Geophysical Prospecting 32,871-891, 1984.

O N ERRORS OF FIT AND ACCURACY I N MATCHING SYNTHETIC SEISMOGRAMS AND

SEISMIC TRACES*

A.T. WALDEN** and R.E. WHITE**

ABSTRACT WALDEN, A.T. and WHITE, R.E. 1984, On Errors of Fit and Accuracy in matching Synthetic Seismograms and Seismic Traces, Geophysical Prospecting 32, 871-891.

A synthetic seismogram that closely resembles a seismic trace recorded at a well may not be at all reliable for, say, stratigraphic interpretation around the well. The most accurate synthetic seismogram is, in general, not the one that displays the smallest errors of fit to the trace but the one that best estimates the noise on the trace. If the match is confined to a short interval of interest or if the seismic reflection wavelet is allowed to be unduly long, there is considerable danger of forcing a spurious fit that treats the noise on the trace as part of the seismic reflection signal instead of making a genuine match with the signal itself. This paper outlines tests that allow an objective and quantitative evaluation of the accuracy of any match and illustrates their application with practical examples.

The accuracy of estimation is summarized by the normalized mean square error (NMSE) in the estimated reflection signal, which is shown to be

where Ps/PN is the signal-to-noise power ratio and n is the spectral smoothing factor. That is, the accuracy varies directly with the ratio of the power in the signal (taken to be the synthetic) to that in the noise on the seismic trace, and the smoothing acts to improve the accuracy of the predicted signal. The construction of confidence intervals for the NMSE is discussed. Guidelines for the choice of the spectral smoothing factor n are given.

The variation of wavelet shape due to different realizations of the noise component is illustrated, and the use of confidence intervals on wavelet phase is recommended.

Tests are described for examining the normality and stationarity of the errors of fit and their independence of the estimated reflection signal.

* Paper read at the 45th meeting of the European Association of Exploration Geophysicists, Oslo, June 1983, revision received January 1984. ** Geophysical Research and Technical Services, BP Exploration Co. Ltd, Britannic House, Moor Lane, London EC2Y 9BU, England.

871

872 A.T. W A L D E N A N D R . E . WHITE

1. I N T R O D U C T I O N Application of partial coherence analysis to matching synthetic seismograms and seismic traces (White 1980) assumes that the recorded seismic trace is a filtered version of the broadband synthetic seismogram computed from the well log, plus some additive noise. Different components of the broadband synthetic seismogram may be differently filtered. The mathematical expression of this picture of the trace is

4

y( t ) = 1 hi(t) * xi(t) + u(t) = signal + noise, i = 1

where * denotes convolution, xi(t) is the ith component of the broadband synthetic reflection spike sequence, hi(t) is the ith wavelet, and u(t) is noise. In the model each input component xi(t) is considered to be a distinct part of the reflection coefficient series, and to be uncontaminated by noise. Suppose q = 2; then, for example, input channel 1 might be attenuated primaries plus internal multiples, and channel 2 surface multiples.

a. stationary and random, b. statistically independent of the other components of the trace, c. normally (Gaussian) distributed with zero mean.

The constraints of stationarity and random noise imply that its mean, variance, correlation, and spectral characteristics do not change with time. Small departures from normality (Gaussianity) are not critical to the estimation procedure, but larger departures could be important.

The reflection sequence is not assumed to be white-it is calculated explicitly from the sonic log-and the individual input components xi(t) can be either non- stationary or nonrandom or both, and correlated (to a limited degree).

The wavelets are not assumed to be minimum phase, but from physical consider- ations each is expected to have a zero d.c. component. Although this fact plays no part in the formulation of the matching procedure, it does provide a useful check on the quality of estimated wavelets.

Since most practical applications concern one-channel matches, the methods for assessing the quality of a match are now discussed in terms of the one-channel case (q = 1) which allows greater simplicity of notation and interpretation. The tech- niques are easily extendable to two or more channels.

The noise u(t) is assumed to be

2. I N F E R E N C E F R O M T H E GOODNESS-OF-FIT The power and cross-spectra employed in matching are smoothed estimates. In conventional spectral analysis (e.g. Jenkins and Watts 1968) the smoothing is carried out explicitly by averaging over a small bandwidth of frequencies by means of a weighting function commonly called the spectral window. The bandwidth b of a window is defined as the width of the ideal rectangular window which would give an estimator with the same variance, and since Fourier analysis of a data segment of

ANALYSIS OF TRACE MATCHING 873

length T gives a frequency separation of 1/T between independent spectral com- ponents, there are b/ ( l /T) such independent components within the spectral window, and smoothing can be usefully envisaged as the averaging of this number of adjacent independent spectral ordinates. Hence the smoothing factor n, henceforth called simply smoothing, is equal to bT, the bandwidth of the smoothing window multiplied by the data gate length T.

The two spectral windows employed by us, the Papoulis (Papoulis 1973) and Daniell (e.g. Bloomfield 1976) windows, are illustrated in fig. 1. For the Papoulis

FREQ (Hz)

Fig. 1. Spectral windows for smoothing of n = 13 and 750 ms analysis gate,

window, the relation of smoothing n to the data gate length T and total width L of the taper applied to the correlations is

n = 3.400 TIL.

The lag window length L is just twice the maximum lag in the taper, i.e. twice the lag at which the taper drops to zero, and it should be chosen long enough to enclose the wavelet-dominated portions of the cross-correlations between the trace and broadband synthetic seismogram. For the Daniell window, which is a tapered sinc function in the time domain, the width L of the main lobe of the sinc function is related to the smoothing by

n = 2T/L.

The goodness-of-fit as a function of frequency can be measured by the estimated The choice of smoothing n is discussed in section 3.

signal-to-noise power ratio of the match at frequencyf:

874 A.T. WALDEN AND R.E. WHITE

where ,. denotes estimate, f , ( f ) is the coherence or proportion of power in the predictable component of the trace, 6Jf) is the smoothed cross-spectrum between the trace y( t ) and input x(t), 6;i(f) is the inverse of the smoothed spectrum of the input, and 6)yy.x(f) is the smoothed spectrum of the residuals to the fit, i.e. the estimatecoise spectrum pN(f). By integrating p s ( f ) and pN(f) over frequency, the overall S/N ratio ps/pN is obtained. It is a measure of goodness-of-fit. (We recall from section 1 that signal refers to the filtered broadband synthetic seismogram).

How reliable is the match? To answer this, let us first of all suppose that well data and seismic data are unrelated. Then the estimates ps and pN are just random positive values and (White 1980) pdpN is approximately distributed as (vl/vZ)FV1, v 2 , where Fvl, v 2 denotes the distribution known as Fishers variance-ratio distribution, with v1 = 2BT/n and v2 = 2BT({n - l ) / n ) degrees of freedom, where B is the sta- tistical bandwidth of the noise and T the duration of data segment employed in matching. If well data and seismic data are related, then the statistic ( v z / v 1 ) p d p N becomes large relative to the F-distribution. The statistic must exceed the main spread of this distribution, say 90% of it, in order to give confidence that the data and the synthetic seismogram are related. This is the role of the 90% confidence level test (White 1980). Most matches pass this test when applied over a data segment long enough (say 500 ms or more) to afford a clear result. However, little can be perceived as to the quality of a detected match, and it is really this that is of primary interest.

When well data and seismic data are related, then (appendix A) pdpN is approx- imately distributed as (vl /vZ)FV1, y 2 , a , where F v l , v 2 , a denotes the noncentral F- distribution with vl, v2 degrees of freedom and noncentrality A, with A = 2BT(Ps/PN). The mean power in the errors of prediction is given by (appendix A)

E @xx I fi - H 1 df = P,/n, (1) is 1 where I? - H denotes the error in estimating the wavelets frequency response H ( f ) , and E{.} denotes expected value. The ratio of signal power to mean power in the errors of prediction is therefore

n ( P S / P N ) =

and the normalized mean square error in the signal estimate is simply the inverse:

NMSE = (l /n)(PN/Ps) = A-. (2) A is a measure of the accuracy of estimation and has the form that one would intuitively expect. The accuracy varies directly with the ratio of the power in the signal to that in the noise on the trace and directly as the smoothing n.

Now A = 2BT(PdPN) = 2BT(l/n)(nP$PN) = (2BT/n)A2 = v,A. Hence a lOO(1 - CI)% confidence interval for A is given by

AI I A I A:,

where A: and A: are defined by-and may be found from-

Pr{Fvi, v 2 , v1A12 2 (v2/v1)(pS/pN)} = d2 = Pr{Fvl, v 2 , v 1 h 2 2 I (vZ/vl)(pS/pN)},

ANALYSIS OF TRACE M A T C H I N G 875

where Pr denotes probability. A lOO(1 - a ) % confidence interval for the NMSE then follows as

A; I NMSE I A;2. Methods for finding Pr{F,,, y 2 , I f *} are discussed in appendix B.

For a white reflection sequence, @Jf) % const = s2, say, so that the mean power in the errors of prediction is given by

E @xxlfi - H I 2 dj-} z s2E{JlH - H I 2 df}, is i.e. s2 multiplied by the expected energy of the errors in the wavelet, and the signal power is

s2 ~ ~ H 1 2 df.

Hence, if the reflection sequence is white the NMSE is the NMSE in the estimate of the wavelet itself, since s2 is eliminated by the normalization.

The theory associated with the NMSE in the signal estimate, outlined above, is derived under the assumption that spectral bias errors are negligible; this is the case when the smoothing is less than, or equal to, the optimal choice, the value of which is considered in section 3. Bias error should not prove to be a practical problem since one should always tend to err on the side of too little smoothing, rather than too much. Poor centering of the cross-correlation between the trace and the syn- thetic seismogram can also cause bias-termed misalignment bias -and the prac- tical procedure includes an automatic scan of alignment to ensure proper centering.

Examples of the confidence intervals obtained for the NMSE in the signal esti- mate from some real data analyses are given in table 1. The NMSE estimates found

Table 1. N M S E results for some synthetic seismogram studies.

m r a t i o 90% NMSE Type of and 90% interval and

smoothing conf. level point estimate Well and factor (brackets) (brackets) %

Well 1 D a n i e 11 1.0 5.4, 16 n = 13.2 (0.23) (7.6)

n = 13.6 (0.30) (11)

n = 13.2 (0.23) (4.2)

n = 13.2 (0.22) ( 3 4

n = 9.4 (0.35) 112)

Well 2 Papoulis 0.65 6.4, 46

Well 3 Daniell 1.80 3.0, 8.0

Well 4 Daniell 2.10 2.7, 6.7

Well 5 D a n i e 11 0.9 7.7, 42

876 A . T . W A L D E N A N D R . E . W H I T E

from substituting the estimated value P^ ,/P^ , in equation (2) are also given, and the skewness of the distribution about this (biased) point estimate can be appreciated. A useful value is the maximum of the confidence interval; with 95% confidence this value is the maximum likely NMSE in the signal estimate. Note that even though well 1 and well 5 both have $% ratios of approximately 1, the different smoothings used lead to very different quality assessments (16% compared with 42%).

3. SMOOTHING The previous section showed that the distribution of the ratio is a function of smoothing through v1 and v 2 . For fixed T and L the number of degrees of freedom associated with the window (n) depends on window type (Daniell or Papoulis) and, hence, the magnitude of the signal-to-noise ratio obtained from matching will vary from one smoothing window to another. The specification of a matching analysis by means of T and L is therefore incomplete if the window type is not also stated.

For statistical purposes the bandwidth of the spectral window is given by b = n/T. The smoothing n should be chosen to give a reasonable bandwidth to the spectral window which, as a rule of thumb, should be somewhat less than half the trace bandwidth. If it is larger than this, the spectral estimates tend to be badly distorted by the smoothing. An unduly large bias from smoothing is called oversmoothing .

While different spectral windows produce similar random errors of estimation for a given smoothing n, they still differ in the distortions and biases they introduce into the estimates. Consider the estimate of spectral power at zero frequency. Any power near zero frequency in the spectrum being smoothed is attenuated by the drop-off of the main lobe of the Papoulis window, but remains undiminished within the bandwidth of the nearly rectangular Daniell window. Hence, a larger d.c. com- ponent will be associated with the use of a Daniell window. In tests for well 1 (table l), for example, the average d.c. level per sample of the wavelet with n = 24 was only about 1% of the peak magnitude wavelet value for Papoulis smoothing, but nearly 10% for Daniell smoothing. Such large values should not arise if the spectra are not oversmoothed, but some oversmoothing becomes inevitable if the match is not good and cannot be extended over a segment of more than 500 ms. The Daniell window therefore is for quick preliminary analyses and searches, and the Papoulis window is generally better for a final estimation.

The smoothing is an important factor in the matching procedure. Since the chosen value n is never more than an educated guess, it is advisable to vary the smoothing over a small range about the chosen value, and examine the relative time alignment of the broadband-synthetic seismogram and the seismic trace for these values. Consecutive-or large-jumps in this timing are indicative of unstable esti- mation. Usually there is a reasonable range of satisfactory values for n over which the error in the wavelet estimate is close to the best attainable; outside of this range the match will be either a forced fit or a strongly biased one. This behavior is illustrated in figs 2a and 2b. In each case the attenuated primaries trace from well 1

ANALYSIS O F TRACE MATCHING

30 - 213 - 26 -

NEE % (-1 2 4 -

22 - 20 - 18 - 16 - 14 - 12 - 10 -

877

26-

2 4 -

2 2 - NEE % (-I 2 0 -

18 - 1 8 - 1 4 -

1 2 -

1 0 -

8 -

6 -

4 -

2 - 0

a. S/N = 0.75

- -.30

--.35 AIC PARAMETER

(---I / --.40 /

/ /

/ / - -.45

/ \ \ / \, - - S O \

\

- - 3 5 SEQ F-TEST

\ / / \ \ \ L \ '. .' / .-- /-' -.45 AIC PARAMETER (---I [ -.50 - -.55 - -.60

'1 SE0 F-TES? , , 4

42.6 25.6 113.3 14.2

SMOOTHING n

Fig. 2a. Normalized error energy in wavelet estimate, and value of AIC parameter, as a function of smoothing n ; S/N = 0.75.

was convolved with a wavelet, and to the result was added random noise filtered by a wavelet with a power spectrum very similar to that of the observed residual trace from the original synthetic study. Scaling was carried out to give a specified S/N ratio. For each of several choices of smoothing the wavelet was estimated by least-

878 A . T . W A L D E N A N D R . E . W H I T E

squares matching (White 1980). The estimated wavelet and the input (known) wavelet were then compared and the relative error, or normalized error energy (NEE), calculated for each smoothing. The plots of NEE for S/N ratios of 0.75 and 1.0 are shown as the heavy lines in figs 2a and 2b, respectively. It is clear that the bias error associated with oversmoothing increases dramatically, while the random error associated with undersmoothing increases more slowly; notice that the penalty incurred by undersmoothing is substantially greater for the lower S/N ratio. It is very dangerous therefore to go into matching without understanding the role of smoothing.

There are statistical criteria for assisting in the choice of lag window length. Two such criteria are the Akaike Information Criterion (AIC) and the Sequential F-test, examined in detail in Bunch (1984). The optimum length corresponds to a minimum of the AIC parameter, or the length corresponding to the crossing of a preset confidence level for the sequential F-test. Examples of the AIC plot for well 1 and well 2 are given in figs 3a and 3b. For well 1 there is a clear minimum, and it is seen that for n < 33.6 there is no shift in the timing alignment of the synthetic seismogram and the seismic trace, and estimates around the minimum, n % 28, are by this test stable. However, for well 2 the AIC plot behaves very poorly, the peak in the AIC plot at n z 14 coinciding with a large timing alignment jump; the best smoothing suggested by the plot is n in the range 10 to 12.

In general, even where the timing alignment is well behaved, there is a clear tendency for these automatic methods to select lag window lengths (in ms) which are too short from bandwidth considerations, i.e. they oversmooth. This behavior is understandable since these automatic methods do not take account of the biases from spectral estimation, and therefore do not penalize short operators sufficiently hard. For example, for well 1 the bandwidth of the seismic trace was 45 Hz, while the bandwidth of the smoothing window, given by b = n/T, was some 33 Hz for both automatic methods. The smoothing actually chosen in the well 1 study corre- sponded to a bandwidth of 17.6 Hz, a much more reasonable value. In figs 2a and 2b the AIC plots (shown as dashed lines) have been superimposed on the NEE (normalized error energy) curves, and the smoothing chosen by the sequential F-test (90% level) has also been marked. For both S/N levels, the two automatic methods select a smoothing which is too large, i.e. they oversmooth.

To summarize the topic of smoothing, it is recommended that in any analysis one should

1. investigate the effects of varying the smoothing n before deciding on a suitable value (cf. tests for deconvolution parameters which vary the operator length);

2. state the smoothing factor and the type of window employed, since this informa- tion, together with T, completely identifies the windowing procedure;

3. adopt a consistent approach, as needed in, say, comparisons of ratios, by applying the same type of smoothing window (say a Papoulis window) in all final analyses ;

4. look for any variations in the relative time alignment of the broadband synthetic seismogram and the seismic trace as the smoothing is varied, since these are

ANALYSIS OF TRACE MATCHING

- -0.30

- -0.35 AIC

PARAMETER

- -0.40

- -0.45

- -0.50

- -0.55

819

r-0.32

--0.34

--0.36 AIC

-0.38 PAR AM ET ER

--0.40

--0.42

Smoothing n 58.1 42.6 33.6 27.8 23.7 22.0 19.4 18.3 16.4

(ma) - 8 - 4 0 0 0 0 0 0 0 Time alignment

Fig. 3a. AIC plot for well 1.

b.

-0.44

28.4 21.3 17.0 14.2 12.2 10.7 9.5 SMOOTHING n

1 SMOOTHING n 18.5 15.8 14.7 13.7 12.9 12.2 10.9 9.9 TIME ALIGNMENT

(me) -12 -12 -12 0 0 0 0 0

Fig. 3b. AIC plot for well 2.

indicative of unstable estimation (as too would be variations in wavelet shape with small shifts in the positioning of the lag window about the optimum alignment), and state the time alignment employed which then leaves no room for doubt about the analysis parameters.

880 A . T . W A L D E N A N D R.E. WHITE

4. WAVELET S I M I L A R I T Y A N D PHASE ERRORS The attenuated primaries trace from well 1 was convolved with the wavelet of fig. 4a. and to the result was added colored noise produced as described in section 3.

U) 0 8 7 0 0

TRUE S/N=1

WAVELET USED IN SIMULATION

WAVELET ESTIMATED FROM 1st SIMULATION

WAVELET ESTIMATED FROM 2nd SIMULATION

2 d DS

Fig. 4. Wavelet used in simulation and two independent estimates of it.

Scaling was then carried out to give a S/N ratio of 1.0 over the 750 ms gate. The wavelet was estimated by least-squares matching with the same parameter values as for well 1 in table 1 ; it is shown in fig. 4b. The procedure was repeated with a different realization of random noise, and the estimated wavelet is shown in fig. 4c.

The wavelet of fig. 4b is less symmetric than that of fig. 4c. It is natural to look at the phases of the two wavelets to see if they differ significantly (in fact, the error energy from matching is on average partitioned equally between the phase and relative amplitude errors). A lOO(1 - a)% confidence interval for phase O ( f ) is given by Jenkins and Watts (1968, p. 434) as

where .it," is the estimated coherence for this one-channel case, and F 2 , 2 n - 2 ; l - a is the lOO(1 - a)% point of the F z , 2 n - 2 distribution. Since 1 sin (x) 1 I 1 it follows that the coherence must exceed a threshold for application of this formula. For a 90% interval, with n = 13.2, as here, F 2 , 24; o.9 is found to be 2.534 and thus it is required that 9; > 0.172. For frequencies f such that ?,"(f) > 0.172, the 90% confidence interval for phase for the wavelet of fig. 4b has been plotted in fig. 5a. Also marked on the diagram is the known phase of the wavelet used in the simulation, i.e. that in fig. 4a. It can be seen that four out of five intervals include the true value. This analysis is repeated in fig. 5b for the wavelet of fig. 4c, and this time all five intervals include the true phase values. (The formula of Jenkins and Watts 1968 given above

ANALYSIS OF TRACE MATCHING 881

CROSS: KNOWN PHASE OF SIMULATION WAVELET LINE 90% CONFIDENCE INTERVAL FROM

ESTIMATED WAVELET

1 st SIMULATION a.

1

- 2 . 0 1 0 20 40

FREQ (Hz)

2.07 2nd SIMULATION b.

- 2 . 0 1 0 20 40

FREQ (Hz)

Fig. 5. 90% confidence intervals for phase from estimated wavelets and known phase of input (simulation) wavelet for (a) first simulation and (b) second simulation.

is not the only one possible; a full discussion of the estimation of confidence inter- vals on gain and phase of frequency response functions is given in Walden 1984.)

The confidence intervals calculated for the phase indicate that there is nothing anomalous about the phase of the wavelet of fig. 4c; indeed, the phase estimates for each wavelet, figs 4b and 4c, are consistent with the phases of the simulation wavelet 4a, even though 4b and 4c look so different.

Before one can look at wavelets and call them different, one has to have some idea of the range of variation likely from the estimation procedure. One way of doing this would be to display a large number of wavelets estimated by simulations like those used to produce figs 4b and 4c. This has been done on a small scale, and the results are displayed in fig. 6. The top and bottom wavelets are the input (known) wavelet, and the intermediate wavelets are 10 independent estimates for different realizations of the noise component. It is interesting to convolve each of these wavelets with the attenuated primaries trace and then compare these filtered synthetics; this has been done in fig. 7, and one now has to look much closer to see dissimilarities. Plots of the phase and its confidence range, and of amplitude too if desired, are obviously much more concise than the displays of figs 6 and 7, they are quantitative, and they can pinpoint the cause of any difference precisely. Another possibility for comparing wavelet shapes would be to compute the mean square difference between the two wavelets (after appropriate scaling) and devise some test related to (2) of section 2, but it would be less diagnostic than the use of confidence intervals.

882 A.T. WALDEN A N D R.E. WHITE

INPUT SIMULATION WAVELET

INDEPENDENT ESTIMATES OF INPUT WAVELET FOR DIFFERENT REALISATIONS OF THE NOISE COMPONENT

INPUT SIMULATION WAVELET

TIME IN SECONDS

Fig. 6. Ten independent estimates of the input wavelet.

Fig. 7. Filtered

2.5 3.0

TIME IN SECONDS

ithetic seismograms corresponding to the wavelets of fig. 6.

ANALYSIS OF TRACE M A T C H I N G 883

5 . TESTING THE MODEL Having estimated a wavelet, a final ideal stage in the analysis would be to check the validity of the model assumptions about the nature of the noise, detailed in section 1.

Properties of the noise are tested by examining the residuals from the match defined by

ii(t) = y(t) - L(t) * x(t). We use two methods to test, in a univariate sense, that the amplitude distribu-

tion of the residuals is normal (Gaussian); one graphical, known as Q-Q plotting (Wilk and Gnanadesikan 1968) and one numerical, employing a statistic called the Cramer-von Mises (CVM) statistic, (Stephens 1974, 1976). The latter appears to be the best quantitative goodness-of-fit test for this purpose.

The Q-Q plot emphasizes visually any departures from normality, especially in the tails of the distribution which is where they are most likely to occur. A sample from a normal distribution will plot as a straight line, and systematic deviations from a straight line indicate non-normality. The Q-Q plot in fig. 8 closely approx-

ORDERED RESIDUALS STANDARDISED

TO VARIANCE OF 1

CORRESPONDING GAUSSIAN QUANTILES

Fig. 8. Q-Q plot for well 1 residuals.

imates a straight line, the deviations for large values being due to only about three points; hence normality is indicated. Of course, any sample shows fluctuations about a straight line, and it is for this reason that a quantitative test is useful to sort out borderline cases.

884 A . T . W A L D E N A N D R.E. W H I T E

The CVM statistic is used to test hypothesis

H, : The sample is from a normal distribution with unknown mean and variance

against hypothesis

H, : The sample is not from such a distribution.

If the value of the statistic exceeds the lOO(1 - CI)% level of its distribution, then at the lOOc(% significance level it is concluded that H, is rejected and H, accepted. It is suggested that a small significance level be chosen for CI so that H, is rejected only in extreme cases.

As mentioned in section 1, the noise is assumed to be stationary, so that the mean, variance, correlation, and spectral characteristics do not change with time. In practice, this assumption is impossible to test rigorously with a single trace without imposing further constraints on the model. However, any long period trends in the mean or variance should show up if one uses moving statistics (Cleveland and Kleiner 1975). In our program, three statistics (essentially the lower quartile, median, and upper quartile) of the distribution of residual values in a moving window of fixed length are plotted as the window moves through the analysis time gate; any longer period trends should be more clearly discernible from these summary sta- tistics than from the raw residual trace. Even when the ideal conditions are fulfilled, sampling fluctuations will give rise to some perturbations in the three lines traced out. By increasing the size of the moving window such random perturbations are reduced, but too large an increase also tends to diminish the ability to detect real trends.

Figures 9a and 9b provide an example of the desirability of considering the constancy of the residual variance. Seismic survey lines A and B intersect near well

a.

2.01 b.

2-01

STATISTICS - - I I

-2.0 J

4.07

-2.oJ

4 a 1

RESIDUALS

TO VARIANCE OF 1 0.0 STANDARDISED ox)

-4.0' -4.0J

I ! I I I I I I , , , t , , 1.8 2.0 2.2 2.4 1.8 2.0 2.2 2.4

TIME IN SECONDS

Fig. 9. Residual trace and moving statistics from matching the synthetic at well 6 to (a) line A seismic trace and (b) line B seismic trace.

ANALYSIS OF TRACE MATCHING 885

MOVING STATISTICS 0.0:

-2.0 -2.0-

"U\.., - -

STANDARDISED o . o y TO VARIANCE OF 1

1 + + 0.0

4.0J 4.0J - - -2.0 0.0 2.0 -2.0 0.0 2.0

SCALED VALUES OF ESTIMATED FILTERED SYNTHETIC

Fig. 10. Two examples of scatter plots and moving statistics of residuals against estimated filtered synthetic seismograms from matching.

wells. In the scatter plot of fig. 10a there is a clear tendency towards positive residuals coincident with negative values of estimated filtered synthetic seismograms and negative residuals coincident with positive values of estimated filtered synthetic seismograms. Such notable behavior is indicative of a poorly fitting model and arises even though the estimated signal and the noise are orthogonal. The moving statistics emphasize the trend. In contrast, the scatter plot behavior in fig. 10b is quite satisfactory, and the moving statistics are really quite parallel and horizontal.

The residuals from matching have been used as approximations to the true unknown errors in order to assess the properties of the latter. Simulations were used to gauge

a. the correctness of making inference about the true errors from the residuals, and b. the utility of the methods used for examining the estimated noise or residuals (i.e.,

886 A.T. W A L D E N A N D R . E . WHITE

the utility of qualitative and quantitative tests for normality, displays of moving statistics to assess stationarity, and scatter plots and displays of their moving statistics to assess correlation between residuals and estimated filtered broadband synthetic seismograms).

These simulations gave no cause for concern with respect to (a), but showed that it is usually difficult to separate sampling fluctuations from real trends, especially where no quantitative test is forthcoming. This difficulty is often due to durations of passable matches between seismic data and synthetics being too short to allow simple clearcut answers. However, when more distinctive behavior-such as seen in fig. 10-does occur, it can be very useful.

Results showed that in matching real data the assumption that the noise is normally distributed was almost always clearly supported by the results of the two tests for normality applied to the residuals from the match. These tests are valid when the error series is stationary.

6. SUMMARY The main statistical points to consider when attempting a match are:

a. Selection and investigation of the right spectral smoothing according to the guidelines in section 3 and knowledge of the likely spectral content of the seismic wavelet from the recording and processing parameters.

b. The measures of accuracy can be helpful when scanning for a good match over several traces and different time gates. The interpretation of the results of such scans demands a careful geophysical assessment; for example, any shift of the best fit trace from the well location has to be reconciled with likely navigational errors and, when matching migrated data, with the possible consequences of incorrect migration velocities. This paper has dealt solely with the statistical accuracy of matching and has made no attempt to cover all its practical rami- fications.

c. The so-called optimal smoothing criteria, the AIC and Sequential F-tests, should be treated with caution. The criteria consistently underestimate the physical wavelet length, or equivalently oversmooth the seismic spectra.

ratio exceeds the 90% confidence level for detectionAhen it is con- cluded that a valid detection has been achieved. The size of the S / N ratio can be used to quantify the accuracy of the estimate, as detailed in section 2.

e. Phase effects can be deceptive in wavelet estimation. Plotting confidence intervals on phase gives one a better appreciation of the magnitude of phase uncertainty. Confidence intervals on amplitude (gain) may also prove useful.

f. Check the residuals from the match for normality, approximately constant variance over time, and lack of correlation with the estimated filtered broadband synthetic seismogram. If any of these assumptions are violated, downgrade the reliability of the estimate. Of course, if it is geophysically desirable or possible to select a substantially different matching gate then this problem may perhaps be effectively overcome.

d. If the

A N A L Y S I S OF T R A C E M A T C H I N G 887

A C K N O W L E D G M E N T S

We thank Dr P.N.S. OBrien for helpful comments on a BP report on which this paper is based, and the Chairman and Board of Directors of the British Petroleum Company plc for permission to publish the work.

A P P E N D I X A THE DISTRIBUTION OF THE ESTIMATED S/N POWER R A T I O

Fourier transform of the single-channel convolutional model

y( t ) = h(t) * x(t) + u(t)

Y ( f ) = W M f ) + W), gives

where H ( f ) is the wavelets frequency response. Least-squares estimation of H ( f ) is equivalent to minimizing at each frequency the smoothed spectrum of the residuals, namely

6yy.x(f) = W f ) * C(1/T) I Y ( f ) - A ( f ) X ( f ) 121, where W ( f ) is the spectral window employed. It is assumed that H ( f ) varies little across this window. Then the least-squares estimate of H ( f ) is

and the estimated residual spectrum & y y . x ( f ) satisfies the analysis-of-variance equa- tion

& y y ( f ) = I A ( f ) 12&xx(f) + &yy.x(f), (A2) where 6xx(f) and & J f ) are the smoothed auto-spectra of x(t) and y(t) and & x y ( f ) is their smoothed cross-spectrum. By writing

Wf)= W ) + Af fC f )X( f ) , AHCf) = - wn and making use of (Al) (orthogonality of input and residuals) one can relate the estimated residual spectrum to the sample noise power spectrum &uu(f) :

= &yy.x(f) + I 12&xx(f). (A31 Equation (A3) is also an analysis of variance; when scaled by the factor

2n/(Duu(f), where n is the spectral smoothing, it represents the decomposition of a xi, variable into two chi-squared components with degrees of freedom, respectively, (2n - 2) and 2 (one for the real and one for the imaginary parts of I A H ( f ) 1 2 ) . The chi-squared distribution of the power spectral estimate &,,,,(f) follows from the assumption of Gaussian noise u(t) that provides the statistical justification for least- squares estimation. The distribution of &uu(f ) is still chi-squared to a reasonable approximation, even if the noise is not Gaussian (Jenkins and Watts 1968, p. 417).

888 A.T. WALDEN AND R.E . WHITE

On taking expectations in (A3) and using E{x:} = v, it follows that

E{I h ~ f ) 16~~(f)I = (1/n)~{6,,(f)} = QUu(f)/n. Equation (1) of section 2, namely,

E{ 1 &xx(f) I A(f ) - H ( f ) 1 df } = P,/n follows on integrating over frequency. This equation also gives the negative bias that results from estimating the total noise power by integrating & ) y y . x ( f ) .

The formation of equations and distributions for estimators obtained through matching has required assumptions about the noise u(t) and the smoothness of H ( f ) . What of the input x(t)? In matching this is supplied and it can be treated as a known driving function. In particular, there is no necessity to regard x(t) as stochas- tic and the use of the notation 6 J f ) here does not imply that & x x ( f ) has a statistical distribution; it simply denotes a known smoothed auto-spectrum. It may be convenient for purposes other than the matching to treat x(t) as stochastic and a brief indication of how matching can be linked to this approach is given after the other derivations that are the aim of this appendix.

The distribution of the estimated signal spectrum I B(f) 1 2 & x x ( f ) follows from the chi-squared decomposition of (A2). The reasoning parallels that related to (A3), but now the distribution of the sum is noncentral chi-squared (Johnson and Kotz 1970, Chap. 28) because any realization of the Gaussian noise has the same signal h(t) * x(t) added to it. The noncentrality comes entirely from the signal spectrum, from the fixed component H ( f ) in A(f). To convert the quantities in (A2) to stan- dardized chi-squared variables, it is multiplied by 2n/@,.,(f) as before and the 2n degrees of freedom associated with 6 J f ) split into 2 for 1 r ? ( f ) 1 and (2n - 2) for 6)yy.x(f). That is,

2n I f i ( f ) 12&xx(f) @,,(f 1

has a xi, a distribution, where the noncentrality parameter is

The value of Alternatively it can be derived from

comes by definition from setting the random components to zero.

E{I R f ) lxx(f)> = @ J f ) - E { & y y . x ( f ) } = I H ( f ) I 2 @ x x ( f ) + @uu(f ) /n by using the relation E{X?, A} = v + A.

The estimated signal-to-noise ratio at frequency f is I f i ( f ) ~&xx(f)/6yy.x(f). A ratio of this kind, in which the numerator has a noncentral chi-squared distribution

ANALYSIS OF TRACE M A T C H I N G 889

and the denominator a central one, follows a noncentral F-distribution. That is to say, the standardized ratio

has a noncentral F-distribution with 2, 2n - 2 degrees of freedom and noncentrality

The results in section 2 concerned the overall signal-to-noise ratio pdP^ , which is the ratio of the integrals over frequency of I d ( f ) 126xx(f) and 6y,,.x(f). Integrating (A2) shows that the total trace power is just the sum of these integrals:

8, denoted F 2 , 2 n - 2 , p *

A A A

Py = Ps + PN. The estimated trace power p,, is distributed approximately as (~"/v)x;, a where v = 2BT and ps and pN are independent since the estimates of signal and noise at any frequency are independent and each is equivalent to a linear summation of independent spectral estimates. Therefore (vPs)/PN is distributed as &, a and ( v P N ) / P N as &, ,, , where v 1 = 2BT/n and v 2 = 2BT(n - l)/n (White 1980). The non- centrality is

1 = VPdPN t

This follows from setting the random components A H ( f ) in

(vpS)/pN = (v/pN) 1 I A(f) I 2 6 x x ( f ) df to zero or from

E{(vpS)/pN} = ( v / p N ) ( p S + [pN/nl) = E { d l , A } = v1 + ,k Thus the ratio (v2 ps)/(vlpN) has the noncentral F-distribution Fvl, Y2r a as stated in section 2.

The variance of a xt, A variable is 41 + 2v. Applying this to the ~ 3 , ~ variable containing 6ss(f) = I I ? ( f ) 1'6~,(f) gives

This expression measures the variations in estimating Q S s ( f ) = I H ( f ) 126,.,(f) caused by the sample of noise on the trace y(t): the signal spectrum is specifically that present in the trace and no allowance is made for any sampling of the signal. If the signal is regarded as a sample from some hypothetical stochastic process, then this approach generates an additional (hypothetical) variance @,,",(f)/n and the total variance becomes

to order l/n. Exactly the same result can be derived for 6ss(f) from the expressions for spectral variances

the large-sample variance of and covariances given by

890 A . T . W A L D E N A N D R . E . W H I T E

Goodman (1957), which are founded on a fully stochastic model for both x(t) and y(t). In a similar way, the x:,, distribution for 6jt,,(f) leads to

and it is only after adding the signal sampling variance (D.,,(f)/n that one obtains the standard expression for the variance of a power spectral estimate

var, {%t,(f)} = @&(f)/n. (47) Equations (A4) and (A6) can be termed noise sampling variances whereas the total stochastic variances (A5) and (A7) include a variance that arises from treating the signal also as a sample from a stochastic process. In matching, the assumption of a stochastic signal is superfluous since the distribution theory for the estimators can be developed from expectations that range solely over the postulated ensemble of noise samples. The smoothed spectrum 6xx(f) in this theory is analogous to the sums of squares and products matrix in regression theory and its appearance does not imply that x(t) is stochastic, although it does restrict the complexity of H ( f ) . Even if a stochastic signal is assumed, the replacement of 6xx(f) by its population value would introduce an unnecessary approximation using an unknown quantity. Note too that in stochastic models of seismic traces containing a common signal, (A6) is a more appropriate measure of power spectral variance than the standard expression (A7) when only one specimen of signal is being considered.

APPENDIX B The noncentral F-distribution can be closely approximated by a standard central F-distribution (which is extensively tabulated and available in most algorithm libraries) using a result of Patnaik (1949), viz.

Pr P V I . V Z , a 5 f * > 7z Pr W V 3 , v2 5 f**>,

f * = (v1 + 1)f**/V1

v3 = (v1 + 1)2/(v1 + 21).

where

and

In the present application, 1 = v l A 2 , so that

f * = (1 + [1/v1])f** = (1 + A2)f** and

v i ( 1 + [1/vJ2 - vl(l + A) v , ( l + [2;l/vl]) - (1 + 2A2) v j =

For example, in order to solve

Pr { F v i , v z , v l A 1 2 (v2/vl)(pS/pN)} =

ANALYSIS O F TRACE M A T C H I N G 891

search over an interval of A2 to find the value A: such that

REFERENCES BLOOMFIELD, P. 1976, Fourier Analysis of Time Series: An Introduction, Wiley, New York. BUNCH, A.W.H. 1984, Predicting the optimal least squares filter using adaptations of standard

statistical theory, BP Report Ext. 25628. CLEVELAND, W. and KLEINER, B. 1975, A graphical technique for enhancing scatterplots with

moving statistics, Technometrics 17, 447454. GOODMAN, N.R. 1957, On the joint estimation of the spectra, co-spectrum and quadrature

spectrum of a two-dimensional stationary Gaussian process, Scientific Paper No. 10, Engineering Statistics Laboratory, New York University, also University of Princeton thesis.

JENKINS, G.M. and WATTS, D.G. 1968, Spectral Analysis and its Applications, Holden-Day, San Francisco.

JOHNSON, N.L. and KOTZ, S. 1970, Continuous Univariate Distributions-2, Wiley, New York.

PAPOULIS, A. 1973, Minimum-bias windows for high resolution spectral estimates, IEEE Transactions on Information Theory IT-19,9-12.

PATNAIK, P. 1949, The non-central x2 and F-distributions and their applications, Biometrika

STEPHENS, M. 1974, EDF statistics for goodness-of-fit and some comparisons, Technometrics

STEPHENS, M. 1976, Asymptotic results for goodness-of-fit statistics with unknown par-

WALDEN, A.T. 1984, Confidence intervals on gain and phase of frequency response functions,

WHITE, R.E. 1980, Partial coherence matching of synthetic seismograms with seismic traces,

WILK, M.B. and GNANADESIKAN, R. 1968, Probability plotting methods for the analysis of

36,202-232.

69,730-737.

ameters, Annals of Statistics 4, 357-369.

BP Report Ext. 25477.

Geophysical Prospecting 28,333-358.

data, Biometrika 55, 1-17.

Table of ContentsON ERRORS OF FIT AND ACCURACY IN MATCHING SYNTHETIC SEISMOGRAMS AND SEISMIC TRACES*ABSTRACT1. INTRODUCTION2. INFERENCE FROM THE GOODNESS-OF-3. SMOOTHING4. WAVELET SIMILARITY AND PHASE ERRORS5. TESTING THE MODEL6. SUMMARYAPPENDIX AAPPENDIX BREFERENCES