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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
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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
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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:
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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)},
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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
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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
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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-
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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
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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.
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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
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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.
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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.
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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.
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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.
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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.,
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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
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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).
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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
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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)} =
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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
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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