Prediction of p-wave velocity and anisotropic property of rock using artificial neural network technique

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1 Journal of Scientific & Industrial Research Vol. 63, January 2004, pp Prediction of velocity and anisotropic property of rock using artificial neural network technique T N Singh*, R Kanchan**, K Saigal** and A K Verma** *Department of Earth Science, Indian Institute of Technology, Powai, Bombay Received: 30 June 2003; accepted: 03 November 2003 Physico-mechanical properties of rocks are significant in all operational parts in mining activities, starting from exploration to final dispatch of material. Compressional wave velocity ( velocity) and anisotropic behaviour of the rocks are two such properties for understanding the rock response behaviour under stress conditions. They also influence the breakage mechanism of rock. There are some known methods to determine the velocity and in in situ as well as in the laboratory. These methods are cumbersome and time consuming. In the present investigation, artificial neural network (ANN) technique is used for the prediction of velocity and, taking chemical composition and other physico-mechanical properties of rocks as input parameters. Cross-validation technique termed as leaving-one-out is used, as the numbers of data sets are limited in number. Network with six input neurons, one hidden layer with five neurons, and two output neurons is designed for sandstone. Similar network for marble with four hidden neurons is also designed. To deal with the problem of overfitting of data, Bayesian regulation is used and network is trained with 1500 training epochs. The coefficients of correlation among the predicted and observed values are high and encouraging and mean absolute percentage error (MAPE) obtained are very low. Keywords: velocity, Anisotropic property, Rocks, Artificial neural network technique Introduction The physico-mechanical properties are important ingredients for planning and design of mining and civil excavations. The present in the rock mass adversely influences the strength of the rock mass. The long-term stability can be only achieved once the compressional wave velocity and anisotropic behaviour of rock mass are completely understood. The compressional wave velocity depends on the density, chemical composition, and hardness of the rock material. It is difficult to determine compressional wave velocity in the field and the laboratory. The anisotropic behaviour of rock controls the bearing capacity of the rock mass. When field and laboratory geophysical data are considered together with the data of geology and geochemistry, it is 1 possible to obtain much information as to the rocks and minerals that are likely to occur in the different deeper layers of the earth. Hence, it is necessary to study and understand the physical properties of the rocks and minerals of which the layers of the earth are composed. ** Author for correspondence Department of Mining Engineering, Institute of Technology, Banaras Hindu University, Varanasi Laboratory measurements of the elastic constants or the elastic wave in the rocks are needed for interpretation of seismic velocities. Numerous measurements of elastic wave velocities have been made in laboratories around the world 1. Measurement of wave velocities in rocks as well as in many other materials is available in literature. The relation of the seismic velocities in rocks of the western region of the central Asia to density and other physical parameters is discussed by Yudborovsky and Vilenskaya 2. The advantages and disadvantages of the static and dynamic methods have been discussed by Volarovich and Fan 3, who suggested the use of field seismic observations and the data obtained by static methods in geodynamics. Aveline et al. 4 have found in granite from the Sidobre massif. Studies, both in the laboratory and the field, revealed some, which is said to be due to microfissuration. These studies revealed the lower velocity in weathered granite, as compared to fresh granite. Berezkin and Mikhaylov 5 have found linear correlation among density and elastic wave velocities in rocks of the central and eastern region of the Russian platform.

2 SINGH et al.: PREDICTION OF P-WAVE VELOCITY & ANISOTROPIC PROPERTY OF ROCK 33 Artificial neural network (ANN) is the simulation of a biological brain. Neural network has the ability to learn from the pattern acquainted before. Once the network has been trained with sufficient number of sample data sets, it can make predictions on the basis of its previous learning about the output related to new input data set of similar pattern. Due to its multidisciplinary nature, ANN is becoming popular among the researchers, planners, and designers as an effective tool for the accomplishment of their work. Therefore, artificial neural network has been successfully used in industrial area as well. Maulenkamp and Grima 6 have developed a model by which uniaxial compressive strength can be predicted from Equotip hardness. They also found that the uniaxial compressive strength of the rock sample is more accurate than the statistic predictions. It is indicated by the consistency of the correlation coefficient for the different test set. Yang and Zhang 7 have made analysis for the point load testing with artificial neural network. Cai and Zhao 8 have used ANN for tunneling design and optimal selection of the rock support measure and to ensure the stability of the tunnel. These applications show that neural network model has superiority in solving problems in which many parameters influence the process and results, when process and results are not fully understood and where historical or experimental data are available. Prediction of the physical properties of the rock is of similar nature and can be more accurately predicted than the methods of correlation. Study aims at predicting the velocity and by artificial neural network model, having higher prediction accuracy, taking physicomechanical properties and chemical composition as inputs. Data Set As already been stated that present investigation aims at predicting the elastic property of the rocks ( velocity and ), taking physicomechanical properties and chemical composition of the rocks as inputs. It is, however, uneconomical to obtain all the parameters because they are expensive and time consuming. On the other hand, some of the parameters are strongly correlated. 9 Hence, it is not important to use all the variables as input parameters. Following parameters have been taken as input parameters for the networks: Parameters Physico-mechanical properties: Chemical composition (): Variables Compressive strength, density, hardness For sandstone: quartz, feldspar, and iron-oxide For marble: calcite, rock. fragments, and iron- oxide Thus, inall six parameters have been taken as input parameters for both the networks. The velocity and have been taken as output parameters for the networks. All the input and output parameters were scaled between 0 and 1. This was done to utilize the most sensitive part of neuron and since output neuron being sigmoid can only give output between 0 and 1 the scaling of output parameter was necessary. Scaled value = max. value unscaled value/max. value min. value Artificial Neural Network Neural network is a branch of the Artificial Intelligence, other than case based reasoning, Expert Systems, and Genetic Algorithms. The Classical Statistics, Fuzzy Logic and Chaos theory are also considered to be related fields. An artificial neural network is a highly interconnected structure that consists of many simple processing elements (called neurons) capable of performing massively parallel computation for data processing and knowledge representation. The paradigms in this field are based on direct modeling of the human neuronal system 10. A neural network can be considered as a black box that is able to predict an output pattern when it recognizes a given input pattern. The neural network must first be trained by having it process a large number of input patterns and showing what sort of output resulted from each input pattern. Once trained, the neural network is able to recognize similarities when presented with a new input pattern resulting from a predicted output pattern. Neural networks are able to detect similarities in inputs, even though a particular input may never have been seen previously. This property allows for excellent interpolation capabilities, especially when the input data is noisy (not exact). Neural networks may be used as direct substitutes for auto-correlation, multi-variable regression, linear regression, trigonometric, and other statistical analysis and techniques.

3 34 J SCI IND RES VOL 63 JANUARY 2004 When a data stream is analyzed using a neural network, it is possible to detect important predictive patterns that were not previously apparent to a nonexpert. Thus the neural network can act as an expert. Particular network can be defined, using three fundamental components: transfer function, network architecture, and learning law 11. One has to define these components, depending upon the problem to be solved. Training a Network with Back-Propagation A network needs first to be trained before interpreting new information. Several different algorithms are available for training of neural networks, but the back-propagation algorithm is the most versatile and robust technique, as it provides the most efficient learning procedure for multilayer neural networks. Also the fact that back-propagation algorithms are especially capable to solve predicting problems, makes them popular 6.The feed-forward back-propagation neural network always consists of at least three layers: input layer, hidden layer and output layer. Each layer consists of many elementary processing units, neurons, and each unit is connected to each unit in the next layer through weights, i.e., neurons in the input layer will send its output as input for neurons in the hidden layer and similar is the connection between hidden and output layer. Number of hidden layers and neurons in the hidden layer changes according to the problem to be solved. Number of input and output neurons is same as the number of input and output variables. To differentiate between the different processing units, values called biases are introduced in the transfer functions. These biases are referred to as the temperature of a neuron. Except for the input layer, all neurons in the back- propagation network are associated with a bias neuron and a transfer function. The bias is much like a weight, except that it has a constant input of 1, while the transfer function filters the summed signals received from this neuron. These transfer functions are designed to map neurons or layers net output to its actual output and they are simple step functions, either linear or non-linear. The application of these transfer functions depends on the purpose of the neural network. The output layer produces the computed output vectors corresponding to the solution. During training of the network, data are processed through the network, until it reach the output layer (forward pass). In this layer the output is compared to the measured values (the true output). The difference or error between both is processed back through the network (backward pass), updating the individual weights of the connections and the biases of the individual neurons. The input and output data are mostly represented as vectors called training pairs. The process, as mentioned above, is repeated for all the training pairs in the data set, until the network error converged to a threshold minimum defined by a corresponding cost function; usually the root mean squared error (RMS) or summed squared error (SSE). In Figure 1, the j th neuron is connected with a number of inputs: X i = (x 1, x 2 x 3 x n ) The net input values in the hidden layer will be, Net j = n i=1 X i W ij + θ j, where X i = Input units, W ij = Weight on the connection of i th input and j th neuron, θ j = Bias neuron (optional), and n = Number of input units. Thus the net output from hidden layer is calculated using a logarithmic sigmoid function O j = f (Net j ) = 1 / 1+ e - (Netj+θj). The total input to the l th unit is, Net l = n k=1 W kl i k + θ l, where θ l = Bias neuron, and W kl = Weight between k th neuron and l th output. Thus the total output from l th unit will be, O l = f (Net l ). In the learning process the network is presented with a pair of patterns, an input pattern and a Figure 1 Back-propagation neural network

4 SINGH et al.: PREDICTION OF P-WAVE VELOCITY & ANISOTROPIC PROPERTY OF ROCK 35 corresponding desired output pattern. The network computes its own output pattern using its (mostly incorrect) weights and thresholds. Now, the actual output is compared with the desired output. Hence, the error at any output in layer W k is: e l = t l O l, where t l is the desired output, and O l is the actual output. The total error function is given by, E = 0.5 n l=1 (t l O l ) 2. Training of the network is basically a process of arriving at an optimum weight space of the network. The descent down error surface is made using the following rule: W jk = -η (δe/δw jk ), where η is the learning rate parameter, and E is the error function. The update of weights for the (n +1) th pattern is given as: W jk (n+1) = W jk (n) + W jk (n). Similar logic applies to the connections between the hidden and output layers. This procedure is repeated with each pattern pair of training exemplar assigned for training the network. Each pass through all the training patterns is called a cycle or epoch. The process is then repeated as many epochs are needed until the error within the user specified goal is reached successfully. This quantity is the measure of how the network has learned. Network Architecture Feed-forward network is adopted here as this architecture is reported to be suitable for problem based on pattern identification. Pattern matching is basically an input/output mapping problem. The closer the mapping better is the performance of network. In the small sample, as we are analyzing the data here, a cross-validation technique termed leaving-oneout, is more appropriate 12. Out of 53 data sets, 52 were taken to train the network and tested on remaining one data set. The procedure is repeated 15- times, leaving one observation randomly chosen out at a time. In the experiment the first observation was predicted, using the outcome of an analysis based on the observations 2, 3, 53, and 53 rd observation was predicted from the observations 1, 2, 3, 52. This method is advantageous as it uses nearly entire data sets for training the network. For both the networks used for sandstone and marble, this cross-validation technique was used. For both the networks, input layer consists of 6 neurons and output layer consists of 2 neurons. Number of hidden layers was decided by training and predicting the training data and testing data by varying the number of hidden layers and neurons in the hidden layer. A suitable configuration has to be chosen for the best performance of the network. Out of the different configurations tested, single hidden layer with 5 hidden neurons produced the best result for sandstone and single hidden layer with 4 neurons for marble. Hence, the final configuration chosen for the network used for sandstone was: 6 input neurons, 1 hidden layer with 5 hidden neurons, and 2 output neurons. The final configuration of network used for marble was: 10 input neurons, single hidden layer with 4 neurons, and 2 output neurons. Suitable numbers of epochs have to be assigned to overcome the problem of over fitting and under fitting of data. In the present paper, to deal with the above mentioned problem, Bayesian regulation 13 was used. Bayesian regulation is an automated regulation the use of which removes the danger of overfitting, as it never lets the data to suffer from overfitting. This eliminates the guess work required in determining the optimum number of epochs of the network. In the network the learning rate assigned was 10-2, number of training epochs given were 1500, and error goal was set to be Results and Discussion To demonstrate the performance of the network the mean absolute percentage error (MAPE) and coefficient of correlation between the predicted and observed values are taken as the performance measures. The prediction was based on the input data sets discussed earlier. For sandstone, observed and predicted values of velocity and, along with the percentage error, are given in Table 1. Using 1 hidden layer with 5 neurons did training for the network. As we have used bayesian regulation 13, there was no danger of overfitting of problems, hence the network was trained with 1500 training epochs. The correlation coefficients for the relationship between predicted and observed values are as high as and for and velocity respectively (Figure 2a-b)

5 36 J SCI IND RES VOL 63 JANUARY 2004 Compressive strength (kgf/cm 2 ) Table 1 Observed and predicted values of velocity and with percentage error for sandstone Hardness Quartz Feldspar Iron-oxide Density (g/cc) Observed Observes Error for Error for ppv value R 2 = Observed value Value 2300 R 2 = Observbed Value Figure 2a Correlation between predicted and observed value of for sandstone Mean absolute percentage errors (MAPE) for these variables are and 0.115, respectively. Figure 2c shows the performance of the network during training process. number of parameters on which output is depending are 18, which is more than 6 (input parameters given for this network), which shows that all the parameters were not included. Figure 2b Correlation among predicted and observed value of velocity for sandstone For marble, training was done using single hidden layer and 4 hidden neurons. As in this network also Bayesian regulation 13 was used the network was trained with 1500 training epochs. The observed and predicted values of velocity and, along with the percentage error, are given in Table 2. The correlation coefficients and training performance are shown in Figure 3a-b. Coefficients of correlation

6 SINGH et al.: PREDICTION OF P-WAVE VELOCITY & ANISOTROPIC PROPERTY OF ROCK 37 Figure 2c Performance graph of neural network used for sandstone Value R 2 = Observed Value Figure 3a Correlation between predicted and observed value of for marble Predited Value R 2 = Observed Value Figure 3b Correlation between predicted and observed value of velocity, for marble between predicted and observed values are and 0.687, respectively for and velocity, which are not as high as observed in prediction of sandstone, but it shows good relationship between predicted and observed values. Figure 3c Performance graph of neural network used for marble MAPE for these variables are 4.98 and 0.44, respectively. It can be seen in the Figure 3c that the network depends on 21 parameters. In the present case, only six important parameters were included. This may be the reason of moderate relation of predicted and observed values in the marble. After observing MAPE, coefficient of correlation and number of predicted parameters, on which output is depending for both the networks, it can be said that prediction made for sandstone is closer to the observed values than for marble. Conclusions Using Bayesian regulation and optimum number of neurons in the hidden layer the mean absolute percentage errors (MAPE) for and velocity are: and 0.115, respectively, for sandstone and 4.98 and 0.44 for marble, respectively. The corresponding coefficients of correlation are: and for sandstone and and for marble. Considering the complexity of the relationship among the inputs and outputs the results obtained are much more encouraging. Since neural network can learn new patterns that are not previously available on the training data sets, and as they can update knowledge over time as long as more training data sets are presented, and process information in parallel way, they result in a greater degree of accuracy, robust and fault tolerance than any other analysis techniques.

7 38 J SCI IND RES VOL 63 JANUARY 2004 Compressive strength (kgf/cm 2 ) Table 2 Observed and predicted values of velocity and with percentage error for marble Hardness Calcite Rock Iron-oxide fragment Density (g/cc ) Observed Observed Error for Error for ppv References 1 Balakrishna S & Ramana Y V, Integrated studies of the elastic properties of some Indian rocks, Geophysical monograph12, American geophysical union (1968), Yudborovsky I & Vilenskaya S M, Some results of study of the elastic properties of rocks of western central Asia, Akad Nauk Turkmen SSR Ser Fiz-Tekh Khim Geol Nauk, 3 (1962) Volarovich M P & Vey-tsin Fan, Investigation of the elastic properties of rocks by static and dynamic methods at high hydrostatic pressure, Akad Nauk SSSR Ins Fiz Zemli Tr, 23(190) (1962) Aveline M, Experimental results on the relation between micro-fissuration and speed of propagation of ultrasounds in the granites of Sidbore, Sci. Terre, 9(4) (1964) Berezkin V M & Mikhaylov I N, On the correlational relationship between the density of rocks and their velocities of elastic wave propagation for the central and eastern regions of Russian platform, Geofiz, Razvedka, 16 (1964) Maulenkamp F & Grima, M A, Application of neural networks for the prediction of the unconfined compressive strength (UCS) from Equotip hardness, Int J Rock Mech. Min Sci, 36 (1999) Yang Y & Zhang Q, Analysis for the results of point load testing with artificial neural network, Comput Methods Adv Geomech, China, (1997) Cai J G & Zhano J, Use of neural networks in rock tunneling, Proc. Computer Methods and Ava in Geomech, China (1997) Hogstrom K, A study on strength parameter for aggregate from south western Sweden rocks, Res Rep (Chamer Univ of Technology, Goteborg, Sweden) Kosko B, Neural networks and fuzzy systems: a dynamical systems approach to machine intelligence (Prentice Hall of India, New Delhi) Simpson P K, Artificial neural system-foundation, paradigm, application and implementation (Pergamon Press, New York) Weiss S M & Kapouleas, I An empirical comparison of pattern recognition, neural sets and machine learning classification methods. 11 th Int Joint Conf on Arti Intell, Kaufmann (1989) MacKay D J C, Bayesian Interpolation, Neural Computation, 4, no.3 (1992)