The use of areal characterization of surface texture with high accuracy in a quality control process requires reliability. Therefore, regular inspection of the measurement systems is needed. Important metrological features of a measure- ment system in dimensional metrology are the ampli cation factor and linearity. This paper presents a simple method for characterizing the axial scanning system of areal topography measuring instruments with little expense and e ort, well suited for industrial routine calibration in the eld. The method is based on employing a single material measure with a range of step heights. It is shown that the ampli cation factor and linearity deviations can be determined and adjusted for large axial measurement ranges. Keywords: surface topography, topography measurement, ampli cation factor, linearity, calibration, step height standards, uncertainty, large range axial scanning 1. Introduction There is growing demand for quality assessment of the dimensional prop- erties of work pieces that have both form and surface texture. In addition, in many cases areal measurements of surface topography (texture and form de- Corresponding author Email address: dorothee.hueser@ptb.de (Dorothee H user) Preprint submitted to Journal of the International Societies for Precision Engineering and NanotechnologyApril 21, 2020 viation) are required rather than simple line pro les, which may not provide 5 enough information needed for characterizing the functional behavior of the surface. A variety of optical measuring instruments is available for areal topog- raphy measurement with high accuracy. The upcoming standard ISO 25178-700 [1] on calibration, adjustment and veri cation of areal topography measuring instruments meets requirements of increasing accuracy and traceability of areal 10 measurement systems and include optical measuring instruments. However, calibration of the instrument axial scanning system ( z-axis) over large scan ranges may be time consuming. Some instruments provide scan ranges up to 70 mm in axial direction [2]. For large scan ranges of the z-axis small sized linearity deviations need to be determined along the complete range. In 15 order to ful ll ISO 9001 [3] regulations, instrument calibration has to be carried out within de ned time intervals. The purpose of such routine calibration in industry is to check, whether instruments are still working appropriately. The routine calibration procedure, therefore, needs to be fast, simple and able to be realized in the eld by the regular sta . In this paper, a method that meets 20 the requirements of being simple and easily performed in the eld has been developed. To obtain the response curve (in the international vocabulary of metrology VIM [4] referred to as calibration curve ) of an axis typically requires costly setups with an interferometer or complex procedures employing material mea- 25 sures being positioned at many locations [5] within the measurement range or representing a number of steps [6]. The method of Kiyono et al. [5] shows that a response curve of the z-axis can be obtained if a relationship of the step heights, which are di erences of positions rather than absolute positions, is maintained. Giusca and Geol have developed a measurement strategy employing a at plane 30 of optical precision with accurately adjusted tilt angles [7]. This method is suitable for instruments with a z-axis in the micrometer range. To overcome the need of a more accurate external positioning stage and to be suitable for instruments with a range of the z-axis of several millimeters, Boedecker et al. [6] employ a material measure representing a number of steps. 35 2 Figure 1: Material measure representing 6 di erent height resp. depth values, here with nominal depth values d= 5;50;450;100;2000;and 5000m. The nominal width at the bottom isw= 300m and the nominal with between neighboring groove top edges is 400 m delivering areas of widths of 100 m and 200m for evaluation of the depths, see Fig. 7. This is realized by a combination of material measures rstly representing coarse step heights covering the complete zrange and secondly, ne step heights cov- ering small sections of the range. The combination of the measurements on the object with the ne heights and those on the object with the coarse heights requires a time consuming calibration process and certi cates for the material 40 measures are often not available. In Leach et al. [8], a review is presented on procedures to measure metrological characteristics (MCs) with a variety of material measures each representing one or more speci c MC. They show step height artefacts, one of the kind employed by Boedecker et al. [6], see Fig. 9 in [8], and the depth setting type used during the investigations of the present 45 paper, see Fig. 8 in [8]. Guisca et al. [9] show linearity deviation studies that are performed by mea- suring four di erent height or depth values. Here, the ampli cation factor de- termination is based on the relationship between the calibrated height values and the measured height values, i.e. solely on the di erential relationship of 50 positions dIn= zdC+r (1) with zin ISO 25178-600 being called the ampli cation factor , if the quantity 3 dCis the actual input quantity anddInthemeasured quantity . In Guisca et al. [9], the values of quantity dCrepresent the calibrated values of the material measure with uncertainties presumed to be signi cantly smaller than those of the 55 measured quantity , in wording of VIM [4] the indicated quantity dIn, allowing the use ofdCas regressor of a linear regression problem. The wording for values of `measured' quantity in ISO 25178-600 refers to the `indicated' values in VIM, while the wording `calibrated values' used in Guisca et al. refers to `measured quantity value' in VIM. 60 With the presumption that the input quantity of the calibrated values has negligible random e ects and rare residuals due to random e ects of the in- struments indication that are normally distributed, the factor zis estimated by linear regression min z(nX i=1r2 i) withri=dIn;i zdC;i (2) and withi= 1;:::;n pairs of measurement and calibration values of the heights 65 or in case of depth setting artefacts, i.e. grooves, pairs of measurement and calibration values of the depths, yielding z=Pn i=1dIn;idC;iPn i=1d2 C;i: (3) The residuals of a regression of this relationship are an estimate of the uncer- tainty of the linearity. However, if systematic non-linear behavior dominates, the absolute position of the location of the step heights with respect to one 70 common origin is required. With this paper, we show that an estimate of linearity deviations can be carried out in the eld. To overcome the lack of an absolute scale zC, a `ladder' is constructed of np` ights' ofnd`rungs', as illustrated in Fig. 2. The construction of the scale zCis achieved by using a small number of steps or grooves on one 75 artefact and with low cost positioning at a small number of locations along the range of the z-axis. In the case of depth setting artefacts (grooves), the `ladders' are built downwards, starting with the top oor and successively going downward. If the distance between two successive oors ts the rung heights, as 4 first flight (floor)second flight (floor)Figure 2: Illustration of the idea to construct an absolute scale by stitching measurements of di erent connected step heights (like rungs/ steps of one ladder) at successive positions ( ights or oors). As the ights are connected by successively using the ladder. illustrated in Fig. 2, then the true ladder length can be assembled from multiples 80 of a previous measured length of a single ladder. The idea of the measurement strategy is to stitch intervals covered by the steps of the material measure to characterize the z-axis of an instrument. The procedure is demonstrated using as example an areal coherence scanning inter- ference measurement system with the z-axis being driven by an electro-magnetic 85 actuator. Furthermore, the determination of the linearity deviation is veri ed with the aid of synthetically generated response curves with sinusoidal linearity deviation. 2. Measurement strategy The German national metrology institute Physikalisch-Technische Bunde- 90 sanstalt (PTB), for instance, provides material measures with ve or six parallel grooves of varying depth in one object with the reference plane for all grooves being at the same level, see Fig. 1. This artefact ensures the grooves have a xed relationship with respect to each other. The material measure used for the investigations presented in this paper has 95 the following nominal depth values d: d=m: 5 50 450 1000 2000 5000 5 𝑧In𝑧C 𝑧C,1 𝑧pos,2𝑧C,1−𝑑C,1 𝑧C,1−𝑑C,5 𝑧In,5(1)=𝑧pos,1−𝑑In,5(1)𝑑1 𝑑3𝑑2 𝑑4𝑑5𝑑6 𝑧pos,1𝑧C,2Figure 3: Principle of relating depth setting artefacts with the response curve of the z-axis of a topography measuring instrument. For each physical translation of the depth setting artefact k= 1;:::;n pexistj(k) = 1;:::;n d(k) di erent depth measurements, with nd(k)6. The plane of the position of the reference plane of the k-th physical translation of the material measure, here indicated by a dashed doted lines, is the value obtained by the measurement of the reference plane zpos;khere withk= 1 andk= 2. The nominal groove width at the bottom is w= 300m and the nominal width between neighboring grooves is 400 m. The material measure is measured at di erent positions along the axial scan- 100 ning system (referred to as z-positionszpos) along the measurement range with the distance between neighboring z-positions approximately being of the same size as the depth of the largest groove. Fig. 3 shows how the concept of stitch- ing or assembling an absolute axis like constructing a `ladder' from successive `ladders' is applied to measurements with a PTB depth setting artefact. Since 105 for the present material measure, each ` ight' of nd`rungs' has the same prop- erties, the start of the `ladder' can be set so that it is convenient for determining the non-linearities, provided this start position is recorded. The ` ights' will be denoted by the index kand the level of a `rung' , complying with a step height or a groove depth d, by the index j. If a ladder connects a pair of ` ights' kand 110 k+ 1 then the measured groove depth, analogous to the `rung level' is labeled with the index j(k) and denoted as dIn;j(k). 6 For a measured groove j= 5 at physical translation position k= 1 the z-position of the bottom of groove as drawn in Fig. 3, the `indicated' z-position of the groove bottom zIn;j(k)is obtained by zIn;j(k)=zpos;kdIn;j(k). Locating 115 the depth setting artefact at di erent z-positions may, for example, be realized by using gage blocks or other plane parallel objects to mount it. The absolute calibrated scale, the dependent quantity, of the response curve characterizing linearity deviations is uncertain mainly due to the lack of knowl- edge of an absolute start position, due to the ambiguities of presumptions on 120 the interpolating model to assemble the `ladder' and ` ights', and to the size of the steps (distance of `rungs'). These uncertainty in uences are dependent on the speci cs of the depth setting artefact (choice of the `rung' distances) and on the spatial wavelength of the linearity deviations of the instrument's z-axis. In Section 5, one example evaluating experimental data and estimating uncertainty 125 in uence factors by a Monte Carlo simulation is given. The number of grooves is nd(here it isnd= 6) and the number of positions in which the material measure is located is np, thenN=ndnpdata points are obtained. For some positions repetitive measurements have been performed, so that the total number Nis larger than ndnp. The PTB calibration certi cate 130 providesj= 1;:::;n ddepth values for each of the grooves referred to by dC;j, the measurements deliver k= 1;:::;n pvalues for the z-positions of the objects' reference plane location zpos;kandNdi erent depth values dIn;j(k). For sim- plicity, the index notation throughout this paper omits dealing with repetitive measurements. 135 3. Analysis method The analysis method consists of two steps 1. Estimation of the adjusted, i.e. calibrated, positions zC;kof the object location due to the calibrated values of the material measure. 2. Estimation of the scaling or ampli cation factor and the linearity deviation 140 curve. 7 𝑧In𝑧C 𝑧C,2 𝑧pos,1𝑧C,1−𝑑C,3 𝑧C,1−𝑑C,5𝑧In,5(1)=𝑧5(1)=𝑧pos,1−𝑑In,5(1) ≈≈𝑧C,1−𝑑C,2𝑧C,1−𝑑C,1 𝑧C,1−𝑑C,4 𝑧C,1−𝑑C,6 𝑧pos,2𝑧C,1 𝑧6(1)𝑧5(1)𝑧4(1)𝑧3(1) 𝑧2(1)𝑧1(1)Figure 4: Illustration that a function fk:z7!fk(z) tted to the measurements of the k-th ensemble of nd(here it isnd= 6) grooves of one object delivers the reference position zC;k+1 of the subsequent, the k+ 1-st position. Step 1 is performed by deriving a value zC;k+1of the position with respect to the calibrated absolute axis from the depth measurements of all grooves performed at the previous location zpos;kand the measured position of the reference plane zpos;k+1. To do this, the indirect quantity is the absolute reference scale zC 145 and the direct experimentally observed quantity, with its values referred to as indicated values, zIn. In Step 1 a t model function fk:zIn7!fk(p;zIn) of the response curve with model parameter vector palong the interval that is covered by the largest depth according to  a priori knowledge is used. Here, a parabola with coecient 150 vector pkhas been chosen fk(zIn) =2X =0pk;(zIn+"In)+"Cwith pk= (pk;0;pk;1;pk;2)T(4) and with ("In;"C) being the deviation vector of data points perpendicular to the curve of the function fkwith estimated parameters pk. To simplify the implementation of the algorithm to estimate the model parameters by linear 8 regression, only residuals "Care considered. This can be justi ed with regard 155 to the fact that the complete model is an approximation that can only be used for linearity deviations of long spatial wavelengths. Future work is planned to compare various models including Bezier splines with total least-squares ts investigating the limitations of the procedure. For all depth measurements carried out at the location with the position of 160 reference plane zpos;k, the observed positions of the groove bottoms respectively the top of step heights are evaluated by zIn;j(k)=8 < :zpos;k+dIn;j(k)for step heights zpos;kdIn;j(k)for grooves:(5) Furthermore, in the case of depth setting artefacts, the sequence of transla- tion positions is chosen such that the highest is the rst and in the case of step heights the lowest is the rst. The rst calibrated position zC;1is set to the 165 measurement value of the rst measured position zpos;1. The calibration value zC;k+1of the successor of the physical translation with number k+ 1 is then obtained by inserting the corresponding measurment value of the translation of the reference plane on the object at zpos;k+1into the model function fkof positionkof the physical translation of the object with k;k+ 12f1;:::;n pg. 170 zC;k+1=fk(zpos;k+1): (6) The parameters pk;of each model function fk, here, for instance, the coecients pk;0,pk;1,pk;2of the parabola, are obtained by tting the target function to the set ofnd(k) points (zC;kdC;j; zpos;kdIn;j(k)) withj(k) = 1;:::;n d(k) and with the target function zC;kdC;j=fk(pk;;zpos;kdIn;j(k)) +rj(k) (7) solving the optimization problem 175 min pk;88 < :X j(k)r2 j(k)9 = ;(8) 9 withbeing the index for the di erent model parameters and pk;8being the notation to search the minimum `for all parameters pk;0;pk;1;:::'. For the model example parabola, Eq. 4, the model parameters are the co- ecentspk;0;pk;1;pk;2of the parabola fk(zIn) =pk;2 zpos;kdIn;j(k)2+ pk;1 zpos;kdIn;j(k) +pk;0are varied untilP j(k) zC;kdC;jfk zpos;kdIn;j(k)2 180 reaches its minimum. Fig. 3 and Fig. 4 illustrate the idea of sequentially tting curves to the sets ofnd= 6 or in case of repetitions of nd(k)>6 points to obtain the response curvezInvs.zC. Step 2 : the estimation of the ampli cation factor zand the linearity devi- 185 ationr(z) is based on the model function zIn= zzC+z0+r(zC) (9) with zdenoting the ampli cation factor, z0a constant o set and the function rdenoting the residual non-linearity depending on the position zC. In ISO 25178-600 [10], the maximum of the residuals is de ned as linearity deviation of thez-axis 190 lz= maxfr(zC)g: (10) The two parameters z,z0are obtained by linear regression tting a straight line to allNpoints (zpos;kdIn;j(k);zC;kdC;j), i.e. using all j(k) = 1;:::;n d(k) measurements for all k= 1;:::;n p. 4. Veri cation of the procedure The procedure is veri ed using synthetic data with known linearity devation 195 functionr:z7!r(z). Here, sinusoidal linearity deviations of amplitudes ad= 1m, 0:5m, and 1:5m and wavelengths = 30 mm and 70 mm along a measuring range of 35 mm have been generated zIn= zztrue+adcos2 (ztrue+ z) (11) with zdenoting the ampli cation factor, adthe amplitude, the wavelength, and zthe phase shift of the linearity deviation. Furthermore, as phase shift  z 200 10 -20 -10 0 10 20 30-1.5-1-0.500.5 zcal in true pos zcal estimated only d zC in mmresiudals in µmFigure 5: Simulation showing the linearity deviation as residuals of tted regression lines, the blue squares obtained from instrumental values zIn;j(k)as of Eq. (11) vs. true values and red circles vs. estimated values zC;kdC;j. The black, lled diamonds are the residuals of a regression line tted to dIn;j(k)vs.dC;j. the values 0 and 10 mm have been chosen. The wavelengths are large compared to the measurement range so that parabolas are appropriate model functions f(zIn) =zCfor intervals of 5 mm, which is the depth of the largest groove of the material measure. Fig. 5 shows the residuals of the line tted to all Npairs of data points 205  zIn;j(k);(zC;kdC;j) as red circles, while the blue squares show the residuals of the regression line tted to Npairs zIn;j(k);ztrue;j(k) . The parameters chosen to generate the simulated data set of Fig. 5 are = 70 mm,  z= 10 mm, ad= 1:5m, and z= 0:99990. The gure illustrates that simply evaluating the regression according to the 210 depth values dIn;ivs.dC;i, which are displayed as black, lled diamonds, does not give a correct characterization of the linearity deviation of the instrument axis. The residuals obtained by our method of successive estimation of calibrated positions, however, deviate very little, less than 104m, from the synthetic residuals. The small nite deviation is mainly caused by the o set between the 215 11 -20 -15 -10 -5 0 5 10-0.3-0.2-0.100.10.2 z-position in mm residuals in um only d estimated zcalFigure 6: Measurement by a Polytec coherence scanning interferometer with blue diamond symbols showing the residuals of a regression line tted to dIn;j(k)vs.dC;jand red circles showing the result using absolute positions with estimated values zC;kdC;j. The systematic trend of non-linearity visible for the red circles is emphasized by a parabola drawn as red curve. rst measured position with respect to the unknown true value. Furthermore, a contribution to the deviation is due to the uncertainty of the model functions fk. Here, the uncertainty is due to the approximation of a segment of a cosine by a parabola. 5. Measurement results 220 The measurement example of Fig. 6 showing a result obtained by routine calibration of the instrument in the non-ideal eld environment illustrates that the procedure presented here can reveal systematic e ects of non-linearity. This would be unrecognizable if only the relationship between measured and cali- brated values of the groove depths is regarded without any estimation of the 225 absolute position within the range of the z-axis. Two contributions to the uncertainty of the result have already been men- tioned above: the ambiguity of the value for the rst position zC;1and the 12 Cw 3w 3w 3w2 3- w2 3-w 3w 3 A Bw dFigure 7: The depth of each groove is evaluated by tting two parallel planes to data sets on the reference plane within regions AandBin the vicinity of the groove and within a region at the center of the groove bottom area C. The nominal width at the groove bottom of the artefact shown in Fig. 1 is w= 300m and the nominal width between neighboring groove top edges is 400 m, such that the two reference planes at the sides of one groove have a distance of 100m to the edges and a width of 200 m. The plane of the bottom center has a width of 100m. uncertainty of the model functions fk. The latter has two components, rst that the choice, here for instance the parabola, does not apply to the behavior 230 of the instrument within each of the intervals sized the depth of the largest groove. Secondly, the uncertainties u(dC) of the depth values given in the calibration certi cate of the material measure as well as the uncertainties u(dIn) of the depth values measured by the instrument under investigation are contributing. 235 Another contribution is the uncertainty of the whole artefact measurement is its tilt within the eld of view, which is mainly in uenced by the atness de- viation1of the reference surface. The atness deviation results in a shift of the reference areas of the grooves with respect to each other. The tilt of the object, uncertainties of the lateral scales together with the coarseness of the depth of 240 grooves contributes to ambiguities of linearity deviations of small spatial wave- length. The uncertainty contributions of the depth measurement uncertainties u(dC) andu(dIn) together with the shifting of reference areas behave like wobbly 1orthogonal deviation of the reference surface from planarity 13 u(di)c ur dC 0:1m0:5:::10:05:::0:1m dIn 0:3m0:4:::1 0:1::0:3m dbow 0:3m 1:3 0:38m Table 1: Uncertainty contributions of the in uencing components with uncertainties given in column 2, the sensitivities of the residuals of a linear response function model in column 3 and the resultant uncertainties of the residuals urin column 4. `rungs' of the ladder. The depth of each groove is evaluated by tting two parallel planes to topo- 245 graphy points of regions on the reference plane in the vicinity of the groove and a region at the center of the groove bottom area as indicated in Fig. 7. The procedure is speci ed in the draft standard ISO 25178-700 as an areal extension close to the method of ISO 5436-1 [11]. The nominal width at the bottom is w= 300m and the nominal width between neighboring grooves is 400 m. 250 At the sides of one groove, the reference plane areas fA;Bghave a distance ofw 3= 100m to the top edges and a width of2w 3= 200m. The plane areaCat the bottom center has a width of 100 m. The reference plane areas fAj;Bjgof di erent grooves jmay not exactly lie on the same level. This type of systematic uncertainty has been modeled as bow with its maximum being 255 ubow= 0:3m. All these contributions have been included into a Monte Carlo simulation using the example of the synthetic data set shown in Fig. 5 and using the uncertainties u(dC) stated in the calibration certi cate. As well, the depth uncertainty u(dIn) of the example of measured data shown Fig. 6 has been 260 chosen as a further contribution to the uncertainty budget. The uncertainty of the certi ed calibration values of the di erent depths di er according to their height, the deepest groove of 5 mm depth has a signi - cantly larger uncertainty of u(dC;6) = 0:1m dominating the e ect of in uences ofu(dC). Table 1 is an example uncertainty budget. 265 14 -20 -10 0 10 20 30-1.5-1-0.500.51 zcal in true pos zcal estimated only d zC in mmresiudals in µmFigure 8: Uncertainty estimation by superimposing the synthetic linearity deviation Simula- tion of Fig. 5 with uncertainty in uences u(dC),u(dIn), andu(dbow) that yield a bias of the estimation of systematic linearity deviations. The quantities of the second column display the uncertainty of the com- ponentsu(di) =u(dC),u(dIn), andu(dbow) respectively with the subscript i representing the subscripts C, In, and bow. The third column states the sen- sitivities by running the Monte Carlo simulation each time for only one of the three components yielding uncertainties urof the residuals of the regression line 270 ofzInvs.zCthat are displayed in column four. The combined uncertainty ur of the linearity deviation is obtained by including all three components to the Monte Carlo simulation. For the example, the combined uncertainty ranges fromur= 0:2m to 0:4m. This value is independent of the choice of the ampli cation factor. We used di erent curvatures, spatial wavelengths of non- 275 linearity of = 30 mm and 70 mm, and used di erent phase shifts  z= 0 and 10 mm. Fig. 8 illustrates the in uence of random uncertainty e ects of u(dC) andu(dIn) together with the systematic e ect of a bow u(dbow) deteriorating the determination of linearity deviations as systematic trend of the instruments axial scanning system. 280 15 6. Conclusion It has been shown that a determination of the ampli cation factor and the linearity deviations is possible using a fast and simple method with little expense and e ort, well suited for industrial routine calibration in the eld. This method is realized by using a small set of grooves on one common depth setting artefact. 285 If the uncertainties of the material measures and the random positioning uncer- tainty of the instrument are signi cantly smaller than the linearity deviation, an adjustment of the non-linearity is possible. Simulations show how to dis- tinguish systematic and random in uences and that a response function can be obtained for small uncertainties of the material measure and the instrument's 290 positioning. A new standard on Calibration and veri cation of areal topography measur- ing instruments is being drafted [1], ISO 25178-700, which allows a simpli ed calibration procedure of the ampli cation factor and non-linearities. If the re- sponse curve resulting from the simpli ed procedure shows irregularities, it is 295 necessary to apply procedures that require more e ort. Acknowledgement We thank Prof. Richard Leach, University of Nottingham, for his fruitful comments and thorough proof reading of the manuscript. We also thank the anonymous reviewers for providing helpful comments on earlier drafts of the 300 manuscript. All the suggestions helped to improve and clarify the paper.