Palmer.04.Digital Processing Of Shallow Seismic Refraction Data.pdf

8/10/2019 Palmer.04.Digital Processing Of Shallow Seismic Refraction Data.pdf

1/28

47

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.


2/28

48

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.


3/28

49

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


4/28

50

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.


5/28

51

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.


6/28

52

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,


7/28

53

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.


8/28

54

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


9/28

55

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.


10/28

56

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


11/28

57

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).


12/28

58

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).


13/28

59

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


14/28

60

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,


15/28

61

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.


16/28

62

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.


17/28

63

Figure 3.7: Convolution section generated by convolving forward and reverse

shot records. The traces are presented at constant gain with no trace

equalization.


18/28

64

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.


19/28

65

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


20/28

66

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.


21/28

67

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.


22/28

68

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


23/28

69

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.


24/28

70

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.


25/28

71

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


26/28

72

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.


27/28

73

Hatherly, P. J., 1982, Wave equation modelling for the shallow seismic refraction

method: Expl. Geophys., 13, 26-34.

Hawkins, L. V., 1961, The reciprocal method of routine shallow seismic refraction

investigations: Geophysics, 26, 806-819.

Heelan, P. A., 1953, On the theory of head waves: Geophysics, 18, 871-893.

Hill, N. R., 1987, Downward continuation of refracted arrivals to determine

shallow structure: Geophysics, 52, 1188-1198.

McQuillan, R., Bacon, M., and Barclay, W., 1979, An introduction to seismic

interpretation: Graham & Trotman Limited.

Nettleton, L. L., 1940, Geophysical prospecting for oil: McGraw-Hill Book

Company Inc.

Palmer, D., 1976, An application of the time section in shallow seismic

refraction studies: Master's thesis, The University of Sydney.

Palmer, D., 1980, The generalized reciprocal method of seismic refraction

interpretation: Society of Exploration Geophysicists.

Palmer, D., 1986, Refraction seismics: the lateral resolution of structure and

seismic velocity: Geophysical Press.

Palmer, D., 1992, Is forward modeling as efficacious as minimum variance for

refraction inversion?: Explor. Geophys. 23, 261-266, 521.

Palmer, D., 2000a, Can new acquisition methods improve signal-to-noise ratios

with seismic refraction techniques?: Explor. Geophys., 31, 275-300.


28/28

Palmer, D., 2000b, Can amplitudes resolve ambiguities in refraction inversion?:

Explor. Geophys., 31, 304-309.

Palmer, D., 2001, Resolving Refractor Ambiguities With Amplitudes: Geophysics

66, 1590-1593.

Rockwell, D. W., 1967, A general wavefront method, inMusgrave, A .W., Ed.,

Seismic Refraction Prospecting: Society of Exploration Geophysicists, 363-415.

Sheriff, R. E., and Geldart, L. P., 1995, Exploration Seismology, 2ndedition:

Cambridge University Press.

Sjogren, B., 1979, Refractor velocity determination - cause and nature of some

errors: Geophys. Prosp., 27, 507-538.

Sjogren, B., 1984, Shallow refraction seismics: Chapman and Hall.

Taner, M. T., Matsuoka, M., Baysal, E., Lu, L., and Yilmaz, O., 1992, Imaging

with refractive waves: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys.

Thornburg, H. R., 1930, Wavefront diagrams in seismic interpretation: AAPG

Bulletin, 14, 185-200.

Werth, G. A., 1967, Method for calculating the amplitude of the refraction arrival,

inMusgrave, A. W., Ed., Seismic refraction prospecting: Society of Exploration

Geophysicists, 119-137.

Documents

Palmer.04.Digital Processing Of Shallow Seismic Refraction Data.pdf