an approximation to sparse-spike reﬂectivity using the gold deconvolution … · 2011-01-22 ·...

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

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 (OZDEMIR, 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 Eylul

University, Baku Street, No:100, 35340 _Inciraltı, _Izmir, Turkey.

E-mail: [email protected]

Pure Appl. Geophys. 167 (2010), 1233–1245

� 2010 Birkhauser/Springer Basel AG

DOI 10.1007/s00024-010-0052-x Pure and Applied Geophysics

of source wavelet from recorded seismic data and,

hence, to obtain the Earth’s reflectivity. Several

deconvolution methods are used for the wavelet com-

pression of the seismic traces to improve the temporal

resolution, wavelet shaping and removal of bubble

oscillations. Some of the most used deconvolution

techniques assume a minimum phase source wavelet

such as Wiener filtering for predictive and spiking

deconvolution, Burg’s method, Kalman filtering, and

other adaptive deconvolution methods (ROBINSON and

TREITEL, 1980; PORSANI and URSIN, 2007).

The most commonly used deconvolution method is

spiking deconvolution which works well for minimum

phase wavelets (PEACOCK and TREITEL, 1969). This

implies that the theory is not directly applicable to the

traces obtained with a zero or mixed phase wavelets. In

these cases, additional effort is required to convert the

wavelet to its minimum phase equivalent, which gen-

erally includes a phase-shifting (GIBSON and LARNER,

1984). In addition to the minimum phase wavelet

assumption, the spiking deconvolution also assumes a

noise-free seismogram, a random reflectivity series and

a stationary seismic wavelet (YıLMAZ, 1987). Underly-

ing theory of the Wiener deconvolution comes from the

one-dimensional convolutional model, which assumes

that the Earth is represented by a set of horizontal layers

of constant acoustic impedance. On the other hand,

minimum phase wavelet and random reflectivity series

assumptions are the most important limitations for the

Wiener deconvolution process (YıLMAZ, 1987).

In this paper, I propose the Gold deconvolution

method to obtain sparse spike reflectivity series from

seismic traces. The method has been successfully

applied for the deconvolution of c-ray spectra in nuclear

data processing (MORHAC et al., 1997; 2003; BANDZUCH

et al., 1997). The Gold deconvolution algorithm itera-

tively solves the one-dimensional convolutional model

equation. I performed several tests on one- and two-

dimensional synthetic data examples using the Gold

deconvolution algorithm to obtain sparse-spike reflec-

tivity series from recorded seismograms. The tests

include noisy and noise-free synthetic seismograms. I

also compared the results obtained by Gold deconvo-

lution to those obtained using conventional Wiener

deterministic and statistical spiking deconvolutions.

2. Gold Deconvolution Method

According to the one dimensional convolutional

model in noisy environments, a normal incidence

seismogram can be obtained by convolving the Earth

reflectivity series and the seismic wavelet,

y tð Þ ¼ w tð Þ � x tð Þ þ n tð Þ; ð1Þ

where y(t) is the seismogram, w(t) is the seismic

wavelet, n(t) is the random noise component and x(t)

is the reflectivity series or impulse response. Omitting

the random noise component n(t), Eq. (1) can be

rewritten in discrete form as

yðiÞ ¼XN

k¼1

wði� kÞ � ðkÞ i ¼ 1; 2; . . .; 2N � 1;

ð2Þ

where N is the number of samples in the wavelet and

reflectivity series. I here assume that both the wavelet

and reflectivity series have the same number of samples.

In matrix notation, Eq. (2) is given as

yð1Þyð2Þ:::::::::

yð2N � 1Þ

2

6666666666666666664

3

7777777777777777775

ð2N�1Þ � ð1Þ

¼

wð1Þ 0 0 . . . 0

wð2Þ wð1Þ 0 . . . 0

: wð2Þ wð1Þ . . . 0

: : wð2Þ . . . 0

: : : . . . 0

: : : . . . wð1ÞwðNÞ : : . . . wð2Þ

0 wðNÞ : . . . :0 0 wðNÞ . . . :0 0 0 . . . :0 0 0 . . . :0 0 0 . . . wðNÞ

2

6666666666666666664

3

7777777777777777775

ð2N�1Þ � ðNÞ

xð1Þxð2Þ:::

xðNÞ

2

6666664

3

7777775

ðNÞ � ð1Þ

: ð3Þ

1234 D. Dondurur Pure Appl. Geophys.

The deconvolution problem in noise-free envi-

ronments is to obtain the reflectivity series x(t) from a

given seismogram y(t) with a known or unknown

seismic wavelet, e.g., deterministic or statistical

deconvolution process, respectively. To solve

Eq. (3), one definitely needs to know the wavelet. In

the case of the conventional Wiener-Levinson algo-

rithm, however, autocorrelogram of the source

wavelet is required instead of the wavelet itself. The

autocorrelogram of the wavelet can be obtained from

input seismogram when the white reflectivity assump-

tion is valid.

In the Gold deconvolution algorithm, it is

assumed that the seismic wavelet in Eq. (3) is known,

and then this overdetermined equation system can be

solved iteratively. If we use matrix notation for the

one-dimensional convolutional model,

y ¼ Wx: ð4Þ

Multiplying both sides by transpose of matrix W,

we obtain

WT y ¼ WTWx: ð5Þ

Equation (5) is also the least-squares solution that

one obtains when minimizing ||Wx - y||2. Rewriting

this equations yields

z ¼ Tx; ð6Þ

where matrix T is a symmetrical Toeplitz matrix. The

equation system in Eq. (6) is a discrete form of the

Fredholm integral equation of the first kind which is

ill-conditioned and the deconvolution operator is

usually stabilized by adding a small perturbation to

the diagonal of the autocorrelation matrix (ROBINSON

and TREITEL, 1980). Its unconstrained least-squares

solution causes enormous oscillations in estimates of

x, since the equation system is very sensitive to noise

present in the vector y (MORHAC et al., 2002; MORHAC,

2006), therefore, direct inversion of this system

cannot produce a stable solution. Different regulari-

zation methods to solve deconvolution problem were

discussed by SACCHI (1997).

Following the method of GOLD (1964), MORHAC

and MATOUSEK (2005) suggested an iterative Gold

deconvolution algorithm to obtain vector x. In Gold

deconvolution, reflectivity series x can be iteratively

obtained by solving

xðkþ1Þi ¼ zi

dixðkÞi i ¼ 1; 2; . . .;N and k ¼ 1; 2; . . .; L;

ð7Þ

where d = Tx(k) and L is the maximum number of

iterations. As an initial solution for vector x, MORHAC

and MATOUSEK (2005) suggest

xð1Þi ¼

1

1

:::1

26666664

37777775

ðNÞ � ð1Þ

: ð8Þ

3. Applications

Gold deconvolution algorithm needs the seismic

wavelet to be known. If this is the case, the method

can be analogous to the deterministic Wiener spik-

ing deconvolution (Fig. 1a) and Earth reflectivity

series can be accurately obtained. In conventional

seismic exploration, however, the seismic wavelet is

generally not known with an exception of Vibroseis

data in which the source signature can be pre-

determined. Therefore, when the wavelet is not

known, it is estimated from seismic trace itself

using an appropriate wavelet extraction method

(Fig. 1b). In this case, the deconvolution is analo-

gous to the statistical spiking deconvolution. In both

cases, the phase of the wavelet in the input seis-

mogram is not a limitation for Gold deconvolution

algorithm. In this study, I refer ‘‘deterministic Gold

deconvolution’’ and ‘‘statistical Gold deconvolu-

tion’’ for the application with known and unknown

seismic wavelet, respectively. To solve the linear

equation system given by Eq. (3), zero padding is

necessary both for input seismogram and wavelet in

order to obtain a reflectivity series with the same

length as input seismogram.

3.1. Deterministic Gold Deconvolution

It is well known that Wiener spiking deconvolu-

tion is an effective method in deconvolution of

minimum phase wavelets. When the wavelet is zero

Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1235

or mixed phase, however, it produces inaccurate

results even if the source waveform is known.

Figure 2 illustrates this phenomenon. Minimum,

mixed and zero phase wavelets and their respective

deterministic Wiener and Gold deconvolution results

are also shown in Fig. 2. 80 ms of deconvolution

operator length is used for all three wavelets to obtain

Wiener deconvolution results. It is clear that the Gold

deconvolution produces superior results for all three

type of wavelets with different phase characteristics,

while Wiener spiking deconvolution produces inap-

propriate results for mixed and zero phase wavelets.

It is also concluded that the Gold deconvolution

preserves amplitude and phase characteristics of the

resultant spikes as it converts the wavelets into the

spikes (e.g., reflectivity series).

3.2. Statistical Gold Deconvolution

In conventional seismic exploration, we generally

do not know the seismic wavelet preserved in the

seismic data, which poses implementation of statis-

tical techniques in deconvolution process. To use the

Gold deconvolution algorithm with a statistical

approach, we need to estimate the wavelet from

seismic data. Spectral analysis of the seismogram can

provide an estimate of the energy spectrum of the

minimum phase wavelet with a random reflectivity

sequence assumption, and then we have the problem

of deriving its phase spectrum. There are three

methods to obtain the phase spectrum: (1) Hilbert

transform (or Kolmogoroff factorization) method, (2)

z transform method, and (3) Wiener-Levinson inverse

method. An overview of these methods can be found

in WHITE and O’BRIEN (1974), LINES and ULRYCH

(1977) and CLAERBOUT (1985).

The general assumption regarding the seismic

source signature is that it is minimum phase for the

impulsive sources such as dynamite or air guns.

WHITE and O’BRIEN (1974) suggest that, even for

noisy environments, Hilbert transform method pro-

duces the best results for minimum phase wavelets.

Therefore, Hilbert transform method, which operates

in frequency domain, is used to obtain the minimum

and zero phase wavelets in this study. An approxi-

mation to mixed phase wavelets is also realized using

zero phase wavelet estimates.

In order to testing the efficiency of the present

method, I perform some tests on a sparse-spike

synthetic reflectivity series. A noise-free seismogram

in Fig. 3c is obtained by convolving a minimum

phase wavelet in Fig. 3a with a synthetic reflectivity

series in Fig. 3b. Their respective amplitude spectra

are also shown on top. Conventional Wiener spiking

deconvolution outputs with 80 ms operator length are

given in Fig. 3d and e for deterministic and statistical

approximations, respectively. The deterministic and

statistical Gold deconvolution results after 5,000

iterations are also given in Fig. 3f and g. Wiener

deterministic deconvolution produces correct results

Figure 1Application of Gold deconvolution method with a known (Deterministic Gold deconvolution), and b unknown (Statistical Gold

deconvolution) seismic wavelet. N the number of samples in the input seismogram


as expected: it resolves the interference between 400

and 500 ms in the input synthetic seismogram and

recovers the accurate reflectivity series, since the

input wavelet is minimum phase. Wiener statistical

deconvolution also reveals the exact locations of the

spikes with their correct polarity, although it also

creates some trailing distortions after each spike,

which is a well-known characteristic of the Wiener

statistical deconvolution. On the other hand, deter-

ministic Gold deconvolution produces good results

as for Wiener deterministic deconvolution, while

statistical Gold deconvolution also gives acceptable

results with less trail distortions as compared to

Wiener statistical deconvolution output. Gold

deconvolution also preserves the relative amplitudes

of the resultant spikes in the output seismogram

(Fig. 3g).

Similar applications are also performed on the

synthetic seismograms by using mixed phase (Fig. 4)

and zero phase (Fig. 5) wavelets, and the results from

Wiener spiking deconvolution and Gold deconvolu-

tion algorithm in noise-free environments are

compared. For both wavelets, Wiener deconvolution

produces inaccurate results as expected (Figs. 4 and

5). For mixed phase wavelet, the output seismograms

of Wiener deconvolution are extremely noisy (see

Fig. 4d and e) and deconvolution process produces no

further resolution improvement. For zero phase

wavelet, the outputs of Wiener deconvolution are

ringy and indicate unstable deconvolution results (see

Fig. 5d and e). The Gold deconvolution results, on

the other hand, seem acceptable for both mixed and

zero-phase wavelets. Deterministic Gold deconvolu-

tion produces noticeably better results: it resolves the

interference effect in the input seismogram between

400 and 500 ms and then it reveals the exact time

locations of the sparsespikes with their correct phase

and polarity attributes, regardless of the wavelet

phase characteristics (see Figs. 4f and 5f). Statistical

Gold deconvolution also produces convenient results

Figure 2a Minimum (top), mixed (middle) and zero phase (bottom) wavelets, b their Deterministic Wiener spiking deconvolution and c Deterministic

Gold deconvolution results


as compared to Wiener deconvolution. Although

statistical Gold deconvolution results are somewhat

noisy, especially for mixed phase wavelet, the output

is for more similar to the input sparse-spike series.

For all three different type of wavelets, the Gold

deconvolution flattens the spectrum of the input

seismogram, indicating the improvement in temporal

resolution.

Figure 3a Minimum phase wavelet with 50 Hz dominant frequency, b sparse-spike synthetic reflectivity series, c their corresponding noise-free

seismogram, and d after Deterministic and e Statistical Wiener spiking deconvolutions with 80 ms operator length. f Deterministic and g

Statistical Gold deconvolution results after 5,000 iterations. Respective amplitude spectra of each seismogram are shown on top

Figure 4a Mixed phase wavelet with 50 Hz dominant frequency, b sparse-spike synthetic reflectivity series, c their corresponding noise-free




3.3. Convergence of the Iteration

Gold deconvolution algorithm carries out an iter-

ative approach in which the subsequent result is

computed using the result obtained during the preced-

ing iteration according to Eq. (7). For the iterative

techniques, convergence to a solution after a finite

number of iterations is essential. In order to test the

performance of the Gold deconvolution to converge to

a solution after a number of iterations, I calculate RMS

errors during the iterations between successive results

estimated by Gold deconvolution using

ek ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XN

i¼1

xðkÞi � x

ðk�1Þi

� �2

vuut k ¼ 2; 3; . . .; L; ð9Þ

where L is the maximum number of iterations. Fig-

ure 6 illustrates the variation in RMS error value with

iteration number for the seismograms shown in

Figs. 3c, 4c and 5c obtained using minimum, mixed

and zero phase wavelets, respectively. Figure 6a

shows the error graphs for Deterministic Gold

deconvolution, whereas Fig. 6b illustrates the errors

obtained during the Statistical Gold deconvolution

process. It is observed from Fig. 6a that, though the

general trend of the RMS error curve slowly

approaches zero with increasing iteration numbers for

all three types of seismograms, some local bursts in

the RMS error curve are obtained as random spikes.

The density of these spikes, however, decreases with

increasing iterations and finally they disappear com-

pletely after a certain number of iterations is reached.

In the deterministic deconvolution case, the RMS

error quickly falls to 0.01 in the first 10 iterations.

Although some local instabilities exist during the

iteration, the overall RMS trend becomes smaller and

smaller as the iteration number increases.

It can be concluded from the error graphs of

deterministic Gold deconvolution (Fig. 6a) that the

algorithm needs at least approximately 5,000 itera-

tions to produce stable results for minimum-and zero-

phase wavelets, whereas a lesser number of iterations

is required for the deterministic Gold deconvolution

of seismogram with mixed-phase wavelet. In all

cases, since the method does not require complex

computations, performing the 5000 iterations takes

only a couple of seconds on an ordinary Pentium IV

microcomputer.

The error graphs of statistical Gold deconvolution

in Fig. 6b, on the other hand, indicate a different

consequence: it appears as if it does not converge to a

stable solution after 5,000 iterations. It is clear that

the RMS values scatter during the iterations which

Figure 5a Zero-phase Ricker wavelet with 50 Hz dominant frequency, b sparse-spike synthetic reflectivity series, c their corresponding noise-free




indicates that some ‘‘local minima’’ and ‘‘local

bursts’’ exist in the convergence error plots. Further-

more, at early stages of the iteration progress, we

observe huge RMS error values for all three seismo-

grams. The RMS error at the beginning is generally

much larger than 106 and it rapidly decreases to

values lower than 1 for the first 10 iterations. As

iteration proceeds, the RMS error between successive

iterations becomes reasonable, although the graphs

are still spiky. Such oscillating RMS errors

sometimes arise during the inversion algorithms,

indicating oscillations between existing local minima.

This type of chaotic behavior of the inversion has

been investigated by COOPER (2000, 2001) and he

suggested that some additional regularization could

further improve the convergence.

In the statistical Gold deconvolution case in

Fig. 6b, one can stop the iteration just after 10 or

15 iterations since the RMS error decrease to 0.1 after

first 10–20 iterations. Our tests, however, showed that

if one continues iteration further, much smaller

‘‘local’’ RMS error values can be obtained. For

instance, after iteration 3,000, the RMS error is about

0.1 for the Statistical Gold deconvolution of the

Figure 6Variation of RMS error with respect to the iteration number for minimum (top), zero (middle) and mixed (bottom) phase wavelets computed

using a Deterministic and b Statistical Gold deconvolution


minimum phase seismogram shown in Fig. 6b,

whereas it is of the order of 105 for the first three

iterations. It is therefore concluded that, though the

error graphs for Statistical Gold deconvolution do not

clearly demonstrate a convergence around a stable

solution, the results are acceptable since RMS error

values becomes significantly smaller after the initial

iterations.

3.4. Effect of Random Noise

In order to examine the effect of random noise on

the performance of Gold deconvolution, I use the

one-dimensional reflectivity model shown in Fig. 7a.

The minimum phase noisy synthetic seismogram is

given in Fig. 7b together with its respective ampli-

tude spectrum on the top, which is obtained by adding

30% of random noise to the noise-free seismogram

shown in Fig. 3c. The amount of random noise used

here is calculated with respect to the maximum

amplitude in the seismogram. The results of Wiener

deterministic and statistical deconvolutions are

shown in Fig. 7c and d, while the results of

Gold deterministic and statistical deconvolutions are

shown in Fig. 7e and f, respectively. Wiener decon-

volution causes boosting in random noise, which is

standard since it generally boosts the amplitudes of

high frequency noise in the data. For this reason, a

conventional Wiener deconvolution process is gen-

erally followed by a band-pass filter to suppress this

boosted high frequency noise. When comparing the

outputs of Wiener and Gold deconvolutions, it is

obvious that both deterministic and statistical Gold

deconvolutions produce clearer results with consid-

erably less noisy deconvolution output. In particular

deterministic Gold deconvolution gives a superior

result: It suppresses most of the random noise and

correctly recovers the reflectivity (see Fig. 7a and e).

3.5. Applications to Real Seismic Data

In conventional reflection seismics, one should

use statistical Gold deconvolution because the wave-

let is unknown. Therefore, I applied the Gold

deconvolution algorithm to real seismic data in a

statistical manner to obtain the sparse-spike reflec-

tivity series. I use stacked seismic data which is an

approximation to zero offset section under certain

Figure 7a Sparse-spike synthetic reflectivity series, b Minimum phase seismogram with an additional random noise of 30%, c after Deterministic and

d Statistical Wiener spiking deconvolutions with 80 ms operator length. e Deterministic and f Statistical Gold deconvolution results after

5,000 iterations. Respective amplitude spectra of each seismogram are shown on top. The minimum phase wavelet used to compute the

synthetic seismogram is shown in Fig. 3a


conditions, since it reflects the subsurface structure

and hence can be used to recover Earth’s reflectivity.

The poststack seismic data that the Gold deconvolu-

tion is applied to should be suitably processed so that

trace-by-trace correlation of the source waveform is

not modified considerably along the line. It is

suggested that any kind of prestack deconvolution

methods and migration process should be avoided

before Gold deconvolution since all these applica-

tions may result in modifications on the source

waveform in varying degrees.

Figure 8 shows a real seismic trace from marine

seismic data. The seismic data used in this study were

preprocessed using almost conventional data

processing steps: editing, geometry definition, 24–

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


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


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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(Received July 17, 2009, revised October 6, 2009, accepted October 22, 2009, Published online February 9, 2010)


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