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O R I G I N A L A R T I C L E

Prediction of unconfined compressive strength of carbonate rocksusing artificial neural networks

Nurcihan Ceryan Umut Okkan Ayhan Kesimal

Received: 7 August 2011 / Accepted: 14 June 2012/ Published online: 3 July 2012

Springer-Verlag 2012

Abstract The unconfined compressive strength (UCS) of

intact rocks is an important geotechnical parameter forengineering applications. Determining UCS using standard

laboratory tests is a difficult, expensive and time consum-

ing task. This is particularly true for thinly bedded, highly

fractured, foliated, highly porous and weak rocks. Conse-

quently, prediction models become an attractive alternative

for engineering geologists. The objective of study is to

select the explanatory variables (predictors) from a subset

of mineralogical and index properties of the samples, based

on all possible regression technique, and to prepare a

prediction model of UCS using artificial neural networks

(ANN). As a result of all possible regression, the total

porosity and P-wave velocity in the solid part of the sample

were determined as the inputs for the LevenbergMarqu-

ardt algorithm based ANN (LM-ANN). The performance

of the LM-ANN model was compared with the multiple

linear regression (REG) model. When training and testing

results of the outputs of the LM-ANN and REG models

were examined in terms of the favorite statistical criteria,

which are the determination coefficient, adjusted determi-

nation coefficient, root mean square error and variance

account factor, the results of LM-ANN model were more

accurate. In addition to these statistical criteria, the non-parametric MannWhitneyU test, as an alternative to the

Studentsttest, was used for comparing the homogeneities

of predicted values. When all the statistics had been

investigated, it was seen that the LM-ANN that has been

developed, was a successful tool which was capable of

UCS prediction.

Keywords Carbonate rock Unconfined compressivestrength Porosity Wave velocity All possible

regression Artificial neural networks LevenbergMarquardt algorithm

Introduction

One of the important rock mechanic parameters for engi-

neering geologists, geotechnical engineers and mining

engineers is the determination of the unconfined com-

pressive strength of rocks (UCS), which is considered by

many researchers to be the most essential rock material

property (Bieniawski 1974). This parameter has great

importance in rock mechanic applications such as tunnel

and dam design, rock blasting and drilling, mechanical

rock excavation and slope stability. There are basically two

methods for assessing the UCS of rocks. One, known as the

direct method, is to test the specimens in the laboratory, the

other, known as the indirect method, is to use previously

derived empirical equations from the literature (Baykaso-

glu et al.2008). Testing procedures for the direct method

have been standardized by both the American Society for

Testing and Materials (ASTM) and International Society

for Rock Mechanics (ISRM). High-quality core specimens

are needed for direct determination of UCS in a laboratory.

N. Ceryan (&)

Department of Geological Engineering,Balkesir University, Balikesir, Turkey

e-mail: [email protected]

U. Okkan

Department of Civil Engineering,

Balkesir University, Balikesir, Turkey


A. Kesimal

Department of Mining Engineering,

Karadeniz Technical University, Trabzon, Turkey


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However, high quality cores in sufficient quantities cannot

always be extracted from weak, highly fractured, weath-

ered and thinly bedded rocks. In addition, careful execution

of this test is very difficult, time consuming, and expensive

and involves destructive tests (Gokceoglu and Zorlu2004;

Baykasoglu et al.2008). To overcome these difficulties, the

indirect method considering simple index parameters and/

or mineralogical analyses and basic mechanical tests of thephysical properties, were developed, including the P-wave

velocity, Schmidt hammer, Point load index (e.g., Kahr-

aman 2001; Chang et al.2006; Ceryan et al.2008; Oyler

et al.2010), block punch test (Ulusay et al.2001; Altindag

et al.2004) and core strangle test (Yilmaz2010). For some

of these studies, rock fabric, mineralogic and petrographic

properties were obtained using image analysis (Akesson

et al. 2001; Lindqvist and Akesson 2001; Jensen et al.

2010). This is because these tests and analyses have smaller

samples and are simpler, faster and more economical (Bell

1978; Fahy and Guccione1979; Brooks1985; Doberenier

and De Freitas 1986; Hawkins and McConnell 1990;Shakoor and Bonelli 1991; Ulusay et al. 1994; Romana

1999; Alvarez Grima and Babuska1999; Singh et al.2001;

Gokceoglu2002; Gokceoglu and Zorlu2004; Sonmez et al.

2004; Chang et al. 2006; Oyler et al. 2010). Indirect

methods for UCS prediction are generally preferred, par-

ticularly when there are limited laboratory facilities

(Baykasoglu et al.2008).

The statistical methods used in rock engineering, for

example the simple and multiple regression techniques, are

conversional predictive models for estimating the

mechanical properties of rock materials including UCS. In

addition to these conventional methods, new techniques

such as artificial neural networks, fuzzy inference systems,

genetic programming, and regression trees are also gaining

considerable attention for estimating UCS (Meulenkamp

1997; Alvarez Grima and Babuska1999; Meulenkamp and

Alvarez Grima 1999; Singh et al. 2001; Kahraman and

Alber2006; Baykasoglu et al.2008; Yilmaz and Yuksek

2009; Sarkar et al. 2010; Yagiz et al. 2012). Since the

1990 s, ANN have recently become more popular, espe-

cially where fewer correlation coefficients of regression

equations with more input variables are required to com-

pletely define rock characteristics and the more flexible

operations between input and output variables (Garret

1994; Huang and Wanstedt1998; Baykasoglu et al.2008).

It is a form of nonlinear analysis which is based on the

understanding of the brain and nervous system (Ghabousi

et al.1991; Ham and Kostanic2001). Furthermore, it is a

fundamentally different approach that has to learn and

generalize interactions between many variables. Conse-

quently, ANN has great potential for modeling material

behavior from experimental data (Ghabousi et al. 1991;

Ellis et al.1992). One of the major advantages of ANN is

its efficient handling of highly non-linear relationships in

data, even when the exact nature of such relationships is

unknown (Dehghan et al. 2010). ANN models are well

suited for UCS predictions, because of the complex nature

of the interrelationships between the various quality

parameters, composition and processing conditions (Deh-

ghan et al. 2010). In the last few years, artificial neural

networks (ANN) have generally been used to establishedpredictive models of UCS for rock engineering applica-

tions (Meulenkamp and Alvarez Grima 1999; Singh and

Dubey2000; Kahraman and Alber2006; Zorlu et al.2008;

Yilmaz and Yuksek2008; Sarkar et al.2010; Cevik et al.

2011; Yagiz et al. 2012). The performance of the ANN

models was also compared with other statistical methods

(e.g., regression analysis). These studies demonstrated that

the results of ANN were more precise than the conven-

tional statistical approaches (Nie and Zhang1994; Tiryaki

2008; Baykasoglu et al.2008; Dehghan et al.2010; Yagiz

et al.2012).

The mechanical strength of carbonate rocks such aslimestone is predominantly governed by five parameters:

porosity, cleavage properties, crystal size, lithification and

micro cracks (Chang et al. 2006, Jensen et al. 2010).

However, porosity in many studies has been considered a

basic input parameter when estimating the UCS of car-

bonate rocks. (Mohd 2009; Asef and Farrokhrouz 2010;

Jensen et al.2010). Furthermore, ultrasonic wave velocity

has often been used in estimation models as a nonde-

structive method that is portable, cheap and easy to use.

(e.g., DAndrea et al.1965; Youash1970; Saito et al.1974;

Lama and Vutukuri1978; Gaviglio1989; Baykasoglu et al.

2008; Diamantis et al. 2009; Yilmaz and Yuksek 2009;

Ameen et al.2009; Moradian and Behnia2009; Kahraman

et al.2009; Dehghan et al.2010, Sarkar et al.2010; Yagiz

et al.2012). However, some significant rock indices such

as the P-wave velocity in the solid parts of rock materials

have not yet been considered in building predictive models

for the UCS of sedimentary rocks.

In this study, the aim is to establish predictive models

for the UCS of carbonate rocks formed from various facies

exposed in the Tasonu quarry, Trabzon, NE Turkey, used

in rock engineering applications. The fabric and its com-

ponents (calcite, clay and, rock and mineral fragments) of

carbonate rocks in the field of study are highly variable due

to the development of different facies. They are mostly

hollow macro- and micro-cracks. The surface structure has

pitted, pitted to vuggy and vuggy shapes. For these reasons,

the rocks studied are known as a group of problematic

rocks such as clay and much-fractured rocks. Conse-

quently, the use of estimation methods was seen to be

useful in determining the UCS. The objective in this study

is to select the explanatory variables (predictors) from a

subset of mineralogical and index properties of the

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samples, based on an all possible regression technique, and

to prepare a prediction model of UCS using ANN. ANN

method was preferred because the main advantage of it

over conventional methods is that it does not require

detailed information about the complex nature of the

underlying process under consideration to be explicitly

described in mathematical form. Some extensive applica-

tion and reviews of such models proposed for the predic-tion of geological variables were reported (Nie and Zhang

1994; Yilmaz and Yuksek2008,2009; Yagiz et al.2012).

The LevenbergMarquardt algorithm based ANN model

was used as the prediction method in this study. The

advantages of LevenbergMarquardt algorithm are that it is

usually faster and more reliable than any other back

propagation algorithms (Hagan and Menhaj1994). Thus it

can handle situations when the relationship between input

and output variables is nonlinear. The performance of LM-

ANN was also compared with the conventional method of

multiple linear regression (REG).

Material and testing procedures

The carbonate rocks samples developed in different facies

and were taken from the Tasonu Quarry, Trabzon, north-

east Turkey (Fig.1). These rocks are used as the raw

materials by Trabzon Askale Cement Factory. They are

part of the Kirechane Formation, which developed in the

Campanian (Fig.2).

The mineralogical composition of the samples from the

Kirechane Formation was studied using X-ray diffraction

(XRD) at Hacettepe University. Semi-quantitative per-

centages of the minerals are calculated by the method

developed by Gundogdu (1982). Details of the method can

be found in Temel and Gundogdu (1996). The some sam-

ples from the quarry were 100 % CaCO3. In other samples,

there were significant variations in other components (clay,

feldspar, biotite and opaque minerals; Table1). In Table1,

the result of XRD analysis, index and strength properties of

the samples examined are given.

In this study, 56 groups of block samples, each sample

having the approximate dimensions of 30 930 930 cm,

were collected in the field for the rock mechanics tests

using core drilling machine of the Rock Mechanics Labo-

ratory in the Engineering Faculty of Karadeniz Technical

University. Core samples were prepared from the rockblocks: they were 50 mm in diameter, and the edges of the

specimens were cut parallel and smooth (ISRM 2007;

Fig.3). Some tests, such as specific density, unit weight,

porosity, effective porosity, P-wave velocity, slake dura-

bility, aggregate impact value and UCS, were carried out in

the laboratory. The physical property and UCS tests were

performed on 15 samples for each sample group. The slake

durability and aggregate impact value test were performed

on three samples for each group. ISRM (1981) suggests

two cycles slake durability. However, Gokceoglu et al.

(2000) proposed four cycles. Similarly, four cycles slake

durability index were used in this study.The total porosity (n) and effective porosity of the rock

were estimated from the following equations:

n 1qsqd

1

neWs Wd

qwV 2

whereqsis the dry density,qdthe density of solid particles,

qw the water density, Wd the weight of the sample in theFig. 1 Location of Tasonu quarry (Trabzon, NE Turkey)

L7

L0 L4b

20

9

23

13

18

11

L4b

L4b

L8c

L8c

L8b

L8b

L8a

L6

L0

L8b

L8b

L1

L1

L0

L0

L2

L2

L2

L3a

8

12

18

25

15

7

21

20

15

L0

15 12

L3b

L0

L5

L5

15

16

L4b 15

L8b

L8b

L4a

L3a

L3a

L3a

584800 585000 585200

N

100 m0

Fig. 2 Geological map of Tasonu quarry (L0basalt, andesit and their

piroklastic, L1 volcanic pebbly red tuff, L2 red tuff alternate with

white limestone, L3 common macro shelly karstic voided limestone

(a), intercalated with red tuff (b), L4 fine grained karstic voided

carbonate mudstone (a) that overlie red sandy clayey limestone (b),

L5 alternate with sandy limestone clayey limestone and marl, L6

volcanic tuff intercalate with clayey limestone and mar, L7 sandypebbly limestone,L8 carbonate cemented sandstone intercalated with

clayey limestone and marl (b). Lower part of the sandstone contain

silicified level (a), Interbedded common macro fossiliferous with

biotite tuffacous carbonate cemented sandstone and sandy limestone

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Table 1 Mineralogical, index and strength properties of the samples examined

Smpl Clt Cly Fld Qrz Qq Bi n ne Id Vp Vm UCS

1a 82 18 16.5 12.9 91.2 3,202 4,245 14.50

1b 72 9 19 16.7 13.4 91.5 3,299 4,458 16.12

2a1 69 26 5 17.3 14.8 2,449 3,475 8.98

2a2 71 20 9 20.3 16.1 88.4 2,650 3,903 13.90

2b1 54 46 18.8 11.8 84.4 2,455 2,834 8.85

2b2 55 45 20.8 13.5 85.2 2,354 2,631 7.77

3a1 95 5 17.9 14.9 3,672 5,418 17.05

3a2 93 4 3 18.2 15.6 3,517 5,507 16.72

3b1 66 34 25.6 22.7 85.9 2,293 3,255 11.34

3b2 60 36 6 25.9 22.5 87.9 2,286 3,249 13.48

3c 62 31 5 1 18.7 14.4 89.5 2,159 3,061 9.51

4a 68 23 8 1 25.0 17.4 92.5 2,991 4,458 15.37

4b 67 20 10 3 26.8 21.7 90.3 2,875 4,422 14.13

4c 51 43 4 2 21.3 12.3 91.3 2,492 2,941 12.79

j1a 56 42 2 25.0 21.6 86.5 2,200 3,362 11.38

j1b 43 51 6 33.9 30.9 83.5 1,963 2,739 9.63

j2 82 12 4 2 15.2 11.6 93.5 3,379 4,648 14.97

j3 76 8 12 2 7.4 5.0 0.0 3,787 5,391 22.69

j4 34 31 7 4 24 24.7 19.7 93.9 3,074 3,914 13.12

j5 77 22 9.6 5.8 92.8 2,913 3,589 11.92

j6a 63 18 9 11 15.8 16.8 93.0 2,780 4,391 13.62

j6b 68 11 5 20 17.6 12.5 92.9 3,263 4,762 14.32

j6c 56 34 8 2 11.8 9.1 92.9 2,652 3,579 12.47

j7 73 15 14 0 10.5 9.4 95.6 3,259 4,641 17.40

jtb8a 43 45 8 4 31.2 27.6 83.3 2,466 3,353 8.31

j8bc 87 13 0 26.6 23.2 85.4 2,522 3,858 9.86

j9a 50 38 5 2 5 12.3 9.8 91.2 2,836 3,440 12.34

j9bc 75 10 11 4 16.7 11.9 93.6 3,329 4,746 13.71j9d 86 14 8.0 6.2 95.9 3,574 5,216 17.92

j10a 83 8 5 4 16.3 14.3 91.6 3,035 4,689 15.49

j10bc 38 22 16 21 9 15.8 13.9 93.3 3,002 4,424 13.80

j11a 100 12.3 10.7 96.3 3,634 5,279 18.62

j11b 67 8 8 5 2 10 8.0 7.3 93.6 3,528 4,851 18.21

j12 100 7.4 4.7 95.2 3,649 5,321 20.76

j13 17.3 14.8 0.0 2,449 8.98

j14 53 19 7 1 10 25.2 19.9 0.0 3,286 4,946 13.16

j15 87 2 3 8 11.5 10.6 93.0 3,527 5,763 15.84

j16 10 69 6 16 33.2 31.6 80.3 1,319 1,401 7.32

j17 90 9 1 9.7 7.2 94.1 3,576 5,168 18.87

j18 10.9 7.5 3,124 4,439 11.70

j19 31 46 23 16.6 13.8 2,736 2,973 10.61

j20 86 14 11.9 9.4 90.8 3,350 4,965 15.41

j21 82 13 5 22.1 16.5 91.3 2,883 53.9 13.42

j22 82 18 17.4 13.7 93.5 3,232 57.6 14.04

j23 87 11 2 17.8 15.5 92.6 2,965 53.3 13.18

j2527 34 41 7 4 14 18.1 17.3 89.4 1,902 41.5 12.21

j26a 95 0 5 9.2 6.7 96.0 4,259 37.8 21.02

j26b 90 5 5 12.4 8.6 95.0 3,521 21.42


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dried condition, Ws the weight of the sample in the satu-

rated condition, andVthe volume of the sample.

In this study, ultrasonic pulse velocity (UPV) tests were

conducted using the first method suggested in ISRM

(1981). UPV measurements were performed on the sam-

ples in the both dried and saturated conditions. The pulsewas generated by a source-transducer: it was transmitted

through the sample and registered by a receiver-transducer.

Using an epoxy polymer between the transducers and the

test sample produced an improvement in the energy

transmission. After measurements, velocity of P-wave of

the V-wave,Vp, was calculated from measured travel time

and the distance between the transmitter and receiver. In

addition to P-wave velocity in the rock samples that lacked

pores and fissures, Vm, were calculated by employing the

Eq.3 (from Barton2007):

1

Vp /

Vfl

1 /

Vm3

where Vp is the P-wave velocity in the sample, Vfl the

velocity in the fluid, / the ratio of the path length in

the fluid to the total path length (i.e., the porosity), andVmthe P-wave velocity in rock samples lacking pores and

fissures (in other words, P-wave in the solid). In this study,

to calculate the Vm value, Vp was used for the P-wave

velocity measured in saturated samples,Vflfor the P-wave

velocity measured in the fluid, and/ for effective porosity.

UCS tests were carried out according to International

Society for Rock Mechanics (ISRM) (2007). Core samples

were prepared in 2.5:1 height/diameter ratio, with adiameter of 50 mm and height of 125 mm. Experiments

were performed on 15 samples in dried condition for each

group. During the test, samples were loaded to be broken in

10 and 15 min.

Aggregate impact tests are included in British Standards

for measurement of the mechanical properties of crushed

rock aggregate, including the aggregate impact value (AIV)

tests (BS 812 1990). In the test, samples ranging in size

from 10 to 14 mm are subjected to shock and static load.

The proportion of material passing BS 2.36-mm sieve after

loading is calculated as a percentage of the original sample

weight, and expressed as the aggregate impact and aggre-

gate crushing values for shock and static loading,

respectively.

Method

Selection of explanatory predictors

The selection of the model inputs depends on the depen-

dent variables generally. Any type of input can be used in

modeling as long as they have acceptable correlation or

determination with the dependent variables. But, the large

number of the predictors may not produce in better results.

Hence, principal component analysis (PCA) and canonical

correlation have been used in some studies to reduce thedimension in the relationship. These types, involve a pro-

cedure that transforms a number of possibly correlated

variables into a smaller number of uncorrelated variables in

order to reduce the number of predictors (Singh and Har-

rison1985; Sharma1996).

There are several ways of selecting predictors if a large

number is available. One common method is a detailed

search in which all possible regressions are tried and one is

selected as the most appropriate predictor according to

statistical performance criteria (Neter et al.1996). Maxi-

mum adjusted determination coefficient (AdjR2), or min-

imum MallowsCpvalues can be used as such performancecriteria (McQuarrie and Tsai1998).

R2 (Eq.4) describes the proportion of the variation in

the dependent variable as explained by the predictors in

the model.R2 increases with the increase in the number of

parameters in the model. Thus, it does not by itself

indicate the correct regression model. AdjR2 (Eq.5) is

the modified version of R2 that has been adjusted for the

number of predictors in the model. AdjR2 is generally

considered a more accurate goodness-of-fit measure than

Table 1 continued

Smpl Clt Cly Fld Qrz Qq Bi n ne Id Vp Vm UCS

j28 83 7 8 2 6.8 3.7 97.4 3,980 31.6 15.24

j29 10.3 6.7 95.7 36,827 19.22

j30 12 49 16 1 22 21.3 19.7 80.5 2,072 68.3 9.75

j31 56 26 6 2 10 18.2 13.9 88.3 3,081 45.7 12.63

j33a 53 41 5 1 9.7 7.2 2,700 36.1 12.49

j33b 88 3 9 0 4.9 3.5 96.3 3,825 21.1 24.06

j33d 82 8 8 2 15.8 13.7 92.5 3,440 37.4 14.57

Clt (%), calsite; Cly (%), clay; Qrt (%), quartz; Qq (%), opacue; Bi (%), biotite; G, spesific density; ck(kN/m3), dry unit weight; n (%), total

porosity; ne (%), effective porosity; Id (%), slake durability index (fourth cycle); Vp (m/s), P-wave velocity in dry samples; Vm (m/s), P-wave

velocity in solid part of the sample; UCS (MPa), unconfined compressive strength


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R2. If Adj R2 is a criterion to select the explanatory

variables, it should only be used for the models with the

same number of parameters. If the selection of explana-

tory parameters is based on R2 and/or AdjR2, then the

problem of collinear variables in the model may arise.

This problem reduces the estimation capability of the

regression model in the verification stage (Hinnes andMontgomery 1990). An alternative criteria proposed by

Mallows (1973) is useful to correlate the predictor set and

to compare the models which have different number of

parameters.

MallowsCp(Eq.6) is a measure of the error in the best

subset model, relative to the error incorporating all vari-

ables. Adequate models are those for whichCp is roughly

equal to the number of parameters in the model (Mallows

1973).

R2 1MSEi

r2 4

AdjR2 1 n1

ni11R2 5

Cp nkMSEi

MSEF n2i1 6

where R2 is determination coefficient, AdjR2 is adjusted

determination coefficient, Cp is Mallows Cp, MSEi is

mean of residual squares in the model with j parameter,

MSEFis mean of residual squares in the full model with

kparameter,r2 is variance of the dependent variable,n is

the number of data, i is the number of parameters in the

model and k is the number of parameters in the full

model.

Fig. 3 Test samples with 50 mm diameter


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Artificial neural networks (ANN)

The basic concept of the ANN is that they are typically

made up of neurons. And in ANN, the neurons are orga-

nized in the form of layers (Fig.4).

The first and last layer of ANN is called the input and

the output layers, respectively. The input layer does not

perform any computations, but only serves to feed the inputdata to the hidden layer which is between the input and

output layers.

In general, there can be any number of hidden layers in

the ANN structures, however, from practical applications,

only one or two hidden layers are used. In addition to this,

the number of hidden layers and also the number of neu-

rons of hidden layers can be determined by the trial and

error (Ham and Kostanic2001).

There are also three important components of an ANN

structure: weights, summing function and activation func-

tion. The importance and functionality of the inputs on

ANN models are obtained with weights (W).So the successof the model depends on the precise and correct determi-

nation of weight values. The summing function (net) acts

to add all outputs; that is each neuron input is multiplied by

the weights and then summed. After computing the sum of

weighted inputs for all neurons, the activation functionf(.)

serves to limit the amplitude of these values. The activation

functions are usually continuous, non-decreasing and

bounded functions. Various types of the activation function

are possible but generally log-sigmoid function is preferred

in applications (Ham and Kostanic2001). This activation

function generates outputs between 0 and 1 as the input

signal goes from negative to positive infinity.

Weight matrix for layer 1W

(1)Weight matrix for layer 2

W(2)

1nxx R

Hidden layer

hneuronsOuput layer

mneurons 1mxx R

Fig. 4 ANN structure

Table 2 Summary of all possible regression analyses

Number of

Inputs

R2 Adj R

2Cp n ne ld Vp Vm

1 0.744 0.738 5.8

1 0.704 0.697 13.0

2 0.777 0.767 1.7

2 0.769 0.758 3.2

3 0.783 0.767 2.7

3 0.779 0.762 3.4

4 0.786 0.765 4.1

4 0.783 0.761 4.7

5 0.786 0.759 6.0

Table 3 LM-ANN and REG performances for the training and testing periods

Models Model structures R2 Adj R2 VAF RMSE(MPa)

Training Testing Training Testing Training Testing Training Testing

LM-ANN n = 2; h = 4; m = 1 0.8837 0.8126 0.8751 0.7814 87.64 81.02 1.1079 1.8970

REG rc = 5.77 ? 0.00225Vm - 0.132n 0.7950 0.7406 0.7798 0.6974 79.21 73.75 1.4298 2.2189

Table 4 Measured and LM-

ANN descriptive statistics forthe training and testing periods

Mean (MPa) Standard

deviation (MPa)

Skewness Maximum

(MPa)

Minimum

(MPa)

Training

Measured 13.52 3.19 0.18 20.76 7.77

LM-ANN 13.41 2.73 0.40 19.74 8.88

REG 13.54 2.63 -0.01 18.94 8.28

Testing

Measured 15.27 4.48 0.38 24.06 7.32

LM-ANN 15.46 4.26 -0.48 21.06 7.39

REG 15.22 4.10 -1.28 19.58 4.97


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f: ffi 1

1e: 7

In addition to thestructure andits components of ANN, the

running procedure is also important which involves typically

two phases; forward computing and backward computing.

In forward computing, each layer uses a weight matrix

[W

(v)

, for v=

1, 2] associated with all the connectionsmade from the previous layer to the next layer (Fig.4). The

hidden layer has the weight matrixW1 2Rhn, the output

layers weight matrix is W2 2Rmh. Given the network

input vector x2 Rn1, the output of the hidden layer

xout;12 Rh1 can be written as

xout;1 f1net1 f1W1x 8

which is the input to the output layer. The output of the

output layer, which is the response (output) of the network

y xout;22Rm1, can be written as

y xout;2f2net2 f2W2xout;1 9

Substituting (Eq.8) into (Eq.9) for xout,1gives the final

output y = xout,2of the network as

y f2W2f1W1x 10

After the phase of forward computing, backward

computing which depending on the algorithms to adjust

weights is used in the ANN. The process of adjusting these

weights to minimize the differences between the actual and

the desired output values is called training or learning

the network. If these differences (error) are higher than the

desired values, the errors are passed backwards through the

weights of the network. In ANN terminology, this phase isalso called the back-propagation. Once the comparison

error is reduced to an acceptable level for the whole

training set, the training period ends, and the network is

also tested for another known input and output data set in

order to evaluate the generalization capability of the ANN

(Ham and Kostanic2001).

Depending on the techniques to train ANN models,

different back propagation algorithms have been used for

modeling of UCS. These modeling studies generally

include the standard feed forward back propagation (FFBP)

algorithms such as gradient-descent (Yilmaz and Yuksek

2008; Sarkar et al.2010), gradient-descent with momentumrate (Singh et al.2001; Kahraman et al.2010; Yilmaz and

Yuksek2009), conjugate gradient (Canakci and Pala2007)

etc. As the standard FFBP algorithms have some disad-

vantages relating to the time requirement and slow con-

vergency in training, LevenbergMarquardt algorithms,

which are alternative approaches to standard FFBP algo-

rithms, were also used in some engineering applications

(Meulenkamp and Alvarez Grima 1999; Tiryaki 2008;

Fistikoglu and Okkan2011; Okkan2011).

In this study LevenbergMarquardt algorithm was used

for training. This algorithm is a second order nonlinear

optimization technique that is usually faster and more

reliable than any other standard back propagation tech-

niques (Meulenkamp and Alvarez Grima 1999; Tiryaki

2008; Fistikoglu and Okkan2011; Okkan2011, Okkan and

Dalkilic2011). It represents a simplified version of New-

tons method (Marquardt 1963) applied to the trainingANN (Hagan and Menhaj 1994).

Consider ANN structure shown in Fig.4, the running of

the network training can be viewed as finding a set of

weights that minimized the error (ep) for all samples in the

training set (Q). If the performances function is a sum of

squares of the errors as

EW 1

2

XPp1

dp yp2

1

2

XPp1

ep2; P mQ 11

where Q is the total number of training samples, m is the

number of output layer neurons, W represents the vector

containing all the weights in the network,yp is the actual

network output, anddp is the desired output.

When training with the LevenbergMarquardt optimi-

zation algorithm, the changing of weights DW can be

computed as follows

DW JTkJk lkI 1

JTkek 12

whereJis the Jacobian matrix,Iis the identify matrix,l is

the Marquardt parameter which is to be updated using the

decay rateb depending on the outcome. In particular,l is

multiplied by the decay rateb (0\b\ 1) wheneverE(W)

decreases, while l is divided by b whenever E(W)increases in a new step (k).

The LM-ANN training process can be illustrated in the

following pseudo-codes,

1. Initialize the weights andl (l = 0.001 is appropriate).

2. Computethe sum of squarederrors over all inputs,E(W).

3. Compute the Jacobian matrixJ.

4. Solve Eq.12 to obtain the changing of weightsDW.

5. Recompute the sum of squared errors E(W) using

Wk1 Wk JTkJk 1

JTkek as the trial W, and

judge


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Results

Explanatory predictors of UCS

To evaluate the strength and direction of the relationships

between the input and output, all possible regression results

of five predictor variables: n (%) total porosity, ne (%)

effective porosity, Id (%) slake durability index (fourth

cycle),Vp (m/s) (P-wave velocity in dry samples), andVm(m/s) (P-wave velocity in solid parts of the sample), and

UCS are given in Table2.

Once the explanatory predictor variables are selected,

based on the MallowsCpcoefficient, theAdjR2 values are

calculated as in Table2. Performance of the model with

n and Vm is nearly the same as that of the full model with

five variables; that is, the explained variance of UCS using

the two effective variables is nearly equal to the variance

explained using the full model. On the other hand, Mal-

lowsCp decreases rapidly with up to two variables (nand

Vm) and then increases with every addition of a variable.

Modeling of UCS using ANN

As a result of the MallowsCpbased all possible regression

analyses, the total porosity (n) and P-wave velocity in the

solid part of the sample (Vm), which are the explanatory

predictor variables of UCS, were selected as inputs for the

LevenbergMarquardt algorithm based ANN model (LM-

ANN). In the implementation of the LM-ANN, MATLAB

code was used. To compare the generalization capabilities

of the LM-ANN model, the inputoutput data were divided

into training and testing in the proportions of 2/3 and 1/3.Before presenting the inputoutput data to the LM-

ANN, all data sets were scaled to the range of 01 so that

the different input signals had the same numerical range.

The training and testing subsets were scaled to the range of

Fig. 5 Determination of number of neurons in hidden layer for the

testing period

y = 0,8564x + 2,3911

R = 0,8126

5

7

9

11

13

15

17

19

21

23

25

5 7 9 11 13 15 17 19 21 23 25

LM-ANN(2,4,

1)(M

pa)

Measured (Mpa)

Test

y = 0,8032x + 2,5519

R = 0,8837

5

7

9

11

13

15

17

19

21

5 7 9 11 13 15 17 19 21

LM-ANN(2,4,

1)(M

pa)

Measured (Mpa)

Training

y = 0,7468x + 3,4416

R = 0,795

5

7

9

11

13

15

17

19

21

5 7 9 11 13 15 17 19 21

c=5.7

7+0.00

255vm-

0.1

32n

Measured (Mpa)

Training

y = 0,7883x + 3,1954

R = 0,7406

5

7

9

11

13

15

17

19

21

23

25

5 7 9 11 13 15 17 19 21 23 25

c=5.7

7+0.00

255vm-

0.1

32n

Measured (Mpa)

Test

Fig. 6 Scatter plots of REG

and LM-ANN models for the

training and testing period


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01 using the equation zt=(xt- xmin)/(xmax - xmin),

wherextis the unscaled data,ztis the scaled data, andxmaxand xmin are the maximum and minimum values of the

unscaled data. The output values of the LM-ANN model,

which were in the range of 01, were then transformed to

real-scaled values using the equation xt= zt (xmax -

xmin) ? xmin. In this study, three widely used transfer

functions, namely tangent sigmoid, linear, and log-sigmoidare evaluated in LM-ANN structure trials. The best results

are achieved by using the log-sigmoid function which

described in Eq. (7). Otherwise, the number of the neurons

in the hidden layer, the initial Marquardt parameter (l0)

and decay rate (b) of the LM-ANN model, were also

determined by trial and error. The suitable network struc-

ture provided the best training result in terms of minimum

E(W) or root of mean E(W). The root mean square error

(RMSE), variance account factor (VAF) and the maximum

determination coefficient value (R2) were also employed in

the testing. RMSE and VAF are calculated as follows:

RMSE

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xni1

diyi2

s 13

VAF 1vardi yi

vardi

100 14

wherenis the number of training or testing samples,yiis the

actual network output, anddiis the measured (desired) data.

A model denoted as LM-ANN (n, h, m) (with

l0 = 0.01; b =0.10; k= 20 iterations) in Tables3and 4

comprisesn = 2 neurons in the input layer,h = 4 neurons

in the hidden layer andm = 1 neuron in the output layer.

The number of neurons in the hidden layer was deter-mined after trying various values (h = 220) for the

selected two input variables (Fig.5).

When the scatter plots of the training and testing data

sets for the LM-ANN and REG are examined, it is

observed that the standard deviations around they = x line

are much lower in the LM-ANN model. The LM-ANN

results for the training and testing periods were compared

with the desired UCS values (Figs.6, 7)

In addition to these preferred statistical criteria, homo-

geneities of the model predictions were also examined with

the MannWhitneyU(MW) test to present more evidence

on the acceptable application and success of the LM-ANN

model. This non-parametric statistical test is used to ana-

lyze two comparison groups to identify whether they have

the same distribution or not (Mann and Whitney 1947).

MW is based on the bringing together and arranging of

two groups (e.g., predicted and measured values). When

these group members are lined up, a line number is

assigned to each member. The membership status of these

members (to which group they belong) is ignored. These

line numbers are then summed up. The sum of the

members of the first group is R1and of the second group is

R2. The Uvalues can then be calculated using

Ui N1N2NiNi1

2 Ri; i 1; 2 15

After the calculation fori =1 andi = 2, U1and U2are

obtained, and the larger is chosen (U*) to determine the test

statistics.

z U N1N2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN1N2N1N21

12

q 16

5

7

9

11

13

15

17

19

21

23

25

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

c

Data Points

TestingMeasured (Mpa)

LM-ANN (2,4, 1) (Mpa)

5

7

9

11

13

15

17

19

21

1 2 3 4 5 6 7 8 910

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

c

Data Points

TrainingMeasured (Mpa)LM-ANN (2,4, 1) (Mpa)

Fig. 7 LM-ANN results with the data points for the training and

testing period

Table 5 MW statistics of LM-ANN and REG models for training

and testing periods

MW test statistics LM-ANN REG

Training Testing Training Testing

MannWhitney U 440 104 439 102

z 0.148 0.353 0.163 0.436

Asymptotic. Sig.

(2-tailed)

0.882 0.744 0.871 0.683


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where N1 and N2 are the quantities of data for the groups

compared.

The z value is compared with a 0.05 significance level

(zcr =1.96). For the values of z\ 1.96, there is no sig-

nificant difference between the measured data and model

predictions. The asymptotic significance of the z test sta-

tistics was also used when making a comparison (Table5).

When MW statistics were examined, it was shown thatboth LM-ANN and REG predictions have homogeneities

for the training and testing set. When the z statistics and

asymptotic significance values are taken as a basis, LM-

ANN again has the best performance.

Conclusions

In this study, a model was developed to estimate, by con-

sidering the index properties, the unconfined compressive

strength value of carbonate rocks formed at the differentfacies. The mineralogical and index properties of the

samples were used for modeling the UCS. The explanatory

predictors from these measured samples were selected by

performing a comprehensive all possible regression anal-

ysis, in which best parameters were determined using the

Mallows Cp value and the UCS was considered as the

dependent variable. As a result of all the possible regres-

sion analyses, total porosity (n) and P-wave velocity in the

solid part of the sample (Vm) were selected as the inputs for

the LevenbergMarquardt algorithm based ANN model

(LM-ANN).

In previous studies, elastic wave velocity measured indried samples, and seismic attenuation or P-wave velocity

ratio measured in dried and fluid conditions were used to

estimate the strength and deformation of the rocks.

P-wave velocity used for determining the mineralogical

composition in the solid part of the rock, namelyVm, was

used as an initial parameter for estimating the UCS value

in this study.

When the model training and testing outputs were

investigated, in terms of the statistics (R2, Adj R2, RMSE,

VAF, descriptive statistics) of the measured and the pre-

dicted values, LM-ANN results fitted well. In addition to

these, the non-parametric MannWhitney U test was alsoused for comparing the homogeneities of predicted values.

When all the statistics were investigated, it was seen that

the LM-ANN that was developed, was a successful ANN

algorithm that was capable of UCS modeling. The authors

also suggest that this approach can be used for the pre-

diction of other geotechnical parameters where rapid

assessment and robustness are essential.

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