Signal and Noise Estimation from Seismic Reflection...

Signal and Noise Estimation from Seismic Reflection Data Using Spectral Coherence Methods

ROY E. WHITE

lnv i ted Paper

Spectral coherence analysis is unrivaled as a quantitative tool over a range of practical problems in seismic interpretation, data processing, quality assessment for data acquisition, and research. Its great virtue is its ability to supply the detailed error information necessary for a thorough interpretation of results. Ordinary coherence analysis is employed in line intersection analysis and the design of filters to cross-equalize differently acquired seismic sections in a given area; both ordinary and partial coherence methods are indispensable in matching synthetic seismograms and seismic data; and multiple coherences are used to estimate the coherent signal and incoherent noise content of seismic sections andgathers. The precise meaning of the signal and noise estimates output by coherent analysis has to be related to the particular technique employed and the type of data input to it. The principles and procedures for analyzing seismic data with these methods are reviewed and illustrated with practical examples.

I. INTRODUCTION

Spectral coherences play a central role in the estimation of the signal and noise content of multiple time series, in the estimation of transfer functions (frequency responses), and in providing measures of statistical accuracy of the estimates. This role is well documented in the literature of spectral analysis (e.g., [24], [36D but, although coherence methods supply information very relevant to seismic reflection work and their potential was appreciated many years ago [14], [63], they have not yet permeated deeply into reflection seismology. This is not due to any lack of applicability on their part. Indeed, they have been applied for over a decade now, during which time their use has steadily increased from their beginnings in research to regular application in a variety of practical projects and service jobs for exploration offices. This paper sets out to demonstrate their power in quantitative seismic data analysis.

A. The Signal- Plus- Noise Model of Coherence Analysis

In coherence analysis, a signal is defined as a component in a multichannel time series that is present as a scaled or filtered version on several channels which causes the series to correlate from channel to channel. The noise is the remnant on each channel that does not correlate with any

Manuscript received May 15,1984; revised May 31, 1984. The author is with British Petroleum Company, London, England

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other component on the same or any other channel. The usual model for the recordings x ~ , ~ i s a two-component additive mixture of signal s ~ , ~ and noise U k , t

X k , t = S k . t i- u k , t (1) where t is discrete time and k is a channel number. A multicomponent model could be set up if need be [39], [44], but it is more sensible in terms of estimation error and ease of analysis to try to reduce the data to a form to which the model (1) applies.

It is not necessary for coherence analysis to assume any stochastic properties for the signal s ~ , ~ (see Section 111). Assumptions about the noise are inescapable and it is assumed to be generated by a stationary random Gaussian process. The Gaussianity is needed to derive sampling distributions for the spectral estimators which, according to Goodman [16], should also be applicable to estimators from many non-Gaussian series. It is not easy to test the Gauss- ianity of seismic noise unambiguously but an attempt to do so using the residuals from matching synthetic seismograms [55] left the assumption unchallenged. Among robust estimation procedures [27l, [30], devised for noise distributions having heavy-tailed deviations from the Gaussian condition, none appear to be appropriate for estimating seismic noise. The remarks of Snee [46] on robust estimation and his quotation from Box [A, that “all models are wrong, but some are useful,” are pertinent here. In reflection seismology there are always too many uncertainties in the signal model to allow any easy rejection of so useful an assumption as Gaussian noise.

What is meant by seismic signal and seismic noise cannot be immediately identified with the definitions of signal and noise in coherence analysis. The appropriate identification depends on the type of analysis employed and on the seismic data at hand. A guide to the meaning of the terms follows, and later sections come back to the identification problem, drawing out a more concrete meaning with the help of examples.

6. Seismic Signal and Noise

The outcome of a seismic reflection survey is a collection of traces x ~ , ~ which can be organized into gathers and sections of various kinds, such as shot-point (SP) and com-

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mon-midpoint (CMP) gathers [41], and constant-offset and stacked sections. As long as the dips of the reflectors below the survey line are not too steep, the traces within a CMP gather can be taken to contain a common reflection signal consisting of primary (single bounce) reflections, all from the same sequence of reflection points as the normal incidence reflections arriving at the CMP. By applying time- variant (normal moveout, NMO) and static time shifts [41], this primary reflection signal can be aligned on the traces of the CMP gather and enhanced by stacking (summation). Thus the stacked section gives a first approximation to the normal incidence primary reflection signal along the survey line.

The sequence of primary reflections recorded on a seismic trace is what seismologists usually intend by the term seismic reflection signal. It is as close as a simple definition can come to the reality, for a full understanding of seismic reflection signals requires some background knowledge of recorded and synthetic seismograms. A study of well logs and the synthetic seismograms constructed from them shows that reflection coefficients in the earth are very finely spaced relative to the wavelengths that can be transmitted through it. The recorded versions of these coefficients are so heavily filtered that, even after the best that deconvolu- t ion can offer, the primary reflections observed on processed seismograms overlap to a considerable degree and are never seen individually. There are regions of the world where there is such a density of strong reflection coefficients that it is not even possible to construct a unique synthetic primary sequence from a well log [43] nor is it easy to see conceptually how to define it. There are, fortunately, many others where the logs produce relatively few large reflec- t ion coefficients and the primary reflectivity can be com- puted without difficulty. Recorded seismograms fit in well with this picture: primary signals can usually be recognized fairly easily in regions where the logs are simple whereas they can be recorded and identified only with great difficulty when they have to penetrate an overburden contain- ing a large number of sizeable velocity contrasts.

Primary reflectivity spectra [54] tend to rise to a corner frequency above which they are fairly flat. This corner frequency and the rate of dropoff below it are diagnostic of how cyclic the rock sequences are and what range of geological epochs they represent. The other determinants of the character of recorded seismograms are the noise and the net filter through which the reflections are seen. The noise is also dependent on the geology and on the source, as well as factors such as the weather and the proximity of human activity. The influence of the net filter acting on recorded reflections is covered in Section 11.

Seismic noise [4], [I21 is everything that is not signal. The most relevant point here is that i t contains components that correlate from trace to trace, the most signal-like being multiple reflections. Consequently, multiples are liable to be confused with the seismic signal. This is more so of short-delay multiples than longer delay ones and the con- ceptual difficulty in defining the primary reflectivity from complex well logs relates to the disappearance of any clear division between primaries and very-short-delay multiples. The statistical properties of reflection coefficients ensure that these multiples reinforce the primaries towards the low-frequency end of the seismic spectrum, producing a low-pass absorption-like transmission response [22], [42].

Without short-delay multiples one would not see "primaries" at all in, say, much of the Middle East.

The discussion so far suggests that an appropriate model for seismic data would be

'k. i = S k . t + ' k , t + U k . r ( 2) where the reflection signal s k , , consists of a filtered reflectivity (sequence of reflection coefficients) with primary and multiple content; v ~ , ~ represents the correlated noise, and uk,, the random noise. The principal requirement on any group of traces subjected to coherence analysis is that s k , , be formed from a common underlying reflectivity r,. The traces in a CMP gather would normally approximate this condition fairly well and, since lateral changes in r, are almost everywhere very slow, so too would a small number of adjacent traces from any gather or seismic section. In the second place, if we wish to apply model (1) to the multitrace estimation of signal and noise, measures must be taken to reduce the correlated noise.

The presence of multiples is not a great hindrance to the application of (1). Because they are so signal-like in the way they correlate across the small groups of adjacent traces that multitrace analysis employs, it is not practical to in- clude them as a separate component in the trace model and the use of (1) is appropriate whenever they are the only significant correlated noise in the data; moreover, the filtering undergone by primaries and multiples is very similar. As a consequence, the reflectivity rt has to be interpreted in terms of the processing through which the data have ad- vanced. When field data are analyzed, rt corresponds to the total primaries-plus-all-multiples (p + am) reflectivity; after deconvolution and stacking, the multiple content is di- minished to largely short-delay and low-frequency components and r, is a primaries-plus-residual-multiples (p + rm) sequence.

The above remarks are directed towards the interpretation of the coherent signal spectrum obtained from the cross-equalization and multitrace techniques of Sections V and VI. In matching synthetic and recorded seismograms (Section IV) a model of the reflectivity is available and it is valuable to split r, into primary and multiple components and to use these as templates for estimating their contribution to the data. Most other noise in the data, whether i t correlates from trace to trace or not, can be treated as random; for i t is effectively independent of these components.

11. SPECTRAL CHARACTERISTICS OF SEISMIC REFLECTION DATA

A. The Seismic Chain

It was noted in Section 1-6 that the geology is only one of the factors that influence the character of seismic signals. The seismic chain [3] also includes the source and the recording and processing systems, all of which introduce filters through which the signal is transmitted. The total filter through which a recorded reflection has passed is called the seismic wavelet [47]. The ghosting and reverberation effects of the immediate subsurface and the time- variant attenuation of amplitudes due to absorption and transmission in the earth, which are strictly part of the reflectivity, are normally included in it and its shape changes, mainly for the better through deconvolution, as processing

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progresses. The seismic signal spectrum is generally dominated by the transfer function of the seismic wavelet.

The outline of the previous section led to the basic idea that the seismic reflection signal is a filtered version of a primary (p) or a primary-plus-reinforcing-multiples (p’) reflectivity, represented by the model

S k , t = W k , t * rt (3) where W k , t i s the seismic wavelet on channel k and the asterisk denotes convolution. Marine sources are so repeat- able that W k , t i s virtually the same from shot to shot; that i s frequently not the case in land work when the standardiza- tion of the wavelet becomes a prime aim of the processing. The wavelet’s time variance is generally weak after 0.5 s or

The characteristics of seismic signal and noise spectra are best seen from examples. A marine test line, which has been thoroughly studied and which passes through a well where synthetic seismograms match the data very clearly, provides the data for most of the examples presented here. In this section, signal and noise spectra estimated by the multiple coherence method of Section VI from data from two sources with approximately the same energy but very different signatures (far-field source pulses) are compared. An alternative decomposition, by means of synthetic seismogram, is presented in Section IV.

B. Field Data Examples

so.

Fig. 1 is an SP gather (or shot record) whose trace amplitudes have been corrected for recording gain and com- pensated for wavefront divergence and absorption. The wedge of strong amplitudes including and following the first breaks contains direct arrivals through the water, critical and super-critical reflections from the sea-bed, and refractions from below it. These are straight away muted out (zeroed) in processing as undesirable noise. There is a strong primary reflection just after 2.6 s on the inner traces and i t is followed by a train of reverberations, due to

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multiple reflection in the 70-m-thick water layer, and other primaries with their own reverberations. Crossing the whole record is some correlated noise with apparent velocity about 2000 m/s which comes from scatterers in the shallow section close to the sea-bed [28].

Equation (2) i s clearly apposite to the data of Fig. 1 and the f-k power spectrum is a suitable tool for analyzing them. Fig. 2 shows the f-k spectrum formed from a set of

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gathers centered on that of Fig. 1 over a 1.0-s gate starting at 2.5 s. The power from the primary and multiple reflections i s concentrated in a wedge to the right of the frequency axis and that from the scattered noise lies on a line corresponding to roughly 2000 m/s and aliases into the left-hand quadrant. The spectral peaks at 16, 27, 38, and 49 Hz come from the reverberations and the spreading of the power along the wavenumber direction, noticeable at 27 Hz particularly, is characteristic of amplitude and timing jitter in the data. Statistical theory and detection criteria have been developed for f-k analysis [IO], [45] but so far their usefulness in reflection seismology has not been demonstrated by applications.

These data were shot with a water-gun array whose source spectrum and signature are shown in Fig. 3 and it is not easy to see its influence in Fig. 2. The signal spectrum estimated by multiple coherence shows i t better, especially if this spectrum is compared with that from a different source, the air-gun array whose signature and spectrum are given in Fig. 4. Since the scattered noise is least noticeable

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Fig. 3. Measured far-field signature from the water-gun source with which the data of Fig. 1 were shot, and its energy spectrum.

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Fig. 4. Measured far-field signature of an air-gun array and its energy spectrum. The line shot with the water-gun array was also shot with this array.

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nom - Fig. 5. Signal and noise spectra estimated from a 1.8-3.8-s time gate on portions of the near-trace sections from the water-gun (top) and air-gun (bottom) arrays. Analysis bandwidth 4 Hz after double-gapped prefilter.

on the inner traces of Fig. 1, multitrace analyses for the signal (p + am) and random noise spectra were carried out on near trace sections and gave the signal and noise spectra shown in Fig. 5. Although i t is the reverberation peaks that dominate, the effect of the hole in the water-gun spectrum at 37 Hz (Fig. 3) is evident.

C. Stacked Data Examples

The signal and noise spectra can be followed through processing, and the preliminary stack used in initial parameter testing (Fig. 6) and the final processed stack (Fig. 7 ) are two obvious markers. The data were phase-corrected for the departure of the source signatures from minimum phase, a negligible correction in the case of the air-gun array, and deconvolved with a spatially averaged whitening operator long enough to make some impression on the reverberations before forming the preliminary stacks. Their signal spectra (Fig. 8) show that some residual reverberation re- mains. Although the scattered noise on the section (Fig. 6) makes the results at low frequency dubious, the peak in the noise spectra near 10 Hz suggests that much of it has been treated as noise, presumably because the correlation lag window excludes a considerable proportion of the cross- power in low-frequency events with such a large moveout per trace.

The main differences between processing the preliminary stack and the final stack consisted of measures taken to reduce the scattered noise seen on Fig. 6, the use of a longer pre-stack deconvolution operator, and post-stack space averaged predictive and whitening deconvolutions, time-variant bandpass filtering, and trace equalization. The section (Fig. 7) shows that the correlated noise has been attenuated and the signal spectra (Fig. 9), that the deconvolutions have, left no significant reverberation. Over the 1.8-3.8-s interval the two signal spectra are now indis- tinguishable, but they did differ at earlier times.

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D. Spectral Estimation

The methods of spectral estimation employed in these examples are those of Blackman and Tukey [6] and Jenkins and Watts [24]. The program for multitrace signal and noise analysis applies a single whitening filter to all the data

before correlation, windowing, and Fourier transforming the correlations. The lag window is normally that of Papoulis [35, note I, = 0.587 MI. Finally, the estimates are corrected for the power transfer of the pre-filter. A very short operator is preferred for automated running of the program as it has been found that, on occasions, there can be deep holes in the power-transfer functions of long ones which give spectral leakage the opportunity to create problems. Where better prefiltering is desirable, a spectrally well-behaved design can be tailor-made. Thus estimation from the near traces (Fig. 5) needed a dereverberating prefilter and the total number of coefficients was kept down by using a double-gapped design with active portions around the two-way and four-way water times. The limitation on the number of coefficients helps the stability of the operator by avoiding often worthless coefficients between these two lags.

The development of improved methods of spectral estimation is a side issue to this paper, but an important one. A major need is for better estimation outside the seismic bandwidth. This is absolutely crucial to the determination of minimum-phase seismic wavelets [62] since the phase calculation requires the signal power spectrum over the complete Nyquist range and is particularly dependent on its low-frequency dropoff [W]. Also, because seismic noise cannot be measured separately from signal and the bandwidth-duration product of analyzable signal segments is severely limited, the main concern of spectral estimation in reflection seismology has to be the balance between reduc- ing statistical variability and controlling residual distortion. The word residual is applied here to indicate that remaining after the adoption of an appropriate analysis model or suitable preconditioning of the data. In neither respect is there much help from modern spectral analysis [25], where the attention paid to algorithmic developments, particularly to estimating narrow spectral peaks by means of autoregressive (AR) and related methods, appears excessive, especially as there is a relative paucity of work on error measures (e&, [33]). Data-windowing techniques [52] and the more elaborate approach of Thomson [49] are promising approaches for seismic analysis.

Ill. THEORETICAL BACKGROUND

The elements of coherence analysis are well covered in several texts (such as [21], [24], [36]) and the reader is referred to them for derivations where no reference is given. Our starting point is (1) in which q t is assumed to be generated by a stationary Gaussian process that is uncw- related with the signal or noise on any other channel. Most authors also take the signal s ~ , ~ to be a stationary random process, which is convenient for the purposes of derivation, but this assumption is not essential [17], 1181, [20]. The condition that any transfer function in s ~ , ~ be smooth over the analysis bandwidth is needed in any frequency-domain derivation for least squares estimation of transfer functions [24, ch. IO] and is sufficient, with the assumption on u ~ , ~ for deriving approximate distributions for transfer function and signal-to-noise estimates [55, app. A]. Avoiding the assumption of a stochastic signal affects expressions for large sample variances since the variability is then due to noise alone, separate from any extra variability due to possible signal sampling (see Sections Ill-C and Ill-D).

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A. Notation and Definitions goodness-of-fit is the proportion of total power on trace k

Let Sx(f) denote the spectral matrix of the traces X k , , ,

@ j k ( f ) i ts elements, and kX( f ) i ts estimate. The notation is simpler if the indication of dependence on frequency f i s dropped when there is no danger of ambiguity. When the signal is stationary and stochastic

S X = $ & S + $ ” (4)

% x = 2 s + %LI. (5)

and when it is a filtered version of a definite reflectivity rt -

Because the noise is incoherent, its power spectral matrix su is diagonal with elements Nk. Ls here is the smoothed auto- and cross-spectral matrix given by

gs = L-s’c-//T ( 6)

where _S(f) is the row vector of the Fourier transforms of the S k , , segments over a specific time gate of duration T, $( f ) is its conjugate transpose, and the bar indicates averaging (its form depending on the constraints of the analysis). From the assumption of smoothness of the wavelet power transfer functions:

- 2 s = B+( f ) 9 r ( f ) _ ~ ( f ) ( 7)

where _H(f) is the vector of wavelet frequency responses and 9 , ( f ) is the averaged autopower of r,

- arr = 1 R( f ) R( f ) J/T. (8)

The signal-to-noise (S/N) power ratio P k on trace k and its reciprocal, the noise-to-signal (N/S) ratio /.bk are

P k E I//.bk IHk125r r /Nk . (9)

The overall S/N ratio of the trace is found by integrating lHk12Srr and Nk over frequency and dividing them.

All the essential features of the theory of coherence functions are contained in the simplest one, the ordinary coherence y h ( f ) , also called the squared coherency [24] and magnitude-squared coherence [34]. It is the squared modulus of the normalized cross spectrum, or complex coherency [36), between traces j and k

y j Z k ( f ) 5 1 @ j k ( f ) 1 2 / @ j j i i ( f ) @ k k ( f ) . (1 0) It measures the proportion of power around frequency f on one of the traces that can be predicted by linearly filtering the other to produce a least squares best fit of the two. The frequency response of the cross-equalizing or matching filter that converts trace j into a least squares best fit to trace k is

Hkl j ( f , = @ j k ( ‘ ) / @ j j ( f , (11)

which for the two-component trace model (1) gives

H k l j ( f , = H k ( f ) / H j ( f ) [ l p j ( f ) ] . (12)

The filter is equivalent to a Wiener noise suppression filter (1 + p j ( f ) ] - l for the input trace in series with a filter that compensates for the difference in wavelet spectra. The coherence is related to the N/S ratios of the two traces by

Clearly, more than cross-equalization is needed to de- termine the S/N of the traces. A useful measure of their

predicted by filtering-trace j , or the effective coherence over the whole seismic band, called here the predictability

The multiple coherence function yl ( f ) of trace k in a set of 9 + 1 traces is likewise related to the multichannel filter H k l q ( f ) that predicts it in a least squares sense from the other q traces

pklj .

B k l q ( f , = &l( f ) $ k ( f , (1 4)

where %-,( f ) is the inverse of the spectral matrix of the q traces and $ k ( f ) is the column vector of their cross spectra with xk+ The residual power on X k , t not predicted by the filtering is

@ k k l q = @ k k - g k q g i $ k = [@“I-’ (1 5)

where akk is the k th diagonal element in the inverse of the full ( q + 1) X (9 + 1) spectral matrix. The multiple coherence is again the proportion of power predicted

[ @ k k - @ k k l q ] / @ k k - [ @ k k @ k k ] -’ (I6)

and it i s related to the S/N ratio on X k , t by [56]

with

For ( q + 1) > 3, the S/N ratios can be determined itera- tively from the multiple coherences; when q = 1 the ordinary and multiple coherences are identical.

Partial coherence analysis is used to decompose the trace X k , t at a well into synthetic seismogram components and noise. Useable segments of seismic data and log are not long enough to permit accurate estimation of more than two input components, the normal pair being a p or p+ sequence and a multiple (m) sequence. In this case, (2) is the data model, with the correlated noise v k , t being represented as the m reflectivity filtered by its wavelet in the same way as the primary signal (3). The wavelet frequency responses are determined by substituting the 2 X 2 spectral matrix of ‘ p , f and rm,f and their cross spectra with X k , t into (14) and the multiple coherence of X k , t is the proportion of trace power that can be predicted from the synthetic. An S/N ratio can be formed for X k , t by dividing the predicted (or signal) power

by the residual (or noise) power

The ratio is a measure of the goodness-of-fit of the filtered synthetic to x k , t and i t depends on the accuracy of the signal model and not just on the amount of noise on X k , , . It can attain the value from multiple coherence analysis of the traces next to and including x k , , only i f the synthetic can explain all the coherent power on X k , f . It never does!

The partial coherence of the input sequence q,t, with the trace X k , t , the output of the multichannel system, IS defined from their so-called conditioned or partial spectra, Q j j I q - , , and @ j x l q - l . These are the spectra of the condi-

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tioned traces formed by removing from 5 , t and x k , t their least squares predictions from the other input(s). Equations like (15) give the conditioned spectra, which for the two-input case are

and the partial coherence is defined by

which is most easily evaluated from [59]

For the two-input case

The left-hand side of this equation is the residual power from the two-channel match, expressed as a fraction of trace power, the first term on the right is the residual fraction after predicting from the first channel only, and the last term, therefore, shows what further reduction in residual power is achieved by including the second channel. That is, small partial coherences are a sign that an input is unimportant. The analysis takes care of any correlation, real or spurious, between the input synthetic sequences.

B. Sources of Error

Estimates of coherences, S/N ratios, and frequency responses are formed by replacing population spectra in the equations of Section Ill-A by estimates. Errors, therefore, propagate through to the quantities we wish to measure and before we can interpret the results we need to know how reliable they are. The great virtue of coherence analysis lies here, in i ts ability to provide clear and detailed error information, and is the main reason for preferring it to correlation techniques. It is hard to overrate the practical value of this information and the ability that it brings to delve into the data. By contrast, errors in correlation and time-domain estimates from colored data tend to be so complicated that they obstruct any detailed error analysis.

Noise on the data is an inescapable source of error. It puts random errors into the spectral estimates. So too can signal sampling in cases where that concept holds. These two sources of error are the ones that the statistics described below provide measures for. Tick [50] points out that two approaches are possible: an approximate solution to the exact problem by means of large-sample expressions, and an exact solution to an approximate problem through a tractable statistical distribution. The latter is the more accurate but large sample variances and covariances bring out some simple dependences and are better than nothing where no distributional solution has been found. Goodman [15], [I61 derives the basic results for both approaches and Jenkins [23] summarises the essentials very well.

Another type of error, bias or distortion, is just as inevita- ble as random error. The spectral window of B-T analysis

introduces smoothing and leakage bias into power and cross-spectral estimates-other methods have other mecha- nisms-and procedures must be adopted to control them. Preconditioning of the data and the constraint on operator length or analysis bandwidth (which estimation of the noise demands) both have a role to play. Preconditioning involves mainly the selection of suitable data, the attenuation of correlated noise, and some whitening of the signal. The selection of filter length or analysis bandwidth is considered after random errors so that their tradeoff with the bias can be appreciated. A sound tradeoff leaves a bias that is a fraction of the random errors and the theory of the next two subsections assumes that is so. This can certainly be achieved within the seismic bandwidth but the strong spectral decays outside i t mean that distortion can never be subordinated there by 6-T analysis.

Truncation of the data introduces some error into estimates made with frequency-domain methods or with time- domain methods that use Toeplitz correlation matrices. It rubs out information about the convolutional contributions to the signal at the edges of any time gate and signal prediction suffers as a result [8]. Truncation errors are most noticeable when the gate is short in relation to wavelet length and the practices of positioning it to start and end in dead portions of data and of tapering its ends help keep them down. A time-domain design that avoids truncation errors (91, a two-channel version of the exact least squares methods of Ulrych and Clayton [SI] and Swingler [a ] , was implemented for matching purposes. No practical case has been found to suggest that truncation errors are important in seismic work and simulations with the program indicate that they are overshadowed by random errors in any segment long enough to afford reasonable accuracy. For best results, the correlations have to be tapered to counteract their increasing corruption by sampling errors at larger lags.

Cross spectra are subject to a further type of bias called misalignment bias. Unless the S/N ratio is so poor that traces cannot be aligned because of cycle skipping, it is not difficult to avoid this automatically, either by picking cross-correlation peaks or finding the alignment for maximum predictability. NMO and static corrections rely essen- tially on the first of these approaches and their application may on occasions be important before multiple coherence analysis; the second approach is used in our programs for matching and cross-equalization.

C. Large-Sample Variances and Covariances

The variance-covariance matrix for dimensionless power and cross-spectral estimators given by Goodman [I51 is the starting point for deriving large-sample expressions for the variability of quantities such as estimated frequency responses

A = l q e x p ( i8) ( 24)

and S/N ratios (6). Goodman’s results assume a stationary stochastic signal; expressions for the case of no signal sampling are given later.

All the standard results follow from Goodman’s matrix; for example, the well-known equation for the variance of a power spectral estimate

var { & k k } E @:k/bT. (25)

Since & k k is equivalent to a linear combination of indepen-

WHITE: SIGNAL A N D NOISE ESTIMATION FROM SEISMIC REFLECTION DATA 1347

dent estimates with variance a&, the bandwidth-duration product bT is an averaging or smoothing factor equal to the effective number of independent estimates averaged into d k k and is often called the number of complex degrees of freedom. The analysis bandwidth b, the bandwidth of the spectral window, is looked at more closely in Section Ill-D.

The following results apply to the estimators of interest in ordinary coherence analysis:

var { fi} = 2 Y i ( 1 - Y j g 2

bT

cov { I Q J } = 0. (28)

Replacing y$ by y i or Y $ , ~ - , in (26) gives the large-sample variance for either of the other coherences. The use of (bT - q) instead of bT in the denominator would improve the accuracy at smaller values of bT. It is reasonable to expect that (28) can be applied to the frequency responses estimated for the q-input channel case, and (27) too i f l / b T is replaced by q/(bT - q ) but the author knows of no proof of this. Multiple and partial coherence analyses provide estimates of signal spectra &ss, noise spectra & I k , and S/N ratios ik . Their variances are

var { dss} 2 ~ ; ~ - I @:S

bT E

Equations (29)-(32) still hold, of course, when a single- channel match is used to provide estimates of these quantities on the assumption that the input is noise-free.

When the signal is considered to come from a fixed sequence, the expressions become more complicated. It is not too difficult to set up the variance-covariance matrix of the sample power, co-, and quadrature spectra. For example, the variance of the sample power spectrum is

(33 )

Equation (28) is still applicable, as is (27) for var {d} , but var { IQ} becomes

Equation (27) approximates this fairly well provided the input channel j is not too noisy. The expression for var { ti} is

(35) It appears that the standard equations for estimates of frequency response are adequate, as long as the input channels are not noisy, but not those relating to the signal such as (29), which is replaced by (42) below.

0. Distributions and Inference

The statistical distributions for the estimates of physical interest all follow from the approximate chi-squared ( x 2 ) distribution of power and power spectral estimates. A chi- squared distribution is described by i ts number of degrees of freedom (d.f.) Y , i.e., the number of squared independent Gaussian random variables (rv) whose sum forms the x 2 N. The coefficients in a sequence of colored random noise are far from independent and its estimated variance or power is not strictly a x 2 rv even if the noise is Gaussian but the x2 distribution can be made to fit that of the power rv well enough by equating the means and variances of the two. Hence an effective number of d.f. can be associated with the power estimate (@. The result is [58]

I - T

where +, is the autocorrelation of the noise and the equation introduces B its equivalent statistical bandwidth. The standard expression for 6 comes from approximating the term in brackets in the denominator by 1. This approximation is generally very close. The statistical bandwidth b of a spectral window can be established from this result [58]

T - l

b = c ( 1 -(l~l/w4 L T I-’ (37)

where w, is the corresponding lag window. The standard expression again results from setting the bracketed term inside the sum to 1 but the approximation becomes noticeable when the number of d.f. is small (bT c IO), an often unavoidable occurrence in seismic work. Simulations [58] have shown that (37 ) fits the statistics of matching better than the standard one.

The distributions for all types of coherence function and some related distributions were worked out by Goodman [15]-[I71 and discussed further by Khattri [26]. That of the ordinary coherence function in particular has been thoroughly studied [ I l l , tables are available [2], and efficient means of computation [29]. All the coherence functions behave similarly. The estimates obtained by substituting the sample spectra into (IO), (16), and (21) are biased by spurious correlations from the noise sequences, i.e., the smoothed cross-products involving noise are not exactly zero. This chance correlation bias can be simply and accu- rately removed by ([SI, [56D

where f 2 is the unbiased coherence and q the number of input channels. Equation (38) follows from the x 2 decomposition of spectral trace power; see (41) below. It should be noted that some authors [34], [a] have proposed more elaborate methods of correcting for bias.

Although the coherence is the fundamental measure of the analysis, i t is the signal and noise spectra and response functions, and their reliability, that are physically interesting and their sampling distributions are more important to us. In matching X k , r to the q traces considered as inputs i ts power is decomposed into a predicted signal power and a residual power

( 39)

1348 PROCEEDINGS O F THE IEEE, VOL. 72, NO. 10, OCTOBER 1984

From this equation, the sample noise power spectrum on trace k can be related to the error power in the predicted signal (due to the filter errors AM and the estimated residual spectrum

f i k = hfl&qABT + & ) k k ( q . (4.0)

Equation (40) is an analysis of variance which, when scaled by 2 b T / N k , represents the decomposition of a x 2 N with 2bT d.f. into two x 2 components with d.f. 2 q and (2bT - 29) . Because E { X ~ 1 = Y , the ratio of the expectations of f i k

and is b T / ( b T - q) and an unbiased estimate of Nk is

and & k k - kk is an unbiased estimate of signal power. The ratio of unbiased signal power to & k k gives the unbiased coherence (38).

Equation (39) is also a x' decomposition but ( 2 b T / N k )

butions with noncentrality 2bTpk = hk [55]. Their ratio, the estimated S / N ratio B k , multiplied by ( b T - q ) / q then has a noncentral F distribution with 2q, 2bT - 2 9 degrees of freedom and noncentrality 2bTpJ. Equation (33) can be derived from the distribution for @,.k by using var { x ; , A } = 4X + 2v while the distribution for QS5 gives the variance of the estimated signal spectrum caused by the noise as

- +& 9 9 - f i r and ( 2 b T / N k ) & k k follow noncentral x 2 distri-

Addition of the hypothetical signal sampling variance @;JbT to this gives (29) to order l / b T .

Because the d.f. associated with the trace power are partitioned at every frequency as q: ( b T - q) between the signal and residual power, the 2BT d.f. of the total trace power are partitioned in the same way and the distributional results just given pass across to the estimates of total signal power, total noise power, and overall S / N ratio if BT replaces bT. Similarly, the predictability &.lj follows the same distribution as fi but with modified d.f.

In matching, the overall S/N is a measure of the goodness-of-fit and its distribution can be used to assess the reliability of the match. The most basic question is whether the synthetic and data are related at all [59]. If not, then, according to model (I), the recorded seismic trace is just uncorrelated noise, H = 0 and AH = f i in (39) and (a), and the estimated S / N ratio has a central F distribution. If the observed estimate does not exceed, say, the %-percent confidence bound of this distribution, there is no reason to suppose that we are not matching random noise and either the data or the synthetic must be rejected as worthless.

Once a match has been detected, by passing the test just described, its assessment can move on to how reliable it is for estimation purposes, which is what we are really inter- ested in. A suitable criterion is the accuracy with which the signal is predicted, which in turn is related to the accuracy of the estimated filter responses. The real and imaginary parts of the filter error, AH = f i - H, are independently normally distributed. Starting from this point, Munk and Cartwright [31] have derived the joint distribution for the normalized gain Imq and the phase error 8 - 8 , from which confidence intervals on the gain and phase can be set up. To do so requires y i , the population coherence, and the best that can be supplied is the unbiased estimate $. There is much to be said for using the measured

coherence and the test from Goodman [I71 based on (40) allows this. It follows from that equation that the ratio

(43)

follows the central F distribution with 29 and 2bT - 2 q d.f. from which Goodman sets up a %-percent confidence region for the response f i k on channel k . it is the interior of a circle in the Re(H), Im(H) plane and reduces to

Fig. 10 shows three such circular confidence regions for a one-channel match having 20 d.f. at each frequency and

4

0.2 0.4 0.6 0.8 1.0 1.2

lMCa e

+l

-1

-3 1 I 1 1 1 I 1 0.2 0.4 0.8 O J 1.0 12 1.4 ndirr e

Fig. 10. Mapping of the circular confidence regions for the real and imaginary parts of the frequency response into confidence regions for logarithmic gain and phase angle. The radii in the top part of the figure are in the ratio 1 : 2 : 4 and correspond to estimated coherences of 0.95, 0.84, and 0.57 when b T - 10 and the regions are chosen for %percent confidence. The p's and 4s mark points of tangency.

corresponding to very good, good, and mediocre coherence such as might be encountered on passing from the middle to the edge of the signal bandwidth. The mapping to joint confidence regions on logarithmic gain and phase is shown below, Usually, confidence bounds on gain and phase are quoted separately, which is equivalent to approximating an oval in Fig. 1 0 by a rectangle that encloses it. Naturally this overestimate produces conservative limits.

There are other ways of specifying these confidence intervals and Walden [53] has reviewed them and elaborated Munk and Cartwright's (M & C) treatment. Walden's com-

WHITE: SIGNAL AND NOISE ESTIMATION F R O M SEISMIC REFLECTION DATA 1349

putations of the variance of the normalized gain from the M & C distribution show that the large-sample expression (27) is satisfactory as long as

y-2 - 1 2bT

after which i t underestimates. The confidence limits on gain and phase allow a very

detailed look at the statistical reliability of the estimates. Often it is convenient to have an overall measure of accuracy and the error power in the predicted signal, the in- tegral of the first term on the right of (40) provides this. Its expectation, or mean, is (q/bT) of the total noise power. Normalizing this mean random error power (NMREP) with respect to the total signal power gives

N M R E P = (9 /bT) (overall N/S ratio). (46) Substitution of the estimated N/S ratio gives a point estimate but the distribution is skewed and confidence limits are a better guide. They can be set from the noncentral F distribution for S/N described above.

0.1 (45)

E. Choice of Analysis Parameters

Equation (46) expresses the way in which the duration of the data segment and noisiness of the data control the accuracy of estimating the signal. This accuracy controls the selection of the data segment for any type of matching; in multiple coherence analysis it is the signal and noise content of the data over some preset interval that is often of interest. Generally, good accuracy needs long segments but practicalities, such as temporal variations in the S/N and the availability of reliable log, intervene in the selection of a time gate.

The increase in NMREP with q indicates that the trace model should not be too elaborate for the capabilities of the data and that the choice of input channels should be restricted to those that really matter. An analysis of variance can be carried out to test whether an extra channel is worth including in a match. If the overall S/N increases from pq-l to pq with the extra channel, then

(bT - 4 ( P q - Pq-1)

1 + Pq-1 (47)

is distributed as F with 2(B/b) and (bT - q)(B/b) d.f. and can be compared with the %-percent point. Considerations for multiple coherence analysis are somewhat different because the use of several traces offers some insurance against the effects of any correlated noise on them. It cannot operate on less than three channels in all ( 9 > 1) and the use of four at a time is a reasonable compromise. The avoidance of large corrections for the chance correla- t ion bias in coherence (38) and S/N estimates is another advantage of limiting q in this case.

The choice of analysis bandwidth b, or equivalently of effective filter length, is a matter of keeping residual bias well below the level of the random errors, at least within the seismic bandwidth. In that band, the main cause of bias is the smoothing action of the spectral window. Papoulis [35] shows that, for small values of b and certain types of windows, including his, the smoothing bias is proportional to b2 and inversely proportional to the local curvature of the power spectrum

B( f ) = Cb2@”( f ) (48)

where the constant C depends on window and is 0.043 for his, and 0.044 for the Parzen window. Combining this with (46) gives a total normalized mean error power (TNMEP) of the form

TNMEP = Cb4 + bT( 4 N/S) (49)

Although this equation is approximate, especially i f carried through to the overall TNMEP, it does embody the essentials of choosing b. In particular, it brings out how sharply the bias rises with increasing bandwidth. It follows that the TNMEP is a minimum when the bias error power is a fraction, a if the equation holds, of the random error power.

The steep rise in bias to one side of the optimum filter length (- l / b ) and the gradual increase in random error on the other are illustrated in Fig. 11, which shows the varia- t ion in the normalized error energy (NEE) of a wavelet measured from some simple simulations. The lowest curve marks the bias, obtained by comparing filters of various lengths estimated by matching a filtered and unfiltered synthetic seismogram with the filter actually applied, and the other curves are the result of adding to this the expected random error energy given by the second term on the right of (49). The minimum in the NEE becomes broader and the optimum filter length increases with increasing S/N. The sharp rise in NEE due to bias has been found from simulations to be purely a function of b [9].

A rough rule of thumb for choosing b i s that i t should never be more than B/2. The simulations of Fig. 11 were run with the time-domain method referred to in Section 111-6 using a Papoulis window and, according to this rule, the filter length should exceed 61 ms. For seismic work, the rule is a safe one and Fig. 11 shows that there would be no sense in risking strong bias by chasing the exact minimum NEE. In practice some “window closing” trials [24] are in order, the wavelets being inspected for signs of distortion or noisiness. Another approach is sometimes possi-

a IO 4

0

SIN mno 1: 4

QD

i SIN = I ”- 4 msunm 03

fig. 11. Variation of filter error with filter length for various signal-to-noise power ratios from simulations in which filters of various lengths were estimated by matching segments of broad-band and noisy filtered synthetic seismogram.

1350 PROCEEDINGS OF THE IEEE, VOL. 72, NO. 10, OCTOBER 1984

ble in cross-equalization where longer segments of match- able data are available. It requires enough data for two matching gates: a design gate, and a test gate. The fit in the test gate after applying operators estimated from the design gate can then act as a guide to the best operator length [8].

The Akaike Information Criterion (AIC, [I]) is often put forward as a means of determining operator lengths. In our experience [55] the application of the AIC to matching operators leads to oversmoothed seismic spectra, the last thing one wants. It appears that order-determining criteria, based as they are on regression theory, take no account of bias in design matrices and therefore miss the main penalty on short operators in matching.

IV. QUANTITATIVE MATCHING OF SYNTHETIC AND RECORDED SEISMOGRAMS

Synthetic seismograms are commonly applied in seismic interpretation to the tying of seismic sections to well logs and coherence techniques are indispensable in establishing reliable identifications of reflection horizons and appreciat- ing the composition of the section. In data acquisition, a properly matched synthetic provides an excellent means of assessing data quality and in data processing i t is the only firm basis for the production of zero-phase sections while the ability to track the seismic signal from the field data to final section through synthetics offers an objective method of selecting processing parameters.

The main drawback to achieving these aims lies in the difficulty of obtaining accurate synthetic models, which in turn relates to the acquisition of long enough segments of reliable well logs. In spite of the snags, synthetic seismograms are generally an aid to interpretation and processing, and we have found an encouraging number good enough for really reliable wavelet processing. A minority, nearly all from regions where the logs are swamped by rapid and large velocity contrasts (Section I-B), are no help at all; most of these show some correlation with the data but not one good enough for estimation purposes (Section Ill-D). This may not be the fault of the logs but of the synthetic seismogram model.

A portion of the calibrated log and plane-wave broad-band synthetic from the well on the test line of Section II is shown in Fig. 12. The computations allow the separation of the p, internal multiple (im), and surface multiple (sm) components of the synthetic. The internal multiples are mainly short delay ones, little attenuated by stacking, whereas stacking and deconvolution do attack the surface multiples, so that after these processes the p or p + im (- p + rm) reflectivity i s the main component in the data.

The techniques and statistics of quantitative matching are described by White [59] and Walden and White [55]. The initial work consists of scanning the data for a suitable time gate, assessing the primary and multiple contributions, and selecting an analysis bandwidth. A scan of traces around the well should be included because navigational errors, reflec- tor dip, and patches of noise can all push the best match a few traces from the well.

A scan across the near-trace air-gun section analyzed in Fig. 5 using the p + am synthetic of Flg. 12 is shown in Fig. 13. A good match is found at the expected match location but there is a correlation between the goodness-of-fit (S/N) from matching and the S/N ratio of the data found by multiple coherence analysis and a slightly better fit occurs

DEPTH B.R.T. 'lo00 METRES 2.5 3.0 3.5

3 0 , . ' ' : - _ I

DENSITY 2.0 - (OMS/CCl l,o , t-

ATTENUATED PRIMARIES

INTERNAL MULTIPLES

SURFACE MULTIPLES

2.0 2.1 2.2 2.3 2.4 2 5 2.6 2.7 2.8 2.9 3.0

n w IN SECONDS FROM DATUM AT M.S.L

Fig. 12 Portions of the density and calibrated velocity logs from a well on the test line and the components of the plane-wave normal incidence broad-band synthetic seismogram constructed from it.

12 _I

-44 764 ?70

SHOTPOINT

Fig. 13. Scan of the signal-to-noise ratios found by matching the full broad-band synthetic seismogram with the air-gun near-trace section around the well and the signal-to-noise ratios estimated by multiple coherence analysis of the same section. The left-hand arrow indicates the SP at which the reflection points for the 2.1-2.85-5 time gate coincide with the well.

nine SP away. The estimated wavelets at the two locations are very similar in shape and differ in timing by 10 ms, as expected from the dip of the reflections in the analysis gate. The wavelet estimated from the best match is in Fig. 14 and Fig. 15 shows a comparison of the filtered synthetic and the near traces. %-percent confidence limits on the power transfer function in Fig. 14 are -2.5, +2.9 dB at 16 Hz, dropping to -1.3, + I .S dB at 27 Hz, and back to -2.5, +3.1 dB at 43 Hz. Outside this range, the coherence drops rapidly and its estimation becomes unreliable. The 90-percent confidence range on the NMREP of the predicted signal is 2 to 6 percent.

The S/N of processed data is generally fairly uniform from trace to trace and is less likely to push the best match

WHITE: S IGNAL AND NOISE ESTIMATION FROM SEISMIC REFLECTION DATA 1351

0 16 3 0 4 6 60 76 FREQUENCY (Hz1

Fig. 14. The seismic wavelet estimated by matching the full synthetic seismogram of Fig. 12 with the air-gun near-trace section over the 2.1-2.85-s time gate at SP773 and its energy spectrum.

away from its expected location. This is certainly the case for the air-gun final section (Fig. 7) and a scan for best match across i t gave a very distinct peak (S/N = 1.7 with p) at SP 766. The 50-m displacement is not serious and within the range of navigational error.

Fig. 16 presents a simple example of how synthetics can be used to assess processing. The data come from the first survey shot on this line and the figure shows the effect of deconvolution before stack (DES) from two different processing sequences. Both were chosen in the usual way according to the processor's opinion of the portions of section output from DBS tests. It would be helpful to know what the primary signal and its autocorrelation should look like and matching at a well supplies this information. Fig. 16 indicates that the choice of the 140-ms whitening DES was a reasonable one in that it would preserve the correlation within the primaries just beyond 140 ms, denoted A . The long predictive operator would be expected to dere- verberate slightly better than the shorter whitening one but to degrade primary correlations between 24 ms and 280-ms

lag such as A , i.e., i t confuses primary and reverberant energy. There is some sign of this on the autocorrelations but it appears that the enhancement of primaries by the stack tends to compensate for any damage by the longer filter. For some examples of the effects of different deconvolution procedures on wavelet shape, see Hampson and Calbraith [19].

The S/N values in Fig. 16 can illustrate the test for the significance of the contribution of the surface multiples to the match: the statistic (47) is more than 99 percent significant for the undeconvolved data but only about 70 percent significant for the other two data sets.

v. CROSS-EQUALIZATION OF SEISMIC TRACES

Cross-equalization [8], [61] is just the matching of traces that arise on an equal footing. Errors do not loom so large as in matching to synthetics since the length of useable data is not usually as restricted as with sonic logs and the operators being estimated are mostly simpler than seismic wavelets. In addition, whereas a 70-percent predictability

Fig. 15. The full synthetic seismogram of Fig. 12 filtered by the wavelet of Fig. 14 spliced into the near-trace section from the air-gun array between SP773 and SP774. Note the bad trace (SP779).

NO DBS SYNTHETICS DES (280/24msl SYNTHETICS DBS 114omrl SYNTHETICS PREDICTIVE W l T E N l N G

i 3 5 I e u)

i m z E

S / N FOR P+lM.SM MATCH = 2.6 SIN FOR PclM .SM MATCH =2.4 S / N FOR P+IM.SM MATCH = 2.3

S / N P * I M ONLY = 1.6 S I N PrlM ONLV =2.1 S/N P 4 M ONLV = 2.0

P + l M = PRIMARIES + INTERNAL MULTIPLES SM = SURFACE MULTIPLES SUM = P - I M * S M

Fig. 16. An example of the application of matched synthetic seismograms in monitoring the performance of deconvolution designs.

1352 PROCEEDINGS O F THE IEEE, VOL. 72, NO. 10, OCTOBER 1%

LINED UNE B

Fig. 17. Line intersection analysis: contours of trace-to-trace predictability (proportion of power predicted by matching trace pairs, one trace from each line) at two line intersections in the same survey.

would be considered good in matching synthetics, it is no more than ordinary in cross-equalizing seismic data.

The standard application of cross-equalization in seismic processing is in the design of filters to convert the sections from one survey to versions having a wavelet response equivalent to that of another survey from the same area. The filters are designed at the intersections of the two sets of survey lines and this calls for some exploratory cross-spectral analyses. These analyses are useful in them- selves whether or not the data come from different surveys: they can help resolve uncertainties in location that sometimes crop up in synthetic seismogram work, they can quantify mis-ties between lines, and the predictability at intersections is a useful quality control for acquisition and processing.

A contour plot of the predictability between trace pairs, one trace from each line, is very useful in line intersection analysis. Provided both lines have uniform N/5 ratios, the peak predictability indicates their point of intersection since only there is rk,t the same for both. The contours around this peak can be very tight if the sections have variable dip. The influence of the N/5 ratios can be evaluated, if need be, by estimating them through multiple coherence analysis along each line. Fig. 17 shows two contour plots of pkl, from stacked data from a marine survey. The elongated contours are indicative of lines that do not intersect at right angles. A map of a network of intersecting lines can be built up from seismic data by picking such Pk,, peaks and the solid lines in Fig. 18 were established in this way here. These lines define the lines of apparent CDPs recorded by the stack and they should l ie beneath a point on the recording cable at roughly one quarter of the maximum offset. The dashed lines give the ship's tracks as inferred from the feathering angle of the cable (in brackets on Fig. 18). These are consistent with the navigational data, according to which the lines A , 6, and C all cut line D at the same point. The positional errors are at least 25 m and the four lines can be brought through the same point with smaller shifts than this.

Cross-equalization also finds application in estimating other filters of seismic interest, such as earth transmission [37] and source calibration [32] responses. Noise is never negligible in estimating from seismic recordings and the cross-equalization operator that matches the input to the output of such a filter is biased by the noise suppression included in it (12). If the recordings have very similar

Q N

/ /

/ /

/

- line of CDP'sfrom

intersection analyses

line of ship from CDP's

531 and feathering angle ' 15.30 (in brackets1

(lo€) LINE C

SCALE - 0 25 50 75 Il*u

Fig. 18. Map of line intersections estimated from con- toured predictabilities such as those shown in Fig. 17.

bandwidths, it is practical to match output to input. The reciprocal of the response of this cross-equalization operator is biased in the opposite sense and a comparison of the two shows where the bias is strong and where the responses are reliable. Fig. 19 illustrates the technique. The data in this case come from an experiment to test the applicability of the scaling law in seismic wavelet estimation [ 6 4 ] and consist of traces shot along the same line by sources of different size which can be sorted into pairs recording the same reflectivity. The ratio of the source spectra is needed in order to estimate the wavelets and it can be estimated by two-way cross-equalization of the trace pairs. Fig. 19 shows

WHITE SIGNAL AND NOISE ESTIMATION FROM SEISMIC REFLECTION DATA 1353

Fig. 19. An exampl? of results from two-way matching: the amplitude response I & ( f ) l compared with l / IHn ( f ) l . The shading marks the %-percent confidence bounds on the estimates.

two oppositely biased estimates from one pair. The error measures from coherence analysis are very useful here in setting up a good averaging and stacking scheme to improve accuracy and in judging its success.

VI. MULTIPLE COHERENCE ESTIMATION OF SIGNAL AND NOISE SPECTRA

The principles and procedures of multiple coherence analysis [56] have been covered in earlier sections. Atten- tion was drawn to the desirability of removing correlated noise in the data in Section 1-6, to prefiltering in Section 11-D, to the estimators, their variances, and distributions in Sections Ill-A, -C, and -D, to the alignment of the reflections in Section 111-8, and to the selection of analysis parameters in Section Ill-E.

Examples of spectra from multiple coherence analysis were presented in Section I I . Here the possibility of using the results in processing is pursued. Bandpass filters for the section can be designed from the signal and noise spectra, but the data length needed for analysis can take in a whole series of reflections. Since time-variant filtering is normally worthwhile, the timing of reflections has to be considered and the usual bandpass filter tests are well suited to handle that. Thus in bandpass filtering, the signal and noise spectra and the usual tests are best seen as complementary. The spectra present the signal and noise in an informative and concise way and can be useful in picking corner frequen- cies and dropoff rates while the tests are needed to take account of the details of the section.

The spectra can be used for the design of deconvolution filters too. It is reasonable to suppose that use of the actual noise spectrum would result in an optimum tradeoff between the expansion of signal bandwidth produced by deconvolution and the noise amplification [I31 it incurs. One possible deconvolution design is

D( f ) = [{I + ac( f ) ) wl f 1 I - l ( 50)

where W ( f ) is the spectrum of a source signature or of a

1354

wavelet derived from the data and a is a noise control parameter. This filter contains a (1 + ap( f ) } noise suppression factor and becomes the Wiener optimum filter when a = 1. Fig. 20 shows i ts noise amplification characteristics when fed an air-gun signature and an N/S spectrum such as might be recorded on good marine field data (overall input N/S of - 6 dB). Two possible designs that do not require the estimation of p ( f ) are: a) a wavelet-whitening filter with added white noise control, and b) a trace-whitening filter with added white-noise control. The control of noise amplification in filter a) means that it is I W( f)12 plus some constant, not IW(f )12 , that is whitened; in b) the trace spectrum plus a constant is whitened. The magnitude of the constant controls the noise amplification and, through it, the expansion of wavelet bandwidth. The characteristic in Fig. 20 is certainly better than that for a) which is about 1 dB higher when the bandwidth ratio is 1.0 and peels away to higher noise amplifications when the bandwidth ratio exceeds 2.0. Filter b) does not do this, because there is a limit to the noise amplification it is capable of. There is nothing to choose between its characteristic and that in Fig. 20 and it appears that the noise amplification behavior of trace-whitening deconvolution, i.e., conventional deconvolution, cannot be noticeably bettered by more elaborate designs.

Because the weaker parts of the signal spectrum that deconvolution amplifies are more prone to measurement error, the deconvolved wavelet contains relatively more error energy than the input one. The "variability multiplier" curve illustrates this effect, but only very schematically since its computation involves a model for the spectral errors which is merely a plausible guess. The variability multiplier curve in Fig. 20 lies about 1 dB above those of filters a) and b). This does not matter for this air-gun signature as i ts error energy measured from repeated recording is about -27 dB down on its total energy but it could be important i f the design wavelet were estimated from the data when error energies would not often be better than -10 dB down.

PROCEEDINGS OF THE IEEE, VOL. 72, NO. 10. OCTOBER 1964

-2 1 u -4-

$ -0 - -0 -

-10 -

- Noise amplification

- Variability multitiplisr

0

6

4

2

0

-2

-4

-8

-0

-10

) 0.m 1.b 1:so 2.00 2w 3.b 3

EWANSION OF BANDWDlH

Fig. 20. Variation of noise amplification and variabiliy multiplier with bandwidth expansion for a signature deconvolution design. These three terms refer to the output-to-input ratios of the N/S power, wavelet error energy, and wavelet bandwidth, respectively.

Just as conventional deconvolution persists as the most dominant by far of all data-dependent methods of deconvolution in practical processing, despite a glut of alterna- tives, so too does stacking as the dominant means of noise suppression. Before traces are stacked their amplitudes are normally equalized from trace to trace in some way and this acts as a very simple means of weighting down noisy traces. A n “optimum” stack, based on the estimated S/N ratios, would downweight them more and should be capable of greater noise suppression. White [57] showed that, as a consequence of estimation error, it requires a rather wide range of input 5 / N ratios for this improvement to be worthwhile and then it can be realized only when the data are good. Fig. 21 compares the noise suppression obtain- able from the conventional straight stack and the optimum weighted stack for the case where the N/S ratios are estimated using a bandwidth-duration product typical of field data, and the variations in input S/N are rather high. Rietsch [38] saw this result as a challenge to devise a better method of S/N estimation but it is really a fundamental property of any estimation scheme and his simulations in fact contained a wide range of input S/N ratios.

1 , 1 : 1 1 . 1 1

0.0 I I I 1 at 1 .o HI0

AVERAGE SIN, p

Fig. 2l. Curves showing the signal-to-noise improvements expected from straight stacking of traces normalized to the same power level and optimally weighted stacking.

WHITE: SIGNAL AND NOISE ESTIMATION F R O M SEISMIC REFLECTION DATA

VII. REVIEW A N D CONCLUSIONS

The principles and procedures of applying spectral coherence methods to estimation problems in reflection seismology and interpreting their results have been reviewed. Their great merit i s their ability to provide a detailed assessment of the reliability of any estimate and it is hoped that the examples presented have given a glimpse of their usefulness.

There is nothing new in the spectral methods applied here and it is probably true that, as in seismic data processing, a sound approach to the data counts for more than algorithmic design. However there is plenty of room for better estimation procedures and it is disappointing that much recent research in spectral estimation misses the main problems of interest in reflection seismology. Two approaches worth pursuing were noted in Section 11-D.

In Section VI it was shown that estimation theory contains the germ of the reason why, in reflection seismology -and surely in many other fields, the direct and simple process is so often more effective than the one that looks better “in principle.” It certainly appears that the term optimum applied to any design that fails to take estimation errors into account is bound to be misleading and frequently at odds with i ts practical performance. A fuller incorporation of estimation statistics into processing theory could well help it to achieve a higher success rate in developing processes that work.

ACKNOWLEDGMENT

The author would like to thank S. A. Raikes, A. T. Walden, B. J. Barley, T. J. Davies, L. A. Hutchins, D. A. James, and A. L. Lucas for their generous assistance in assembling examples: P. N. S. O’Brien, S. A. Raikes, and A. T. Walden for their help in reviewing the manuscript: and the Chair- man and Board of Directors of The British Petroleum Com- pany for permission to publish the paper.

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