presentasi raphael (final)

8/11/2019 Presentasi Raphael (Final)

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Identification of StratigrFormation Interfaces

Wavelet and Fourier TranRaphael Ar


2/25

Objective interpretation?

Formation boundary

criteria is subjective

Different interpretation for

each interpreter

well log is treated with

signal processing methods

Objective,apparently


3/25

Modification to Fourier transform was implemented for

identification of lithologic boundaries

Short-time Fourier transform

(STFT):Segments signals using

windows

Combination of Wavelet transform

and Fourier transform was studied in

this paper to analyze SP and GR log toidentify stratigraphic interfaces

Wavelet transform (WT):

Uses wavelet functioncontaining both time and

frequency information

simultaneously


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Discrete Fourier Transform

=

/

x = /

Discrete Fourier

transform

Inverse discrete Fourier

transform

The depth in well log can be treated as time inprocessing the SP or GR log as a signal

Transformed data can be used to select frequency bandsto filter SP or GR log

In this paper, Fast Fourier Transform is used to efficiently

calculate discrete Fourier transform of SP or GR log


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Wavelet Transform

Provides varying time and frequency resolutions using

windows of different lengths

Consists of two kernel variables, phase (or location) andscale, by utilizing wavelet function

, 1

1

1 ,

is called wavelet coefficients

is conjugatecomplex of

a and b are scale and p

location variables)

Continuous wavelet tran

Shifting the wavelet func

phase (b) and stretching

wavelet function from th


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Wavelet Transform

Types of wavelet transform depends on wavelet

functions used

Haar function

Daubechies function


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Wavelet Transform

Discrete wavelet transform:

Signal x(t) is decomposed into several lower-resolution components,

approximation (cA) and detail wavelet coefficients (cD)

+

. ( )

()

()

1

2

1

2

1212

is a sorthog

functio

is eqphase

Reconstruction of the original signal

x is obtained by the sum of inverse

WT of approximation (cA) and its

detail (cD)


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Discrete Wavelet Transform

. Inverse WT of detail wavelet coefficient(cD) is used to reconstruct high-frequency

signal RcD

Approximation response (low-pass filtering) and

detail response (high-pass filtering) can be

calculated at the same time

Approximation (cA) represents low-frequency

component of signal x

Detail wavelet coefficient (cD) represents high-frequency

component of signal x


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Wavelet transform method

Well log data

(SP or GR)

Scale distributions

of waveletcoefficients Approximation

coefficients

(cA)

Detail

coefficients

(cD)

Stratigraphic

interfaces

Stratigraphic

interfaces

Identification of

interfaces

Continuous

wavelet transform

Discrete wavelet

transform

clo

c

h

H

a

tin


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Combination of wavelet

transform and Fourier

transform methods

Well log data(SP or GR)

Result of the discrete wavelet transform

Noisydata?

Detail coefficients (cD)

Reconstructed data(RcD)

Spectrum ofreconstructed data(RcDF)

Modified spectrum ofreconstructed data (MRcD

Modified reconstructeddata (MRcD)

Logarithmic distributiondata (LRcD)

Approximationcoefficients (cA)

Approximationcoefficients (cA)

Detailcoefficients (cD)

Approximationcoefficients (cA) Detailcoefficients (cD)

Stratigraphic interfaces

Level 1

Level 2

Level 3

NoYes

Inverse wav

Fourier trans

Filtering

Inverse Four

log transform

Interface identification

Decomposition

Decomposition


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Ideal well log data application of wavelet transform

method

Clean sand

formation

between 7420 and

7440 ft

Shale formation

above 7420 andbelow 7440 ft Ideal data has nonoise.Therefore the method

of combining WT and FT

is not necessary to be

applied


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Ideal well log data - application of wavelet

transform method

Applying continuous

wavelet transform

Wavelet coefficients


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Ideal well log data - application of wavelet

transform method

Applying Discrete

wavelet transform

Detail coefficients cD


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Field data

From co

were fo

units:

A-sand

B-sand

C-sandD-sand

SP log s

curve re

zone 1,

GR logfour for

of noise


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SP log data analysis

Applying continuous

wavelet transform

Four formations ofsand cannot be

clearly observed

Even the three

formation zones

observed in

conventional log

analysis cannot be

observed


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Applying discrete

wavelet transform

Formation

stratigraphy areshown from three

positive-negative

pairs of detailcoefficients cD


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Applying WT and FT

combination

(1)Data was less noisy, cD

from single level

decomposition were

calculated

(2)High-frequency

component is obtained

using inverse WT ofwavelet coefficient cD


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Applying WT and FT

combination

(3)RcD is transformed using

FFT to be analyzed

Dominant frequencies

are between 20-40 nHz

Chosen frequency

band is the center of the

spectrum, 30-31 nHz


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Applying WT and FT

combination

(4)Inverse FFT is applied to

chosen frequency

band, obtaining

modified reconstructed

data MRcDF

(5)Logarithmic transform

data is applied to

reconstructed data

The logarithmic distribution

shows four clear formation

stratigraphic zones


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GR log data analysis

Applying continuous

wavelet transform

To much noise causing

the wavelet coefficientsbecome sparse

Not possible to

determine the formation

stratigraphic from these

sparse scales

GR l d t l i


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Applying discrete

wavelet transform

The number of positive-

negative pairs of detailcoefficients is difficult to

be analyzed

If there is no core data, it

would be a painstaking

process to analyze this

noisy data

GR l d t l i


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Applying combination of

WT and FT

(1)Three level decomposition

is applied to filter out thenoise and obtain detail

coefficients cD

(2)Reconstructed signal RcD

is obtained using inverse

wavelet transform



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WT and FT

(3) Dominant frequency is

chosen after FFT isapplied to RcD

Dominant frequencies

are between 0-20 and

30-45 nHz

(4)The strongest frequency

band is chosen, 8.5-10 nHz



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WT and FT

(5)Inverse FT is applied to the

chosen frequency bands

to reconstruct GR signal

(6)Logarithmic transform is

then applied to the

reconstructed GR signal

The logarithmic distribution

shows four clear formationstratigraphic zones

Concluding remarks


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Concluding remarks

Well log can be treated with signal

processing methods to minimize subjective

results

Result of wavelet transform is similar to

conventional well log interpretation, which

causes difficulties in identifying formation

boundaries

Combination of wavelet transform and

Fourier transform provides objective

identification of boundaries

presentasi raphael (final)

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