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Chapter 3
Imaging Refractors with theConvolution Section
3.1 - Summary
Seismic refraction data are characterized by large moveouts between adjacent
traces and large amplitude variations across the refraction spread. The
moveouts are the result of the predominantly horizontally traveling trajectories of
refraction signals, while the amplitude variations are the result of the rapid
geometric spreading factor, which is at least the reciprocal of the distance
squared.
The large range of refraction amplitudes produces considerable variation in
signal-to-noise (S/N) ratios. Inversion methods which use traveltimes only,
employ data with a wide range of accuracies, which are related to the variations
in the S/N ratios.
The time section, generated by convolving forward and reverse seismic traces,
addresses both issues of large moveouts and large amplitude variations.
The addition of the phase spectra with convolution effectively adds the forward
and reverse traveltimes. The convolution section shows the structural features of
the refractor, without the moveouts related to the source to detector distances.
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Unlike the application of a linear moveout correction or reduction, a measure of
the refractor wavespeed is not required beforehand.
The multiplication of the amplitude spectra with convolution, compensates for the
effects of geometric spreading and dipping interfaces to a good first
approximation, and it is sufficient to facilitate recognition of amplitude variations
related to geological causes. These amplitude effects are not as easily
recognized in the shot records.
The convolution section can be generated very rapidly from shot records without
a detailed knowledge of the wavespeeds in either the refractor or the overburden.
3.2 - Introduction
In this study, I propose the application of full trace processing as one method of
addressing the fundamental issue of the large variations in signal-to-noise (S/N)
ratios with seismic refraction data.
I begin with a discussion of the effects of geometric spreading on two shot
records from a shallow seismic refraction survey. The data demonstrate that the
spreading is large, it is not adequately described with the reciprocal of the
distance squared expression and it dominates any geological effects. These
large variations in amplitudes result in large variations in S/N ratios and in turn, in
large variations in the accuracies of the measured traveltimes.
Next, I briefly review various methods of full trace processing and then propose
the generation of a refraction time cross-section by the convolution of forward
and reverse traces. I demonstrate that convolution provides very good
compensation for geometric spreading and for the variations in amplitudes
caused by changes in the dip of the refracting interface.
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Figure 3.1: Field record for shot point at station 1, presented at constant gain.
The large drop in amplitudes from about station 51 can be clearly seen.
Finally, I present a convolution section across a complex refractor in which there
are large variations in depths and wavespeeds. The image presents the same
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time structure that would be obtained with the standard methods of processing
traveltime data, while the amplitudes are a function of the head coefficient, which
is the expression relating the refraction amplitudes to the petrophysical
parameters of the upper layer and the refractor.
3.3 - The Large Variations in Signal-to-Noise Ratios withRefraction Data
A long standing problem with the acquisition of seismic refraction data is the
relatively high source energy requirements, which are necessary to compensate
for the rapid decrease of signal amplitudes with distance. For signals which have
traveled several wavelengths within a thick refractor with a plane horizontal
interface, the geometrical spreading factor is approximately the reciprocal of the
distance squared (Grant and West, 1965), and it is much more rapid than the
equivalent function for reflected signals which is the reciprocal of the distance
traveled.
Figures 3.1 and 3.2 are two shot records presented at a constant gain, and
illustrate the large variations in S/N ratios. The shot points are offset
approximately 120 m from each end of a line of 48 detectors, which are 5m apart.
Qualitatively, each shot record exhibits high amplitudes close to the shot point,
followed by greatly reduced amplitudes from about station 51 onwards. Figure
3.3 shows the amplitudes of the first troughs of the forward shot data, normalized
to the value at station 50. As expected, the amplitudes show the rapid fall with
distance from the shot, with the variation between the near and far traces being a
factor of 20, or 26 decibels. The reduction with distance is much more rapid than
the reciprocal of the distance squared spreading function, which is also shown in
Figure 3.3, and the reciprocal of the cube of the distance appears to be a much
closer approximation.
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Figure 3.2: Field record for shot point at station 97, presented at constant gain.
The large drop in amplitudes from about station 51 is even more pronounced
than on the previous record.
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Figure 3.3: Amplitudes of the first trough measured on the forward shot record,
together with the reciprocals of the distance squared and distance cubed
geometric effects.
A similar result occurs with the reverse shot data in Figure 3.4. The amplitudes
decrease much more rapidly than a reciprocal of the distance squared function,
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and in this case, the variation between the near and far traces is a factor of 60, or
36 decibels. Again, a reciprocal of the distance cubed function is a better
approximation, although the fit with the low amplitude values is not particularly
close.
Figures 3.3 and 3.4 demonstrate that the reduction in amplitude with distance is
large, and that it dominates any secondary effect caused by geological
variations. An interpretation of the traveltime data derived from these shot
records is presented in Chapter 5 (Palmer, 2000c), and it shows rapid changes in
the depth to the main refractor, which in this case is the base of the weathering,
as well as large variations in the wavespeed of the refractor. Accordingly, the
challenge is to effectively separate the amplitude variations related to geological
factors from those caused by geometrical spreading.
In addition, Figures 3.3 and 3.4 demonstrate the difficulties in employing
corrections for geometrical spreading based on widely accepted theoretical
treatments. The reciprocal of the distance squared function only applies to
homogeneous media separated by plane horizontal interfaces, and only after the
signal has traveled 5-6 times the predominate wavelength of the pulse (Donato,
1964). These latter results are in keeping with model studies (Hatherly, 1982),
and are the norm, rather than the exception in most shallow refraction surveys.
Furthermore, this example highlights the very large variations in S/N ratios at
each detector for the usual ensemble of shot points and in turn, the considerable
range of accuracies in the measured traveltime data for most refraction surveys.
At any given location, a detector will be close to a source, and the measured
traveltimes will be comparatively accurate, because of the high S/N ratio.
However for the traveltime in the reverse direction, the source-to-receiver
distance will be much larger, and the accuracy will be greatly reduced, because
of the lower S/N ratio. Such large variations in accuracies adversely affect the
quality of data processing with anymethod.
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Figure 3.4: Amplitudes of the first trough measured on the reverse shot record,
together with the reciprocals of the distance squared and distance cubed
geometric effects.
Most methods for the processing of seismic refraction data use simple scalar first
arrival traveltimes, and the problem is normally perceived as achieving
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satisfactory, rather than uniform S/N ratios. Commonly, a simple gain function is
applied to adjust amplitudes to a convenient level, but this still does not alter the
large variations in S/N ratios. With statics corrections for reflection surveys,
typically a limited source-to-detector interval over which the refraction data are of
sufficient quality, is selected. For geotechnical, groundwater and environmental
studies, the source energy levels are usually increased as far as environmental
and cultural factors permit, or vertical stacking with repetitive sources is
employed.
The following section reviews full trace processing and the issue of the large
variations in S/N ratios.
3.4 - Full Trace Processing Of Refraction Data
Perhaps the simplest approach to full trace processing, is the application of a
linear moveout (LMO) correction to each shot record. With this approach, which
is also known as reduction, each refraction trace is shifted or reduced by a time
equal to the source-to-detector distance, divided by a velocity, which is usually
the known or estimated wavespeed in the target of interest, (Sheriff and Geldart,
1995, Fig. 11.10). The result is normally presented as a set of traces for which
the first arrivals occur at the sum of the source point and detector delay times.
One benefit of this presentation is that it maps any variations in the target depth
in terms of the delay times.
However, this process does not address the basic issue of the large variation inS/N ratios across the refraction recording spread. The degradation of the arrivals
at the more distant detectors is usually very significant, particularly with crustal
and earthquake studies. Furthermore, it is usually inconvenient to include any
reverse shot records within the same presentation, and therefore to readily
accommodate any lateral variations in wavespeed with irregular refractors.
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Other approaches are the broadside and fan shooting methods, in which the
source is usually located at an offset point, orthogonal to the center of a linear or
circular array of detectors. Since the source-to-detector distances are essentially
constant, the geometric spreading effects are also constant, and there are much
smaller variations in the S/N ratios from trace to trace. Furthermore, corrections
for the source-to-detector distances, such as with an LMO, in order to emphasize
any structural anomalies in the target refractor, are not essential because such
time shifts are virtually constant also. Examples of the imaging or migration of
broadside data (Mcquillan et al, 1979, Figure 7/15), indicate some of the
possibilities of full trace processing of refraction data.
These methods represent the first true 3D seismic methods for exploration and
pre-date the current reflection 3D methods by many decades (Sheriff and
Geldart, 1995). As such, they will eventually be incorporated into the routine
refraction methods of the future. However, the methods described above do
have two major limitations. They do not determine wavespeeds in the refractor,
nor are they able to separate source and receiver delay times without additional
information, such as borehole control, or the simultaneous recording of a
conventional in-line profile orthogonal to the broadside pattern.
A recent method of imaging refractors with forward and reverse data, is
downward continuation using the tau-p transform (Hill, 1987). It can achieve
good resolution by accommodating diffraction and shadow zone effects. Like all
wavefront methods, it requires an accurate knowledge of the wavespeed of the
upper layer, but this is probably one of the least reliable parameters determined
in most refraction surveys (Chapter 2; Palmer, 1992; Appendix 2).
In this study, I describe the generation of a refraction time section through the
convolution of forward and reverse traces as an effective method of addressing
the fundamental issues of large S/N variations and large moveouts with refraction
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data. The result, the refraction convolution section (RCS), is similar in
appearance to the familiar reflection time cross section, in which the results are
displayed for example, as a series of wiggle traces.
There are several benefits to processing with this approach. The first is that it is
extremely rapid, avoiding in particular the familiar time consuming tasks of
determining first arrival traveltimes. The second is that little, if any, a priori
information on overburden or refractor wavespeeds is required, although of
course such information is essential for the generation of final depth cross
sections. Accordingly, the convolution section is a very convenient presentation
for an assessment of the quality of processing using other detailed methods,
such as tomography.
In addition, the approximate compensation for large variations in the S/N ratios
facilitates the vertically stacking of refraction data, in a manner analogous to the
common midpoint method with reflection data. This in turn, suggests more
efficient methods of data acquisition with lower environmental impact, particularly
for geotechnical investigations (Palmer, 2000a).
The benefits to interpretation are that the amplitudes obtained through
convolution are essentially a function of the refractor wavespeeds and/or
densities, rather than the source to detector separation. In general, high
wavespeeds and/or densities in the refractor produce low amplitudes. This
relationship between amplitudes and contrasts in the parameters of the refractor
and the overburden provides an additional valuable method for resolving
ambiguities, especially with model-based methods of refraction inversion
(Palmer, 2000c).
The concept of the convolution section was first proposed by Palmer (1976), but
initial tests with Vibroseis data were not especially encouraging, because of
correlation noise before the first breaks (K B S Burke, pers. comm., circa 1982).
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However, the method was later successfully applied to synthetic data (Taner et
al, 1992).
3.5 - Imaging The Refractor Interface Through The Addition ofForward And Reverse Traveltimes
The unambiguous resolution of dip with plane interfaces or structure with
irregular interfaces, and variable wavespeed within the refractor, usually requires
forward and reverse traveltime data, or off-end data with a high density of source
points, from which the equivalent reversed traveltime data can be generated.
Accordingly, the majority of refraction processing methods explicitly identify and
use forward and reverse traveltimes within their algorithms. These methods
include the wavefront construction methods (Thornburg, 1930; Rockwell, 1967;
Aldridge and Oldenburg, 1992), the conventional reciprocal method (CRM),
(Hawkins, 1961), which is also known as the ABC method in the Americas,
(Nettleton, 1940; Dobrin, 1976), Hagiwara's method in Japan, (Hagiwara and
Omote, 1939), and the plus-minus method in Europe, (Hagedoorn, 1959), Hales'
method, (Hales, 1958; Sjogren, 1979; Sjogren, 1984), and the generalized
reciprocal method (GRM), (Palmer, 1980; Palmer, 1986).
There are minor differences in detail between the algorithms for each of these
methods. These differences include whether the reciprocal time, the time from
the forward shot point to the reverse shot point, is used, the inclusion of the
factor of a half, or whether the offset distance, which is the horizontal separation
between the point of refraction on the interface and the detector position on the
surface, is accommodated through the operation known as refraction migration
(Palmer, 1986, p.74-80).
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Figure 3.5: Traveltime data for a line crossing a major shear zone in
southeastern Australia. The station interval is 5 m. The traveltimes for the offset
shots which are offset 120 m from either end at stations 1 and 97, are shown in
bold.
Nevertheless, each of these methods includes an algorithm in which the forward
and reverse traveltimes are added, in order to obtain a measure of the depth to
the refractor in units of time. This process of addition averages most of the dip
effects to the horizontal layer approximations and replaces the moveout with a
constant value for all detectors between the forward and reverse source points.
With the CRM and GRM, this constant is then removed by subtracting the
reciprocal time. Finally, the result is halved to derive a parameter which is
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essentially the mean of the forward and reverse delay times. The result is known
as the time-depth, where
time-depth = (tforward+ treverse- treciprocal)/2. (3.1)
Figure 3.5 presents the traveltime data recorded across a major shear zone in
southeastern Australia with a set of collinear shots and receivers. The station
interval is 5 m, and the shot points are at stations 1 which is offset 120 m to the
left, 25, 49, 73 and 97 which is offset 120 m to the right. The traveltimes indicate
a three layer model consisting of a thin surface layer of friable soil with a
wavespeed of about 400 m/s, a thicker layer of weathered material with a
wavespeed of approximately 700 m/s, and a main refractor with an irregular
interface.
Figure 3.6: Time-depths computed from traveltime data with shot points offset
120 m from each end of the geophone array at stations 1 and 97.
An example of the application of equation 3.1 is shown in Figure 3.6, using the
traveltime data measured from the shot records shown in Figures 3.1 and 3.2,
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and summarized in bold in Figure 3.5. The time-depths have been computed
with a reciprocal time of 147 ms, (Palmer, 1980, equation 33), and an optimum
XY value of 5 meters.
The XY value is the separation between the pairs of forward and reverse
traveltimes used in equation 3.1, and it is usually a multiple of the detector
spacing. The optimum XY value is obtained with the minimum variance criterion
described elsewhere (Palmer, 1980, p.31-35) and it is the sum of the forward and
reverse offset distances. This sum is essentially independent of the dip angles,
unlike the individual forward and reverse components. At the optimum XY value,
the forward and reverse rays are refracted from near the same point on the
refractor and the smoothing effects of other XY values are minimized.
3.6 - The Addition of Traveltimes With Convolution
The traditional methods for the inversion of refraction data, can be categorized by
how the addition of the forward and reverse traveltimes is implemented. The
wavefront construction and Hales' methods achieve it graphically, while the CRM
and GRM achieve it with the simple addition of two numbers.
In this study, I demonstrate the use of convolution of forward and reverse traces
to effectively achieve the addition.
The convolution process has usually been associated with filtering. Its effect can
be described in the frequency domain, as the multiplication of the amplitudespectra and the addition of the phase spectra of the two functions.
A similar result occurs with the convolution of two seismic refraction traces. The
amplitude spectra are multiplied, and the arrival times, which are contained within
the phase spectra, are added.
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Alternatively, the addition of first arrival times with convolution can be
demonstrated with the z transform notation (Sheriff and Geldart, 1995). The
digitized seismic trace can be represented as a polynomial in z, in which the
exponent represents the sample number. The forward trace F(z) is given by
F(z) = fmzm+ fm+1 z
m+1 + fm+2 zm+2 + .... (3.2)
where fj= 0 for j < m.
The forward traveltime is m, since fmis the first non-zero amplitude for the
forward trace and therefore represents the onset of seismic energy. Similarly,
the reverse trace R(z) is given by
R(z) = rnzn+ rn+1 z
n+1 + rn+2 zn+2 + .... (3.3)
where rj=0 for j < n. In this case, the reverse traveltime is n, since rnis the first
non-zero amplitude.
Convolution in the z domain is achieved by polynomial multiplication, ie.
F(z) * R(z) = fmrnzm + n + (fmrn+1 + fm+1 rn) z
m + n + 1
+ (fmrn+2+ fm+1 rn+1 + fm+2 rn) zm+n+2 + .... (3.4)
It can be seen that the first non-zero coefficient is fmrnand it occurs at the time m
+ n, which is at the sum of the forward and reverse traveltimes.
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Figure 3.7: Convolution section generated by convolving forward and reverse
shot records. The traces are presented at constant gain with no trace
equalization.
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The convolution section generated with the shot records in Figures 3.1 and 3.2
and an XY separation of 5 m, is shown in Figure 3.7. Each trace in fact
represents the time-depth, as both the subtraction of the reciprocal time and the
halving of the time scale have been carried out. (These operations were readily
achieved with software for processing seismic reflection data, by treating the
reciprocal time as a static correction and by halving the sampling interval in the
trace headers.)
It is immediately apparent that the moveout has been removed by the
convolution process. The convolution section shows the same structure on the
refractor interface as that obtained in Figure 3.6 with the traveltime data.
In addition, perhaps the other striking effect of the convolution section is the
convenient presentation of the amplitude information. It is clear that convolution
has compensated for the very large amplitude variations related to geometrical
spreading and other factors with the shot records, and that the signal-to-noise
ratios of the convolved traces are very similar. Although the compensation is not
exact, as will be shown below, it is still sufficient to permit the recognition of
amplitude variations related to geological factors.
However, the interface computed using traveltimes in Figure 3.6 is about 10 ms
shallower than that recognizable from the convolution section in Figure 3.7. This
discrepancy arises from the various gain functions used with each approach.
The time-depths in Figure 3.6 were computed with traveltimes at which the first
onset of seismic energy was detected on the shot records, using as high a gain
as was possible without the background noise causing any detectable deflections
before the first breaks. This gain is usually sufficient to cause clipping of most of
the seismic data after the first arrivals. On the other hand, the presentation gain
in Figure 3.7 is much lower, and it has been selected to permit the examination of
the first few cycles after the computed time-depth.
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3.7 - The Effects of Geometrical Spreading on the ConvolutionSection Amplitudes
The shot record amplitudes shown in Figures 3.3 and 3.4 demonstrate the very
large variations due to geometrical spreading, as well as the difficulties in
selecting an appropriate mathematical description. Figure 3.8 shows normalized
theoretical amplitudes for reciprocal distance squared and reciprocal distance
cubed functions for a shot at station 1. The values are normalized to that at
station 72, which is the most distant detector from the shot at station 1. The
variation in amplitude between the first and last detectors is about 19 db for
reciprocal distance squared spreading, while it is 28.6 db for the reciprocal
distance cubed case, with an average of about 24 db.
Figure 3.8 also shows the geometrical effects for the convolved traces, obtained
with equation 3.5, viz.
Geometric factor convolved trace= 1 / (Xn(L-X)n) (3.5)
where, X is the distance from one shot point to the detector, L is the shot point to
shot point distance, which in this case is 480 m, and n is 2 for the reciprocal
distance squared and 3 for the reciprocal distance cubed cases. The convolved
amplitudes have been normalized to the minimum values which are at station 49,
the midpoint of the shot point to shot point distance. The maximum variation in
the convolved amplitudes is between the ends and the midpoint of the detector
array, and is 5 db for n equal to 2 and 7.5 db for n equal to 3, with an average of
about 6 db.
It is clear that convolution has reduced the effects of geometrical spreading by
approximately 18 db, but that a residual geometric effect of about 6 db still
remains. However, the reduction is sufficient to be able to recognize amplitude
variations related to geological effects. This is shown in Figure 3.9, with the
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convolved amplitudes as well as the convolved amplitudes which have been
corrected for the residual geometric spreading with equation 2.5 for n equal to
both 2 and 3 and normalized to the value midway between the two shot points.
The first positive amplitudes are low and erratic, and so the absolute values of
the following first negative which are much larger and more consistent, are used.
Figure 3.8: Geometric spreading factors for shot records with the shot point at
station 1, and the convolution section for shot points at stations 1 and 97, for
reciprocal distance squared and cubed functions.
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Figure 3.9: First positive and negative normalized amplitudes measured on the
convolution section. The first negative amplitudes are also shown with inverse
distance squared and inverse distance cubed geometric corrections.
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Figure 3.10: The product of the forward and reverse amplitudes of the first
trough measured on the shot records, together with the product corrected for
inverse distance squared and inverse distance cubed geometric effects.
Figure 3.10 shows the product of the forward and reverse amplitudes presented
in Figures 3.3 and 3.4, together with the values corrected for the geometric effect
with equation 3.5. The pattern of amplitude variations is similar to that in Figure
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3.9, confirming that convolution has in fact multiplied the amplitudes, and that the
product has greatly reduced the geometrical effect.
In both Figures 3.9 and 3.10, it is possible to separate the convolved and
multiplied amplitudes into four regions which correlate well with those recognized
in chapter 5, (Palmer, 2001), using wavespeed and depth. Correction of the
convolved and multiplied amplitude products with the theoretical geometrical
effects improves the ease in recognizing the four regions, but does not alter the
general features of the amplitudes.
3.8 - Effects Of Refractor Dip On Convolution Amplitudes
The convolution of forward and reverse traces provides an approximate
correction for the effects of a dipping interface on the amplitudes measured with
vertical component geophones. Suppose the angle from the vertical at which a
critically refracted ray approaches the surface is for a horizontal refractor. The
vertical component measured with the standard geophone will be the forward or
reverse amplitude multiplied by cos. Therefore, the convolved amplitude will be
multiplied by cos2, ie.
Convolved Amphorizontal refractor= cos2AmpforwardAmpreverse (3.6)
Next, suppose the refractor has a dip of . The vertical component measured will
be the shot amplitude multiplied by cos(+) in one direction, cos(-) in the
reverse direction.
Vertical Shot Amp dipping refractor= cos() Amp (3.7)
The vertical component of the convolved amplitude is given by equation 3.8, viz.
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Convolved Ampdipping refractor=(cos2cos2- sin2sin2) AmpforwardAmpreverse
(3.8)
For small dip angles, say less than about fifteen degrees, the second order terms
in sincan be neglected, while the cos2term is approximately one. Therefore,
to sufficient accuracy the product of the forward and reverse amplitudes achieved
with convolution is given by
Convolved Ampdipping refractor= cos2AmpforwardAmpreverse (3.9)
Accordingly, amplitudes computed for plane horizontal refractors (Heelan, 1953;
Werth, 1967) can still be usefully applied to dipping layers when convolution is
employed.
3.9 - Conclusions
Seismic refraction acquisition techniques are characterised by large source to
receiver distances. Commonly, these distances are greater than about four
times the depth of the target, whereas for reflection methods, the equivalent
distances are less than the target depth. The large distances produce
commensurately large moveouts between adjacent traces and large amplitude
variations between the near and far traces.
The wide range of refraction amplitudes is the result of the rapid geometric
spreading factor, which is at least the reciprocal of the distance squared, and it
produces considerable variation in S/N ratios. Accordingly, most refraction
inversion methods use traveltime data with widely varying accuracies, which are
related to the large variations in signal-to-noise ratios.
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The time section, generated by convolving forward and reverse seismic traces
together with a static shift equal to the reciprocal time, addresses both issues of
large moveouts between adjacent traces and large amplitude variations.
The addition of the phase spectra with convolution effectively adds the forward
and reverse traveltimes. This process of addition is common to most of the
standard techniques for the inversion of refraction data. The convolution section
after shifting by the reciprocal time, shows the same structural features of the
refractor in units of time, as is obtained with the standard approaches.
Furthermore, the convolution section can be generated without a prior knowledge
of the wavespeeds in either the upper layer, as is required with the downward
continuation methods, or in the refractor, as is required with the application of a
linear moveout correction or reduction. This latter is especially important where
there are significant lateral variations in the wavespeed of the refractor.
The multiplication of the amplitude spectra with convolution, to a good first
approximation, effectively compensates for the effects of geometric spreading,
which can be significantly larger than the commonly assumed reciprocal of the
distance squared function. This compensation is generally sufficient to be able to
recognize amplitude variations related to geological causes, which are not as
easily detected in the shot records. The correlation of any amplitude variations
with the structural variations on the interface of the refractor can be more
conveniently and more rapidly carried out using the convolution section, than for
example by multiplying amplitudes measured on the shot records.
If necessary, a geometric correction based on the product of a reciprocal of the
distance power function in the forward and reverse directions, can be applied to
the convolution section. This correction exhibits a much reduced variation
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compared with those for the individual shot records, and it is most useful near the
shot points where it can have a value of up to a factor of about 2, or 6 decibels.
The ease and convenience of generating the convolution section facilitate its
inclusion in the routine processing of seismic refraction data using anymethod.
3.10 - References
Aldridge, D. F., and Oldenburg, D. W., 1992, Refractor imaging using an
automated wavefront reconstruction method: Geophysics, 57, 378-385.
Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edition:
McGraw-Hill Inc.
Donato, R. J., 1964, Amplitude of P head waves: J. Acoust. Soc. Am., 36, 19-25.
Grant, F. S., and West, G. F., 1965, Interpretation theory in applied geophysics:
McGraw-Hill Inc.
Hagedoorn, J. G., 1959, The plus-minus method of interpreting seismic refraction
sections: Geophys. Prosp, 7, 158-182.
Hagiwara, T., and Omote, S., 1939, Land creep at Mt Tyausa-Yama
(Determination of slip plane by seismic prospecting): Tokyo Univ. Earthquake
Res. Inst. Bull., 17, 118-137.
Hales, F. W., 1958, An accurate graphical method for interpreting seismic
refraction lines: Geophys. Prosp., 6, 285-294.
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