[Gwyddion-users] RMS calculated by "statistical quantities" and HHCF function

[Frederik Edler]

Hello Gwyddion users,

I have a question concerning the RMS(sigma) values one can calculate
with Gwyddion.

It is possible to calculate the RMS(sigma) with the "Statistical
Quantities" tool. If I calculate the RMS(sigma) for the complete scan
area, I get for example a value of around 150pm.
However, if I take the HHCF for the same scan (again using the complete
scan area) and fit the the curve with the "Exponential (HHCF)" function,
I get a value of around 112pm.

In the manual
(http://gwyddion.net/documentation/user-guide-en/statistical-analysis.html)
it is written in the "Autocorrelation Function / Height-Height
Correlation Function" part that "σ denotes the root mean square
deviation of the heights" so I would expect the same value in both cases.

I am probably overlooking something. What is the reason for the difference?

Thanks for any help,

Frederik

------------------------------------------------------------------------------

[David Nečas (Yeti)]

Post by Frederik Edler
What is the reason for the difference?

The reason is that one value is calculated by summing differences from
the mean value, whereas the other is calculated by fitting
one-dimensional HHCF using an assumed theoretical curve. Why do you
expect exactly the same value? The values have to agree mutually and
with the true value only if the surface is infinite and corresponds
perfrectly to the theoretical HHCF (including no global distortion, row
mismatch etc.).

Value from HHCF will be typically smaller because roughness with small
spatial frequencies (comparable to √image size) contributes less,
row mismatch increases the plain σ estimate while having no influence on
1D HHCF, etc.

Regards,

Yeti


------------------------------------------------------------------------------

[David Nečas (Yeti)]

See for instance chapter 5 in Zhao & Wang & Lu, Characterization of
Amorphous and Crystalline Rough Surface -- Principles and Applications,
for a discussion of the convergence of HHCF obtained from sampled
surface to the true HHCF.

Regards,

Yeti


------------------------------------------------------------------------------

[Frederik Edler]

Hi Yeti,
Thanks a lot for the fast help. I don't know why I was assuming exactly
the same values yesterday. Your explanation just makes sense. Thank you
very much for the reference to the interesting book chapter of Zhao,
Wang & Lu.

Best regards and hopefully a nice weekend,

Frederik

------------------------------------------------------------------------------