The basic measurement equation r = B + αd + n is solved for α (the weight or abundance of the spectral target vector d) by two methods: (a) by subtracting the stochastic spectral background vector B from the spectral measurement’s vector r (subtraction solution) and (b) by orthogonal subspace projection (OSP) of the measurements to a subspace orthogonal to B (the OSP solution). The different geometry of the two solutions and in particular the geometry of the noise vector n is explored. The angular distribution of the noise angle between B and n is the key factor for determining and predicting which solution is better. When the noise-angle distribution is uniform, the subtraction solution is always superior regardless of the orientation of the spectral target vector d. When the noise is more concentrated in the direction parallel to B, the OSP solution becomes better (as expected). Simulations and one-dimensional hyperspectral measurements of vapor concentration in the presence of background radiation and noise are given to illustrate these two solutions.
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