Methods for Temperature Profile Reconstruction from Incomplete Measurements in Incoherent Raman OFDR

Optical distributed temperature sensing usually relies on the Brillouin, Raman or Rayleigh response of the fiber under test to obtain a spatial temperature profile [1]. Strain insensitivity is guaranteed when relying on the Raman response of the fiber, which makes such interrogators popular. The Raman response can be acquired in the frequency domain, as depicted in Fig 1(a), or in the time domain. Although time-domain interrogators are popular, Raman optical frequency-domain reflectometry (ROFDR) interrogators remain interesting for their low hardware requirements and overall simplicity [2], [3], It has been shown that the acquisition speed of incoherent ROFDR interrogators can utilize compressed sensing (CS) to reconstruct the temperature profile along the fiber from incomplete frequency measurements when the variation of that profile is sparse in a certain transformed domain [4]. This work, based on experimental data, demonstrates other means to retrieve such a profile without the need to express the signal in a wavelet basis. Instead, the sought-after signal is parameterized to be blocky directly in the spatial domain, and this constraint is enforced by total variation regularization (TVR). The reconstructed signal expresses the temperature changes between two successive measurements, a satisfying target model in applications where sudden temperature variations are expected spatially. Following the approach of Ref [4], the recovery of temperature profiles can be indirectly attempted by piecewise constant or piecewise smooth approximation of the difference signal of two measurements of the Stokes or anti-Stokes backscattering data in the time domain. When TVR is used, the differential spatial profile is found by solving argmin <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\frac{1}{2}\Vert d-Gm\Vert_{2}^{2}+\alpha\Vert\nabla m\Vert_{1}$</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex>, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> is the observed data, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$G$</tex> the non-uniform Fourier transform operator, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> the model. TVR promotes the blockiness of the difference signal. The Chambolle-Pock Primal Dual solver is used for minimization [5]. Alternatively, piecewise smooth profiles can be reconstructed using relaxed Mumford-Shah (MSR) regularization, of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\min(\alpha\Vert x\Vert_{2}^{2}, \kappa)$</tex>. This method combines TVR and Tikhonov regularization to allow jumps in the signal (with penalty <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\kappa)$</tex> as well as smoothness [6], useful for spatially uneven temperature changes. Fig. 1(b) depicts the results of these techniques for a signal composed of a series of hotspots of varying sizes, for a reconstruction with 25% of the full set of frequencies. A too small number of acquired frequencies will affect the spatial resolution of the system.