Getting Started with Transmission Spectroscopy ============================================== Last update: March 15th (2025) Hajime Kawahara for v2.0 In this getting started guide, we will use ExoJAX to simulate a high-resolution **transmission** spectrum from an atmosphere with CO molecular absorption and hydrogen molecule CIA continuum absorption as the opacity sources. We will then add appropriate noise to the simulated spectrum to create a mock spectrum and perform spectral retrieval using NumPyro’s HMC NUTS. First, we recommend 64-bit if you do not think about numerical errors. Use jax.config to set 64-bit. (But note that 32-bit is sufficient in most cases. Consider to use 32-bit (faster, less device memory) for your real use case.) .. code:: ipython3 from jax import config config.update("jax_enable_x64", True) The following schematic figure explains how ExoJAX works; (1) loading databases (``*db``), (2) calculating opacity (``opa``), (3) running atmospheric radiative transfer (``art``), (4) applying operations on the spectrum (``sop``) In this “getting started” guide, there are two opacity sources, CO and CIA. Their respective databases, ``mdb`` and ``cdb``, are converted by ``opa`` into the opacity of each atmospheric layer, which is then used in the radiative transfer calculation performed by ``art``. Finally, ``sop`` convolves instrumental profiles, generating the emission spectrum. ``mdb``/``cdb`` –> ``opa`` –> ``art`` –> ``sop`` —> spectrum This spectral model is incorporated into the probabilistic model in NumPyro, and retrieval is performed by sampling using HMC-NUTS. .. figure:: https://secondearths.sakura.ne.jp/exojax/figures/exojax_get_started_transmission.png :alt: Figure. Structure of ExoJAX Figure. Structure of ExoJAX 1. Loading a molecular database using mdb ----------------------------------------- ExoJAX has an API for molecular databases, called ``mdb`` (or ``adb`` for atomic datbases). Prior to loading the database, define the wavenumber range first. .. code:: ipython3 from exojax.utils.grids import wavenumber_grid nu_grid, wav, resolution = wavenumber_grid( 22920.0, 23000.0, 3500, unit="AA", xsmode="premodit" ) print("Resolution=", resolution) .. parsed-literal:: xsmode = premodit xsmode assumes ESLOG in wavenumber space: xsmode=premodit Your wavelength grid is in *** descending *** order The wavenumber grid is in ascending order by definition. Please be careful when you use the wavelength grid. Resolution= 1004211.9840291934 .. parsed-literal:: /home/kawahara/exojax/src/exojax/spec/unitconvert.py:82: UserWarning: Both input wavelength and output wavenumber are in ascending order. warnings.warn( Then, let’s load the molecular database. We here use Carbon monoxide in Exomol. ``CO/12C-16O/Li2015`` means ``Carbon monoxide/ isotopes = 12C + 16O / database name``. You can check the database name in the ExoMol website (https://www.exomol.com/). .. code:: ipython3 from exojax.spec.api import MdbExomol mdb = MdbExomol(".database/CO/12C-16O/Li2015", nurange=nu_grid) .. parsed-literal:: /home/kawahara/exojax/src/exojax/utils/molname.py:197: FutureWarning: e2s will be replaced to exact_molname_exomol_to_simple_molname. warnings.warn( /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( /home/kawahara/exojax/src/exojax/utils/molname.py:91: FutureWarning: exojax.utils.molname.exact_molname_exomol_to_simple_molname will be replaced to radis.api.exomolapi.exact_molname_exomol_to_simple_molname. warnings.warn( .. parsed-literal:: HITRAN exact name= (12C)(16O) radis engine = pytables => Downloading from http://www.exomol.com/db/CO/12C-16O/Li2015/12C-16O__Li2015.def => Downloading from http://www.exomol.com/db/CO/12C-16O/Li2015/12C-16O__Li2015.pf => Downloading from http://www.exomol.com/db/CO/12C-16O/Li2015/12C-16O__Li2015.states.bz2 => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__H2.broad => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__He.broad => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__air.broad => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__self.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__self.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__Ar.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__Ar.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__CH4.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__CH4.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__CO.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__CO.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__CO2.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__CO2.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__H2.broad => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__H2O.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__H2O.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__N2.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__N2.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__NH3.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__NH3.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__NO.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__NO.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__O2.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__O2.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__NH3.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__NH3.broad and save. => Downloading from http://www.exomol.com/db/CO/12C-16O/12C-16O__CS.broad Error: Couldn't download .broad file at http://www.exomol.com/db/CO/12C-16O/12C-16O__CS.broad and save. Summary of broadening files downloaded: Success: ['H2' 'He' 'air' 'H2'] Fail: ['self' 'Ar' 'CH4' 'CO' 'CO2' 'H2O' 'N2' 'NH3' 'NO' 'O2' 'NH3' 'CS'] Note: Caching states data to the pytables format. After the second time, it will become much faster. Molecule: CO Isotopologue: 12C-16O ExoMol database: None Local folder: .database/CO/12C-16O/Li2015 Transition files: => File 12C-16O__Li2015.trans => Downloading from http://www.exomol.com/db/CO/12C-16O/Li2015/12C-16O__Li2015.trans.bz2 => Caching the *.trans.bz2 file to the pytables (*.h5) format. After the second time, it will become much faster. => You can deleted the 'trans.bz2' file by hand. Broadener: H2 Broadening code level: a0 .. parsed-literal:: /home/kawahara/miniconda3/lib/python3.12/site-packages/radis-0.16-py3.12.egg/radis/api/exomolapi.py:687: AccuracyWarning: The default broadening parameter (alpha = 0.07 cm^-1 and n = 0.5) are used for J'' > 80 up to J'' = 152 warnings.warn( 2. Computation of the Cross Section using opa --------------------------------------------- ExoJAX has various opacity calculator classes, so-called ``opa``. Here, we use a memory-saved opa, ``OpaPremodit``. We assume the robust tempreature range we will use is 500-1500K. .. code:: ipython3 from exojax.spec.opacalc import OpaPremodit opa = OpaPremodit(mdb, nu_grid, auto_trange=[500.0, 1500.0], dit_grid_resolution=1.0) .. parsed-literal:: /home/kawahara/exojax/src/exojax/spec/opacalc.py:348: UserWarning: dit_grid_resolution is not None. Ignoring broadening_parameter_resolution. warnings.warn( .. parsed-literal:: OpaPremodit: params automatically set. default elower grid trange (degt) file version: 2 Robust range: 485.7803992045456 - 1514.171191195336 K OpaPremodit: Tref_broadening is set to 866.0254037844389 K # of reference width grid : 2 # of temperature exponent grid : 2 max value of ngamma_ref_grid : 9.450919102366303 min value of ngamma_ref_grid : 7.881095721823979 ngamma_ref_grid grid : [7.88109541 9.4509201 ] max value of n_Texp_grid : 0.658 min value of n_Texp_grid : 0.5 n_Texp_grid grid : [0.49999997 0.65800005] .. parsed-literal:: uniqidx: 0it [00:00, ?it/s] .. parsed-literal:: Premodit: Twt= 1108.7151960064205 K Tref= 570.4914318566549 K Making LSD:|####################| 100% cross section (xsvector/xsmatrix) is calculated in the closed mode. The aliasing part cannnot be used. wing cut width = [15.12718787427093, 15.23298725175755] cm-1 .. parsed-literal:: Then let’s compute cross section for two different temperature 500 and 1500 K for P=1.0 bar. opa.xsvector can do that! .. code:: ipython3 P = 1.0 # bar T_1 = 500.0 # K xsv_1 = opa.xsvector(T_1, P) # cm2 T_2 = 1500.0 # K xsv_2 = opa.xsvector(T_2, P) # cm2 Plot them. It can be seen that different lines are stronger at different temperatures. .. code:: ipython3 import matplotlib.pyplot as plt plt.plot(nu_grid, xsv_1, label=str(T_1) + "K") # cm2 plt.plot(nu_grid, xsv_2, alpha=0.5, label=str(T_2) + "K") # cm2 plt.yscale("log") plt.legend() plt.xlabel("wavenumber (cm-1)") plt.ylabel("cross section (cm2)") plt.show() .. image:: get_started_transmission_files/get_started_transmission_16_0.png 3. Atmospheric Radiative Transfer --------------------------------- ExoJAX can solve the radiative transfer and derive `the transmission spectrum <../userguide/rtransfer_transmission.html>`__. To do so, ExoJAX has ``art`` class. ``ArtTransPure`` means Atomospheric Radiative Transfer for **Transmission** with Pure absorption. So, ``ArtTransPure`` does not include scattering. You can choose either the trapezoid or Simpson’s rule as the integration scheme. The default setting is ``integration="simpson"``. We set the number of the atmospheric layer to 200 (nlayer) and the pressure at bottom and top atmosphere to 1 and 1.e-11 bar. .. code:: ipython3 from exojax.spec.atmrt import ArtTransPure art = ArtTransPure( pressure_btm=1.0e1, pressure_top=1.0e-11, nlayer=200, ) .. parsed-literal:: integration: simpson Simpson integration, uses the chord optical depth at the lower boundary and midppoint of the layers. .. parsed-literal:: /home/kawahara/exojax/src/exojax/spec/dtau_mmwl.py:13: FutureWarning: dtau_mmwl might be removed in future. warnings.warn("dtau_mmwl might be removed in future.", FutureWarning) /home/kawahara/exojax/src/exojax/spec/atmrt.py:53: UserWarning: nu_grid is not given. specify nu_grid when using 'run' warnings.warn( Let’s assume the power law temperature model, within 500 - 1500 K. :math:`T = T_0 P^\alpha` where :math:`T_0=1200` K and :math:`\alpha=0.1`. .. code:: ipython3 art.change_temperature_range(500.0, 1500.0) Tarr = art.powerlaw_temperature(1200.0, 0.1) Also, the mass mixing ratio of CO (MMR) should be defined. .. code:: ipython3 mmr_profile = art.constant_mmr_profile(0.01) Surface gravity is also important quantity of the atmospheric model, which is a function of planetary radius and mass. Unlike in the case of the emission spectrum, the transmission spectrum is affected by the opacity from the lower to the upper layers of the atmosphere. Therefore, it is better to calculate gravity as a function of altitude. To achieve this, the gravity and radius at the bottom of the atmospheric layer are specified as ``gravity_btm`` and ``radius_btm``, respectively, and the layer-by-layer gravity profile is computed using ``art.gravity_profile``. .. code:: ipython3 import jax.numpy as jnp from exojax.utils.astrofunc import gravity_jupiter from exojax.utils.constants import RJ gravity_btm = gravity_jupiter(1.0, 1.0) radius_btm = RJ mmw = 2.33*jnp.ones_like(art.pressure) # mean molecular weight of the atmosphere gravity = art.gravity_profile(Tarr, mmw, radius_btm, gravity_btm) When visualized, it looks like this. .. code:: ipython3 plt.plot(gravity, art.pressure) plt.plot(gravity_btm, art.pressure[-1], "ro", label="gravity_btm") plt.yscale("log") plt.xlim(2300,2600) plt.gca().invert_yaxis() plt.xlabel("gravity (cm/s2)") plt.ylabel("pressure (bar)") plt.legend() plt.show() .. image:: get_started_transmission_files/get_started_transmission_27_0.png In addition to the CO cross section, we would consider `collisional induced absorption `__ (CIA) as a continuum opacity. ``cdb`` class can be used. .. code:: ipython3 from exojax.spec.contdb import CdbCIA from exojax.spec.opacont import OpaCIA cdb = CdbCIA(".database/H2-H2_2011.cia", nurange=nu_grid) opacia = OpaCIA(cdb, nu_grid=nu_grid) .. parsed-literal:: H2-H2 Before running the radiative transfer, we need cross sections for layers, called ``xsmatrix`` for CO and ``logacia_matrix`` for CIA (strictly speaking, the latter is not cross section but coefficient because CIA intensity is proportional density square). See `here `__ for the details. .. code:: ipython3 xsmatrix = opa.xsmatrix(Tarr, art.pressure) logacia_matrix = opacia.logacia_matrix(Tarr) Convert them to opacity .. code:: ipython3 dtau_CO = art.opacity_profile_xs(xsmatrix, mmr_profile, mdb.molmass, gravity) vmrH2 = 0.855 # VMR of H2 dtaucia = art.opacity_profile_cia(logacia_matrix, Tarr, vmrH2, vmrH2, mmw[:, None], gravity) Add two opacities. .. code:: ipython3 dtau = dtau_CO + dtaucia .. code:: ipython3 gravity_btm .. parsed-literal:: 2478.57730044555 Then, run the radiative transfer. As you can see, the emission spectrum has been generated. This spectrum shows a region near 4360 cm-1, or around 22940 AA, where CO features become increasingly dense. This region is referred to as the band head. If you’re interested in why the band head occurs, please refer to `Quatum states of Carbon Monoxide and Fortrat Diagram `__. .. code:: ipython3 Rp2 = art.run(dtau, Tarr, mmw, radius_btm, gravity_btm) Rp = jnp.sqrt(Rp2) .. code:: ipython3 fig = plt.figure(figsize=(15, 4)) plt.plot(nu_grid, Rp) plt.xlabel("wavenumber (cm-1)") plt.ylabel("planet radius (RJ)") plt.show() .. image:: get_started_transmission_files/get_started_transmission_39_0.png To examine the contribution of each atmospheric layer to the transmission spectrum, one can, for example, look at the optical depth along the chord direction. This can be done as follows: .. code:: ipython3 from exojax.spec.opachord import chord_geometric_matrix from exojax.spec.opachord import chord_optical_depth normalized_height, normalized_radius_lower = art.atmosphere_height(Tarr, mmw, radius_btm, gravity_btm) cgm = chord_geometric_matrix(normalized_height, normalized_radius_lower) dtau_chord = chord_optical_depth(cgm, dtau) By plotting the data, it becomes clear that in the case of transmitted light, information from a wide range of atmospheric layers, from the upper to the lower layers, is included. .. code:: ipython3 from exojax.plot.atmplot import plottau plottau(nu_grid, dtau_chord, Tarr, art.pressure) .. parsed-literal:: /home/kawahara/exojax/src/exojax/plot/atmplot.py:51: SyntaxWarning: invalid escape sequence '\m' plt.xlabel("wavenumber ($\mathrm{cm}^{-1}$)") /home/kawahara/exojax/src/exojax/plot/atmplot.py:68: SyntaxWarning: invalid escape sequence '\m' labelx["um"] = "wavelength ($\mu \mathrm{m}$)" /home/kawahara/exojax/src/exojax/plot/atmplot.py:70: SyntaxWarning: invalid escape sequence '\A' labelx["AA"] = "wavelength ($\AA$)" /home/kawahara/exojax/src/exojax/plot/atmplot.py:71: SyntaxWarning: invalid escape sequence '\m' labelx["cm-1"] = "wavenumber ($\mathrm{cm}^{-1}$)" /home/kawahara/exojax/src/exojax/plot/atmplot.py:24: UserWarning: nugrid looks in log scale, results in a wrong X-axis value. Use log10(nugrid) instead. warnings.warn( .. image:: get_started_transmission_files/get_started_transmission_43_1.png 4. Spectral Operators:　instrumental profile, Doppler velocity shift and so on, any operation on spectra. --------------------------------------------------------------------------------------------------------- The above spectrum is called “raw spectrum” in ExoJAX. The effects applied to the raw spectrum is handled in ExoJAX by the spectral operator (``sop``). Then, the instrumental profile with relative radial velocity shift is applied. Also, we need to match the computed spectrum to the data grid. This process is called ``sampling`` (but just interpolation though). Below, let’s perform a simulation that includes noise for use in later analysis. .. code:: ipython3 from exojax.spec.specop import SopInstProfile from exojax.utils.instfunc import resolution_to_gaussian_std sop_inst = SopInstProfile(nu_grid, vrmax=1000.0) RV = 40.0 # km/s resolution_inst = 30000.0 beta_inst = resolution_to_gaussian_std(resolution_inst) Rp2_inst = sop_inst.ipgauss(Rp2, beta_inst) nu_obs = nu_grid[::5][:-50] from numpy.random import normal noise = 0.001 Fobs = sop_inst.sampling(Rp2_inst, RV, nu_obs) + normal(0.0, noise, len(nu_obs)) .. code:: ipython3 fig = plt.figure(figsize=(12, 6)) ax = fig.add_subplot(111) plt.errorbar(nu_obs, Fobs, noise, fmt=".", label="RV + IP (sampling)", color="gray",alpha=0.5) plt.xlabel("wavenumber (cm-1)") plt.legend() plt.show() .. image:: get_started_transmission_files/get_started_transmission_48_0.png 5. Retrieval of a Transmission Spectrum --------------------------------------- Next, let’s perform a “retrieval” on the simulated spectrum created above. Retrieval involves estimating the parameters of an atmospheric model in the form of a posterior distribution based on the spectrum. To do this, we first need a model. Here, we have compiled the forward modeling steps so far and defined the model as follows. The spectral model has six parameters. .. code:: ipython3 def fspec(T0, alpha, mmr, radius_btm, gravity_btm, RV): """ computes planet radius sqaure spectrum Args: T0 (float): temperature at 1 bar alpha (float): power law index of temperature mmr (float): Mass mixing ratio of CO radius_btm (float): radius at the bottom in cm gravity_btm (float): gravity at the bottom in cm/s2 RV (float): radial velocity in km/s Returns: _type_: _description_ """ Tarr = art.powerlaw_temperature(T0, alpha) gravity = art.gravity_profile(Tarr, mmw, radius_btm, gravity_btm) #molecule xsmatrix = opa.xsmatrix(Tarr, art.pressure) mmr_arr = art.constant_mmr_profile(mmr) dtau = art.opacity_profile_xs(xsmatrix, mmr_arr, opa.mdb.molmass, gravity) #continuum logacia_matrix = opacia.logacia_matrix(Tarr) dtaucH2H2 = art.opacity_profile_cia(logacia_matrix, Tarr, vmrH2, vmrH2, mmw[:, None], gravity) #total tau dtau = dtau + dtaucH2H2 Rp2 = art.run(dtau, Tarr, mmw, radius_btm, gravity_btm) Rp2_inst = sop_inst.ipgauss(Rp2, beta_inst) mu = sop_inst.sampling(Rp2_inst, RV, nu_obs) return mu Let’s verify that spectra are being generated from ``fspec`` with various parameter sets. .. code:: ipython3 fig = plt.figure(figsize=(12, 3)) plt.plot(nu_obs, fspec(1200.0, 0.09, 0.01, RJ, gravity_jupiter(1.0, 1.0), 40.0),label="model") plt.plot(nu_obs, fspec(1400.0, 0.12, 0.01, RJ, gravity_jupiter(1.0, 1.3), 20.0),label="model") .. parsed-literal:: [] .. image:: get_started_transmission_files/get_started_transmission_53_1.png NumPyro is a probabilistic programming language (PPL), which requires the definition of a probabilistic model. In the probabilistic model ``model_prob`` defined below, the prior distributions of each parameter are specified. The previously defined spectral model is used within this probabilistic model as a function that provides the mean :math:`\mu`. The spectrum is assumed to be generated according to a Gaussian distribution with this mean and a standard deviation :math:`\sigma`. i.e. :math:`f(\nu_i) \sim \mathcal{N}(\mu(\nu_i; {\bf p}), \sigma^2 I)`, where :math:`{\bf p}` is the spectral model parameter set, which are the arguments of ``fspec``. .. code:: ipython3 from numpyro.infer import MCMC, NUTS import numpyro.distributions as dist import numpyro from jax import random .. parsed-literal:: /home/kawahara/miniconda3/lib/python3.12/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html from .autonotebook import tqdm as notebook_tqdm .. code:: ipython3 def model_prob(spectrum): #atmospheric/spectral model parameters priors logg = numpyro.sample('logg', dist.Uniform(3.0, 4.0)) RV = numpyro.sample('RV', dist.Uniform(35.0, 45.0)) mmr = numpyro.sample('MMR', dist.Uniform(0.0, 0.015)) T0 = numpyro.sample('T0', dist.Uniform(1000.0, 1500.0)) alpha = numpyro.sample('alpha', dist.Uniform(0.05, 0.2)) radius_btm = numpyro.sample('rb', dist.Normal(1.0,0.05)) mu = fspec(T0, alpha, mmr, radius_btm*RJ, 10**logg, RV) #noise model parameters priors sigmain = numpyro.sample('sigmain', dist.Exponential(1000.0)) numpyro.sample('spectrum', dist.Normal(mu, sigmain), obs=spectrum) Now, let’s define NUTS and start sampling. .. code:: ipython3 rng_key = random.PRNGKey(0) rng_key, rng_key_ = random.split(rng_key) num_warmup, num_samples = 500, 1000 #kernel = NUTS(model_prob, forward_mode_differentiation=True) kernel = NUTS(model_prob, forward_mode_differentiation=False) Since this process will take several hours, feel free to go for a long lunch break! .. code:: ipython3 mcmc = MCMC(kernel, num_warmup=num_warmup, num_samples=num_samples) mcmc.run(rng_key_, spectrum=Fobs) mcmc.print_summary() .. parsed-literal:: sample: 100%|██████████| 1500/1500 [2:23:08<00:00, 5.73s/it, 127 steps of size 1.14e-02. acc. prob=0.94] .. parsed-literal:: mean std median 5.0% 95.0% n_eff r_hat MMR 0.01 0.00 0.01 0.01 0.01 309.22 1.00 RV 39.79 0.16 39.79 39.53 40.05 709.96 1.00 T0 1130.71 53.36 1126.14 1044.55 1215.44 396.74 1.00 alpha 0.09 0.01 0.09 0.08 0.11 309.49 1.00 logg 3.37 0.03 3.37 3.32 3.42 402.09 1.00 rb 1.00 0.05 1.00 0.91 1.09 670.37 1.00 sigmain 0.00 0.00 0.00 0.00 0.00 760.18 1.00 Number of divergences: 0 .. parsed-literal:: After returning from your long lunch, if you’re lucky and the sampling is complete, let’s write a predictive model for the spectrum. .. code:: ipython3 from numpyro.diagnostics import hpdi from numpyro.infer import Predictive import jax.numpy as jnp .. code:: ipython3 # SAMPLING posterior_sample = mcmc.get_samples() pred = Predictive(model_prob, posterior_sample, return_sites=['spectrum']) predictions = pred(rng_key_, spectrum=None) median_mu1 = jnp.median(predictions['spectrum'], axis=0) hpdi_mu1 = hpdi(predictions['spectrum'], 0.9) .. code:: ipython3 fig, ax = plt.subplots(nrows=1, ncols=1, figsize=(15, 4.5)) ax.plot(nu_obs, median_mu1, color='C1') ax.fill_between(nu_obs, hpdi_mu1[0], hpdi_mu1[1], alpha=0.3, interpolate=True, color='C1', label='90% area') ax.errorbar(nu_obs, Fobs, noise, fmt=".", label="mock spectrum", color="black",alpha=0.5) plt.xlabel('wavenumber (cm-1)', fontsize=16) plt.legend(fontsize=14) plt.tick_params(labelsize=14) plt.show() .. image:: get_started_transmission_files/get_started_transmission_64_0.png You can see that the predictions are working very well! Let’s also display a corner plot. Here, we’ve used ArviZ for visualization. .. code:: ipython3 import arviz pararr = ['T0', 'alpha', 'logg', 'MMR', 'radius_btm', 'RV'] arviz.plot_pair(arviz.from_numpyro(mcmc), kind='kde', divergences=False, marginals=True) plt.show() .. image:: get_started_transmission_files/get_started_transmission_66_0.png Further Information ------------------- Correlated noise can be introduced using a Gaussian process, and parameter estimation can be performed using SVI or Nested Sampling, just as in the case of emission spectra. See below for details. - `Including GP in an emission spectrum `__ - `SVI for an emission spectrum `__ - `Neste Sampling for an emission spectrum `__ Not enough GPU device memory? In that case, you can perform wavenumber splitting. See below for details. - `wavenumber stitching `__ Want to analyze JWST data? The Gallery and the following repositories may be helpful. - `ExoJAX gallery <../examples/index.html>`__ - `exojaxample_WASP39b `__ by Shotaro Tada (@sh-tada) That’s it.