G 017 V Alternativna metoa TESTIRANJA premikov v GEODETSKI mreži GEODETSKI VESTNIK letn. / Vol. 61 št. / No. 3 An Alternative Approach to Testing Displacements in a Geoetic Network 61/3 Simona Savšek UDK: 58.0:58.48 Klasifikacija prispevka po COBISS.SI: 1.01 Prispelo: 5. 6. 017 Sprejeto: 31. 8. 017 DOI: 10.159//geoetski-vestnik.017.03.387-411 SCIENTIFIC ARTICLE Receive: 5. 6. 017 Accepte: 31. 8. 017 IZVLEČEK V geoeziji s postopki statističnega testiranja hipotez z ocenjevanjem in testiranjem značilnih parametrov ugotavljamo izpolnjevanje nekaterih kriterijev in zahtev pri merskih in računskih postopkih. V praksi so se za presojo značilnih premikov uveljavili nekateri kriteriji, ki testno statistiko primerjajo s konstantama T = 3 ali T = 5. V članku prelagamo alternativni postopek za oločitev empirične porazelitvene funkcije s simulacijami v Din 3D-geoetski mreži, saj se testna statistika T = / ne porazeljuje po nobeni o znanih porazelitvenih funkcij. Kritična vrenost je spremenljivka, ki pri različni imenziji mreže ob enakem tveganju ni enaka. Na več testnih primerih pokažemo, a je kritično vrenost T mogoče izračunati natančneje, kar zagotavlja zanesljivo ugotavljanje značilnih premikov glee na izbrano stopnjo značilnosti testa. V vseh testnih primerih T oseže vrenost 3 pri tveganju, ki je manjše o 1 % za poljubno imenzijo mreže. Če ocenimo, a je sprejemljivo tveganje 5 %, je kritična vrenost bistveno manjša o 3 ali 5. Značilne premike je torej smiselno obravnavati glee na sprejemljivo tveganje in ne glee na približno ocenjeno kritično vrenost. ABSTRACT In geoes, statistical testing ais in etermining the extent to which the eria an requirements neee in the measurement an calculation proceeings have been fulfille. A rule of thumb metho that compares test statistics to constants T = 3 or T = 5 has been establishe. The test statistic T is the ratio between the isplacement an its precision. Since it is not istribute through an of the known istribution functions, (statistical) simulations are use to assess the empirical istribution in D an 3D geoetic networks. The propose alternative proceure leas to a more precise etection of significant isplacements at a given test significance level α. Regarless of the network's imensionalit, T obtains the value of 3 at a risk level below 1%. When 5% is consiere to be an acceptable risk level, the ical value can be lower than 3 or 5. Thus, significant isplacements shoul be consiere with regar to the acceptable risk level an not accoring to the usual rule of thumb. KLJUČNE BESEDE statistično testiranje hipotez, stopnja značilnosti, simulirana porazelitvena funkcija, kritična vrenost, ejansko tveganje, premiki točk KEY WORDS statistical hpothesis testing, significance level, simulate istribution function, ical value, actual risk, point isplacement 387
61/3 GEODETSKI VESTNIK 1 INTRODUCTION In their research, professionals working in the fiel of geosciences (civil engineers, geologists, miners an others) efine the expecte isplacements. Surveors can etermine the actual isplacements base on fiel measurements such as Marjetič et al. (01), Kregar et al. (015), an asses their accurac using the statistical methoolog such as Kregar et al. (01), Urbančič et al. (016). This task can also be reverse. We can efine the threshol that etermines whether the isplacement is significant or not with respect to the isplacement etermination accurac, which is obtaine through measurements an calculations. Some of the approaches can be foun in literature. In the literature overview the same smbols as use b the original authors were use. The classical approach for etermining whether the isplacement is statisticall significant or not is use an escribe in numerous articles (Caspar, 1987; Dognan et al., 013; Heck et al., 1977; Koch, 1999; Kuang, 1996; Niemer, 1985, Pelzer, 1971; Savšek-Safić et al., 003; Sütti an Török, 1996). The ical value at a selecte risk level α is compare to the quotient of the quaratic form calculate as the prouct of the isplacement vector an the corresponing cofactor matrix T = T Σ -1, which is istribute b istribution (Koch, 1999). χ α Statistical significance of isplacement can be teste with conventional eformation analsis (CDA methos) or robust methos (IWST methos). The choice of the metho usuall epens on the kin of the eformation that is investigate an the tpe of a control network that is esigne for the certain object of stu. CDA methos are base on the least squares estimation (LSE) such as Hannover an Karlsruhe metho where global congruenc test of two epoch ajustment results is use (Pelzer, 1971; Niemeier, 1981). IWST metho (Iterative Weighte Similarit Transformation) is a goo estimator of the single-point isplacement vector in the process of robust S-transformation (Chen et al. 1990). An alternative metho M-estimation has been propose to etermine the isplacement vector irectl from ifferences in raw unajuste observations (Nowel, 015; Nowel, 016). An alternative metho is even more efficacious when law values of isplacements that slightl excee measurement errors are expecte. A ver goo empirical measure of the efficac of outlier tests an robust estimation methos are propose b Hekimoglu an Koch (000). The measure is calle the mean success rate (MSR) an is compute base on man thousans of ifferent simulate observation sets b the Monte Carlo metho as the ratio of the number of sets for which outliers were correctl etecte to the number of all sets. The propose measure of efficienc MSR can be goo alternative to statistical measures such as global an local measures of internal reliabilit in traitional eformation analsis methos. Rueger (1999) treats the require accurac of isplacement etermination similarl to Welsch et al. (000) an Pelzer et al (1987). If is the estimate value of minimal isplacement, then the require accurac s can be assesse with a rule of thumb esign equation s /5. Other authors claim that isplacements are significant when the ratio between the isplacement an its accurac is greater than 3 (Klein an Heunecke, 006). The U.S. Arm Corps of Engineers (USACE Arm, 00) recommens that the stanar eviation of measurements for isplacement etection shoul be 9 times lower than the greatest expecte isplacement. If one uses 95 % confience level to escribe the measurement accurac, then it shoul be 4 times lower than the greatest expecte isplacement. 388
GEODETSKI VESTNIK 61/3 Hekimoglu et al. (010) calculate the raius of the corresponing isplacement circle r from the elements of the respective sub-matrix of the cofactor matrix Q an α-fractile of the χ -istribution for egrees of freeom χ. The observe what percentage of points has move in the interval between r,α an r or between r an 3r for a chosen value of error probabilit α. Ramos et al. (01) evelope software for 3D movements of olive trees ue to erosion of the terrain. 3D movements were analze in two parts: as planimetric an altimetric movements. The ecie that the planimetric isplacements were significant if the 99 % error ellipses of iniviual points from each campaign o not intersect. To etermine the altimetric movements, the have taken a similar ecision: isplacement is significant, if the altimetric isplacement vector for each point from each campaign is higher than the error interval as calculate b Error _ C1 + Error _ C. 99 % 99 % So far the literature overview ealt preominantl with isplacements in a D plane. Berber (006) an Berber et al. (009) calculates the threshol value in a 3D space which is an approximation of the 3D confience ellipsoi using the following equation δ = + +, in which the semi-axes of the i a95 b i 95 h i 95i 95 % confience ellipsoi an the vertical interval are obtaine as =.795 a,b,h. The stanar 95i a,b,hi 3D ellipsoi at χ = 1 has a confience level (1-α) of about 0 %. In orer to achieve a 95 % confience level, a multiplication factor.795 must be use (Stauinger, 1999). 3;1-α Displacements an eformations appear on artificial objects such as ams, embankments, briges, their surrounings (e.g. in reservoir valles, on the banks of accumulation lakes) as well as in organic areas such as lanslies, tectonic faults an marshes. The reasons for isplacements can be attribute to external forces (temperature changes, win, tectonic or seismic effects), mechanical properties of the construction materials an elements, inaequate knowlege of the groun s geomorphologic composition, mechanic properties an hrological conitions at the time of projecting the object. Establishing the isplacements an eformations of artificial an natural objects is one of the most complex tasks in surveing. Inirectl obtaine quantities such as the shift an the corresponing stanar eviation (statisticall teste with the test statistic T = / ) are usuall use when assessing the statistical significance of isplacements. Since the test statistic T is compare to the ical value T which is etermine on the basis of the appropriate istribution function an the chosen significance level it is important to etermine the istribution function correctl. The statistical test etermines the subset of a sample space known as the ical value approach or the null hpothesis rejection interval. In practice the erion of T > 3 or T > 5, also known as the rule of thumb is often use in the assessment of significant isplacements. However, as simulation proceures allow us to etermine the ical value T much more accuratel this erion is not precise enough. In this article we will eal with the specialities use in the calculation of isplacements with corresponing stanar eviations in 1D, D an 3D geoetic networks. The proceures use to etermine the istribution function for the test statistic T = / are propose accoring to the imensionalit of the network. For a D an 3D geoetic network the test statistic in not istribute through an analtical istribution function, which is wh the appropriate istribution function is etermine empiricall, through simulations. Since the simulations allow us to etermine the ical value T for each iniviual point within a network, the propose alternative proceure leas to higher qualit of the etection of significant isplacements at a given test significance level α. 389
61/3 GEODETSKI VESTNIK DISPLACEMENT AND DISPLACEMENT ACCURACY ESTIMATION In orer to calculate the isplacement an its stanar eviation, the positions of ientical points nee to be etermine in two time perios. If we have a sufficient number of stable points within a geoetic network, geoetic atum of the network is ensure well. If the calculate isplacement values are significantl higher than the corresponing precision, a ecision can be reache without an statistical testing. It is often har to etermine the sufficient number of stable points an etect significant isplacements in geotechnical stuies, for their value can often be foun in the rang of their stanar eviation. Statistical testing proceures are use in orer to reuce the risk of creible etection of significant isplacements. Consier the position of a point P in time t an t+δt. The isplacement is calculate as follows: in 1D network: 1D = H = H t+ t H t, (1) in D network: x ( ) ( x x ) D t+ t t t+ t t = + = - + - in () in 3D network: ( x H ) ( ) ( x x ) ( H H ) 3D t+ t t t+ t t t+ t t = + + = - + - + -, (3) in which t, x t, H t an t+δt, x t+δt, H t+δt are ajuste coorinates of the same point in ifferent time perios. In orer to calculate the isplacement precision, we not onl nee to know the point coorinates, but also their variance-covariance matrix. Let s assign point P t in the first time perio with the variance-covariance matrix Σ P t an the same point P t+δt in the secon time perio with the variance-covariance matrix Σ P t+ t. Let s assume that the coorinates of points P in time t are not correlate with the coorinates in time t+δt. The variance-covariance matrix of the point in both time perios can be written as follows: ΣP 0 t Σ = + Σ, PP t t t 0 P t+ t (4) H 0 t in 1D network: Σ PP = t t+ t, 0 Ht+ t (5) 0 0 t x t t in D network: x 0 0 t t xt Σ = in PP t t+ t 0 0 t+ t t+ txt+ t 0 0 t+ txt+ t xt+ t (6) 0 0 0 t x t t H t t x 0 0 0 t t x t xh t t H 0 0 0 t t xh t t Ht in 3D network: PP =. t t+ t 0 0 0 t t t tx t t t th + + + + t+ t 0 0 0 t tx t t x t t xt th + + + + t+ t 0 0 0 t th t t xt th + + + t+ t Ht+ t (7) Taking into account the error propagation law, the variance of point P isplacement is 390
GEODETSKI VESTNIK 61/3 T = D Σ D PP t t+ t D J J, (8) in which the Jacobi matrix J D equals: in 1D: 1D 1D J = 1 1D = Ht H - t + t 1, (9) in D: D D D D = D = - - t xt t+ t xt+ t D D D D (10) in 3D: 3D 3D 3D 3D 3D 3D H H = 3D = - - - t xt Ht t+ t xt+ t Ht+ t 3D 3D 3D 3D 3D 3D When we put eqs. (9), (10) an (11) into (8) the variance of point P s isplacement is written as follows: in 1D: = 1D H +, (1) t Ht + t in D: = ( + ) + ( x + x ) + ( x + x ) D t t+ t t t+ t t t t+ t t+ t D D D D H x 3D t t+ t xt xt+ t Ht Ht+ t x t t t+ txt+ t 3D 3D 3D 3D ( ) ( ) ( ) ( ) in (13) = + + + + + + + + in 3D: (14) H x H + ( H + ) ( ) t t t th + t t xh + + + t t xt+ tht+ t 3D The variance is use for testing the statistical significance of isplacements. 3D 3 DETERMINING THE DISTRIBUTION FUNCTION OF THE TEST STATISTICS WITH SIMULATIONS Following the ajustment of the measurements for both time perios, the isplacement of points an their accuracies are etermine. The test statistic for the etection of significant isplacements is written as T =. (15) The test statistic can be istribute through known istribution functions (normal, stuent, Fischer etc.) or the istribution function can be empiricall etermine through simulations. It is important to accuratel etermine the istribution function, since the ical value T epens on it. The ical value also epens on the significance level α an how the null hpothesis H 0 is set. The null hpothesis has to be set for statistical testing. This etermines the subset of the sample space known as the ical value approach or the null hpothesis rejection interval. An alternative hpothesis nees to be set in case the null hpothesis is rejecte. The alternative hpothesis is the opposite of the null hpothesis. The test statistic is teste in relation to the null an alternative hpothesis: H 0 : = 0; point has not move between the two time perios, H a : 0; point has move between the two time perios. 391
61/3 GEODETSKI VESTNIK The test statistic T is compare to the ical value T which is etermine on the basis of the istribution function. If the test statistic is lower than the ical value at a chosen significance level α, the risk for rejecting the null hpothesis becomes too high. In this case it can be conclue that the isplacement is not statisticall significant. On the other han, when the test statistic is higher than the ical value, it can be conclue that the risk for rejecting the null hpothesis is lower than the chosen significance level α. In this case the null hpothesis is rejecte an followe b the conclusion that the teste isplacement is statisticall significant. Two cases are consiere: T T : null hpothesis cannot be rejecte; point isplacement is not statisticall significant, T > T : null hpothesis must be rejecte; point isplacement is statisticall significant. xˆ xˆ 3.1 Calculation of the ical value in a 1D network We assume that the ranom measurement errors are normall istribute ε N(0, ). If this is the case, then it can be safel assume that the quantities that are linear combinations of these measurements are also normall istribute x ˆ~ N ( µ, ). Since 1D is calculate as the ifference of two normall istribute variables, the height ifference of the same point between two time perios is also normall istribute. 3. Calculation of the ical value in a D network Since the isplacement in D networks is a non-linear function of two normall istribute variables Δ an Δx it is not istribute through the normal istribution function. The isplacement istribution is empiricall etermine through simulations (Savšek-Safić, 00). To come up with an accurate istribution function we suggest the use of simulations with epenent ranom variable samples obtaine through linear transformation. The proceure allows for a precise etermination of the ical value at a chosen risk level (for null hpothesis rejection). In orer to generate a sample of epenent normall istribute ranom variables we nee to generate a sample of inepenent normall istribute ranom variables upon which a linear transformation can be applie. The Box an Müller metho (Box et al., 1958; Press et al., 199) is use to generate inepenent normall istribute ranom variables. Let u 1i an u i, i = 1,,n be two samples of inepenent ranom variables U 1 an U evenl istribute on interval (0,1). A sample of two inepenent normall istribute ranom variables Z 1 an Z is calculate as: ( π ) ( π ) z - lnu sin u 1i 1i i z i =, i 1... n z = =. (16) i - lnu1 i cos ui The linear transformation is use to convert the sample of inepenent normall istribute ranom variables into a sample of epenent normall istribute ranom variables i = U T z i, i = 1,..., n (17) in which matrix U is obtaine through Cholesk s ecomposition of the variance covariance matrix Σ, Σ = U T U 39
GEODETSKI VESTNIK 61/3 x U =. (18) x 0 1- When we want to generate a sample of normall istribute coorinate ifferences Δ i an Δx i we use eq. (17) an assume that the mean values µ Δ an µ Δx equal 0, which gives us: = z i 1i x. (19) x xi = z1 i + zi 1- Variances an covariances are: = + t t+ t = + xt xt+ t = + x x t t t+ txt+ t in which t, x t, t x t, t+ t, x t+ t, t+ t x t+ t are variances an the covariances of coorinates t, x t, t+ t, x t+ t. Since the isplacement an stanar eviations of point coorinates iffer in each time perio, the istribution function of the test statistic T is ifferent for each point. Displacement an its stanar eviation can be calculate with the use of simulate normall istribute variables. The simulation is use for generating a sample of inepenent normall istribute ranom variables an through linear transformation we obtain epenent normall istribute ranom variables. In n simulations the proceure allows us to etermine the empirical cumulative probabilit istribution of the test statistic an corresponing ical value accoring to the chosen significance level α for each point (Savšek-Safić et al., 006). (0) 3.3 Calculation of the ical value in a 3D network Since the isplacement in 3D networks is a non-linear function of three normall istribute variables Δ, Δx an ΔH it is not istribute through the normal istribution function. The isplacement istribution is empiricall etermine through simulations. The istribution function is etermine similarl as in D networks. We start off b generating a sample of inepenent normall istribute ranom variables. Let u 1i, u i, i = 1,,n an u 3i, u 4i, i = 1,,n be two pairs of samples of inepenent ranom variables U 1, U an U 3, U 4 evenl istribute on the interval (0,1). A sample of three inepenent normall istribute ranom variables Z 1, Z an Z 3 is calculate as follows: z ln 1 sin( ) 1i - u i πu i z z i = i = - lnu1 cos( ), i 1... n i πu i = (1) z 3i - lnu3i sin( π u4i) 393
61/3 GEODETSKI VESTNIK The linear transformation (17) is use to transform the sample of inepenent normall istribute variables into a sample of epenent normall istribute variables, for which we use the ecompose matrix U: x H x x H - H x () U = 0 1-. x 1- H x H - H x 0 0 H - - x 1- The sample of epenent normall istribute variables Δ i, Δx i an ΔH i is generate as follows: = z i 1i x x xi = z1 i + zi 1- H = z + z + z - - H x H - H x H x H - H x i 1i i 3i H x x 1-1-. (3) The variances an covariances of the ifference between the coorinates of a specific point in two time perios are calculate as follows: for Δ an Δx through eq. (0) while for ΔH eq. (4) is use: = + = + = + H Ht Ht+ t H H t t t+ tht+ t x H xh t t xt+ tht+ t, (4) in which t, x t, H t, t x t, t H t, xt H t, t+ t, x t+ t, H t+ t, t+ t x t+ t, t+ t H t+ t, xt+ t H t+ t are variances an covariances of the spatial coorinates of a point t, x t, H t, t+ t, x t+ t, H t+ t. In 3D an D networks the istribution of the test statistic T varies for each iniviual point. 394
GEODETSKI VESTNIK 61/3 4 RESULTS Several control geoetic networks of ifferent imensions, shapes, number of reference an control points establishe for the purpose of eformation monitoring of ams an geological faults in the Slovene area are analze in the article. Stabilit monitoring networks tpicall have specific geometr (see Appenix). All observations were mae using precise total station (Leica Geosstems TS30 R1000: ISO-THEO HZ,Z = 0.5 an ISO-EDM = 0.6 mm; 1ppm) in 7 sets of angels. Observations for height etermination with geometric levelling are mae using precise igital level Leica Geosstems DNA03: ISO-LEV = 0.3 mm/km. The ical value T for a normal istribution function is calculate at a selecte significance level α. In practice the most common values are α = 0.10, α = 0.05, α = 0.01 an α = 0.001. Besies these values in Table 1 also shows significance levels α that provie ical values T 3 an more. Table 1: Critical value T with respect to significance level α. 1D D 3D α T T min T max T mean T min T max T mean 0.10 1.64 1.94.13.07 1.67 1.81 1.71 0.05 1.96..43.36 1.98.11.0 0.01.58.79 3.01.93.59.69.6 0.007 3.00 0.0081.86 3.08 3.00 0.0084.97 3.07 3.00 0.001 3.9 3.43 3.68 3.59 3.4 3.35 3.8 0.00001 4.4 4.47 4.93 4.76 4.06 4.1 4.08 Table 1 shows that in a 1D network the ical value T of the normal istribution function is uniquel efine at a given significance level α. When the significance value ecreasing (i. e. the risk of an unjustifie rejection of the null hpothesis error tpe I. is reuce) the ical value rises accoringl. We can observe that for ical value T = 3.9 the risk is onl 0.1 %. In practice a 95 % probabilit is usuall acceptable when eciing whether the isplacement is significant or not. In this case we have to compare the test statistic T with the ical value T = 1.96 an not with constants 3 or 5. If the client consiers that a 5 % risk is acceptable, the given isplacement can be ientifie as significant much sooner. This is crucial for testing as the aim is to etect significant isplacement with the greatest possible probabilit. In orer to obtain the most probable ical value at a chosen significance level we teste several D an 3D geoetic networks in Slovenia with ifferent geometries (see Tables in Appenix). Table 1 lists the minimum, maximum an average ical values T mean for test networks base on a simulate empirical istribution function for chosen significance level α=0.10; α=0.05; α=0.01 an α=0.001. Table 1 shows a significance level α, that provie an average ical values T mean = 3, which often occurs in practice for testing significant isplacement. Level α for the ical value of 5 is not liste since its value is lower than 0.00001 an can be treate as a non-risk ecision. Table 1 inicates that when the significance value ecreasing (i. e. the risk of an unjustifie rejection of the null hpothesis error tpe I. is reuce) the ical value rises. This can be clearl seen when we 395
61/3 GEODETSKI VESTNIK compare the test statistic T with the ical value T mean mean = 3.59 for D geoetic networks an T = 3.8 for 3D geoetic networks where the risk is onl 0.1 %. In practice a 95 % probabilit is usuall acceptable when eciing whether the isplacement is significant or not. In such an event we have to compare the test statistic T with the ical value T mean mean =.36 for D geoetic networks an T =.0 for 3D geoetic networks an not with constants 3 or 5. If the client consiers a 5 % risk to be acceptable, then the given isplacement can be ientifie to be significant much sooner. This is crucial for testing as the aim is to etect significant isplacement with the greatest possible probabilit. The client etermines the acceptable risk in accorance to the consequences brought forth b a wrong ecision. The aim of statistical testing is to etect significant isplacement as reliabl as possible. Simulation proceures can be of great help since a properl etermine istribution function is of utmost importance for a ical value etermination. 5 CONCLUSION A reliable estimate of measurements, coorinates an isplacements is of great importance since the eformation analsis eals with large amounts of observations in multiple time perios. Statistical testing represents an inispensable tool for evaluating parameters, ientifing the aequac of observations an mathematical moels, etecting gross errors in observations, ientifing the compliance of supposel stable points uring a time perio an etecting statisticall significant isplacements of unstable points. Statistical testing proceures help us eliminate the possibilit of a specific point isplacement being wrongl treate as significant ue to inaequate observations, a wrongl selecte mathematical moel or the non-compliance between the measurements in two time perios. In orer to ensure reliable isplacement etection it is important to select a representative test statistic when the homogeneous precision of two perios is ensure. This article iscusses the common test statistic T = /. In D an 3D geoetic networks the isplacement is a non-linear function of variables Δ, Δx an ΔH an therefore it is not istribute through an of the analtical istribution functions. The propose proceure was teste on multiple geoetic networks varing in size, imensionalit (1D, D, 3D), geometr, number of probable stable points an number of repeate measurements (perios). In all cases the empirical istribution function was simulate an ical values were calculate for commonl use significance levels α = 0.10, α = 0.05, α = 0.01 an α = 0.001. Special attention was pai to eria T > 3 an T > 5, known as the rule of thumb. In the same wa as the istribution function iffers for each point, it also oes for the ical value. In aition, the ical values that correspon to the same risk level iffer for a 1D, D or 3D geoetic network. The ical value is a variable so it shoul not be treate as a constant (3 or 5). All test cases have shown that the ical value 3 correspons to an extremel low risk level (less than 1 %). Risk level of 1% is use when statistical tests are applie for research, risk management or in case of eformation measurement on ical infrastructure (nuclear power plants, etc.). For most geotechnical objects (briges, ams etc.) a risk level of 5 % is sufficient. The client has greater benefits if the empirical istribution function an accurate ical values are etermine accoring to the chosen significance/risk level. The constant risk in all network imensions is important for the isplacement evaluation as this ensures the comparabilit between perios. 396
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Robust M-estimation in analsis of control network eformations: classical an new metho. Journal of Surveing Engineering, 141 (4), 0401500. DOI: https://oi.org/10.1061/(asce)su.1943-548.0000144 Nowel, K. (016). Investigating efficac of robust M-estimation of eformation from observation ifferences. Surve Review, 48 (346), 1 30, DOI: https://oi.org /10.1080/0039665.015.1097585 Pelzer, H. (1987). Ingenieurvermessung: Deformationsmessungen, Massenberechnung, Vol. 15. Stuttgart: Verlag Konra Wittwer GmbH. Pelzer, H. (1971). Zur Analse geoätischer Deformationsmessungen. Deutsche Geoätische Kommission, Reihe C, No. 164. München: Verlag er Baerischen Akaemie er Wissenschaften. Press, W. H., Teukolsk, S. A., Vetterling, W. T., Flanner, B. P. (199). Numerical Recipes in Fortran, The Art of Scientific Computing. Cambrige: Cambrige Universit Press. Ramos, M. I., Cubillas, J. J., Feito, F. R., Gil, A. J. (01). Software for etecting 3D movements: The case of olive tree isplacements in an olive grove locate on sloping lan. Computers & Geosciences, 4, 143 151. DOI: https://oi. org/10.1016/j.cageo.011.09.009 Rueger, J. M. (1999). Overview of Geoetic Deformation Measurements of Dams in Australian Committee on Large Dams. ANCOLD 006 Conference. Savšek-Safić, S., Ambrožič, T., Stopar, B., Turk, G. (006). Determination of Point Displacements in the Geoetic Network. Journal of Surveing Engineering, 13 (), 58 63. DOI: http://ascelibrar.org/oi/10.1061/%8asce%90733-9453%8006%913%3a%858%9. Savšek-Safić, S., Ambrožič, T., Stopar, B., Turk, G. (003). Determining point isplacements in a geoetic network. Geoetski vestnik, 47 (1&), 7 17. http:// www.geoetski-vestnik.com/47/1/gv47-1_007-017.pf. Schmit, U. M. (003). Objektorientierte Moellierung zur geoätischen Deformationsanalse. Dissertation. Karlsruhe, Universitätsverlag Karlsruhe. Stauinger, M. (1999). A cost Orientate Approach to Geoetic Network Optimisation. Dissertation. Vienna, Universit of Technolog, Facult of Technical Science. Sütti, J., Török, C. (1996). Testing 3D isplacement vectors b confience ellipsois. Acta Montanistica Slovaca, 1 (4), 301 310. https://pfs.semanticscholar. org/661/331bc70998f095154b847b38133b5fe.pf. Urbančič, T., Vrečko, A. an Kregar, K. (016). Zanesljivost metoe RANSAC pri oceni parametrov geometrijskih oblik. Geoetski vestnik, 60 (1), 69 97. DOI: https:// oi.org/10.159/geoetski-vestnik.016.01.69-97 Sace, A. (00). Engineering an Design Structural Deformation Surveing. Engineer Manual EM 1110--1009. Washington, Department of the Arm. Welsch, W., Heunecke, O., Kuhlmann, H. (000). Auswertung geoätischer Überwachungsmessungen, Hanbuch er Ingenieurgeoäsie. Heielberg, Herbert Wichmann Verlag. Savšek S. (017). An alternative approach to testing isplacements in a geoetic network. Alternativna metoa testiranja premikov v geoetski mreži. 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61/3 GEODETSKI VESTNIK Appenix Network shape Number of irections Number of istances Num. of zenith istances Number of unknowns Number of points Average p (mm) α T min T max T mean D 10 5 1 4 0. 0.10.01.1.07 0.05.30.4.63 0.01.86 3.00.93 0.008.9 3.08 3.00 0.001 3.49 3.68 3.59 0.00001 4.60 4.95 4.75 3D 10 5 10 16 4 44.8 0.10 1.64 1.64 1.64 0.05 1.96 1.96 1.96 0.01.57.57.57 0.0034 3.00 3.00 3.00 0.001 3.3 3.3 3.3 0.00001 4.06 4.06 4.06 D 93 11 99 41 0.3 0.10 1.77.14.04 0.05.06.44.33 0.01.65 3.0.91 0.0075.75 3.1 3.00 0.001 3.33 3.69 3.56 0.00001 4.47 4.97 4.80 3D 93 11 16 140 41.1 0.10 1.70.18 1.81 0.05.01.46.10 0.0171.61 3.01.69 0.01.93 3.31 3.00 0.001 3.7 3.64 3.34 0.00001 4.06 4.75 4.14 D 10 5 1 4 0.5 0.10.06.14.10 0.05.35.43.39 0.01.93 3.0.98 0.0093.95 3.05 3.00 0.001 3.58 3.70 3.64 0.00001 4.55 4.85 4.70 D 10 5 1 4 0.5 0.10.06.1 3D 10 5 10 16 4 30.7 0.10 1.64 1.64 1.64 0.05 1.96 1.96 1.96 0.01.57.57.57 0.007 3.00 3.00 3.00 0.001 3.3 3.3 3.3 0.00001 4.06 4.06 4.06 398
GEODETSKI VESTNIK 61/3 Network shape Number of irections Number of istances Num. of zenith istances Number of unknowns Number of points Average p (mm) α T min T max T mean D 44 34 9 1 0. 0.10 1.89.11.00 0.05.18.41.9 0.01.74.99.85 0.006.88 3.15 3.00 0.001 3.37 3.64 3.50 0.00001 4.51 4.90 4.67 3D 44 34 44 41 1 3.1 0.10 1.66 1.67 1.66 0.05 1.97 1.99 1.98 0.01.58.59.58 0.0063 3.00 3.01 3.00 0.001 3.3 3.5 3.4 0.00001 4.06 4.07 4.06 D 39 9 9 1 0. 0.10.0.13.10 0.05.30.43.39 0.01.86 3.01.97 0.0090.90 3.04 3.00 0.001 3.50 3.68 3.6 0.00001 4.45 4.95 4.8 3D 39 8 36 41 1 1.5 0.10 1.71 1.94 1.79 0.05.0..09 0.01.6.77.67 0.003.94 3.08 3.00 0.001 3.7 3.41 3.33 0.00001 4.07 4.1 4.09 D 76 45 51 19 0. 0.10 1.87.14.08 0.05.16.44.37 0.01.71 3.03.95 0.0085.76 3.08 3.00 0.001 3.34 3.70 3.61 0.00001 4.5 4.97 4.80 3D 76 45 76 70 19.4 0.10 1.66 1.76 1.7 0.05 1.97.06.0 0.0188.58.65.6 0.01.96 3.04 3.00 0.001 3.3 3.31 3.9 0.00001 4.03 4.10 4.07 399
61/3 GEODETSKI VESTNIK SI EN Alternativna metoa testiranja premikov v GEODETSKI mreži OSNOVNE INFORMACIJE O ČLANKU: GLEJ STRAN 387 1 uvo Strokovnjaki, ki elujejo na poročju geoznanosti (grabeniki, geologi, ruarji in rugi), v svojih raziskavah opreelijo pričakovane premike. Geoeti na polagi terenskih meritev oločimo ejanske premike kot na primer Marjetič et al. (01), Kregar et al. (015), in natančnost oločitve premikov z uporabo statističnih meto kot na primer Kregar et al. (01), Urbančič et al. (016). Nalogo lahko tui obrnemo. Glee na natančnost oločitve premikov, ki je posleica merjenja in izračunov, lahko izračunamo mejo, o katere lahko trimo, a točka miruje ali se je značilno premaknila. V literaturi je opisanih nekaj pristopov. V pregleu literature ohranjamo oznake tako, kot jih zapisujejo posamezni avtorji. Klasični pristop ocenjevanja, ali je premik statistično značilen ali ne, je uporabljen in opisan v številnih člankih (Caspar, 1987; Dognan et al., 013; Heck et al., 1977; Koch, 1999; Kuang, 1996; Niemer, 1985; Pelzer, 1971; Savšek-Safić et al., 003; Sütti in Török, 1996). Kritična vrenost pri izbrani stopnji značilnosti α se primerja s kvocientom kvaratne forme, ki se izračuna kot proukt vektorja premika točk s pripaajočo matriko kofaktorjev T = T Σ -1, ta se porazeljuje po porazelitvi χ (Koch, 1999). α Statistično značilne premike lahko testiramo s klasičnimi postopki eformacijske analize (angl. CDA conventional eformation analsis) ali z robustnimi metoami (angl. IWST iterative weighte similarit transformation). Izbira metoe je običajno ovisna o vrste eformacij, ki jih raziskujemo, in o vrste kontrolne mreže, ki jo vzpostavimo za obravnavani objekt. CDA-metoe temeljijo na oceni najmanjših kvaratov (LSE), kot sta Hannover in Karlsruhe, kjer uporabimo globalni kongruenčni test me vema terminskima izmerama (Pelzer, 1971; Niemeier, 1981). IWST-metoe zagotavljajo obro oceno vektorja premikov za posamezno točko v postopku robustne S-transformacije (Chen et al., 1990). V literaturi je prestavljena alternativna M-ocena za oločitev vektorja premikov neposreno iz razlik»surovih«neizravnanih meritev (Nowel, 015; Nowel, 016). Metoa je še posebej učinkovita, kaar se pričakujejo mali premiki, ki le malo presežejo merske pogreške. Hekimoglu in Koch (000) prelagata zelo obro empirično mero za učinkovito testiranje grobih pogreškov in oceno robustnih meto, imenovano povprečna ocena uspešnosti (angl. MSR mean success rate). Izračunamo jo na polagi velikega števila simuliranih meritev po metoi Monte Carlo kot razmerje me številom meritev, pri katerih okrijemo grobe pogreške, in številom vseh meritev. Prelagana mera MSR je lahko obra alternativa statističnim meram, kot so globalne in lokalne mere notranje zanesljivosti v traicionalnih pristopih eformacijske analize. Rueger (1999) obravnava zahtevano natančnost oločitve premikov poobno kot Welsch et al. (000) in Pelzer et al. (1987). Če je ocenjena velikost minimalnega premika, potem lahko zahtevano natančnost meritev s ocenimo s približno enačbo s /5. V rugih člankih avtorji prepostavijo, a je premik značilen, če je kvocient me premikom in stanarno eviacijo premika večji kot 3 (Klein in Heunecke, 006). 400
GEODETSKI VESTNIK 61/3 V ameriški agenciji za vojno inženirstvo USACE (USACE Arm, 00) priporočajo, naj bo stanarna eviacija meritev ( 39% ) za oločitev eformacij evetkrat manjša o največje pričakovane vrenosti eformacij. Če uporabimo 95 % stopnjo zaupanja, pa mora biti stanarna eviacija meritev ( 95% ) štirikrat manjša o največje pričakovane vrenosti eformacij. Hekimoglu et al. (010) izračuna raij pripaajočega kroga premikov r iz elementov pripaajoče pomatrike matrike kofaktorjev Q in izbrane stopnje značilnosti α porazelitve χ za prostostni stopnji χ. Ob izbrani stopnji značilnosti α ugotavlja, koliko ostotkov točk se je premaknilo v intervalu me,α r in r oziroma me r in 3r. Ramos et al. (01) so razvili program za 3D-premike oljk, ki jih povzroča erozija terena. Premike so analizirali ločeno za horizontalno in vertikalno komponento. Premike v ravnini so obravnavali kot značilne, kaar se 99 % elipsi pogreškov iz veh zaporenih izmer ne prekrivata. V višinskem smislu so premik označili kot značilen, kaar je večji o intervala natančnosti Error _ C1 + Error _ C. 99 % 99 % Do seaj opravljen pregle literature obravnava premike pretežno v D-ravnini. Berber (006) in Berber et al. (009) izračunajo mejno vrenost 3D-elipsoia zaupanja z enačbo δ = + +, kjer so i a95 b i 95 h i 95i posamezne polosi 95 % elipsoia zaupanja in vertikalnega intervala izračunane kot a,b,h95i =,795 a,b, hi. Stanarni 3D-elipsoi ima pri χ stopnjo zaupanja (1-α) približno 0 %. Da osežemo 95 % verjetnost, je treba osnovno enačbo stanarnega 3D-elipsoia pomnožiti s faktorjem,795 (Stauinger, 1999). 3;1-α Premiki in eformacije nastanejo tako na umetnih objektih, kot so jezovi, nasipi, mostovi, kot tui v njihovi okolici, na primer v olinah jezov, na obrežjih umetnih akumulacij, pa tui na naravnih območjih, kot so plazovi, ob tektonskih prelomnicah, na barjanskih tleh. V splošnem lahko vzroke za nastanek premikov in eformacij pripisujemo elovanju zunanjih vplivov (spremembi temperature, vetru, tektonskim in seizmičnim vplivom), mehaničnim lastnostim grabenega materiala in konstrukcijskih elementov ter nezaostnemu upoštevanju geološke sestave, mehanskih lastnosti ter hiroloških pogojev tal pri projektiranju objekta. Določitev premikov in eformacij naravnih in umetnih objektov je ena zahtevnejših nalog geoetske stroke. Pri presoji statistično značilnih premikov navano uporabimo inirektno oločene količine, na primer premik in pripaajočo stanarno eviacijo, ki jih z zapisom testne statistike T = / testiramo s postopki statističnega testiranja hipotez. Ker testno statistiko T primerjamo s kritično vrenostjo T, ki jo oločimo na polagi ustrezne porazelitvene funkcije in izbrane stopnje značilnosti testa α, je zelo pomembno, a pravilno oločimo porazelitveno funkcijo, po kateri se testna statistika porazeljuje. Statistični test prepisuje pravilo za oločitev neke pomnožice prostora vzorcev, ki jo imenujemo kritično območje preizkusa ali območje zavrnitve ničelne hipoteze. V praksi se pri presoji o značilnih premikih pogosto uporabi približni kriterij T > 3 ali T > 5, ki pa je pogosto preohlapen, saj je s postopki simulacij mogoče natančneje izračunati kritično vrenost T porazelitvene funkcije. V članku obravnavamo posebnosti pri izračunu premika in pripaajoče stanarne eviacije v 1D-, Din 3D-geoetskih mrežah. Zapišemo postopek za oločitev porazelitvene funkcije za testno statistiko T = / glee na imenzijo mreže. V ravninski in prostorski mreži se izračunana testna statistika ne porazeljuje po nobeni o znanih porazelitvenih funkcij, zato pripaajočo porazelitveno funkcijo oločimo empirično s simulacijami. S simulacijami lahko oločimo kritično vrenost T porazelitvene 401
61/3 GEODETSKI VESTNIK funkcije za vsako točko v mreži, zato lahko prelagani alternativni postopek zagotovi večjo občutljivost pri ugotavljanju značilnih premikov pri izbrani stopnji zaupanja α. ocena PREMIKOV IN NJIHOVE NATANČNOSTI Pogoj za izračun premika in pripaajoče stanarne eviacije premika je, a so ientične točke izmerjene najmanj v veh terminskih izmerah. Če obstaja ovolj stabilnih točk, ki zagotavljajo kakovosten geoetski atum v obeh izmerah, izračunani premiki pa so značilno večji o stanarne eviacije, se za premik lahko opreelijo iz razlike koorinat v veh terminskih izmerah in oatno statistično testiranje ni potrebno. V geotehničnih raziskavah je pogosto težavno oločiti ovolj stabilnih točk, kakor tui značilne premike, ki so pogosto le malo večji o stanarne eviacije premika. Da bi zmanjšali tveganje pri veroostojni obravnavi značilnih premikov, uporabimo postopke statističnega testiranja, ki nam olajšajo oločitve. Prepostavimo, a obravnavamo položaj točke P v času t in t + t. Premik točke izračunamo: v 1D-mreži: 1D = H = H t+ t H t, (1) SI EN v D-mreži: x ( ) ( x x ) D t+ t t t+ t t = + = - + - in () v 3D-mreži: ( x H ) ( ) ( x x ) ( H H ) 3D t+ t t t+ t t t+ t t = + + = - + - + -, (3) kjer so t, x t, H t in t+δt, x t+δt, H t+δt izravnane koorinate ene točke v veh terminskih izmerah. Da bi lahko izračunali natančnost premika točke, moramo poleg koorinat točke poznati tui kovariančno matriko koorinat točke za posamezno terminsko izmero. Naj ima točka P t v prvi izmeri pripaajočo variančno-kovariančno matriko Σ Pt in ientična točka P t+ t v rugi izmeri pripaajočo variančno-kovariančno matriko Σ Pt+ t. Prepostavimo, a so koorinate točke P v času t nekorelirane s koorinatami v času t+δt. Variančno-kovariančno matriko ientične točke v veh časovno neovisnih izmerah lahko zapišemo: ΣP 0 t Σ PP = t t+ t 0, ΣP t+ t (4) H 0 t v 1D-mreži: Σ PP = t t+ t, 0 Ht+ t (5) 0 0 t x t t 0 0 t t t v D-mreži: x x Σ PP = in t t+ t 0 0 t+ t t+ txt+ t 0 0 t tx + t+ t xt+ t (6) 0 0 0 t x t t H t t x 0 0 0 t t x t xh t t H 0 0 0 t t xh t t Ht v 3D-mreži: PP =. t t+ t 0 0 0 t t t tx t t t th + + + + t+ t 0 0 0 t tx t t x t t xt th + + + + t+ t 0 0 0 t th t t xt th + + + t+ t Ht+ t (7) 40
GEODETSKI VESTNIK 61/3 Po zakonu o prenosu varianc in kovarianc oločimo natančnost premika T = D Σ D PP t t+ t D J J, (8) kjer je Jacobijeva matrika J D enaka v 1D: 1D 1D J = 1 1D = Ht H - t + t 1, (9) v D: D D D D = D = - - t xt t+ t xt+ t D D D D (10) v 3D: 3D 3D 3D 3D 3D 3D H H = 3D = - - - t xt Ht t+ t xt+ t Ht+ t 3D 3D 3D 3D 3D 3D Če vstavimo enačbe (9), (10) in (11) v enačbo (8), varianco premika točke P lahko zapišemo v 1D: = 1D H +, (1) t Ht + t v D: = ( + ) + ( x + x ) + ( x + x ) D t t+ t t t+ t t t t+ t t+ t D D D D H x 3D t t+ t xt xt+ t Ht Ht+ t x t t t+ txt+ t 3D 3D 3D 3D 3D ( ) ( ) ( ) ( ) 3D in (13) = + + + + + + + + v 3D: (14) H x H + ( H + ) ( ) t t t th + t t xh + + + t t xt+ tht+ t Varianco premika uporabimo za testiranje statistično značilnih premikov. 3 oločitev PORAZDELITVENE FUNKCIJE S SIMULACIJAMI Po izravnavi najmanj veh terminskih izmer je mogoče oločiti premik točke in stanarno eviacijo premika. Zapišimo ustrezno testno statistiko, s katero skušamo zaznati značilne spremembe točk v mreži, ki nas zanimajo: T =. (15) Testna statistika se lahko porazeljuje po znanih porazelitvenih funkcijah (normalna, Stuentova, Fisherjeva ir.) ali pa je porazelitveno funkcijo treba oločiti analitično ali empirično s simulacijami. Natančna oločitev porazelitvene funkcije, po kateri se porazeljuje testna statistika T, je zelo pomembna, saj se glee na porazelitveno funkcijo izračuna kritična vrenost T. Za izračun kritične vrenosti T je pomembno tui, kakšno vrenost stopnje značilnosti testa α izberemo in kako postavimo ničelno hipotezo H 0. V postopku testiranja hipotez je treba postaviti ničelno hipotezo, s katero se prepisuje pravilo za oločitev pomnožice prostora vzorcev, ki jo imenujemo kritično območje preizkusa ali območje zavrnitve ničelne hipoteze. V primeru zavrnitve ničelne hipoteze zapišemo ustrezno alternativno hipotezo. Testno statistiko testiramo glee na postavljeno ničelno H 0 in alternativno hipotezo H a : 403
61/3 GEODETSKI VESTNIK H 0 : = 0; točka v obobju veh terminskih izmer miruje, H a : 0; točka se je v obobju veh terminskih izmer značilno premaknila. Testno statistiko T primerjamo glee na kritično vrenost T, ki jo izračunamo na polagi porazelitvene funkcije. Ko je testna statistika manjša o kritične vrenosti ob izbrani stopnji značilnosti testa α, je tveganje za zavrnitev ničelne hipoteze preveliko. V tem primeru ugotovimo, a premik ni statistično značilen. Če je vrenost testne statistike večja o kritične vrenosti porazelitvene funkcije, pa ugotovimo, a je tveganje za zavrnitev ničelne hipoteze manjše o izbrane stopnje značilnosti testa α. Zato upravičeno zavrnemo ničelno hipotezo in tako potrimo, a je obravnavani premik statistično značilen. Obravnavamo torej va primera: T T : ne zavrnemo ničelne hipoteze; premik točke ni statistično značilen, T > T : zavrnemo ničelno hipotezo; premik točke je statistično značilen. SI EN 3.1 Izračun kritične vrenosti v 1D-mreži Če prepostavimo, a so pogreški opazovanj normalno porazeljeni ε N(0, ), se enako porazeljujejo tui količine, ki so linearne funkcije opazovanj x ˆ~ N ( µ xˆ, xˆ ). Ker 1D izračunamo kot razliko veh normalno porazeljenih slučajnih spremenljivk (glej poglavje ), je tui razlika višin ientične točke me vema terminskima izmerama normalno porazeljena količina. 3. Izračun kritične vrenosti v D-mreži Premik je v D-mrežah nelinearna funkcija normalno porazeljenih spremenljivk in, zato se ne porazeljuje po normalni porazelitveni funkciji. Empirično porazelitveno funkcijo oločimo s simulacijami (Savšek-Safić, 00). Za oločitev natančne porazelitvene funkcije prelagamo uporabo simulacij za priobitev vzorca ovisnih normalno porazeljenih slučajnih spremenljivk s pomočjo linearne transformacije. Postopek omogoča natančno oločitev kritične vrenosti pri izbrani stopnji tveganja (za zavrnitev ničelne hipoteze). Osnovna ieja za generiranje vzorca ovisnih normalno porazeljenih slučajnih spremenljivk je, a najprej generiramo vzorec neovisnih normalno porazeljenih spremenljivk, potem pa uporabimo linearno transformacijo za priobitev vzorca ovisnih slučajnih spremenljivk. Za generiranje vzorca normalno porazeljene slučajne spremenljivke uporabimo metoo Box in Müller (Box et al., 1958; Press et al., 199). Naj bosta u 1i in u i, i = 1,..., n va vzorca slučajnih spremenljivk U in U, ki sta neovisni in 1 porazeljeni enakomerno na intervalu (0,1). Vzorec veh neovisnih stanarizirano normalno porazeljenih slučajnih spremenljivk Z 1 in Z izračunamo ( π ) ( π ) z - lnu sin u 1i 1i i z i =, i 1... n z = =. (16) i - lnu1 i cos ui Za pretvorbo vzorca neovisnih normalno porazeljenih spremenljivk v vzorec ovisnih normalno porazeljenih slučajnih spremenljivk uporabimo linearno transformacijo i = U T z i, i = 1,..., n, (17) 404
GEODETSKI VESTNIK 61/3 kjer matriko U obimo s Cholesk razcepom variančno-kovariančne matrike Σ, Σ = U T U x U =. (18) x 0 1- Če želimo generirati vzorec normalno porazeljenih koorinatnih razlik Δ i in Δx i, izračunamo po (17) in prepostavimo, a sta srenji vrenosti µ Δ in µ Δx enaki 0, obimo: = z i 1i x. (19) x xi = z1 i + zi 1- Variance in kovariance izračunamo = + = + = + t t+ t xt xt+ t x x t t t+ txt+ t, (0) kjer so t, x t, t x t, t+ t, x t+ t, t+ t x t+ t variance in kovariance ravninskih koorinat točke t, x t, t+ t, x t+ t. Porazelitvena funkcija testne statistike T je za vsako točko rugačne oblike, saj sta premik in stanarna eviacija koorinat točk v posamezni terminski izmeri za različne točke različna. Z uporabo simuliranih normalno porazeljenih spremenljivk izračunamo premik in stanarno eviacijo premika. Simulacije uporabimo za generiranje vzorca neovisnih normalno porazeljenih slučajnih spremenljivk in z linearno transformacijo priobimo ovisne normalno porazeljene slučajne spremenljivke. V n simulacijah nam postopek omogoča oločitev empirične kumulativne verjetnostne porazelitvene funkcije testne statistike in izračun pripaajoče kritične vrenosti glee na izbrano stopnjo značilnosti testa α za vsako posamezno točko (Savšek-Safić et al., 006). 3.3 Izračun kritične vrenosti v 3D-mreži Premik je tui v 3D-mrežah nelinearna funkcija normalno porazeljenih spremenljivk Δ, Δx i in ΔH, zato se ne porazeljuje po normalni porazelitveni funkciji in moramo empirično porazelitveno funkcijo oločiti s simulacijami. Porazelitveno funkcijo oločimo poobno kot v D-mrežah, tako a najprej generiramo vzorca neovisnih normalno porazeljenih slučajnih spremenljivk. Naj bosta u 1i, u i, i = 1,,n in u 3i, u 4i, i = 1,,n va para vzorcev slučajnih spremenljivk U 1, U in U 3, U 4, ki sta neovisna in porazeljena enakomerno na intervalu (0,1). Vzorec treh neovisnih stanarizirano normalno porazeljenih slučajnih spremenljivk Z 1, Z in Z 3 izračunamo po enačbi: 405