palmer.05.digital processing of shallow seismic refraction data.pdf

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    Chapter 4

    Starting Models For RefractionInversion

    4.1 - Summary

    The algorithms of the generalized reciprocal method (GRM) are applied to a set

    of reversed traveltime data for a two layer model with a synclinal refractor

    interface, in order to generate a family of starting models. Each starting model

    shows much the same depths as the original model, but each has a narrow zone

    in the refractor with an anomalous wavespeed. The traveltimes through each of

    the starting models differ from those for the original model by less than a

    millisecond. If any were used as starting models for tomographic or model-based

    inversion, then the final result would show only minor differences. This example

    demonstrates the non-uniqueness of model-based inversion.

    In order to address the issues of non-uniqueness with model-based inversion, it

    is recommended that a range of starting models, such as those which can be

    generated with the GRM, be used.

    Alternatively, other approaches, which aim to resolve these ambiguities, can be

    employed. In many cases, the minimum variance criterion of the GRM can

    resolve whether lateral variations in the refractor wavespeeds are genuine, or

    whether they are artifacts of the inversion algorithm. In addition, it is proposed

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    that the amplitudes of the refraction convolution section can indicate where there

    are genuine changes in the wavespeed of the refractor, because the amplitudes

    are a function of the contrasts in wavespeeds between the refractor and the layer

    above.

    4.2 - Introduction

    The inversion of seismic refraction data with model-based or tomographic

    methods consists of deriving a starting model of the subsurface with standard

    algorithms, and then testing it by comparing the computed traveltimes of the

    model with the observed data. If there are differences, then the model is

    adjusted until an acceptable agreement is achieved. Commonly, several

    iterations may be required.

    While most geophysicists are satisfied to generate a model which reproduces the

    observations, there is the fundamental theoretical reality that an infinite number

    of solutions can reproduce the data (Oldenburg, 1984; Treitel and Lines, 1988),

    although not all of these solutions will be geologically plausible. This non-

    uniqueness becomes more significant where the data are inaccurate and

    incomplete as is often the case with field data, and where models, which do not

    fit the data precisely, are accepted.

    The issues of non-uniqueness are not usually considered with shallow refraction

    tomography. The non-uniqueness includes the wavespeeds in both the

    overburden and the refractor and they are often inter-related. One compellingexample is the somewhat paradoxical situation of the poor determination of

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

    from that layer (Lanz et al, 1998, Figure 8). This situation is at variance with the

    experiences of most seismologists using more traditional methods of refraction

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    processing, and it is probably related to the use of a linear wavespeed function

    with a very high gradient in the upper layer.

    Previous studies (Hagedoorn, 1955) have demonstrated the ambiguities in

    determining the wavespeed stratification within a single layer above the refractor.

    Even in the absence of undetected layers, generally known as hidden layers

    within the blind zone and reversals in wavespeed, it is not possible to accurately

    specify the mathematical function which describes the wavespeed in the

    overlying layer. As a result, there is a large range in the depths to the refractor

    computed with the various mathematical functions which can be fitted to the first

    arrival traveltime data with acceptable accuracy.

    Palmer (1992, Appendix 2) has demonstrated, that when the refractor interface is

    sufficiently irregular in relation to its depth, the generalized reciprocal method

    (GRM) (Palmer, 1980; 1986), can significantly improve the accuracy of the depth

    computations for a wide range of mathematical functions in the upper layer. The

    mathematical functions include wavespeed reversals and transverse isotropy,

    which are not adequately addressed with other approaches.

    This study examines the non-uniqueness in the determination of the wavespeeds

    in the refractor. In these cases, the non-uniqueness is usually related to the

    starting model for the inversion process and in turn, to the selection of the

    inversion algorithm used to generate that model.

    I demonstrate that a range of geologically plausible starting models can be

    readily generated from the one set of reversed traveltime data with the algorithms

    of the GRM, and that each of these models fits the data to an acceptable

    accuracy of a few milliseconds. I conclude that the selection of the initial starting

    model is critical with model-based methods of refraction inversion. I further

    conclude that the issues of non-uniqueness, which currently are not adequately

    examined with most model-based methods for inverting refracting data, can be

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    addressed by testing a family of starting models which can be generated for

    example, with the GRM. Finally, I propose the use of two methods for resolving

    ambiguities, namely, the minimum variance criterion of the GRM and the use of

    amplitudes in the refraction convolution section.

    4.3 - Inversion Of A Two Layer Model With The GRM Algorithms

    Figure 4.1 shows a simple two layer model with isotropic homogeneous seismic

    wavespeeds separated by a synclinal interface. It represents an obvious step for

    increasing the complexity of the interpretation model over the simple two layer

    case with plane interfaces. The dips of the sloping interfaces are 9.2, which

    are relatively large. This model was used to generate the traveltime data shown

    in Figure 4.2, which in turn were processed or inverted using the two algorithms

    of the GRM for computing time-depths and refractor wavespeeds.

    Figure 4.1: Two layer model with a synclinal interface.

    The time-depth tG, at G is given by equation 4.1, viz.

    tG= (tAY+ tBX- tAB- XY/Vn)/2 (4.1)

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    where A, X, G, Y, and B are collinear, A and B are source points, X and Y are

    detectors and G is midway between X and Y, tAYis the traveltime from A to Y, tBX

    is the traveltime from B to X, tABis the reciprocal time, the traveltime from the

    source at A to the source at B, and Vnis the wavespeed in the refractor.

    Figure 4.2: Traveltimes generated for two layer model with a synclinal interface

    shown in Figure 4.1. The station spacing is 5 m.

    Figure 4.3 shows the time-depths computed for XY values from zero to 30 m in

    increments of 5 m, which is the detector spacing. Each set of time-depths shows

    the synclinal structure of the refractor, although there are minor differences in

    detail around the hinge point at station 12.

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    Figure 4.3: Time-depths computed for the synclinal model in Figure 4.1 for a

    range of XY values. The reciprocal times have been systematically decreased

    with increasing XY value, in order to separate each set of graphs for clarity.

    The second function computed with the GRM is the refractor wavespeed analysis

    function tV, given by equation 4.2, viz.

    tV= (t

    AY- t

    BX+ t

    AB)/ 2 (4.2)

    Two parameters can be derived from this function. The first is the wavespeed in

    the refractor Vn, from the reciprocal of the gradient, ie

    d/dx tV= 1 / Vn (4.3)

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    Figure 4.4: Wavespeed analysis function computed for the synclinal model in

    Figure 4.1 for a range of XY values.

    Figure 4.4 shows the wavespeed analysis function for the same range of XY

    values used in Figure 4.3. Each set of graphs for a given XY value shows the

    same wavespeed in the refractor of 2820 m/s, except for a short interval around

    the hinge point at station 12. Here the wavespeed ranges from as low as 2000

    m/s to as high as 4800 m/s.

    The second parameter determined from the wavespeed analysis function is the

    intercept of tVat the source point, which is the time-depth tAat a distance of XY

    from the source point, ie

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    tA= tV|x=0 (4.4)

    For this two layer model, the time-depths presented in Figure 4.3 can be

    converted into depths zG, with equation 4.5, viz.

    zG= tG/ DCF (4.5)

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

    depth, is given by:

    DCF = V Vn/ (Vn2- V2)

    (4.6)

    or

    DCF = V / cos i (4.7)

    V is the average wavespeed above the refractor and

    sin i = V / Vn

    (4.8)

    Figure 4.5: A summary of the starting models which can be generated from the

    traveltime data for the synclinal model in Figure 4.1. The region with the variable

    wavespeeds near the hinge point of the interface is an artifact.

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    Figure 4.5 is a summary of the range of depth models which can be generated

    with the XY values from zero to 30 m. Although the depth sections reproduce the

    synclinal structure of the original model, there is an additional segment in the

    second layer with wavespeeds from 2000 m/s to 4800 m/s which is not present in

    the original model. This additional segment could represent a weathered dyke or

    a shear zone for the low wavespeed cases or an unweathered dyke or a silicified

    shear zone for the high wavespeed cases. Therefore, all models are geologically

    both plausible and significant. Nevertheless, they are artifacts generated by

    equation 4.2, the refractor wavespeed analysis algorithm.

    4.4 - Time Differences Between Starting Models

    Figure 4.6 shows the time-depths for the range of XY values from zero to 30 m

    plotted without the vertical separation obtained by changing the reciprocal time

    tAB, in equation 4.1. This presentation, which emphasizes the subtle variations

    between different XY values, shows that the time-depth values are identical for

    the planar sloping surfaces, but diverge by less than 2 ms in the vicinity of the

    hinge point.

    The smaller values are associated with the XY values which are less than the

    optimum of 15 m and in turn are associated with the zone of lower wavespeeds

    in the refractor. Although there is a slightly higher DCF computed with equation

    4.6 in this narrow region, there is still a reduction in depth at the hinge point, and

    in turn a small decrease in traveltimes in the upper layer. This decreaseapproximately compensates for the slight increase in traveltimes in the region of

    lower wavespeeds in the refractor.

    The larger time-depth values are associated with the XY values which are

    greater than the optimum of 15 m and in turn are associated with the zone of

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    higher wavespeeds in the refractor. Although there is a slightly lower DCF

    computed with equation 4.6 in this narrow region, there is still an increase in

    depth at the hinge point, and in turn a small increase in traveltimes in the upper

    layer. This increase approximately compensates for the slight decrease in

    traveltimes in the region of higher wavespeeds in the refractor.

    Figure 4.6:Time-depths computed for the synclinal model in Figure 4.1 for a

    range of XY values. The reciprocal times are identical for each XY value, and itresults in an emphasis of the subtle variations between different XY values.

    Therefore, while there are small differences in depths at the hinge point, they are

    matched with compensating changes in the wavespeeds in the refractor. The

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    final result is that there are few significant differences in the traveltimes in final

    depth models.

    Figure 4.7: Refractor wavespeed analysis function computed for the synclinal

    model in Figure 4.1 for a range of XY values. The reciprocal times are identical

    for each XY value.

    Figure 4.7 is a similar presentation in which the wavespeed analysis function in

    equation 4.2 is presented with identical reciprocal times for all XY values. Again

    the aim is to emphasize the subtle variations between each set of values. It can

    be seen that the maximum difference between the values computed for XY

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    values of zero and 30 m is 2.2 ms and that maximum difference between the any

    set and those computed with optimum XY value of 15 m is less than 1.1 ms.

    Figure 4.7 also demonstrates a fundamental problem in determining wavespeeds

    in narrow intervals of the refractor with seismic refraction methods. The variation

    in wavespeed in the refractor of 2000 m/s to 4800 m/s is very large and

    geologically significant. However, these variations in wavespeed do not result in

    commensurately large changes in traveltimes. An inspection of Figure 4.4 shows

    that a single wavespeed can be fitted to each set of points with an accuracy of

    better than a millisecond.

    The significance of Figures 4.6 and 4.7 is that the time differences between each

    model of the refractor computed with the selected range of XY values are subtle

    and are generally within 1 ms of that computed with the optimum XY value of 15

    m. These differences are typical of the acceptable residuals for most model-

    based or tomographic methods of inversion.

    4.5 - Agreement Between Starting Models And Traveltime Data

    The small time differences between the various models as shown in Figures 4.6

    and 4.7 suggest that each model should closely honor the original traveltime

    data. Such a result is in fact the norm with the GRM, because the algorithms

    seek to separate or analyze the traveltimes into the source point and detector

    time-depths, together with the traveltime in the refractor, while still preserving the

    original traveltime data. This is demonstrated by the simple addition of equations4.1 and 4.2, viz.

    tAY= tG+ tV+ XY/Vn (4.9)

    From equations 4.3 and 4.4, it is readily shown that

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    tV= tA+ AG/Vn (4.10)

    Equations 4.9 and 4.10 can be combined to obtain

    tAY= tG+ tA+ AY / Vn (4.11)

    4.6 - Discussion

    This study illustrates some of the inherent problems of non-uniqueness with

    determining wavespeeds in the refractor. Using a simple model and the GRM

    algorithms, it is possible to generate a family of starting models, each of which

    has much the same depths to the refractor as the original model but each of

    which includes a narrow zone in the refractor with an anomalous wavespeed.

    Even with the noise-free model data used in this study, the time differences are

    generally less than one millisecond, which is the error commonly assigned to the

    measurement of traveltimes from field data, and which is within the range of

    acceptable residuals for tomography. Therefore, if any were used as starting

    models for tomography, then there would be minimal differences with the final

    result of the inversion process. Furthermore, all of these models are geologically

    meaningful and hence cannot be readily discarded.

    (As an aside, geologically meaningless models can also be generated with the

    GRM, simply by increasing the XY value in the wavespeed analysis function in

    equation 4.2, until negative wavespeeds are obtained with equation 4.3.)

    Frequently, the algorithms of the standard reciprocal method (SRM) (Hawkins,

    1961), which is a special case of the GRM with a zero XY value, are used to

    generate starting models. These algorithms are probably the most commonly

    used throughout the world for shallow seismic refraction investigations, because

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    of their simplicity and robustness. These algorithms, which are 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), can be viewed as simple extensions of the

    slope/intercept method (Ewing et al, 1939), whereby computations are extended

    from the source points to each detector location (Palmer, 1986).

    However, this study demonstrates that the starting model generated by any

    single method, such as an SRM analysis of the traveltime data need not

    necessarily converge to the correct model. Therefore, in order to address the

    issues of non-uniqueness, it is recommended that model-based methods of

    inversion test a family of starting models such as those which can be readily

    derived with the GRM.

    The results over the Elura orebody (Hawkins and Whiteley, 1980) demonstrate

    the significance of artifacts. The claim, that the massive sulphide orebody was

    characterized by a low wavespeed, attracted considerable debate (Emerson,

    1980), and was at variance with laboratory tests on hand specimens (MacMahon,

    1980). An alternative analysis with the GRM indicated that the low wavespeed

    was probably an artifact which coincided with an increase in the depth of the

    regolith over the orebody (Palmer, 1980b). Many of the qualitative aspects of the

    model study above can be recognized in the Elura case history.

    The inability of model-based inversion methods to recognize artifacts can also

    have important legal implications. There are instances where the combination of

    the SRM and ray tracing is a contractual requirement of major geotechnical

    investigations, in order to obviate claims for compensation by construction

    companies for unexpected variations in site conditions.

    Numerous model studies and case histories (Palmer, 1980; Palmer, 1986,

    Palmer, 1991) demonstrate that the minimum variance criterion of the GRM is

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    frequently able to resolve whether lateral variations in wavespeeds in the

    refractor are genuine or are artifacts. For the model study above, the wavespeed

    analysis function in Figure 4.4, shows that the optimum XY value of 15 m, that

    the measured wavespeed is same as the model and that no artifacts are

    generated.

    However, the effective application of the GRM is not always possible, often

    because the detector interval is too large. In these cases, alternative methods

    are required. Other studies demonstrate that the amplitudes of the refraction

    convolution section (Palmer, 2001a; Chapter 5) can indicate where there are

    genuine changes in the wavespeed of the refractor. These amplitudes are a

    function of the contrasts in wavespeeds between the refractor and the layer

    above, and therefore provide another approach which is independent of the

    traveltime data.

    4.7 - Conclusions

    The inversion of seismic refraction data with model-based methods or

    tomography consists of deriving a starting model of the subsurface with standard

    algorithms, and then testing it by comparing the computed traveltimes of the

    model with the observed data. If there are differences, then the model is

    adjusted until an acceptable agreement is achieved. Commonly, several

    iterations may be required.

    However, a simple model study illustrates the inherent problems of non-uniqueness with this approach. The GRM is able to generate a family of starting

    models, all of which are geologically meaningful and all of which are compatible

    with the original traveltime data. If any were used as starting models for

    tomography, then there would be minimal differences with the final result of

    inversion.

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    In view of the significance of the starting model, it is recommended that model-

    based methods of inversion test a range of starting models such as those which

    can be readily generated with the GRM. In general, the models which can be

    derived with the GRM tend to be compatible with the original traveltime data.

    In many cases, the minimum variance criterion of the generalized reciprocal

    method (GRM) can resolve whether lateral variations in the refractor wavespeeds

    are genuine, or whether they are artifacts of the inversion algorithm.

    In those cases where the effective application of the GRM is not possible, then

    alternative methods are required. It is proposed that the amplitudes of the

    refraction convolution section (Palmer, 2001a; Chapter 5) frequently can indicate

    where there are genuine changes in the wavespeed of the refractor.

    4.8 - References

    Dobrin, M. B., 1976, Introduction to geophysical prospecting, 3rd edn.: McGraw-

    Hill Inc.

    Emerson, D. W., 1980, The geophysics of the Elura orebody, Cobar, NSW: Bull.

    Aust. Soc. Explor. Geophys., 11, 347.

    Ewing, M., Woollard, G. P., and Vine, A. C., 1939, Geophysical investigations in

    the emerged and submerged Atlantic Coastal Plain, Part 3, Barnegat Bay, NewJersey section: Bull. GSA, 50, 257-296.

    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.

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    Hagedoorn, J. G., 1955, Templates for fitting smooth velocity functions to seismic

    refraction and reflection data: Geophys. Prosp., 3, 325-338.

    Hagedoorn, J. G., 1959, The plus-minus method of interpreting seismic refraction

    sections: Geophys. Prosp., 7, 158-182.

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

    investigations: Geophysics, 26, 806-819.

    Hawkins, L. V., and Whiteley, R. J., 1980, The seismic signature of the Elura

    orebody: Bull. Aust. Soc. Explor. Geophys., 11, 325-329.

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

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

    MacMahon, B. K., 1980, Discussion inEmerson, D. W., ed., The geophysics of

    the Elura orebody, Cobar, NSW: Bull. Aust. Soc. Explor. Geophys., 11, 346.

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

    Company Inc.

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

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

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

    interpretation: Society of Exploration Geophysicists.

    Palmer, D., 1980b, Comments on "The seismic signature of the Elura orebody":

    Bull. Aust. Soc. Explor. Geophys., 11, 347.

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    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.

    Palmer, D., 2001a, Resolving refractor ambiguities with amplitudes: Geophysics

    66, 1590-1593.

    Palmer, D., 2001b, Model determination for refraction inversion: Geophysics,

    submitted.

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

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