UNCERTAINTY IN REFRACTION INVERSION – A GRM STRATEGY · Uncertainty in Refraction Inversion – A GRM Strategy Derecke Palmer Copyright © Derecke Palmer 2011 – All rights reserved

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Uncertainty in Refraction Inversion – A GRM Strategy Derecke Palmer

Copyright © Derecke Palmer 2011 – All rights reserved 1

UNCERTAINTY IN REFRACTION INVERSION – A GRM STRATEGY

Derecke Palmer

The University of New South Wales Sydney 2052, Australia.

Email: [email protected]

Abstract Uncertainty in the tomographic inversion of near-surface seismic refraction data can be

separated into aleatory variability, which describes the misfit errors and epistemic uncertainty, which describes the suite of acceptable models. Common implementations of refraction tomography usually focus on reducing aleatory variability and frequently disregard epistemic uncertainty.

In this study, the tomograms generated with three models of the seismic velocities in both the weathering and in the sub-weathering, using the generalized reciprocal method (GRM), are consistent with the traveltime data. However, only one tomogram is consistent with the optimum XY value and the attributes derived from the head wave amplitudes and seismic velocities. This study demonstrates that epistemic uncertainty can be explicitly addressed with the GRM, because the most probable tomogram can be selected objectively from a number of acceptable alternatives.

The GRM based tomogram successfully detects, defines and differentiates narrow regions with low seismic velocities which represent shear zones and a massive sulphide ore body. None of these zones is detected with the tomogram generated with the default starting model using smooth vertical velocity gradients.

It is concluded that minimizing epistemic uncertainty through the use of the most appropriate starting model is more important than minimizing aleatory variability.

Introduction The Ubiquity of Non-uniqueness

Refraction tomography (Lanz et al., 1998; Stefani, 1995; Zhang and Toksoz, 1998; Zhu et al., 1992), is widely employed to invert seismic refraction data recorded for geotechnical and environmental investigations. Refraction tomography is one example of model-based inversion, in which an initial starting model is systematically updated through iteratively comparing the modeled response with the field data.

However, model-based inversion, whether it be traveltime or full waveform inversion, is inherently non-unique: an infinite number of solutions can satisfy a given set of data (Ivanov et al, 2005a, 2005b; Oldenburg and Li, 2005; Treitel and Lines, 1988). Furthermore, the resolution of the final tomogram is usually directly related to the resolution of the starting model. Therefore, if a low resolution starting model is used, such as the smooth vertical velocity gradient which is the default with many common implementations of refraction tomography, then the most likely result is a low resolution tomogram. Even regions in excess of 50 m in width with low seismic velocities characteristic of major shear zones are neither delineated nor even detected with low resolution starting models (Palmer 2007, 2008a, 2008b, 2009a, 2010b, 2010c, 2010d, 2010e).



Aleatory Variability and Epistemic Uncertainty Non-uniqueness in the inversion of geophysical data can be equated with uncertainty,

which in most engineering treatments of risk, is generally separated into aleatory variability and epistemic uncertainty (Helton et al, 2004a, 2004b, Oberkampf, 2005). Aleatory variability is the natural randomness in a process, and it can be equated with misfit errors in refraction tomography. Epistemic uncertainty is the scientific uncertainty in the model of the process, due to limited data and knowledge, and it is characterized by alternate models.

The concepts of aleatory variability and epistemic uncertainty correspond with precision and accuracy. Precision refers to how closely individual measurements agree with each other. Accuracy refers to how closely a measured value agrees with the correct value. Figure 1 presents a common dart board analogy to illustrate the difference between aleatory variability and epistemic uncertainty and its relevance to the tomographic inversion of near-surface seismic refraction data.

Figure 1: A common dart board analogy which illustrates the differences between precision or aleatory variability and accuracy or epistemic uncertainty. A major conclusion of this study is that minimizing epistemic uncertainty through the use

of the most appropriate starting models is more important than minimizing misfit errors or aleatory variability. In fact, this study demonstrates that geologically more useful refraction tomograms can often exhibit marginally larger misfit errors than the default alternatives. It is concluded that an emphasis on precision and aleatory variability with common implementations of refraction tomography can often result in an increase in epistemic uncertainty and a decrease in geological verisimilitude.

The GRM as Backus-Gilbert Appraisal

Much of the current theory on the inversion of geophysical data had its genesis with the three landmark publications of Backus and Gilbert (1967, 1968, 1970). They showed that linear inverse problems can have either a single unique solution or infinitely many solutions. In the latter case, Backus and Gilbert (1967, 1968) showed that it is possible to generate linear combinations of the data, which represent unique averages of the model. This process is known as appraisal, whereas model-based inversion, such as refraction tomography, is known as construction.

The inversion algorithms of the generalized reciprocal method (GRM) (Palmer, 1980, 1981, 1986, 1992), constitute a version of Backus-Gilbert appraisal, because they consist of simple linear combinations of the traveltime data. Few, if any, other geophysical methods share



this fortuitous coincidence. Accordingly, the use of the GRM can overcome a common source of non-uniqueness or epistemic uncertainty, namely the selection of the inversion algorithm used to compute the starting model (Oldenburg 1984).

Strategies for Managing Epistemic Uncertainty

Despite the reality of non-uniqueness, most strategies for the inversion of near-surface seismic refraction data fail to separate aleatory variability from epistemic uncertainty. In fact, a common but incorrect strategy has been to assume that minimizing misfit errors and therefore aleatory variability demonstrates that epistemic uncertainty has also been minimized. However, that strategy does not explicitly address epistemic uncertainty through testing a suite of acceptable refraction tomograms generated from a set of starting models which represent viable alternate models of the near surface.

In this study, epistemic uncertainty is explicitly addressed by generating starting models which consist of three models of the seismic velocities in the weathering and three models of the seismic velocities in the sub-weathering. The models in the weathering include (i) uniform seismic velocities without any increases in depth obtained with an uncritical acceptance of the traveltime data, (ii) vertical velocity gradient equivalents generated with the hyperbolic velocity function, and (iii) a probable velocity reversal. The models in the sub-weathering are generated with optimum and non-optimum implementations of the GRM refractor velocity analysis function. All models produce tomograms with misfit errors which are comparable to, but marginally larger than the default starting model of smooth vertical velocity gradients.

However, only one model is comprised of seismic velocities which are consistent with the optimum XY value, and with the a priori attributes derived from the head wave amplitudes. This study demonstrates that epistemic uncertainty can be explicitly addressed with the GRM, because the most probable tomogram can be selected from a number of acceptable alternatives. By contrast, common implementations of automatic refraction tomography neither facilitate the routine generation of a range of acceptable models, nor do they employ objective criteria for validating the single model so obtained, ray coverage diagrams notwithstanding.

The data used in this study were recorded over a narrow massive sulphide ore body and an adjacent narrow shear zone (Palmer, 2006, 2010d). Although they do not constitute a common geotechnical target, the ore body and shear zone are both narrow vertical features with low seismic velocities. Therefore, they provide a crucial test of the spatial resolution of refraction tomography, and in turn, the importance of epistemic uncertainty. This study demonstrates that the GRM is able to provide good estimates of their lateral extent, and to differentiate the ore body from the adjacent shear zone through the computation of various useful geotechnical attributes. By contrast, the ore body and shear zone remain undetected, undefined and undifferentiated with the smooth velocity gradient tomogram.

Traveltime Inversion

Summary of Geology Mt Bulga is located near Orange in southeastern Australia. It is the site of a small narrow

(5 – 10 m) massive sulphide ore body which consists of syngenetic fine grained banded massive pyrite-galena-sphalerite-chalcopyrite in steeply dipping unfolded altered Silurian meta-sediments, and which is associated with several major shear zones. The ore body underwent extensive investigations several decades ago, but was subsequently abandoned because it was considered to be uneconomic.

Figure 2 shows a cross section taken from Whiteley (1986) (see also Whiteley and Leung, 2006, Figure 2). It shows the massive sulphide ore body and the adjacent shear zone.



Figure 2: Geological cross-section showing the massive sulphide ore body and the adjacent shear zone.

The set of traveltime data in Figure 3, was recorded across a small ridge, which also

marks the approximate location of the massive sulphide ore body. A more extensive profile was recorded earlier in approximately in the same location by Whiteley et. al. (1984).

Figure 3: The traveltime graphs recorded across a narrow massive sulphide ore body at Mt Bulga in southeastern Australia. The surface conditions along the profile were quite variable, and accordingly, they

represent challenges for the application of most approaches to the inversion of near-surface seismic refraction data. Outcrop occurs between stations 31 and 49, where there were difficulties in planting the receivers and with auguring shot holes. Gossan and recently restored historical mine workings occur in the vicinity of stations 49 to 53. Soil occurs between stations 53 and 70, where planting of receivers and the auguring of shot holes was much easier. At each end of the profile, there were areas of considerable disruption to the surface layers, caused by ongoing forestry activities. In these areas, the soil contained a large biogenic component. The variations in the surface conditions are reflected in the seismic velocities which exhibit significant lateral variations in both the weathering and the sub-weathering.

Sections of the ore body were subjected to mining of the enriched supergene layer in the historical past. These historical workings were restored in the recent past. The seismic profile was offset by ~10 m from the areas disturbed by mining and restoration.



Smooth Vertical Velocity Gradient Tomography Figure 4 shows the smooth vertical velocity gradient starting model which is the default

with many implementations of refraction tomography. In general, smooth velocity gradient starting models emphasize the vertical resolution of many layers, whereas the inversion algorithms of the GRM emphasize the lateral resolution of individual layers.

Figure 4: The smooth vertical velocity gradient starting model. It is the default in many implementations of refraction tomography. Figure 5 presents the wavepath eikonal traveltime (WET) (Schuster and Quintus-Bosz,

1993) tomograms for five, ten and twenty iterations. A comprehensive summary of the misfit errors is shown in Table 1.

Figure 5: Three tomograms generated from the smooth vertical velocity gradient starting model after five, ten and twenty iterations.



The seismic velocities of approximately 1800 m/s in the vicinity of station 37 correspond with weathered outcrop, the values of approximately 750 m/s in the vicinity of station 49 correspond with gossan, while the values of approximately 1000 m/s in the vicinity of stations 53 to 70 correspond with soil and completely weathered rock. The tomograms systematically grade into largely featureless sets of quasi parallel contours to a depth of approximately 50 m to a maximum value of approximately 4000 m/s below station 61. It is not clear where the base of the weathering occurs, but the region of the closer spacing of the 1500 m/s and 2000 m/s contours would be a common interpretation. Alternatively, the region of the 3000 m/s contour might also be interpreted as the approximate base of the weathering. However, there are no indications of any distinctive features which might correspond with the steeply dipping massive sulphide ore body or any associated shear zones.

As the number of iterations increases, the contours in the vicinity of station 31 exhibit greater curvature, whereas those for 3000 m/s and higher seismic velocities exhibit greater smoothing. The smooth velocity gradient WET tomogram generated after twenty iterations will be used in this study, as recommended by the vendor of the tomography software, even though that generated after ten iterations exhibits minimal significant differences.

Seismic Velocities in the Sub-weathering with the GRM

Figure 6 presents the GRM refractor velocity analysis function (Palmer, 2010b, equation A1) computed with the traveltimes for SP1 and SP97, using XY values from -2.5 m to 10 m in increments of 2.5 m, which is the station spacing (-2.5 m ≤ XY(2.5) ≤ 10 m), as well as the average. (The traveltime data have been partially corrected for the equivalent of a short wavelength static correction (Palmer, 2009b), which originates in the surface soil layers, using the method described in Palmer (2006)). Although there are intervals where the graphs for different XY values are the same, such as between stations 55 and 66, there are regions where this is obviously not the case, such as between stations 45 and 55. Accordingly, the computation of seismic velocities with the reciprocal of the spatial derivative will produce values which in general, will depend upon the selection of the XY value.

Figure 6: The GRM refractor velocity analysis function generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97.

Detailed seismic velocities in the sub-weathering for a range of XY values are derived at

each station, with a novel application of the Hilbert transform described Palmer (2010b, equation A8). The algorithm computes an average of the reciprocal of the seismic velocities derived over



a range of distances, in this study 2.5 m, 5 m, 7.5 m, and 10 m, which are multiples of the station spacing of 2.5 m, and which are centred on the reference station. However, the refractor velocity analysis function is first averaged over a range of XY values, in this case ten or eleven XY values, in order to further minimize the effects of the short-wavelength statics in the surface soil layers. The effective XY value is the average of all of the XY values in the window, or equivalently, the average of the two end values.

Figure 7: The GRM seismic velocities for a range of effective XY values 0 ≤ XY(1.25 m) ≤ 7.5 m. The optimum XY value is 3.75 ± 1.25 m. Figure 7 presents the seismic velocities computed with a range of XY values from zero to

7.5 m in increments of 1.25 m, which is half the station spacing. Figure 7 demonstrates that it is possible to derive effective XY values in half station increments by employing an appropriate selection of the range of XY values in the average. Accordingly, the error in determining the optimum XY value can be taken as half the station spacing. The XY value for which the seismic velocity model is the simplest with the least extreme values is taken as the optimum, which is 3.75 m ± 1.25 m in Figure 7.

Where the determination of an optimum XY value is unambiguous, other studies (Palmer, 2010c, 2010d) have recommended that epistemic uncertainty be explicitly addressed by computing three starting models in which the seismic velocities in the sub-weathering have been computed with the optimum XY value and plus/minus half the station spacing, which is 3.75 m ± 1.25 m in this study. However, this study employs effective XY values of zero, 3.75 m and 7.5 m, that is, the optimum XY value of 3.75 m ± 3.75 m, in order to emphasize the significance of epistemic uncertainty. Furthermore, these somewhat extreme examples would be appropriate where the practitioner is unable to determine an optimum XY value with confidence.

Time Models for the Sub-weathering with the GRM

The GRM time model algorithm (Palmer, 2010b, equation A3) provides a measure of the depth to the refracting interface in units of time. The time model for -2.5 m ≤ XY(2.5 m) ≤ 10 m, as well as the average for 0 ≤ XY(2.5 m) ≤ 7.5 m, are presented in Figure 8. In general, there is not a great deal of variation with the XY value, apart from around station 49. Although it is possible to determine an optimum XY value(s) from the time model, experience with many case studies suggests that the use of the GRM refractor velocity analysis function and its derivatives is usually a more sensitive and a more reliable approach.



Figure 8: The GRM time model of the base of the weathering generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97.

Uniform Seismic Velocities in the Weathering

Figure 9: A compilation of three starting models which employ constant seismic velocities, a velocity reversal, and vertical velocity gradients in the weathering, and which employ the GRM seismic velocities in the sub-weathering for the optimum XY value of 3.75 m.



Three GRM starting models for which the seismic velocities in the sub-weathering have

been derived with the optimum XY value of 3.75 m are shown in Figure 9. The first starting model, which uncritically accepts the traveltime data in the weathering and ignores the possibility of any velocity reversals or vertical velocity gradients, has been used to generate the upper WET tomogram in Figure 10.

Figure 10: A compilation of three GRM WET tomograms which represent three models of the seismic velocities in the weathering. They are uniform seismic velocities derived from uncritical acceptance of the traveltime data, the average vertical velocities which accommodates a likely velocity reversal, and the hyperbolic velocity gradients. The GRM WET tomograms presented in Figure 10 have been derived after five iterations.

An inspection of the default outputs after 5, 10, 15 and 20 iterations indicated that five iterations represented the best compromise between minimizing the smoothing effects of tomography and minimizing misfit errors.

Unlike the smooth vertical velocity WET tomograms in Figure 5, there is little ambiguity in recognizing the base of the weathering in the GRM WET tomogram. There are large gradients in the seismic velocities in vicinity of the 2000 m/s contour.

The shear zone between stations 45 and 49 with a seismic velocity of ~3000 m/s can be differentiated from the adjacent massive sulphide orebody between stations 49 and 53 with a



seismic velocity of ~2500 m/s. Furthermore, a major shear zone with a low seismic velocity of ~2500 m/s between stations 61 and 67 is readily apparent. None of these features can be recognized in the smooth vertical velocity WET tomograms in Figure 5.

Velocity Reversal in the Weathering

A unique feature of the GRM is the computation of an average vertical seismic velocity in the weathering (Palmer, 2010b, equation A7), where an optimum XY value can be recovered from the refractor velocity analysis function. The GRM average vertical velocity is computed with arrivals which are refracted from the base of the weathering. It overcomes the incomplete sampling of the direct arrivals, which essentially determine the seismic velocities within a thin section of the top of the weathering in the horizontal direction (Palmer, 2010a). Therefore, it can provide a useful average vertical velocity, irrespective of whether constant velocities, vertical velocity gradients, velocity reversals or seismic anisotropy occur in the weathering (Palmer, 1992). Furthermore, Palmer (2010a, equation 12) provides a simple measure of the errors in terms of the errors in determining the optimum XY value.

The average vertical seismic velocity in the weathering computed with an average time model of 17 ms, average refractor velocity of 3500 m/s and optimum XY value of 3.75 m± 1.25 m is ~600 ± 100 m/s. This value, which is comparable to that obtained several hundred metres away (~500 ± 250 m/s) (Palmer, 2010d), shows that vertical velocity gradients are unlikely. Instead, these values indicate the probable occurrence of a reversal in the seismic velocities as has been identified nearby (Palmer, 2009a, 2010b) or even the possibility of vertical seismic anisotropy.

The upper tomogram in Figure 10 shows that there are significant lateral variations in the seismic velocities in the weathering. These lateral variations can also be accommodated with the GRM average vertical velocity simply by computing the average vertical velocity at each station. Figure 11 demonstrates that the lateral variations in the average vertical velocity due to the lateral variations in the seismic velocity in the sub-weathering (-7.5 m ≤ XY(2.5 m) ≤ 15 m, that is, 3.75 m) are more significant than the vertical variations due to any uncertainties in the optimum XY value.

Figure 11: A compilation of seismic velocities in (i) the sub-weathering (-7.5 m ≤ XY(2.5 m) ≤ 15 m), (ii) the weathering as measured from the GRM WET tomograms and (iii) the weathering as computed with the GRM average vertical velocity using XY values of 2.5 m, 3.75 m and 5 m.



The middle GRM WET tomogram shown in Figure 10 has been derived from the starting

model for the velocity reversal shown in Figure 9, in which the average seismic velocity in the weathering has been computed with an optimum XY value of 3.75 m. Also, the traveltime data have been altered to reflect the average vertical seismic velocities.

The use of the lower seismic velocities in the weathering facilitate a clearer differentiation of the shear zone between stations 45 and 49 with a seismic velocity of ~3000 m/s from the adjacent massive sulphide orebody between stations 49 and 53 with a seismic velocity of ~2500 m/s. Furthermore, the major shear zone with a low seismic velocity of ~2500 m/s between stations 61 and 67 is still unambiguous.

The Hyperbolic Velocity Function

The hyperbolic velocity function (Slichter, 1932; Healy, 1963; Berry, 1971; Aki and Richards, 2002, p.422) is the maximum vertical velocity gradient which is consistent with linear traveltime graphs. It represents the most general description of the undetected layer problem (Merrick et al, 1978), except where velocity reversals occur (Palmer, 2010a).

Figure 9 shows the starting model obtained by approximating the uniform seismic velocities in the weathering derived from the traveltime graphs with the hyperbolic velocity equivalents, together with the GRM model of the seismic velocities in the sub-weathering for the optimum XY value. It effectively represents a combination of the image in Figure 4 with the top image in Figure 9, and demonstrates that the GRM and vertical velocity gradients are not necessarily mutually exclusive as suggested by Rohdewald et. al. (2010).

Although the bottom WET tomogram in Figure 10 exhibits similar lateral variations in the seismic velocities as the other WET tomograms above, the contours are more evenly spaced, resulting in considerably greater ambiguity in the determination of the base of the weathering. Furthermore, the differentiation of the shear zone between stations 45 and 49 with a seismic velocity of ~3000 m/s from the adjacent massive sulphide orebody between stations 49 and 53 with a seismic velocity of ~2500 m/s, is questionable. However, the major shear zone with a low seismic velocity of ~2500 m/s between stations 61 and 67 can still be recognized.

Over-Processing

Many implementations of refraction tomography frequently recommend increasing the number of iterations from the default, commonly twenty, to fifty or one hundred, in order to improve resolution and reduce artifacts, even if the RMS error does not decrease (Rohdewald et al, 2010). Figure 12 confirms that the various measures of misfit errors exhibit minimal significant change after about ten iterations for the starting model employing uniform seismic velocities in the weathering in the upper part of Figure 9.

Figure 12: The misfit errors for the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data for a range of iterations.



However, Figure 13 demonstrates that the resolution is significantly reduced after fifty

iterations, while the tomograms generated after one and two hundred iterations are virtually identical to those generated with the smooth vertical velocity gradient starting model in Figure 5. Figure 13 demonstrates that refraction tomography is effectively a smoothing operation which rarely, if ever, improves the spatial resolution of the seismic velocities in the sub-weathering.

Figure 13: Three tomograms generated from the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data, after ten, fifty and two hundred iterations.

Seismic Velocities in the Sub-weathering Another demonstration of the importance of epistemic uncertainty is provided in Figure

15, in which the WET tomograms are generated from three the GRM starting models in Figure 14 using the seismic velocities in the sub-weathering computed with XY values of zero, 3.75 m and 7.5 m and uniform spatially varying seismic velocities in the weathering. All three tomograms exhibit two narrow regions with low seismic velocities that are generally consistent with the



spatial variations shown in Figure 7. Furthermore, all three tomograms have comparable misfit errors of ~1.87 ms after ten iterations.

However, the exact location and lateral extent of the narrow region with the low seismic velocity in the vicinity of station 49 varies with the selection of the XY value of the starting model. In particular, the differentiation of the shear zone between stations 45 and 49 with a seismic velocity of ~3000 m/s from the adjacent massive sulphide orebody between stations 49 and 53 with a seismic velocity of ~2500 m/s, is critically dependent on the XY value. However, the major shear zone with a low seismic velocity of ~2500 m/s between stations 61 and 67 can still be recognized. It can be concluded that the starting models generated with both optimum and non-optimum GRM parameters (3.75 m ± 3.75 m) further demonstrate the significance of epistemic uncertainty and that it is quite separate from aleatory variability.

Figure 14: Three starting models generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m.



Figure 15: Three WET tomograms generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m after ten iterations.

Ray Coverage

Figure 16: The wavepath eikonal traveltime (WET) tomogram and the corresponding ray coverage image derived from a default starting model consisting of smooth vertical velocity gradients.



A common output of many refraction tomography programs is a diagram which shows the

ray coverage. These diagrams are widely used as validation of the corresponding tomogram, and in turn, the starting model. In many geotechnical and environmental investigations, ray coverage images are required to demonstrate that the target has been investigated to a specified depth.

Figure 16 presents the ray coverage for the smooth vertical velocity gradient tomogram in Figure 5. It shows ray coverage to a depth of more than 60 m with ray concentrations in the regions of the 1500 m/s to 2000 m/s and the 3000 m/s contours. In general, it can be concluded that the ray coverage ostensibly supports the tomogram and the interpretation.

Figure 17: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities, a velocity reversal and velocity gradients in the weathering.

Figure 17 presents the ray coverage for the three WET tomograms in Figure 10. Each ray

coverage image indicates significantly less depth of investigation than is the case with the smooth vertical velocity gradient in Figure 16. Although each image is consistent with the respective tomogram, it is not possible to determine the most probable tomogram on the basis of the ray coverage alone.



Figure 18: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities in the weathering, after five, fifty and two hundred iterations. Figure 18 presents the ray coverage images for the three GRM WET tomograms in Figure

13. With increasing numbers of iterations, the ray coverage demonstrates increased depth of penetration, but at the expense of decreased spatial resolution. As with Figures 10 and17, Figures 13 and 18 are internally consistent, but they do not provide diagnostic information with which to determine the most likely tomogram. Furthermore, although the ray coverage for the default smooth vertical velocity gradient tomogram in Figure 16 has many similarities with that for two hundred iterations in Figure 18, nevertheless, there are several differences in detail which suggest that ray coverage images should not be over interpreted.

It can be concluded that ray coverage images are an inevitable result of both the tomogram, and in turn, the starting model, as well as the number of iterations. Furthermore, ray coverage images, which demonstrate large and unrealistic depths of investigation, are usually the result of low resolution tomograms, which are generated either with low resolution starting models such as smooth vertical velocity gradients or by over-processing detailed starting models. Just as many tomograms which have comparable misfit errors, can be generated from the one set of travelime data, so can many corresponding ray coverage images can be produced. It can be concluded that ray coverage images are an example of circulus in probando, that is, they assume that which they seek to demonstrate. Accordingly, they rarely constitute a useful tool with which to validate any tomogram, whether it be in relation to the starting model or the depth of investigation.



Attributes Derived from Head Wave Amplitudes

The Scaled Density Ratio Head wave amplitudes constitute half of the volume of refraction data recorded with all

seismic surveys. Regrettably, these data are normally under utilized. A simple 1D analysis of the shot amplitudes often can indicate whether uniform seismic velocities, vertical velocity gradients or even velocity reversals are appropriate (Palmer, 2009a, 2010b).

Geometrical spreading dominates the observed head wave amplitudes. Palmer (2001a) demonstrates that a 2D analysis of shot amplitudes with the multiplication of the forward and reverse amplitudes largely compensates for geometrical spreading and that the resulting amplitude products are approximately proportional to the square of the head coefficient. Palmer (2001b) also demonstrates that the head coefficient, which is the refraction analogue of the Zoeppritz transmission coefficient in reflection seismology, is approximately proportional to the ratio of the specific acoustic impedance in the overburden to that in the refractor.

The head wave amplitudes increase in accordance with the transmission coefficient of the Zoeppritz equations, where an additional soil layer occurs, (Palmer, 2006). Since the seismic velocities and densities of the uppermost few centimetres of surface soil layers can be very low, it is not unusual for the transmission coefficient to be almost two. Accordingly, the computed head coefficient for the profile across the ore body has a “noise” envelope of a factor of two, due to the occurrence and disappearance of soil and completely weathered rock.

Figure 19: The scaled density ratio computed from the head coefficient and the seismic velocities.

Figure 19 is a cross-section which presents the scaled density ratio computed from the

head coefficient and the seismic velocities. In this model, the density of the weathered layer has been assumed to be uniformly one. In the absence of any density values measured on discrete samples, no account has been taken of the likely higher densities of the outcropping rocks between stations 31 and 49, the lower densities over the recently restored historical mine workings in the vicinity of stations 49 to 53, or the intermediate densities between stations 53 and 70, where soil and completely weathered rock occur. Nevertheless, Figure 19 clearly shows the increased density of the massive sulphide ore body between stations 49 and 53, as well as a decrease in density of the shear zone between stations 45 and 49.



The P-Wave Modulus The head coefficient and the seismic velocities can also be combined to obtain a model of

the P-wave modulus (Palmer, 2010c, 2010d), which is shown in Figure 20. This attribute appears less sensitive to lateral variations in the densities in the weathered layer and there is generally quite good correlation with the seismic velocities.

Figure 20: The P-wave modulus computed from the head coefficient and the seismic velocities.

The P-wave modulus is quite high for the region of high density between stations 49 and

53, where the massive sulphides occur. In this case, the inclusion of the density with the head coefficient has effectively counteracted the effect of the low seismic velocity, and generated a more realistic measure of the rock strength of the massive sulphides. It demonstrates that the accommodation of the density with the head coefficient can often result in more representative estimates of rock strength. In fact, it is likely that the P-wave modulus will eventually prove to be a more useful measure of rock strength than the seismic velocity alone.

The density ratio and the P-wave modulus facilitate the separation of the narrow region with the low seismic velocities between stations 45 and 53 into two distinct zones, one of which is the massive sulphide ore body, whereas the other is a shear zone.

Discussion Aleatory Variability

Table 1 presents the misfit errors for the six starting models used in this study. The average error for all eighteen tomograms is 2.09 and the standard deviation is 0.36. The comparable figures for the six tomograms produced after ten iterations are 1.96 and 0.24. Although these figures indicate that the misfit errors are much the same for the six starting models, nevertheless those for two models merit further comment.

The misfit error for the tomogram generated with smooth vertical velocity gradients in Figure 5 is the smallest at 1.76 ms for ten iterations. This tomogram, which exhibits no features characteristic of any vertical shear zones or ore bodies with low seismic velocities, is considered to be the least geologically likely.

The misfit error for the tomogram which accommodates the velocity reversal in the weathering in Figure 10 is the largest at 2.44 ms for ten iterations. This tomogram is considered



to be the most geologically likely. It can be concluded that a simplistic comparison of misfit errors is not a reliable measure of geological verisimilitude.

Table 1 - RMS Misfit Errors

Model Starting Model 5 Iterations 10 Iterations 20 Iterations Default – smooth vertical velocity gradient

4.54 ms 2.05 ms 1.76 ms 1.63 ms

GRM XY = 0 Uniform velocities

20.17 ms 2.39 ms 1.89 ms 1.81 ms

GRM XY = 3.75 m Uniform velocities

20.23 ms 2.40 ms 1.86 ms 1.78 ms


20.25 ms 2.41 ms 1.87 ms 1.77 ms

GRM XY = 3.75 m Velocity reversal

9.52 ms 2.94 ms 2.44 ms 2.45 ms

GRM XY = 3.75 m Hyperbolic velocity gradient

10.93 ms 2.36 ms 1.93 ms 1.80 ms

Furthermore, the relatively minor reduction in the magnitude of the misfit errors with

increasing numbers of iterations can be associated with a significant reduction in the resolution of important geological features. For example, the narrow (~10 m) region between stations 45 and 49 with the seismic velocity of ~3500 m/s between the adjacent values of ~4000 m/s and ~2500 m/s is not well defined after more than ten iterations (see Figure 13). In fact, Figure 13 demonstrates that any significant spatial variability in the sub-weathering can be effectively removed given the application of a sufficient number of iterations. It can be concluded that minimizing misfit errors through increasing the number of iterations beyond between five and ten represents an emphasis on aleatory variability, usually at the expense of increasing the epistemic uncertainty.

Epistemic Uncertainty

The major focus of this study has been on the importance of the starting model and epistemic uncertainty. The primary objective has been the resolution and differentiation of the narrow massive sulphide ore body which exhibits low seismic velocities and high densities from the adjacent shear zone.

The first task has been to generate a family of viable starting models which can be tested with refraction tomography. This task can be usefully implemented with the GRM because the algorithms honour the traveltime data independently of the XY value selected (Palmer, 1980, p.50; Palmer, 1986, p.106-107). Since the lateral resolution of the GRM time model does not vary significantly with the XY value, the major source of non-uniqueness is usually in the determination of the lateral variations in the seismic velocities. Figure 8 presents a suite of models derived with different effective XY values.

Three of these models, which have been tested in Figure 15, demonstrate that significantly different detailed models of the seismic velocities in the sub-weathering can exhibit comparable misfit errors. The second task has been to determine which tomogram represents the most likely geological image of the subsurface. In this study, the tomogram generated with the optimum XY value has been taken as the most likely because it represents the simplest model of the seismic velocities in the sub-weathering, and because it is consistent with the a priori information in Figures 19 and 20.

The emphasis on spatial resolution with the GRM contrasts with the emphasis on vertical resolution with the smooth vertical velocity gradients in Figures 3 and 4. Contrary to statements by Sheehan et al (2005), this study demonstrates that both the GRM starting models and the GRM tomograms are able to successfully accommodate and define lateral variations in both the



weathering and the sub-weathering. In particular, the GRM and the RCS are able to generate good estimates of the lateral extent and various useful petrophysical and geotechnical attributes of the ore body between stations 49 and 53. This is not the case with the smooth velocity gradient tomograms.

Paradoxically, the emphasis on spatial resolution with the GRM can result in an improvement in vertical resolution through the determination of more representative vertical seismic velocities in the weathering with the average vertical velocity. Figure 10 shows that the use of vertical velocity gradients in the weathering can result in both an increase in the thickness of the weathering, as well as increased uncertainty in identifying the base of the weathering, using the reduced separation of the velocity contours. At station 37, the depth to the 4000 m/s contour ranges from > 720 m for the hyperbolic velocity model in the weathering, to ~ 730 m for the tomogram which uncritically accepts the traveltime data, to < 740 m for the velocity reversal.

The determination of the tomogram which is considered to exhibit the best geological verisimilitude in Figure 10 is probably a matter of personal preference, or possibly even prejudice. Nevertheless, only the tomogram which accommodates the probable velocity reversal contains seismic velocities in both the weathering and sub-weathering which are consistent with the optimum XY value, and which are supported by the a priori information in Figures 19 and 20.

Conclusions Uncertainty in the tomographic inversion of near-surface seismic refraction data can be

separated into aleatory variability, which describes the inherent random or stochastic variation associated with misfit errors and epistemic uncertainty which is due to a lack of knowledge of acceptable starting models for inversion. Standard implementations of refraction tomography generally disregard the importance of the starting model or epistemic uncertainty, and usually focus on reducing aleatory variability or misfit errors.

The GRM based tomograms generated with three models of the seismic velocities in the weathering (Figure 10) and three models of the seismic velocities in the sub-weathering (Figure 15) are consistent with the traveltime data. However, only one tomogram consists of seismic velocities which are consistent with the optimum XY value in both the weathering and the sub-weathering and which are consistent with the a priori amplitude attributes (Figure 20). It can be concluded that epistemic uncertainty can be explicitly addressed with the GRM, because the most probable tomogram can be objectively selected from a number of acceptable alternatives.

The GRM based tomograms successfully detect, define and differentiate narrow regions with low seismic velocities which represent shear zones and a massive sulphide ore body. By contrast, refraction tomograms generated from low resolution starting models, such as those employing smooth vertical velocity gradients, are not able to provide a reliable measure of the occurrence or otherwise of virtually any significant spatial variations in the seismic velocities in the sub-weathering. It can be concluded that inversion algorithms, which generate detailed starting models, such as those of the GRM, are essential for the majority of routine geotechnical investigations.

Refraction tomography is effectively a smoothing operation which rarely, if ever, improves the spatial resolution of the seismic velocities in the sub-weathering. In fact, it is possible to generate essentially featureless tomograms with minimal misfit errors, even using detailed starting models, simply by employing a sufficient number of iterations.

It can be concluded that refraction tomography should be employed thoughtfully, and that minimizing epistemic uncertainty is more important than minimizing aleatory variability. Refraction tomograms which are both accurate and precise are always preferable. Nevertheless, imprecisely accurate tomograms are invariably more useful than precisely inaccurate tomograms.



References

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Oldenburg, D. W., and Li, Y, 2005, Inversion for applied geophysics: a tutorial, in Near-surface geophysics, Dwain K Butler (ed.) Investigations in geophysics no. 13, 89-150, SEG, Tulsa. Palmer, D., 1980, The generalized reciprocal method of seismic refraction interpretation. Society of Exploration Geophysicists, 104p. 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. 1992. Is forward modeling as efficacious as minimum variance for refraction inversion? Exploration Geophysics 23, 261-2, 521. Palmer, D., 2001a, Imaging refractors with the convolution section. Geophysics, 66, 1582-1589. Palmer, D., 2001b, Resolving refractor ambiguities with amplitudes. Geophysics, 66, 1590-1593. Palmer, D., 2006, Refraction traveltime and amplitude corrections for very near-surface inhomogeneities. Geophysical Prospecting 54, 589-604. Palmer, D., 2007. Is it time to re-engineer geotechnical seismic refraction methods? 19th ASEG Conference and Exhibition, Perth (Extended Abstract). Palmer, D., 2008a. Is it time to re-engineer geotechnical seismic refraction methods? First Break 26(8), 69-77. Palmer, D., 2008b. Non-Uniqueness in near-surface refraction inversion. In: Xu, Y. X., Xia, J. H., eds., Proceedings of the 3rd International Conference on Environmental and Engineering Geophysics, Wuhan, China. Science Press, Beijing. 42–54. Palmer D. 2009a. Maximising the lateral resolution of near-surface seismic refraction methods: Exploration Geophysics 40, 85–90; Butsuri-Tansa 62, 85–90; Mulli-Tamsa 12, 85–90. Palmer, D., 2009b. Integrating long and short wavelength time and amplitude statics. First Break 27(6), 57-65. Palmer, D., 2010a, Non-uniqueness with refraction inversion – a synclinal model study: Geophysical Prospecting 58, 203-218. Palmer, D., 2010b, Non-uniqueness with refraction inversion – the Mt Bulga shear zone. Geophysical Prospecting 58, 561-575. Palmer, D., 2010c, Are refraction attributes more useful than refraction tomography?: First Break 28, 43–52. Palmer, D., 2010d, Characterizing the near surface with detailed refraction attributes. in R. D Miller, J. H. Bradford and K. Hollinger, eds., Advances in near-surface seismology and ground-penetrating radar: SEG Geophysical Development Series No. 15, Chapter 14, 233-250.



Palmer, D., 2010e, Is visual interactive ray trace an efficacious strategy for refraction inversion? Exploration Geophysics 41, 260-267. Rohdewald, S., Sheehan, J., and Burton, B., 2010, Processing of seismic refraction tomography data, SAGEEP, Keystone, Colorado. Schuster, G. T., and Quintus-Bosz, A., 1993, Wavepath eikonal traveltime inversion: theory: Geophysics 58, 1314-1323. Sheehan, J. R., Doll, W. E., and Mandell, W. A., 2005, An evaluation of methods and available software for seismic refraction tomography analysis. Journal of Engineering and Environmental Geophysics 10(1), 21-34. Slichter, L.B., 1932, Theory of the interpretation of seismic travel-time curves in horizontal structures: Physics 3, 273–295. Stefani, J. P., 1995, Turning-ray tomography: Geophysics 60, 1917-1929. Treitel, S., and Lines, L., 1988, Geophysical examples of inversion (with a grain of salt): The Leading Edge 7, 32–35. Whiteley, R. J., 1986, Electrical and seismic responses of shallow volcanogenic massive sulphide ore deposits. Ph D dissertation, University of New South Wales, Sydney, 393 p. Whiteley, R. J., and Leung, T. M., 2006, Mt Bulga revisited: http://rayfract.com/pub/Mt_Bulga_Revisited.pdf. Whiteley, R. J., Hawkins, L. V., and Govett, G. J. S., 1984, The seismic, electrical, and electrogeochemical character of the Mount Bulga massive sulphide orebody, NSW., Australia. SEG Expanded Abstracts 310 – 314. Zhang, J., and Toksöz, 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.



Table 1 - RMS Misfit Errors

Model Starting Model 5 Iterations 10 Iterations 20 Iterations Default – smooth vertical velocity gradient

4.54 ms 2.05 ms 1.76 ms 1.63 ms

GRM XY = 0 Uniform velocities

20.17 ms 2.39 ms 1.89 ms 1.81 ms


20.23 ms 2.40 ms 1.86 ms 1.78 ms


20.25 ms 2.41 ms 1.87 ms 1.77 ms

GRM XY = 3.75 m Velocity reversal

9.52 ms 2.94 ms 2.44 ms 2.45 ms

GRM XY = 3.75 m Hyperbolic velocity gradient

10.93 ms 2.36 ms 1.93 ms 1.80 ms



Figure Captions

Figure 1: A common dart board analogy which illustrates the differences between precision or aleatory variability and accuracy or epistemic uncertainty. Figure 2: Geological cross-section showing the massive sulphide ore body and the adjacent shear zone. Figure 3: The traveltime graphs recorded across a narrow massive sulphide ore body at Mt Bulga in southeastern Australia. Figure 4: The smooth vertical velocity gradient starting model. It is the default in many implementations of refraction tomography. Figure 5: Three tomograms generated from the smooth vertical velocity gradient starting model after five, ten and twenty iterations. Figure 6: The GRM refractor velocity analysis function generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97. Figure 7: The GRM seismic velocities for a range of effective XY values 0 ≤ XY(1.25 m) ≤ 7.5 m. The optimum XY value is 3.75 ± 1.25 m. Figure 8: The GRM time model of the base of the weathering generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97. Figure 9: A compilation of three starting models which employ constant seismic velocities, a velocity reversal, and vertical velocity gradients in the weathering, and which employ the GRM seismic velocities in the sub-weathering for the optimum XY value of 3.75 m. Figure 10: A compilation of three GRM WET tomograms which represent three models of the seismic velocities in the weathering. They are uniform seismic velocities derived from uncritical acceptance of the traveltime data, the average vertical velocities which accommodates a likely velocity reversal, and the hyperbolic velocity gradients. Figure 11: A compilation of seismic velocities in (i) the sub-weathering (-7.5 m ≤ XY(2.5 m) ≤ 15 m), (ii) the weathering as measured from the GRM WET tomograms and (iii) the weathering as computed with the GRM average vertical velocity using XY values of 2.5 m, 3.75 m and 5 m. Figure 12: The misfit errors for the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data for a range of iterations. Figure 13: Three tomograms generated from the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data, after ten, fifty and two hundred iterations. Figure 14: Three starting models generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m. Figure 15: Three WET tomograms generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m after ten iterations. Figure 16: The wavepath eikonal traveltime (WET) tomogram and the corresponding ray coverage image derived from a default starting model consisting of smooth vertical velocity gradients.



Figure 17: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities, a velocity reversal and velocity gradients in the weathering. Figure 18: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities in the weathering, after five, fifty and two hundred iterations. Figure 19: The scaled density ratio computed from the head coefficient and the seismic velocities. Figure 20: The P-wave modulus computed from the head coefficient and the seismic velocities.



Figure 1: A common dart board analogy which illustrates the differences between precision or aleatory variability and accuracy or epistemic uncertainty.



Figure 2: Geological cross-section showing the massive sulphide ore body and the adjacent shear zone.



Figure 3: The traveltime graphs recorded across a narrow massive sulphide ore body at Mt Bulga in southeastern Australia.



Figure 4: The smooth vertical velocity gradient starting model. It is the default in many implementations of refraction tomography.



Figure 5: Three tomograms generated from the smooth vertical velocity gradient starting model after five, ten and twenty iterations.



Figure 6: The GRM refractor velocity analysis function generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97.



Figure 7: The GRM seismic velocities for a range of effective XY values 0 ≤ XY(1.25 m) ≤ 7.5 m. The optimum XY value is 3.75 ± 1.25 m.



Figure 8: The GRM time model of the base of the weathering generated with -2.5 m ≤ XY(2.5 m) ≤ 10 m, together with the average for SP1 and SP97.



Figure 9: A compilation of three starting models which employ constant seismic velocities, a velocity reversal, and vertical velocity gradients in the weathering, and which employ the GRM seismic velocities in the sub-weathering for the optimum XY value of 3.75 m.



Figure 10: A compilation of three GRM WET tomograms which represent three models of the seismic velocities in the weathering. They are uniform seismic velocities derived from uncritical acceptance of the traveltime data, the average vertical velocities which accommodates a likely velocity reversal, and the hyperbolic velocity gradients.



Figure 11: A compilation of seismic velocities in (i) the sub-weathering (-7.5 m ≤ XY(2.5 m) ≤ 15 m), (ii) the weathering as measured from the GRM WET tomograms and (iii) the weathering as computed with the GRM average vertical velocity using XY values of 2.5 m, 3.75 m and 5 m.



Figure 12: The misfit errors for the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data for a range of iterations.



Figure 13: Three tomograms generated from the GRM starting model for the optimum XY value of 3.75 m which uncritically accepts the traveltime data, after ten, fifty and two hundred iterations.



Figure 14: Three starting models generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m.



Figure 15: Three WET tomograms generated from the seismic velocities in the sub-weathering using the GRM refractor velocity function with effective XY values of zero, 3.75 m, and 7.5 m after ten iterations.



Figure 16: The wavepath eikonal traveltime (WET) tomogram and the corresponding ray coverage image derived from a default starting model consisting of smooth vertical velocity gradients.



Figure 17: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities, a velocity reversal and velocity gradients in the weathering.



Figure 18: Ray coverage images for three WET tomograms derived from the GRM model of the seismic velocities in the sub-weathering and uniform seismic velocities in the weathering, after five, fifty and two hundred iterations.



Figure 19: The scaled density ratio computed from the head coefficient and the seismic velocities.



Figure 20: The P-wave modulus computed from the head coefficient and the seismic velocities.

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