R.E. White (Birkbeck University of London) & E. Zabihi Naeini* (Ikon Science)

Transcription

1 Tu ELI2 10 Broad-band Well Tie R.E. White (Birkbeck University of London) & E. Zabihi Naeini* (Ikon Science) SUMMARY By extending the low-frequency content, broad-band seismic data lessens the dependence of seismic inversion on a background model. Despite that, the merging of the inverted seismic with a very low frequency background model poses a new problem: the duration of a well-log synthetic seismogram is often seriously inadequate for defining the sharp spectral decay to zero frequency. To address this issue, a practical approach based on multi-taper spectral analysis is proposed to determine the low-frequency spectral decay of the seismic well tie wavelet. Furthermore the impact on seismic inversion is demonstrated.

2 Introduction By extending the low-frequency content, broad-band seismic data lessens the dependence of seismic inversion on a background model. Despite that, the merging of the inverted seismic with a very low frequency background poses a new problem: the duration of a well-log synthetic seismogram, 500 ms say, is seriously inadequate for defining the sharp spectral decay to zero frequency. The demands from recording higher frequencies are more tractable, requiring precise timing of the well-log synthetic seismogram and meticulous imaging. Between the low and high frequencies, the broad bandwidth may also strain the trade-off between stability and resolution if the seismic spectrum is not fairly flat. For reliable results all well ties require accurate modelling of the synthetic seismogram and good quality seismic data of sufficient duration relative to the length of the seismic wavelet. The purpose of this paper is to propose a practical approach to determining the low-frequency spectral decay of the seismic wavelet when tying well-log synthetic seismograms to broad-band seismic data. Well tie principles The principles of making a well tie are essentially the principles of system identification adapted to the characteristic properties of seismic data and limited time duration of well-log synthetic seismograms. The tutorial of White and Simm (2003) covers these principles in more detail with examples. The following considers only those that particularly affect ties with broad-band data. A routine least-squares best-fit well tie assumes that the synthetic seismogram is error-free. As a consequence the best-fit filter is not the seismic wavelet but the wavelet combined with a Wiener filter for attenuating the noise in the synthetic. The frequency domain solution can be adapted to account of errors in the synthetic seismogram as well as noise in the data (Walden and White (1998)). The frequency domain approach also has the major advantage of providing diagnostics of the accuracy of a well tie, such as error bars or confidence bounds on the amplitude and phase spectrum of the seismic wavelet. In the time domain sample-to-sample correlations make confidence bounds so multi-dimensional that the only credible recourse is to produce Monte Carlo realisations of the wavelet. White and Simm (2003) emphasise the need for these diagnostics, rather than goodness-of-fit which may simply indicate overfitting. The frequency domain approach has another advantage when estimating the sharp spectral decays at the edges of seismic spectra: spectral analysis has tools specifically for spectra with a large dynamic range. As with most estimation procedures, a well tie involves a trade-off between stability and resolution. Insofar as noise in the system is largely random, this becomes a trade-off between noise-induced random errors and bias or distortion. The limited time duration of well-log synthetic seismograms accentuates this trade-off. In the time domain a finite data duration induces sampling errors into both auto- and cross-correlations. Assuming a compact wavelet and a reasonable signal-to-noise (S/N) in the data, the signal dominates the short lag correlations whereas sampling errors affect all lags. With increasing lag the sampling errors tend to dominate. The application of a lag window to the correlations then has to balance the control of sampling errors with avoiding serious truncation of the signal correlation. In the frequency domain this becomes a balance between choosing a spectral window narrow enough to capture the spectral shape and wide enough to allow sufficient spectral smoothing to attenuate random errors. The spectral window can be characterised by its analysis bandwidth b. The spectral smoothing is then bt where T is the duration of the data segment. Since T and the data bandwidth B are both limited, the latter by the S/N, this forces a need to balance the demand of stability that bt be as large as possible with the requirement of resolution that b is a fraction of B. In the past an acceptable well tie to the North Sea data typically required BT 25, corresponding to data having a bandwidth B~50 Hz and a T~500 ms synthetic seismogram. This assumes that the data spectrum is reasonably flat or smooth over the 50 Hz bandwidth and that the proportion of trace energy predicted (PEP) by the well tie is 0.6 or more. While such estimation is reliable over the seismic bandwidth, the low-frequency response of the seismic wavelet is not estimated well. This is because the

3 decay of seismic spectra towards zero frequency is too sharp to be estimated reliably from 500 ms data segments. This lack of spectral resolution becomes acute with broad-band data whose spectrum extends down to frequencies of 4 Hz or less. Estimation of the low-frequency spectral decay demands the use of longer data segments. The wavelet amplitude spectrum is then obtained from the square root of the seismic power spectrum divided by the reflectivity power spectrum (or a model for the reflectivity colouring). This division assumes that the data and synthetic have the same S/N (Walden and White, 1998). The reflectivity colouring is unlikely to have much effect on the low-frequency decay. The estimated decay is then merged onto the wavelet spectrum. At low frequencies it is important to minimise spectral leakage, i.e. leakage of power through the side lobes of the spectral window from the nearby pass-band. Multi-taper spectral analysis using prolate spheroidal tapers (Percival and Walden, 1993) is designed for this task. In this method the spectral averaging comes from the number of tapers 2nw 1 where nw is the taper s time-bandwidth product. The spectral resolution is proportional to 1/T. Since smoothing bias from the main lobe of the spectral window is unavoidable, the choice of nw and T is a critical decision. The example in the next section indicates that stable estimation of a 18 db/oct decay from 4 Hz needs a 4 s trace segment with nw = 4. For practical reasons use of a shorter data segment centred on the well tie would be desirable. Since sacrificing resolution would blur the decay, use of 2 s of data, say, demands that nw is reduced to 2. One then has to rely on areal averaging of spectra around the well to reduce sampling fluctuations. At high-frequencies estimation of a 18 db/oct roll-off is not a problem but the same approach may be needed for sharper roll-offs. The main problem at high frequencies comes from timing errors in the synthetic seismogram which increase estimation errors and introduce a false amplitude decay if the synthetic is assumed to be error-free. Direct estimation of the low-frequency phase is not possible from a well tie. The estimated phase at zero frequency is either 0 or ±180, depending on whether the sum of wavelet coefficients happens to be positive or negative. Really the phase at zero frequency jumps between -m 90 and m 90 when the low frequency decay is m 6 db/oct. Thus estimation of the amplitude decay becomes doubly important since it gives a clue to the phase at zero frequency. The phase can then be interpolated to the nearest reliable low-frequency estimate. When there is not enough well log available to satisfy the requirement of BT 25, it is unrealistic to expect to estimate a phase spectrum. The pragmatic option is to use a constant phase wavelet. This uses fewer degrees of freedom than estimating a phase spectrum and has some empirical basis in that, after processing, the phase of seismic wavelets is often approximately constant across the seismic bandwidth. If need be, the constant phase φ C can be modified by simple extrapolation to the phase φ 0 at zero frequency (e.g. to φ C + (φ 0 φ C ) exp( f/f c ) where f c is a corner frequency). The amplitude spectrum can be computed by spectral division as described above. Simulated example To simulate a well tie, an artificial seismic reflection signal was constructed by filtering a sequence of Laplacian random numbers with a minimum phase 4 80 Hz Butterworth impulse response. The amplitude spectrum of this wavelet decays at 18 db/oct outside its passband Hz random noise was added to the filtered reflectivity. The rms signal-to-noise (S/N) ratio of the simulated data was 2. Since the S/N power ratio is 4, the ratio of signal power to trace power is 0.8, which corresponds to the proportion of trace power (PEP) that can ideally be predicted by least-squares filtering of the primary reflectivity. A 500 ms segment of reflectivity was then matched to the simulated data using an analysis bandwidth b of 8.4 Hz. Although this is too broad a spectral window to detect the low-cut notch below 4 Hz, the spectral smoothing, bt = 4.2, cannot be reduced further without incurring undesirable estimation errors, resulting in a noise-prone estimate of the seismic wavelet. Figure 1 compares the estimated and input wavelet spectra. Clearly the estimated amplitude spectrum gives no indication of the 18 db/oct decay below 4 Hz and the estimated phase misses the trend to 270 at zero frequency. Beyond 80 Hz the amplitude spectrum captures the 18 db/oct decay down to -30 db, which in

4 practice would often fall below the noise floor. While the estimates at high frequency are not seriously amiss, the simulation is very idealised. In practice small timing misalignments would at least inflate the error bars. Figure 1 Simulation of a well tie showing results from estimating a seismic wavelet by matching a 500 ms reflectivity sequence to noisy 4-80 Hz seismic data. left: amplitude spectra of the input and estimated wavelets; right: phase spectra of the input and estimated wavelets. To detect the low frequency decay, varying lengths of the noisy seismic trace were analysed by the multi-taper method using a time-bandwidth product of nw=4. It was not until a 4 s segment of data is analysed (Figure 2A) that the low-frequency decay is detected with reasonable accuracy. This decay was merged into the wavelet spectrum below 5.86 Hz. The phase at zero frequency inferred from the 18 db/oct rate of decay is ±270 which was interpolated to the 5.86 Hz phase estimate (Figure 2B) at neighbouring frequencies. Modelling the acquisition system may help infer other "guesstimates" of the low-frequency phase. It is not easy to discriminate between different guesstimates from inspection of the wavelet because the broad low-amplitude side lobes characteristic of a sharp low-cut are obscured by noise-induced fluctuations (Figure 2C). Figure 2 A: amplitude spectra at low frequencies from the well tie wavelet and multitaper analysis using varying lengths of trace compared with the Butterworth lowcut response; B: phase spectra of the input and estimated wavelets at low frequencies and an extrapolation of the estimated phase to 270 ; C: comparison of the phaseadjusted wavelet estimate with the Butterworth impulse response. Estimation of the high-frequency spectral decay was not a problem. The multi-taper method using a 500 ms data segment captures the 18 db/oct high-frequency decay down to -50 db and from longer segments follows it easily to below -100 db. Practical applications This example comes from conventional data that gave a very close well tie. The low-cut corner frequency is ~8 Hz. The effect of the low-frequency gap is emphasised because it is wider than in broad-band data. Figure 3 shows the seismic wavelet and its amplitude spectrum before and after including a lowcut response. The phase was extrapolated to 360 at zero frequency. Figure 4 shows the Figure 3 Left: the wavelet estimated from the well tie. Right: the modified wavelet.

5 data and band-limited impedance sections obtained using the well-tie wavelet and the modified wavelet. The striations in the centre panel correspond to a random walk component caused by integrating low-frequency energy that is not part of the seismic signal. Figure 5 shows the well tie wavelet and modified wavelet estimated from broad-band seismic data. Modification of the amplitude spectra is obvious at the low frequency end of the spectrum. These two wavelets were both used in AVO type inversion to derive the impedances. A good QC for this is to see how the inverted impedance models the seismic i.e. the difference between modelled and actual seismic should be minimal. Figure 5C shows that the modified wavelet has reduced the residual energy. Conclusions Figure 4 Left: seismic data. Centre: impedance section from inversion using the wavelet estimated from the well tie. Right: impedance section from inversion using the modified wavelet. (A) (B) Although broad-band seismic data lessens the dependence of seismic inversion on a background model, it is still desirable to merge the inverted seismic with a low frequency background model. Well ties do not offer the spectral resolution needed to define the low frequency cut-off of the broadband data. We propose application of multitaper spectral analysis to determine this cutoff. The estimated cut-off can also help define the phase spectrum at low frequencies. We demonstrated the application of this technique on conventional and broad-band seismic data inversion. (C) 1 km Acknowledgements Figure 5 A: Original well tie wavelet, B: modified wavelet and C: residual energy from modelling the seismic with wavelet A and B. We thank Ikon Science for support of this research and Mike Bacon for useful discussions. References Percival, D. and Walden, A. [1993] Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press, Cambridge, U.K., 1st edition. Walden, A. and White, R. [1998] Seismic wavelet estimation: a frequency domain solution to a geophysical noisy input output problem. IEEE Transactions on Geoscience and Remote Sensing, 36(1), White, R.E. and Simm, R. [2003] Tutorial: Good practice in well ties. First Break, 21(10), th EAGE Conference & Exhibition 2014