An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method

Transcription

1 Pure Appl. Geophys. 167 (2010), Ó 2010 Birkhäuser/Springer Basel AG DOI /s x Pure and Applied Geophysics 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 addition 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 overcomes 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 seismograms. The method can be extended into a two-dimensional case to obtain time-and-space dependent reflectivity, although I use onedimensional 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, signal processing. 1 Institute of Marine Sciences and Technology, Dokuz Eylül University, Bakü Street, No100, _Inciraltı, _Izmir, Turkey. derman.dondurur@deu.edu.tr 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 coefficient from surface seismics have been proposed. These include maximum likelihood method (ÖZDEMIR, 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

2 1234 D. Dondurur Pure Appl. Geophys. of source wavelet from recorded seismic data and, hence, to obtain the Earth s reflectivity. Several deconvolution methods are used for the wavelet compression 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 generally 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). Underlying 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 iteratively solves the one-dimensional convolutional model equation. I performed several tests on one- and twodimensional synthetic data examples using the Gold deconvolution algorithm to obtain sparse-spike reflectivity series from recorded seismograms. The tests include noisy and noise-free synthetic seismograms. I also compared the results obtained by Gold deconvolution 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, yt ðþ¼wt ðþxt ðþþnt ðþ; 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 ¼ XN k¼1 wði kþðkþ 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 2 3 yð1þ yð2þ yð2n 1Þ ð2n 1Þ ð1þ 2 ¼ 6 4 wð1þ wð2þ wð1þ wð2þ wð1þ... 0 wð2þ wð1þ wðnþ... wð2þ 0 wðnþ wðnþ wðnþ ð2n 1Þ ðnþ 2 3 xð1þ xð2þ xðnþ ðnþ ð1þ ð3þ

3 Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1235 The deconvolution problem in noise-free environments 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 algorithm, 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 assumption 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 W T y ¼ W T Wx 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 regularization 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 x ðkþ1þ i ¼ z i d i x ðkþ i 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 2 3 x ð1þ i 1 1 ¼ ðnþ ð1þ 3. Applications ð8þ Gold deconvolution algorithm needs the seismic wavelet to be known. If this is the case, the method can be analogous to the deterministic Wiener spiking 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 predetermined. 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 analogous to the statistical spiking deconvolution. In both cases, the phase of the wavelet in the input seismogram is not a limitation for Gold deconvolution algorithm. In this study, I refer deterministic Gold deconvolution and statistical Gold deconvolution 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 Deterministic Gold Deconvolution It is well known that Wiener spiking deconvolution is an effective method in deconvolution of minimum phase wavelets. When the wavelet is zero

4 1236 D. Dondurur Pure Appl. Geophys. Figure 1 Application 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 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 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 inappropriate 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) Statistical Gold Deconvolution In conventional seismic exploration, we generally do not know the seismic wavelet preserved in the seismic data, which poses implementation of statistical 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 produces 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 approximation 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

5 Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1237 Figure 2 a Minimum (top), mixed (middle) and zero phase (bottom) wavelets, b their Deterministic Wiener spiking deconvolution and c Deterministic Gold deconvolution results 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, deterministic 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 deconvolution 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 deconvolution 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

6 1238 D. Dondurur Pure Appl. Geophys. Figure 3 a 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 4 a Mixed 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 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.

7 Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1239 Figure 5 a Zero-phase Ricker 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 3.3. Convergence of the Iteration Gold deconvolution algorithm carries out an iterative approach in which the subsequent result is computed using the result obtained during the preceding 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 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u1 e k ¼ t N X N i¼1 x ðkþ i x ðk 1Þ 2 i k ¼ 2; 3;...; L; ð9þ where L is the maximum number of iterations. Figure 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 completely 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 iterations to produce stable results for minimum-and zerophase 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

8 1240 D. Dondurur Pure Appl. Geophys. Figure 6 Variation 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 indicates that some local minima and local bursts exist in the convergence error plots. Furthermore, at early stages of the iteration progress, we observe huge RMS error values for all three seismograms. The RMS error at the beginning is generally much larger than 10 6 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 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

9 Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1241 minimum phase seismogram shown in Fig. 6b, whereas it is of the order of 10 5 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 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 amplitude 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 deconvolution 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 generally 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 considerably 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) Applications to Real Seismic Data In conventional reflection seismics, one should use statistical Gold deconvolution because the wavelet is unknown. Therefore, I applied the Gold deconvolution algorithm to real seismic data in a statistical manner to obtain the sparse-spike reflectivity series. I use stacked seismic data which is an approximation to zero offset section under certain Figure 7 a 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

10 1242 D. Dondurur Pure Appl. Geophys. conditions, since it reflects the subsurface structure and hence can be used to recover Earth s reflectivity. The poststack seismic data that the Gold deconvolution 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 applications 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, Hz bandpass filtering, gain recovery (t 2 ), 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 deconvolution result and extracted minimum phase wavelet. The seismic source was a generator/injector (GI) gun with 45? 45 inch 3 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 wavelet 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 = 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 deconvolution 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 deconvolution 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 8 a 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

11 Vol. 167, (2010) An Approximation to Sparse-Spike Reflectivity Using the Gold Deconvolution Method 1243 Figure 9 a 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 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-bytrace 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 nonstationarity 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 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 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 seismograms obtained with minimum, mixed or zero phase wavelets. Especially for Deterministic Gold

12 1244 D. Dondurur Pure Appl. Geophys. deconvolution, as compared to the Wiener deterministic 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 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. To overcome the nonstationarity of the wavelet in the input seismograms, the Gold deconvolution algorithm should be applied via constant-length windows both in time and space axes. The algorithm and the background mathematics of Gold deconvolution are very easy to apply, however, 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 reflectivity. 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. REFERENCES BANDZUCH, P., MORHAC M., and KRISTIAK, J. (1997), Study of the Van Cittert and Gold iterative methods of deconvolution and their application in the deconvolution of experimental spectra of positron annihilation, Nuclear Instr. Meth. Phys. Res. A 384, BERKHOUT, A.J. (1977), Least-squares inverse filtering and wavelet deconvolution, Geophysics 42, BILGERI, D., and CARLINI, A. (1981), Non-linear estimation of reflection coefficients from seismic data, Geophys. Prosp. 29, CLAERBOUT, J.F., Fundamentals of Geophysical Data Processing (Blackwell Sci. Pub. 1985). COOPER, G.R.J. (2000), Fractal convergence properties of geophysical inversion, Comp. Graphics 24, COOPER, G.R.J. (2001), Aspects of chaotic dynamics in the leastsquares inversion of gravity data, Comp. 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