Target detection algorithms for hyperspectral remote sensing have been studied for decades. The Least Square (LS) approach is one of the most widely used algorithms. It has been proved that the Noise Whitened Least Square (NWLS) can outperform the original version. But in order to have good results, the estimation of the noise covariance matrix is very important and still remains a great challenge. Many estimation methods have been proposed in the past, including spatial and frequency domain high-pass filter, neighborhood pixel subtraction, etc. In this paper, we further adopt the Fully Constrained Least Square (FCLS), which combine sum-to-one and non-negative constraints, with the NWLS and we also conduct a quantitative comparison with computer simulation of material spectrum from AVIRIS data base on the detection performance and the difference from the designed noise covariance matrix. We will also compare the results with real AVIRIS image scene.