CHAPTER VI

INVERSION OF RESISTIVITY DATA BY ARTIFICIAL NEURAL

NETWORK (ANN)

6.1 Introduction

Interpretation of the Geophysical data is necessary to understand the concept of

reality behind the earth system and there should be an efficient tool to estimate and

predict the parameters which are relevant to the system. Thus computational method

involves the intelligent network to justify all the non linear problem is necessary. Neural

network can have the power to understand the factors related to the variations of the sub

surface geology. Conventional mathematical analysis is difficult when there are uncertain

and partially defined systems with a degree of vagueness. The use of intelligent systems

to evaluate such complex system is ideal.

Computational applications in the field of Geophysics is more advantageous

nowadays to understand the non linear behavior of the earth. Artificial neural network

(ANN) plays a major role to dig out the mystery of earth science by means of the

previous learned examples. An Artificial Neural Network (ANN) usually called Neural

Network (NN) is a mathematical model or computational model that tries to simulate the

structure and functional aspects of biological neural networks by training. It consists of

interconnected group of artificial neurons and also processes information using a

connectionist approach to computation. The concept of Neural network diagram is shown

in Figure (6.1).

Figure 6.1 Concept of Neural network

Neural networks are non-linear statistical data modeling tools. They can be used

to model complex relationships between inputs and outputs or to find patterns in data.

A computing system is built from a large number of simple processing elements dealing

individually with parts of a large problem. This net may have several layers of processing

elements that are massively interconnected. In this, adaptively adjusted weights are

applied to the inputs and to the connections between the processing elements. There are

many applications of ANN which includes oil and gas exploration, speech and pattern

recognition, financial forecasting and health care cost reduction. Thus ANN is a general-

purpose model with a limited set of variables and is used as a universal functional

approximation. Usually, ANNs are trained by adjusting the values of these connection

weights between the network elements. These networks have self-learning capability and

are fault-tolerant as well as noise-immune and have applications in various fields

(Sreekanth et al, 2009).

A notable feature of these networks is their adaptive nature, whose learning by

example replaces programming in solving problems. This feature makes such

computational models very appealing in application domains where one has little or

incomplete understanding of the problem to be solved but where training data is readily

available. ANNs are now being increasingly recognized in the area of classification and

prediction, where regression model and other related statistical techniques have

traditionally been employed. Introducing the layer of neurons involved in the training is

shown in Figure (6.2 and 6.3). The most widely used learning algorithm in an ANN is the

back propagation algorithm, others are multilayered perception, radial basis function and

kohonen networks. In fact, majority of the network are more closely related to the

traditional, mathematical and statistical models such as non-parametric pattern classifiers,

clustering algorithms, non-linear filters, and statistical regression models. ANNs have

been used for a wide variety of applications where statistical methods are traditionally

employed.

Figure 6.2 A neuron with a single scalar input and no bias appears on the left and

with bias on the right.

Figure 6.3 A neuron with a single R-element vector is shown above

In ANN, the sigmoid function is used as activation function. It can produce

outputs with reasonable discriminating power and its output functions are differentiable,

which is essential for the back propagation of errors (Yegnanarayana, 2005).Its Gaussian

shaped derivatives help to stabilize the network and compensate for over-correction of the

weights (Caudill, 1988). During ANN training, each hidden and output neuron process

inputs by multiplying each input by its weights. The products are summed and processed

using the activation function. The expression for logistic sigmoid function and its

derivative are given below.

The number of neurons in the hidden layer varies as per the requirement of

optimum performance, which could be decided on trial and error basis. Initial weights are

assigned at random in the range suitable for the activation function at neurons. (Jimmy

Stephen et al,2004).In this iterative non-linear feed forward network Back-Propagation

algorithm, uses learning rate (µ) for training the network; if µ is very high, the network

may not meet the desired minimum and if µ is very small, the network takes so much

time to meet the desired minimum. The mentioned algorithm adds the momentum for

decreasing the probability of the error accumulation and also reducing the number of

training.

An adaptive network consists of a structure of nodes and the directional links

through which these nodes are connected. The outputs of the nodes depend on the

parameters associated with these nodes. The change in these parameters is brought about

by a learning rule to minimize the error. The learning techniques can automate this

process and substantially reduce development time and cost while improving

performance. For neural networks, the knowledge is automatically acquired by the back

propagation algorithm and will be a more efficient tool used for the interpretation of

Geophysical data extensively.

Artificial neural networks have been successfully applied to a number of

geophysical problems, including parameter estimation, classification, filtering and

optimization (Poulton et al,1992;Sheen,1997;Vander Baan and Jutten, 2000;Calderon-

Macias et al, 2000; El-QadyandUshijima, 2001). The adoption of the neural network

approach holds several advantages over more conventional modeling techniques,

allowing the effective solution of non-linear problems from complex, incomplete and

noisy data. The computational expense of the inversion using neural networks, being

dependent upon the dimension of the space of unknown parameters rather than the

physical dimensions of the model space, is also comparatively low (Spichak and Popova,

2000).

6.2 Significance of Artificial Neural network

ANN finely tune to tasks other than number crunching; the human brain possesses

many advantages over a digital computer. One can quickly recognize a face, even when

seen from the side in bad lighting in a room full of other objects. One can easily

understand speech, even that of an unknown person in a noisy room. Despite years of

focused research, computers are far from performing at this level. The brain is also

remarkably robust; it does not stop working just because a few cells die. Compare this

with a computer, which will not normally survive any degradation of the central

processing unit( CPU). But perhaps the most fascinating aspect of the brain is that it can

learn. No programming is necessary; one does not need a software upgrade, just because

one wants to learn to ride a bicycle. (Yegnanarayana 2005 )The computations of the

brain are done by a highly interconnected network of neurons, which communicate by

sending electric pulses through the neural wiring consisting of axons, synapses and

dendrites.

In some cases, a neuron is modeled as a switch that receives input from other

neurons and, depending on the total weighted input, is either activated or remains

inactive. The weight, by which an input from another cell is multiplied, corresponds to

the strength of a synapse—the neural contacts between nerve cells. These weights can be

both positive (excitatory) and negative (inhibitory). In the 1960s, it was shown that

networks of such model neurons have properties similar to the brain: they can perform

sophisticated pattern recognition, and they can function even if some of the neurons are

destroyed. The demonstration, in particular by Rosenblatt, that simple networks of such

model neurons called perceptrons could learn from examples stimulated interest in the

field, showed that simple perceptrons could solve only the very limited class of linearly

separable problems, activity in the field diminished. Nonetheless, the error back-

propagation method, which can make fairly complex networks of simple neurons learn

from examples, showed that these networks could solve problems that were not linearly

separable. Net talk, an application of an artificial neural network for machine reading of

text, was one of the first widely known applications, and many followed soon after. In the

field of biology, the exact type of network as in Net talk was also applied to prediction of

protein secondary structure; in fact, some of the best predictors still use essentially the

same method. Another big wave of interest in artificial neural networks started, and led to

a fair deal of hype about magical learning and thinking machines.

Artificial neural networks are inspired by the early models of sensory processing

by the brain. An artificial neural network can be created by simulating a network of

model neurons in a computer. By applying algorithms that mimic the processes of real

neurons, one can make the network to ‘learn’ and to solve many types of problems It

receives input from a number of other units or external sources, weighs each input and

adds them up. If the total input is above a threshold, the output of the unit is one;

otherwise it is zero. Therefore, the output changes from 0 to 1 when the total weighted

sum of inputs is equal to the threshold. The points in input space satisfying this condition

define a scaled hyper plane. In two dimensions, a hyper plane is a line, whereas in three

dimensions, it is a normal plane. Points on one side of the hyper plane are classified as 0

and those on the other side as 1. It means that a classification problem can be solved by a

threshold unit if the two classes separated by a hyper plane, are said to be linearly

separable.

6.3 Back-propagation

Back-propagation learning algorithm works for feed-forward networks with

continuous output. Training starts by setting all the weights in the network to small

random numbers. Now, for each input example, the network gives an output, which starts

randomly. The sum of all these numbers over all training examples is called the total error

of the network. If this number is zero, the network would be perfect, and the smaller the

error, the better the network. By choosing the weights that minimize the total error, one

can obtain the neural network that best solves the problem at hand. This is the same as

linear regression, where the two parameters characterizing the line are chosen such that

the sum of squared differences between the line and the data point is minimal. This can be

done analytically in linear regression, but there is no analytical solution in a feed-forward

neural network with hidden units. In back-propagation, the weights and thresholds are

changed each time an example is presented, such that the error gradually becomes

smaller. This is repeated, often hundreds of times, until the error no longer changes. An

illustration can be found at the Neural Java site, following the link “Multi-layer

Perceptron (with neuron outputs in {0;1})”. In back-propagation, a numerical

optimization technique called gradient descent makes computing particularly simple; the

form of the equations gave rise to the name of this method. There are some learning

parameters called learning rate and momentum that need tuning when using back-

propagation, and there are other problems to consider. For instance, gradient descent is

not guaranteed to find the global minimum of the error, so the result of the training

depends on the initial values of the weights. However, one problem overshadows the

others: that of over-fitting. Over-fitting occurs when the network has too many

parameters to be learned from the number of examples available, that is, when few points

are fitted with a function with too many free parameters. Although this is true for any

method of classification or regression, neural networks seem especially prone to over

parameterization. For instance, a network with 10 hidden units for solving the example

problem would have 221 parameters: 20 weights and a threshold for the 10 hidden units

and 10 weights and a threshold for the output unit. That is too many parameters to be

learned from 100 examples. A network that over fits the training data is unlikely to

generalize well to inputs that are not in the training data. There are many ways to limit

over-fitting apart from simply making small networks, but the most common include

averaging over several networks, regularization and using methods from Bayesian

statistics. To estimate the generalization performance of the neural network, one needs to

test it on independent data, which have not been used to train the network. This is usually

done by cross-validation, where the data set is split into, for example, and ten sets of

equal size. The network is then trained on nine sets and tested on the tenth, and this is

repeated ten times, so all the sets are used for testing. This gives an estimate of the

generalization ability of the network; that is, its ability to classify inputs that it was not

trained on. To get an unbiased estimate, it is very important that the individual sets do

not contain examples that are very similar. Goh (1995) developed back propagation

neural network (BPNN) models to evaluate the shaft resistance of piles driven in cohesive

soil. Teh et al,(1997) developed a BPNN model to estimate the static pile capacity from

the dynamic stress wave data.

In an artificial neural network several processing units are interconnected

according to some topology to accomplish a pattern recognition task. Therefore, the

inputs to a processing unit may come from the outputs of other processing units, and from

external sources. The output of each unit may be given to several units including itself.

The amount of the output of one unit received by another unit depends on the strength of

the connection between the units, and it is reflected in the weight value associated with

the connection link. If there are N units in a given ANN, then at any instant of time each

unit will have a unique activation value and a unique output value. The set of the N

activation values of the network defines the activation state of the network at that instant.

Likewise, the set of the N output values of the network defines the output state of the

network at that instant. Depending on the discrete or continuous nature of the activation

and output values, the state of the network can be described by a discrete or continuous

point in an N-dimensional space. In operation, each unit of an ANN receives inputs from

other connected units and from an external source. A weighted sum of the inputs is

computed at a given instant of time. The activation value determines the actual output

from the output function unit and the output state of the unit. The output values and the

other external inputs in turn determine the activation and output states of the other units.

Activation Dynamics determines the activation values of all the units, and also the

activation state of the network as a function of time. The activation dynamics also

determines the dynamics of the output state of the network. The set of all output states

defines the output state space of the network. Activation dynamics determines the

trajectory of the path of the states in the state space of the network. For a given network,

defined by the units and their interconnections with appropriate weights, the activation

states determine the short term memory function of the network.

Generally, given an external input, the activation dynamics is followed to recall a

pattern stored in a network. In order to store a pattern in a network, it is necessary to

adjust the weights of the connections in the network. The set of all possible weight

vectors define the weight space. When the weights are changing, then the synaptic

dynamics of the network determines the weight vector as a function of time. Synaptic

dynamics is followed to adjust the weights in order to store the given patterns in the

network. The process of adjusting the weights is referred to as learning. Once the learning

process is completed, the final set of weight values corresponds to the long term memory

function of the network. The procedure to incrementally update each of the weights is

called a learning law or learning algorithm. Artificial neural networks are useful only

when the processing units are organized in a suitable manner to accomplish a given

recognition task (Anderson and Rosenfeld 1988). These networks are normally organized

into layers of processing units. The units of a layer are similar in the sense that they all

have the same activation dynamics and output function. Connections can be made either

from the units of one layer to the units of another layer (interlayer connections) or among

the units within the intralayer connections or both interlayer and interlayer connections.

Further, the connections across the layers and among the units within the layer can be

organized either in a feed forward manner or in a feedback manner. In a feedback

network, the same processing unit may be visited more than once.

6.3.1 Feed forward Back propagation Algorithm

Back propagation learning involves propagation of the error backwards from the

output layer to the hidden layers in order to determine the update for the weights leading

to the units in a hidden layer. The error at the output layer itself is computed using the

difference between the desired output and the actual output at each of the output units.

The actual output for a given input training pattern is determined by computing the

outputs of units for each hidden layer in the forward pass of the input data

(Yegnanarayana, 2005). A trained multilayer feed forward neural network is expected to

capture the functional relationship between the input-output pattern pairs in the given

training data. It is implicitly assumed that the mapping function corresponding to the data

is a smooth one. The function approximation interpretation of a single layer feed forward

neural network enables one to view different hidden layers of the network performing

different functions. Feed forward back propagation algorithm is used here to invert the

Vertical Electrical Sounding (VES) Data.

6.4 Learning

Synaptic dynamics is attributed to learning in a biological neural network. The

synaptic weights are adjusted to learn the pattern information in the input samples.

Typically, learning is a slow process, and the samples containing a pattern may have to be

presented to the network several times before the pattern information is captured by the

weights of the network. A large number of samples are normally needed for the network

to learn the pattern implicit in the samples. Pattern information is distributed across all the

weights, and it is difficult to relate the weights directly to the training samples. The only

way to demonstrate the evidence of learning pattern information is that, given another

sample from the same pattern source, the network would classify the new sample into the

pattern class of the earlier trained samples. Another interesting feature of learning is that

the pattern information is slowly acquired by the network from the training samples, and

the training samples themselves are never stored in the network. That shows a preference

to learn from examples, not store the examples themselves.

The adjustment of synaptic weights is represented by a set of learning equations,

which describe the synaptic dynamics of the network. The learning equation describing a

synaptic dynamics model is given as an expression for the first derivative of the synaptic

weight wij connecting the unit j to the unit i. The set of equations for all the weights in the

network determine the trajectory of the weight states in the weight space from a given

initial weight state. Learning laws refer to the specific manner in which the learning

equations are implemented. Depending on the synaptic dynamics model and the manner

of implementation, several learning laws have been proposed.

6.4.1 Requirements of learning laws

a. The learning law should lead to convergence of weights.

b. The learning or training time for capturing the pattern information from samples

should be as small as possible.

c. An on-line learning is preferable than to an off-line learning. That is, the weights

should be adjusted on presentation of each sample containing the pattern information.

d. Learning should use only the local information as far as possible. That is, the change

in the weight on a connecting link between two units should depend on the states of

these two units only. In such a case, it is possible to implement the learning law in

parallel for all the weights, thus speeding up the learning process.

e. Learning should be able to capture complex nonlinear mapping between input-output

pattern pairs, as well as between adjustment patterns in a temporal sequence of

patterns.

f. Learning should be able to capture as many patterns as possible into the network. That

is, the pattern information storage capacity should be as large as possible for a given

network.

6.4.2 Categories of learning

Learning can be viewed as searching through the weight space in a systematic

manner to determine the weight vector that leads to an optimum (minimum or maximum)

value of an objective function. The search depends on the criterion used for learning.

They include minimization of mean squared error, relative entropy, maximum likelihood,

gradient descent, etc. There are several learning laws in use, and new laws are being

proposed to suit a given application and architecture. At first, the learning or weight

adjustment could be supervised or unsupervised. In supervised learning the weight

adjustment is determined based on the deviation of the desired output from the actual

output. Supervised learning may be used for structural learning or for temporal learning.

Structural learning is concerned with capturing in the weights the relationships between

the given input-output pattern pairs. Temporal learning is concerned with capturing in the

weights the relationship between neighboring patterns in a sequence of patterns.

Unsupervised learning discovers features in a given set of patterns, and organizes

the patterns accordingly. There is no externally specified desired output in the case.

Unsupervised learning uses mostly local information to update the weights. The local

information consists of signal or activation values of the units at either end of the

connection for which the weight update is being made.

Learning methods may be off-line or on-line. In an off-line learning all the given

patterns are used together to determine the weights. On the other hand, in an on-line

learning the information in each new pattern is incorporated into the network by

incrementally adjusting the weights. Thus an on-line learning allows the neural network

to update the information continuously. In practice, the training patterns can be

considered as samples of random processes. Accordingly, the activation and output states

could also be considered as samples of random processes. Randomness in the output state

could also result if the output function is implemented in a probabilistic manner rather

than in a deterministic manner. These input, activation and output variables may also be

viewed as fuzzy quantities instead of crisp quantities. Thus we can view the learning

process as deterministic or stochastic or fuzzy or a combination of these characteristics.

Finally, in the implementation of the learning methods, the variables may be

discrete or continues. Likewise the update of weight values may be in discrete steps or in

continuous time. All these factors influence not only the convergence of weights, but also

the ability of the network to learn from the training samples. Yegnananarayana, (2005).

Neural network is also gaining importance in liquefaction susceptibility evaluation of

foundation soils (Goh 1995, Hanna et al, 2007).

6.5 Application of ANN on interpreting Vertical Electrical Sounding Data

Computational neural networks provide much more than just a set of novel, and

valuable data driven mathematical tools , with respect to geophysical analysis and

modeling.These tools provide an appropriate frame work for re-engineering many well

established spatial analysis and environmental modeling techniques to meet the large

scale data processing. Application of ANN models to spatial data set holds the potential

for fundamental advances in empirical understanding across a broad spectrum of sub-

surface features of the study area.

The study of 1D resistivity inversion procedure using the ANN system was carried

out because the procedure works well for the observed data at Abishekapatti study area.

Best training performance can be achieved after the iterations has been successfully

completed the goal using the number of epochs for the synthetic data. Whenever the goal

has been achieved in a particular number of epochs, the training stops and the plot

showing the performance and training state, synthetic and trained and the regression plot

data for the respective VES 2, 6, 8,14, as shown in Figure( 6.4 to 6.12). The test using

trained data indicates that the ANN system can converge to the target rapidly and

accurately. This inversion thus represents the layer parameters of subsurface geology like,

thickness and resistivity. A successful interpretation was produced only after the training

and determines the model parameters with the trained data. The results of interpretation of

the near-surface features by means of this ANN technique is satisfactory and are more

efficient and the error percentage before and after training is less than 11% (Table

6.1).Here, the adaptability of ANN in direct parameter estimated from VES data on hard

rock area involves a variety of curve types with a wide range of model parameter values.

Fig

ure

6.4

S

yn

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ure

6.5

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Fig

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6.6

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sta

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f V

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6.7

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2

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or

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T=

Th

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1

VE

S 2

ρ ρρρ

1=

10

.38

ρ ρρρ2

=6

4.6

88

ρ ρρρ3

=7

66.2

T1

=1

5

T2

=1

4.4

41

2

10

.71

12

2

VE

S 6

ρ ρρρ

1=

18

.362

9

ρ ρρρ2

=9

4.8

32

9

ρ ρρρ3

=7

96

T1

=1

4

T2

=1

5

11

.43

3

VE

S 8

ρ ρρρ

1=

7.6

91

5

ρ ρρρ2

=6

1.9

ρ ρρρ3

=2

65

T1

=1

8

T2

=1

7

10

.4

4

VE

S 1

4

ρ ρρρ1

=2

5.1

60

4

ρ ρρρ2

=5

1.3

09

6

ρ ρρρ3

=7

52

T1

=2

9.0

9

T2

=2

8.0

0.1

73

3

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6.1

3

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4

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6

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6

.19

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ure

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3

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In

terp

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ay

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od

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f fi

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=T

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Err

or

per

cen

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1

S3

ρ1

=1

7.4

ρ2

=1

.2

ρ3

=0

.56

5

T1

=3

.6,

T2

=1.5

7

1.3

63

3

2

S7

ρ1

=1

7.4

ρ2

=3

.2

ρ3

=0

.56

50

T1

=1

.0

T2

=0

.36

1.2

39

4

3

S11

ρ1

=4

.18

69

ρ2

=5

0.1

1

ρ3

=7

51

.73

T1

=5

.6,

T2

=4

5.6

4.0

01

3

4

S13

ρ1

=2

7.1

ρ2

=8

4.1

ρ3

=7

83

T1

=2

3,

T2

=1

77

6.4

04

6

Similarly, the study of 1D resistivity inversion procedure using the ANN system

was carried out at Thuthukudi study area for the sounding data S1 to S20. Best training

performance can be achieved after the iterations has been successfully completed the goal

using the number of epochs for the synthetic data. Whenever the goal has been achieved

in a particular number of epochs, the training stops. The plot showing the performance

and training state and also the regression of the VES of S3 - S13 data as shown in Figure

(6.13 to 6.20) for the respective VES of S3 - S13. The test using trained data indicates that

the ANN system can converge to the target rapidly and accurately. The synthetic and

ANN trained synthetic data along with layer model is plotted and. This inversion thus

represents the layer parameters of subsurface geology. The interpretation was produced

with the trained data. The results of interpretation were satisfactory and efficient and

show less than 6% (Table 6.2) error percentage before and after training. The present

approach of two step net work parameter estimation performs effectively and once

trained, the model parameters are calculated fast on simulating the net work with saved

weights and bias on new examples. The network also achieved the generalization

capability but within the limits of the presented model parameters in the study area .This

will help to a greater extent for onsite interpretation of VES data.