Abstract

An earlier work¹ described a generalization of two-wavelength differential absorption lidar to the case of multiple wavelength channel returns for estimating the path-integrated concentration (CL) for multiple materials. A maximum likelihood (ML) estimator was constructed for the N-dimensional vector CL = [CL₁,…,CL_N]^T of path-integrated concentration values in terms of lidar returns p_i(k) collected simultaneously at i = 1,…1, M ≥ N+1 wavelengths at time step k. Those returns were used to define an observation vector s(k), Eq. (1), which could be expressed approximately as Eq. (2) in terms of the M-1 × N-dimensional matrix of absorptivity differences ${\text{R}}_{i\mathcal{l}}={\rho }_1^{\mathcal{l}}-{\rho }_{1+1}^{\mathcal{l}}$ of material ℓ at wavelength i. The measurement noise n_s(k) was taken to be additive, uncorrelated, and zero-mean with covariance Λ_s (k,k′) = <n n^T >δ_kk′. The ML estimator for CL was shown to be given by Eq. (3) in terms of the ML estimate for the CL covariance ${\Lambda }_{\text{CL}}^{\text{ML}}$, Eq. (4).

© 1987 Optical Society of America