3 Mathematical Background of Statistical PIV Evaluation

3Mathematical Background of StatisticalPIV EvaluationA detailed mathematical description of statistical PIV evaluation has been given by Adrian[78].This early work from1988concentrated on autocor-relation methods and was later expanded to cross-correlation analysis[84]. Most of the characteristics and limitations of the statistical PIV evaluation have been described therein.To date the most complete and careful mathe-matical description of digital PIV has been given by Westerweel[51].In this chapter,a simplified mathematical model of the recording and subse-quent statistical evaluation of PIV images will be presented.For this purpose the two-dimensional spatial estimator for the correlation will be referred to as the correlation.First,we analyze the cross-correlation of two frames of singly exposed recordings.Then we expand the theory for the evaluation of doubly exposed recordings.The motivation behind employing auto-and cross-correlation methods are employed in PIV evaluation will be given in chapter5.3.1Particle Image LocationsTypically,PIV recordings are subdivided into interrogation areas during eval-uation.These areas are called interrogation spots–in the case of optical interrogation–or interrogation windows1when digital recordings are con-sidered.Due to reasons stated afterwards,for cross-correlation analysis those interrogation areas need not necessarily be located at the same position of the PIV recording.Their geometrical back-projection into the light sheet will be referred to as interrogation volumes in the following(seefigure3.1).Two interrogation volumes used for statistical evaluation together define the mea-surement volume.Now,a single exposure recording is considered.It consists 1The local sample of a PIV image from which a velocity vector is determined is referred to as the interrogation spot or window.Its size determines to what degree the recovered velocityfield is spatially smoothed.In optical interrogation systems it is defined by the diameter of the probing laser beam;in digital systems the rectangular pixel grid imposes a rectangular window.803Mathematical Background of Statistical PIV Evaluationof a random distribution of particle images,which correspond to the following pattern of N tracer particles inside theflow:Γ=⎛⎜⎜⎜⎜⎜⎜⎝X1X2···X N⎞⎟⎟⎟⎟⎟⎟⎠with X i=⎛⎝X iY iZ i⎞⎠being the position of a tracer particle in a3N-dimensional space.Γdescribes the state of the ensemble at a given time t.X i is the position vector of the particle i at time t.For more details about the mathematical description of the tracer ensemble,see[51].The lower case letters refer to the coordinates in the image plane(figure3.1)such thatx=x yis the image position vector in this plane.In the remainder of this section we will assume that the particle position and the image position are related by a constant magnification factor M for simplicity,such that:X i=x i/M and Y i=y i/M.As already described in section2.6.3,a more complex model of imaging geom-etry has to be used to take the effect of perspective projection intoaccount.Fig.3.1.Schematic rep-resentation of geometric imaging.3.2Image Intensity Field81 3.2Image Intensity FieldIn this section a mathematical representation of the intensity distribution in the image plane is given.It is assumed that the image can best be described by a convolution of the geometric image and the impulse response of the imaging system,the point spread function.For infinite small particles and perfectly aberration-free,well focused lenses the amplitude of the point spread function can mathematically be described by the square of thefirst order Bessel function also known as Airy function(see section2.6).A more complex model of imaging has to include imperfections of lenses and photographicfilms or sensors.For lenses and photographicfilms an es-timation of the main effects besides diffraction can be obtained by analyzing their modulation transfer functions(MTF’s)(see section2.6.2).For CCD sen-sors,a careful analysis requires more complex models which have not yet been described in the PIV literature sufficiently.The description of digital imaging of very small objects is especially important,because the systematic arrange-ment of sensor elements can cause significant bias errors in statistical particle image displacement estimation(q.v.peak locking;in section5.5).In the following analysis we assume the point spread function of the imag-ing lensτ(x)to be Gaussian versus x and y(see appendix A.2),which is a common practice in literature and a good approximation for the point spread function of real lens systems[51,78].The convolution product ofτ(x)with the geometric image of the tracer particle at the position x i therefore describes the image of a single particle located at position X i.Furthermore,we restrict the analysis to infinitely small geometric particle images which would be the case for small particles imaged at small magnifications.Therefore,we use the Dirac delta-function shifted to position x i to describe the geometric part of the particle image.As schematically illustrated in Fig3.2,the image intensity field of thefirst exposure may be expressed by:I=I(x,Γ)=τ(x)∗Ni=1V0(X i)δ(x−x i)(3.1)where V0(X i)is the transfer function giving the light energy of the image of an individual particle i inside the interrogation volume V I and its conversion into an electronic signal or optical transmissivity2.τ(x)is considered to be identical for every particle position.The visibility of a particle depends on many parameters as for example the scattering properties of the particle,the light intensity at the particle position,the sensitivity of the recording optics and the sensor orfilm at the corresponding image position.However,we adopt that the particles at every position have the same scattering properties and 2Strictly speaking equation(3.1)is valid only for incoherent light.For coherent light a term considering the interference of overlapping particle images has to be included[51].In most practical situations the particle images do not overlap.Therefore,we use equation(3.1)also for coherent illumination.823Mathematical Background of Statistical PIV Evaluationthe recording optics and media have a constant sensitivity over the image plane.In many situations different weight is assigned to different locations inside the interrogation area.This can be done by a multiplication of the recorded image intensity with weight kernels in the case of digital evaluation or implic-itly due to the spatial intensity distribution of the interrogating laser beam in the case of optical evaluation.Further on,we presume that Z is the viewing direction,the light intensity inside the interrogation volume is only a function of Z and the image intensityfinally analyzed depends on X and Y only due to the weight function.Therefore,V0(X)just describes the shape,extension and location of the actual interrogation volume:V0(X)=W0(X,Y)I0(Z)(3.2) where I0(Z)is the intensity profile of the laser light sheet in the Z direc-tion and W0(X,Y)is the interrogation window function geometrically back-projected into the light sheet.This is mathematically not correct,because it does not consider the convolution with the point spread function.For rectan-gular interrogation windows this means that in our mathematical description we neglect the effects of partially cropped images at the edges of the interroga-tion area.However,we will use this simple model of the interrogation volumes in theflow,because it also simplifies the description of PIV evaluation:I0(Z)=I Z exp−8(Z−Z0)2ΔZ20might be used to describe the Gaussian intensity profile of the laser light sheet, whereΔZ0is the thickness of the light sheet measured at the e−2points and I Z is the maximum intensity of the light sheet.W0(X,Y)can be described in a similar way if a Gaussian window function with a maximum weighting W XY at position X0,Y0has to be considered:W0(X,Y)=W XY exp−8(X−X0)2ΔX20−8(Y−Y0)2ΔY20.Since many pulsed lasers used for PIV have an intensity distribution which is closer to a top-hat function than to a Gaussian function and since digitized recordings are commonly interrogated with rectangular windows,V0(X)can also be defined as a rectangular box:I0(Z)=I Z if|Z−Z0|≤ΔZ0/20elsewhere(3.3)W0(X,Y)=W XY if|X−X0|≤ΔX0/2and|Y−Y0|≤ΔY0/20elsewhere.(3.4)3.3Mean Value,Autocorrelation and Variance of a Single Exposure Recording83Fig.3.2.Example of an intensity field I (single exposure).The factor I 0(Z i )represents the amount of light received from the particle i inside the flow,and located at distance |Z i −Z 0|from the center plane of the laser light sheet.ΔZ 0is the light sheet thickness and therefore the extension of the interrogation volume in the Z direction.ΔX 0=Δx 0/M and ΔY 0=Δy 0/M are the extension of the interrogation volume in the X -and Y -direction respectively.With τ(x −x i )=τ(x )∗δ(x −x i )(see appendixA.1)and the assumption that the particle images under consideration do not overlap,equation (3.1)can alternatively be written as:I (x ,Γ)=Ni =1V 0(X i )τ(x −x i ).(see appendix A)(3.5)This expression for the image intensity field will intensively be used in the following sections.In the following we will illustrate different representations of the intensity field and their correlation by giving an example for the recording of three arbitrarily located particles.3.3Mean Value,Autocorrelation and Varianceof a Single Exposure RecordingIn this section we will determine spatial estimators for the mean value and the variance of the image intensity field,because these quantities will be used for the normalization of the cross-correlation.Furthermore,autocorrelation and auto-covariance of a single exposure intensity field will be introduced.The main equations used in the following are taken from Papoulis [20,21].The spatial average is defined as:I (x ,Γ) =1a I a II (x ,Γ)d x where a I is the interrogation area.Employing equation (3.5)yields:I (x ,Γ) =1a I a I N i =1V 0(X i )τ(x −x i )d x .843Mathematical Background of Statistical PIV EvaluationThe mean value of the intensity field can be approximated by:μI = I (x ,Γ) =1a I N i =1V 0(X i ) a Iτ(x −x i )d x .We can now derive the autocorrelation of the single exposure intensity field in a similar way:R I (s ,Γ)= I (x ,Γ)I (x +s ,Γ) =1a I a I N i =1V 0(X i )τ(x −x i )N j =1V 0(X j )τ(x −x j +s )d x where s is the separation vector in the correlation plane.By distinguishing the i =j terms which represent the correlation of different particle images and therefore randomly distributed noise in the correlation plane,and the i =j terms which represent the correlation of each particle image with itself,we come to the following representation:R I (s ,Γ)=1a I N i =jV 0(X i )V 0(X j ) a I τ(x −x i )τ(x −x j +s )d x +1a I N i =j V 02(X i ) a Iτ(x −x i )τ(x −x j +s )d x .Following the decomposition proposed by Adrian ,we can write:R I (s ,Γ)=R C (s ,Γ)+R F (s ,Γ)+R P (s ,Γ)where R C (s ,Γ)is the convolution of the mean intensities of I and R F (s ,Γ)is the fluctuating noise component both resulting from the i =j terms.R P (s ,Γ)finally is the self-correlation peak located at position (0,0)in the correlation plane.It results from the components that correspond to the correlation of each particle image with itself (i =j terms).The autocorrelation of actual particle image data is provided in Fig 3.3and clearly shows a strong central self-correlation peak surrounded by a noise floor.We will now concentrate on this central peak in order to evaluate its fea-tures.For a Gaussian particle image intensity distributionτ(x )=K exp −8|x |2d 2τit can be shown that the autocorrelation R τ(s )is again a Gaussian function with a width that is √2d τ(see appendix A.3).Consequently R P (s ,Γ)may be rewritten as follows:3.3Mean Value,Autocorrelation and Variance of a Single Exposure Recording 85R PR +R C Fs y 00s x position of peaks in the auto-correlation function.R P (s ,Γ)=Ni =1V 02(X i )exp −8|s |2(√2d τ)2 1a I a I τ2 x −x i +s 2 d x .In the remainder of this book we will always use the representation:R τ(s )=exp−8|s |2(√2d τ)2 1a I a I τ2 x −x i +s 2d x taking into account that its features are mainly the same also for non-Gaussian τ(x ):the maximum of R τ(s )is located at |s |=0and the characteristics of its shape is given by the particle images shape.Therefore,we will write R P asR P (s ,Γ)=R τ(s )Ni =1V 02(X i ).In figure 3.4the schematic of the autocorrelation of the example intensity field I is given.The correlation peaks (R P and R F )occur at locations which are given by the vectorial differences between particle image locations.Their strength is proportional to the number of all possible differences which resultx 3x 32R -= = 0Fig.3.4.Schematic representation of theautocorrelation of the intensity field Igiven in figure 3.2.863Mathematical Background of Statistical PIV Evaluationin that location.For intensityfields with zero mean value the autocorrelation equals the auto-covariance.For nonzero mean values of the intensityfield the auto-covariance C I(s)can be obtained by[20]:C I(s)=R I(s)−μI2.An estimator of the variance of the intensityfield can be obtained by:σ2I=C I(0,Γ)=R I(0,Γ)−μI2=R P(0,Γ)−μI2.3.4Cross-Correlation of a Pair of Two SinglyExposed RecordingsAs already mentioned before,PIV recordings are most often evaluated by locally cross-correlating two frames of single exposures of the tracer ensem-ble.The mathematical background of this technique will be described in the following.In the remainder of this section,a constant displacement D of all parti-cles inside the interrogation volume is assumed,so that the particle locations during the second exposure at time t =t+Δt are given by:X i =X i+D=⎛⎝X i+D XY i+D YZ i+D Z⎞⎠.Furthermore,we assume that the particle image displacements are given by:d=MD X MD Ywhich is a simplification of the perspective projection that is valid only for particles located in the vicinity of the optical axis(see section2.6.3).We come to the following representation of the image intensityfield for the time of the second exposure(see equation3.5):I (x,Γ)=Nj=1V0 (X j+D)τ(x−x j−d)where V0 (X)defines the interrogation volume during the second exposure. If wefirst consider identical light sheet and windowing characteristics,the cross-correlation function of the two interrogation areas can be written as:R II(s,Γ,D)=1a Ii,jV0(X i)V0(X j+D)a Iτ(x−x i)τ(x−x j+s−d)d x3.4Cross-Correlation of a Pair of Two Singly Exposed Recordings87 IFig.3.5.The intensityfield I recorded attime t and the intensityfield I recordedafter a time delay ofΔt at t .where s is the separation vector in the correlation plane.Analogous to the procedure used in the previous section we arrive at:R II(s,Γ,D)=i,jV0(X i)V0(X j+D)Rτ(x i−x j+s−d).By distinguishing the i=j terms which represent the correlation of different randomly distributed particles and therefore mainly noise in the correlation plane and the i=j terms,which contain the displacement information de-sired,we obtain:R II(s,Γ,D)=i=jV0(X i)V0(X j+D)Rτ(x i−x j+s−d)+Rτ(s−d)Ni=1V0(X i)V0(X i+D).Again,we can decompose the correlation into three parts:R II(s,Γ,D)=R C(s,Γ,D)+R F(s,Γ,D)+R D(s,Γ,D)where R D(s,Γ,D)represents the component of the cross-correlation function that corresponds to the correlation of images of particles obtained from theR + RR DFCssyxpositionof peaks in the cross-correlation function.883Mathematical Background of Statistical PIV Evaluationx3=-dFig.3.7.Schematic representation of thecross-correlation of the intensityfields Iand I given infigure3.5.first exposure with images of identical particles obtained from the second exposure(i=j terms):R D(s,Γ,D)=Rτ(s−d)Ni=1V0(X i)V0(X i+D).(3.6)Hence,for a given distribution of particles inside theflow,the displacement correlation peak reaches a maximum for s=d.Therefore,as already antici-pated,the location of this maximum yields the average in-plane displacement, and thus the U and V components of the velocity inside theflow.Infigure3.7the schematic of the cross-correlation of the example intensity fields I and I is given.Nearly the same correlation peaks occur as in the autocorrelation shown infigure3.4,but at locations which are displaced by d.Correlations of x 2do not appear here,because this image is located outside the interrogation window(seefigure3.5).It can be seen from equation(3.6)that the displacement correlation is a function of the random variables(X i)i=1···N.Consequently it is a random variable itself and for different realizations at the same overall conditions we will obtain different qualities of the displacement estimation depending on the state of the tracer ensemble.In order to derive rules for a general optimization of the displacement estimation,we will determine the expected value of the displacement correlation in section3.6.3.5Correlation of a Doubly Exposed RecordingThe correlation function of a doubly(or multiply)exposed recording can be derived by analogy to the correlation for single exposed recordings.Instead of cross-correlating I with I ,we will consider the correlation of the intensity field I+=I+I with itself.Assuming identical light sheets and windowing characteristics,the intensityfield of both exposures I+can be written as:3.5Correlation of a Doubly Exposed Recording89Fig.3.8.The sum of the intensity fields I and I (see figure 3.5)as obtained by a recording of the tracer ensemble at t and t on the same frame.I +(x ,Γ)=I (x ,Γ)+I (x ,Γ)=Ni =1(V 0(X i )τ(x −x i )+V 0(X i +D )τ(x −x i −d )).It can be shown that the autocorrelation of I +consists of four terms:R I +(s ,Γ,D )=R I (s ,Γ)+R I (s ,Γ)+R II (s ,Γ,D )+R II (−s ,Γ,D ).It is therefore appropriate to decompose the estimator into the following terms:R I +(s ,Γ,D )=R C (s ,Γ,D )+R F (s ,Γ,D )+R P (s ,Γ)+R D +(s ,Γ,D )+R D −(s ,Γ,D )(3.7)where R C (s ,Γ,D )is the convolution of the mean intensity of I +and R F (s ,Γ,D )is the fluctuating noise component.R P (s ,Γ)is the self-correlation peak lo-cated at the center of the correlation plane.It results from the compo-nents that correspond to the correlation of each particle image with itself.R D +(s ,Γ,D )and R D −(s ,Γ,D )represent the components of the correlation function which correspond to the correlation of images of particles obtained D DR R R PR + R FC +-0s y x s ponents of the autocorrelationfunction.903Mathematical Background of Statistical PIV Evaluationfrom the first exposure with that of images of identical particles obtained from the second exposure and vice versa.When comparing the correlation of a doubly exposed recording with the correlation of a pair of two singly exposed recordings,the following statements can be made:R I +is symmetric with respect to its central peak R P .Two identical displacement peaks R D +and R D −appear and as a consequence the sign of the displacement cannot be determined.Therefore,the correlation of a doubly exposed recording is not conclusive if the displacement field of the whole recording is not unidirectional.Another problem appears if the field contains displacements close to zero,which would lead to an overlap between the displacement peaks with the central peak.However,these problems have to be solved during recording.Precautions have to be made so that the images of identical particles due to the different exposures do not overlap and the sign of their displacement is determined.If the flow field under investigation contains areas of reverse flow or of relative slow velocities image shifting has to be used (see section 4.3).It can be seen from figure 3.10that the correlation of doubly exposed recordings contains more than twice the number of randomly distributed noise peaks.The example given in figure 3.10shows that in situations for which the cross-correlation of single exposure yields good results,the correlation of doubly exposed recordings contains noise peaks of the strength of the displacement peak.Hence,the evaluation of multiply exposed recordings has to be per-formed with more particle image pairs in order to get the same performance as that of single exposure evaluation.This can be done by different methods:the seeding density,the number of exposures or the light sheet thickness can be increased.Besides other problems related to these methods their appli-cation is restricted due to the limited number of particle images that can be stored on the sensor without a significant overlap.Therefore,in most cases the size of the interrogation areas has to be increased compared to the evaluation of single exposures resulting in a lower spatial resolution of the measurement at the same sensor size.1x x 3(R )F--=x 1x x 3--= x (R )F x 3x 3-= -33x x 0= 1x 12x 2x = -= --= x x x 31x (R +)D --= x (R -)D31x -= -x x Fig. 3.10.Schematic representation of the autocorrelation of the intensity field I +I given in figure 3.8.3.6Expected Value of Displacement Correlation91 3.6Expected Value of Displacement CorrelationIn order to derive rules for a general optimization of the displacement esti-mation we will determine the expected value of the displacement correlation E{R D}for all realizations ofΓ.More concretely:we want to calculate the mean correlation function of all possible“patterns”that can be realized withN particles.From equation(3.6),it follows thatE{R D}=ERτ(s−d)Ni=1V0(X i)V0(X i+D)=Rτ(s−d)ENi=1V0(X i)V0(X i+D)Defining f l(X)=V0(X)V0(X+D)yields:E{R D}=Rτ(s−d)ENi=1f l(X i).(3.8)We prove in appendix A.4that:ENi=1f l(X i)=NV FV Ff l(X)d XwhereV Ff l(X)d X is the volume integralf l(X,Y,Z)d X d Y d Z.Thus:E{R D}=NV FRτ(s−d)V Ff l(X)d X.(3.9)Since we defined N to be the number of all particles of the ensemble,V F has to be interpreted as the whole volume offluid that has been seeded with particles. According to the above definition of f l(X)we can say in a more practical sense that the integration has to be performed over the volume which contained all particles that were inside the interrogation volumes during thefirst or second exposure.We can rewrite the integral over f l(X)as:V F f l(X)d X=I0(Z)I0(Z+D Z)d Z×W0(X,Y)W0(X+D X,Y+D Y)d X d Y =V FV20(X)d X·F O(D Z)F I(D X,D Y)923Mathematical Background of Statistical PIV Evaluation withF I(D X,D Y)=W0(X,Y)W0(X+D X,Y+D Y)d X d YW20(X,Y)d X d Y(3.10)andF O(D Z)=I0(Z)I0(Z+D Z)d ZI20(Z)d Z.(3.11)Keane&Adrian[82,83,84]have defined F I as a factor expressing the in-plane loss-of-pairs,and F O as a factor expressing the out-of-plane loss-of-pairs.When no in-plane or out-of-plane loss-of-pairs are present the latter two are unity.Finally equation(3.9)yields:E{R D(s,D)}=C R Rτ(s−d)F O(D Z)F I(D X,D Y)(3.12) where the constant C R is defined as:C R=NV FV FV20(X)d X.3.7Optimization of CorrelationThefirst parameter that has to be optimized during a PIV measurement is the pulse separation time between the successive light pulses.Besides techni-cal limitations some general effects have to be considered.According to the principle of PIV the measured velocity is determined by the ratio of two com-ponents of the measured particle displacement between successive light pulses D X and D Y respectively,and the pulse separation timeΔt.Since the particle displacement–which is considered to be a function ofΔt in the following–is determined by the particle image displacement with D X(Δt)=d x(Δt)/M and D Y(Δt)=d y(Δt)/M respectively,and the measured image displacements contain certain residual errors,εresid,we can define the following equation for the magnitude of the locally measured velocity:|U|=|d(Δt)|MΔt+εresidMΔt.(3.13)Since the particle image displacement for a given recording configuration re-duces linearly with the pulse separation time,thefirst term of the above equation stays constant for vanishing pulse separations:3.7Optimization of Correlation93lim Δt→0|d(Δt)|MΔt=|U|.In contrast to that,the residual error contained in the measured image dis-placement will not be reduced below a certain limit by a reduction of the pulse separation,because the uncertainty in determining the particle image positions will be unaffected.Therefore,the second term of equation(3.13)–which states that the measurement error is weighted with1/Δt–increases rapidly with decreasing pulse separation:lim Δt→0εresidMΔt=∞.From these considerations it can be seen that the accuracy of PIV measure-ments can be increased by increasing the separation time between the ex-posures at least within certain limits.However,for high values ofΔt the measurement noise increases.This becomes clear when looking at the expec-tation of the displacement correlation given in equation(3.12).It can be seen that the average signal strength is weighted with the loss of pairs due to the particle displacement D(Δt).For a very large separation time the particle displacement,which increases linearly withΔt,will exceed the extent of the interrogation volume.Then,no particle will be illuminated twice and no image correlation would be obtained.What can be done to improve the situation? First of all the pulse separation time can be reduced.This directly reduces the particle displacement and the loss of pairs.Infigure3.11we have tried to illustrate the two aspects of the choice ofΔt on the quality of the PIV data:the dotted line,curve g,represents the effect of the weighting of the residual error withΔt,the solid line,curve f,representsthe influence of the loss of pairs.The optimumΔt could therefore befoundQoptFig.3.11.Schematic representation for the optimization of the pulse delay time.943Mathematical Background of Statistical PIV Evaluationby determining the maximum of a quality function Q PIV,for example the product of curves f and g which is represented by the dashed line.However,the shape of curve f has been chosen arbitrarily,since a general value for the quality of a measurement is difficult to define.When using digi-tal equipment,which allows immediate feedback during the measurement,the optimum can be found interactively by slowly increasing the pulse separation until the number of obvious outliers3within the vector map increases.How-ever,the number of valid data yield is only one parameter of the obtained quality,but not an exact measure of it.Another parameter,which can be used for optimization,if it is made available from the evaluation software,is the normalized strength of the displacement correlation.The cross-correlation coefficient given by:c II=C IIσIσI=R II−μIμIσIσI.While using photographic recording the choice of all recording parameters merely depends on the experiences of the experimentalist,because the valid data yield,the cross-correlation coefficient,or,in the case of optical evaluation, the visibility of the Young’s fringes,can be assessed only after several hours.Another way to reduce the loss of pairs is to change the size of the inter-rogation volumes and/or to displace them slightly with respect to each other in order to compensate for the mean particle displacement.The extension of the interrogation volume in the out-of-plane direction is given by the light sheet thickness.This parameter can be increased only if adequate laser power is available.If one of the two possible out-of-plane directions is predominant, the light sheet can be displaced between the successive illuminations towards the meanflow.While using double oscillator systems,this can be achieved by a slight“misalignment”of the beam combining optics.In the case of CW lasers a displacement requires additional equipment(see e.g.section9.5).The extension and location of the interrogation volumes in the in-plane directions is given by the size of the interrogation areas during evaluation and the magni-fication during recording.In the case of cross-correlation analysis the location of the interrogation windows with respect to each other can be changed.This is one main advantage of cross-correlation and the reason why it is frequently applied also for the evaluation of single frame recordings instead of autocor-relation.The effects of the interrogation volume locations during thefirst and sec-ond exposure X0=(X0,Y0,Z0)and X 0=(X 0,Y 0,Z 0)respectively,can best be described by presenting equation(3.12)in a more generalized form:3Outlier is a common term describing data whose validity is questionable on the basis of a certain acceptance criterion.。