International Journal of Spectroscopy

Volume 2017, Article ID 9265084, 29 pages

https://doi.org/10.1155/2017/9265084

## Fourier Spectroscopy: A Bayesian Way

^{1}CCFE, Culham Science Centre, Abingdon OX14 3DB, UK^{2}Max-Planck-Institut für Plasmaphysik, Teilinstitut Greifswald, Wendelsteinstraße 1, 17491 Greifswald, Germany

Correspondence should be addressed to Stefan Schmuck; ti.rnc.pfi@kcumhcs

Received 6 February 2017; Revised 11 July 2017; Accepted 27 July 2017; Published 31 October 2017

Academic Editor: Wei Kong

Copyright © 2017 Stefan Schmuck and Jakob Svensson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 zero-path 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 zero-path 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.

#### 1. Introduction

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 high-temperature plasma, being magnetically confined [1].

Commonly, an interferometer diagnostic, let us say of Michelson [2] or Martin-Puplett [3] design type, probes the Fourier transform of a spectral quantity. The corresponding interferometric data is a discrete set with finite length and includes noise contributions. Standard Fourier data analysis techniques [4–6] have been developed. These techniques lack describing and capturing properly several fundamental aspects, like the noisy nature of measured data and possible spectral limits and their impact on the spectral quantity to be inferred.

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 [7].

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 [8, 9]. For a spectroscopic problem, a model is formulated via Bayes’ theorem which allows to state prior information about model parameters. Furthermore, a Gaussian likelihood connects functionally the parameters with the noisy interferometric data. Then, after having measured a noisy data set and framing the reality by a certain model, the knowledge about model parameters is expressed in probabilistic terms by the posterior. This approach [8, 9] demonstrated that the uncertainty on the posterior mean of a frequency to be estimated for a single-frequency problem can be orders of magnitude below the FWHM width of the equivalent convolution function. On top of that, a criterion has been derived how far frequencies need to be separated, so that the Bayesian approach is still able to make a distinction. This separation can be well below the FWHM width of the convolution function. These findings are a direct consequence of the use of probabilistic theory.

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 so-called hyperparameters, entering merely in the prior.

In Section 2, the basic equations for Fourier spectroscopy and their implications are generally investigated, and the main concepts of the standard analysis and their drawbacks are pointed out. Section 3 of this paper presents a Bayesian formalism, so that it lends itself to applications in the field of Fourier spectroscopy for Gaussian (white) noise. The fundamental information about the spectral quantity, that it must vanish at a lower and upper spectral limit, can be stated by the covariance function of a Brownian bridge process as described in Section 4. This covariance is used as prior in the example application of the Bayesian approach presented in Section 5 to infer continuous but band-limited spectral quantities given an actually measured interferometric data set. Thereby, some diagnostic imperfections like a drifting signal offset, the zero-path difference, and a nonuniform spatial sampling are also taken into account. Section 6 discusses the strategy for a plausibility study of models, using different priors for the spectral quantities, and attempts to compare results and computational efforts obtained with the Bayesian model and with a standard model. The last section presents the conclusions.

#### 2. Fourier Spectroscopy

##### 2.1. General Definitions

Commonly, the complementary coordinates used for Fourier transformations are the frequency and time , or the wavenumber and a spatial coordinate . For the moment, the latter pair is used to state the basic operations of Fourier transformation. Afterwards, the wavenumber is replaced via for convenience.

The real-valued continuous functions and form a Fourier transformation pair stated byThe inverse operation readsNote that so far holds. To find a relation, one inserts (1) in the above expression. After applying trigonometric identities, the spatial integral becomesbecause the sinusoidal contribution vanishes. Hence,follow, because the two wavenumber coordinates equal each other. In fact, the delta distribution occurs, because is a distribution.

Replacing with gives in the spectral domain the quantity . This givesand rescales the inverse operation liketo match the units /Hz.

##### 2.2. Selection of Representation

The spectral domain over which the integration is performed in (5) includes the whole negative and the positive ranges. While the cosine transform acts on the even part of , the sine transform is linked only to the odd part . Thus, an alternative formulation readsand it becomes clear that the even and the odd parts of and are connected.

Another representation uses the amplitude and the phase by settingwhich givesBecause the model representations (5) and (7) are linear in , , and , both are favoured over the formulation (9), when linear inversion techniques are to be applied. Of both favoured representations, the one, using even and odd parts, has preferred properties. Since the orders of magnitude of the amplitudes can be quite different for the even and odd functions, a separation is logical.

##### 2.3. Finite Bandwidth

If the spectral domain is band-limited, such that and are finite for the range from and with the bandwidth and the centre frequency , (7) becomesHere, the assumption must be mentioned that both functions have the same spectral limits. This is assumed in the following but not mandatory. Furthermore, for any combination of and the relation must be fulfilled to be meaningful.

###### 2.3.1. Relation: Fourier Transform-Fourier Coefficients

When the bandwidth is finite, one can express the spectral functions by Fourier coefficients multiplied each with the associated sinusoidal basis function of order (). These coefficients are defined here by the integralswhich carry the unit as and . The coefficients label the mean values of and in the spectral domain covered. Then, one can replace the even and odd functions withwhich allows performing the spectral integration in (10) analytically. Since the resultwithfollows, it becomes clear that the Fourier transform of a band-limited function can be expressed by Fourier coefficients scaled with the bandwidth and multiplied each with the associated continuous basis function in the spatial domain. These basis functions have two contributions. The first is a sum/difference of two functions which depend on the order , the spatial coordinate, and the bandwidth. The latter quantity determines the spatial width of . Furthermore, the localisation is permitted at , where a coefficient for a given mainly acts. Hence, increasing the order implies the localisation at a larger distance from the spatial origin. This explains the occurrence of factor 2 for for which both sinc functions coincide.

The second contribution causes a modulation of and is given by a sine/cosine with the centre frequency and spatial coordinate in the argument. This dependency makes the basis function for vanish at the spatial origin.

With respect to the spatial origin, the transformed basis functions of the coefficients for and are symmetric and antisymmetric, respectively.

Some basis functions in the spatial domain are shown in Figures 1(a) and 1(b) for = 500 GHz and = 1000 GHz.