A simulated comparison between profile and areal surface parameters: $R_a$ as an estimate of $S_a$

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.