An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method DERMAN DONDURUR 1 Abstract—Wiener deconvolution is generally used to improve resolution of the seismic sections, although it has several important assumptions. I propose a new method named Gold deconvolution to obtain Earth’s sparse-spike reflectivity series. The method uses a recursive approach and requires the source waveform to be known, which is termed as Deterministic Gold deconvolution. In the case of the unknown wavelet, it is estimated from seismic data and the process is then termed as Statistical Gold deconvolution. In addi- tion to the minimum phase, Gold deconvolution method also works for zero and mixed phase wavelets even on the noisy seismic data. The proposed method makes no assumption on the phase of the input wavelet, however, it needs the following assumptions to produce satisfactory results: (1) source waveform is known, if not, it should be estimated from seismic data, (2) source wavelet is stationary at least within a specified time gate, (3) input seismic data is zero offset and does not contain multiples, and (4) Earth consists of sparse spike reflectivity series. When applied in small time and space windows, the Gold deconvolution algorithm over- comes nonstationarity of the input wavelet. The algorithm uses several thousands of iterations, and generally a higher number of iterations produces better results. Since the wavelet is extracted from the seismogram itself for the Statistical Gold deconvolution case, the Gold deconvolution algorithm should be applied via constant-length windows both in time and space directions to overcome the nonstationarity of the wavelet in the input seismo- grams. The method can be extended into a two-dimensional case to obtain time-and-space dependent reflectivity, although I use one- dimensional Gold deconvolution in a trace-by-trace basis. The method is effective in areas where small-scale bright spots exist and it can also be used to locate thin reservoirs. Since the method produces better results for the Deterministic Gold deconvolution case, it can be used for the deterministic deconvolution of the data sets with known source waveforms such as land Vibroseis records and marine CHIRP systems. Key words: Reflectivity series, wavelet, deconvolution, sig- nal processing. 1. Introduction Earth’s reflectivity series depends on velocity and density distribution in the subsurface and it is considered as the connection between seismic data and the geology. Different techniques in estimating the reflection coeffi- cient from surface seismics have been proposed. These include maximum likelihood method (O ¨ ZDEMIR, 1985; URSIN and HOLBERG, 1985), Kalman filtering (MENDEL and KORMYLO, 1978), frequency domain methods (BIL- GERI and CARLINI, 1981), singular value decomposition (URSIN and ZHENG, 1985;LEVY and CLOWES, 1980), matched-filter approach (SIMMONS and BACKUS, 1996), sparse-spike inversion (OLDENBURG et al., 1983) and minimum entropy or blind deconvolution methods (Wiggins, 1978;VAN dER BAAN and PHAM, 2008), all of which have their own advantages and limitations regarding the assumptions they make. The impulse response, or Earth’s reflectivity, is generally obtained by least-squares iterative approximation with a known or estimated seismic wavelet designing a wavelet inverse filter (BERKHOUT, 1977;BILGERI and CARLINI, 1981;LINES and TREITEL, 1984;URSIN and HOLBERG, 1985). The sparse-spike deconvolution is also used to obtain reflectivity series, which seeks the least number of spikes in the input so that, when convolved with the seismic wavelet, it fits the data within a given tolerance (VELIS, 2008). Temporal resolution of the seismic data limits the accuracy of detailed mapping of geology, which is quite important in mapping of thin reservoirs for hydrocarbon exploration (URSIN and HOLBERG, 1985). Temporal resolution and its relation to the spectral bandwidth are discussed in detail by OKAYA (1995). For many years, deconvolution techniques have been widely used in seismic exploration to remove the effect 1 Institute of Marine Sciences and Technology, Dokuz Eylu ¨l University, Baku ¨ Street, No:100, 35340 _ Inciraltı, _ Izmir, Turkey. E-mail: [email protected]Pure Appl. Geophys. 167 (2010), 1233–1245 Ó 2010 Birkha ¨user/Springer Basel AG DOI 10.1007/s00024-010-0052-x Pure and Applied Geophysics
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An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method
DERMAN DONDURUR1
Abstract—Wiener deconvolution is generally used to improve
resolution of the seismic sections, although it has several important
assumptions. I propose a new method named Gold deconvolution
to obtain Earth’s sparse-spike reflectivity series. The method uses a
recursive approach and requires the source waveform to be known,
which is termed as Deterministic Gold deconvolution. In the case
of the unknown wavelet, it is estimated from seismic data and the
process is then termed as Statistical Gold deconvolution. In addi-
tion to the minimum phase, Gold deconvolution method also works
for zero and mixed phase wavelets even on the noisy seismic data.
The proposed method makes no assumption on the phase of the
input wavelet, however, it needs the following assumptions to
produce satisfactory results: (1) source waveform is known, if not,
it should be estimated from seismic data, (2) source wavelet is
stationary at least within a specified time gate, (3) input seismic
data is zero offset and does not contain multiples, and (4) Earth
consists of sparse spike reflectivity series. When applied in small
time and space windows, the Gold deconvolution algorithm over-
comes nonstationarity of the input wavelet. The algorithm uses
several thousands of iterations, and generally a higher number of
iterations produces better results. Since the wavelet is extracted
from the seismogram itself for the Statistical Gold deconvolution
case, the Gold deconvolution algorithm should be applied via
constant-length windows both in time and space directions to
overcome the nonstationarity of the wavelet in the input seismo-
grams. The method can be extended into a two-dimensional case to
obtain time-and-space dependent reflectivity, although I use one-
dimensional Gold deconvolution in a trace-by-trace basis. The
method is effective in areas where small-scale bright spots exist
and it can also be used to locate thin reservoirs. Since the method
produces better results for the Deterministic Gold deconvolution
case, it can be used for the deterministic deconvolution of the data
sets with known source waveforms such as land Vibroseis records
180 Hz bandpass filtering, gain recovery (t2), sort to
12-fold CDP, velocity analysis, NMO corrections and
stacking. The seismic trace in Fig. 8a contains two
distinctive bright spot reflections with a polarity
reversal indicated by B, its statistical Gold deconvo-
lution result and extracted minimum phase wavelet.
The seismic source was a generator/injector (GI) gun
with 45 ? 45 inch3 total volume. GI guns do not
produce bubble oscillations and their near-field
signature is a very narrow minimum phase wavelet
with a wide frequency spectrum. The minimum phase
wavelet produced with the GI gun was estimated
using Hilbert method from seismic trace. In this
process, the length of the extracted wavelet is an
important parameter, and our experiences show that a
wavelet length which is equal to the length of input
seismic trace produces suitable results. In Fig. 8a, a
small portion of the trace consisting of 400 samples
with 1 ms sample rate is used as input to Gold
deconvolution. The extracted minimum phase wave-
let and output of Gold deconvolution are illustrated in
Fig. 8b and c, respectively. The deconvolution result
was obtained using 5,501 iterations and a maximum
RMS error of e = 0.05207. Sparse-spike series output
of Gold deconvolution determines the time locations
of bright spot reflections correctly with their correct
phase characteristics relative to the seabed reflection.
The Gold algorithm is also applied to a stacked
seismic section (Fig. 9a) and the result is shown in
Fig. 9c together with its Wiener spiking deconvolu-
tion result in Fig. 9b with 80 ms operator length for a
comparison. The seismic source and the processing
sequence were the same as those for the trace in
Fig. 8a. The input data have 75 traces and 500
samples with 1 ms sample rate. The Gold deconvo-
lution is applied to the data as trace-by-trace basis
with a wavelet length of 500 samples. I estimate a
separate wavelet for each individual trace and then
use it for the deconvolution of that trace.
The output of Gold deconvolution for 2-D real
dataset in Fig. 9c indicates that it produces a two-
Figure 8a A stacked trace from a marine seismic line with two distinct bright spot reflections indicated by B, b extracted minimum phase wavelet, and
c Statistical Gold deconvolution result. The seismic source was a generator/injector (GI) gun with a minimum phase near-field signature
1242 D. Dondurur Pure Appl. Geophys.
dimensional sparse-spike reflectivity section. Wiener
spiking deconvolution results in a section with a wider
bandwidth wavelet (Fig. 9b) than those in the output of
Gold deconvolution (Fig. 9c), hence its output section
is less spiky as expected. However, in place to place,
the output of Gold deconvolution suffers from trace-by-
trace discontinuity, which is especially evident after
1.5 s (Fig. 9c). This is due to several reasons: (1) The
data set becomes somewhat chaotic after 1.5 s and
shows poor lateral continuity in Fig. 9a, (2) because of
the attenuation effect of the Earth, the seismic wavelet
loses its high frequency components and its amplitude
decays as it travels into the Earth. This, in turn, results
in a nonstationary seismic wavelet, which means that
the recorded waveform is time-dependent. The nonsta-
tionarity of the source waveform suggests a gated
application of the Gold deconvolution algorithm.
Assuming a stationary wavelet within small time
windows along the temporal axis of seismic data, one
can apply the proposed algorithm along this specified
time gate in order to obtain a stationary wavelet
approach within the input trace. Several tests suggest
that a time gate of 200 ms produces satisfactory results.
It should be noted that a gate along the space axis
consisting of 10–20 traces may also be useful to avoid
nonstationarity effects along the line. (3) At the very
end of time axis, there are insufficient samples to match
the estimated wavelet and the actual seismic trace,
resulting in a somewhat coarser estimate of the
reflectivity at the deeper parts.
For noise-free synthetic examples, it is sufficient
to run the algorithm for the maximum 5,000 iterations
as shown in Fig. 6. For real data examples, on the
other hand, the maximum number of iterations
required to get an RMS error of 0.001 may be huge.
For instance, to obtain the result shown in Fig. 9b,
approximately 55,000 iterations must be done for
each trace. As a rule of thumb, the larger the
iterations, the smaller the RMS error.
4. Conclusions
Gold deconvolution is an effective method to
obtain sparse-spike reflectivity series from surface
seismic data. It produces good results for the seis-
mograms obtained with minimum, mixed or zero
phase wavelets. Especially for Deterministic Gold
Figure 9a Stacked section from a portion of a marine seismic line, b its Statistical Wiener spiking deconvolution result with 80 ms operator length, and
c its Statistical Gold deconvolution result. The seismic source was a generator/injector (GI) gun
Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1243
deconvolution, as compared to the Wiener deter-
ministic deconvolution output, the results are superior
even for the noisy seismograms.
Although the method does not require an
assumption regarding the phase of the input wavelet,
it is necessary to fulfill the following assumptions: (1)
source waveform is known, if not, it should be esti-
mated from seismic data, (2) source wavelet is
stationary at least within a specified time gate, (3)
input seismic data is zero offset and does not contain
multiples and (4) Earth consists of sparse-spike
reflectivity series. To overcome the nonstationarity of
the wavelet in the input seismograms, the Gold
deconvolution algorithm should be applied via con-
stant-length windows both in time and space axes.
The algorithm and the background mathematics
of Gold deconvolution are very easy to apply, how-
ever, since it is a recursive approach, it may become a
time consuming process especially for long seismic
traces. Therefore, it is recommended that the method
can be used in the areas of small-scale bright spots to
determine thin reservoirs both on stack and amplitude
envelope sections. The method has the ability to
improve the interpretability of the envelope sections,
increasing their frequency bandwidth and, hence,
their temporal resolution since the envelope sections
have rather low dominant frequency content due to its
computational nature.
I apply 1-D Gold deconvolution to seismic data on a
trace-by-trace basis nonetheless the method can easily
be extended into a 2-D case to obtain both time- and
space-dependent reflectivity directly. Applications and
tests on the real seismic data with a well log control
may also indicate the effectiveness of the method by
comparing the results with sonic log-derived reflec-
tivity. Because the method works best for a known
seismic waveform, it can also be applied to Vibroseis
data for land seismics and controlled-source very high
resolution marine CHIRP subbottom profiler data.
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