b. l. n. kennett

Geophys. J. R. astr. SOC. (1973) 32,389-408.

The Effects of Scattering on Seismic Wave Pulses

B. L. N. Kennett

(Received 1972 December 1)

Summary

This paper presents a detailed study of the scattering effect of a single, sand-lens like, obstacle embedded in a uniform medium. The problem is first considered theoretically using the first order perturbation technique previously developed by the author and this theory is then used to compute scattered pulse forms for different receiver positions and obstacle shapes. These pulse forms are subsequently compared with results from two- dimensional seismic modelling experiments. This comparison shows the utility of th is theory as an aid in improving seismic interpretation in the presence of scatterers.

1. Jntroduction There is now increasing evidence as to the complicated fine structure of the

upper layers of the Earth. On a small scale in work on crustal structure and on a larger scale in studies relating to the Upper Mantle it is no longer possible to interpret the observational results in terms of horizontally layered structures (e.g. Mack 1969). Increasing attention is therefore being paid to the problems of understanding seismic wave propagation in laterally varying media so that improvements may be made in our approach to the interpretation of seismic data.

Considerable attention has been focused on the effect of scattering from inhomogeneities as a complicating factor in such interpretation. Aki (1969) has looked at the possible generation of the P-wave coda by scattering, and it has been invoked by Greenfield (1971) to explain multipath effects observed from nuclear explosions at Novaya Zemlya. Further the study of simple scatterers provides at least some improvement in our knowledge of the characteristics of the wave field associated with some types of stratigraphic traps in crustal exploration.

Nearly all the previous work which has been done on the scattering of seismic waves by obstacles or interfacial irregularities has concentrated on the effect of scattering on a monochromatic incident wave. Though Gilbert & Knopoff (1960) did consider approximate expressions for the pulse forms to be expected on scattering by a surface obstacle on a half space.

We present here a theoretical and experimental study of the effect of a single scatterer on a seismic pulse. The form of the scatterer has been chosen to represent a sand lens or a small reef type structure. The treatment is based on the methods developed by the author (Kennett 1972a, b) for handling scattering problems for localized lateral inhomogeneities, coupled with numerical integration to perform Fourier inversion integrals. The results obtained from this approximate scattering theory are compared with those obtained using model seismic experiments.

389

1

390 B. L. N. Kennett

2. The effect of a single scatterer We consider the scattering produced by a plane wave incident upon a region whose

properties differ from those of the surrounding matrix, e.g. a sand lens or a reef lying in an otherwise homogeneous layer. For convenience in exposition and in the sub- sequent comparison with experiment we shall here confine our attention to two- dimensional scattering problems.

The scatterer is assumed to be bounded by the horizontal planes z = 0 and z = H, (Fig. 1). Then provided that the velocity contrast across the surface of the scatterer is not too severe and its horizontal size is relatively modest, we may as a good approximation use the first-order scattering formation developed by Kennett (1 972a).

Thus we define a stress-displacement vector B(x, z, t) as

B(x, z, 0 = col W, z, t ) , w(x, z, t ) , d x , ~ , t ) , ~ ( x , z, t)l

where u, w are the horizontal and vertical displacement components and ri, is the stress tensor. The scattering effect of an obstacle may then be introduced as a volume source distribution in terms of the spatially and temporally Fourier transformed vector B(k, z, o) as (Kennett 1972a, equation (4.26))

H w

where B’ is the scattered part of the wave field and Bo is the wave field which would be present in the absence of the scatterer. P(k,H,O) is the ‘propagator matrix’ appropriate to the dependence of the elastic parameters outside the obstacle on the depth co-ordinate z (for a homogeneous medium this is just the Haskell layer matrix). C(k, 5, y) depends on the Fourier transform of the departures of the elastic parameters from lateral homogeneity and thus gives rise to the dependence of the scattered field on the shape of the scatterer and the parameter contrasts at the boundary.

To concentrate on the effects due to scattering rather than complications due to the propagation paths of the scattered waves we consider the scatterer to be embedded in an infinite homogeneous medium and take a single plane wave to be incident upon it from z < 0 with horizontal wave number ko. Then the source problem to be

i Fro. 1. Configuration of scatterer.

Seismic wave pulsea 391

solved is

0 and in z,zo Q 0; Z,ZO 3 H

B’(k, z, 0) = P(k, z, zo) B’(k, zo, 0)

with the imposed physical condition that the scattered wave field should consist only of waves moving out from the scatterer.

Because of the importance of the reflection method in seismic exploration we shall confine our attention to an analysis of the back scattered waves, though similar calculations could be performed for forescattering. We shall consider a scatterer within which the elastic wave speeds a‘, /3’ and density p’ are constant surrounded by a medium with properties a, fi, p. We may describe the shape of the scatterer as the envelope of a suite of cross-sections on planes z = const. through a rectangular function g(x,z) of unit height within the scatterer and zero outside. Consider an incident plane wave travelling at an angle 8 to the z axis, then on solving the source problem representing the scattering the solution may be cast into a convenient form where the dependence on the parameter contrasts between the scatterer and its surrounding medium is entirely through non-dimensional combinations of the parameters.

We define v, = (w2/a2 -k2)*, Im v,, vg 2 0

and V# = ( 0 2 / f i 2 4 2 ) + ,

r, = a‘/a, r, = f i ’ / /3, rp = p‘/p.

Then the Fourier transforms of the scattered displacements observed at the plane z = - D are, for an incident pulse 4(t),

and those parts of the displacements associated with fl and fz correspond to P- and SV-wave contributions respectively. The dependence of the scattering on the shape of the scatterer and its properties is through the somewhat complicated functions fi, f2 which have the form

fi (k, 0) = 1 dy exp {i [v, + (w/a> cos ely} g[k - (w/a) sin e, y l R

0

- (T 1 -1) 4k sin0 cose


I1 o a

and

fi (k , o) =

H

d y exp { i [vs + (cola) cos 81 y } g [k - (cola) sin 8, y] 0

+ ((1- $) - r p r i ( l - 5)) 4k sin28]

o a

where g(x, z) is the envelope function defked above. To find the scattered displacement we perform a double Fourier inversion, for example

w o o

u’(x, - D , t ) = - I I exp ( ikx- iwt) ii’(k, - D, o) dk do. -00 --m

4;r2

The results presented below in Sections 3 and 5 have been obtained by performing this inversion numerically, to reduce the computation times required we have made use of the Fast Fourier transform algorithm to evaluate our inverse transforms. This has the consequence of imposing temporal and spatial periodicity on the computed solutions and also spatial periodicity on the scatterer model itself. However by appropriate choice of the time and space windows used we were able to arrange that the effects of aliasing were minimized whilst still providing adequate resolution of the shape of the scatterer. The integrations in the expressions for fi, f 2 were evaluated by using a simple trapezoidal approach which works well in the case of an oscillatory integrand.

From the form of the inversion integral above we see that it is most convenient to evaluate the scattered wave forms at different horizontal positions along planes z = const. Such a scheme has the advantage that it closely represents the normal pattern of observations in a field experiment.

The scattering formalism developed above is expected to provide quite a good approximation to the scattered wave field if the following condition is satisfied for all significant frequencies in the incident pulse (cj. Kennett 1972a, b)

Seismic wave pulses 393

where A is the included area of the scatterer and q is a non-dimensional contrast measure, e.g. a reflection coefficient.

3. Theoretical results In this section we present the results of a number of computations using the

theory developed in Section 2 for a simple plane P-wave pulse incident upon a single scatterer. The scatterer was taken to be lens shaped as illustrated in Fig. 2(a) and to have a P-wave speed of 4*00kms-', S-wave speed 2*00kms-' and density 2.5 g cm-3 whilst the surrounding medium had P-wave speed 3-00 km s-l, S-wave speed 1*50kms-' and density 2-00gcm-'. These parameters were chosen to represent an example of a sand lens in a shale layer. The incident waveform used was of Ricker-type and is shown in Fig. 2(b), the dominant wavelength in the incident pulse is approximately 120 m.

The results presented here are obtained by taking a 128-point FFT in the co- ordinate x and a 64-point transform in time. Thus each trace is synthesized froin 128 plane waves at each of 64 frequencies. To enable visual comparisons to be made the waveforms at each position have been plotted after cubic interpolation between the points in the time series. The x-step used was 7.5 m which proved sufficient to give resolution whilst yielding a fairly long spatial periodicity of 960 m, roughly five times as long as the widest scatterer considered.

In Figs 3 and 4 we illustrate the dependence of the spatial amplitude distribution on the height and width of a scatterer for a normally incident P wave. These diagrams show the scattered waveforms for the Pwave part of the vertical displacement evaluated in the plane 200m above the scatterer (i.e. in the fairly near field) at a horizontal interval of 55 m.

We shall first consider the effect of the width of a scatterer on the nature of the scattered wave field. At the position above the centre of the scatterer (i.e. in the direction of specular reflection) we may show that the scattered amplitude should

0 0.05 0-10 Time ( 5 )

FIG. 2. (a) Typical scatterer shape, simulating a sand lens; (b) Input pulse used in theoretical calculations.


I 'd

FIG. 3. Theoretical traces of vertical displacement, at different horizontal separations from the centre line of the scatterer (xo) in the plane 200 m above the scatterer, showing the effect of scatterer width (predominant wavelength h f; 120 m). - - - width 200 m, thickness 25m; - width 100 m, thickness

25 m; _._._ width 50 m, thickness 25 m.

be roughly proportional to the included area of the scatterer; this effect is well illustrated in Fig. 3, where an amplitude increase of 3 . 3 : 1 accompanies an increase in included area of 4 : 1.

The representation of the scattering effect of an obstacle which we have developed in Section 2 corresponds to a distribution of field dependent source terms on the perimeter of the scatterer. The duration of the scattered waveform at a point due to the interference of the radiation from these sources will increase as the angle sub- tended by the scatterer at the point increases. Thus the scattered pulse length will increase as the width of a scatterer increases and also as the observation point moves away from the centre line of the scatterer on a horizontal plane. Both these effects are clearly seen in Fig. 3.

In those regions where no reflected wave from the top of the scatterer is anticipated the onset of the scattered wave corresponds to a ray path from the tip of the scatterer nearest to the source (i.e. the nearest virtual source). In some cases, for suitable elastic parameters, one may see a definite indication of a second pulse emerging from the tail of the main arrival which apparently comes from the further end of the scatterer, this effect is most pronounced for thin scatterers.

From Fig. 3 we see that the scattered amplitude for a wide scatterer drops con- siderably from its maximum value even for observing points before the end of the scatterer. The rate of amplitude decay increases with the width of the scatterer so


FIG. 4. Theoretical traces showing effect of scatterer thickness. - - - width 100 m, thickness 50 m; - width 100 m, thickness 25 m; -*- width 100 m,

thickness 12.5 m.

that examination of the scattered waveforms will give some indications of its size but detailed migration calculations would be needed to determine the end points.

In Fig. 4 we present a series of wave forms for scatterers of the same width but varying thickness; to enable comparisons to be made with Fig. 3 the solid line in each diagram represents the same model scatterer (width 0.812, thickness 0.24). We can see immediately a striking difference between the effect of doubling the width and doubling the thickness; in the former case the amplitude very nearly doubled at the centre line of the scatterer whereas in the latter the amplitude has hardly increased. This indicates significant departures from the linear relation between amplitude and thickness for thin scatterers (see e.g. Gilbert t Knopoff 1960). We may also note that the length of the scattered pulse increases as the thickness of the scatterer increases. These effects may be explained in terms of interference between the waves scattered from different parts of the scatterer and introduction of more widely separated virtual sources, thereby giving increased time lags as the thickness increases.

Though we have above confined our attention to the P-wave part of the scattered wave form there are also significant scattered S waves and these may be important, particularly in the near field of a scatterer. In Fig. 5 we show both the P- and S-wave parts of the vertical scattered displacement for observation points in the horizontal plane 200 m above the scatterer (width 0.812, thickness 0.112). At this height we are able to look at the scattered field at wide angles from the scatterer and we see that the S-wave field may be more important well away from the centre line of the scatterer. At larger distances from the obstacle the scattered S-wave field will be widely separated in time from the P wave and will be an important contributor to the background noise.


I W

I , # I t I I I I , I I ~ " " " " " " ' " 0 0-05 0-10 0.15 0.20 0.25

Time (s)

FIG. 5. Generation of P and S waves by scattering; scattered P pulse, solid line; S pulse, broken line (vertical displacement).

So far we have only considered waves at normal incidence but scattering effects for oblique incidence are also important. Fig. 6 illustrates scattered P-wave forms for incident plane P pulses of the form shown in Fig. 2(b) as the angle of incidence increase, for a scatterer of width 0.82, thickness 0.22. The observation points all lie in the same horizontal plane as before (200 m above the scatterer) and are such that the distance from the centre line of the scatterer xo = 200 tan 8 for an angle of incidence 8. The main differences between these wave forms are in the relative amplitudes of the main peaks, the earlier one being enhanced as the angle of incidence increases.

A series of trials with differing contrasts (reflection coefficients in the range -0.25 to 0.25) between the scatterer and its surroundings suggests that the spatial amplitude distribution of the scattered wave field is largely controlled by the overall shape of the scatterer though there are small contrast dependent effects. In general the scattering effect of a negative parameter contrast is found to be larger than that of a corresponding positive contrast.

For scatterers of the same general shape and size it is found that although differences in the amplitude distribution are detectable for changes in the detail of the shape they are not really significant. Thus interpretation of scattered wave data will be somewhat insensitive to the detailed shape of the scatterer.

4. Experimental study We have also measured the effect of a single scatterer on a seismic pulse using the

techniques of two-dimensional model seismology (O'Brien & Symes 1971). Experi-


A

0 0 05 0 10 0 15 0 20 2-' ' ' ' ' ' ' ' '

Time ( 5 )

FIG. 6. The effect of varying the angle of incidence (0) of the incident wave. The arrow indicates arrival time of a wave normally reflected at the top of the scatterer.

ments were carried out on a number of different sizes of scatterer with the intention of comparing the predictions of the approximate theory developed in Section 2 with results from analogue models. The comparison will be discussed in the next section.

The experimental medium consisted of a 1.2mx2.4m plate of 3mm thick Perspex mounted in a tension frame. The scatterers used were pieces of lead foil of thickness 0.8 mm rolled flat and stuck onto the plate with double-sided adhesive tape. We used a piezoelectric ceramic PZT5H bar (15 x 3 x 3 mm) as a source excited by a repetitive pulse generator.

The scatterers were mounted in the far field of the source so that the wavefront incident upon them was very closely plane and normal to the scatterer. The waveform across the width of the widest scatterer was reproducible to within 4 per cent. The disturbances in the plate were detected using a biomorph piezoelectric transducer (a ' bender ') sensitive to displacement when observing on the side of the plate and a commercially available detector (Ultrasonoscope type 10/0/23s) when working on the top edge of the plate. The signals were recorded using a pass band of 20-60 kHz and a wave form translator system for visual display. The predominant wavelength in the pulse used is about lOcm so that pulse dispersion effects due to the nature of wave propagation in a plate should be very small (G0.01 per cent). A range of scattering models were considered, all had the same general form as that shown in Fig. 2(a), but the width and thickness were varied. The scatterer thicknesses were between 0.15 and 0.41 and widths between 0.3 and 1.51.

Unfortunately because of the duration of the incident pulse it did not prove possible to find an effective time window in which to observe the near-field scattering, since the scattered energy started to arrive before the tail of the incident pulse had

3 98 B. L. N. Kennett

Table 1

Properties of experimental materials a(km s- I) j(km s- I) p(g cm-3)

Perspex 2.35 1.35 1.2 LeadIPerspex sandwich:

Experimental 1.81 1 -08 Theoretical 1.77 1-04 3.36

completely passed. However we were able to make measurements at about 61 above the scatterer, and at the surface ( w 91jexcept of course in the immediate vicinity of the source. The recordings were made at equally spaced intervals along horizontal lines on the vertically mounted plate to enable comparisons to be made between the experimental and theoretical results. (Fig. 7(a)).

The velocity of P and S waves in the Perspex plate and also in the lead foil/ Perspex sandwich were determined from the slopes of time-distance curves made using compressional and shear sources respectively. The results are summarized in Table 1 together with the values expected for a composite plate from the relations (Healy & Press 1960)

wheref, is the fraction of material of type n, pn is its density and cn the velocity of either P or S waves. In the theoretical value no allowance has been made for im- perfect bonding or for the double-sided adhesive tape whose properties are rather difficult to determine.

The pulse form incident upon the scatterer is illustrated in Fig. 7(b) recorded with a ' bender ' using a wide pass band (200 Hz-70 kHz), and it is this pulse which has been used as the input to the theoretical calculations of the scattered waveforms. This procedure unfortunately suffers from the defect that as we are unable to fully

(0) ,J Source l l Y Y Y " "

\ Rocording points

T

I 'Scotterer

FIG. 7(a). Experimental configuration.


I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15

Time (,us)

FIG. 7(b). The experimental input pulse.

allow for the effect of the coupling of the detector to the plate we are unable to deconvolve the recorded signal to recover the true incident pulse. This is not important if all the recordings are made with the same transducer but causes problems when as at a free edge a different transducer is used.

Visual records of the experimental results were made using the waveform translator system described in detail by O’Brien & Symes (1971) and examples of such records can be seen below in Figs 8(a)-11 (a).

The experimental results for different sizes of scatterer showed very good qualitative agreement with the theoretical results presented in Section 3. For example, the measured effect of doubling the width of a scatterer was to increase the amplitude in the reflected direction by very nearly a factor of 2 and also to give more rapid amplitude decay as the observation point moved horizontally away from the scatterer. Similarly we find increased pulse lengths associated with thicker scatterers. Thus such characteristics are a property of the shape of the scatterer rather than any elastic parameter contrasts; since for the previous calculations the scattering was largely due to velocity contrasts, whereas experimentally it is determined by density contrasts.

5. Comparison of theory and experiment

As described in the previous section we have made observations on seismic plate models, both on the face of the plate and at the free surface. In order to make comparisons between the theory of Section 2 and observations on the surface we have to introduce correction factors in the transform domain before performing the Fourier inversion, this process is described in detail in the Appendix. In addition the surface detector used is sensitive to velocity so that on the surface we compare the experimental results with computed velocity traces. Rather than perform a numerical differentiation of the computed displacement we made use of the result that the displacements may be expressed as a convolution in time and that for a given input pulse 44)

400 B. L. N. KeMett

Thus we use the derivative of the experimental input pulse as the input to the procedure in Section 2, and as we need the spectrum of the input this derivative is taken by simply performing a multiplication by io in the frequency domain. For observations on the face of the plate comparisons may be made directly with the theorectical results.

The Fourier inversions have been performed on a 64-point time transform and a 256-point x-co-ordinate transform. The spacing of the points in the x direction was chosen to be a submultiple of the sampling separation in the experimental situation and thus experimental and theoretical traces could be obtained for the same positions. A zero-phase band-pass filter with a pass band of 20-60 kHz was applied to the theoretical results in the frequency domain before performing the Fourier inversion, to allow direct comparison of the experimental and theoretical results. The final 64-point time series were interpolated for visual display as described above (Section 3). At the positions used the temporal separation of scattered P and S is such that only scattered P waves have been considered.

The scatterers used in the comparisons made below are all such that the condition for the use of first-order perturbation theory is barely obeyed; since this provides a more critical test of the theoretical method’s capabilities. Throughout this section we shall be considering waves hitting the scatterers at normal incidence, as this corresponds to the experimental situation. In this configuration one expects on theoretical grounds that the scattered P-wave field should possess the same vertical plane of symmetry (if any) as the scatterer. For an obstacle of the type shown in Fig. 2(a), this is found to be quite well obeyed experimentally, the differences in amplitude between image points in the symmetry plane being about 5 per cent, experimental error through variability of transducer model coupling probably accounting for most of this.

The most direct comparison between theory and experiment can be made for observations on the face of the plate. In Figs 8 and 9 we compare vertical displace-

I A 0-

10-

2 0 .

30 ~

40 -

0 5 10. I 5 20 25 Time (ps)

FIG. S(a). Comparison of vertical displacement traces for different horizontal separations from the centre line of the scatterer (xo). Scatterer: width 71.5 mm,

thickness 17 mm. (a) experimental.

Seismic wave pulses 40 1

30

I I I I I I I I I I I I I I I I I 1 1 I I 5 10 15 20 0

Time (ps)

Fig. 8(b). Theoretical.

ment traces obtained at a height of 66 cm above the scatterer; sampled at 5-cm intervals horizontally, starting at the centre line of a symmetric scatterer. Fig. 8 compares the results for a 71.5 mm wide, 17 mm thick scatterer and Fig. 9 is for a scatterer of the same width but twice the thickness. The experimental results (a) are tracings of the output of the waveform translator, and the theoretical (b) come from computer plots, no particular significance should be attached to the absolute magnitudes. We see that there is quite good qualitative agreement between the two sets of traces (a) and (b); the pulse shapes in both Figs 8 and 9 compare very well and the sense of initial motion is the same. However the rate of amplitude decay horizontally is more rapid in the theoretical results. On comparing Figs 8 and 9 we see the pulse lengthening effects associated with increasing scatterer thickness discussed above.

In Figs 10 and 11 we compare experimental results observed at the surface of the plate with results obtained using the free-surface modified theory. The surface is 91 cm above the scatterer and the horizontal sampling interval again 5 cm, the traces here start lOcm displaced from the source (this is to reduce the interference due to scattered Rayleigh waves generated at the source transducer in the experimental set-up). In Fig. 10 we consider a scatterer of width 143 mm and thickness 17 mm, i.e. twice as wide as in Fig. 8, and in Fig. 11 a scatterer of the same width but twice as thick. The agreement between experiment and theory is here unfortunately not as good as for the side of plate positions. However the theoretical traces do show many of the main features of the scattering, e.g. the double-pulse effect in Fig. 11, but the tail of the pulses is somewhat different. The theoretical results would be improved if we could make full allowance for the coupling between the detector and the plate and apply a frequency dependent impedance; in the traces in Figs 10 and 11 a constant impedance has been assumed.

402

0 .

10-

20

Xn I

30

0 5 10 15 20 Time (ps)

FIG. 9. Comparison of vertical displacement traces. Scatterer: width 71.5 mm, thickness 34 mm. (a) experimental; (b) theoretical.


Time ( p s )

FIG. 10. Comparison of free surface traces. Scatterer: width 143 mm, thickness 17 mm. (a) experimental; (b) theoretical.


1 1 1 1 1 1 1 1 1 I l l I I l l I I I I I I I I L

20 -

I I I I I I I I I 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15 20

Time (ps)

FIG. 11. Comparison of free surface traces. Scatterer: width 143 mm, thickness 34 mm. (a) experimental; (b) theoretical.


0 0

I 1 I I I I I I I I I 0 10 20 30 40

*O

FIG. 12. Comparison of relative amplitude variations in vertical displacement: solid symbols, experimental; open symbols, theoretical. A width 71 - 5 mm. thickness 34 mm; 0 width 71.5 mm, thickness 17 nun; 0 width 143 mm,

thickness 17 mm.

For the scatterers considered above we are working at the extreme limit of the domain in which we would expect first-order perturbation theory to be useful. Since the theoretical scattering effects then increase more rapidly with frequency than those observed, we will tend to get a bias towards higher frequency components in the theoretical results.

Above we have considered a comparison of predicted and observed waveforms; we now consider in more detail the relation between theoretical and experimental amplitudes. In Figs 12 and 13 we consider a comparison of first swing (peak-to- trough) amplitude variations with horizontal position. The experimental results were measured directly on a CRO display and then normalized to give unit amplitude directly above the centre line of the scatterer. The theoretical results have been derived from measurements on computed plots and allowance has also been made for the attenuative effect of the Perspex, these results have similarly been normalized. In Fig. 12 we show amplitude curves for vertical displacement recorded on the face of the plate at 66 cm above the scatterer; three different scatterers are considered and the curves show the contrast between doubling the width of a scatterer and doubling its thickness. The broken line in this diagram is a cosine curve modified to allow for attenuation along the wavepath, we thus see that the decay in the vertical displacement amplitude with horizontal position cannot solely be attributed to the increasing angle to the vertical made by the wavepath. Fig. 13 shows amplitude curves for the same three scatterers as before but now observed at the free surface, and it may be seen that the effect of the free surface has been to enhance the amplitude at greater distances from the centre line. From both these sets of plots we see that in general the

2

406

10”

W

e

u - ._ -

W >

0

E

u “ -

0 0 5 - - - -

B. L. N. Kennett

A A

- 8 A e A a

0 A

0 0 a A

A A I . 0 A 0 0 .

a

8

a . A

A 0

I 0

I A

O . A 0

0

8 0 0

0 8

0 0 0 8

0

0 = o 8

0 0

I I I I I I I I I I I0 20 30 40 50

Fro. 13. Comparison of relative amplitude variations at free surface (Symbols defined as for Fig. 12).

first-order theory somewhat underestimates the amplitude scattered at wide angles from the incident beam.

If however we compare the absolute magnitudes of the scattered waves in the reflected direction we find quite good agreement; for example for the scatterer in Fig. 8, experimentally the scattered wave amplitude was 0.08 of the incident amplitude and theoretically we find the ratio to be 0.073. The size of the scattered wavefield for this, the smallest of the scatterers considered shows how close we are to the limits on the use of a simple first-order approximation, which requires that the ratio of scattered field to incident field should be very much smaller than 1.

6. Conclusions We have seen above that the use of first-order scattering theory can provide good

qualitative agreement and fair quantitative agreement with experimental results even when used at the upper limit of its range of application.

The method can therefore be of considerable utility in providing an understanding of scattering processes. Though, in this paper, we have considered a uniform medium surrounding the scatterer, to isolate effects due to the scatterer alone; this formulation of the theory may be applied conveniently to scatterers in multilayered media by using conventional matrix methods (e.g. the Thomson-Haskell technique). Thus first-order scattering methods may be used to help improve seismic interpretation in the presence of scatterers.


Acknowledgment

I would like to thank Dr E. R. Lapwood and Dr P. N. S. O’Brien for advice and encouragement, and Mr M. P. Symes for assistance with the experimental work. I am grateful to the Chairman and Directors of the British Petroleum Company Limited for a Research Studentship and permission to publish this paper and to the B. P. Research Centre for provision of laboratory facilities.

Department of Applied Mathematics and Theoretical Physics, University of Cambridge,

Silver Street, Cambridge CB3 9E W

References

Aki, K., 1969. Coda of near earthquakes as back-scattering from non-uniform

Gilbert, F. & Knopoff, L., 1960. Seismic scattering from topographic irregularities,

Greenfield, R., 1971. Short period P wave generation by Rayleigh wave scattering

Healy, J. H. & Press, F., 1960. Two-dimensional seismic models with continuously

Kennett, B. L. N., 1972a. Seismic waves in laterally inhomogeneous media,

Kennett, B. L. N., 1972b. Seismic wave scattering by obstacles on interfaces,

Knopoff, L., Fredricks, R. W., Gangi, A. F. & Porter, L. D., 1957. Surface ampli-

Mack, H., 1969. Nature of short period P wave signal variations at LASA, J.

O’Brien, P. N. S. & Symes, M. P., 1971. Model Seismology, Rep. prog. Phys., 34,

structure, J. geophys. Res., 74, 615-631.

J. geophys. Res., 65, 3437-3444.

at Novaya Zemlya, J. geophys. Res., 76,7988-8002.

variable velocity-depth and density functions, Geophysics., 25, 987-997.

Ceophys. J. R. astr. SOC., 21, 301-325.

Geophys. J. R. astr. SOC., 28, 249-266.

tudes of reflected body waves, Geophysics, 22, 842-847.

geophys. Res., 74, 3161-3170.

697-764.

Appendix

Correction factors for free surface displacements In the theoretical development above (Section 2) we have considered the problem

of the scattered wave field generated by a scatterer in an infinite medium on which impinges a known incident wave. However, in order to make experimental comparisons we have to know the displacement which would be observed at a free surface.

If we consider the displacements due to a plane wave, with horizontal wave- number k and frequency w, on a plane z = const. in an infinite medium, the displacements expected at a notional free surface introduced at that plane may be calculated using plane wave reflection formulae. We find that for an incident P wave, the vertical displacement at a free surface is obtained from that in an infinite medium by multiplying by a factor


where K~~ = 02/j2 and v,, vg are defined as in Section 2. Similarly the horizontal displacement at the free surface may be found by multiplying the infinite medium displacement by

4V, Vg KpZ cz(k) = ____ - ..-__

4kZ v, vs+ ( K ~ ’ -2k2)’ ‘

For an incident SV wave, the appropriate factors are simply cz(k) for vertical displacements and c1 (k) for horizontal displacements. These factors differ from those given by Knopoff et al. (1957) by dealing with displacements rather than potentials.

We may thus allow for the effect of the free surface by applying the above correction factors to the displacement representations in the transform domain ii,,(k,z, o), iT,,(k,z, o) (G = P or S), before performing the Fourier inversion procedure to obtain the displacements.

b. l. n. kennett

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