Background elimination in three-dimensional spectroscopy using the intensian method

The method to eliminate background in the case of quantitative multidimensional spectroscopy, chromatography or any analytical 3-dimensional technique is shown. The 3-dimensional signal is required to be proportional to the concentration of determined substance and the additivity of signals should be obeyed. Eliminated background is assumed to be a low-order polynomial of two variables. The intensian method 1] is a generalization of the Beer-Lambert law, where a certain determinant called intensian replaces absorption and absorptivity. In practice there will be no need to use determinants, since usually they are replaced by expressions of few terms. Some details on the practical use of the method are given.Index of used symbols x, y UV, IR, GC, NMR or other scale. - A (x, y) intensity (absorption) of the 3-dimensional band of interest. - a (x, y) standard intensity (absorption) of the 3-dimensional band of interest. - B(x, y) intensity (absorption) of 3-dimensional background. - S(tx, y) intensity (absorption) of 3-dimensional multicomponent spectrum. - f (x, y) auxiliary function:f (x, y) =A (x, y), B(x, y), a(x, y), S(x, y). - (x _i, Yi) selected point,i positive integer number. - f(xi, yi) value off (x, y) in point (xi, yi). - S _i value ofS(x, y) in point (x _i, y_i). - b pathlength, measurement coefficient,c concentration. - , , , real numbers. - ij, _ij real coefficients of power expansions. - x ⁱy^j monomial of degree (i +j). - F(·) linear functional acting on 3-dimensional spectral functions. - J^3-dim(·) 3-dimensional intensian acting on 3-dimensional spectral functions. - J _n ^3-dim (·) 3-dimensionaln-points intensian. - d _i ith intensian coefficient, cofactor of expansion of J _n ^3-dim (f(x, y)) according to its first row (eq. (10)). - (.) absolute error. - r _i,, R random variables: eqs. (13) and (14). - G(.) (normal) distribution function. - z ordinate axis. - a - f , h abreviations for some arguments. - d _ijk,D _mnp abreviations defined in eq. (22).