This paper presents a linear least squares method for fiber-longitudinal power profile estimation (PPE), which estimates the optical signal power distribution throughout a fiber-optic link at a coherent receiver. The method finds the global optimum in the least squares estimation of the longitudinal power profiles; thus, its results closely match the true optical power profiles and locate loss anomalies in a link with high spatial resolution. Experimental results show that the method achieves accurate PPE with an RMS error of 0.18 dB from OTDR. Consequently, it successfully identifies a loss anomaly as small as 0.77 dB, demonstrating the potential of a coherent receiver in locating even splice and connector losses. The method is also evaluated under WDM conditions with optimal system fiber launch power, highlighting its feasibility for use in practical operations. Furthermore, the fundamental limit for stable estimation and the spatial resolution of least-squares-based PPE are quantitatively discussed in relation to the ill-posedness of the PPE by evaluating the condition number of the nonlinear perturbation matrix.