A minimum curvature algorithm for tomographic reconstruction of atmospheric chemicals based on optical remote sensing

Abstract

Abstract. Optical remote sensing (ORS) combined with the computerized tomography (CT) technique is a powerful tool to retrieve a two-dimensional concentration map over an area under investigation. Whereas medical CT usually uses a beam number of hundreds of thousands, ORS-CT usually uses a beam number of dozens, thus severely limiting the spatial resolution and the quality of the reconstructed map. The smoothness a priori information is, therefore, crucial for ORS-CT. Algorithms that produce smooth reconstructions include smooth basis function minimization, grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is promising because of the fast speed. However, its theoretical basis must be clarified to better understand the characteristics of its smoothness constraints. Moreover, the computational efficiency and reconstruction quality need to be improved for practical applications. This paper first treated the LTD algorithm as a special case of the Tikhonov regularization that uses the approximation of the third-order derivative as the regularization term. Then, to seek more flexible smoothness constraints, we successfully incorporated the smoothness seminorm used in variational interpolation theory into the reconstruction problem. Thus, the smoothing effects can be well understood according to the close relationship between the variational approach and the spline functions. Furthermore, other algorithms can be formulated by using different seminorms. On the basis of this idea, we propose a new minimum curvature (MC) algorithm by using a seminorm approximating the sum of the squares of the curvature, which reduces the number of linear equations to half that in the LTD algorithm. The MC algorithm was compared with the non-negative least square (NNLS), GT-MG, and LTD algorithms by using multiple test maps. The MC algorithm, compared with the LTD algorithm, shows similar performance in terms of reconstruction quality but requires only approximately 65 % the computation time. It is also simpler to implement than the GT-MG algorithm because it directly uses high-resolution grids during the reconstruction process. Compared with the traditional NNLS algorithm, it shows better performance in the following three aspects: (1) the nearness of reconstructed maps is improved by more than 50 %, (2) the peak location accuracy is improved by 1–2 m, and (3) the exposure error is improved by 2 to 5 times. Testing results indicated the effectiveness of the new algorithm according to the variational approach. More specific algorithms could be similarly further formulated and evaluated. This study promotes the practical application of ORS-CT mapping of atmospheric chemicals.