Title: A simulated comparison between profile and areal surface parameters: $R_a$ as an estimate of $S_a$,
Abstract: Direct comparison of areal and profile roughness measurement values is not
advisable due to fundamental differences in the measurement techniques. However
researchers may wish to compare between laboratories with differing equipment,
or against literature values. This paper investigates how well the profile
arithmetic mean average roughness, $R_a$, approximates its areal equivalent
$S_a$. Simulated rough surfaces and samples from the ETOPO1 global relief model
were used. The mean of up to 20 $R_a$ profiles from the surface were compared
with surface $S_a$ for 100 repeats. Differences between $\bar{R_a}$ and $S_a$
fell as the number of $R_a$ values averaged increased. For simulated surfaces
mean % difference between $\bar{R_a}$ and $S_a$ was in the range 16.06% to
3.47% when only one $R_a$ profile was taken. By averaging 20 $R_a$ values mean
% difference fell to 6.60% to 0.81%. By not considering $R_a$ profiles parallel
to the main feature direction (identified visually), mean % difference was
further reduced. For ETOPO1 global relief surfaces mean % difference was in the
range 52.09% to 22.60% when only one $R_a$ value was used, and was 33.22% to
9.90% when 20 $R_a$ values were averaged. Where a surface feature direction
could be identified, accounting for reduced the difference between $\bar{R_a}$
and $S_a$ by approximately 5% points. The results suggest that taking the mean
of between 3 and 5 $R_a$ values will give a good estimate of $S_a$ on regular
or simple surfaces. However, for some complex real world surfaces discrepancy
between $\bar{R_a}$ and $S_a$ are high. Caveats including the use of filters
for areal and profile measurements, and profile alignment are discussed.