1. Numerical evaluation of parameter correlation in the Hartmann-Tran line profile
Erin M. Adkins, Zachary D. Reed, and Joseph T. Hodges
Material Measurement Laboratory
National Institute of Standards & Technology
Gaithersburg, MD 20899, USA

2. Advanced Lineshapes Voigt Profile is characterized by:
Doppler Broadening - ΓD
Collisional Broadening – Γ0, Δ0
Advanced lineshapes can also account for higher order effects such as:
Collisional induced velocity changes (VC) - νVC
Hard collision models
Soft collision models
Speed dependence of relaxation rates (SD) – Γ2, Δ2
Correlation between velocity and rotations state changes due to collisions - η
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0
Recommended isolated-line profile for representing high-resolution spectroscopic transitions (IUPAC Technical Report). Pure and Applied Chemistry 2014, 86 (12),

3. Advanced Lineshapes: HTP
partially –Correlated, quadratic speed-dependent, hard-collision profile (pCqSDHCP)  Hartmann-Tran Profile
Profile
Parameters
Limit of HTP
Voigt (VP)
ΓD, Γ0, Δ0
Γ2 = Δ2= νvc = η = 0
Rautian (RP)
Nelkin-Ghatak (NGP)
ΓD, Γ0, Δ0, νvc
Γ2 = Δ2 = η = 0
Speed-dependent Voigt (SDVP)
ΓD, Γ0, Δ0, Γ2, Δ2
νvc = η = 0
Speed-dependent Rautian (SDRP)
Speed-dependent Nelkin-Ghatak (SDNGP)
ΓD, Γ0, Δ0, Γ2, Δ2, νvc
η = 0
Hartmann-Tran (HTP)
ΓD, Γ0, Δ0, Γ2, Δ2, νvc, η
Stick with RP or alternatively NGP
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0, 𝑝𝑡𝑠 Γ 𝐷 =5
Recommended isolated-line profile for representing high-resolution spectroscopic transitions (IUPAC Technical Report). Pure and Applied Chemistry 2014, 86 (12),

4. Application of Advanced Lineshapes to Experiment
Focus of study
Ability to accurately retrieve advanced lineshape parameters from noisy HTP spectra
I = HTP(ΓD, Γ0, Δ0, Γ2, Δ2, νvc, η)
Ability to accurately fit simulated spectra in the limits of the HTP
I = HTP(ΓD, Γ0, Δ0, Γ2, Δ2, νvc, η)
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0 , 𝑝𝑡𝑠 Γ 𝐷 =5

5. Methodology: Simulation and Fitting
Γ 0 Γ 𝐷
0.1
0.5 1 5 10 50
100
Line Shape Parameters
ΓD: 1
Γ0: 0.1ΓD, 0.5ΓD, 1ΓD, 5ΓD, 10ΓD, 50ΓD, 100ΓD
Γ2: 0.01 Γ0, 0.1 Γ0, 0.5 Γ0
νvc: 0.01 Γ0, 0.1 Γ0, 0.5 Γ0
η: 0, 0.1, 0.33, 0.66, 0.9
Spectral Parameters
Spectral Density (Pts/ ΓD): 2, 5, 10, 20
Spectral width: -10ΓV - 10ΓV
Baseline: ±(10ΓV - 7.5ΓV)
Noise: 0%, 0.001%, 0.01%, 0.1%, 1%, 10%
Statistical/Fit Parameters
3 replicates of each combination of spectral and lineshape parameters
5 fits of each spectra with variable initial guess within bounds
Γ0: 0.9 Γ Γ0
Γ2: 0.1 Γ Γ2
νvc : 0.1 νvc - 10 νvc
η: 0 – 1
Mechanics
HITRAN Application Programming Interface (HAPI) for HTP definition
LMFIT: Non-linear Least-Square Minimization and Curve-Fitting for Python
Amplitude
Residuals
420 generated spectra * 3 for replicates
Frequency
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0 , 𝑝𝑡𝑠 Γ 𝐷 =5, Noise = 0.1 %
An isolated line-shape model to go beyond the Voigt Profile in spectroscopic databases and radiative transfer codes. Journal of Quantitative Spectroscopy and Radiative Transfer 2013, 129,
Efficient computation of some speed-dependent isolated line profiles. Journal of Quantitative Spectroscopy and Radiative Transfer 2013, 129,
HITRAN Application Programming Interface (HAPI): A comprehensive approach to working with spectroscopic data. Journal of Quantitative Spectroscopy and Radiative Transfer 2016, 177,
HITRAN Application Programming Interface (HAPI) - V , LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python, 2014.

6. Trends in Parameter Fractional Error using HTP
Normalized Fractional Error Separation
SNR
More important factor for νvc and η
Spectral Density
Only a significant factor for Γ0, where the fractional error difference is small between variables and the average fractional error is small Γ2
Significant for all parameters
νvc and η
Mostly significant for themselves
SNR
Spectral Density Γ2
νvc η
Average Fractional Error (%)
Fractional Error Standard Deviation (%) Γ0
7.2%
11.6%
55.0%
3.6%
22.6%
0.08
0.24
2.1%
0.1%
93.8%
0.4% 14 21
31.4%
1.1%
48.4%
12.8%
6.4% 18
18.7%
0.3%
49.2%
0.6%
31.2% 8 15
Fractional Error = | 𝑝 𝑠𝑖𝑚 − 𝑝 𝑓𝑖𝑡 | 𝑝 𝑠𝑖𝑚

7. νvc : Fractional Error η νvc Γ2 SNR 0.01 0.1 0.5 0.01 0.1 0.5 0.1 0.33
0.66
0.9
0.01
0.1
0.5
0.01
0.1
0.5
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
100,000
10,000
1,000
SNR

8. νvc : Fractional Error η νvc η Γ2 = 0.1 and SNR = 10,000 νvc Γ2 SNR
0.01
0.1
0.5
0.01
0.1
0.5
0.1
0.33
0.66
0.9
Γ2 = 0.1 and SNR = 10,000
0.01
0.1
0.5 η
0.01
0.1
0.5
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
νvc
100,000
10,000
1,000
SNR

9. η: Fractional Error η νvc Γ2 SNR 0.01 0.1 0.5 0.01 0.1 0.5 0.1 0.33
0.66
0.9
0.01
0.1
0.5
0.01
0.1
0.5
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
100,000
10,000
1,000
SNR

10. η: Fractional Error η νvc η Γ2 = 0.1 and SNR = 10,000 νvc Γ2 SNR 0.01
0.5
0.01
0.1
0.5
0.1
0.33
0.66
0.9
Γ2 = 0.1 and SNR = 10,000
0.01
0.1
0.5 η
0.01
0.1
0.5
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
νvc
100,000
10,000
1,000
SNR

11. vvc: Fractional Uncertainty versus Fractional Error
Percent difference between fit value and simulated value for a parameter
Unknown in typical experiments
Fractional Uncertainty:
Ratio of the reported fit uncertainty to the fitted parameter value
Fit uncertainty calculated from the diagonal of the covariance matrix neglects the correlation of terms and can lead to fractional uncertainties that don’t include the true value
Fractional Error
In fitting data we generally will only know the fractional uncertainty based on the fit and not the fractional error based on the true value, since that is unknown. Want at least a some idea of if the a fit uncertainty is x what is the actual uncertainty in the measurement
Actually find the same order of magnitude trend across the board for the various fitted parameters
Looking at the converged cases only
Can see that Noise is obviously the driving force in the convergence of Gamma0
In reality we won’t know if Gamma0 is too high or too low, so we will look at the correlation between Fractional uncertainty and fraction error based on the absolute value.
Fractional Uncertainty

12. Correlation between Fit ParametersΓ0
Markov Chain Monte-Carlo ensemble sampler
Provides visualization of correlation between parameter by looking at one dimensional and two dimensional probability distributions of parameters Γ2
νVC
Markov chain Monte Carlo ensemble sampler takes the initial guesses of the least squares fit. Sampling around that solution and giving the  one and two dimensional projections of the posterior probability distributions of your parameters. This is useful because it quickly demonstrates all of the covariances between parameters.  That means that the corner plot shows the marginalized distribution for each parameter independently in the histograms along the diagonal and then the marginalized two dimensional distributions in the other panels. η Γ0
νVC η Γ2
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0, 𝑝𝑡𝑠 Γ 𝐷 =20

13. Correlation Between νvc and η
An isolated line-shape model to go beyond the Voigt Profile in spectroscopic databases and radiative transfer codes. Journal of Quantitative Spectroscopy and Radiative Transfer 2013, 129,

14. Correlation Between νvc and η
𝜈 𝑣𝑐 𝑣 = 𝜈 𝑣𝑐 −𝜂 Γ 𝑣 −𝑖Δ(𝑣)
An isolated line-shape model to go beyond the Voigt Profile in spectroscopic databases and radiative transfer codes. Journal of Quantitative Spectroscopy and Radiative Transfer 2013, 129,

15. QF Ratio and Lineshape HTP HTP – νvc fixed SDNGP SDVP NGP VP
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
SNR
100,000
10,000
1,000
100 10

16. QF and Lineshape Γ2 = νvc = η = 0.1 SNR = 100,000 SNR = 10,000
𝑄 𝐹 𝑓𝑖𝑡 𝑄 𝐹 𝑠𝑖𝑚
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.
HTP
HTP – νvc fixed
NGP VP
SDVP
SDNGP
HTP
HTP – νvc fixed
NGP VP
SDVP
SDNGP
HTP
HTP – νvc fixed
NGP VP
SDVP
SDNGP

17. Conclusions
The Hartmann-Tran profile accounts for collisional broadening, collision-induced velocity changes, speed-dependence, and the correlation between velocity- changing and rotational-state changing collisions.
The ability to accurately determine HTP parameters from noisy spectra and to accurately reproduce noisy spectra with the HTP lineshape is primarily related to (in order):
Spectral noise
parameter space
Spectral density
Correlations between η and νvc are significant and generally following the relationship 𝜈 𝑣𝑐 −𝜂 Γ 0 .
Next steps: Simulating spectra based on parameters generated from fits of real experimental data to explore parameter error and correlations in conditions matching experiment
Stick with RP or alternatively NGP

18. Correlation between Noise and SNR
Looking at cases where the noise was not equal to zero
These cases only look at where Noise !=0 (because then the SNR is infinity), but for the most part Gaussian noise of 0.01% correlated to a SNR of and then there is an order of magnitude decrease in the in the SNR with each order of magnitude increase in the percent Noise level.
Additionally increasing the spectral density seems to shrink the distribution of SNR and there does seem to be some trend at the lower noise levels with the SNR actually being smaller at the higher Spectral density.
SNR was calculated as the Max peak in the spectra with Noise added, divided by the standard deviation in the residuals evaluated at the simulated lineshape (over the outside +/- 7.5 Lineshape)
Discuss use of SNR for an analogue to experiments, but will also use QF to describe the ability of a combination of parameters to define a given lineshape

19. Correlation between QF Ratio and Lineshape
HTP
HTP – νvc fixed
SDNGP
SDVP
NGP VP
𝑄 𝐹 𝑓𝑖𝑡 𝑄 𝐹 𝑠𝑖𝑚
Second figure shows that the QF will only increase if you are at a noise level that it is appropriate. But also that there you can overfit your spectra and get larger QF than what we know in this case is the QF that is intrinsic to that spectra.

20. Signal to Noise Ratio (SNR) and Quality of Fit (QF)
Spectrum Defined
Fit Defined
NGP
Looking at cases where the noise was not equal to zero
These cases only look at where Noise !=0 (because then the SNR is infinity), but for the most part Gaussian noise of 0.01% correlated to a SNR of and then there is an order of magnitude decrease in the in the SNR with each order of magnitude increase in the percent Noise level.
Additionally increasing the spectral density seems to shrink the distribution of SNR and there does seem to be some trend at the lower noise levels with the SNR actually being smaller at the higher Spectral density.
SNR was calculated as the Max peak in the spectra with Noise added, divided by the standard deviation in the residuals evaluated at the simulated lineshape (over the outside +/- 7.5 Lineshape)
Discuss use of SNR for an analogue to experiments, but will also use QF to describe the ability of a combination of parameters to define a given lineshape
Γ 0 Γ 𝐷 = 1, 𝜈 𝑣𝑐 Γ 0 = Γ 2 Γ 0 =𝜂=0.1, Δ 0 Γ 0 = Δ 2 Γ 2 =0 , 𝑝𝑡𝑠 Γ 𝐷 =5, Noise = 0.1 %

21. Γ2: Fractional Error η νvc Γ2 SNR 0.01 0.1 0.5 0.01 0.1 0.5 0.1 0.33
0.66
0.9
0.01
0.1
0.5
0.01
0.1
0.5
100,000
10,000
1,000
SNR

22. Γ2: Fractional Error η νvc η Γ2 = 0.1 and SNR = 10,000 νvc Γ2 SNR 0.01
0.5
0.01
0.1
0.5
0.1
0.33
0.66
0.9
Γ2 = 0.1 and SNR = 10,000
0.01
0.1
0.5 η
0.01
0.1
0.5
νvc
100,000
10,000
1,000
SNR

23. Fractional Error and Noise
Restricted to only SNR above 1000
Both nuVC shows no trend, eta shows that the case where eta is zero is the outlier in the general trend and after that the uncertainty decreases with an increase in the eta parameter. However these effects are very small.
Only aw and the sampling parameters Noise and spectral density actually significant differences.