Well log data inversion using radial basis function network Kou-Yuan Huang* ,National Chiao Tung University, Liang-Chi Shen, University of Houston, Li-Sheng Weng, National

Chiao Tung University

Summary

We adopt the radial basis function network (RBF) for well log data

inversion. We propose the 3 layers RBF. Inside RBF, the first layer

is the K-means clustering method and PFS-test. Then the 1-layer

perceptron is replaced by 2-layer perceptron. It can do more

nonlinear mapping. The gradient descent method is used in the

back propagation learning rule at 2-layer perceptron. The input of

the network is the apparent conductivity (Ca) and the output of the

network is the true formation conductivity (Ct). 25 simulated well

log data are used in the training. From experimental results, the

network with 10 input data, first layer with 27 nodes, second layer

with 9 hidden nodes and 10 output nodes can get the smallest

average mean absolute error in the training. After training in the

network, we apply it to do the inversion of the 6 simulated well log

data and 1 real field well log data to get the inverted Ct. Result is

good. It shows that the RBF can do the well log data inversion.

Introduction

The inversion of the well log is a difficult problem to get the

properties of the true formation around the well. The well log

inversion were ever done using the method of least-square error by

Lin et al. (1984), the maximum entropy method by Dyos (1987), and

the neural network with Levenberg-Marquardt (L-M) optimization

algorithm by Martin et al. (2001). They were computationally

intensive for the well logs with high relative dip angle. Goswami et

al. (2004) used the evolution algorithm to invert the well log, but it

was a less efficient method. As compared with these methods, the

neural networks could obtain a faster operational speed and a

robustness result. Powell (1985) proposed the radial basis function

network (RBF), it had computational efficiency. Here we adopt the

RBF to do the inversion of well log data.

We modify the two-layer RBF to the three-layer RBF in the well

log data inversion. There are two steps in the network. The first

step is the K-means clustering algorithm and Pseudo F-statistic

(PFS) test to determine the number of nodes in the first layer. The

second step is a 2-layer perceptron. The 1-layer perceptron is

replaced by 2-layer perceptron. It has a more non-linear mapping.

RBF Network and Learning

Conventional 2-layer RBF Network

Figure 1 is a 2-layer RBF network. It is a supervised training

model (Huang, 2003). First, classify all training samples to I

clusters by unsupervised K-means clustering algorithm (Tou and

Gonzales, 1974). Center and variance of each cluster is

recorded in the first layer.

∑

(1)

where denotes each input sample.

Figure 1: Conventional RBF network.

The first layer is to calculate the statistic distance between input

sample and each cluster center, and then calculate the response of

Gaussian basis function:

[

] (2)

The second layer is the 1-layer perceptron. The output of the first

layer becomes the input of the second layer. Then do the linear

combination in inputs and weights, and get the output of node :

∑

Output of node is ( ) is the activation function. We use

sigmoidal activation function, and the output of perceptron is:

( )

(4)

where is the output of node of the second layer. The error

function E of network:

∑

(5)

where denotes the node number of second layer, denotes the

desired output of node of second layer, and is the actual

output of node of the second layer. The weighting adjustment is

by gradient descent method (Rumelhart et al., 1986).

(6)

where is the learning rate of network, and is the count of

execution. And the formula of network adjustment:

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(( ) ( ) )

( ) ( )=( )

(7)

(8)

To determine the node number of first layer, we use K-means

clustering algorithm and Pseudo F-statistic (PFS) test (Vogel and

Wang, 1979). The PFS is:

(9)

where is the number of samples, is the number of clusters; tr is

the trace of a matrix, i.e., the summation of diagonal elements. Do

K-means clustering algorithm for each K and calculate PFS value.

Find the maximum value of PFS and the corresponding K as the

optimal number of clusters.

Three-layer RBF Network

Figure 2: Three-layer RBF.

To have more nonlinear mapping, we propose three-layer RBF.

Figure 2 is a three-layer RBF network. The first layer is the same

as the conventional RBF in Figure 1. Then it is a 2-layer

perceptron. The weights at the second and the third layer are

adjusted by back-propagation learning rule with gradient descent

method and momentum term (Rumelhart et al., 1986).

The output of first layer is the same as (2):

[

] (10)

where denotes each input sample. Activation function of network

is sigmoidal function, the output of the second layer , and the

output of third layer are:

( ) ∑

∑

(11)

(12)

The formula of weighting adjustment between second and third

layer:

( )

(13)

(14)

The formula of weighting adjustment between first and second

layer:

( (∑

) ( ) )

(∑

) ( )

(15)

(16)

Besides, we adopt the momentum term to consider the effect of

the previous step. Then the final adjustment are:

(17)

(18)

is the learning rate of network, is the momentum

coefficient.

Determination of the Number of Hidden Nodes

From the following experimental results of conventional 2-layer

RBF, the network has the minimum error when we use 10 input

features, 27 nodes in the first layer, and 10 output nodes. We

expand this kind of 2-layer RBF to the three-layer RBF. And we

use the theorem developed by Michandani and Cao (1989) to

determine the number of the hidden nodes in the two-layer

perceptron.

When the inputs of the network are d dimensions and the number

of hidden nodes is H, then the maximum number of linear

separable regions M is:

∑

(20)

If d>H, M=H2 . In the experiment, the input nodes are d=10, the

number of the training patterns is 500. In order to have one pattern

almost in one region, M=500. According to the theorem, we can

get the maximum regions M as follows:

Each region has almost one training pattern, so we may have 8 or

9 hidden nodes (H) in the network.

Training Process

In our experiments, the inverse of apparent resistivity (Ra), which

is called apparent conductivity (Ca), is the input of the network.

And the desired output is the true formation conductivity (Ct). The

transition diagram is shown in Figure 3.

Figure 3: Transition of apparent properties and true formation

properties in the processing.

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There are 31 simulated well log datasets and 1 real field log. We

use the 1st to the 25th well logs for training, and others are for

testing. The depth of simulated well is from 490 to 589.5 feet and

the sample interval is 0.5 feet. Each well log has 200 Ca data and

200 Ct data. We divide the 200 data into every 1, 2, 4, 5, 10, 20, 40,

50, 100, and 200 data as the input features to the networks.

Because we do the inversion, the number of output nodes is the

same as the number of input nodes. We compare the mean absolute

error of each network in the training process to determine the best

network. The mean absolute error (MAE) is defined:

∑ ∑|( )|

(19)

where P is the number of training patterns, K is the number of

output nodes, and and are the desired output and real

output of pattern p at the output node k. The number of iterations is

set to 20,000. is set to 0.6. is set to 0.4. The error threshold for

stopping training is set to 0.002.

Experimental Results

Results of Conventional 2-layer RBF

We use the 2-layer RBF on the well log inversion. The numbers of

input nodes are 1, 2, 4, 5, 10, 20, 40, 50, 100, and 200 respectively.

We have 25 training well logs. The training results are shown in

Table 1. From Table 1, the network has the minimum inversion

error when we use 10 input features, 27 nodes in the first layer, and

10 nodes in output layer. Figure 4 shows 27 is the maximum in

PFS value versus cluster number in 10 input features. After

training, we use the network to do inversion on the 26th-31st well

log data and the results are shown in Table 2. Figure 5 shows the

error MAE versus iteration in 10-27-10 RBF. Figure 6 shows the

inverted Ct and desired Ct of the 26th well log with 10-27-10 RBF.

Network

size

Number of

training patterns

Error MAE at

20,000 iterations

Error MAE of 6 well

log data inversion

1-2426-1 5000 0.018645 0.123119

2-14-2 2500 0.045808 0.071876

4-7-4 1250 0.049006 0.069823

5-44-5 1000 0.032716 0.058754

10-27-10 500 0.031394 0.048003

20-7-20 250 0.048767 0.073768

40-2-40 125 0.164247 0.174520

50-2-50 100 0.160658 0.165190

100-2-100 50 0.185587 0.188436

200-4-200 25 0.277741 0.252023

Table 1: Experimental results of conventional RBF.

Well Log Data Error MAE of well log data inversion

#26 0.051753

#27 0.055537

#28 0.041952

#29 0.040895

#30 0.047587

#31 0.050294

Table 2: Inversion error of RBF (10-27-10).

Figure 4: Result of PFS value versus cluster number at 10 input features.

Figure 5: Mean absolute error vs. iteration in 10-27-10 network learning.

Figure 6: Inverted Ct and desired Ct of the 26th well log with 10-27-10

network.

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Results of Three-layer RBF with Original Features

In the experiments, we adopt the three-layer RBF with 7, 8, 9,

10, 11, and 12 hidden nodes in the 2-layer perceptron respectively.

We have 25 training well logs. The inversion errors in training are

in Table 3. When the number of iteration is set to 20,000, we find

that the 10-27-9-10 network can get the smallest error MAE than

that of the conventional RBF. The error versus iteration in the

training process is shown in Figure 7. We try the hidden nodes

from 12 to 24 in the 2-layer perceptron, but the errors do not

decrease. After training, we use the network to do the 26th-31st

well log data inversion and the results are shown in Table 4. The

inverted real output Ct of the 26th well log with 10-27-9-10

network is shown in Figure 8. Because the MAE error between the

inverted Ct and the desired Ct of the 26th well log is 0.041526 that

is the smallest error in Table 4, the inverted Ct and the desired Ct

on the Figure 8 are very close.

Network size Number of

training

patterns

Error MAE at 20,000

iterations

Error MAE of 6 well log data

inversion

10-27-7-10 500 0.017231 0.050430

10-27-8-10 500 0.017714 0.050313

10-27-9-10 500 0.015523 0.046625

10-27-10-10 500 0.015981 0.048452

10-27-11-10 500 0.019848 0.048173

10-27-12-10 500 0.021564 0.053976

Table 3: Inversion error of each kind of three-layer RBF.

Well Log Data Error MAE of well log data inversion

#26 0.041526

#27 0.059158

#28 0.046744

#29 0.043017

#30 0.046546

#31 0.042763

Table 4: Inversion error of three-layer (10-27-9-10) RBF.

Figure 7: Mean absolute error vs. iteration in 10-27-9-10 network learning.

Figure 8: Inverted Ct and desired Ct of the 26th well log with 10-27-9-10

network.

Result of Real Well Log Data Inversion

We adopt the 10-27-9-10 RBF to do the real well log data

inversion. The well depth is from 5,577.5 to 6,722 feet, and the

sampling interval is 0.5 feet. So, there are total 2,290 input features

from this well. Every 10 data are the inputs. After the previous 25

well log training, the real data are fed to the network for inversion.

Figure 9 shows the output, the inverted Ct.

Figure 9: Result of real well log data inversion.

Conclusions

From experiments we find that the three-layer RBF can get the

less mean absolute error between the actual output and the desired

output. The reason is that the 2-layer perceptron has the property of

more non-linear mapping. The three-layer RBF with 10 input

features, 27 nodes in the first layer, 9 hidden nodes in the second

layer, and 10 output nodes can get the smallest average mean

absolute error on simulated well log data. In the final step, the

network with the smallest error in training is applied to the

inversion of real field log. The result is good. It shows that the

RBF network can do the well log data inversion.

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EDITED REFERENCES

Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2011

SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for

each paper will achieve a high degree of linking to cited sources that appear on the Web.

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