-7 4.48941e-7 5.09545e-7 5.76747e-7 6.50529e-7 7.30566e-7 8.16142e-7 9.06065e-7 9.98608e-7 1.09148e-6 1.18184e-6 1.26645e-6 1.34179e-6 1.40437e-6 1.45104e-6 1.47926e-6 1.48747e-6 1.47519e-6 1.44311e-6 1.39304e-6 1.32763e-6 1.25016e-6 1.16412e-6 1.07301e-6 9.79992e-7 8.87808e-7 7.98633e-7 7.14087e-7 6.35257e-7 5.62776e-7 4.969e-7 4.37598e-7 3.84632e-7 3.37628e-7 2.96128e-7 2.59636e-7 2.27648e-7 1.99673e-7 1.75248e-7 1.53947e-7 1.3538e-7 1.19199e-7 1.05096e-7 9.2798e-8 8.20664e-8 7.26933e-8 6.44979e-8 5.73236e-8 5.10349e-8 4.55146e-8 4.06617e-8 3.6389e-8 3.26212e-8 2.92932e-8 2.6349e-8 2.37401e-8 2.14245e-8 1.93659e-8
1.50747e-9 1.60684e-7 1.82028e-7 2.06259e-7 2.33709e-7 2.64711e-7 2.99596e-7 3.38663e-7 3.82163e-7 4.30257e-7 4.8298e-7 5.40191e-7 6.0152e-7 6.66321e-7 7.3362e-7 8.02093e-7 8.70061e-7 9.35527e-7 9.96254e-7 1.04989e-6 1.09415e-6 1.12698e-6 1.14677e-6 1.15251e-6 1.14391e-6 1.12141e-6 1.08615e-6 1.03985e-6 9.846e-7 9.22739e-7 8.56599e-7 7.8838e-7 7.20018e-7 6.53125e-7 5.88952e-7 5.28403e-7 4.72069e-7 4.20267e-7 3.73099e-7 3.30502e-7 2.92293e-7 2.58211e-7 2.27945e-7 2.01166e-7 1.77538e-7 1.56734e-7 1.38446e-7 1.22384e-7 1.08288e-7 9.5919e-8 8.50649e-8 7.55371e-8 6.71688e-8 5.98135e-8 5.33426e-8 4.76439e-8 4.26197e-8 3.81846e-8 3.42645e-8 3.07949e-8 2.77197e-8 2.49903e-8 2.25642e-8 2.04045e-8 1.84792e-8
1.58322e-9 1.43463e-7 1.61601e-7 1.82028e-7 2.04969e-7 2.3065e-7 2.59278e-7 2.9103e-7 3.26032e-7 3.64333e-7 4.05881e-7 4.50485e-7 4.97788e-7 5.47237e-7 5.98055e-7 6.49237e-7 6.99553e-7 7.47583e-7 7.91777e-7 8.30538e-7 8.62333e-7 8.85811e-7 8.99921e-7 9.04011e-7 8.97886e-7 8.81836e-7 8.56599e-7 8.23296e-7 7.83322e-7 7.38233e-7 6.89625e-7 6.39028e-7 5.87827e-7 5.3721e-7 4.88135e-7 4.41334e-7 3.97318e-7 3.5641e-7 3.18768e-7 2.84423e-7 2.53308e-7 2.25285e-7 2.00169e-7 1.77748e-7 1.57798e-7 1.40091e-7 1.24404e-7 1.10528e-7 9.82651e-8 8.74349e-8 7.78725e-8 6.94296e-8 6.19734e-8 5.53856e-8 4.95616e-8 4.4409e-8 3.98462e-8 3.58018e-8 3.22132e-8 2.90252e-8 2.61898e-8 2.36649e-8 2.14137e-8 1.94037e-8 1.76069e-8
1.65897e-9 1.28052e-7 1.43463e-7 1.60684e-7 1.7987e-7 2.01166e-7 2.24699e-7 2.50565e-7 2.78813e-7 3.09431e-7 3.42323e-7 3.77291e-7 4.14015e-7 4.52035e-7 4.90742e-7 5.29372e-7 5.67024e-7 6.02681e-7 6.35257e-7 6.63652e-7 6.86825e-7 7.03869e-7 7.14087e-7 7.17044e-7 7.12614e-7 7.00988e-7 6.82654e-7 6.58359e-7 6.29042e-7 5.95761e-7 5.59619e-7 5.21693e-7 4.8298e-7 4.44355e-7 4.0655e-7 3.70145e-7 3.3557e-7 3.03121e-7 2.72972e-7 2.45202e-7 2.19809e-7 1.96732e-7 1.75869e-7 1.57088e-7 1.40241e-7 1.25173e-7 1.11727e-7 9.9749e-8 8.90932e-8 7.96225e-8 7.12103e-8 6.37407e-8 5.71084e-8 5.12187e-8 4.59867e-8 4.13365e-8 3.72009e-8 3.35201e-8 3.02414e-8 2.73181e-8 2.47091e-8 2.2378e-8 2.02932e-8 1.84263e-8 1.67526e-8To momdel the power spectrum we also need to construct our execution plan for the given grid. This will be used to construct the actual correlated realization of the RF given some initial white noise.
julia
pl = StationaryRandomFieldPlan(grid)StationaryRandomFieldPlan
Base type: Float64
size: (64, 64)
exec: ComradeBase.Serial()For the stationary random field prior we also need to define the noise prior. Luckily VLBIImagePriors provides a helper function to do this for us.
julia
cprior = std_dist(pl)VLBIImagePriors.StdNormal{Float64, 2}(dims=(64, 64))For the coefficients of the spectral expansion we will use a uniform prior between 0.01 and the maximum image dimension. This prior is rather uninformative and allows for a wide range of power spectra. Additionally, we truncate the expansion at order 3 for simplicity.
julia
skym = sky(grid; mimg, pl, cprior, ρmax = max(size(grid)...))SkyModel
with map: _sky_sky
on grid:
RectiGrid(
executor: ComradeBase.Serial()
Dimensions:
(↓ X Sampled{Float64} LinRange{Float64}(-3.1134128583753017e-9, 1.6589718150466935e-9, 64) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-1.6589718150466935e-9, 3.1134128583753017e-9, 64) ForwardOrdered Regular Points)
)
)Since we are fitting closures we do not need to include an instrument model, since the closure likelihood is approximately independent of gains in the high SNR limit.
julia
using Enzyme
post = VLBIPosterior(skym, dlcamp, dcphase)VLBIPosterior
ObservedSkyModel
with map: _sky_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}(-3.1134128583753017e-9, 1.6589718150466935e-9, 64) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-1.6589718150466935e-9, 3.1134128583753017e-9, 64) ForwardOrdered Regular Points)
)
Visibility Domain: UnstructuredDomain(
executor: ComradeBase.Serial()
Dimensions:
378-element StructArray(::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}, ::Vector{Float64}) with eltype @NamedTuple{U::Float64, V::Float64, Ti::Float64, Fr::Float64}:
(U = -4.4188694399948233e8, V = 6.442995108563881e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -3.78109695999557e8, V = 2.404476519997183e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -1.6174918577758828e8, V = 2.091189688886439e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 9.460673599988917e7, V = 1.4597793955538455e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 4.143100711106257e7, V = 2.4368619911082566e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.681022811423088e8, V = 3.255453348567615e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.04600675999526e8, V = -7.829091899990828e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -1.8850678399977916e8, V = -1.0960438666653827e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -2.676056888885754e7, V = -3.187235448885155e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -1.1683855399986314e7, V = -8.610983839989913e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 6.765242959992075e7, V = -1.7278293599979758e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 1.460444819998289e7, V = -7.504541679991208e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -5.279119817136673e8, V = 3.868989211424039e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.6431755199945605e8, V = -1.69033972499802e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -2.4813381688859823e8, V = -4.818409022216578e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -8.63869279998988e7, V = -2.5730307911080968e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -7.121938799991657e7, V = -2.4656214799971115e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 8.116499049990492e6, V = -1.1132941039986959e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.4930432799947366e7, V = -1.3591557799984077e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -5.9534249199930266e7, V = 6.145408239992801e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.5658577828517944e8, V = 4.1167499885666054e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -3.9302742799953955e8, V = 7.832072562490825e6, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -1.7687868622201502e8, V = -2.348792577775026e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -1.5132090222204497e7, V = -2.326071839997275e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 7.933609519990706e7, V = -8.667305919989847e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 2.6288769799969204e7, V = 1.1064483399987038e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -2.7955152914252967e8, V = 4.3536144457091856e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -2.160720879997469e8, V = 3.1417642749963194e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 2.0317771377753976e8, V = 3.456715155551506e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -5.36343026285086e8, V = 4.9821439999941635e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.726417839994463e8, V = 9.441682299988939e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -2.5635488177747747e8, V = 6.314076755548159e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -5.304724079993786e7, V = 9.77376079998855e7, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.193364479995088e8, V = -3.261677046871179e6, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 6.339297657135431e7, V = -4.040669165709552e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = 4.829051611422914e8, V = -4.005662308566736e8, Ti = 5.208333492279053, Fr = 4.297575e10)
(U = -4.643065417137418e8, V = 1.7539614628550884e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -5.856890788564568e8, V = 5.652382445707664e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -2.340685119997258e8, V = 1.7856432399979082e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 7.03348459999176e7, V = 1.5872099799981406e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -1.5644589999981672e7, V = 2.4569112599971217e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.165620022852263e8, V = -1.4171867428554827e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -5.379452617136556e8, V = 2.481230422854236e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -1.865852279997814e8, V = -1.3857540799983767e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 4.747231649994439e7, V = -3.171404159996285e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 3.8281999722177377e6, V = -8.671816977767618e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 4.65004945777233e8, V = -1.6164001422203287e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 1.1764090577763996e8, V = -1.5848405333314767e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 3.1769406222185e7, V = -7.146645866658294e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.983829211422733e8, V = -9.125757028560737e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -6.197659977135597e8, V = 2.985842559996502e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -2.6833825599968565e8, V = -8.806972199989682e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -3.427801049995985e7, V = -2.6663290599968764e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -7.785369777768657e7, V = -3.616638399995763e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -8.168113333323765e7, V = 5.0552354222163e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 3.59598462221801e7, V = -1.0793158488876244e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.991167822216375e7, V = -2.0913955555531055e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 3.8331529599955094e8, V = -1.110912035554254e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.2050363428522164e8, V = -5.5004157714221284e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -5.418866651422223e8, V = 3.348376594281792e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -1.9047134999977687e8, V = -5.185894149993925e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 4.358830599994894e7, V = -2.3042411199973005e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 4.611824817772375e8, V = -7.491916355546778e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 1.1381460177764446e8, V = -7.176511111102705e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 2.7941857777745046e7, V = 1.5252010888871022e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -8.827262811418229e8, V = 1.939996071426299e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -1.0041099611416808e9, V = 4.0924160457094914e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -6.52185247999236e8, V = 2.2797592999973293e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.181021759995102e8, V = -1.5575599999981755e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -3.4736175644403756e8, V = 3.1570244999963017e6, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.332327502217147e8, V = 9.017483466656102e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -2.3001099657115912e8, V = -3.0409466071392945e6, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -3.513941074281598e8, V = 3.8680095085668975e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 2.1841527399974412e8, V = 6.712692449992135e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -5.345386011422309e8, V = 1.6634966571409082e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -6.559209051420888e8, V = 4.0647706971380955e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -3.043985719996434e8, V = 1.9841789749976754e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -8.587155733323273e7, V = 8.701759999989806e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -4.484491428566175e8, V = -7.028746971420337e7, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 1.2138311657128637e8, V = -3.8984174171382904e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = 5.698323657136182e8, V = -3.1955434057105416e8, Ti = 6.304166793823242, Fr = 4.297575e10)
(U = -5.0767591199940526e8, V = 1.1186584099986896e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -2.806592359996712e8, V = 1.447865699998304e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 4.512995377772491e7, V = 1.6627380266647187e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -6.321947866659261e7, V = 2.405447519997182e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -6.7047689955477e8, V = 4.834726897772114e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -3.998379279995316e8, V = -1.9508904399977145e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -1.7282156199979752e8, V = -1.6216807799981e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.0759534755542952e8, V = -3.070130844440848e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.683563733331361e7, V = -8.53624879999e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 5.842921955548711e8, V = -9.280812799989128e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.527223182220433e8, V = -1.4074111644427955e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 4.437457777772579e7, V = -6.646895333325546e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -5.628807679993405e8, V = 1.764665973331266e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -4.951860959994199e8, V = -1.56205809999817e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -2.6816959799968582e8, V = -1.2328506999985558e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.2293010222207822e7, V = -2.6807796266635263e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -7.846569066657475e7, V = -4.642661644439006e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -9.530096622211057e7, V = 3.893632755550995e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -5.092639644438478e7, V = -2.7532399777745526e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -6.581810133325622e8, V = 2.1540606222196987e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 5.741955155548829e7, V = -1.0180552799988073e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 4.889784924438716e8, V = -5.3880541333270214e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -4.1672766399951184e8, V = -1.0973549099987145e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -1.8971105599977776e8, V = -7.681472999991001e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 9.075861244433813e7, V = -2.2165064888862923e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 5.674594204437796e8, V = -7.443534166657947e6, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.358875039998408e8, V = -5.5378022222157344e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 2.7539288888856627e7, V = 1.8893785777755644e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -5.797166079993209e8, V = 2.6182703288858217e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -9.845734639988465e8, V = -1.0260762899987979e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -7.575565039991126e8, V = -6.968656499991836e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -4.76706972443886e8, V = -2.1421213511086017e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -5.399142328882564e8, V = 2.634179799996914e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -1.1471686684431007e9, V = 2.692803093330179e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -4.3156413511060554e8, V = -4.792854577772163e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -2.2701645599973404e8, V = -3.2920671749961436e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 2.172452879997455e8, V = 9.572384599988785e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -3.90097303999543e8, V = 3.383187359996036e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -5.526931199993526e8, V = -5.443342049993623e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -3.256765719996185e8, V = -2.15123779999748e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -1.0834697155542864e8, V = 7.427270577769077e7, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -7.156016355547173e8, V = 3.1721091199962837e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -4.4426163199947953e8, V = -1.2864479599984929e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 1.6308092799980894e8, V = -3.712394319995651e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = 6.072545351103997e8, V = -2.429384462219376e8, Ti = 7.237499713897705, Fr = 4.297575e10)
(U = -1.1816283999986157e8, V = 2.2536240999973598e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -5.192849031105028e8, V = 2.640569022219129e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -3.1531223466629726e8, V = 9.520118666655514e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 9.398909222211212e6, V = 1.7083167288868874e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -7.211011626658219e8, V = 3.672656675551253e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 1.7466396599979538e8, V = -2.83378303999668e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -3.4457826799959636e8, V = -2.5725569599969864e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 3.1828453749962714e7, V = -8.128587799990478e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -1.4071481199983516e8, V = -1.8834890999977934e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 1.839429859997845e8, V = -1.1254049599986815e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 5.649372699993382e7, V = -5.8018704999932036e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -5.464922239993598e8, V = 8.349447199990219e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -1.0432894099987778e8, V = 2.221318199997398e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -7.250190899991506e7, V = -5.90708904999308e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -4.783602649994396e7, V = -3.580435599995805e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 7.015271466658448e7, V = -2.6120300266636065e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -2.451594951108239e8, V = -1.6600183111091664e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -4.491356764439183e8, V = -2.347934915552805e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 7.955103022212903e7, V = -9.03718817776719e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -6.509510399992374e8, V = 1.0607116355543129e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 1.428288099998327e8, V = -2.0209578199976325e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -3.7641529999955904e8, V = -1.7596062599979386e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -1.7254704599979785e8, V = -1.0705869799987458e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 1.521129399998218e8, V = -3.125842824996338e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 2.4665898249971103e7, V = 2.3266529749972746e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -5.783249679993225e8, V = 1.6479448999980694e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 1.9721293599976897e8, V = 1.3032520299984732e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -2.0397260799976104e8, V = -6.879557422214164e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -4.0578894577730244e8, V = 2.7206469688857013e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -1.2744710399985069e8, V = 5.4525057999936126e7, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -3.247114488885085e8, V = -7.563159555546695e7, Ti = 8.4208345413208, Fr = 4.297575e10)
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(U = -7.30501617776922e8, V = 1.9644128355532542e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -4.0108113599953014e8, V = -1.9922702999976662e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 6.029908319992936e8, V = -1.415280859998342e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = 2.0181466844420803e8, V = -3.408622542218229e8, Ti = 8.4208345413208, Fr = 4.297575e10)
(U = -5.044512274279805e8, V = -2.0316156857119057e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -3.216957851424803e8, V = 6.613430914277967e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.0843828999987295e7, V = 1.7076975771408564e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.435999337141175e8, V = 2.134905462854642e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -7.195767222848712e8, V = 3.015084251425039e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 2.044574514283319e8, V = -2.6618689599968815e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 1.9361438857120177e8, V = -9.541684685703108e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 6.085639257135729e7, V = -5.269822114279541e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 3.8780291999954574e7, V = -7.80848902856228e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.1739754099986246e8, V = -1.9998526399976572e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -3.003083879996482e8, V = -2.8632846799966455e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 7.121871919991657e8, V = 7.63985009999105e7, Ti = 9.06666612625122, Fr = 4.297575e10)
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(U = 9.942183428559782e7, V = -2.5346967999970308e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 8.857865371418194e7, V = -8.26999222856174e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.0503607085701981e8, V = 1.2720007857127955e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -4.0504509257095414e8, V = -2.7377478399967927e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -6.625508399992239e7, V = -6.536459599992342e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 6.072593737135743e8, V = 8.951413371418086e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -2.2228189714259672e8, V = -1.8732938742835197e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -4.4179355428519666e7, V = -3.9977918857096024e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -6.201675062849878e8, V = 4.8055539428515136e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 1.656746765712345e8, V = -1.881032502854939e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 1.5483133714267573e8, V = -1.733330428569398e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 2.2075877714259855e7, V = 2.5386576571398832e7, Ti = 9.06666612625122, Fr = 4.297575e10)
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(U = 6.735132434277824e8, V = 1.5487275657124713e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.5602869942838866e8, V = -1.2196330742842855e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -5.53915574856494e8, V = 1.1342506628558142e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -5.0784346971369076e8, V = -3.430004617138839e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -5.186869759993924e8, V = -1.7223217371408394e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -6.514378422849511e8, V = -1.2948837599984832e8, Ti = 9.06666612625122, Fr = 4.297575e10)
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(U = 1.7810388799979135e8, V = 1.4735038857125595e8, Ti = 9.06666612625122, Fr = 4.297575e10)
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(U = -3.979663159995338e8, V = 2.35601075999724e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -4.9360717714227885e8, V = -1.910867977140619e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -3.1085210514249295e8, V = -1.0463594399987742e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -1.3275661257127306e8, V = 4.2720647999949954e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -7.087333119991697e8, V = 1.307372525712754e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -3.608683291424344e8, V = -2.3379509257115468e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 5.75990189713611e8, V = -8.803663999989687e7, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = 2.1505936399974805e8, V = -3.219517359996228e8, Ti = 9.06666612625122, Fr = 4.297575e10)
(U = -4.832396239994339e8, V = -5.466561249993596e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -3.202766799996248e8, V = 4.381262799994867e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -2.6099658749969423e7, V = 1.6948726399980146e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.6034738199981216e8, V = 2.0292514199976227e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -7.043232879991748e8, V = 2.519832999997048e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 2.2337291599973834e8, V = -2.5131303599970558e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.972723919997689e8, V = -8.182673299990414e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -2.5988891799969554e8, V = -3.0597056799964154e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 4.3369911499949194e7, V = -7.522843499991187e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -9.691528999988647e7, V = -2.0749766399975693e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 6.3023974499926165e7, V = -4.8390785999943316e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 7.194769519991572e8, V = 1.2654065399985176e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -4.8098037199943656e8, V = 695596.8671866851, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.198423809998596e8, V = -2.45848897999712e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 9.374230699989018e7, V = -7.636194699991055e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -3.6342183199957424e8, V = -3.00500911999648e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -6.016138849992952e7, V = -6.975972499991828e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -2.004473059997652e8, V = -2.020285299997633e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -4.050722749995255e7, V = -4.2922115499949716e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.0353117399987872e8, V = 5.469035562493593e6, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 6.159461759992784e8, V = 1.3201844299984534e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -5.845055199993153e8, V = 6.159489374992784e6, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.8000005399978912e8, V = -1.760866939997937e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.5390048199981973e8, V = -6.600191749992268e6, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -3.0326036799964476e8, V = -2.307411959997297e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 6.76108511999208e8, V = 2.017811279997636e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.4028592199983567e8, V = -1.3226882199984506e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.9654173249976978e7, V = 2.683761024996856e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -5.243414759993858e8, V = 7.591646899991107e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -4.961064959994188e8, V = -3.778941119995573e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -5.222059799993883e8, V = -2.0840416399975586e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -9.793684319988527e8, V = -4.325216439994933e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -8.163938559990436e8, V = -3.3404995999960864e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -6.56453567999231e8, V = -1.7494348199979505e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.2004590719985938e9, V = -1.2584203899985258e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.629742679998091e8, V = -9.847251599988464e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 1.5994008199981263e8, V = 1.5910643799981362e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -3.840555959995501e8, V = 2.081853379997561e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -4.5714295599946445e8, V = -2.241507979997374e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -2.9417828799965537e8, V = -1.2567329399985278e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -1.3424756599984273e8, V = 3.343843074996083e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -6.782259439992055e8, V = 8.249920299990335e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -3.229146159996217e8, V = -2.5757877799969825e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 2.2108154399974102e8, V = -3.0665814799964076e8, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = 5.439956719993627e8, V = -4.9078929999942504e7, Ti = 9.566667079925537, Fr = 4.297575e10)
(U = -4.0564616533285815e8, V = -1.2518950133318669e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.971729333329852e8, V = -5.162910583327285e6, Ti = 10.6875, Fr = 4.297575e10)
(U = -5.866126266659794e7, V = 1.6277091466647598e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.8789818133311322e8, V = 1.753108666664613e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -6.263160533325996e8, V = 1.4640058133316183e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 2.518573226663716e8, V = -2.1362742933308306e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 6.908169599991907e8, V = 2.384054506663874e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 1.9323766857120222e8, V = -5.095684399994031e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.5417208685696223e8, V = -3.387350262853175e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.550101828566098e7, V = -2.1876564342831516e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 5.093679999994033e7, V = -6.777325371420631e7, Ti = 10.6875, Fr = 4.297575e10)
(U = 6.396568457135364e7, V = -3.835120457138365e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -3.749157714281322e8, V = -6.700784685706436e7, Ti = 10.6875, Fr = 4.297575e10)
(U = 1.581427253331481e8, V = -2.2380814933307114e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 5.971011839993005e8, V = 2.2822551999973264e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 9.947633028559776e7, V = -6.108900171421415e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.4793231314256668e8, V = -3.4886703542816275e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.39262969142694e8, V = -2.28896633142589e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.28268925713784e7, V = -7.790286742848016e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.9798065142822236e7, V = -4.848080971422892e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -9.376469371417587e7, V = -1.012875571427385e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.686637211423081e8, V = -7.714477828562391e7, Ti = 10.6875, Fr = 4.297575e10)
(U = 2.009015066664313e8, V = -1.4588172799982908e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 6.398598719992504e8, V = 3.0615193599964136e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 1.4229960228554758e8, V = 1.6815108142837442e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.0510824457118827e8, V = -2.70963163428254e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -9.643771199988702e7, V = -1.5099316342839453e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 1.3028805999984737e7, V = 2.9422059142822675e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.258294765709297e8, V = 754111.4129455452, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.3895796799948573e8, V = -4.520332373328038e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -8.446033386656772e8, V = -5.772231359993237e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -7.361312426658043e8, V = -4.5719596799946445e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -4.9761979199941707e8, V = -2.892623786663278e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -6.26856533332599e8, V = -2.767228213330092e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.0652715413320853e9, V = -3.056350346663086e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.0867421028558697e8, V = -1.199688914284309e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 1.0946581485701463e8, V = 1.8041537142836007e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -3.2938291657104266e8, V = 1.517420891426794e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -3.474024274281645e8, V = -2.877799314282343e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.3873229028543463e8, V = -1.678097439998034e8, Ti = 10.6875, Fr = 4.297575e10)
(U = -1.2927212342841998e8, V = 1.2607282999985231e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -5.68106212570763e8, V = -1.6074099428552598e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -2.18135956571173e8, V = -3.0038555428536236e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 2.2070518628545573e8, V = -2.717077485711103e8, Ti = 10.6875, Fr = 4.297575e10)
(U = 4.388583177137715e8, V = 2.866794114282356e7, Ti = 10.6875, Fr = 4.297575e10)
(U = -3.3336450488849837e8, V = -1.6438386844425187e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.670391839996872e8, V = -3.5077149333292246e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -7.781758311101994e7, V = 1.555403679998178e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.9735635377754655e8, V = 1.5489184177759632e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -5.429836728882527e8, V = 8.438350311101227e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 6.370161188563966e8, V = 3.095338697139231e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 2.5882449955525234e8, V = -1.865575644442259e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -8.2240279999903655e6, V = -2.2163478222196257e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 1.8101063288867682e8, V = -3.1021847777741432e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 5.359582133327055e7, V = -6.2204485333260454e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 6.146968622215021e7, V = -3.166849622218512e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -7.457743911102374e7, V = -3.509384888884778e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.8420369777744484e8, V = -1.0216118399988033e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 5.546519405707787e8, V = 2.8991581257108897e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 1.7632290666646013e8, V = -2.0608245511086968e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -9.07292177776715e7, V = -2.411572959997175e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 9.850591466655128e7, V = -5.0544542666607454e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.891075888885502e7, V = -8.172587822212648e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.103736711108647e7, V = -5.1189565333273366e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -8.250754311101446e7, V = -1.9520718222199354e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.5708550222203818e8, V = -3.7045874133289933e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -3.6671223111068153e8, V = -1.2168134666652411e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 5.834028251421736e8, V = 3.716755199995646e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 2.052278773330929e8, V = -1.24355314666521e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -6.182017733326091e7, V = -1.5943100088870212e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 1.2741463822207294e8, V = 3.1181786444407914e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.2816685244429429e8, V = -2.887342755552173e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 7.873743388879665e6, V = 3.0536226222186446e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -3.377918435551598e8, V = -3.99588471110643e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -3.781708205709855e8, V = -4.957998765708478e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -7.10693860570596e8, V = -6.605588205706547e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -6.448329782849588e8, V = -5.31177298285092e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -4.5617755885660845e8, V = -3.4034685714245844e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -5.755874834278971e8, V = -3.4113023542817175e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -9.201519725703506e8, V = -4.12027181713803e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -6.633659955547784e7, V = -1.293050497776263e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 6.969327022214058e7, V = 1.899672942219997e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.759590719996767e8, V = 1.194676435554156e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -2.5557062044414505e8, V = -3.19917745777403e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.8923413155533388e8, V = -1.9061289777755448e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.195415964443044e8, V = -645454.2239575773, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -4.651939626661217e8, V = -7.114511555547221e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -1.360441715553962e8, V = -3.192698488885149e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 3.4566988799959505e8, V = 7.049351644436187e7, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = 2.0962463288864332e8, V = -2.4877746133304188e8, Ti = 11.445834159851074, Fr = 4.297575e10)
(U = -9.207263359989214e7, V = 1.4769042239982697e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.993114655997665e8, V = 1.3655433919984e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -2.5953787199969596e8, V = -1.9181637119977528e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -2.3217219199972802e8, V = -5.815901679993187e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 5.704450879993317e8, V = 3.650183439995724e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 1.6456286719980723e8, V = -1.5035685399982387e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 5.732435439993285e7, V = -2.6171716799969338e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 2.5663634559969935e8, V = -1.6272916159980935e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -2.8920358499966124e6, V = -3.5454680319958466e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 5.4189763999936506e7, V = -5.7218954399932966e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 2.4469552399971332e7, V = -2.208932927997412e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 5.004670559994137e8, V = 3.3839647199960357e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 1.8657538559978142e8, V = -1.8930432319977823e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -7.005882719991793e7, V = -2.658328919996886e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -7.292603679991457e7, V = -3.811318527995535e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.586919919998141e7, V = -8.380051999990183e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -4.557854239994661e7, V = -2.4747529919971007e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 9.450004319988929e7, V = -4.161575279995125e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.273544939998508e7, V = -5.2753907999938205e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 5.1625703199939525e8, V = 4.222054079995054e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 1.1037319839987071e8, V = 4.218259759995058e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 3.1346761999963275e6, V = 3.104657599996363e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 2.0244657919976282e8, V = -1.0551020799987641e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -5.708186079993312e7, V = -2.973294015996517e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -2.9720188799965184e7, V = -1.6367388799980825e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -3.1384717599963236e8, V = -5.275974959993819e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -5.728808319993289e8, V = -7.195693759991571e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -5.457757759993607e8, V = -5.858931199993136e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -4.0599993599952435e8, V = -3.799596799995549e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -5.131514079993988e8, V = -3.9115746399954176e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = 3.285070159996152e7, V = 1.9472075839977187e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -2.7374415599967934e7, V = -1.3365462559984343e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.6744523839980385e8, V = -3.3951283199960226e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.4009061119983587e8, V = -2.0585586879975882e8, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -1.0723841439987437e8, V = -1.1136108399986956e7, Ti = 12.104166984558105, Fr = 4.297575e10)
(U = -6.020668319992946e7, V = -3.283766911996153e8, Ti = 12.104166984558105, Fr = 4.297575e10)
)
)
IdealInstrumentModelData Products: Comrade.EHTLogClosureAmplitudeDatumComrade.EHTClosurePhaseDatumReconstructing the Image
To reconstruct the image we will first use the MAP estimate. This is approach is basically a re-implentation of regularized maximum likelihood (RML) imaging. However, unlike traditional RML imaging we also fit the regularizer hyperparameters, thanks to our interpretation of as our imaging prior as a hierarchical model.
To optimize our posterior Comrade provides the comrade_opt function. To use this functionality a user first needs to import Optimization.jl and the optimizer of choice. In this tutorial we will use the Adam optimizer. We also need to import Enzyme to allow for automatic differentiation.
julia
using Optimization, OptimizationLBFGSBtpost = asflat(post)
julia
xopt, sol = comrade_opt(post, LBFGSB(); initial_params = prior_sample(rng, post), maxiters = 5000)
using CairoMakie┌ Warning: Using fallback BLAS replacements for (["cblas_zdotc_sub64_"]), performance may be degraded
└ @ Enzyme.Compiler ~/.julia/packages/Enzyme/9OkvN/src/compiler.jl:5340The image we actually fit is a continuous object so we can easily refine the image to produce a higher resolution rendering. Here we refine the image by a factor of 3 in each dimension.
julia
g = refinespatial(grid, 3)RectiGrid(
executor: ComradeBase.Serial()
Dimensions:
(↓ X Sampled{Float64} LinRange{Float64}(-3.1134128583753017e-9, 1.6589718150466935e-9, 192) ForwardOrdered Regular Points,
→ Y Sampled{Float64} LinRange{Float64}(-1.6589718150466935e-9, 3.1134128583753017e-9, 192) ForwardOrdered Regular Points)
)Now to produce the intensity map we just do
julia
imgmap = intensitymap(skymodel(post, xopt), g)
fig = imageviz(imgmap, colorscale = log10, colorrange = (1.0e-8, 1.0e-4), size = (650, 500));
To see how well the MAP estimate fits the data we can plot the residuals.
julia
res = Comrade.residuals(post, xopt)
fig = Figure(; size = (800, 300))
plotfields!(fig[1, 1], res[1], :uvdist, :res);
plotfields!(fig[1, 2], res[2], :uvdist, :res);
fig |> DisplayAs.PNG |> DisplayAs.Text
Overall, the image looks reasonable. However, the MAP is not a robust estimator of the image morphology. For high dimensional problems the MAP is often not representative of the entire image posterior. For this reason Comrade's main goal is to sample the posterior of the image given the data.
To sample from the posterior we will use HMC and more specifically the NUTS algorithm similar to the other imaging tutorials. For this tutorial we will also show how to use the DiskStore functionality that save the chain to disk to reduce memory usage. This is especially useful for high-dimensional imaging problems where the chain can easily reach multiple GBs in size. This also allows us to restart sampling from a previous chain if needed. by using the keyword argument restart=true in the sample function.
julia
using AdvancedHMC
mc = sample(
rng, post, AdvancedHMC.NUTS(0.8), 300 + 400, n_adapts = 400,
initial_params = xopt, saveto = DiskStore(; stride = 10, name = "VLBA_2025")
);
chain = load_samples(mc)PosteriorSamples
Samples size: (700,)
sampler used: AHMC
Mean
┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky │
│ @NamedTuple{c::Matrix{Float64}, ρs::Tuple{Float64, Float64, Float64}, σimg::Float64, fb::Float64} │
├─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = [-0.268327 -0.65633 … 0.218209 -1.33258; -1.5833 1.46442 … 0.0838931 1.15533; … ; -0.233124 -0.467236 … 1.12855 -0.115986; -0.0243204 -0.692557 … 0.108726 0.16931], ρs = (31.5287, 14.9692, 7.70329), σimg = 7.56722, fb = 0.0996375) │
└─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘
Std. Dev.
┌───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐
│ sky │
│ @NamedTuple{c::Matrix{Float64}, ρs::Tuple{Float64, Float64, Float64}, σimg::Float64, fb::Float64} │
├───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┤
│ (c = [1.03334 0.406827 … 0.532951 0.466327; 0.517392 0.788915 … 0.412595 0.745802; … ; 0.706521 0.484458 … 0.65881 0.661904; 0.354281 0.463243 … 0.924084 0.503428], ρs = (8.32314, 9.97474, 1.20142), σimg = 1.35826, fb = 0.154842) │
└───────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┘Warning
This should be run for longer!
Now that we have our posterior, we can assess which parts of the image are strongly inferred by the data. This is rather unique to Comrade where more traditional imaging algorithms like CLEAN and RML are inherently unable to assess uncertainty in their reconstructions.
To explore our posterior let's first create images from a bunch of draws from the posterior
julia
msamples = skymodel.(Ref(post), chain[501:5:end]);
k = range(1 / size(grid)[1], π / 2, length = 512)
fig = Figure()
ax = Axis(fig[1, 1], xscale = log10, yscale = log10)
for i in 501:10:length(chain)
lines!(ax, k, VLBIImagePriors.ampspectrum.(Ref(MarkovPS(chain.sky.ρs[i])), tuple.(k, 0)))
end
fig
The mean image is then given by
julia
using StatsBase
gpl = refinespatial(grid, 3)
imgs = intensitymap.(msamples, Ref(gpl))
mimg = mean(imgs)
simg = std(imgs)
fig = Figure(; size = (500, 300));
crange = (5.0e-6, 5.0e-2)
axs = [Axis(fig[i, j], xreversed = true, aspect = DataAspect()) for i in 1:1, j in 1:2]
image!(axs[1, 1], mimg, colormap = :afmhot, colorscale = log10, colorrange = crange); axs[1, 1].title = "Mean"
image!(axs[1, 2], simg ./ (max.(mimg, 1.0e-12)), colormap = :afmhot);axs[1, 2].title = "Fractional Uncertainty"
hidedecorations!.(axs)
fig |> DisplayAs.PNG |> DisplayAs.Text
We can also compare the Comrade reconstruction to the CLEAN reconstruction of the same data.
julia
cleanf = Base.download("https://www.bu.edu/blazars/VLBA_GLAST/1308/1308+326Q.2021-03-19.IMAP.gz")"/tmp/jl_tgT0zaxG3c"By default this will load the clean components with the beam defined in the FITS header.
julia
mcl = load_clean_components(cleanf)VLBISkyModels.MultiComponentModel{VLBISkyModels.ModifiedModel{VLBISkyModels.Gaussian{Float64}, Tuple{VLBISkyModels.Stretch{Float64, Float64, Float64}, VLBISkyModels.Rotate{Float64}}}, Vector{Float32}, Vector{Float32}}(ModifiedModel
base model: VLBISkyModels.Gaussian{Float64}()
Modifiers:
1. VLBISkyModels.Stretch{Float64, Float64, Float64}
2. VLBISkyModels.Rotate{Float64}, Float32[0.8002344, 0.42969742, 0.07445484, 0.06368931, 0.02217047, 0.01861643, 0.02657656, 0.02344725, 0.02024817, 0.00761945, 0.03046582, 0.00726002, 0.00563375, 0.00645834, 0.01627629, 0.06097463, 0.02168821, 0.02517034, 0.00513046, 0.00820482, 0.00453077, 0.02388651, 0.01229127, 0.00830369, 0.01919385, 0.01770286, 0.02719101, 0.01437432, 0.01668327, 0.00800299, 0.00437514, 0.00740964, 0.00203799, 0.00196259, 0.00541239, 0.01453394, 0.01154794, 0.0057647, 0.00416458, 0.00356502, 0.01970395, 0.00934755, 0.00110652, 0.00262081, 0.00360425, 0.01713799, 0.00271393, 0.00328356, 0.00679949, 0.00558265, 0.00692711, 0.00366, 0.00783088, 0.00134895, 0.00117046, 0.00256492, 0.00428372, 0.00154532, 0.00259595, 0.00186204, 0.00249551, 0.00155613, 0.00023076, 0.00083379, 0.00276655, 0.00022331, 0.00191691, 0.00076527, 0.00019861, -0.00356351, 0.00017628, 0.0013579, -0.0012659, -0.00583323, -0.00684484, 0.00043194, -0.00273778, 0.00282735, 0.00513306, -0.0030624, 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1.6580698f-8, 3.3597484f-8, -2.9379878f-8, 3.3306613f-8, -5.9631864f-9, -1.5999008f-8, -9.453742f-9, 3.1415908f-8, 1.8180668f-8, 1.9925716f-8, -3.6651805f-8, -6.8358754f-9, 2.1816579f-8, 1.803519f-8, -2.0943855f-8, 1.4544762f-9, 1.3671978f-8, 4.2178607f-9, 2.6616114f-8, 2.501654f-8, 4.363191f-9, 3.03977f-8, 2.5743608f-8, 1.1490125f-8, -3.0979525f-8, -2.0071445f-8, -7.854057f-9, 2.7636322f-9, 1.2507968f-8, -2.385286f-8, -3.6942712f-8, 1.6289743f-8, 2.7052545f-8, 1.4253542f-8, -1.2508188f-8, -2.196184f-8, -5.8181504f-10, -1.2362655f-8, -2.4725427f-8, 3.0397715f-8, 1.38171465f-8, -1.3526079f-8, 1.7889631f-8, -1.9198467f-8, 9.162834f-9, -1.7744199f-8, -3.2579184f-8, 2.3998236f-8, -1.4253716f-8, 3.0543294f-8, -1.774425f-8, 2.0216802f-8, 2.9525202f-8, -3.374301f-8, 3.054277f-9, 1.3817141f-8, -2.777973f-8, 2.545271f-8, 1.2071861f-8, 2.3852865f-8, 3.5633605f-8, -3.5779223f-8, 1.745341f-8, 3.2143237f-8, 2.9088788f-8, 3.563382f-8, -3.6506247f-8, -2.108944f-8, 3.4761154f-8, -1.2508182f-8, 3.2870272f-8, -2.3707399f-8, 3.4470432f-8, -4.3655948f-10, -3.1270485f-8, 1.6871434f-8, 1.1198973f-8, 1.7598705f-8, -2.7925301f-8, -2.7197865f-8, 2.327076f-8, -2.4580144f-8, -8.1450535f-9, -3.447021f-8, -2.676174f-8, -4.508819f-9, 6.108633f-9, -5.819527f-10, -1.3235481f-8, -3.025235f-8, -6.981431f-9, -3.40339f-8, -2.9816213f-8, -2.1816604f-9, 1.4253226f-8, 1.17810375f-8, 3.0833974f-8, 3.228859f-8, -1.3235343f-8, -2.9524907f-8, 3.0107078f-8, -3.5633782f-8, -3.1415944f-8, 2.9379713f-8, 2.3270381f-9, 3.05433f-8, 3.0688568f-8, -2.99616f-8, 2.3416453f-8, -3.3597583f-8, -1.09082245f-8, 3.4615592f-8, -3.068878f-8, -2.908882f-8, -7.563066f-9, -2.9087335f-9, 5.33295f-9, 4.8481368f-11, -1.1635528f-8, 9.841718f-9, -4.1693977f-9, 5.33295f-9, 5.332951f-10, -5.5753575f-9, 1.0956789f-8, -1.1102233f-8, -7.950945f-9, 7.029798f-9, 1.1053752f-8, 1.1684009f-8, 1.1732491f-8, -3.9754724f-9, -6.302578f-10, 7.223724f-9, -4.4118047f-9, 3.151289f-9, -2.230143f-9, 2.230143f-9, 3.8785095f-9, -8.581202f-9, -4.363323f-10, -9.502348f-9, 3.684584f-9, -6.787391f-10, -5.235987f-9, -2.8119194f-9, 3.684584f-9, -1.134464f-8, -3.8785095f-10, 8.0963884f-9, 3.1997702f-9, -7.6115745f-9, -3.3936956f-10, 4.1209165f-9, 6.884354f-9, -1.0811345f-8, 9.890199f-9, 1.5029225f-9, 3.1997702f-9, 8.338795f-9, -9.841718f-9, 6.2540964f-9, -7.950945f-9, 7.029798f-9, 7.272205f-9, -1.0665901f-9, -1.1635528f-8, -6.157134f-9, 8.338795f-9, 1.5514038f-9, 3.296733f-9, 4.314842f-9, 7.660057f-9, 8.775127f-9, -8.338795f-9, 2.2786244f-9, -1.0665901f-9, 6.6419474f-9, 5.429913f-9, 2.9573635f-9, -5.526876f-9, -7.660057f-9, 2.375587f-9, -1.0956789f-8, 8.0963884f-9, 8.435758f-9, -1.1150715f-8, -6.4965033f-9, -1.0956789f-8, 8.872091f-9, -8.338795f-9, -1.8907733f-9, 2.908882f-10, 2.7149565f-9, -6.1086523f-9, 1.1635528f-8, 1.1635528f-9, -7.999426f-9, 5.4783946f-9, -2.5210312f-9, -8.0963884f-9, -3.8785095f-10, -7.272205f-9, 1.0229568f-8, -1.10052705f-8, 3.2482517f-9, 2.666475f-9, 6.932835f-9, 7.6115745f-9, -8.678165f-9, -1.0665901f-9, -9.599311f-9])We can also choose the load the clean components with a user-defined beam.
julia
mcl_25 = load_clean_components(cleanf, modify(Gaussian(), Stretch(beamsize(dlcamp) / 4 / fwhmfac)))VLBISkyModels.MultiComponentModel{VLBISkyModels.ModifiedModel{VLBISkyModels.Gaussian{Float64}, Tuple{VLBISkyModels.Stretch{Float64, Float64, Float64}}}, Vector{Float32}, Vector{Float32}}(ModifiedModel
base model: VLBISkyModels.Gaussian{Float64}()
Modifiers:
1. VLBISkyModels.Stretch{Float64, Float64, Float64}, Float32[0.8002344, 0.42969742, 0.07445484, 0.06368931, 0.02217047, 0.01861643, 0.02657656, 0.02344725, 0.02024817, 0.00761945, 0.03046582, 0.00726002, 0.00563375, 0.00645834, 0.01627629, 0.06097463, 0.02168821, 0.02517034, 0.00513046, 0.00820482, 0.00453077, 0.02388651, 0.01229127, 0.00830369, 0.01919385, 0.01770286, 0.02719101, 0.01437432, 0.01668327, 0.00800299, 0.00437514, 0.00740964, 0.00203799, 0.00196259, 0.00541239, 0.01453394, 0.01154794, 0.0057647, 0.00416458, 0.00356502, 0.01970395, 0.00934755, 0.00110652, 0.00262081, 0.00360425, 0.01713799, 0.00271393, 0.00328356, 0.00679949, 0.00558265, 0.00692711, 0.00366, 0.00783088, 0.00134895, 0.00117046, 0.00256492, 0.00428372, 0.00154532, 0.00259595, 0.00186204, 0.00249551, 0.00155613, 0.00023076, 0.00083379, 0.00276655, 0.00022331, 0.00191691, 0.00076527, 0.00019861, -0.00356351, 0.00017628, 0.0013579, -0.0012659, -0.00583323, -0.00684484, 0.00043194, -0.00273778, 0.00282735, 0.00513306, -0.0030624, -0.00383175, 0.00058704, -0.00042813, -0.00119051, -0.00064053, -0.00163516, 0.00054966, 0.00084859, -7.381f-5, 0.0006635, -0.00044083, 0.00037928, -6.607f-5, -0.00012819, 0.00048191, -0.00035963, 0.00011693, -5.959f-5, 0.00022961, -5.729f-5, -0.00013392, -0.00079685, 0.00072197, 0.00034334, -0.00097948, -0.00015774, 0.00051691, -0.0006402, 0.00044609, 0.00087918, 3.652f-5, -0.00055121, 9.933f-5, -6.465f-5, -0.00032134, -3.137f-5, 5.849f-5, -2.748f-5, -0.00010579, 5.2f-5, -5.127f-5, 0.00035554, -0.00029238, -3.006f-5, 0.0002672, -2.731f-5, -4.819f-5, 8.595f-5, -2.157f-5, 0.00010204, -1.844f-5, -1.722f-5, -3.119f-5, 3.983f-5, -4.165f-5, 4.214f-5, 2.588f-5, -0.00016791, 8.986f-5, 4.892f-5, 3.631f-5, -4.12f-5, -4.098f-5, 2.426f-5, -7.95f-6, 0.00091278, -0.00057268, -0.00029619, -0.00119706, 0.00015178, 0.00085392, -0.00061533, -0.00112126, 0.00112285, 0.00070315, 0.00142955, 0.0003729, -0.00037157, 0.00076261, -0.00013381, 0.00036936, 0.00046957, -0.00045138, 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0.00048657, 0.00017201, -9.844f-5, -0.00067348, -0.0007629, -0.00066883, -0.00025644, -0.00062516, -0.00040877, 0.00018744, -9.715f-5, 0.00024938, -0.00018529, -0.00049863, 0.00018962, -9.584f-5, 0.00026124, -0.00041277, 9.594f-5, -0.00053801, 0.00017567, -0.00048683, -9.466f-5, -0.00026424, 9.391f-5, -0.00017536, 0.00079613, -0.00025201, -0.00017694, -9.283f-5, -0.00031651, -0.0003118, 0.00017335, 9.212f-5, 0.00023269, 0.00017921, 0.00038753, -0.00062605, 0.00039291, 0.00025095, -0.00016638, 0.00083793, -0.00028106, -9.091f-5, -0.00054764, -0.00025152, -0.00024645, -0.00077056, 0.00042359, -0.00063193, 0.00025265, -0.00015849, 0.00029153, -8.829f-5, -0.00024277, -8.821f-5, 0.00015983, -0.0001532, 0.00014799, -0.00045303, -0.00027441, -0.00022904, -8.69f-5, -0.00023067, 0.00034039, -0.0001637, 0.00089268, -8.594f-5, 0.00024313, -0.00023577, 0.00037906, 8.53f-5, 0.0002559, 0.00015593, 0.00063096, 0.00023243, 8.493f-5, -0.00031506, -0.00030899, -8.482f-5, -0.00027777, -0.00038965, 0.00033585, -0.00030175, 0.00016042, -0.00015951, 0.00060314, 0.00054915, -0.00046574, -0.00059426, -0.00035636, 0.00031237, 0.00036061, 0.00015628, 0.00075616, 0.00055138, 0.00040534, -8.218f-5, 0.00033437, -0.00016058, 0.00052278, -0.00040659, -8.138f-5, -0.00057287, 0.00043958, -0.00041426, 8.077f-5, -0.00014554, 8.03f-5, -0.00041413, -0.0003432, 0.0001531, -0.00028394, -7.962f-5, 7.959f-5, 0.00014633, -7.913f-5, -0.00015216, 0.00053488, 0.00014864, 0.00028178, 7.819f-5, -7.801f-5, 0.00031215, -0.0003279, 0.00042652, 0.00014139, 0.00019918, -0.00031758, -0.00044056, -0.00078313, 0.00085577, -0.00039111, -7.738f-5, 0.00014875, -7.664f-5, 7.666f-5, -0.00020745, -0.00014306, -0.0001433, 7.61f-5, -0.00043319, 0.00014372, 0.00074174, -7.573f-5, 7.56f-5, 0.00076941, 0.00048902, -7.51f-5, -7.496f-5, -7.483f-5, -7.512f-5, -0.00036722, -7.457f-5, 0.00014375, 0.00042547, -0.00036333, 7.372f-5, 0.00039367, -7.363f-5, 0.00013721, 0.00014022, -0.00020466, 0.00038185, 0.0002512, -0.00013948, -0.00032638, 0.00030389, 0.00032645, -0.00031466, 0.00018735, 0.00045417, -0.00037161, 0.00031854, -7.194f-5, 0.00030934, 7.195f-5, 0.0003567, -7.152f-5, 0.0004618, 0.00023852, -7.084f-5, 0.000137, 0.00018962, 0.00041872, -0.00019352, 0.00029904, -7.046f-5, -7.029f-5, 0.00040229, -0.00035505, 7.008f-5, -0.00028297, 0.00030176, 0.00046359, -6.96f-5, -0.00013396, 0.00046304, -6.938f-5, 0.00024163, -0.0003055, -0.0001984, -0.00045323, -6.897f-5, -0.00013068, 0.00018227, 0.00052098, -0.00040877, 6.827f-5, 0.00040236, -0.00017673, -0.00017151, -0.00018665, -6.75f-5, 6.742f-5, -0.00013109, -0.00033806, -0.00039632, -0.000294, -0.0001878, 0.00012158, -6.628f-5, 0.00039945, -0.00029432, -0.00023444, 0.00024312, 0.00040298, -0.00018249, -0.00024034, 6.541f-5, -6.566f-5, 0.00017751, 0.00012515, 0.0002779, 0.00012305, 0.00044798, 0.00060066, 0.00023271, 0.00022838, 0.00044229, -6.391f-5, -6.417f-5, 6.347f-5, -0.0002184, 6.354f-5, 6.345f-5, 0.00012326, 0.00039329, -0.00043708, 0.0001203, 6.279f-5, 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-1.0665901f-9, -9.599311f-9])Now we can produce the CLEAN images on the same grid as our Comrade reconstruction.
julia
cleanimg = intensitymap(mcl, gpl)
cleanimg25 = intensitymap(mcl_25, gpl)
fig = Figure(; size = (900, 350));
axs = [Axis(fig[1, j], xreversed = true, aspect = DataAspect()) for j in 1:3]
image!(axs[1], mimg, colormap = :afmhot, colorscale = log10, colorrange = crange); axs[1].title = "Comrade Mean"
image!(axs[2], max.(cleanimg, 1.0e-20), colormap = :afmhot, colorscale = log10, colorrange = crange); axs[2].title = "CLEAN (Nominal beam)"
image!(axs[3], max.(cleanimg25, 1.0e-20), colormap = :afmhot, colorscale = log10, colorrange = crange); axs[3].title = "CLEAN (25% beam)"
hidedecorations!.(axs)
fig |> DisplayAs.PNG |> DisplayAs.Text
From the plot you can see that the Comrade reconstruction is significantly superresolved compared to the CLEAN reconstruction with the nominal beam. If we use a smaller beam for CLEAN we see a reconstruction that is more similar to Comrade. However, unlike CLEAN Comrade automatically infers the effective resolution from the data itself and does not require a restoring beam.
Additionally, Comrade allows us to fully explore the distributions of images that are consistent with the data. For example, we can plot a few random samples from the posterior to see the variety of images that are consistent with the data.
julia
fig = Figure(; resolution = (800, 450))
axs = [Axis(fig[i, j], xreversed = true, aspect = DataAspect()) for i in 1:2, j in 1:3]
map(enumerate(axs)) do (i, ax)
hidedecorations!(ax)
image!(ax, sample(imgs), colormap = :afmhot, colorscale = log10, colorrange = crange)
text!(ax, 0.05, 0.9, text = "χ²= $(round(mean(chi2(post, chain[51:5:end][i]; reduce = true)); digits = 2))", space = :relative, color = :white)
end
axcl = Axis(fig[1:2, 4], xreversed = true, aspect = DataAspect())
hidedecorations!(axcl)
image!(axcl, max.(cleanimg25, 1.0e-20), colormap = :afmhot, colorscale = log10, colorrange = crange)
axcl.title = "CLEAN (25% beam)"
Label(fig[0, 1:3], "Comrade Post. Samples", tellheight = true)
rowgap!(fig.layout, 1, 0.0)
fig
In summary, we have demonstrated how to use Comrade to reconstruct VLBA data of an AGN using only closure quantities. Additionally, we have shown how to use a Markov Random Field expansion to model the power spectrum of the AGN. This allows us to model more complex structures in the AGN jet and infer the power spectrum directly from the data.
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