M.R. RetamosoH.F. Campos Velho, M.T. Vilhena (2000): Estimation of Boundary Conditions from Different Experimental Data using LTSn Method and Tikhonov Regularization, Integral Methods in Science and Engineering (IMSE-2000), 12-15 June, Banff, Alberta, Canada. (submitted).

Abstract: A general methodology for estimating many properties in natural waters have been established by using an technique for parameter and function estimation from in situ radiometric measurements. The algorithm is formulated as a constrained nonlinear optimization problem, in which the direct problem is iteratively solved for successive aproximations of the unknown parameters. Iteration proceeds until an objective-function, representing the least-squares fit of model results and experimental data, converges to a specified small value. A regularization term is added to the objective function with the help of a Lagrange multiplier. In this paper the first-order Thikhonov regularization was used to get smooth solution. The implicit algorithm described above for solving the radiative transfer inverse problem, it has been applied to identifing the bioluminescence source term [5], absortion and scattering coefficients, as well as the unified approach of these properties [6], and phase function [3]. The goal of the present study is to analyse different strategies used to obtain experimental data for a good inversion for estimating bottom boundary condition in hydrological optics from "in situ" radiance and irradiance data. A key aspect to get a good reconstruction is played by the quality of the observed data. The reconstruction strategy is examined for many arrangements of the experimental grid of the measurement devices, in order to plan good designs for experimental works. The associated direct problem is tackled with the LTSn method [7]. This model solves numerically the time-independent, one-dimensional radiative transfer equation in natural water bodies using an analytical inversion of the Laplace transform of the discrete ordinate equations (Sn equations). A proof of convergence of the LTSn method have been derived using the semi-group theory [3, 4]. This scheme appeared in early nineties in the neutron transport context [1, 8].

[1] L.B. Barichello and M.T. Vilhena, M.T. (1993a): A General Approach to One Group One Dimensional Transport Equation, Kerntechnik, 58, pp. 182-184.

[2] E.S. Chalhoub, H.F. Campos Velho (2000): Simultaneous Estimation of Radiation Phase Function and Albedo in Natural Waters, Journal of Quantitative Spectroscopy & Radiative Transfer (JQSRT) - submitted.

[3] R.P. Pazos and M.T. Vilhena (1999a): Convergence of the LTSn Method: Approach of C_0 Semigroups, Progress in Nuclear Energy, 34, pp. 77-86.

[4] R.P. Pazos and M.T. Vilhena (1999b): Convergence in Transport Theory, Applied Numerical Mathematics, 30, pp. 79-.

[5] S. Stephany, F.M. Ramos, H.F. Campos Velho, C.D. Mobley (1998): A Methodology for Internal Light Sources Estimation, Computer Modeling and Simulation in Engineering (CMSE), 3(3), pp. 161-165.

[6] S. Stephany, F.M. Ramos, H.F. Campos Velho, C. D. Mobley (1999): Identification of Inherent Optical Properties and Bioluminescence Source Term in a Hydrologic Optics Problem, JQSRTr - to appear.

[7] C.F. Segatto, M.T. Vilhena (1994): Extention of the LTSn Formulation for Discrete Ordiantes Problem without Azimuthal Summetry, Annals of Nuclear Energy, 21, pp. 701-710.

[8] M.T. Vilhena, L.B. Barichello (1991): A New Analytical Approach to Solve the Neutron Transport Equation, Kerntechnik, 56, pp. 334-338.