The algorithm adopted here for retrieving the optical properties is the OE method43–45 that is a Bayesian approach. This method can include inside the inversion procedure a priori information both on fit parameters and on fixed forward model parameters. The forward model, , simulates the experimental measurements . The vector is the measured DTOF, where the time dependence is represented by the vector components. The forward model does not reproduce the measurements exactly; thus, and are linked by the expression Display Formula
(1)where is a vector containing the actual retrieval unknowns (fit parameters), i.e., the “retrieved” parameters such as and , that usually are the objective of the measurements. The vector contains the fixed forward model parameters (e.g., the refractive index, ), which are not the objective of the retrieval and for which a reasonable estimate of their values and of their associated uncertainties already exists. Some parameters, depending on the kind of retrieval and application, can take both the roles, either of fit parameters or of fixed forward model parameters. The vector is an error term accounting for the overall measurement accuracy but also for the estimated error on . The retrieval involves the determination of the vector that minimizes the cost function, . In the constrained nonlinear least-square fit approach based on the OE,43,44 is equal to Display Formula
(2)where is the complete variance-covariance matrix (VCM) of the residuals of the fit, is the best estimate (decided by the experimentalist) of the fixed forward model parameters, is the a priori estimate of (in our program set as “initial values” of the fit parameters) with the VCM equal to . Two sources of uncertainty affect : the standard deviation due to the measurements noise of the observations, and the standard deviation of (uncertainty of the fixed forward model parameters). Therefore, we have Display Formula
(3)with , the VCM of the measurements’ noise, given by the diagonal matrix Display Formula
(4)and , the VCM of the forward model standard deviations, is Display Formula
(5)where the vector is Display Formula
(6)and where is evaluated for each parameter of the forward model, (component of ), affected by a priori standard deviations . Similarly, given the a priori information on the fit parameters by an expectation value and by a standard deviation , the VCM is Display Formula
(7)The retrieval procedure (fitting) determines the fit or target parameters, , minimizing , by using the iterative Gauss-Newton procedure44 [see Eq. (7) of Ref. 44]. For the fit parameters , the a priori information is taken into account by assuming Gaussian probability distributions (centered in with standard deviations ). Thus, the feasible range of the fit parameters of the inversion procedure (i.e., the range where the fit parameter is expected to fall) is essentially delimited by the width of the Gaussian distributions, i.e., about three standard deviations (), and centered at their initial expected values . For the fixed forward model parameters , the a priori information is given by estimating a standard deviation, , for their expected true values. We note that affects the iterative procedure of the retrieval and the error budget of the fit parameters. In the inversion procedures, for the most commonly used methods such as LM, the parameters are assumed to be exactly known and in principle do not affect the overall accuracy of the forward model. Indeed, their assumed values may be affected by systematic errors that, if estimated, can be considered as a priori information in minimizing the cost function. Therefore, the OE can also account for the systematic errors affecting the estimation of the fixed forward model parameters and also the retrieved fit parameters.