A spectrometer records a spectral line profile

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

A spectrometer records a spectral line profile b\mathbf b, related to the true intensity profile x\mathbf x by a known blurring (instrument response) matrix AA, b=Ax+noise\mathbf b=A\mathbf x+\text{noise}, with independent Gaussian measurement noise of standard deviation δ=1.00×10−3\delta=1.00\times10^{-3} (in the same normalised intensity units as b\mathbf b) on each measured channel. Software has produced the singular value decomposition A=UΣVTA=U\Sigma V^{\rm T} and reports, for the three singular components that matter here, σ1=1.00,σ2=0.100,σ3=2.00×10−4,\sigma_1=1.00,\quad \sigma_2=0.100,\quad \sigma_3=2.00\times10^{-4}, together with the measured projections onto the left singular vectors, c1=u1Tb=0.900,c2=u2Tb=0.0500,c3=u3Tb=1.50×10−5.c_1=\mathbf u_1^{\rm T}\mathbf b=0.900,\quad c_2=\mathbf u_2^{\rm T}\mathbf b=0.0500,\quad c_3=\mathbf u_3^{\rm T}\mathbf b=1.50\times10^{-5}. Because UU has orthonormal columns, the noise on each cic_i has the same standard deviation δ\delta as the noise on b\mathbf b. ((a)) Compute the naive (unregularised) deconvolved coefficients xi=ci/σix_i=c_i/\sigma_i for i=1,2,3i=1,2,3 and their standard errors δ/σi\delta/\sigma_i. Identify which component's naive estimate is dominated by noise rather than signal. ((b)) Compute the Tikhonov-regularised coefficients xiλ=fi ci/σix_i^\lambda=f_i\,c_i/\sigma_i, with filter factor fi=σi2/(σi2+λ2)f_i=\sigma_i^2/(\sigma_i^2+\lambda^2), for λ=3.00×10−3\lambda=3.00\times10^{-3}. ((c)) State, with a one-sentence justification comparing λ\lambda with σ2\sigma_2 and σ3\sigma_3, why this value of λ\lambda is a reasonable choice, and what would go wrong if λ\lambda were instead 1.00×10−51.00\times10^{-5} or 11.

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