Governing equations of a Raman fiber distributed temperature sensor supported by a counter propagating pumped Raman amplifier are written in the form of integral equations. Forward pump and backward going probe are in the pulsed and continuous wave modes of operation respectively. It is shown that powers of the forward and backward Raman signals (stokes and anti stokes) approximately are the convolution of the input forward pump pulse and corresponding optical fiber impulse responses. The Fourier Wavelet Regularized Deconvolution (ForWaRD) method is employed to improve the spatial resolution in fiber Raman distributed temperature sensor without reducing the pulse width of the laser source.
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