Well Log Data Inversion Using Radial Basis Function Network

Kou-Yuan Huang, Li-Sheng WengDepartment of Computer Science

National Chiao Tung UniversityHsinchu, Taiwan

kyhuang@cs.nctu.edu.twand

Liang-Chi ShenDepartment of Electrical & Computer

EngineeringUniversity of Houston

Houston, TX

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Real well log data: Apparent conductivity vs. depth

Inversion to get the true layer effect?

Review of well log data inversion

Lin, Gianzero, and Strickland used the least squares technique, 1984.

Dyos used maximum entropy, 1987.Martin, Chen, Hagiwara, Strickland,

Gianzero, and Hagan used 2-layer neural network, 2001.

Goswami, Mydur, Wu, and Hwliot used a robust technique, 2004.

Huang, Shen, and Chen used higher order perceptron, IEEE IGARSS, 2008.

Review of RBF Powell, 1985, proposed RBF for

multivariate interpolation.

Hush and Horne, 1993, used RBF network for functional approximation.

Haykin, 2009, summarized RBF in Neural Networks book.

Conventional two-layer RBFHush and Horne, 1993

Training in conventional two-layer RBF

Properties of RBFRBF is a supervised training model.

The 1st layer used the K-means clustering algorithm to determine the K nodes.

The activation function of the 2nd layer was linear. f(s)=s. f ’(s)=1.

The 2nd layer used the Widrow-Hoff learning rule.

Output of the 1st layer of RBF

Get mean & variance of each cluster from K-means clustering algorithm.

Cluster number K is pre-assigned.Variance

Output of the 1st layer: response of Gaussian basis function

k

kT

kk

k n groupin

2 )()( 1

xmxmx

Training in the 2nd layerWidrow-Hoff’s learning rule.Error function

Use gradient descent method to adjust weights

f(s)=s. =1

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Modification of two-layer RBF

Training in modified two-layer RBF

Optimal number of nodes in the 1st layer

• We use K-means clustering algorithm & Pseudo F-Statistics (PFS) (Vogel and Wong, 1979) to determine the optimal number of nodes in the 1st layer.

• PFS: n is the pattern number. K is the cluster number.• Select K when PFS is the maximum. K becomes the node

number in the 1st layer.

Perceptron training in the 2nd layer

Activation function at the 2nd layer: sigmoidal Error Function

Delta learning rule (Rumelhart, Hinton, and Williams, 1986): use gradient descent method to adjust weights

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Proposed three-layer RBF

Training in proposed three-layer RBF

Generalized delta learning rule (Rumelhart, Hinton, and Williams, 1986)Adjust weights between the 2nd layer and

the 3rd layer

Adjust weights between the 1st layer and

the 2nd layer,

Adjust weights with momentum term:

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Experiments: System flow in simulation

Apparent resistivity (Ra)

Apparent conductivity (Ca)

True formation resistivity (Rt)

Radial basis function network (RBF)Scale Ca to 0~1 (Ca’)

Desired true formation conductivity (Ct’’)

Re-scale Ct’ to Ct

True formation conductivity (Ct’)

Ct

1

Ra

1

Experiments: on simulated well log data

In the simulation, there are 31 well logs.

Professor Shen at University of Houston worked on theoretical calculation.

Each well log has the apparent conductivity (Ca) as the input, and the true formation conductivity (Ct) as the desired output.

Well logs #1~#25 are for training.

Well logs #26~#31 are for testing.

Simulated well log data: examples

Simulated well log data #7

Simulated well log data #13

Simulated well log data #26

What is the input data length? Output length?

• 200 records on each well log. 25 well logs for training. 6 well logs for testing.

• How many inputs to the RBF is the best?

Cut 200 records into 1, 2, 4, 5, 10, 20, 40, 50, 100, and 200 data, segment by segment, to test the best input data length to RBF model.

• For inversion, the output data length is equal to the input data length in the RBF model.

• In testing, input n data to the RBF model to get the n output data, then input n data of the next segment to get the next n output data, repeatedly.

Example of input data length at well log #13 If each segment (pattern vector) has 10 data, 200

records of each well log are cut into 20 segments (pattern vectors).

Input data length and # of training patterns from 25 training well logs

Input data

lengthNumber of training patterns

1 5000

2 2500

4 1250

5 1000

10 500

20 250

40 125

50 100

100 50

200 25

Optimal cluster number of training patternsExample: for input data length 10

PFS vs. K. For input N=10, the optimal cluster number K is 27.

Optimal cluster number of training patterns in 10 cases

Set up 10 two-layer RBF models. Compare the testing errors of 10 models to select the

optimal RBF model.

N features Training patterns K clusters

1 5000 24262 2500 144 1250 75 1000 44

10 500 2720 250 740 125 250 100 2

100 50 2200 25 4

Experiment: Training in modified two-layer RBF

Parameter setting in the experiment

Parameters in RBF training

Learning rate : 0.6 Momentum coefficient : 0.4 (in 3-layer RBF) Maximum iterations: 20,000

Error threshold: 0.002.

Define mean absolute error (MAE):

P is the pattern number, K is the output nodes.

MAE= 1𝑃𝐾 ∑

𝑝=1

𝑃

∑𝑘=1

𝐾

|𝑑𝑝𝑘− 𝑜𝑝𝑘|

Testing errors at 2-layer RBF models in simulation

10-27-10 RBF model gets the smallest error in testing.

Network size

Number of training patterns

MAE at 20,000 iterations

Average MAE of 6 well log

data inversion

Training CPU Time(H:M:S)

1-2426-1 5000 0.018645 0.123119 01:43:15

2-14-2 2500 0.045808 0.071876 00:22:25

4-7-4 1250 0.049006 0.069823 00:11:18

5-44-5 1000 0.032716 0.058754 00:11:07

10-27-10 500 0.031394 0.048003 00:05:26

20-7-20 250 0.048767 0.073768 00:03:22

40-2-40 125 0.164247 0.174520 00:01:28

50-2-50 100 0.160658 0.165190 00:01:45

100-2-100 50 0.190826 0.185587 00:01:33

200-4-200 25 0.191159 0.277741 00:01:13

Training result: error vs. iterationusing 10-27-10 two-layer RBF

Inversion testing using 10-27-10 two-layer RBF

Inverted Ct of log #26 by network 10-27-10 (MAE= 0.051753).

Inverted Ct of log #27 by network 10-27-10 (MAE= 0.055537).

Inverted Ct of log #28 by network 10-27-10 (MAE= 0.041952).

Inverted Ct of log #29 by network 10-27-10 (MAE= 0.040859).

Inverted Ct of log #30 by network 10-27-10 (MAE= 0.047587).

Inverted Ct of log #31 by network 10-27-10 (MAE= 0.050294).

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Experiment: Training in modified three-layer RBF.

Hidden node number?

Determine the number of hidden nodes in the 2-layer perceptron

On hidden nodes for neural nets (Mirchandani and Cao,1989) It divides space to maximum M regions when input space is d dimensions and there are H hidden nodes.

T: number of training patterns. Each pattern is in one region. From T ≈ M, we can determine H hidden nodes.

Hidden node number and optimal 3-layer RBF

• 10-27-10 2-layer RBF gets the smallest error in testing. We extend it to 10-27-H-10 in the 3-layer RBF. H=?

• For original 10 inputs, the number of training patterns is 500. T=500.

• For a 27-H-10 two-layer perceptron, the number of input nodes is 27.

When d=27, H=9 (=500), we select hidden node number H=9.

• Finally, we get 10-27-9-10 as the optimal 3-layer RBF model.

Training result: error vs. iterationusing 10-27-9-10 three-layer RBF

Inversion testing using 10-27-9-10 three-layer RBF

Inverted Ct of log 26 by network 10-27-9-10 (MAE= 0.041526)

Inverted Ct of log 27 by network 10-27-9-10 (MAE= 0.059158)

Inverted Ct of log 28 by network 10-27-9-10 (MAE= 0.046744)

Inverted Ct of log 29 by network 10-27-9-10 (MAE= 0.043017)

Inverted Ct of log 30 by network 10-27-9-10 (MAE= 0.046546)

Inverted Ct of log 31 by network 10-27-9-10 (MAE= 0.042763)

Testing error of each well log using 10-27-9-10 three-layer RBF model

Average error: 0.046625

Well Log Data MAE of well log data inversion

#26 0.041526

#27 0.059158

#28 0.046744

#29 0.043017

#30 0.046546

#31 0.042763

Average testing error of each three-layer RBF model in simulation

Experiments using RBFs with different number of hidden nodes.

10-27-9-10 get the smallest average error in testing. So it is selected to the real data application.

Network sizeNumber of

training patterns

Error MAE at 20,000

iterations

Training CPU Time(H:M:S)

Average Error MAE of 6 Well

log data inversion

10-27-7-10 500 0.017231 00:31:02 0.05043010-27-8-10 500 0.017714 00:29:40 0.05031310-27-9-10 500 0.015523 00:29:45 0.046625

10-27-10-10 500 0.015981 00:30:37 0.04845210-27-11-10 500 0.019848 00:29:59 0.04817310-27-12-10 500 0.021564 00:29:27 0.053976

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Real well log data: Apparent conductivity vs. depth

Application to real well log data inversion

Real well log data:• Depth from 5,577.5 to 6,772 feet.• Sampling interval 0.5 feet.• Total 2,290 data in one well log.

Select 10-27-9-10 optimal RBF model for real data inversion.

After convergence in training, input 10

real data to the RBF model to get the 10 output data, then input 10 data of the next segment to get the next 10 output data, repeatedly.

Inversion of real well log data: Inverted Ct vs. depth

Outline Introduction

Proposed Methods• Modification of two-layer RBF• Proposed three-layer RBF

Experiments• Simulation using two-layer RBF• Simulation using three-layer RBF• Application to real well log data inversion

Conclusions and Discussion

Conclusions and Discussion We have the modification of 2-layer RBF and propose

3-layer RBF for well log data inversion. 3-layer RBF has better inversion than 2-layer RBF

because more layers can do more nonlinear mapping. In the simulation, the optimal 3-layer model is 10-27-

9-10. It can get the smallest average mean absolute error in the testing.

The trained 10-27-9-10 RBF model is applied to the real well log data inversion. The result is acceptable and good. It shows that the RBF model can work on well log data inversion.

Errors are different at experiments because initial weights are different in the network. But the order or percentage of errors can be for comparison in the RBF performance.

Thank you for your attention.