The concepts of standard analysis techniques applied in the field of Fourier spectroscopy treat fundamental aspects insufficiently. For example, the spectra to be inferred are influenced by the noise contribution to the interferometric data, by nonprobed spatial domains which are linked to Fourier coefficients above a certain order, by the spectral limits which are in general not given by the Nyquist assumptions, and by additional parameters of the problem at hand like the zeropath difference. To consider these fundamentals, a probabilistic approach based on Bayes’ theorem is introduced which exploits multivariate normal distributions. For the example application, we model the spectra by the Gaussian process of a Brownian bridge stated by a prior covariance. The spectra themselves are represented by a number of parameters which map linearly to the data domain. The posterior for these linear parameters is analytically obtained, and the marginalisation over these parameters is trivial. This allows the straightforward investigation of the posterior for the involved nonlinear parameters, like the zeropath difference location and the spectral limits, and hyperparameters, like the scaling of the Gaussian process. With respect to the linear problem, this can be interpreted as an implementation of Ockham’s razor principle.
Fourier spectroscopy is a diagnostic application which reveals information about spectral quantities like refractive index, absorption, and transmission of a medium under test. In addition, the characterisation in absolute terms is possible for broadband spectra, for example, emitted by electrons of a hightemperature plasma, being magnetically confined [
Commonly, an interferometer diagnostic, let us say of Michelson [
One misconception, arising from the standard formulation, is that certain spectral information must be lost inherently, because only a finite amount of data is acquired. This is proven in standard literature by evaluating the convolution function which has a finite full width at half maximum (FWHM), implying that only a finite amount of Fourier coefficients is accessible via measurements. While this conclusion remains valid when a continuous spectrum is probed, the reasoning does not hold in general for a discrete spectrum. This fact was exploited to develop a (self)deconvolution procedure, so that some discrete lines which were separated by less than the FWHM width of the convolution function have been inferred [
Opposed to the standard data analysis techniques, a probabilistic ansatz was introduced to estimate model parameters, like amplitude spectra and frequencies, in the field of Fourier spectroscopy [
One of the advantages of the Bayesian approach is that different models and, hence, their assumptions can be compared with each other also known as Ockham’s razor. This enables the identification of the best, that is, the most likely, model, complying with the data. Given this context, the (self)deconvolution procedure mentioned above is interpreted here as an optimisation to find a minimising set of discrete frequencies to describe the data sufficiently. In general, the most fundamental issue is whether the spectrum to be inferred is discrete or continuous. If a discrete spectrum is more likely or follows by a physics model, how many discrete frequencies are involved and what are their estimates including uncertainties? Each frequency is associated with an amplitude and a phase which need to be estimated as well. If a continuous spectrum is at hand, then the spectral limits are of interest. In addition, in case, the underlying physics process is understood, what is the uncertainty on the spectrum and the phase following from experimentally inaccessible regions in the data domain.
The investigation of the fundamental issues listed above is quite challenging from numerical point of view. However, the computational effort is largely reduced when even and odd amplitudes are used instead of the phase and amplitude. This ensures a linear dependence between the even and odd amplitude parameters and the data. Formulating the prior information about all even and odd amplitudes as a multivariate normal with a specific prior mean and covariance gives straightforwardly the posterior mean and covariance. Furthermore, the marginalisation can be carried out analytically for these linear parameters. The remaining posterior quantity carries information about the nonlinear model parameters like frequencies or spectral limits and socalled hyperparameters, entering merely in the prior.
In Section
Commonly, the complementary coordinates used for Fourier transformations are the frequency
The realvalued continuous functions
Replacing
The spectral domain over which the integration is performed in (
Another representation uses the amplitude
If the spectral domain is bandlimited, such that
When the bandwidth is finite, one can express the spectral functions by Fourier coefficients multiplied each with the associated sinusoidal basis function of order
The second contribution causes a modulation of
With respect to the spatial origin, the transformed basis functions of the coefficients for
Some basis functions in the spatial domain are shown in Figures
Basis functions in spatial domain of Fourier coefficients (of order
The functions
In the spatial domain, the basis functions for
For
Basis functions in spatial domain of Fourier coefficients
Parseval’s theorem states abstractly that the length of a function in the spectral domain equals the length of its Fourier transform counterpart in the spatial domain. The length
The function
The Fourier transform can be performed by an interferometer, achieving an optical path difference between two partial beams, and the realvalued function
In the spatial domain,
The finite spatial sampling leaves
A diagnostic limitation is that the distance
Since any measurement has a noise contribution, the noisy data value can be written as
The combination of the relation (
The basic model is a starting point and must be amended by diagnostic imperfections and specifics to the interferometer design type.
To infer the spectral quantities
The model used and stated by (
Several steps are carried out to deduce the quantities of main interest
Two fundamental assumptions, called Nyquist assumptions in the following, are made by setting the spectral limits to 0 and the Nyquist frequency
The uncertainty of a Fourier coefficient, relying on the Nyquist assumptions, scales like
The spatial origin or zeropath difference
With
Having only a finite amount of Fourier coefficients probed causes the Gibbs phenomenon to appear for the spectral quantities inferred. To reduce this ringing feature, window functions are applied in the spatial domain to bring the interferometric data smoothly to zero towards the sampling limits. More precise, probed Fourier coefficients of higher orders are damped out, and a window function corresponds to a certain convolution function in the spectral domain. Hence, a weighted averaging of the spectral quantities is carried out which reduces the ringing. This approach can give a good global approximation of
Implicitly, the application of window functions excludes the investigation of the uncertainty on
The joint probability density function (pdf)
The product rule
Bayes’ rule can be extended to
Let the joint pdf for the random vector
If the dependency between the data and the parameters of interest is linear, and the likelihood and the prior can be expressed by multivariate normals, then the evaluation of the posterior is analytically straightforward. Such a model is the starting point for investigating a more complex model which includes parameters with a nonlinear mapping to the data domain and/or hyperparameters.
The linear model is amended by hyperparameters, entering in some way in the prior, and parameters with a nonlinear connection to the data domain. Such a model is then applicable in the field of Fourier spectroscopy.
Unfortunately, a general analytical expression is not available for this posterior, and, thus, it needs to be investigated numerically for the problem at hand. In order to do so, the quantity
The continuous even and odd spectral functions to be inferred can be modelled each by a Gaussian process [
The Brownian bridge is a continuous stochastic process for an interval, say from 0 to
The even and odd spectral functions, being finite for the interval
with the unit
The parameters
The Brownian bridge covariance function for the spectral domain can be studied in the domain of the Fourier coefficients via the coordinate transform stated in (
For orders greater than 0,
According to Parseval’s theorem (see (
For the even process, the signal level can be estimated by the envelope
For the envelope
The MartinPuplett interferometer diagnostic [
The data set
(a) Difference interferogram
The components
The diagnostic is set up, so that the sampling of the interferogram is triggered ideally when the optical path difference has changed by the increment
The upward trend of
The kernel of the spectral integration in (
Finally, the joint prior
In the following, the linear parameters are summerised by the set
The matrix
To give some insight, the conditional posterior for the amplitudes is evaluated given the specific set of values for
The values for
The spectral domain is covered from
Examination of Gaussian prior for even and odd spectra modelled by Brownian bridge process. The spectral discretisation
To match the maximum order (330) of probed Fourier coefficient, the discretisation is set to
The hyperparameters are set to large values like
With the chosen spectral priors, sample functions/vectors
For
Investigation of conditional amplitude posterior
The offset
Since the quantity
For the spectral quantities, the square root of the main diagonal elements of
For the doublesided region
With the specific values of the settings one gets the number
As the problem is formulated, the settings posterior is proportional to
Tendimensional quantity
To ease the computational effort but still being able to characterise
Characterisation of maxima of


4  3  2  1  1/2 

33.499  33.243  33.277  32.817  32.445  32.425 

913.572  913.243  912.277  912.817  913.445  913.425 

0.9152  1.0127  1.0623  1.1002  1.1224  1.1299 

5.7381  5.7731  5.7776  5.797  5.8126  5.8145 

−5.115293  −5.115297  −5.115288  −5.115291  −5.115295  −5.115295 

−4.814  −4.814  −4.814  −4.814  −4.814  −4.814 

62.150  62.148  62.146  62.143  62.142  62.142 

6.031  6.003  6.003  6.000  5.998  5.998 

11.541  9.884  9.913  9.725  9.575  9.573 

1052035.05  1052037.54  1052038.52  1052039.15  1052039.48  1052039.55 
Odds  0.011  0.134  0.357  0.668  0.929  1 
Odds for maxima of
For
For
Posterior standard deviations dependent on


4  3  2  1  1/2 

1.742  1.795  1.746  1.762  1.754  1.718 

3.101  3.155  2.927  2.869  2.858  2.810 

0.1064  0.1179  0.1242  0.1290  0.1316  0.1327 

1.5912  1.6064  1.6112  1.6164  1.6199  1.6211 

1.207  1.207  1.207  1.206  1.205  1.205 

0.170  0.170  0.170  0.170  0.170  0.170 
In the following, the amplitude posterior is investigated for the maximising settings given
(a) Posterior means
The posterior means
The histogram of the residuals (see Figure
The standard deviations
The covariances
Characterisation of posterior covariance for amplitude parameters obtained at maximum of settings posterior for
At the condition
The correlations
(a) Individual posterior sample functions
The samples mapped to the data domain give
For each sample, the sum
With the samples
Since only the marginal posterior
To reduce the numerical efforts, only the most important parameters
From the sixdimensional Gaussian posterior
(a) Means
For each settings sample, 100 samples are drawn from
Characterisation of posterior covariance
For the ideal MartinPuplett interferometer, the odd spectral function, and, hence, the odd process must vanish from theoretical point of view. However, imperfections of a realworld interferometer leave the odd contribution finite in general. This imperfection is captured by the scaling
The results obtained in the previous section rely on the model
In principle, the plausibility of a model relative to an alternative can be investigated within the Bayesian framework by rising the abstraction level. Starting from (
The signal envelope, corresponding to the used prior covariance for the even and odd spectral functions, is expected to be an indicator for a competitive model. This envelope should be able to resemble the global trend of the interferometric data (see Figure
For the even and odd spectral functions, an alternative prior choice could be
A competitive model could use a prior covariance
The standard analysis approach for the interferometer investigated here relies on a different model
Comparison of standard model
Model 






0  0  0  33.614 

3747.4  3747.4  3747.4  911.912 

N/A  19.8182  19.8182  1.0304 

N/A  129.8670  129.8670  5.7641 






0  0 



N/A  61.735  61.735  62.146 

N/A  5.822  5.822  6.007 

N/A  0.000087  0.000087  10.102 

N/A  1051396.55  1051400.41  1052037.88 
Odds  N/A 


1 
Aliasing feature  Yes  Yes  No  No 
Overfitting data  Yes  Yes  Yes  No 
The only window function applied here in the standard analysis weighs the singlesided data domain twice as large as the doublesided domain. Furthermore, the settings of the standard approach
Comparison of even and odd spectral quantities inferred from standard model
Keeping fixed all settings in
The odds 1 :
The algorithms for the model
For the model
Measured computational times


4  3  2  1  1/2 

313  443  589  883  1765  3527 

0.072  0.097  0.180  0.421  2.084  11.231 

0.079  0.108  0.200  0.471  2.406  13.648 
The characterisation of the settings posterior (see Section
The numerical marginalisation described in Section
The Fourier transform is the heart of Fourier spectroscopy applications. Thereby, the interferometric data has a linear dependence on the even and odd continuous spectra to be inferred. Standard analysis techniques lack appropriate handling of fundamental aspects like noisy measurements, the influence of nonprobed spatial domains linked to Fourier coefficients above a certain order, the estimation of spectral limits, and the propagation of uncertainties of additional parameters like the zeropath difference onto the inferred spectra. For instance, the Nyquist assumption implies the fundamental misconception that the upper spectral limit of spectra to be inferred would depend on the spatial sampling. In addition, a broad spectral bandwidth would follow which increases artificially the number of Fourier coefficients necessary to describe the data. On the contrary, it can be shown analytically that a bandlimitation causes spatially extended basis functions (modulated sinc functions) assigned to the Fourier coefficients in the data domain. Thus, several nearby data points are captured sufficiently by less coefficients. This example demonstrates that interferometric data contains more information than usually extracted.
As an alternative to the standard analysis techniques, a probabilistic ansatz, relying on Bayes’ theorem, was proposed which is able to capture the fundamental aspects listed above. In general, Bayes’ theorem relates the posterior probability density function of model parameters to the product of the likelihood and the prior probability density function for these parameters. The ansatz presented here uses multivariate normal distributions for the likelihood and the prior for parameters which map linearly to the data domain. This gives straightforwardly an analytical solution for the posterior of these linear parameters in form of a multivariate normal. Though, this amplitude posterior is conditional on the settings parameters, summarising all nonlinear model parameters and hyperparameters. After the trivial marginalisation over the linear parameters, the remaining quantity can be scanned in the settings parameters to investigate their joint posterior. This can be understood as a means of applying Ockham’s razor for the linear problem. With the settings posterior at hand, the marginalisation projects the uncertainties in the settings onto the linear parameters.
The example application for the Bayesian approach infers even and odd spectra, which qualify as linear parameters, in the microwave and farinfrared spectral domain and several settings parameters, like the spectral discretisation increment, the spectral limits, the scalings of the even and odd processes, the zeropath difference, and a shift correction to the spatial sampling, given a measured interferometric data set. Each spectrum is modelled by a scaled Brownian bridge process which is able to capture a bandlimitation, and the associated covariance is used in the Gaussian prior. This covariance assigns a broadband correlation, but its transform to the domain of Fourier coefficients reveals no correlation (vanishing offdiagonal elements except in connection with the zerothorder term) between the coefficients. Furthermore, the diagonal elements drop with the square of the order of the coefficients. Hence, the prior information stated by the Brownian bridge covariance considers functions which are squareintegrable and, thus, converge globally in the limit when the discretisation increment approaches zero and the order of the Fourier coefficients tends to infinity. In addition, these functions vanish smoothly at the lower and upper spectral boundaries. In the data domain, a signal envelope follows from the Brownian bridge process. This envelope decays with the optical path difference and the spectral bandwidth.
For the linear parameters like the even and odd spectra, a conditional amplitude posterior was briefly examined, relying on the Nyquist assumptions. Due to the large upper spectral limit, all noise contributions to the interferometric data are captured by the posterior mean of the linear parameters. This implies an overfitting. Because large and equal values are taken for the two Brownian bridge scalings (large signal envelops), the mapped posterior means of the spectra describe the even and odd parts of the interferometric data to equal parts in the singlesided domain, while the even part dominates in the doublesided domain. This is an indicator that the Fourier coefficients located in the singlesided domain are underestimated and overestimated for the even and odd spectra, respectively. The posterior samples for both spectra show large deviations from the means, and the even and odd contributions, obtained by mapping the samples, form much wider bands than the measurement uncertainty, especially in the singlesided domain. This indicates an unnecessary expanded solution space for the problem. Only by the posterior covariance of the linear parameters, the sum of the mapped samples complies with the data and its uncertainty band. The listed features mark a very unlikely conditional amplitude posterior which is revealed by the settings posterior.
The settings posterior for the most important settings is well approximated by the product of individual normal distributions, because no significant correlations could be found. The corresponding posterior means and standard deviations take reasonable values. These values are affected little by the discretisation increment which tends to be small, confirming the proposition that continuous spectral quantities are probed. The upper spectral limit is about a factor four smaller than the Nyquist frequency, and the lower limit is well separated from zero. This reduction of the bandwidth implies that the interferometric data can be described by a number of Fourier coefficients with associated spatial basis functions which is about onequarter of the amount of data points. The scaling of the even process exceeds the one for the odd process by about two orders of magnitude.
For values corresponding to the maximum of the settings posterior at a small discretisation increment, the conditional amplitude posterior was investigated. By the discretisation, the number of the linear parameters exceeds the one of the Fourier coefficients, which is mandatory to describe the data points within the measurement uncertainty, by one order of magnitude. However, the Ockham’s razor principle implemented by the settings posterior limits the solution space, so that the posterior means and samples for the linear spectral parameters, mapped to the data domain, have a smooth transition between the double and singlesided regions. Due to the much larger scaling with respect to the odd process, the even process and, thus, the even spectrum describe most of the singlesided and doublesided interferogram region. While the probed interferometric data is well described within the uncertainty by the means and samples, the nonprobed data domain, corresponding to Fourier coefficients above a certain order, is filled broadly by these mapped samples. This filling decays with increasing optical path difference and is limited by the signal envelopes which follow from the estimated scalings of the Brownian bridge processes and the bandwidth. Because the spread of the mapped samples in the nonprobed domain exceeds the one in the probed region, the main uncertainties for the even and odd spectra originates in nonprobed Fourier coefficients.
The numerically costly marginalisation over some of the settings shows that the zeropath difference changes the covariance for the odd spectral parameters significantly. Basically, a broad increase of the posterior uncertainties and correlation was found.
A figure of merit was introduced which states the deviation of a realworld interferometer from an ideal diagnostic. By relating the scalings of the even and odd processes, the used interferometer is characterised as being close to the ideal case for which the odd process must vanish.
The views and opinions expressed herein do not necessarily reflect those of the European Commission.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The first author acknowledges the institutions CCFE (Abingdon, United Kingdom), IPP (Greifswald and Garching, Germany), IFP (Milan, Italy), and KTH (Stockholm, Sweden) and thanks C. Marchetto, Drift, U. May, C. and J. Hastie, C. Giroud, S. Jachmich, K. AnkePense and E. Pense, A. Dinse, M. Domin, W. and T. Schlett, W. Mühlenbeck and J. Vieweg, D. and E. and H. Förster, S. and J. Ferdinand, and K. and U. Schmuck for the continuous support and help, especially during a period of severe sickness of his mother Sabine. This work has been carried out within the framework of the Contract for the Operation of the JET Facilities and has received funding from the European Unions Horizon 2020 research and innovation programme.