AP02_02.vp 1 Introduction Nominally flat surfaces are widely used in practice. These can be mathematically described by the composition of the deterministic and random components of irregularities in the given Cartesian coordinate system: � � � � � �h x y S x y x y, , ,� � � (1) where S ( x, y) is the deterministic function of the surface ( x, y) coordinates and �( x, y) is the homogeneous random normal field. The parameters of surface irregularities measured over the whole surface (the topographic parameters) characterize more positively the functional properties of the surface than the h profile parameters. Surface deviations are functions of two coordinates ( x, y) and therefore the profile evaluation gives incomplete information about the surface. 2 Surface topography parameter measurement For surface topography parameter measurement it is nec- essary to determine the actual value of the parameter and to know the accuracy of the measurements. In this case the analogue mean value is taken for the actual value of the parameter, and[s] the measurement error is determined by the systematic and random components. The series of the surface topographic parameters can be represented as an averaging operator of the generalized transformation G { h ( x, y)} of the surface coordinates on the given rectangular area of the surface L1 × L2 with sides L1 and L2 [1]: � �� �P L L G h x y x ys LL � �� 1 1 2 00 21 , d d (2) Since there is a random component on the measured sur- face, the topographic parameter measured is a random value, which is characterized by the mathematical expectation E (Ps) and the variance D (Ps). Therefore, one of the problems in measuring the topographic parameter is the determination of its probability characteristics, i.e., the mathematical expec- tation E (Ps) and the variance D (Ps). It is known that the mathematical expectation of the parameters given by equa- tion (2) can be derived by integration of the mathematical expectation E(G) of the transformation G{h ( x, y)} in equation (2) by the x and y variables. As an example of the application E(G) of the transformation G{h ( x, y)} in equation (2) by the x and y variables. As an example of the application of mea- suring methods of the topographic parameters the Ps � Ras parameter is used. This is the arithmetic mean deviation of the surface coordinates of the mean plane. � � � �� �E R L L E h x y x y LL as d d� �� 1 1 2 00 21 , (3) where � �h x y, is the absolute value of the h( x, y) surface coor- dinate. This expression can also be extended for the h( x, y) surface: � �� � � �� � � � � � E h x y S x y S x y S x y , exp , , , � � � � � � � � � � � � � � � � 2 2 1 2 2 2� �� � �21 2 � � � �� � � � �� (4) where � � � �� z t t z � �� 2 1 2 2 0 � exp d is the Laplace function. The generalized transformation G{h ( x, y)} is the random field which has the correlation function [3] defined as � � � � � � � � � �� � K x x y y G h G h f h h h h E G h G h G 1 2 1 2 1 2 1 2 1 2 1 , , , ,� � � �� �� �� �� �� d d � �� �2 , (5) where � �h h x y1 1 1� , , � �h h x y2 2 2� , are the coordinates of the surface at the� �x y1 1, and� �x y2 2, points, � �f h h1 2, distribution density expands into a series in terms of Hermite polynomi- als. The correlation function (5) can be represented as [3]: � � � � � � � �� � K x x y y C x y C x y n G n n n 1 2 1 2 1 1 1 1 1 2 1 2 1 , , , , , , ! � � � � �� � � � � � (6) � � � � � � � � C x y G h H h S h S hn 1 1 1 2 2 1 2 2 , exp� � � � � � �� � � �� � � � ��� � � � n d �� �� � (7) �1 2 1� �x x �2 2 1� �y y � is the r.m.s. deviation of the random component � ( x, y) and � �� � �1 2, is the correlation coefficient of the random compo- nents � ( x, y) , � �h h x y� , , � �S S x y� , . For the generalized transformation � �� � � �G h x y h x y� �, , , which determines the Ras parameter, the coefficients Cn in equation (7) are written as follows after the transformation: 35 Acta Polytechnica Vol. 42 No. 2/2002 A Mathematical Approach for Evaluation of Surface Topography Parameters A. K. Haghi The probability characteristics of surface topography parameters described by the composition of the deterministic component and the homogeneous random normal field were analysed. Formulae for the calculation of the mathematical expectation of the Ras parameter and the evaluation of its variance are given. Keywords: mathematical approach, surface topography, deterministic component. � � � � � �� � C x y S x x S x y S x 1 1 1 1 2 1 2 1 1 2 2 2 , , exp , � � � � � � � � � � � � � � � � �� � �1 1 1 22 , y � � � � � � � � � � �� � C x y S x y H S x y n 1 1 1 2 2 2 2 1 2 2 , exp , , � � � � � � � � � � � � � ! �� � � � n � � � � � �1 1 1 1 1 � � � � � � � � � � � � � � � � � � � � � � � � � � S x y H S x y, , n 1 (8) where Hn( z) are Hermite polynomials. The variance of the parameter PS is determined by integration of the correlation function � �K x x y yG 1 2 1 2, , , of the generalized transformation G{h(x, y)} using variables x1, x2, y1, y2: � � � �D P L L K x x y y x x y y LLL S G d d d d� ��� 1 1 2 2 2 1 2 1 2 1 2 1 2 000 211 , , , . (9) Calculation of the integral in equation (9) involves consid- erable difficulties; therefore, in the general case, formula (9) in not suitable for calculations. However, having � �S x y, " � and � �S x y, # � offers simplified evaluations of the variance. Thus, having � �S x y, " � from equation (8), C1 $ � Cn $ 0 n � %2 3, , (10) The correlation function of the transformation G{h ( x, y)} from equation (6) is reported approximately as � � � �K G � � � � � ��1 2 1 2, ,$ . (11) By using � �S x y, # � from equation (8) we obtain C1 0$ � �C Hn n$ � � � � 2 0 1 2 2 � � . The correlation function of the transformation G{h ( x, y)} from equation (6) is � � � �� � K C n G n n n � � � � � 1 2 2 1 2 2 , , ! $ � �� . (12) In both cases the correlation function � �K G � �1 2, is a func- tion of only two variables �1 and �2. Thus, for the above-men- tioned approximations, D ( PS) can be calculated by using the formula: � � � �D P L L L L K G L S d d� � � � � � � � �� 4 1 1 1 2 1 1 2 2 1 2 1 2 0 2 � � � � � �, 0 1L � . (13) The following notation is now used: � � � �S K KK GG G d d� �� �� 1 0 0 1 2 1 2 00 , ,� � � � (14) with L L SK1 2! " G equation (13) may be written approxi- mately: � � � �D P L L K SKS G G$ 4 0 0 1 2 , . (15) � �KG 0 0, is the variance of G{h ( x, y)}. It is not possible to carry out analogue measurements of the topographic parameter PS. Therefore the analogue-discrete and discrete methods are the only ones that can be used to measure the topographic parameters. With analogue-discrete measurements the aver- aging operator is: � �� �� ,P N L G h i y y L i N S d� � � 1 1 2 1 01 21 & (16) where &1 is the sampling in the sampling interval between the profiles and N1 is the number of profiles. The evaluation of the parameter in equation (2) in the general case is therefore shifted together with the task of determining the parameter probability characteristics, i.e., for its mathematical expecta- tion and variance it is necessary to determine the shift in the value of the evaluation of equation (16) with respect to that of equation (2). We can regard the probability characteristics of the evaluation (16) for that particular transformation as � �� � � �G h i y h i y& &1 1, ,� . (17) �Ras represents the analogue-discrete evaluation of the top- ographic parameter Ras. The mathematical expectation of the transformation (17) will be determined by the expression in equation (4) with � � � �S x y S i y, ,� &1 . The mathematical expectation of the �Ras parameter is identically: � � � �� �E R N L E h i y y L i N � ,as d� � � 1 1 1 2 1 01 21 & . (18) The integrals in equation (18) are not taken into account, but if the deterministic component � �S i y&1, is linearized with a small error using the y argument on the terminal num- ber of intervals the length of which is 2 then it is possible to obtain an exact expression for the evaluation � �E R�as of the topographic parameter Ras. For this, we expand the deter- ministic component � �S i y&1, in its Taylor series by using the y argument up to the linear terms at the point j = 1, 2, …, n: � � � � � � � �S i y S i j S i j y y& & & & & &1 1 2 1 2 2, , , $ � � . (19) By substituting equation (19) into equation (18) and tak- ing into account equation (4), we obtain: � � � � E R N L j j j n i N � exp as � ' ' � � � � � ��� � 1 1 2 1 2 111 1 2 2 21 & & � � � � � � � � � � � � � � ij ij ij ij ij ij y y y �� � � �� � � � �� � � � � � � � � 2 2 1 2 2 2 � � � � � � � dy (20) where � � � � � � � � ij ij S i j y S i j S i j y j � � � & & & & & & & 1 2 1 2 1 2 2 , , , . (21) The integrals in both components of expression (20) can be tabulated [4]. Then after transformation we obtain: 36 Acta Polytechnica Vol. 42 No. 2/2002 � �E R N L C A C A CD j n i N ij ij ij ij ij � as � � � �� 1 1 12 1 4 2 1 2 11 1 � � � ��� � � � � � � � �� � � � �� � (A Cij (22) � �( � � � � � � � � � � � B C B CD B C Aij ij ij ij ij ij ij ij � � � 1 4 2 1 2 exp� � � � � � � � C A B C B ij ij ij ij 2 2 1 2 2 2 � � exp where Aij ij ij� �� �&1 � �B jij ij ij� � �� �1 1& C � 1 2 1 2� Dij ij � 2 2 1 2 � �� The � �� z and exp( )�z2 and the standard calculation pro- gram for the � �� z and exp( )�z2 functions can be used to calculate E R( � )as on the computer. Formula (22) shows that the mathematical expectation of the parameter �Ras depends on the characteristic of the � � S i y&1 , the deterministic compo- nent S i y( , )&1 and the random component � 2. With the definite relationship between the deterministic and the ran- dom components, an approximate evaluation of E R( � )as can be obtained. Thus, having � �S x y, # � and linearizing equation (4) we obtain: � � � �E R N L S i y y L i N � ,as d$ � � 1 1 1 2 1 01 21 & . (23) By using � �S x y, # � and linearizing equation (4): � �� � � � � � � � E h x y S x y , , $ � � � � 2 2 1 2 2 1 2� � � � . (24) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 37 Acta Polytechnica Vol. 42 No. 2/2002 5,06 mm 5,06 mm μm 20 0 0 1 2 3 4 mm 0 90 μm Fig. 1: Representation of surface roughness measurement 2,9 mm 2,5 mm 0 0 1 2 mm 20 10 μm Fig. 2: Representation of surface roughness measurement 38 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 2/2002 Figs. 3 and 4: A 3D topographical image © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 39 Acta Polytechnica Vol. 42 No. 2/2002 F ig . 5: R ep re se n ta ti on of 3D ro u gh n es s te st in a la rg e sc al e Then � � � � � �� �E R L L S x y x y LL � ,as $ � � � � �� 2 1 2 1 2 1 2 1 2 2 00 21 � � � � d d . (25) To analyse the E R( � )as dependence on S � and to deter- mine what relationship with S � is needed in order to make formulae (23) and (25) applicable to the determination of the mathematical expectation E R( � )as , calculations were carried out using formulae (22), (23) and (25). The mathematical models of the composition surface and the experimental calculation of the parameter �R hij j N i N i as � �� 11 1 (26) where hij is the deviation of the h (x, y) surface from the mean plane at the discrete points. The random component was modelled with the help of the random number generator. The deterministic component was designated by formula (19). 3 Discussion � If the deterministic component is a piecewise linear func- tion and the random component is a homogeneous normal field, then the mathematical expectation of parameter �Ras increases with increasing S �. � If S � � 1, when E R( � )as is calculated the influence of the de- terministic component can be ignored and the error is less than 10 %. � When S � � 3, formula (25) can be used to calculate E R( � )as with an error of less than 10 %. � If S � � 4, E R( � )as can be evaluated with an error of less than 10 % by using the deterministic component (for- mula 23). � Formula (22) holds for the whole range of S � changes. � Full agreement with the theoretical expression can be demonstrated by the experimental calculations using for- mula (26). � Some optical profilometers based on these principles are shown in figures (1–6) to measure the statistical parameters of rough surfaces. The contrast is found to be related to sur- face roughness when the length of coherence of the light [are] is comparable in magnitude. 4 Conclusion The theoretical dependence of the mathematical expecta- tion of parameter �Ras has been determined for the case when the deterministic component S ( x, y) can be linearized on the separate sections. The dependence of the evaluation bias in parameter �Ras on the relationship between the random and deterministic components has been determined. References [1] Lukyan, V.: Measurement of the surface topography. ICTE, 2000, p. 103–154. [2] Lewis, J.: Fundamentals of Microgeometry. RPI, 2001, p. 32–65. [3] Zahedi, M.: Mesure Sans Contact de la Topographie d’une Surface. IUT de Belfort, 2000, p. 43–78. [4] Galat, T.: Optical Surface Roughness Determination. Ap- plied Optic, 2000. Dr. A. K. Haghi e-mail: haghi@kadous.gu.ac.ir Guilan University P.O. Box 3756 Rasht, IRAN 40 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 42 No. 2/2002 Fig. 6: Microtopograph of a rough surface