Performing The Deconvolution; Regularization - Micromeritics ASAP 2020 Confirm Operator's Manual

ASAP 2020 Confirm

Performing the Deconvolution

The integrations in equations (2) and (3) are carried out over all surface energies or pore sizes
in the model. The functions q(p,e) and q(p,H), which we call the kernel functions, are con-
tained in numeric form as model isotherms. Because, in general, there is no analytic solution
for equation (1), the problem is best solved in a discrete form; the integral equation for any
distributed property Z becomes a summation:
Q p  
Given a set of model isotherms, q(p,Z), from a model chosen from the models library and an
experimental isotherm, Q(p), contained in a sample information file, the software determines
the set of positive values f(Z) that most nearly, in a least squares sense, solves equation (4).
The distributed property, surface energy or pore size, is then displayed on the Report Options
dialog box as a selection of tables or graphs.

Regularization

DFT reports allow a selectable regularization (also referred to as smoothing) constraint to be
applied during the deconvolution process to avoid over-fitting in the case of noisy data or ill-
fitting models. The method used is based on co-minimization of the second derivative of the
distribution. The relative weight given to this term is determined by the value of the regular-
ization parameter, which is set on the DFT Pore Size or Surface Energy dialog box and also is
shown in the header of reports. The value of the regularization parameter varies from zero
(for no second derivative constraint) to ten (indicating a weight equal to minimizing the resid-
uals), or even larger. When the distribution and residuals obtained change little with the value
of the regularization parameter, it indicates that the chosen model provides a good represen-
tation of the data. Conversely, a large sensitivity to the regularization parameter might
indicate inadequate data or a poor choice of model to represent the data.
C-42

   f Z  
= q p Z
i i
i
Appendix C
(4)
202-42811-01 - Mar 2011