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198 Appendix 1 Comments on “A brief study of the generalized reciprocal method and some of the limitations of the method” by Bengt Sjögren. A.1 - Introduction Despite the implications of the title, Sjögren (2000) is essentially a rejection of the generalized reciprocal method (GRM). It is clear that there are very few aspects of the GRM, which Sjögren finds acceptable, if in fact there are any at all. I consider that most of the substance of his critique of the GRM is either wrong or ill informed, while other aspects are simply matters of opinion. A thorough response to his paper would be quite lengthy and somewhat technical, and so I will restrict my response to three main issues only. They are the following: 1. Do we always need to define all layers above the target refractor, or are there situations where the use of an average wavespeed is more appropriate? I accept that Sjogren’s re-interpreted depth sections for the two case histories provide a better estimate of total depths across the complete profiles. However, the original depth sections in Palmer (1991) generated with the average wavespeeds are appropriate to the objectives of each case history and to those of the paper as discussed in the third issue.

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Page 1: Palmer.12.Digital Processing Of Shallow Seismic Refraction Data.pdf

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Appendix 1

Comments on “A brief study of thegeneralized reciprocal method and

some of the limitations of themethod” by Bengt Sjögren.

A.1 - Introduction

Despite the implications of the title, Sjögren (2000) is essentially a rejection of the

generalized reciprocal method (GRM). It is clear that there are very few aspects

of the GRM, which Sjögren finds acceptable, if in fact there are any at all.

I consider that most of the substance of his critique of the GRM is either wrong or

ill informed, while other aspects are simply matters of opinion. A thorough

response to his paper would be quite lengthy and somewhat technical, and so I

will restrict my response to three main issues only. They are the following:

1. Do we always need to define all layers above the target refractor, or are there

situations where the use of an average wavespeed is more appropriate? I

accept that Sjogren’s re-interpreted depth sections for the two case histories

provide a better estimate of total depths across the complete profiles. However,

the original depth sections in Palmer (1991) generated with the average

wavespeeds are appropriate to the objectives of each case history and to those

of the paper as discussed in the third issue.

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2. There are no fundamental mathematical differences between the GRM and

the collection of methods used by Sjögren. From both an assessment of his

descriptions of the various methods and an examination of some of his figures, it

is clear that both approaches generate essentially the same processed data, and

that his mean-minus-T method is identical to the wavespeed analysis function of

the GRM. The main differences are found in how the processed data are

interpreted. The GRM provides a systematic and objective framework, whereas

in my opinion, Sjögren's approach is neither consistent nor objective.

3. Sjögren has not addressed the fundamental aims of Palmer (1991), namely

the demonstration of an objective approach for recognizing and defining narrow

zones with low wavespeeds in the refractor. Sjogren’s approach is not

systematic and it frequently relies on personal judgment, and as a consequence,

it can result in the generation of artifacts, such as those shown in Sjögren (2000,

Fig. 5(a)).

A.2 - The Use of Average Wavespeeds

Sjogren’s assertion of the need to define all layers in geotechnical investigations

rather than use an average wavespeed, is not cognizant of the objectives of each

field study in Palmer (1991). In the case of the collapsed doline, the objective

was to determine its depth. Drilling had not been completely successful because

the loss of circulation in the rubble had stopped progress at depths of about 50 m

and before solid rock was encountered. Therefore, the issue is not whether

Sjogren’s detailed layer by layer approach is more useful than the average

wavespeed approach, but rather why both approaches, which give much the

same maximum depths of about 15 m, are clearly at variance with the drilling.

Irrespective of which interpretation approach is used, it is obvious that the

original objectives have not been achieved, that the refraction method is

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inappropriate for solving the problem, and that the differences in depth

computations between stations 56 and 71 are peripheral to the survey objectives.

Furthermore, Sjogren’s reasoning for his rejection of my explanation that the

refracted energy propagates around rather than under the doline is convoluted,

not convincing and ignores the implications of a genuine three-dimensional

structure.

The second case history was across a fault. An earlier high-resolution reflection

survey had been carried out in order to test whether the method was efficacious

in detecting known faulting in the underlying coal seams. These results were

poor, possible due to the proximity of the line to a busy road and the use of small

explosive charges. The refraction survey was then carried out in order to

generate a more accurate set of statics corrections. The component of the

statics corrections, which effectively replaces the weathered layer with

unweathered material prior to the application of the elevation component, was

generated with the approach described in Dobrin (1976, p.215), Palmer (1995),

and Palmer et al (2000). This method simply scales the time-depths by a factor

which is a function of the average wavespeed in the weathering and the

wavespeed in the refractor.

Therefore, while I accept that Sjogren’s depth computations may be more

appropriate to many types of geotechnical investigations, the use of average

wavespeeds in Palmer (1991) was entirely compatible with the aims of both the

field studies and the paper. Furthermore, the differences in total depths between

Palmer (1991) and Sjogren (2000) do not alter the major conclusions of the

paper with respect to the study of narrow zones with low wavespeeds.

In Palmer (1981, 1992, 2000a, 2000b) I demonstrate the use of the average

wavespeed in accommodating undetected layers, wavespeed reversals and

transverse isotropy. At present, there are no other published approaches to

solving these problems which are commonly encountered in many parts of the

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world, especially those with deep regoliths. Furthermore, Sjogren appears to

have overlooked other case studies (Palmer, 1980) in which the GRM has

effectively defined all layers above the target refractor.

A.3 - The Similarities Between The GRM and Sjogren’s Approach

Sjogren's indication that he does not accept the usefulness of this average

wavespeed is surprising, especially since he also uses an average overburden

wavespeed with Hales' method where there are multiple layers (Sjogren, 2000,

p.819). In that same paragraph, he also describes varying the XY distances, in a

manner analogous to the GRM. It is clear that there are many similarities

between the two average wavespeeds, but that the GRM has extended the

concept to include several important benefits such as the ability to accommodate

undetected layers, as well as minor differences in the interpretation of the

traveltime graphs. Even though Sjogren has emphasized such differences, there

are only minor differences in depth computations between Palmer (1991) and

Sjogren (2000) at the points where the average wavespeeds were determined.

While, it is acknowledged that the accuracy of the average wavespeed can be

reduced with distance from the point of computation, this is not necessarily a

problem, as was the case with the two field studies.

Furthermore, the similarities extend beyond the average wavespeeds. In Palmer

(1986), I describe Hales’ method and conclude that fundamentally, it is very

similar to the GRM. The similarities are that both methods obtain a measure of

the depth to the refracting interface in units of time through the addition of

forward and reverse traveltimes, and a measure of the wavespeeds in the

refractor from the differencing of the same forward and reverse traveltimes. The

equations describing these two operations for each method are virtually identical.

Furthermore, both methods employ refraction migration in order to accommodate

the offset distance, which is the horizontal distance between the point of

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refraction on the interface and the point of detection at the surface. The major

difference between the two methods is that Hales’ method achieves the addition,

subtraction and migration with a graphical approach, while the GRM performs

these same operations arithmetically with the scalar traveltimes prior to their

graphical presentation.

Sjogren’s determination of refractor wavespeed in the main refractor begins with

a version of the ABC method, which is a special case of the GRM with a zero XY

spacing. In the first case history but not the second, he then applies what is

clearly another version of the GRM wavespeed analysis algorithms with finite XY

values. Sjogren (2000, p.825) refers to curves 1 and 2 in his Fig. 4 as having

been computed with migrations of 5 m and 7.5 m. These curves bear a

remarkable resemblance to those computed with the GRM with similar XY values

in Palmer (1991, Fig.16). The minor differences are due to Sjogren’s editing of

the traveltime data. He then applies yet another method, namely Hales’ method,

to further refine his wavespeed determinations.

Accordingly, Sjogren’s approach with a number of methods and my approach

with the GRM are essentially generating the same set of computations for the

determination of refractor wavespeeds. Where Sjogren uses a succession of

different techniques, all of which employ addition and subtraction of forward and

reverse traveltimes, together with accommodation of the offset distance with

migration, the GRM achieves the same results within a single presentation, such

as in Palmer (1991, Fig.16).

Are the differences between Sjogren’s approach and the GRM important? An

examination of the wavespeed analysis function in Palmer (1991, Fig.16), reveals

that the graph for an XY value of 5 m has intervals of steeper gradient which

would correspond with lower wavespeeds, and which occur over the same

intervals with low wavespeeds shown in Sjogren (2000, Fig 5(a)). Therefore,

where Sjogren interprets virtually every change in slope in a single graph as

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evidence of a lateral change in wavespeed, I have concluded that there is

insufficient evidence for the existence of those intervals. This conclusion is

based on an assessment of all of the graphs in Palmer (1991, Fig. 16), and the

recognition of systematic changes in patterns, rather than the detailed

interpretation of a single graph.

The issue then, is whether the approach of Palmer (1991) under-interprets the

data and therefore overlooks intervals with low wavespeeds, or whether Sjogren

(2000) over-interprets the data and generates artifacts which do not really exist.

The issue is important because such features can be very significant in most

geotechnical, groundwater and environmental applications.

A.4 - Recognizing And Defining Narrow Zones With LowWavespeeds In Refractors

This difference between the two approaches introduces the fundamental

question which Palmer (1991) seeks to address. Is there an objective and

systematic approach, which is independent of individual interpretation styles, for

recognizing and defining narrow zones with low wavespeeds in refractors?

At the present time, I am still largely of the opinion that most narrow zones with

low wavespeeds are simply artifacts of the inversion algorithms and individual

interpretation styles. I have several reasons for holding this view.

1. Numerous model studies have shown that the algorithms which seek to

determine the wavespeeds in the refractor through the differencing of forward

and reverse traveltimes, can readily produce narrow zones with alternating high

and low wavespeeds, where there are significant changes in depths to the

refractor. This pattern can be seen for example, in the vicinity of the doline in

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Sjogren (2000, Fig. 5(a)), and it immediately raises doubts about the veracity of

the lateral changes in wavespeed.

2. The traveltime differences through these zones are small, and frequently they

are within the acceptable errors of the data, commonly plus or minus one

millisecond. For example, in Palmer (1991, Fig. 16), the differences between the

computed points and the line representing the fitted wavespeed function are

generally less than a millisecond for the optimum XY value of 5m. (Also see Fig.

2 to be discussed later.) I question whether any minor changes in slope are

statistically significant.

3. The issue is not resolved with forward modeling with either ray tracing or the

eikonal equation which are employed, for example, with tomographic and other

model-based methods of inversion. The GRM is able to generate a family of

geologically acceptable starting models in which the wavespeeds range from low

to high in narrow zones (Palmer, 2000c; 2000d) and which essentially satisfy the

original traveltime data (Palmer, 1980, p.49-52; 1986, p.106-107) to better than a

millisecond. This is simply another statement of the fundamental problem of non-

uniqueness with all inversion processes (Oldenburg, 1984; Treitel and Lines,

1988), but it is rarely if ever, addressed satisfactorily with refraction methods.

Therefore all of the refractor wavespeed models generated with different XY

spacings in Palmer (1991, Fig. 16), satisfy the traveltime data. Although some of

these can be rejected on simple geological grounds, such as those with negative

wavespeeds which obviously are not geologically realizable, there still remains a

range of models which fit the data to an acceptable accuracy.

Therefore, while I accept that Sjogren has a methodology for determining

wavespeeds in his Figures 1, 2, 4 and 7, I do not accept that he has addressed

the fundamental issues of non-uniqueness, that is of recognizing the artifact from

the real. I consider his approach relies heavily on his familiarity with Hales’

method and modified versions of the ABC method which include a migration

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process identical to the GRM, and that there is a significant subjective

component. Furthermore, his approach is poorly explained and the use of

different interpretation methods in a non-systematic manner is confusing.

Accordingly, I do not accept that he has demonstrated he has an objective

approach, or that there are narrow zones with low wavespeeds in the refractor in

Figure 5(a).

Sjogren's non-systematic approach can also be demonstrated with the model

data. Sjogren (2000, Fig. 2) changes the inclinations of the slope lines in the

Hales time loop on the basis of the wavespeeds derived from curve 1 for which

the XY value is zero, in order to improve the resolution of the narrow zone with

the low wavespeed. If that approach were to be employed with the first model

(Palmer, 1991, Fig. 2), using the wavespeed analyses for zero XY shown in

Palmer (1991, Fig. 5), then artifacts with both high and low wavespeeds at the

sloping interface, would be generated.

A.5 - Use Of Alternative Presentations And Amplitudes ForDetermining Wavespeeds In Refractors

In Palmer (1991), I present a systematic and objective criterion, generally known

as minimum variance. It is clear that an important aspect of this approach is to

determine a gross model of the refractor wavespeeds, and then to systematically

fit this model as has been done in Palmer (1991, Fig. 16). This is usually an

iterative process, simply because it is difficult to obtain the correct wavespeeds at

the first attempt. It can be somewhat challenging because of the need to

recognize the pattern of the departure of the computed points from the fitted line

as shown in Palmer (1991, Fig. 5), while at the same time accommodating the

normal errors in field data.

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Recently, I have been experimenting with averaging the wavespeed analysis

function for a range of XY values which range from less than to greater than the

optimum value, with the range of XY values being symmetrical about the

optimum, in order to derive a gross wavespeed model. This process minimizes

many of the apparent changes in wavespeeds due to structure where there are

no narrow lateral changes in wavespeed such as is shown in Palmer (1991, Fig.

5), and it averages many of the errors in picking traveltimes.

Figure 1. Refractor wavespeed analysis function, averaged for XY values from

zero to 10 m.

Using the traveltime data for the doline field study, Figure 1 shows a graph in

which the points computed with the wavespeed analysis function for XY values

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between zero and 10 m have been averaged. While Figure 1 risks introducing

yet another model of the refractor wavespeeds, namely about 2150 m/s between

stations 39 and 58, and about 3240 m/s elsewhere, there is no indication of any

narrow zones with low wavespeeds.

Figure 2. Differences between the averaged refractor wavespeed analysis

function in Figure 1, and the individual refractor wavespeed analysis functions for

XY values from zero to 10 m.

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Figure 2 shows the differences between the average and the computed

wavespeed analysis function for XY values between zero and 10 m. The

patterns of these differences are also consistent with there being no narrow

zones with low wavespeeds. The minimum differences occur for an XY value of

5 m, they are essentially random, and they are generally less than a millisecond.

Figure 3. Amplitudes of the forward and reverse offset shots.

I have also been investigating the use of amplitudes as an additional approach to

addressing this fundamental problem of non-uniqueness. In Palmer (2000e,

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2000f), I demonstrate that the amplitude of the refracted head wave, after

correction for geometrical spreading and refractor dip with either convolution or

multiplication, is essentially a function of the head coefficient. I further

demonstrate that the head coefficient is approximately proportional to the ratio of

the specific acoustic impedances (which is the product of the wavespeed and

density) in the overburden to that in the refractor. Therefore, arrivals from zones

in the refractor with low wavespeeds should exhibit high amplitudes, and vice

versa.

The amplitudes for the shot records are shown in Figure 3. For the shot at

station 1, the amplitudes shown a strong decay between stations 24 and 40,

which is interpreted as the geometric effect, together with an interval with

extremely low amplitudes between stations 46 and 59. The amplitudes for the

reverse shot at station 97 show a much less pronounced geometric effect, but

again there is an interval with very low amplitudes between stations 38 and 44.

These very low amplitudes were a major limitation on the measurement of

accurate and consistent traveltimes.

There are very few model studies on the effects of structure on the refraction

amplitudes. Nevertheless, it is unlikely that the very low amplitudes are

compatible with the simple refraction of energy from under the survey profile, but

rather with some form of scattering. They are compatible with energy

propagating around the doline and being scattered back to the surface through

the highly attenuating medium of the rubble in the collapsed doline, as has been

proposed in Palmer (1991).

Figure 4 shows the product of the amplitudes computed with an XY separation of

5 m. The results can be broadly separated into three regions which correspond

approximately with those recognized in Figure 1 with the wavespeed analysis

function. They include high values between stations 24 and 37, very low values

between stations 37 and 59, and higher values between stations 59 and 71. The

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lower values between stations 37 and 59 suggest higher rather than lower

wavespeeds. The slightly higher values between stations 37 and 42 suggest

lower wavespeeds, but do not correlate with those determined by Sjogren (2000,

Fig5(a)). However, these results should be used with considerable caution

because of the severe attenuation of seismic energy within the rubble of the

collapsed structure, and because the feature is three-dimensional.

Figure 4. Product of the forward and reverse shot amplitudes shown in Figure 3

with an XY value of 5m in arbitrary units.

Therefore, it seems probable that there are no narrow zones with low

wavespeeds associated with the collapsed doline and that Sjogren has

generated artifacts through an over-interpretation of the processed data.

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A.6 - A Systematic Approach With The GRM

In summary, Sjogren concludes that “the GRM must be regarded as being of

limited use for detailed and accurate interpretations of refraction seismics for

engineering purposes.” This conclusion is surprising given the very close

similarities between the GRM, Hales’ and the mean-minus-T methods, the latter

two of which he clearly favours. The fact that key features of his figures bear a

remarkable resemblance to parts of the GRM presentations emphasizes the

essential similarities between the two approaches. Instead, he has sought

differences where none really exist, and as a corollary, he has not recognized

similarities where in reality there are many. On the basis of the fundamental

similarities between the GRM and Sjogren's processed data, I conclude that

Sjogren (2000) is more a demonstration of his interpretation style and experience

using a collection of methods rather than a cohesive assessment of the GRM.

However, his approach is neither systematic nor entirely objective, and as a

result it is prone to the generation of artifacts.

It seems that the aim of his paper is to emphasize minor differences, mainly in

the assignment of layers to the traveltime graphs, and then to illogically imply

fundamental shortcomings of the GRM. As such, his paper lacks balance and

objectivity, and it is more in keeping with seeking a conviction in an adversarial

court system than with a scientific journal.

Sjogren (2000) has produced nothing of substance which requires any

fundamental re-assessment of the main features of the GRM in general, or of

Palmer (1991), in particular. He has not satisfactorily addressed the aims of

Palmer (1991), namely an objective method for the recognition and definition of

narrow zones with low wavespeeds. His conclusion that the GRM is unsuitable

for geotechnical applications is not substantiated, and it is based on an

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incomplete understanding of the GRM and of Palmer (1991), rather than any

genuine shortcomings with the method.

A.7 - The Need To Promote Innovation In Shallow RefractionSeismology

Sjogren’s rejection of the GRM as a useful method for processing and

interpreting shallow seismic refraction data, does little to encourage others to

present new approaches through fear of biased criticism. It is irresponsible and

does not advance the science through balanced and objective debate. In the last

fifty years, innovation in shallow refraction seismology has been rather modest at

best and it has focused predominantly on the various competing methods for

inverting field data. Until there is widespread consensus through a recognition of

fundamental similarities between these inversion methods, there will be little

advancement in other equally important aspects of the science. By comparison,

reflection seismology has achieved major advances through the development of

common midpoint methods, digital signal processing, three-dimensional methods

and sophisticated computer interpretation programs, over the same period of

time. It is now time to move on to the refraction techniques which will be

appropriate to the requirements and the technology of the new millennium.

In Palmer (2000g), I demonstrate the generation of the refraction convolution

section (RCS) through the convolution of forward and reverse traces. The

addition of the traveltimes with convolution is equivalent to that achieved

graphically with Hales’ method and arithmetically with the GRM. The RCS

facilitates full trace processing of seismic refraction data and in turn, the

examination of many important issues such as signal-to-noise ratios, “amplitude

statics”, 3D refraction methods and azimuthal anisotropy, signal processing to

enhance second and later events and stacking data in a manner similar to CMP

reflection methods. A major advantage of the RCS is that it incorporates

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amplitudes and time structure within a single presentation, facilitating the

resolution of many of the non-uniqueness issues discussed here. It is extremely

rapid and suitable for use with any volume of data, and therefore it can be readily

included in the processing of refraction data with any method.

The RCS is a new approach to obtaining more and better information from

shallow seismic refraction data, and in time it may even supercede the GRM as

well as Hales’ method. However, that possibility can only occur if there is a shift

in culture from one of conflict and an emphasis on minor differences to one of

consensus and an emphasis on fundamental similarities which traditionally, has

characterized the scientific method.

Sjogren’s critique of the GRM does not seek the consensus essential for the

advancement of the science of shallow refraction seismology. Regrettably, it is

neither balanced nor objective, it shows minimal insight into the fundamental

similarities of most methods of refraction inversion, it does not provide an

alternative systematic approach to refraction interpretation, it does not address

the important issues of non-uniqueness, and it does not provide a vision for

future innovation.

A.8 - References

Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edition:

McGraw-Hill Inc.

Oldenburg, D. W., 1984, An introduction to linear inverse theory: Transactions

IEEE Geoscience and Remote Sensing, GE-22(6), 666.

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

interpretation: Society of Exploration Geophysicists.

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Palmer, D., 1981, An introduction to the generalized reciprocal method of seismic

refraction interpretation: Geophysics 46, 1508-1518

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

seismic velocity: Geophysical Press.

Palmer, D., 1991, The resolution of narrow low-velocity zones with the

generalized reciprocal method: Geophysical Prospecting 39, 1031-1060.

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

refraction inversion?: Exploration Geophysics, 23, 261-266, 521.

Palmer, D., 1995, Can linear inversion achieve detailed refraction statics?:

Exploration Geophysics, 26, 506-511.

Palmer, D., Goleby, B., and Drummond, B., 2000a, The effects of spatial

sampling on refraction statics: Exploration Geophysics, 31, 270-274.

Palmer, D., 2000a, Model determination for refraction inversion: submitted.

Palmer, D., 2000b, The measurement of weak anisotropy with the generalized

reciprocal method: Geophysics, 65, 1583-1591.

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

Exploration Geophysics 31, 304-309.

Palmer, D., 2000d, Starting models for refraction inversion: submitted.

Palmer, D., 2000e, Imaging refractors with the convolution section: Geophysics,

in press.

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215

Palmer, D., 2000f, Resolving refractor ambiguities with amplitudes: Geophysics,

in press.

Palmer, D, 2000g, Digital processing of shallow seismic refraction data with the

refraction convolution section, PhD thesis, UNSW, submitted.

Sjogren, B., 2000, A brief study of the generalized reciprocal method and of

some limitations of the method: Geophysical Prospecting 48, 815-834.

Treitel, S., and Lines, L., 1988, Geophysical examples of inversion (with a grain

of salt): The Leading Edge 7, 32-35.

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Appendix 2

Model Determination For RefractionInversion

A.1 - Summary

In recent years, tomography or model-based inversion, has been used to

construct a model of the subsurface from seismic refraction data, mainly to

determine statics corrections for reflection data. With these methods, the

parameters of a model of the subsurface are refined by comparing the

traveltimes of the model with the field data. When the differences between the

computed and field traveltimes are a minimum, the model and parameters are

taken as an accurate representation of the wavespeeds in the subsurface.

In this study I demonstrate the sensitivity of model-based inversion to the

selection of the inversion model, using eleven geological models. In general, the

residuals between the original and modeled traveltimes are better than 0.4%,

indicating that virtually any model can be fitted to the data with high accuracy.

However, the errors in depth computations are between 5% and 10% for simple

monotonic increases of wavespeed with depth. For other models such as a

reversal in wavespeed, the errors are indeterminate, but much larger. The depth

errors are least when the inversion model and the original geological model are

similar, and greatest when the models are different.

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I also determine the parameters for the same eleven inversion models with the

generalized reciprocal method (GRM). With the GRM, the parameters of each

model are constrained with the optimum XY value. The errors in depth

computations are generally about one third of those for the equivalent model-

based case. Furthermore, the GRM is able to produce reasonable results with

the wavespeed reversal and transverse isotropy models, unlike the model-based

methods. However, the residuals between the direct traveltimes for the inversion

model and the original geological model are up to a factor of four greater than

with the equivalent model-based case. The residuals are least when the models

are similar, and greatest when the models are different.

The GRM determines model parameters by interpolation with the optimum XY

value. This parameter is determined from the refracted traveltimes from the

underlying layer, and therefore it is a function of all wavespeeds and thicknesses

in the overlying layer(s). By contrast, the model-based methods determine the

model parameters from the direct traveltimes from the upper part of the layer(s),

and then extrapolate those parameters throughout the remainder of the layer(s)

I conclude that suitable starting models for tomographic or model-based

inversion have parameters which are determined with optimum XY values, and

which have minimum residuals between the field and modeled traveltimes of the

direct arrivals.

A.2 - Introduction

In recent years, model-based inversion or tomography, has been used to process

seismic refraction data, often to determine statics corrections for reflection data

(Zhu et al., 1992). With these methods, the parameters of a model of the

subsurface are refined by comparing the traveltimes of the model with the field

data. When the differences between the computed and field traveltimes are a

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minimum, the model and parameters are taken as an accurate representation of

the wavespeeds in the subsurface.

While the performance of refraction tomography has been continually improved

through more efficient inversion and forward modeling routines, (see Zhang and

Toksoz, 1998 for an overview of these advances), one issue, which has yet to

receive widespread attention, is the choice of the model for the inversion

process. This situation is not surprising, since the role of model-based inversion

is to provide information about the unknown numerical parameters which go into

the model, not to provide the model itself (Menke, 1989, p3).

Perhaps the most common model has been the linear increase of wavespeed

with depth (Zhu et al.,1992; Stefani, 1995; Miller et al., 1998; Lanz et al., 1998),

possibly because of mathematical convenience. However, this model is of

questionable validity as most theoretical (Iida, 1939; Gassman, 1951, 1953;

Brandt, 1955; Paterson, 1956; Berry1959), laboratory (Birch, 1960; Wyllie et al.,

1956, 1958), and field studies (Faust, 1951, 1953; White and Sengbush, 1953;

Acheson, 1963, 1981; Hall, 1970; Hamilton, 1970, 1971; Jankowsky, 1970),

suggest a more gentle increase for clastic sediments, such as a one sixth power

of depth function.

Furthermore, the gradients obtained range from 0.342 and 2.5 m/s per metre

(Stefani, 1995), and 2.68 and 4.67 m/s per metre (Zhu et al., 1992), to as high as

40 m/s per metre (Lanz et al, 1998). These values are generally larger than

those applicable to the compaction of clastic sediments (Dobrin, 1976), but they

are rarely justified on geological or petrophysical grounds.

The combination of the linear increase of wavespeed with depth and the high

gradients probably contributes to instability in the inversion process. The

example of the somewhat paradoxical situation of the poor determination of

wavespeeds in the refractor, despite the fact that over 90% of traveltimes are

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from that layer (Lanz et al., 1998), is at variance with the experiences of most

seismologists using more traditional methods of refraction processing.

In this study, I demonstrate the effects of the choice of the model on refraction

inversion using both the model-based approach and the generalized reciprocal

method (GRM), (Palmer, 1980, 1986). With the model-based approach, I show

that a range of models can be fitted to noise-free traveltime data with acceptable

accuracy, but that there is a wide range in the depths computed to the main

refractor. However, the accuracy of depth computations improves as the

inversion model approaches the subsurface or geological model. In addition,

only a limited number of models can be examined with model-based methods of

inversion.

With the GRM approach, I show that a wider range of models can be addressed,

with generally greater accuracies in depth computations. However, in contrast

with the methods of model-based inversion, the agreement with the traveltime

data is usually much poorer.

I conclude that the appropriate starting model for tomographic inversion is one for

which the depths are similar to those obtained with the GRM, and for which the

field and model traveltimes agree.

A.3 - Model and Inversion Strategies

The model of the subsurface used in this study is shown in Figure A.1, and

consists of two layers with the upper layer having a parabolic variation in

wavespeed with depth, viz.

V(z) = 1750(1 + 0.001 z)½ (A.1)

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where V(z) is the wavespeed and z is the depth. The rate of increase of the

wavespeed is 0.875 m/s per metre at the surface, while the average is about

0.73 m/s per metre over a depth of 1000m. These values are comparable with

those caused by compaction of clastic sediments (Dobrin, 1976). The parabolic

wavespeed function is a compromise between the more commonly used linear

wavespeed function and the theoretically derived function which is a one sixth

power of depth.

Figure A.1: Two layer model for which the traveltime data shown in Figure A.2

were computed. The seismic wavespeed in the upper layer is a parabolic

function of depth.

The depth to the refractor changes from 750m to 1000m over a horizontal

distance of 500m, and the wavespeed in the refractor is 5000 m/s.

The traveltimes are shown in Figure 2, and are taken from Palmer (1986).

In this study, I examine the fundamental issue of what is the appropriate model

for the wavespeed of the first layer, given that the traveltime data indicate that

there are two layers with a single interface. I approximate the wavespeed in the

first layer with eleven inversion models. They are the constant wavespeed model,

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the two layer model, including a reversal in wavespeed, the Evjen function, and

transverse isotropy.

Figure A.2: Traveltime data for the model shown in Figure A.1. The direct

arrivals from the shot points at 2400m and 4800m penetrate 85m for a range of

1200m and 300m for a range of 2400m.

In the model-based approach, I determine the wavespeed stratification in the

layers above the refractor from the direct arrivals from those layers. I derive the

parameters for each model from the traveltime graphs for the one and two layer

models, by iteratively adjusting for the Evjen models until the traveltimes for the

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data and the model agree, or simply by arbitrarily selecting wavespeeds for the

wavespeed reversal and the transverse isotropy models.

With the GRM approach, I determine the wavespeeds in the layers above the

refractor using the optimum XY value, which I derive with the traveltimes from the

refractor. The optimum XY value is determined in the following way.

The refractor wavespeed analysis function tV, is given by equation A.2, viz.

tV = (tforward - treverse}+ treciprocal)/ 2 (A.2)

Equation A.2 was evaluated for a range of XY values, which is the separation

between the detectors at which the forward and reverse traveltimes are recorded,

from 500 m to 1150 m.

I describe the method of determining optimum XY values in some detail

elsewhere (Palmer, 1991). Essentially, the procedure involves the determination

of the overall trends of the wavespeed analysis function, from which the refractor

wavespeed is derived, and then the deviations from those trends, from which the

optimum XY value is obtained. In this study, I use averaging to obtain the

wavespeed in the refractor, and then differencing to obtain the optimum XY

value.

Figure A.3 shows the average of the refractor wavespeed analysis function

computed at each location for the full range of XY values from 500 m to 1150 m.

The reciprocal of the gradient of this function is the refractor wavespeed Vn, viz.

5046 m/s. This value is higher than the true value for the model of 5000 m/s, and

is the result of the non-planar refractor interface. A correction for dip (Palmer,

1980, equation 9), when the refractor interface is approximated with a planar

interface with an average dip of about six degrees (tan-1(250/2400)), improves

the wavespeed estimate to 5018 m/s.

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Figure A.3: The averaged wavespeed analysis function, computed for the range

of XY values from 500 m to 1150 m in increments of 50 m, the station spacing.

The reciprocal of the gradient of this function is the wavespeed in the refractor,

namely 5046 m/s.

Figure A.4 presents the differences between the computed values of tV for a

selected range of XY values and the averaged values. A visual inspection

indicates that the average optimum XY value is 800 m, because the differences

are the closest to zero. This value is consistent with the values of 700 m and 950

m for each side of the model obtained with a more detailed analysis, (Palmer,

1986).

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Figure A.4: The differences between the wavespeed analysis function

computed for the range of XY values from 650 m to 1000 m, and the averaged

values shown in Figure A.3. The values for an XY value of 800 m are assessed

as being closest to zero, thereby indicating that the optimum XY value is 800m.

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Depths for both the model-based inversion and GRM approaches are computed

from the time-depths, tG, where

tG = (tforward + treverse - treciprocal – XY / Vn)/2 (A.3)

The reciprocal time is the traveltime between the two shot points.

As the term implies, the time-depth is a measure of the depth to the refracting

interface in units of time. It is analogous to the one-way reflection time, and it is

often, but incorrectly identified with the delay time. The time-depth is, in fact, an

average of the forward and reverse delay times.

The time-depths range from 342 ms to 429 ms, which correspond to the

horizontal sections of the interface. The average value used in the GRM

calculations is 384 ms.

The time-depth is related to the depth zG, by:

zG = tG DCF (A.4)

where the DCF, the depth conversion factor relating the time-depth and the

depth, is given by:

DCF = V Vn / (Vn2 - V2)½ (A.5)

or

DCF = V / cos i (A.6)

where V is the average wavespeed above the refractor and

sin i = V / Vn (A.7)

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Figure A.5: The difference in traveltimes between the original data and the

modeled response for the single layer with a constant wavespeed inversion

model, for the model-based and GRM approaches.

A.4 - Single Layer Constant Wavespeed Inversion Model

The simplest model has a single layer with a constant wavespeed above the

refractor. As the traveltime graphs with the shot points at distances 2400 m and

4800 m show some curvature, the fitting of one straight line is easily recognized

as an approximation. However, for completeness with the model-based

approach, I use a line which passes through the origin and through the graph at a

traveltime of 1103ms and at a distance of 2000m from the shot point, to obtain an

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average wavespeed of 1813 m/s. The differences in the traveltimes for the data

and the constant wavespeed model range from -16 ms to 19 ms, and are shown

in Figure A.5. These residuals are comparable to those obtained by Stefani

(1995) with the Tibalier Trench data which is at approximately the same depth.

Accordingly, I view the fit between the computed and original data as acceptable.

Figure A.6: Unmigrated depth sections for the single layer with a constant

wavespeed inversion model, for the model-based and GRM approaches.

The depths computed with this wavespeed are shown in Figure A.6, and they

have been plotted vertically, rather than orthogonally to the refractor interface.

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An additional step, equivalent to the migration operation of reflection data, is

required to accurately image the refractor interface (Palmer, 1986, p.196-198).

The depths computed over the horizontal sections of the refractor are 659.4m

(for an error of -12.1%), and 833.4m (for an error of -16.7%), and they are listed

in Table A.1.

The GRM approach uses the average velocity formula (Palmer, 1980, p.42, eqn.

27; 1986, p.147, eqn. 10.4), for the single layer, constant wavespeed model,

viz.

V = [(XY Vn2) / (XY + 2 tG Vn)]½ (A.8)

The resultant average wavespeed is 2107 m/s, which in turn produces depths

over the horizontal sections of 770.2 m (for an error of 2.7%) and 973.6 m (for an

error of -2.6%). The differences in the traveltimes for the data and the constant

wavespeed model range from zero to -140 ms, and are shown in Figure A.5.

This inversion model emphasizes the appropriateness of the selection of the

starting model with model-based inversion. The constant wavespeed model is

applicable to a wide range a subsurface targets especially those in the near

surface, and it has proven to be efficacious since the earliest days of refraction

seismology. Furthermore, it is often used for field data, where the curvature of

the traveltime graph is not obvious (Palmer, 1983; Zhu et al., 1992). This

situation is not uncommon because the amplitudes at the larger offsets are

usually much weaker than those on the near offsets, thereby resulting in the

arrival times being measured later than is the case with noise-free data.

Even though the differences in the traveltimes between field and the modeled

data are an order of magnitude smaller with the model-based inversion approach

than with the GRM, there is not a commensurate improvement in the depth

computations.

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Figure A.7: The difference in traveltimes between the original data and the

modeled response for the two layers with constant wavespeeds inversion model,

for the model-based and GRM approaches.

A.5 - Two Layer Constant Wavespeed Inversion Model

Clearly, the single layer constant wavespeed model is not a completely

satisfactory approximation, and the next step is to consider a two segment

approximation to the traveltime graphs. This model is especially realistic when

the curvature of the traveltime graphs is not obvious with field data, and as a

result, the graphs are approximated with two straight line segments. In this case

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the two wavespeeds are 1775 m/s and 1932 m/s, and the intercept time is 65ms

for the second layer. Figure A.7 shows that the modeled traveltimes are within

±4 ms of the original data, a result which would normally be considered an

excellent approximation. Figure A.8 shows the computed depths for the

horizontal sections are 695.2 m (for an error of -7.3%), and 882.7 m (for an error

of -11.7%).

Figure A.8: Unmigrated depth sections for the two layers with constant

wavespeeds inversion model, for the model-based and GRM approaches.

For the GRM approach to the two layer approximation, I solve two simultaneous

equations which relate layer thicknesses and wavespeeds to the time-depth, and

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to the optimum XY value, (Palmer, 1980, p.47: 1986, p.136-137). The results are

the layer thicknesses z1 and z2, after values for the wavespeeds have been set.

The equations are:

tG = z1 cos i1n / V1 + z2 cos i2n / V2 (A.9)

XY = 2 z1 tan i1n + 2 z2 tan i2n (A.10)

where

ijn = sin-1 (Vj / Vn) (A.11)

The values used for the wavespeeds are 1800 m/s for V1, which is suggested by

the traveltime graphs, and 2400 m/s for V2. The computed depths are 735.7 m

(for an error of -1.9%), and 981.2 m (for an error of -1.9%).

The traveltime differences are shown in Figure A.7 and range from -10 ms to +25

ms. As with the single layer inversion model, these differences are much larger

than those for the model-based inversion approach, but the depth computations

are still more accurate.

A.6 - Two Layer Wavespeed Reversal Inversion Model

Suppose that there were sufficient reasons for considering the existence of a

surface layer with a wavespeed of 3600 m/s. In this case, the traveltime graphs

with the shot points at 2400m and 4800m, would be ignored because of

additional, more compelling reasons, such as the existence of a local surface

lens of limestone or permafrost with a high wavespeed, or even extrapolation

from an adjacent area. The model-based inversion approach can only assume

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that this wavespeed occurs throughout the upper layer, resulting in computed

depths of 1758.6 m (for an error of 137%) and 2222.8 m (for an error of 123%).

The GRM approach to the wavespeed reversal model is similar to the two layer

constant wavespeed model described previously. In that case, the selection of

the second layer wavespeed of 2400 m/s might have appeared to be rather

arbitrary. In fact, almost any wavespeed will suffice, provided it is above a

certain minimum, because the thickness of the corresponding layer is

automatically adjusted to produce an effective average wavespeed similar to that

obtained with equation A.4. Furthermore, it will be noted that there is no

requirement for the wavespeeds to increase monotonically with depth with this

approach, and therefore it can be applied equally validly to the wavespeed

reversal model. Using wavespeeds of 3600 m/s and 1800 m/s, solution of

equations A.9 to A.11 produces depths of 731.81 m (for an error of -2.4%), and

904.7m (for an error of -9.5%).

The wavespeed reversal model is generally acknowledged as one of the most

difficult if not impossible to address satisfactorily with refraction tomography

(Lanz et al., 1998). In the absence of other geological or geophysical data, the

GRM approach provides a measure of constraint for a normally intractable

problem.

A.7 - The Evjen Inversion Model

If in fact, it was recognised that the traveltime data represented a variable

wavespeed medium, then the wavespeed function in the overburden can be

approximated with the Evjen function, viz.

V(z) = V0 (1 + q z)1/m (A.12)

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For the model-based inversion approach, I determine the variable q, by iteratively

matching the traveltime graphs, once the value of V0 is set at 1750 m/s, and m is

set at 1,2,3,4 or 6.

Figure A.9: The difference in traveltimes between the original data and the

modeled response for the single layer with a linear function of the depth

wavespeed inversion model, for the model-based and GRM approaches.

The GRM approach to the variable wavespeed model is described in detail in

Palmer (1986, p.175-181), and is outlined in the Appendix.

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Figure A.9 shows the results for the linear wavespeed function. As with other

inversion models, the model-based inversion is able to achieve excellent

agreement with the traveltime data (-2 ms to + 0.5 ms), while depth computations

are 766.7 m (2.2% error) and 1026.2 m (2.6% error). With the GRM approach,

the differences in traveltimes are up to 12 ms, while the depths are 751.0 m

(0.1% error) and 996.7 m (-0.3% error).

Figure A.10: The difference in traveltimes between the original data and the

modeled response for the single layer with a parabolic function of the depth

wavespeed inversion model, for the model-based and GRM approaches.

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Figure A.10 shows the results for the parabolic wavespeed function. The errors

in the traveltimes with model-based methods are zero, and the depth estimates

are 757.2 m (1% error) and 1002.5 m (0.25% error). The GRM depth

computations are of comparable accuracy being 749.5 m (-0.4% error) and 989.6

m (-1% error). The differences in the traveltimes are up to 6 ms.

Figure A.11: The difference in traveltimes between the original data and the

modeled response for the single layer with a one third power of the depth

wavespeed inversion model, for the model-based and GRM approaches.

The one third power of the depth function produces excellent agreement with the

traveltimes for both the model-based inversion and the GRM approaches. As is

shown in Figure A.11, traveltimes differences are generally less than 1 ms.

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Depth estimates are also quite accurate with 749.9 m(-0.01% error) and 985.7 m

(-1.5% error) for the model-based inversion, and 751.2 m (0.16% error) and

987.9 m (-1.2% error) for the GRM.

Figure A.12: The difference in traveltimes between the original data and the

modeled response for the single layer with a one fourth power of the depth

wavespeed inversion model, for the model-based and GRM approaches.

For the one fourth power of the depth function, traveltimes are generally within ±1

ms (see Figure A.12), while the depths are 745.0 m (-0.7% error) and 966.5 m (-

3.4% error) for the model-based inversion. For the GRM approach, the

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traveltime differences are as much as -5 ms, while the depths are 752.7 m (0.4%

error) and 986.4 m (-1.4% error).

Figure A.13: The difference in traveltimes between the original data and the

modeled response for the single layer with a one sixth power of the depth

wavespeed inversion model, for the model-based and GRM approaches.

The results for the one sixth power of the depth function are similar to those

described above. For the model-based inversion approach, the traveltime

differences are between -2 ms and +4 ms (see Figure A.13) while the depths are

735.0 m (-2% error) and 954.8 m (-4.5% error). For the GRM approach, the

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traveltime differences are up to -15 ms, while depths are 755.2 m (0.7% error)

and 984.4 m (-1.6% error).

A.8 - Transverse Isotropy Inversion Model

Finally, there is the case of seismic anisotropy, in which the horizontal

wavespeed is different from the vertical value. This is one area where the

geophysical idealization of the earth can differ substantially from the

petrophysical reality. While there is overwhelming evidence that wavespeeds are

more likely to be anisotropic than isotropic, because of intrinsic anisotropy, cyclic

layering, etc, it is rare for the basic model for most refraction inversion routines to

include this property. No doubt the complexities of the mathematical treatment,

together with the lack of accepted methods for determining anisotropy

parameters from surface measurements, contribute to this situation.

In most cases, the horizontal wavespeed is greater than the vertical value. This

is usually the case with undisturbed sedimentary rocks, because of the effects of

compaction and cyclic layering. However, the reverse condition may apply, such

as with steeply dipping sedimentary or metamorphic rocks, or with non-

hydrostatic stress. Accordingly, simply assuming an anisotropy factor with the

vertical wavespeed being less than the horizontal can sometimes be

inappropriate.

Unless there is additional information, such as bore hole control, seismic

anisotropy is not easily recognised or accommodated with the standard refraction

methods, and any depth and wavespeed computations can have indeterminate

errors. However, for completeness of the comparison, I use the commonly

assumed anisotropy factor for P waves in sedimentary rocks of 1.10, together

with the horizontal wavespeeds of 1800 m/s, 2100 m/s and 2300 m/s, for the

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model-based inversion approach. The vertical wavespeeds used for depth

conversion are then 1636 m/s, 1909 m/s and 2091 m/s.

The GRM approach to anisotropic overburdens is described in detail in Palmer

(Palmer, 1986). With this approach, I seek to determine the anisotropy factor,

which is the horizontal wavespeed divided by the vertical wavespeed, for which

the wavespeed given by the Crampin equation, viz.

V2(φ) = A + B cos 2φ + C cos 4φ (A.9)

(where φ is the angle from the vertical), is equal to the average wavespeed

modified for anisotropy, viz.

V = [(XY Vn2) / (XY + 2 c tG Vn)]½ (A.10)

where

c = (A - B + C - 8 C cos4 φ) / (A + B + C - 8 C sin4 φ) (A.11)

The value of φ used in equations A.13 and A.15, is that for the critical angle for

an horizontal refractor. I examine three models of anisotropy.

The first model uses a horizontal wavespeed of 1800 m/s, which can be

recovered from the traveltime graphs. For the model-based inversion approach

using an anisotropy factor of 1.1, the computed depths are 664.3 m (-11.4%

error) and 839.7 m (-16% error).

The GRM approach produces an anisotropy factor of 0.855, and depths of 789.7

(5.3% error) and 998.1 m (-0.2% error). The computed value of the anisotropy

factor is unusual in that it is less than unity. Nevertheless, the computed depths

are reasonably close to the true values.

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The second model uses a horizontal wavespeed of 2100 m/s, which is

approximately the average wavespeed in the first layer. For the model-based

inversion approach using an anisotropy factor of 1.1, the computed depths are

700.0 m (-6.7% error) and 885.1 m (-11.5% error).

The GRM approach produces an anisotropy factor of 1.025, and depths of 763.0

m (1.7% error) and 965.5 m (-3.5% error). In this case, the anisotropy factor is

close to unity, which is compatible with the selection of a horizontal wavespeed

similar to the average computed with equation A.4. Therefore, the GRM can

produce an isotropic result if indeed that is appropriate.

The third model uses a horizontal wavespeed of 2400 m/s. For the model-based

inversion approach, the computed depths are 780.4 m (4.1% error) and 986.4 m

(-1.4% error). This is the best result for this approach and it is the result of a

fortuitous combination of the horizontal wavespeed and the assumed anisotropy

factor producing a vertical wavespeed which is comparable with the average

computed with equation A.4.

The GRM approach produces an anisotropy factor of 1.21, and depths of 706.9

m (-5.8% error) and 893.6 m (-10.6% error).

Seismic anisotropy, like wavespeed reversals, emphasizes the fact the model-

based inversion is only efficacious when the traveltime data are an accurate and

complete reflection of the subsurface wavespeed stratification, and therefore

when an inversion model can be determined.

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A.9 - Errors Related to the Optimum XY Value

In general, the errors in the depth computations related to the choice of the

choice of the inversion model, are between 2% to 3% with the GRM approach.

For this study, a single or average optimum XY value is used. I demonstrate

further improvements in accuracy when the optimum XY values, which are

applicable to each side of the sloping segment of the refractor, are used (Palmer,

1986, 1992).

The errors in the depth computations which are related to the errors in the

measurement of the XY value, can be estimated in the following way.

The errors in the optimum XY value are one half of the detector spacing, ie ± 25

m. Therefore the possible range of optimum XY values is 800 m ± 25 m, ie 775

m to 825 m. (An inspection of Figure A.4 suggests that the latter value of 825 m

is probably the best estimate of a single average value. It is also the mean of the

values applicable to either side of the sloping interface (Palmer, 1986).)

The single layer constant wavespeed will be used for convenience. The average

wavespeed computed with equation 4 using optimum XY values of 775 m and

825 m, are 2079 m/s and 2133 m/s. They differ from the value of 2107 m/s used

in the single layer inversion model above, by an average of ± 27 m/s, or about

1.3%. The resulting error in depth calculations is approximately ± 11.4 m or

1.3%.

Alternatively, it can be demonstrated by differentiation of equation A.4 that:

∆V / V ≈ ½ ∆XY / XY (A.16)

Therefore, an error of 3% (25/800) in the optimum XY value, results in an error of

1.5% in the average wavespeed.

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Figure A.14: Summary of the errors in depth computations for all inversion

models, for the model-based and GRM approaches. The errors for the

wavespeed reversal model using the model-based approach have been

arbitrarily set at 22% and 25%.

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INVERSION MODEL MODEL-BASED GRM

InversionModel

TrueDepth

Wavespeed Depth Error Wavespeed Depth Error

One Layer 750 m

1000 m

1813 m/s

1813 m/s

659 m

834 m

-12%

-17%

2107 m/s

2107 m/s

770 m

974 m

2.7%

-2.6%

Two Layers 750 m

1000 m

1775 m/s

1932 m/s

1175 m/s

1932 m/s

146 m

695 m

146 m

883 m

-7%

-12%

1800 m/s

2400 m/s

1800 m/s

2400m/s

382 m

736 m

429 m

981 m

-1.9%

-1.9%

Wavespeed

Reversal

750 m

1000 m

3600 m/s

3600 m/s

1759 m

2223 m

137%

123%

3600 m/s

1800 m/s

3600 m/s

1800 m/s

119 m

732 m

174 m

906 m

-2.4%

-9.5%

Evjen, m=1 750 m

1000 m

q=.00048

q=.00048

767 m

1026 m

2.2%

2.6%

q=.000422

q=.000422

751 m

997 m

0.1%

-0.3%

Evjen, m=2 750 m

1000 m

q=.001

q=.001

757 m

1026 m

1 %

0.3%

q=.000941

q=.000941

750 m

990 m

0%

-1.0%

Evjen, m=3 750 m

1000 m

q=.001577

q-.001577

750 m

986 m

0%

-1.5%

q=.00156

q=.00156

751 m

988 m

0.2%

-1.2%

Evjen, m=4 750 m

1000 m

q=.00218

q=.00218

745 m

967 m

-0.7%

-3.4%

q=.00235

q=.00235

753 m

986 m

0.4%

-1.4%

Evjen, m=6 750 m

1000 m

q=.0035

q=.0035

735 m

955 m

-2%

-4.5%

q=.004413

q=.004413

755 m

984 m

0.7%

-1.6%

Anisotropy

VH=1800 m/s

750 m

1000 m

k=1.10

k=1.10

664 m

840 m

-11.4%

-16%

k=0.855

k=0.855

790 m

998 m

-5.3%

-0.2%

Anisotropy

VH=2100 m/s

750 m

1000 m

k=1.10

k=1.10

700 m

885 m

-6.7%

-11.5%

k=1.025

k=1.025

763 m

966 m

1.7%

-3.5%

Anisotropy

VH=2400 m/s

750 m

1000 m

k=1.10

k=1.10

780 m

986 m

4.1%

-1.4%

k=1.21

k=1.21

707 m

894 m

-5.8%

-10.6%

Table A.1: Comparison of depth estimates using model-based and GRM

approaches.

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244

A.10 - Discussion and Conclusions

This study demonstrates that virtually any inversion model can be fitted to

traveltime data to a high accuracy. With the noise-free data used in this study,

the accuracy is generally better than ±5 ms for traveltimes of up to 1300 ms or

0.4%. The errors are about four times larger for the single layer constant

wavespeed inversion model, but they are still acceptable.

However, the superior agreement with the traveltime data is not associated with

more accurate depth estimates. The average errors in depth computations are

about 5% to 10% and they are approximately three times larger than those

obtained through a GRM analysis. Figure A.14 summarizes the results.

In those cases where the inversion models are similar to the test model as with

the Evjen functions, the accuracy of the depth computations is quite high. The

accuracy is highest with the Evjen function with exponents of 2 and 3, which are

close to the test model, and it is least for exponents of 1 and 6, which represent

less similar models.

The lower accuracy of depths computed with the one and two layer

approximations, which are valid and widely applicable inversion models,

indicates the significance of the inversion model rather than the superiority of

variable wavespeed models, such as the linear function. As a corollary, it

indicates that the use of linear functions where constant wavespeed models are

applicable will also result in comparable significant errors in depth computations.

The cases of a reversal in the wavespeed and transverse isotropy illustrate a

fundamental shortcoming of model-based inversion. With a wavespeed reversal,

no traveltimes are recorded from the layer, and therefore it is simply not possible

to achieve agreement between the observed data and the computed traveltimes.

With seismic anisotropy, the traveltimes are obtained in the wrong direction. The

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245

field data are a measure of the horizontal wavespeed, while the vertical or near-

vertical wavespeed is required for depth conversion. These models demonstrate

that layers or wavespeeds not represented in the traveltime data can only be

modeled after an empirical assignment of parameters, or only if other data are

available, such as borehole control.

By contrast, the differences between the original traveltime data and the

computed traveltimes for models whose parameters are determined with the

GRM, are much larger. In several cases, these differences can be many tens of

milliseconds or up to 11%. However, the computed depths are generally more

accurate than those computed with the model-based approach. Furthermore, the

GRM is able to address the valid and important inversion models of wavespeed

reversal and transverse isotropy, unlike the model-based methods.

The consistent depth computations with the GRM are related to the constraint of

the wavespeed models with the optimum XY value. This parameter is a function

of both the thicknesses and wavespeeds of all layers above the refractor.

Therefore, the accuracy of the GRM approach is due to the inherent accuracy of

interpolation.

By contrast, the maximum depth of penetration of the direct arrivals used in the

model-based approach is less than 300 m (Palmer, 1986, equation 13.18). As a

result, wavespeeds throughout the first layer are determined by extrapolation

from the upper part of that layer. The various inversion models represent

different extrapolation functions and the errors are due to the inherent instability

of extrapolation. This instability is aggravated where there are undetected layers.

Furthermore, the application of linear wavespeed functions to constant

wavespeed layering can result in large gradients (Lanz et al., 1998), which in turn

can result in the ubiquitous ray path diagrams demonstrating almost complete

coverage of the subsurface. These diagrams are misleading when the inversion

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246

model does not accurately represent the subsurface, because the shortcomings

of extrapolation are not overcome.

In general, the direct traveltimes through a layer are unable to provide a reliable

wavespeed model of that layer, except in the trivial cases of isotropic constant

wavespeeds. This casts doubts on the benefits of very high precision of the

traveltimes. Although the data used in this study are noise-free and accurate to

0.1 ms, there is not a corresponding increased accuracy in the depth

computations with the model-based approach. Accordingly, it is questionable

whether it is necessary to measure traveltimes to an accuracy of 0.1 ms (Lanz et

al., 1998), especially when the acceptable residuals appear to be between 2 ms

and 5 ms for most near surface targets.

A more reliable measure of the accuracy of depth computations is the optimum

XY value, because it constrains the parameters for each inversion model. This in

turn, is a function of the detector interval. Therefore, trace spacing is at least as

important as the accuracy of the traveltime data in assessing of the accuracy of

refraction inversion.

While it is clear that the quality of the field data will have some effect on the

measurement of the optimum XY value, it is not as critical as with the model-

based methods. This single parameter is obtained through the recognition of a

distinctive pattern of minimum residuals between the computed and averaged

wavespeed analysis function, as shown in Figure A.4. This pattern is less

sensitive to errors in the individual traveltimes.

The major conclusion to be drawn from this study is that the choice of the

inversion model is an important, if not the most important factor in the

performance of model-based inversion. However, the wavespeed model for

each layer cannot be determined uniquely from the traveltime data from that

layer, because the data generally do not provide a complete, an accurate nor a

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247

representative sample of that layer. By contrast, the optimum XY value does

provide a complete sample, because it is determined from the traveltimes for the

underlying refractor, and it is able constrain an almost complete range of

inversion models. Therefore, the appropriate model for model-based inversion

has parameters determined with a GRM analysis, together with minimum

residuals between observed and modeled traveltimes.

The results of this study suggests a two stage inversion strategy. Firstly, the

GRM is employed where the refractor is sufficiently irregular, in order to

determine an appropriate model for inversion. Experience indicates that this

situation occurs in about 30% of the data. Secondly, model-based inversion

methods such as tomography are then employed to process the full set of data,

using the wavespeed model determined with the GRM analysis.

Alternatively, if depth estimates obtained with the GRM using the single layer

constant wavespeed model are similar to those obtained with model-based

inversion, then the inversion model can be taken as appropriate.

A.11 - References

Acheson, C. H., 1963, Time-depth and velocity-depth relations in Western

Canada: Geophysics, 28, 894-909.

Acheson, C. H., 1981, Time-depth and velocity-depth relations in sedimentary

basins - a study based on current investigations in the Arctic Islands and an

interpretation of experience elsewhere: Geophysics, 46, 707-716.

Berry, J. E., 1959, Acoustic velocity in porous media: Petroleum Trans. AIME,

216, 262-270.

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248

Birch, F., 1960, The velocity of compressional waves in rocks at 10 kilobars: J.

Geophys. Res., 65, 1083-1102.

Brandt, H., 1955, A study of the speed of sound in porous granular media: J.

Appl. Mech., 22, 479-486

Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edition:

McGraw-Hill Inc.

Faust, L. Y., 1951, Seismic velocity as a function of depth and geologic time:

Geophysics, 16, 192-206.

Faust, L. Y., 1953, A velocity function including lithologic variation: Geophysics,

18, 271-288.

Gassman, F., 1951, Elastic waves through a packing of spheres: Geophysics,

16, 673-685.

Gassman, F., 1953, Note on Elastic waves through a packing of spheres:

Geophysics, 16, 269.

Hall, J., 1970, The correlation of seismic velocities with formations in the

southwest of Scotland: Geophys. Prosp., 18, 134-156.

Hamilton, E. L., 1970, Sound velocity and related properties of marine sediments,

North Pacific: J. Geophys. Res., 75, 4423-4446.

Hamilton, E. L., 1971, Elastic properties of marine sediments: J. Geophys. Res.,

76, 579-604.

Page 52: Palmer.12.Digital Processing Of Shallow Seismic Refraction Data.pdf

249

Iida, K., 1939, Velocity of elastic waves in granular substances: Tokyo Univ.

Earthquake Res. Inst. Bull., 17, 783-897.

Jankowsky, W., 1970, Empirical investigation of some factors affecting elastic

velocities in carbonate rocks: Geophys. Prosp., 18 103-118.

Lanz, E., Maurer, H., and Green, A. G., 1998, Refraction tomography over a

buried waste disposal site: Geophysics, 63, 1414-1433.

Menke, W., 1989, Geophysical data analysis: discrete inverse theory: Academic

Press, Inc.

Miller, K. C., Harder, S. H., and Adams, D. C., and O'Donnell, T., 1998,

Integrating high-resolution refraction data into near-surface seismic reflection

data processing and interpretation: Geophysics, 63, 1339-1347.

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

interpretation: Society of Exploration Geophysicists.

Palmer, D., 1983, Comment on "Curved raypath interpretation of seismic

refraction data" by S.A. Greenhalgh and D.W. King: Geophys. Prosp., 31, 542-

543.

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

seismic velocity: Geophysical Press.

Palmer, D., 1991, The resolution of narrow low-velocity zones with the

generalized reciprocal method: Geophys. Prosp., 39, 1031-1060.

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

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

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250

Paterson, N. R., 1956, Seismic wave propagation in porous granular media:

Geophysics, 21, 691-714.

Stefani, J. P., 1995, Turning-ray tomography: Geophysics, 60, 1917-1929.

White, J. E., and Sengbush, R. L.,, 1953, Velocity measurements in near surface

formations: Geophysics, 18, 54-69.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1956, Elastic wave velocities in

heterogeneous and porous media: Geophysics, 21, 41-70.

Wyllie, W. R., Gregory, A. R.,and Gardner, L. W., 1958, An experimental

investigation affecting elastic wave velocities in porous media: Geophysics, 23,

459-593.

Zhang, J., and Toksoz, M. N., 1998, Nonlinear refraction traveltime tomography:

Geophysics, 63, 1726-1737.

Zhu, X., Sixta, D. P., and Andstman, B. G., 1992, Tomostatics: turning-ray

tomography + static corrections: The Leading Edge, 11, 15-23.

A.12 - Appendix: Definition of Variable Wavespeed Media withthe GRM

The GRM approach to the variable wavespeed model is described in detail in

(Palmer, 1986, p.175-181). Essentially the approach assumes that the Evjen

function in equation A-1a, is applicable, ie.

V(z) = V0 (1 + q z)1/m (A.1a)

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The wavespeed of the upper layer at the surface, V0, is obtained from the

traveltime graphs. The refractor wavespeed, Vn, the time-depth, tG, and the

optimum XY value are obtained from the application of the GRM wavespeed

analysis and time-depth algorithms.

These parameters are substituted into the left hand side of equation A-2a, which

is a standard integral for integer values of m, the exponent in equation A-1a.

tG Vn / XY = i0∫i1 (sinm-2 i - sin2 i) di / i0∫i1 sinm i (A.2a)

sin i0 = V0 / Vn (A.3a)

sin i1 = V1 / Vn (A.4a)

and V1 is the wavespeed in the variable wavespeed medium, immediately above

the refractor.

The sine of the critical angle, i1, is obtained from equation A-2a, once a value of

m has been selected, either from a cross-plot of V0 / Vn and tG Vn / XY, or by

iteration.

The critical angle is then used to evaluate the relationship between the time-

depth tG and the angles i0 and i1, in equation A-5, to obtain the parameter q (see

equation A-1a), viz.

tG Vn = (m / q V0 sinm-1 i0) i0∫i1 (sinm-2 i - sin2 i) di (A.5a)

Finally the depth, zG, is obtained from

sin i1 = V0(1 + q zG)1/m / Vn (A.6a)

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Appendix 3

Surefcon.c

/* Copyright (c) Colorado School of Mines, 1999.*/

/* All rights reserved. */

/*SUCONV: $Revision: 1.12 $ ; $Date: 1996/09/05 19:24:26 $ */

"

"

#include "su.h"

"

#include "segy.h"

"

#include "header.h"

"

"

/*********************** self documentation **************************/

char *sdoc[] = {

" ",

" SUREFCON - Convolution of user-supplied Forward and Reverse ",

" refraction shots using XY trace offset in reverse shot ",

"

", surefcon <forshot sufile=revshot xy={trace offseted} >stdout ",

" ",

" Required parameters: ",

" sufile= file containing SU trace to use as reverse shot ",

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253

" xy = Number of traces offseted from the 1st trace in sufile ",

" ",

" Optional parameters: ",

" none ",

" ",

" Trace header fields accessed: ns ",

" Trace header fields modified: ns ",

" ",

" Notes: It is quietly assumed that the time sampling interval on the",

" output traces is the same as that on the traces in the input files.",

" ",

" Examples: ",

" suconv<DATA sufile=DATA xy=1 | ...

","

",

" Here, the su data file, \"DATA\", convolved the nth trace by ",

"

" (n+xy)th trace in the same file ",

" ",

" ",

NULL};

/* Credits:

* CWP: Jack K. Cohen, Michel Dietrich

* UNSW: D. Palmer, K.T. LEE

*

* CAVEATS: no space-variable or time-variable capacity.

* The more than one trace allowed in sufile is the

* beginning of a hook to handle the spatially variant case.

*

* Trace header fields accessed: ns

* Trace header fields modified: ns

*/

/**************** end self doc ********************************/

segy intrace, outtrace, sutrace;

int

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254

main(int argc, char **argv)

{

int nt; /* number of points on input traces */

int ntout; /* number of points on output traces */

int xy; /* the offset number for GRM */

float *forshot; /* forward shot */

int nforshot; /* length of input wavelet in samples */

cwp_String sufile; /* name of file of forward SU traces */

FILE *fp; /* ... its file pointer */

int itr; /* trace counter */

/* Initialize */

initargs(argc, argv);

requestdoc(1);

/* Get info from first trace */

if (!gettr(&intrace) ) err("can't get 1st reverse shot trace");

nt = intrace.ns;

/* Default parameters; User-defined overrides */

if (!getparint("xy", &xy) ) xy = 0;

/* Check xy values */

if (xy < 0) err("xy=%d should be positive", xy);

if (!getparstring("sufile", &sufile)) {

err("must specify sufile= desired forward shot");

} else {

/* Get parameters and set up forshot array */

fp = efopen(sufile, "r");

for (itr = 0; itr <= xy; ++itr) {

if (!fgettr(fp, &sutrace) ) {

err("can't get 1st requested forward trace");

};

};

nforshot = sutrace.ns;

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255

forshot = ealloc1float(nforshot);

/* Set output trace length */

ntout = nt + nforshot - 1;

/* Main loop over reverse shot traces */

do {

fprintf(stderr,"rev==%d\t , for=%d\t", intrace.tracf,

sutrace.tracf);

memcpy((void *) forshot,

(const void *) sutrace.data, nforshot*FSIZE);

/* Convolve forshot with revshot trace */

conv(nforshot, 0, forshot,

nt, 0, intrace.data,

ntout, 0, outtrace.data);

/* Output convolveed trace */

memcpy((void *) &outtrace, (const void *) &intrace,

HDRBYTES);

outtrace.ns = ntout;

outtrace.dt = outtrace.dt/2;

/*outtrace.cdp = 2*intrace.tracf + xy;*/

fprintf(stderr,"out_cdp=%d\n", 2*intrace.tracf + xy);

puttr(&outtrace);

} while ( gettr(&intrace) && fgettr(fp, &sutrace) );

} ;

return EXIT_SUCCESS;

}

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Appendix 4

The Effects of Spatial Sampling onRefraction Statics

Palmer, D., Goleby, B., and Drummond, B., 2000a, The effects of spatial

sampling on refraction statics: Exploration Geophysics, 31, 270-274.