Performing The Deconvolution; Regularization - Micromeritics Gemini VII 2390a Operator's Manual

Appendix C

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,ε) and q(p,H), which we call the kernel functions, are contained in numeric
form as model isotherms. Because, in general, there is no analytic solution for equation (1), the prob-
lem is best solved in a discrete form; the integral equation for any distributed property Z becomes a
summation:

Q p   
=
q p Z
i
Given a set of model isotherms, q(p,Z), from a model chosen from the models library and an experi-
mental 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 prop-
erty, surface energy or pore size, is then displayed on the Report Options window as a selection of
tables or graphs.

Regularization

DFT allows 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 regularization parameter, which is set on the
DFT Pore Size or Surface Energy dialog 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 residuals), 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 representation 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-38
  f Z  
i i
Gemini VII
(4)
239-42828-01 - Dec 2012