Iranian Journal of Oil & Gas Science and Technology
,
Vol. 4 (2015), No. 2, pp. 2739
http://ijogst.put.ac.ir
An Improvement in Temporal Resolution of Seismic Da
ta Using
Logarithmic Timefrequency Transform Method
Amin Roshandel Kahoo
*
and Saman Gholtashi
1
School of Mining, Petroleum and Geophysics Enginee
ring, University of Shahrood, Shahrood, Iran
Received:
December 05, 2014
; revised:
February 18, 2015
; accepted:
March 14, 2015
Abstract
The improvement in the temporal resolution of seism
ic data is a critical issue in hydrocarbon
exploration. It is important for obtaining more det
ailed structural and stratigraphic information. Man
y
methods have been introduced to improve the vertica
l resolution of reflection seismic data. Each
method has advantages and disadvantages which are d
ue to the assumptions and theories governing
their issues.
In this paper, we improve the temporal resolution o
f reflection seismic data using the
logarithmic timefrequency transform method. This m
ethod has minimum userdefined parameters.
The algorithm uses valuable properties of both the
timefrequency transform and the cepstrum to
extend the frequency band at each translation of th
e spectral decomposing window. In this method,
the displacement of amplitude spectrum by its logar
ithm is the basic idea of the algorithm. We tested
the mentioned algorithm on both synthetic and real
data. The results of the both tests show that the
introduced method can increase the temporal resolut
ion of seismic data.
Keywords:
Seismic Temporal Resolution, Timefrequency Transfo
rm, Logarithmic Method,
Enhancing Temporal Resolution
1. Introduction
There are two types of resolutions in surface refle
ction seismic data, namely horizontal resolution an
d
vertical resolution. The vertical or temporal resol
ution is expressed by the tuning thickness and the
horizontal or spatial resolution is expressed by th
e Fresnel zone (Badley, 1985). The improvement in
the temporal resolution of seismic data is a critic
al subject in hydrocarbon exploration and
characterizing thinlayered hydrocarbon reservoirs.
It is used for obtaining more detailed structural
and stratigraphic information.
Tuning thickness is defined as a quarter of the dom
inant wavelength at the position of the target laye
r
(Sheriff and Geldart, 1995). The tuning thickness i
s related to the interval velocity of target layer
and
dominant frequency of the traveling wave at the dep
th of the target layer. Therefore, the increase in
the dominant frequency of seismic data can help to
improve the temporal resolution.
Many methods have been introduced to increase the v
ertical resolution of reflection seismic data.
Inverse Qfilter (Wang, 2008), different deconvolut
ion methods (Yilmaz and Doherty, 2001) and
timevariant spectral whitening (Yilmaz and Doherty
, 2001) are the basic methods of the resolution
*
Corresponding Author:
Email: roshandel@shahroodut.ac.ir
28
Iranian Journal of Oil & Gas Science and Technolog
y,
Vol. 4 (2015), No. 2
improvement. In the deconvolution procedure, the ba
nd limited seismic source signature is
compressed by various methods to increase the frequ
ency band of seismic source wavelet.
The wavelet transform and timefrequency representa
tion are the basis of many methods of vertical
resolution improvement to seismic data (Matos and M
arfurt, 2014; Sajid and Ghosh, 2014; Shang and
Caldwell, 2003; Zhou et al., 2014). Shang and Caldw
ell (2003) improved the bandwidth of seismic
data based on highorder cumulant wavelet analysis.
Matos and Marfurt (2014) broadened the seismic
trace spectrum by creating a high resolution seismi
c trace guided by the complex continuous wavelet
transform ridges detected along the scales. Zhou et
al. (2014) proposed an improved timefrequency
spectral modeling deconvolution method to enhance t
he seismic temporal resolution.
Cepstrum analysis is one of the mathematical tools
frequently used in seismic data processing. Herra
and van der Baan (2012) applied the cepstrum theory
to estimate the seismic wavelet. In this paper,
the cepstrum theory (Oppenheim et al., 1997) was us
ed to improve the temporal resolution of
reflection seismic data. The inverse Fourier transf
orm of the logarithm of the amplitude spectrum of a
signal is named the cepstrum (Sajid and Ghosh, 2014
). Herein, the cepstrum was extended to time
frequency representation. The mentioned algorithm i
s applied to synthetic and real seismic data.
2. Methodology
First, the method of resolution improving based on
cepstrum in Fourier domain is introduced. This
method consists of three steps.
In the first step, the time domain signal,
x
(
t
), is transformed to
frequency domain (
f
) by Fourier transform formula (Proakis and Manolak
is, 2007) as given in
Equation 1. Then, the amplitude and phase spectrum
are calculated from the Fourier transform of
signal as denoted in Equation 2.
( )
( )
j t
X f
x t e
dt
ω
+∞
−
−∞
=
∫
(1)
(
)
(
)
( )
( )
Amplitude Spectrum
Phase Spectrum
A f
X f
f
X f
=
=
= Φ
=
∡
(2)
where,
(
)
x t
is the time domain signal,
(
)
X f
is the Fourier transform of the time domain signal
, and
(
)
A f
and
(
)
f
Φ
are the amplitude and phase spectrum of time domai
n signal respectively. In the
next step, the amplitude spectrum of the signal is
extended by replacing it with its logarithm. The
phase spectrum of the signal remains unchanged in t
his procedure.
In the third step, the minimum value of the logarit
hmic amplitude spectrum subtracted from all
spectral values to make the logarithmic amplitude s
pectrum positive. In order to make the total energy
of logarithmic amplitude spectrum equal before and
after the whitening, the logarithmic amplitude
spectrum is normalized by its total energy (Sajid a
nd Ghosh, 2014). The high resolution seismic trace
can be reconstructed by using the normalized logari
thmic amplitude spectrum and the unchanged
phase spectrum.
The mentioned algorithm was tested on a 15 Hz Ricke
r wavelet (Sheriff and Geldart, 1995) and the
results are shown in Figure 1. Figure 1 (a, b, c) s
hows the time domain original Ricker wavelet and it
s
amplitude and phase spectrum respectively. The norm
alized logarithmic amplitude spectrum and the
original phase spectrum are shown in Figure 1 (e, f
). The inverse Fourier transform of the modified
amplitude and the original phase spectra was calcul
ated to gain the improved resolution seismic trace
A.
Roshandel Kahoo
which is shown in Figure 1 (d). Comparing Figures 1
(a) and (d)
can improve the temporal resolution of seismic trac
e. Since the amplitude spectrum can be changed in
the
above mentioned process, the obtained wavelet canno
t
However, the high resolution wavelet is also close
to the Ricker wavelet and increasing the resolution
of the data is more important than this.
Figure 1
(a)
Input wavelet: 15 Hz Ricker wavelet in time domain
and its (b) amplitude and (c) phase spectrum. (d)
Reconstructed high resolution wavelet from (e, f) t
he normalized logarithmic amplitude spectrum and th
e
original phase spectrum of input wavelet.
Because o
f the Fourier transform limitations in analyzing no
n
the Fourier transform
There are various types of time
transform
(STFT)
1946). The
STFT of a time domain signal,
(
)
( )
,
X t f
x
g
t e
d
τ
τ
τ
+∞
−∞
=
−
∫
where,
( )
g t
is a Gaussian window and
signal. In general, the
reads:
(
)
(
[
(
)
1
,
real
,
imag
,
imag
,
,
tan
real
,
A t f
X t f
X t f
t f
−
=
+
Φ
=
where,
(
)
,
A t f
representation of time domain signal.
both Fourier and STFT transform
representation, the Gaussian window with a length e
qual to one quarter of the signal length
The
Gaussian window is shifted by one sample along the
time axis.
Roshandel Kahoo
and S. Gholtashi /
An Improvement in Temporal Resolution of
which is shown in Figure 1 (d). Comparing Figures 1
(a) and (d)
can improve the temporal resolution of seismic trac
e. Since the amplitude spectrum can be changed in
above mentioned process, the obtained wavelet canno
t
However, the high resolution wavelet is also close
to the Ricker wavelet and increasing the resolution
of the data is more important than this.
Input wavelet: 15 Hz Ricker wavelet in time domain
and its (b) amplitude and (c) phase spectrum. (d)
Reconstructed high resolution wavelet from (e, f) t
he normalized logarithmic amplitude spectrum and th
e
original phase spectrum of input wavelet.
f the Fourier transform limitations in analyzing no
n
the Fourier transform
was replaced with time
frequency transform in
There are various types of time
frequency
transforms
(STFT)
is used herein
which is the usual and
STFT of a time domain signal,
( )
x t
, can be calculated as
(
)
2
i
f
X t f
x
g
t e
d
π τ
τ
τ
τ
−
=
−
is a Gaussian window and
(
)
,
X t f
is the time
signal. In general, the
(
)
,
X t f
is a complex value and can be represented by amplit
ude and phase as
(
))
]
(
)
(
)
[
]
(
)
(
)
(
)
(
)
2
2
,
real
,
imag
,
imag
,
real
,
A t f
X t f
X t f
X t f
X t f
=
+
and
(
)
,
t f
Φ
are the amplitude and phase spectrum of the time
representation of time domain signal.
T
he time domain Ricker wavelet and its amplitude spe
ctrum of
both Fourier and STFT transform
s are show
n in Figure 2
representation, the Gaussian window with a length e
qual to one quarter of the signal length
Gaussian window is shifted by one sample along the
time axis.
An Improvement in Temporal Resolution of
which is shown in Figure 1 (d). Comparing Figures 1
(a) and (d)
shows
that the employed algorithm
can improve the temporal resolution of seismic trac
e. Since the amplitude spectrum can be changed in
above mentioned process, the obtained wavelet canno
t
be classified
in Ricker wavelet family.
However, the high resolution wavelet is also close
to the Ricker wavelet and increasing the resolution
Input wavelet: 15 Hz Ricker wavelet in time domain
and its (b) amplitude and (c) phase spectrum. (d)
Reconstructed high resolution wavelet from (e, f) t
he normalized logarithmic amplitude spectrum and th
e
f the Fourier transform limitations in analyzing no
n

stationary signals such as seismic trace,
frequency transform in
the
above mentioned algorithm
transforms
(Boashash, 2003), but the
short time
which is the usual and
common time
frequency transform
, can be calculated as
follows:
is the time

frequency representation of time domain
is a complex value and can be represented by amplit
ude and phase as
are the amplitude and phase spectrum of the time
he time domain Ricker wavelet and its amplitude spe
ctrum of
n in Figure 2
. To calculate the time
representation, the Gaussian window with a length e
qual to one quarter of the signal length
Gaussian window is shifted by one sample along the
time axis.
An Improvement in Temporal Resolution of
Seismic...
29
that the employed algorithm
can improve the temporal resolution of seismic trac
e. Since the amplitude spectrum can be changed in
in Ricker wavelet family.
However, the high resolution wavelet is also close
to the Ricker wavelet and increasing the resolution
Input wavelet: 15 Hz Ricker wavelet in time domain
and its (b) amplitude and (c) phase spectrum. (d)
Reconstructed high resolution wavelet from (e, f) t
he normalized logarithmic amplitude spectrum and th
e
stationary signals such as seismic trace,
above mentioned algorithm
.
short time
Fourier
frequency transform
(Gabor,
(3)
frequency representation of time domain
is a complex value and can be represented by amplit
ude and phase as
(4)
are the amplitude and phase spectrum of the time
frequency
he time domain Ricker wavelet and its amplitude spe
ctrum of
. To calculate the time
frequency
representation, the Gaussian window with a length e
qual to one quarter of the signal length
is used.
30
Iranian Journal of Oil & Gas Science and Technology
,
Figure 2
(a) 40 Hz Ricker wavelet in
transform.
The method of seismic resolution improving based on
cepstrum in STFT domain also consists of three
steps
(Sajid and Ghosh, 2014)
by using the Gaussian window as described in Eq
logarithmic amplitude of the
(
)
(
(
,
ln
,
LF t f
A t f
=
Similar to
the method based on Fourier transform, the logarith
mic amplitude spectrum is made purely
positive by subtracting its minimum value from all
normalized logarithmic amplitude of the trace spect
rogram is calculated
(
)
(
,
,
min
,
LFP t f
LF t f
LF t f
=
−
(
)
,
,
LFPE t f
LFP t f
=
×
Now, one
can obtain the high resolution seismic trace by
the original phase spectrum as described in
Iranian Journal of Oil & Gas Science and Technology
,
(a) 40 Hz Ricker wavelet in
time domain, (b) its spectrum of Fourier transform,
and (c) short time Fourier
The method of seismic resolution improving based on
cepstrum in STFT domain also consists of three
(Sajid and Ghosh, 2014)
.
In the first step, the trace spectrogram is calcula
ted through the STFT
by using the Gaussian window as described in Eq
uation
logarithmic amplitude of the
trace spectrogram as
given
)
)
,
ln
,
LF t f
A t f
the method based on Fourier transform, the logarith
mic amplitude spectrum is made purely
positive by subtracting its minimum value from all
normalized logarithmic amplitude of the trace spect
rogram is calculated
)
(
)
(
)
,
,
min
,
LFP t f
LF t f
LF t f
=
−
(
)
(
)
(
)
,
,
,
,
A t f df
LFPE t f
LFP t f
LFP t f df
=
×
∫
∫
can obtain the high resolution seismic trace by
the original phase spectrum as described in
Equations 8 and 9
Iranian Journal of Oil & Gas Science and Technology
,
Vol. 4 (2015
), No.
time domain, (b) its spectrum of Fourier transform,
and (c) short time Fourier
The method of seismic resolution improving based on
cepstrum in STFT domain also consists of three
In the first step, the trace spectrogram is calcula
ted through the STFT
uation
3. The second step is the calculation of the
given
below:
the method based on Fourier transform, the logarith
mic amplitude spectrum is made purely
positive by subtracting its minimum value from all
the
spectral values. In the final step, the
normalized logarithmic amplitude of the trace spect
rogram is calculated
by Equations 6 and 7
can obtain the high resolution seismic trace by
using the modified amplitude spectrum and
Equations 8 and 9
:
), No.
2
time domain, (b) its spectrum of Fourier transform,
and (c) short time Fourier
The method of seismic resolution improving based on
cepstrum in STFT domain also consists of three
In the first step, the trace spectrogram is calcula
ted through the STFT
3. The second step is the calculation of the
(5)
the method based on Fourier transform, the logarith
mic amplitude spectrum is made purely
spectral values. In the final step, the
by Equations 6 and 7
:
(6)
(7)
using the modified amplitude spectrum and
A.
Roshandel Kahoo
Figure 8
(a) Noisy synthetic seismic section and (b) its ave
rage amplitude spectrum; (c)
after the application of the propose method and (d)
its average amplitude spectrum; (e) and (f) the am
plitude
spectrum of 31
st
and 10
Figure 9
(a) Free noise synthetic seismic section (Figure 7a
) after the application of the frequency domain dec
onvolution
and (b) its average amplitude spectrum; (c) noisy s
ynthetic seismic section (Figure 8a) after the appl
ication of
the frequency domain deconvo
Roshandel Kahoo
and S. Gholtashi /
An Improvement in Temporal Resolution of
(a) Noisy synthetic seismic section and (b) its ave
rage amplitude spectrum; (c)
after the application of the propose method and (d)
its average amplitude spectrum; (e) and (f) the am
plitude
and 10
th
traces before (blue line) and after (red dashed lin
e) resolution enhancement
(a) Free noise synthetic seismic section (Figure 7a
) after the application of the frequency domain dec
onvolution
and (b) its average amplitude spectrum; (c) noisy s
ynthetic seismic section (Figure 8a) after the appl
ication of
the frequency domain deconvo
lution and (d) its average amplitude spectrum
An Improvement in Temporal Resolution of
(a) Noisy synthetic seismic section and (b) its ave
rage amplitude spectrum; (c)
noisy synthetic seismic section
after the application of the propose method and (d)
its average amplitude spectrum; (e) and (f) the am
plitude
traces before (blue line) and after (red dashed lin
e) resolution enhancement
(a) Free noise synthetic seismic section (Figure 7a
) after the application of the frequency domain dec
onvolution
and (b) its average amplitude spectrum; (c) noisy s
ynthetic seismic section (Figure 8a) after the appl
ication of
lution and (d) its average amplitude spectrum
.
An Improvement in Temporal Resolution of
Seismic...
35
noisy synthetic seismic section
after the application of the propose method and (d)
its average amplitude spectrum; (e) and (f) the am
plitude
traces before (blue line) and after (red dashed lin
e) resolution enhancement
.
(a) Free noise synthetic seismic section (Figure 7a
) after the application of the frequency domain dec
onvolution
and (b) its average amplitude spectrum; (c) noisy s
ynthetic seismic section (Figure 8a) after the appl
ication of
36
Iranian Journal of Oil & Gas Science and Technology
,
T
he sensitivity to noise of the seismic resolution i
mproving method based on cepstrum in STFT
domain
was investigated
10 shows the results of
12.5,
and 9.5 dB. It can be seen that
ratio is 14 dB.
Figure 10
Results of applying the employed method for noisy s
ynthetic data with different values of the signal t
o noise
ratio: (a) 18 dB, (b) 14 dB, (c) 12.5 dB, and (d) 9
.5 dB
3.2. Field data
In addition, the
method
southwest of Iran. The real seismic section
Figures 11
a and 11b.
application of the propose
frequency bandwidth of
Moreover, the
obtained results of the real data
deconvolution shown in Figure 11
deconvolved section
real seismic data
look, three windows of
Figure 12.
When comparing their amplitude spectra as shown in
Figure 13, it can be
seismic data after applying the algorithm have a br
oader
data.
Iranian Journal of Oil & Gas Science and Technology
,
he sensitivity to noise of the seismic resolution i
mproving method based on cepstrum in STFT
was investigated
by varying the level of the signal to noise ratio i
n
10 shows the results of
the
method for synthetic seismic data with four level
and 9.5 dB. It can be seen that
t
he results are almost acceptable as long as the sig
nal to noise
Results of applying the employed method for noisy s
ynthetic data with different values of the signal t
o noise
ratio: (a) 18 dB, (b) 14 dB, (c) 12.5 dB, and (d) 9
.5 dB
.
method
was applied to
a field seismic data from one of hydrocarbon fields
in
southwest of Iran. The real seismic section
s
before and after the application of method are show
n in
a and 11b.
The time
frequency representation of real seismic data befor
e and after th
application of the propose
d algorithm is
shown in Figure
frequency bandwidth of
the seismic data is
expanded during applying the proposed algorithm.
obtained results of the real data
were compared
deconvolution shown in Figure 11
c.
It is clear that the
deconvolved section
reduces
the quality of data. It can be easily observed that
the resolution of the
real seismic data
is consi
derably increased and many hidden features
look, three windows of
the
data before and after applying the algorithm
When comparing their amplitude spectra as shown in
Figure 13, it can be
seismic data after applying the algorithm have a br
oader
Iranian Journal of Oil & Gas Science and Technology
,
Vol. 4 (2015
), No.
he sensitivity to noise of the seismic resolution i
mproving method based on cepstrum in STFT
by varying the level of the signal to noise ratio i
n
the
synthetic data. Figure
method for synthetic seismic data with four level
s
of noise
he results are almost acceptable as long as the sig
nal to noise
Results of applying the employed method for noisy s
ynthetic data with different values of the signal t
o noise
a field seismic data from one of hydrocarbon fields
in
before and after the application of method are show
n in
frequency representation of real seismic data befor
e and after th
shown in Figure
s 11d and 11e. It
can be seen
expanded during applying the proposed algorithm.
were compared
with the
results of frequency domain
It is clear that the
presence of high frequency noise in the
the quality of data. It can be easily observed that
the resolution of the
derably increased and many hidden features
are
discovered. For a closer
data before and after applying the algorithm
were chosen
When comparing their amplitude spectra as shown in
Figure 13, it can be
seen that the field
seismic data after applying the algorithm have a br
oader
amplitude spectrum than the original real
), No.
2
he sensitivity to noise of the seismic resolution i
mproving method based on cepstrum in STFT
synthetic data. Figure
of noise
, namely 18, 14,
he results are almost acceptable as long as the sig
nal to noise
Results of applying the employed method for noisy s
ynthetic data with different values of the signal t
o noise
a field seismic data from one of hydrocarbon fields
in
the
before and after the application of method are show
n in
frequency representation of real seismic data befor
e and after th
e
can be seen
that the
expanded during applying the proposed algorithm.
results of frequency domain
presence of high frequency noise in the
the quality of data. It can be easily observed that
the resolution of the
discovered. For a closer
were chosen
and magnified in
seen that the field
amplitude spectrum than the original real
A.
Roshandel Kahoo
Figure 11
Real seismic data (a) before and (b) after the appl
ication of the proposed algorithm; (c) real seismic
data after
the
application of the frequency domain deconvolution;
the time
(d) before and (e) after the application of the pro
posed algorithm
Roshandel Kahoo
and S. Gholtashi /
An Improvement in Temporal Resolution of
Real seismic data (a) before and (b) after the appl
ication of the proposed algorithm; (c) real seismic
data after
application of the frequency domain deconvolution;
the time
(d) before and (e) after the application of the pro
posed algorithm
An Improvement in Temporal Resolution of
Real seismic data (a) before and (b) after the appl
ication of the proposed algorithm; (c) real seismic
data after
application of the frequency domain deconvolution;
the time

frequency representation of real seismic data
(d) before and (e) after the application of the pro
posed algorithm
.
An Improvement in Temporal Resolution of
Seismic...
37
Real seismic data (a) before and (b) after the appl
ication of the proposed algorithm; (c) real seismic
data after
frequency representation of real seismic data
38
Iranian Journal of Oil & Gas Science and Technology
,
Figure 12
(a, b) The blue window in Figure 11 before and afte
r the application o
d) the green window in Figure 11 before and after t
he application of the proposed algorithm respective
ly; (e, f)
the black window in Figure 11 before and after the
application of the proposed algorithm respectivel
Figure 13
Average amplitude spectrum of (a) the blue window i
n Figure 9 before (blue line) and after (red dashed
line)
resolution enhancement, (b) the green window in Fig
ure 9 before (blue line) and after (red dashed line
)
resolution enhancement and
resolution enhancement
4. Conclusions
A
new algorithm
logarithmic time

time
frequency transform and the cepstrum to extend the
frequency band at each translation of the
spectral decomposing window. The result
real seismic data show that the introduced method c
an increase the temporal resolution of
data. Fur
thermore
improve the temporal resolution of the seismic data
w
employed method makes
Iranian Journal of Oil & Gas Science and Technology
,
(a, b) The blue window in Figure 11 before and afte
r the application o
d) the green window in Figure 11 before and after t
he application of the proposed algorithm respective
ly; (e, f)
the black window in Figure 11 before and after the
application of the proposed algorithm respectivel
Average amplitude spectrum of (a) the blue window i
n Figure 9 before (blue line) and after (red dashed
line)
resolution enhancement, (b) the green window in Fig
ure 9 before (blue line) and after (red dashed line
)
resolution enhancement and
(c) the black window in Figure 9 before (blue line)
and after (red dashed line)
resolution enhancement
.
new algorithm
is introduced which
improves the temporal resolution of seismic data by
using the

frequency transform
method. The algorithm uses valuable properties of b
oth the
frequency transform and the cepstrum to extend the
frequency band at each translation of the
spectral decomposing window. The result
s
of the application of the algorithm
real seismic data show that the introduced method c
an increase the temporal resolution of
thermore
, the results of the algorithm in the presence of n
oise show that the algorithm can
improve the temporal resolution of the seismic data
w
employed method makes
a
little change in the wavelet shape which can be neg
lected.
Iranian Journal of Oil & Gas Science and Technology
,
Vol. 4 (2015
), No.
(a, b) The blue window in Figure 11 before and afte
r the application o
f the proposed algorithm respectively; (c,
d) the green window in Figure 11 before and after t
he application of the proposed algorithm respective
ly; (e, f)
the black window in Figure 11 before and after the
application of the proposed algorithm respectivel
Average amplitude spectrum of (a) the blue window i
n Figure 9 before (blue line) and after (red dashed
line)
resolution enhancement, (b) the green window in Fig
ure 9 before (blue line) and after (red dashed line
)
(c) the black window in Figure 9 before (blue line)
and after (red dashed line)
improves the temporal resolution of seismic data by
using the
method. The algorithm uses valuable properties of b
oth the
frequency transform and the cepstrum to extend the
frequency band at each translation of the
of the application of the algorithm
to
both synthetic and
real seismic data show that the introduced method c
an increase the temporal resolution of
, the results of the algorithm in the presence of n
oise show that the algorithm can
improve the temporal resolution of the seismic data
w
ithout greatly boosting noise.
little change in the wavelet shape which can be neg
lected.
), No.
2
f the proposed algorithm respectively; (c,
d) the green window in Figure 11 before and after t
he application of the proposed algorithm respective
ly; (e, f)
the black window in Figure 11 before and after the
application of the proposed algorithm respectivel
y.
Average amplitude spectrum of (a) the blue window i
n Figure 9 before (blue line) and after (red dashed
line)
resolution enhancement, (b) the green window in Fig
ure 9 before (blue line) and after (red dashed line
)
(c) the black window in Figure 9 before (blue line)
and after (red dashed line)
improves the temporal resolution of seismic data by
using the
method. The algorithm uses valuable properties of b
oth the
frequency transform and the cepstrum to extend the
frequency band at each translation of the
both synthetic and
real seismic data show that the introduced method c
an increase the temporal resolution of
the seismic
, the results of the algorithm in the presence of n
oise show that the algorithm can
ithout greatly boosting noise.
However, the
A. Roshandel Kahoo and S. Gholtashi / An Improvemen
t in Temporal Resolution of Seismic...
39
5. Nomenclature
t
: Time
f
: Frequency
(
)
x t
: Seismic trace in time domain
(
)
X
f
: Seismic trace in frequency domain
(
)
A f
: Amplitude spectrum of seismic trace
(
)
f
Φ
: Phase spectrum of seismic trace
(
)
,
X t f
: Timefrequency transform of seismic trace
(
)
g t
: Gaussian window for timefrequency transform comp
uting
(
)
,
A t f
: Amplitude spectrum of timefrequency transform of
seismic trace
(
)
,
t f
Φ
: Phase spectrum of timefrequency transform of sei
smic trace
(
)
,
LF t f
: Logarithm of amplitude spectrum of timefrequency
transform of seismic trace
(
)
,
LFP t f
:
(
)
,
LF t f
which is made purely positive
(
)
,
LFPE t f
:
(
)
,
LFPE t f
which is normalized
(
)
ˆ
,
X t f
: Modified timefrequency transform of seismic trac
e
(
)
ˆ
x t
: Estimated high resolution seismic trace