Quantitative and Qualitative Characterization of Liquid Materials - Broadband Reflectometry for Enhanced Diagnostics and Monitoring Applications - page 70

4.3.3.3 Validation of the Robustness of the Procedure

It is important to point out that the adopted optimization procedure converges to a
final solution that depends on the initial estimates and on the intermediate conver-
gence point of the random part of the optimization procedure itself. To assess the
robustness of the overall procedure, it was repeatedly applied to the same sample,
slightly varying the initial guesses for the Cole-Cole parameters of water and fuel.
The obtained results, in terms of extrapolated fuel levels, show dispersion within
0.1 mm, thus demonstrating the robustness of the procedure. To summarize the ob-
tained results and to compare the different methods, Table 4.7 compares the fuel
levels extrapolated from the time-domain and from the two frequency-domain ap-
proaches, with the reference values.

Ta b l e 4 . 7 Reference and extrapolated fuel levels, in time-domain and frequency-domain ap-
proaches, for the three samples

Fuel level (mm)

reference time frequency frequency

domain domain domain

(VNA) (TD/FD combined approach)

Sample #1 283.0 281 283.5 282.7

Sample #2 219.0 219 219.5 219.3

Sample #3 335.0 331 335.1 334.7

The total time required for the optimization (encompassing the steps outlined)
was of the order of 10 minutes.
Indeed, the time required for optimization might be dramatically reduced by us-
ing a customized software routine rather than resorting to the commercial circuit
simulator.
It is worth mentioning that, in a practical monitoring application, after the first
run of the optimizer has been completed and initial Cole-Cole parameters and liquid
levels have been estimated, the continuous monitoring simply requires the optimizer
to track the small changes in dielectric parameters and level that occur in time. This
means that the further optimizations will only require a few iterations of the simplex
algorithm, which however should not take more than a few seconds 2 .

2

As a matter of fact, different minimization methods, such as the globalized bounded
Nelder-Mead (GBNM), can be alternatively used. Nevertheless, this would probably im-
prove the efficiency of the first run, but would probably have no relevant impact in the sub-
sequent continuous monitoring, when the optimization routine should just track smooth
temporal variations in liquid levels and permittivities.