V16N1C.indb EARTH SCIENCES RESEARCH JOURNAL GEOTECHNICAL ANALYSIS Earth Sci Res SJ Vol 16, No 1 (June, 2012): 65 - 74 Multicriteria decision-making analysis based methodology for predicting carbonate rocks’ uniaxial compressive strength Hakan Ersoy and Derya Kanik Karadeniz Technical University, Geological Engineering Department, 61080, Trabzon, Turkey, phone: + 90 462 377 35 06, fax: +90 462 325 7405 E-mail: blavetirraa@hotmail com; kanik@ktu edu tr ABSTRACT Uniaxial compressive strength (UCS) deals with materials’ to ability to withstand axially-directed push- ing forces and especially considered to be rock materials’ most important mechanical properties How- ever, the UCS test is an expensive, very time-consuming test to perform in the laboratory and requires high-quality core samples having regular geometry Empirical equations were thus proposed for pre- dicting UCS as a function of rocks’ index properties Analytical hierarchy process and multiple regres- sion analysis based methodology were used (as opposed to traditional linear regression methods) on data-sets obtained from carbonate rocks in NE Turkey Limestone samples ranging from Devonian to late Cretaceous ages were chosen; travertine-onyx samples were selected from morphological environ- ments considering their surface environmental conditions Test results from experiments carried out on about 250 carbonate rock samples were used in deriving the model While the hierarchy model focused on determining the most important index properties affecting on UCS, regression analysis established meaningful relationships between UCS and index properties; 0 85 and 0 83 positive coefficient cor- relations between the variables were determined by regression analysis The methodology provided an appropriate alternative to quantitative estimation of UCS and avoided the need for tedious and time consuming laboratory testing RESUMEN La  resistencia a la compresión uniaxial  (RCU) trata con la capacidad de los materiales para soportar fuerzas empujantes dirigidas axialmente y, especialmente, es considerada ser uno de las más importantes propiedades mecánicas de los materiales rocosos Sin embargo, una prueba de RCU es costosa, lleva mucho tiempo para hacerlo en el laboratorio y  requiere muestras de núcleos de alta calidad que tienen una geometría regular Por lo tanto, ecuaciones empíricas fueron propuestas para la predicción de RCU como una función de las propiedades índice de las rocas Las metodologías de proceso analítico jerárquico (PAJ) y análisis de regresión múltiple fueron utilizados (en vez de los métodos tradicionales de regresión lineal) en con- juntos de datos obtenidos de las rocas carbonatadas en el noreste de Turquía Muestras de rocas calizas que van desde el Devónico hasta finales del Cretácico fueron escogidas;  muestras de travertino y  ónix fueron seleccionadas de ambientes morfológicos teniendo en cuenta sus  condiciones ambientales de superficie Los resultados de los experimentos llevados a cabo en alrededor de 250 muestras de rocas carbon- atadas fueron utilizados para derivar un modelo Mientras que el modelo de jerarquía se centró en de- terminar las propiedades índice más importantes afectados por la RCU, el análisis de regresión establece relaciones significativas entre la RCU y las propiedades del índice; coeficientes de correlación positivas de 0,85 y 0,83 fueron determinadas por análisis de regresión entre las variables La metodología proporciona una alternativa adecuada para la estimación cuantitativa de la RCU y evita la necesidad de realizar pruebas del laboratorio las cuales son tediosas y dispendiosas Palabras claves: Resistencia a la compresión, ecuaciones empíricas, rocas carbonatadas, jerarquía analítica Keywords: Compressive strength, empirical equations, car- bonate rock, analytical hierarchy Record Manuscript received: 06/02/2012 Accepted for publications: 01/06/2012 Hakan Ersoy and Derya Kanik66 Introduction Intact rocks’ uniaxial compressive strength (UCS) is the main param- eter used for almost all rock mechanical studies in most civil, geological and mining projects (Bieniawski 1974; Cargill and Shakoor 1990); however, regular geometry, high-quality core samples are necessary for determining UCS Standard cores cannot always be extracted from weak, highly-frac- tured, thinly-bedded, foliated rocks This test is thus expensive, time-con- suming and requires well-prepared rock samples Resent trends in estimating UCS from simple laboratory index tests has been improved to overcome such difficulties, and simple prediction models have become attractive for engineering geologists using non-destructive and easily-applied techniques such as rocks’ ultrasonic wave velocity (UWV) and index properties Many attempts have been made to predict intact rocks’ UCS (Kahraman 2001; Katz et al., 2000; Koncagul and Santi 1999; Chau and Wong 1996) Some researchers have found that sound velocity is closely related to rock proper- ties (Gaviglio 1989; Chang et al., 2006; Yalçınalp et al., 2008; Babacan et al., 2009; Moradian and Behnia, 2009) whilst other have correlated UCS with index properties such as porosity, density and UWV (Ramana and Venka- tanarayana 1973; Yasar and Erdogan 2004; Kanik 2010) Simple statistical methods-based multiple regression techniques have been used to establish predictive models; new techniques such as artificial neural networks and fuzzy inference systems have been used for devel- oping predictive models for estimating the required parameters during recent years (Grima and Babuska 1999; Kayabasi et al., 2002, Gokceoglu and Zorlu 2004; Sonmez et al., 2003; Karakus and Tutmez 2006; Deh- figure 1. Map showing the location of the area being studied Multicriteria decision-making analysis based methodology for predicting carbonate rocks’ uniaxial compressive strength 67 ghan et al., 2010, Yagiz et al., 2011) Different evaluation methods have been developed to establish predictive models; such methods include lin- ear vector approach, matrix method, fuzzy set theory, checklist methods, parametric ranking methods and multi criteria decision analysis based methodologies Analytic hierarchy process (AHP), was developed as a type of multi-criteria analysis, to standardise multi-criteria decision-mak- ing (Saaty 1980) However, AHP uses a quantitative comparison method based on pair-wise comparisons of decision-making criteria, rather than utility and weighting functions Many engineering geological projects have adopted such as approach (Cook et al., 1984; Siddiqui et al., 1996; Ersoy and Bulut 2009) The paper presents multi-criteria decision-making and multiple re- gression analysis-based methodology for predicting UCS regarding car- bonate rocks’ index properties The model is based on non-destructive and relatively easy to apply laboratory tests research methodology Sampling location and geology The morphology of mountainous region being studied in north-east- ern Turkey (Figure 1) consist of rough, irregular land having steep slopes and peaks reflecting the eastern Black Sea region’s geology and tectonic fea- tures The main morphological units (mainly faults and folds) have been shaped by structural elements in the region trending NE-SW The region is drained by the Harşit and Çoruh rivers, forming the most significant fluvial system Deep incision forms like v-shaped valleys are characterised by deep gorges and steep slopes in these drainage systems Turkey is located in the Alpine-Himalayan orogenic belt having the world’s richest natural stone formations; it is the country having 5 bil- lion tons of reserves and almost 35% of the world’s natural stone reserves The eastern Black Sea region has rich potential in terms of the variety of carbonate- bearing rocks The region has around 450 million tons natu- ral stone reserves almost 70% of production today consists of travertine and limestone, the remaining 30% being volcanic rocks The rock samples for this study were collected from 5000 km2 in the southern zone of the eastern Pontides (NE Turkey) The Eastern Pontide Belt is a major metal- logenetic province in the eastern Black Sea coastal region and forms a 500 km long and 100 km wide mountain chain along the Black Sea coast The Eastern Pontides may be subdivided into northern and eastern zones on structural and lithological differences (Özsayar et al., 1981; Okay and Sahinturk 1997) The northern zone is dominated by Late Cretaceous and Middle Eocene volcanic and volcaniclastic rocks, whereas pre-Late Creta- ceous sedimentary rocks are widely exposed in the Southern Zone (Arslan et al., 1997; Eyuboglu 2006; Sen 2007) (Figure 2) The limestone samples were selected from pre-Late Cretaceous plat- form carbonates which are widely exposed in the area and Liassic-aged for- mations; travertine and onyx samples were selected from the Gümüşhane, Bayburt and Giresun area Three rock types from ten rock formations were sampled and tested for this study Table 1 gives the rock type, age, location and description of the samples Sample preparation and laboratory tests Several carbonate rock samples were collected in this study to deter- mine physical and strength properties Seven oriented block samples for each group of carbonates were collected from the field for laboratory test- ing; each block was represented by at least ten core specimens 225 NX- size core specimens having a 2 5:1 length to diameter ratio were prepared from the block samples Rock samples were carefully inspected before testing, to obtain the most representative stone samples for performing laboratory tests figure 2. Main tectonic units and zones in the Eastern Pontides NAF: North Anatolian Fault; NEAF: Northeast Anatolian Fault (Eyüboğlu 2006) Hakan Ersoy and Derya Kanik68 UWV, unit weight, water absorption and content are the rock materi- als’ most important index properties and they are often related to porosity Porosity is the ratio of the non-solid volume to the total volume of mate- rial and it also describes how densely the material is packed In the study, firstly unit weight, water content, apparent porosity and water absorp- tion by weight were determined with respect to the description criteria of ISRM (1981) A Pundit Plus ultra-sonic pulse (USP) instrument giving more precise rock sample measurements and two 54 KHz transducers hav- ing piezoelectric properties were used in this study to calculate ultrasonic longitudinal wave velocity The transducers were located parallel to the surface of the sample, transit time was measured and ultrasonic wave ve- locities were calculated from transit time UCS was determined for the study according to ISRM (1981) descrip- tion criteria Core surface flatness was supplied for the UCS test to obtain an even load distribution; specimens were loaded axially up to failure Establishing decision-making methods Regression analysis includes any techniques for modelling and ana- lysing several variables when the focus is on the relationship between a dependent variable and one or more independent variables (Freedman 2005) Multiple regression is aimed at learning more about the relation- ship between several independent or predictable variables and a dependent or criterion (Davis 1986) Regression analysis was used in this with ana- lytic hierarchy process (AHP), a widely accepted decision making method Constraints are compared to each other in AHP, to designate each variable’s relative importance in accomplishing an overall goal Numerical values were assigned to each pair of constraints using established guidelines and a constrained matrix is built The sum of each column within the matrix was then normalised and weighting was calculated Mathematic formula- tion (simple additive weighting) was defined following an equation for calculating final grading values in multiple criteria problems (Saaty 1980); where, v(y) was area’s suitability index, w was a criterion’s weighting or importance factor of a, y was a criterion’s degree or compliance level, i was the criterion number and q was the number of criteria Pairwise comparisons were used to determine each criterion’s rela- tive importance; AHP is based on such approach Decision makers can quantify their opinions about the criteria magnitude by using a verbal scale when comparing pairs of criteria The pairwise comparison matrix (PCM) constructed by decision-makers in the previous had to have the following attributes; w ij = w ji -1 The next step was to calculate the criteria’s relative importance weight- ing implied by previous comparisons Saaty (1980) proposed estimating PCM’s right principal eigenvector which can be approximated using the geometric mean for each row of the PCM (by multiplying the elements in each row and then taking the nth root, where n is the number of criteria) This mode is known as multiplicative AHP (Saaty and Millet 2000) and was used in the present work The calculated geometric means were then normalised and relative importance weighting extracted (2) Table 1. Location and general properties of the carbonate rocks Rock type Rock description Age Location 1a. Travertine (28 samples) Porous and soft White-pale cream in colour Quaternary Bahçecik (Gümüshane) 1b. Travertine (25 samples) Porous and soft Pale cream in colour Quaternary Kalecik (Gümüshane) 1c. Travertine (25 samples) Porous and soft White-pale cream in colour Quaternary Çamoluk (Giresun) 1d. Travertine (10 samples) Compact and soft White-pale cream in colour Quaternary Çamoluk (Giresun) 1e. Travertine (13 samples) Compact and soft White-pale cream in colour Quaternary Gölova (Giresun) 2a. Onyx (23 samples) Compact and soft White-pale grey and cream in colour Contained cream colored band Quaternary Yaylapınar (Bayburt) 2b. Onyx (15 samples) Compact and soft Light green, pale cream in colour Contained green colored band Quaternary Yaylapınar (Bayburt) 3a. Limestone (20 samples) Hard and compact Light grey in colour Contained some micro fossils L. Cretaceous Dogger-Malm Kelkit (Gümüshane) 3b. Limestone (24 samples) Fractured and slightly metamorphosed Reddish in colour Contained a number of fossils Liassic Esenyurt (Gümüshane) 3c. Limestone (20 samples) Hard and compact Light grey and white in colour Cretaceous Jurassic Kemah (Erzincan) 3d. Limestone (20 samples) Hard and slightly metamorphosed light Purple and white in colour Contained light brown colored band Cretaceous Devonian Tercan (Erzincan) v(y)= w j y ij q ∑ j-1 (1) Multicriteria decision-making analysis based methodology for predicting carbonate rocks’ uniaxial compressive strength 69 Table 2. Water absorption, water content and apparent porosity values of the rock groups studied Table 3. Unit weight, P wave velocity and UCS values for the rock groups being studied Sample location and type Water content (%) Water absorption by weight (%) Apparent porosity (%) Max Min SD Mean Max Min SD Mean Max Min SD Mean Kalecik travertine (1a: 30 samples) 0.20 0.08 0.04 0.13 3.33 0.88 0.73 1.39 7.37 2.16 1.36 3.29 Bahçecik travertine (1b: 15 samples) 0.77 0.12 0.15 0.34 1.20 0.48 0.24 0.77 2.88 1.18 0.57 1.90 Çamoluk travertine (1c: 25 samples) 2.48 0.49 0.43 1.40 2.90 0.98 0.48 1.88 6.65 2.37 1.01 4.28 Çamoluk (2) (1d: 10 samples) 0.62 0.45 0.06 0.59 1.30 0.91 0.12 1.07 3.03 2.44 0.28 2.61 Gölova travertine (1e: 13 samples) 0.31 0.18 0.04 0.23 0.62 0.37 0.063 0.47 1.55 0.94 0.15 1.19 Bayburt onyx (2a and b: 21) 0.79 0.01 0.18 0.07 0.72 0.02 0.02 0.10 1.95 0.06 0.32 0.26 Kelkit limestone (3a: 12 samples) 0.06 0.03 0.01 0.05 0.18 0.13 0.01 0.15 0.48 0.34 0.04 0.40 Esenyurt limestone (3b: 15 samples) 0.46 0.04 0.09 0.14 0.31 0.06 0.07 0.19 0.08 0.16 0.19 0.51 Kemah limestone (3c: 21 samples) 0.14 0.01 0.03 0.03 0.07 0.02 0.01 0.04 0.18 0.05 0.03 0.11 Tercan limestone (3d: 25 samples) 0.09 0.01 0.01 0.06 0.12 0.04 0.02 0.09 0.31 0.12 0.05 0.24 Sample location and type Unit weight (kN/m3) velocity (m/s) P wave UCS (MPa) Max Min SD Mean Max Min SD Mean Max Min SD Mean Kalecik travertine (1a: 30 samples) 24.7 22.2 0.8 24.0 4541 3813 220 4050 65 34 8 53 Bahçecik travertine (1b: 15 samples) 25.1 24.1 0.3 24.6 5072 4091 297 4598 44 25 5 37 Çamoluk travertine (1c: 25 samples) 24.6 24.1 0.2 24.4 3693 3506 61 3590 44 30 5 38 Çamoluk (2) travertine (1d: 10 samples) 25.4 24.9 0.2 25.2 4727 3913 268 4323 43 39 1.2 41 Gölova travertine (1e: 13 samples) 29.2 22.2 0.9 25.0 5072 3813 374 4430 66 20 10 40 Bayburt onyx (2a and b: 21 samples) 27.4 26.5 0.3 27.1 6800 4121 805 5240 70 20 13 45 Kelkit limestone (3a: 12 samples) 27.1 26.8 0.1 27.0 6321 5988 98 6200 154 30 38 80 Esenyurt limestone (3b: 15 samples) 27.4 26.6 0.2 27.0 6103 5562 169 5760 117 40 24 70 Kemah limestone (3c: 21 samples) 27.1 26.8 0.1 27.0 6353 6127 59 6250 162 25 39 90 Tercan limestone (3d: 25 samples) 27.3 25.3 0.6 26.8 6057 4327 371 5414 101 79 9 85 Hakan Ersoy and Derya Kanik70 Integrating site selection criteria was based on multi-criteria assess- ment methods (Eastmen, 1995): = ∑ (Sk*wk) where, Sk was the land’s suitability for landfill for objective k (priority groups), (fi) k was factor i (discriminating features) for objective k, (wi)k was the weighting for factor i (score given experimental studies) for objective k, (rj) k was constraint j for objective k (value 0 or 1), S was multi-objective suitability and wk was the weighting for objective k results The rocks’ index and strength properties UCS is an expensive and very time-consuming test The core surfaces have to be ground down to make them parallel at a specified tolerance and a high load capacity loading frame is usually required Thus, in the study, ten groups of carbonate rock samples were collected in the Southern Zone of the eastern Pontides (NE Turkey) to establish some relationships between UCS and index properties In the study the index and strength properties of the rocks were de- termined in accordance with ISRM (1981) The results of the laboratory studies are given Table 2 and Table 3 When taking into consideration the mean values of the water content, water absorption by weight, apparent porosity, P wave velocity and UCS value data, the highest values were observed for Erzincan limestone and the lowest values were observed for Gümüşhane travertines However, the values of the UCS were obtained between 20 and 66 MPa for travertines, 20 and 70 MPa for onyx and 25 and 162 for different limestones Depending on the mineral shapes, weathering features and occurrence of the microcracks, the UCS values varied in a wide range as found in the study Empirical relationship between index and strength properties Rock materials’ uniaxial compressive strength greatly depends on their index properties such as porosity, water content, water absorption and mineral composition P wave velocity and porosity are the most pre- ferred properties for predicting UCS amongst such properties (Sachpazis 1990; Tugrul and Zarif 1999; Palchik 1999) Regression analysis is usually preferred for establishing statistical rela- tionships between different variables Relationship intensity of between vari- ables is defined by regression values and correlation coefficients; explanatory variables must thus be weighted regarding dependent variables before mul- P w av e ve lo ci ty ( m /s ) 6500 R2=0 70 6000 5500 5000 4500 4000 3500 0 50 100 UCS (mPa) A p p ea ra n ce p or os it y (% ) 5 R2=0 45 4 3 2 1 0 0 50 100 UCS (mPa) W at er a bs ro rb ti on b y w ei gh t (% ) 2 0 R2=0 45 1 5 1 0 0 5 0 0 50 100 UCS (mPa) U n it w ei gh t (g r/ cm ) 2 8 R2=0 40 2 7 2 6 2 5 2 4 0 50 100 UCS (mPa) W at er c on te n t (% ) 1 5 R2=0 32 1 0 0 5 0 0 50 100 UCS (mPa) figure 3. The empirical relationship between UCS and index parameters for different carbonate rock groups Table 4. Pairwise comparison matrix and relative importance weighting for index properties A B C D E Eigenvector Weighting A 1 2 3 3 4 3.23 0.45 B 1/2 1 2 3 3 2.34 0.33 C 1/3 1/2 1 2 3 0.73 0.10 D 1/3 1/3 1/2 1 2 0.60 0.08 E 1/4 1/3 1/3 1/2 1 0.30 0.04 A: ultrasonic wave velocity, B: apparent porosity, C: water absorption by weight, D: water content, E: unit weight 1: equal importance, 2: weak or slight (two parameters contributed equally to the objective) 3: moderate importance, 4: moderate plus Experience and/or analysis slightly favoured one parameter over another S k = ( ) k ∑ fi*wi i S - ∏ rj j (3) (4) Multicriteria decision-making analysis based methodology for predicting carbonate rocks’ uniaxial compressive strength 71 figure 4. The empirical relationship between UCS and Vp/n for different travertines and onyx (a: Bayburt onyx, b: Bahçecik and Kalecik, c: Çamoluk and d: Gölova travertines) figure 5. The empirical relationship between UCS and Vp/n for the different limestones (a: Esenyurt, b: Kelkit, c: Tercan and d: Kemah) U C S (m P a) 80 70 60 50 40 30 20 10 0 25000 50000 75000 100000 Vp/n U C S (m P a) 160 140 120 100 80 60 40 20 14000 15000 16000 17000 18000 Vp/n U C S (m P a) 60 55 50 45 40 35 30 25 2600 3000 3400 3800 4000 Vp/n U C S (m P a) 70 60 50 40 30 20 10 500 1500 2500 3500 4500 5500 Vp/n U C S (m P a) 160 120 60 40 30000 50000 70000 90000 110000 130000 Vp/n U C S (m P a) 85 75 65 55 45 10000 20000 30000 40000 50000 Vp/n U C S (m P a) 110 90 70 50 30 6000 10000 14000 18000 22000 Vp/n U C S (m P a) 50 40 30 20 10 500 750 1000 1250 1500 1750 Vp/n UCS = 14 In (Vp/n)-96 R2=0 83 n: 20 UCS = 0 0184 (Vp/n)-22 R2=0 80 n: 12 UCS = 0 0049 (Vp/n)+12 R2=0 84 n: 14 UCS = 0 001 (Vp/n)+30 R2=0 71 n: 25 UCS = 0 001 (Vp/n)+26 R2=0 81 n: 21 UCS = 0 0032 (Vp/n)-427 R2=0 86 n: 12 UCS = 0 022 (Vp/n)+0 R2=0 84 n: 33 UCS = 19 5 In (Vp/n)-107 R2=0 87 n: 35 (a) (a) (c) (c) (b) (b) (d) (d) Hakan Ersoy and Derya Kanik72 Rock description Formulas R2 Onyx (21 samples) UCS= 14 ln (Vp/n) - 96 83 Travertine (76 samples) UCS= 15 ln (Vp/n) - 73 86 Limestone (60 samples) UCS= 0.0009 (Vp/n) + 38 83 Table 5. The formulas used for estimating UCS in this study tiple regression analysis to establish an empirical model AHP used the data obtained from linear regression analysis results concerning ten rock groups (Table 1 and Figure 3) to decide which material properties should be selected as explanatory variables from rock materials’ index properties This represents a tool which reduces time and the cost involved in many engineering models Table 4 lists the priority vectors for all criteria; relative importance weightings are included in the final column of this table AHP parameters are also shown in the table, indicating that the judgments (and therefore final relative importance weightings) seemed to be reasonable The method needed a scale of numbers indicating how many times more important or dominant one element was over another figure 6. The empirical relationship between UCS and Vp/n for all travertine (a) and limestone (b) samples figure 7. Line-scatter plot diagrams for measured compared to calculated UCS values and histograms showing frequency versus difference in measured and calculated UCS (a and b: travertines, c and d: limestones) U C S (m P a) 60 50 40 30 20 10 0 1000 2000 3000 4000 5000 Vp/n U C S (m P a) 140 120 100 80 60 40 20 0 0 20000 40000 60000 80000 100000 Vp/n UCS = 15 In (Vp/n)-73 R2=0 8671 n: 276 UCS = 0 0009 (Vp/n)+38 R2=0 83 n: 60 (a) (a) (b) (c) (d) (b) C al cu la te d U C S (m P a) 70 60 50 40 30 20 10 10 20 30 40 50 60 70 measured UCS (mPa) fr eq u en ci e (n u m be r) 25 20 15 10 5 0 -9 -6 -3 0 3 6 9 measured UCS Calculated UCS (mPa) fr eq u en ci e (n u m be r) 16 14 12 10 8 6 4 2 0 -16 -12 -8 -4 0 4 8 12 16 measured UCS Calculated UCS (mPa) C al cu la te d U C S (m P a) 140 120 100 80 60 40 20 20 40 60 80 100 1200 140 measured UCS (mPa) Multicriteria decision-making analysis based methodology for predicting carbonate rocks’ uniaxial compressive strength 73 element regarding the criterion or property to which they were being com- pared The AHP analysis showed that the most important criteria affecting on UCS for the carbonate rocks being studied were UWV (45% weight- ing) and apparent porosity (33 % weighting) The other index properties’ weighting did not surpase %10 As regression analysis provides a means of summarising the relation- ship between variables, multiple regression analysis-based methodology was used to establish some numerical relationships between rock materials’ ultrasonic wave velocity, apparent porosity and UCS Furthermore it is known that Vp decreases with increased porosity Vp and n-1 were thus used in the equations for predicting UCS in this study Index properties were considered to be explanatory variables, and the UCS a depending variable Figure 4 and 5 show multiple regression analysis results, involv- ing some positive correlations between UWV rate regarding apparent po- rosity and UCS: such relationships were characterised by 0 71 and 0 87 regression coefficients However, Figure 6 indicates meaningful empirical relationships for all travertines and limestones and the relationships are represented by the formulas given in Table 5 A goodness of fit test establishes whether an observed frequency dis- tribution differs from a theoretical distribution; however, a model’s suit- ability is tested using the difference between observed and expected values Normal distribution is a continuous probability distribution which is often used as a first approach for describing real-valued random variables tend- ing to cluster around a single mean value A normal distribution would thus be expected in the histograms showing difference between observed and expected values Line-scatter plot diagrams of observed-expected val- ues (Figure 7a and c) and histograms of difference between observed and expected values (Figure 7b and d) were prepared and normal distributions were observed Histograms having a normal distribution showed that these equations did fairly well in estimating UCS using Vp/n The results showed that the model being tested was suitable for the study Conclusions This study involved three types of carbonate rocks being collected from ten rock formations in the north-eastern Turkey Laboratory studies were conducted in line with ISRM (1980) to establish some relationships between P wave velocity, apparent porosity and UCS and AHP-based multiple regression analyses methodology was used for statistically analyz- ing the suggested methods and results AHP analysis indicated that UWV (45% weighting) and apparent porosity (33% weighting) were the most important index properties affecting on UCS regarding the carbonate- bearing rocks being studied Multiple regression analysis correlations in- dicated by the 0 81-0 87 regression coefficient were determined for UWV rate regarding apparent porosity and UCS The equations obtained by such analysis results were practical, simple and accurate enough to apply and may be recommended for use in practice Acknowledgments This study was partly supported by the Karadeniz Technical Univer- sity Scientific Research Fund (Project No: 2008 112 005 10) The authors would like to thank Bülent Yalçınalp and Ali Babacan from Karadeniz Technical University in Trabzon (Turkey) for field and laboratory work The authors would also like to thank Nazmiye Yazıcı from Güvencem Marble Limited Company for helping during all stages of the study references Arslan, M , Tuysuz, N , Korkmaz, S , Kurt, H (1997) Geochemistry and petrogenesis of the eastern Pontide volcanic rocks, Northeast Turkey Chemie der Erde, 57: 157–187 Babacan, A E , Ersoy, H , Gelisli, K (2009) Determination of physical and mechanic properties of rocks with direct and indirect methods: A case study on the beige limestones in the Eastern Pontides, Proceed- ings of the 21st International Mining Congress and Exhibition of Turkey, Antalya, 123-130 Bieniawski, Z T (1974) Estimating the strength of rock materials Journal of the South African Institute Mining Metallurgy, 74: 312-320 Cargill, J S , Shakoor, A (1990) Evaluation of empirical methods for measuring the uniaxial compressive strength International Journal of Rock Mechanic and Mining Science, 27: 495-503 Chang, C , Mark, D , Zoback, A , Abbas, K B (2006) Empirical relations between rock strength and physical properties in sedimentary rocks Journal of Petroleum Science and Engineering, 51: 223-237 Chau, K T , Wong, R H C (1996) Uniaxial compressive strength and point load strength International Journal of Rock Mechanic and Mining Science, 33: 183–188 Cook, T , Falchi, P , Mariano, R (1984) An urban allocation model com- bining time series and analytic hierarchical methods Management Science, 30 (2): 198-208 Davis, J C (1986) Statistics and Data Analysis in Geology John Wiley and Sons, New York Dehghan, S , Sattari, G H , Chelgani, C , Aliabadi, M (2010) Prediction of uniaxial compressive strength and modulus of elasticity for traver- tine samples using regression and artificial neural networks Mining Science and Technology, 20: 41-46 Ersoy, H , Bulut, F (2009) Spatial and multi-criteria decision analysis- based methodology for landfill site selection in growing urban re- gions Waste Management and Research, 27(5):489-500 Eyuboglu, Y (2006) Description and Geotectonic Important of the Alas- kan-Type Ma c–Ultrama c Rocks in the Eastern Pontide Magmatic Arc (NE Turkey), PhD Thesis, Karadeniz Technical University, Trab- zon, Turkey (unpublihed) Freedman, D (2005) Statistical models theory and practice London: Cambridge University Press Gaviglio, P (1989) Longitudinal waves propagation in a limestone: the relationship between velocity and density Rock Mechanic and Rock Engineering, 22: 299-306 Gokceoglu, C , Zorlu, K (2004) A fuzzy model to predict the uniaxial compressive strength and the modulus of elasticity of a problematic rock Engineering Applications of Artificial Intelligent, 17(1): 61–72 Grima, A M , Babuska, R (1999) Fuzzy model for the prediction of un- confined compressive strength of rock samples International Journal of Rock Mechanic and Mining Science, 22: 339-349 ISRM (International Society for Rock Mechanics) suggested methods: rock characterization, testing and monitoring E T Brown (ed ), Per- gamon Press, London; 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