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Modelling seismic reflections from D″ using the Kirchhoff method
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Physics of the Earth and Planetary Interiors 90 (1995) 273-281
ELSEVIER
Modelling seismic reflections from D" using the Kirchhoff method Jiirgen Neuberg
*, T i m P o i n t e r
Department of Earth Sciences, Unieersity of Leeds, Leeds LS2 9JT, UK
Received 30 October 1993; accepted 24 August 1994
Abstract
The observation of seismic reflections from the top of D" is restricted to the epicentral distance range near thc critical angle where the amplitude is large and a phase distortion occurs. This phase advance in the reflected wavelet is examined and implications for commonly used seismic processing techniques such as phase picking, stacking and deconvolution are made. Kirchhoff synthetics are generated 1o study several models of D" reflectors, such as steeply inclined interfaces and laterally varying velocity contrasts. Observations of strong amplitude variations in seismic reflections, the 'bounce points' of which lie well within a Fresnel zone, are successfully explained. In a search for reflections from D" investigators have exclusively used models of the D" layer parallel to the core-mantle boundary. We speculate about a steep slope as it may be produced by deposited slab material and model the influence of such features on travel-time and waveform.
I. Introduction
Several seismological investigations of the lower mantle and the c o r e - m a n t l e boundary (CMB) have contributed significantly to an interdisciplinary programme to study this area in detail. The transition zone right above the CMB, now commonly referred to as D" (Bullen, 1950), is of major importance for the understanding of the dynamic behaviour and the interaction between the core and the mantle. For example, electromagnetic coupling mechanisms across the CMB require a conducting layer at the base of the mantle. Likewise, the temperature contrast
* Corresponding author.
across the CMB could produce a boundary layer whose heterogeneities may be related to flow patterns on the core side and the formation of plumes on the mantle side. Furthermore, core flow anomalies and heterogeneities in D" may be linked to anomalous features in the Earth's magnetic dipole field. Single geophysical and mineralogical studies over the last few years have provided important but somewhat isolated clues, interpretations of which led to three distinct models of D": (1) a chemical boundary layer, (2) a phase transition zone, and (3) a thermal boundary layer. A discussion about these models and about the obvious possibility of combining them is given in Lay (1989). Seismological studies utilise two major tools to investigate a certain area:
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(95)05089-2
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
274
(1) travel-times of seismic phases provide information on the seismic velocity of a region penetrated by a seismic wave and therefore, on related physical properties such as density, temperature and anisotropy. In the case of an interface the travel-time is essential for the determination of the interface level; (2) waveforms of seismic phases bouncing off an interface reveal information about its character such as dip, extent and impedance contrast, and for phases propagating through a medium the waveform carries information on scattering mechanisms which can cause dispersion and damping. Both tools have been extensively applied to the D" layer and the CMB in investigations of global extent (e.g. Lay and Helmberger, 1983; Doornbos and Hilton, 1989) and on a regional scale (e.g. Weber, 1993; Neuberg and Wahr, 1991). These investigations have given evidence towards the existence of a seismic reflector at the top of D", whose level and sharpness exhibit strong lateral variations. In some regions such a reflector has only been detected by using S waves (SdS) rather than P waves (PdP), and vice versa. In some areas it cannot be identified at all (Houard and Natal, 1993). One motivation for this study was provided by this surprisingly strong variation of detected seismic reflectors as described by Weber and Davis (a)
(b)
P-velocity contrast = 1.0, 2.0 & 3.0% from PWDK
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(1990) and Weber (1993), and more recently by Kendall and Shearer (1994). These studies present maps of D" reflectors showing regions where reflected phases (PdP and SdS) have been identified and also regions where no reflection could be detected. The puzzling fact is, however, that the so-called 'bounce points' of seismic rays showing and not showing a reflection are located within the same Fresnel zone; the area that constructively contributes to the reflected phase. Therefore, the size of the Fresnel zone restricts the spatial resolution of identifiable areas with or without a reflector. The second topic of this paper deals with steeply inclined reflectors and was stimulated by Christensen (1987) who suggested from geodynamical modelling that piles of slab material may be deposited at the CMB. Such a feature could imply steep and laterally varying reflectors in the lower mantle. All of the models proposing a reflection from the top of D" have been restricted so far to single 'footprints' on a spherical interface even if adjacent 'footprints' have been on significantly different reflector levels. Here we allow for topography. In the next section we explain why D" reflections can be more easily detected at an epicentral distance larger than 70° and comment on some interesting implications of a phase shift in the seismic wavelet to commonly applied processing
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J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281 i
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techniques such as beam-forming and phase picking. Section 3 is dedicated to the application of Kirchhoff synthetics, our major modelling tool, and the analysis of the Fresnel zone. There we address the problem of adjacent bounce points with varying reflectivity. Potential features of D" reflectors such as a steep slope and a laterally varying impedance contrast are described and modelled in Section 4. In the last section we conclude with a short discussion and suggest future work.
2. Wide angle reflections from D" The amplitude and therefore, the possible detection of PdP and SdS is controlled by the respective reflection coefficient of the interface. Lay and co-workers (e.g. Lay and Helmberger,
275
1983) proposed an S-velocity contrast of 2.8% for a reflector at the top of D". Subsequently, Weber and Davis (1990) used P- and S-velocity contrasts of 3% and 2% in their respective D" models P W D K and SWDK. The physical interdependence between the P- and S-velocity contrasts under pressure and temperature conditions prevalent in D" is still an open question but it is most likely that one constrains the other (Li et al., 1991). Fig. 1 shows the amplitude of the reflection coefficient versus incidence angle for PdP and SdS (SH) and a velocity jump ranging from 1 to 3% which covers all existing models. The steep amplitude increase between 70 ° and 75 ° demonstrates that the favourable epicentral distance range lies, depending on the earth model and the focal depth beyond approximately 70 ° and 60 ° for P and S waves, respectively. This explains why almost all observations of PdP and SdS have been made in this distance range. The steep increase of the reflection coefficient near the critical angle indicates the high sensitivity of the amplitude to the angle of incidence and therefore, to any inclination of the reflector. This point will be further examined when different reflector models are described in Section 4. Owing to the existence of inhomogeneous waves beyond the critical angle the reflection coefficient becomes complex and results in a phase advance of the reflected wave with respect to the incident wave. Fig. 2 shows the phase distortion for an example of PdP generated by the Kirchhoff method for a spherical boundary 290 km above the CMB and a P velocity increase of 3%. For longer epicentral distances, depending on the velocity gradient beneath the interface, the diving waves (PdP, PDDP etc.) are superimposed. Fig. 3 shows this part of the travel-time triplication for the model PWDK. We restrict our investigation to reflections only, i.e. we model the seismic wave in an epicentral distance range where the reflected phase (PdP) is dominant. A phase shift such as the phase advance described above has important implications for any method that uses a reference wavelet for the detection and identification of seismic phases. This is obvious for straightforward phase picking where a high amplitude section of a wavelet pop-
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
276
PWDK model, source depth = 100 km
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ping out of the noise is correlated without allowing for the phase shift. But even more adequate techniques like matched filters (Schlittenhardt, 1986, Neuberg and Wahr, 1991) or phase stripping (Kendall and Shearer, 1994) which work properly for a wave with an undisturbed phase spectrum such as PcP or subcritical PdP would suffer from a phase difference between the wavelet being analysed and the reference wavelet unless the phase shift is taken into account. Depending on the wavelength of the seismic signal under consideration this leads to a systematic bias towards earlier or later arrivals. For a shortperiod (1 Hz) PdP wavelet the error can be as large as +0.5 s and for a long-period (10 s) SdS wavelet + 5 s. This leads to an equally systematic bias in the determination of the reflector level of + 15 to + 50 kin. Even though this might not be considered significant in view of other error sources in travel-time determination, the phase advance can affect the determination of amplitudes as well. As PdP and SdS are very weak phases, several authors have applied array techniques to enhance the signal-to-noise ratio. Stacking and beam-forming methods are likewise affected by a phase advanced wavelet if the aperture o f the seismic array or network covers an epicentral distance
range over which the waveform changes significantly. Weber and Davis (1990) use the Gr~ifenberg Array with an aperture of only 1° with respect to their source-receiver constellation and do not face this problem when applying beamforming methods to their data. However, other investigations (e.g. Houard and Nataf, 1993; Shibutani et al., 1994) use networks covering a larger distance range of 8 ° to 10° and possibly destroy their signal by stacking differently phaseshifted seismic wavelets rather than enhance it. An appropriate way would be to stack subsets of seismic traces recorded over a smaller epicentral distance range, and subsequently to account for the phase advance. Although the phase shift seems to be only an obstacle for seismic data processing it can also be used to further constrain the reflecting interface. Rather than using only the amplitude of a wavelet to characterise the boundary, incorporating the phase provides the additional constraint for the determination of both the impedance contrast and the angle of incidence. This is used in a deconvolution scheme described in Pointer and Neuberg (1995) where the incident P wave is phase advanced before it is used as a reference wavelet to retrieve the differential travel-time and the amplitude ratio. 3. Kirchhoff synthetics and the effective Fresnel zone
Kirchhoff synthetics are generated according to Huygens' principle where a wavefield is represented by the superposition of elementary sources located at any surface in the confined volume under consideration. The Kirchhoff method has often been applied to those seismological problems that involve the modelling of seismic waves interacting with an interface of irregular topography. For a full description of the method see, e.g. Scott and Helmberger (1983). We used the formulation corresponding to Neuberg and Wahr (1991) in the frequency domain 1 c R(~b) u(r, to)=~--~Jso ll' itof( ~o)
~ l+l') dS
× (cos 4' + cos 4") exp/lw---77-,
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
where the scalar function u is a component of displacement, f(o)) the Fourier transform of an arbitrary incident wavelet, l and l' the ray paths through a homogeneous mantle with velocity V, ~b the angle of incidence and R(~b) the complex reflection coefficient. The numerical integration is carried out over an area of S O= 70 ° x 40 ° representing even for very large epicentral distances at least one Fresnel zone at the D" reflector for a spherical earth. For a 1 Hz wavelet and an epicentral distance range between 83 ° and 101 ° the Fresnel zone ranges from, approximately, 600 × 300 to 1250 x 350 km. The epicentral distances for the homogeneous mantle are determined in such a way that the seismic rays through the homogeneous mantle match the angle of incidence of rays propagating through the more realistic P W D K earth model. The complex reflection coefficient R(~b) plays the key role in generating the reflected wavefield. It follows from its amplitude behaviour (Fig. 1) that the contributions to the reflected wavefield beyond the critical angle of incidence are higher than for smaller angles. Hence, the complex reflection coefficient represents a weighting function over the integration area, i.e. the Fresnel zone. In the case where the integration area contains rough topography the concept of a confined Fresnel zone breaks down as the contributions to the reflected wavefield emerge from isolated patches of piecewise smooth reflectors. But even for a completely smooth reflector with the impedance characteristic of D" the part of the Fresnel zone which we refer to as the effective Fresnel zone yielding the major contribution to the seismic reflection is by far smaller than the classical Fresnel zone defined by the limits of constructive interference. Its size is determined by the magnitude of the complex reflection coefficient together with the source-receiver geometry. The resolution of any seismic investigation is restricted by the size of the Fresnel zone even if the seismic ray concept and corresponding 'bounce points' suggest an unlimited one. A smaller, effective Fresnel zone, however, results in a higher resolution and furthermore, in the separation of 'footprints' on different reflector
277
levels situated within one classical Fresnel zone. This is further explored in the presence of lateral inhomogeneities.
4. Potential D" features Three different representations of lateral inhomogeneities in D" are considered in detail: (1) a laterally varying impedance contrast on a constant reflector level, (2) a sharp edge or cliff, and (3) a steeply inclined reflector linking two different reflector levels. When using Kirchhoff synthetics to generate the seismic reflections from these features we need to restrict the epicentral distance to a range where the reflection provides 3pc 104 103
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278
J. Neuberg, T. Pointer~Physics of the Earth and Planetary Interiors 90 (1995) 273-281
the dominant part of the wavefield rather than a larger distance where other seismic phases of the travel-time triplication such as the head and diving wave could be superimposed. For a constant velocity or a negative velocity gradient beneath the interface the reflection remains dominant over the entire distance range. The simplest case of a lateral heterogeneity at the top of D" is a varying density or seismic velocity contrast as it could be produced by a lateral change in chemical composition or temperature. The impedance contrast has been increased from 0 to 3% over a lateral transition distance of 200 km while the reflector level has been kept constant. Fig. 4 shows the reflected wavelets. They exhibit a remarkably strong amplitude variation over a distance which is short compared with the size of a Fresnel zone. This is due to the effective Fresnel zone as discussed above and is also reinforced by the simultaneous change in reflectivity. Seismic noise in real data further disguises weak phases and observations could be made that show a reflection from one area and none from an adjacent one, even though the Fresnel zones are overlapping. In this way a patchy distribution of reflectors and a lack of those within confined areas could be explained by the rapidly increasing reflection coefficient whose effect on the amplitude is even enhanced by a laterally varying impedance contrast. Such a feature is not unlikely for a heterogeneous boundary layer like D". The model of a sharp edge or cliff is from a seismological point of view the extreme case of a varying impedance contrast where the distance of lateral transition is reduced to zero. Such an abrupt change in chemical composition or reflector level (rather than t e m p e r a t u r e change) would produce a similarly sharp variation in the amplitude of the reflected wavefield. The striking point is, however, that such an abrupt lateral change is not necessary to explain the seismic observations. Therefore, one does not need to use a somewhat unrealistic model when a smoother and hence, more realistic lateral transition produces the same effect as demonstrated above. Steeply inclined reflectors a t the base of the mantle could be an indication for deposited slab
material forming the heterogeneous D" layer where the seismic impedance contrast represents an interface between two completely different chemical compositions. The emphasis here is on the orientation of the interface and its interaction with an incident seismic wave. Several models of reflector topography linking two horizontal levels are depicted in Fig. 5. The difference in elevation
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J. Neuberg, T. Pointer/Physics o]"the Earth and Planetary, Interiors 90 (1995) 273-281
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between the two levels and the transition range are the two parameters that control the curvature of the intermediate steeply inclined reflector. Fig. 6 shows the corresponding set of reflections. As the epicentral distance increases and the effective Fresnel zone moves over this feature, a second seismic wavelet with a different phase, amplitude and travel-time emerges. The differences in travel-time and phase correspond to the different reflector levels. The waveform of both wavelets is again determined by the change of reflection coefficient with epicentral distance but is also strongly affected by focusing and defocusing according to the curvature of the reflector linking both levels. The inverse problem to obtain the location of the reflector from seismic traveltimes is a classical migration problem and well established in exploration seismology. However, the data coverage needed to carry out such an inversion cannot be achieved yet in global seismology. So far only horizontal reflectors, or spherical reflectors in the case of a global earth model, have been considered where the calculation of travel-times for a given earth model is straightforward. It is essential for the detection of steeply inclined reflectors however, to allow for the possibility of two wavelets rather than one in a time window which is big enough to cover an entire depth range in D". Weber and K6rnig (1992) searched seismic bulletins for additional seismic arrivals between P and PcP and identified regions of D" where many additional seismic phases have been reported. If some of these anomalous phases were caused by steep reflectors but were interpreted as being caused by spherical interfaces the corresponding reflection amplitudes and levels would naturally show a large scatter.
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Fig. 6. Reflected wavelets for the models in Fig. 5(a)-(c). The epicentral distance is given for a homogeneous mantle (left axis, 83°-101 °) and the corresponding angle of incidence is also indicated (right axis). This corresponds to an epicentral distance range from 76 ° to 86 ° for PWDK. The two wavelets originate from the two reflector levels; their waveforms differ owing to different phase advances. The additional late arrival on the three lower traces is an artefact of the Kirchhoff method.
280
J. Neuberg, 1". Pointer~Physics of the Earth and Planetary Interiors 90 (1995) 273-281
The increasing number and quality of observations might enable us in future to look for seismic reflectors of any kind in a way similar to the migration problem in exploration seismology, including amplitude and phase information. For now, the approach could be to study D" in an area such as beneath northern Siberia where the data coverage for D" reflections is sufficiently good; start at a well-identified reflector and work through several Fresnel zones towards areas without a reflector while searching for phase changes, amplitude variations and secondary seismic arrivals.
5. Conclusions A phase advance affecting overcritical reflections from the top of D" has certain implications for commonly applied seismological methods. (1) Stacking of seismic traces for the enhancement of the signal-to-noise ratio should only be carried out over a short epicentral distance where the phase advance is negligible. Stacking over a longer range in the presence of a phase shift spoils the signal rather than enhances it. (2) Travel-time determinations made by phase picking or cross-correlation methods need to allow for the phase advance in order to avoid a systematic bias which would otherwise lead to a systematic error in obtaining reflector levels. When applying cross-correlation techniques both the travel-time and the amplitude determination would be affected if the phase shift were ignored. (3) The phase advance also carries valuable information about the complex reflection coefficient and can be used to place an additional constraint on the determination of a reflector model. The amplitude of the reflection coefficient for commonly accepted models of D" acts as a weighting function over the area which constructively contributes to the reflected wave field and reduces it to a smaller effective Fresnel zone. By using Kirchhoff synthetics a set of reflected waves from a laterally varying impedance contrast has been generated. The combination of such a feature and the reduced size of the Fresnel zone
increases the spatial resolution of the reflected wavefield. In this way we explain strong amplitude variations from reflections originating from adjacent areas within one Fresnel zone. Steeply inclined reflectors produce secondary wavelets with strong amplitude variations and phase distortions which may be difficult to identify if a detection procedure is set up for a single horizontal (or spherical) reflector. If such a wavelet is found in seismic data the interpretation as a horizontal reflector will result in a strong amplitude and travel-time scatter.
Acknowledgements The authors are grateful to Michael Weber for helpful discussions, and to Michael Kendall and Peter Shearer for providing a preprint of their paper. Leo Lisapaly is thanked for providing some of the figures and checking the Kirchhoff code thoroughly. Constructive comments by Michael Kendall and Hanneke Paulssen are gratefully acknowledged.
References Bullen, K.E., 1950. An earth model based on a compressibility-pressure hypothesis. Mon. Not. R. Astron. Soc., Geophys. Suppl., 6: 50-59. Christensen, U.R., 1987. Formation and dynamics of a compositional boundary layer at the base of the mantle. Eos, Trans. Am. Geophys. Union, 68: 1488. Doornbos, D.J. and Hilton, T., 1989. Models of the coremantle boundary and the travel times of internally reflected core phases. J. Geophys. Res., 94: 15741-15751. Houard, S. and Nataf, H.-C., 1993. Laterally varying reflector at the top of D" beneath northern Siberia. Geophys. J. Int., 115: 168-182. Kendall J.M. and Shearer, P.M., 1994. Lateral variations in D" thickness from long-period shear wave data. J. Geophys. Res., 99: 11575-11590. Lay, T., 1989. Structure of the core-mantle transition zone: a chemical and thermal boundary layer. Eos, Trans. Am. Geophys. Union, 70: p. 49. Lay, T. and Helmberger, D.V., 1983. A lower mantle S-wave triplication and the shear velocity structure of D". Geophys. J. R. Astron. Soc,, 75: 799-837. Li, X.-D., Giardini, D. and Woodhouse, J.H., 1991. The
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281 relative amplitudes of mantle heterogeneity in P velocity, S velocity and density from free-oscillation data. Geophys. J. Int., 105: 649-657. Neuberg, J. and Wahr, J., 1991. Detailed investigation of a spot on the core-mantle boundary using digital PcP data. Phys. Earth Planet. Inter., 68: 132-143. Pointer, T. and Neuberg, J., 1995. Deconvolution of overcritical P-wave reflections from the top of D". Phys. Earth Planet. Inter., 90: 283-301. Schlittenhardt, J., 1986. Investigation of the velocity- and Q-structure of the lowermost mantle using P c P / P amplitude ratios from arrays at distance of 700-84 °. J. Geophys., 60: 1-8. Scott, P. and Helmberger, D.V., 1983. Applications of the
281
Kirchhoff-Helmholtz integral to problems in seismology. Geophys J.R. Astron. Soc., 72: 237-254. Shibutani, Tanaka, T., A., Kato, M. and Hirahara, K., 1994. A study of P-wave velocity discontinuity in D" layer with J-array records: preliminary results. J. Geomag. Geoect., 45: 1275-1285. Weber, M., 1993. P and S wave reflections from anomalies in the lowermost mantle. Geophys. J. Int., 115: 83-210. Weber, M. and Davis, J.P., 1990. Evidence of a laterally variable lower mantle structure from P- and S-waves, Geophys. J. Int., 102: 231-255. Weber, M. and K6rnig, M., 1992. A search for anomalies in the lowermost mantle using seismic bulletins. Phys. Earth Planet. Inter., 73: 1-28.
Physics of the Earth and Planetary Interiors 90 (1995) 273-281
ELSEVIER
Modelling seismic reflections from D" using the Kirchhoff method Jiirgen Neuberg
*, T i m P o i n t e r
Department of Earth Sciences, Unieersity of Leeds, Leeds LS2 9JT, UK
Received 30 October 1993; accepted 24 August 1994
Abstract
The observation of seismic reflections from the top of D" is restricted to the epicentral distance range near thc critical angle where the amplitude is large and a phase distortion occurs. This phase advance in the reflected wavelet is examined and implications for commonly used seismic processing techniques such as phase picking, stacking and deconvolution are made. Kirchhoff synthetics are generated 1o study several models of D" reflectors, such as steeply inclined interfaces and laterally varying velocity contrasts. Observations of strong amplitude variations in seismic reflections, the 'bounce points' of which lie well within a Fresnel zone, are successfully explained. In a search for reflections from D" investigators have exclusively used models of the D" layer parallel to the core-mantle boundary. We speculate about a steep slope as it may be produced by deposited slab material and model the influence of such features on travel-time and waveform.
I. Introduction
Several seismological investigations of the lower mantle and the c o r e - m a n t l e boundary (CMB) have contributed significantly to an interdisciplinary programme to study this area in detail. The transition zone right above the CMB, now commonly referred to as D" (Bullen, 1950), is of major importance for the understanding of the dynamic behaviour and the interaction between the core and the mantle. For example, electromagnetic coupling mechanisms across the CMB require a conducting layer at the base of the mantle. Likewise, the temperature contrast
* Corresponding author.
across the CMB could produce a boundary layer whose heterogeneities may be related to flow patterns on the core side and the formation of plumes on the mantle side. Furthermore, core flow anomalies and heterogeneities in D" may be linked to anomalous features in the Earth's magnetic dipole field. Single geophysical and mineralogical studies over the last few years have provided important but somewhat isolated clues, interpretations of which led to three distinct models of D": (1) a chemical boundary layer, (2) a phase transition zone, and (3) a thermal boundary layer. A discussion about these models and about the obvious possibility of combining them is given in Lay (1989). Seismological studies utilise two major tools to investigate a certain area:
0031-9201/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0031-9201(95)05089-2
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
274
(1) travel-times of seismic phases provide information on the seismic velocity of a region penetrated by a seismic wave and therefore, on related physical properties such as density, temperature and anisotropy. In the case of an interface the travel-time is essential for the determination of the interface level; (2) waveforms of seismic phases bouncing off an interface reveal information about its character such as dip, extent and impedance contrast, and for phases propagating through a medium the waveform carries information on scattering mechanisms which can cause dispersion and damping. Both tools have been extensively applied to the D" layer and the CMB in investigations of global extent (e.g. Lay and Helmberger, 1983; Doornbos and Hilton, 1989) and on a regional scale (e.g. Weber, 1993; Neuberg and Wahr, 1991). These investigations have given evidence towards the existence of a seismic reflector at the top of D", whose level and sharpness exhibit strong lateral variations. In some regions such a reflector has only been detected by using S waves (SdS) rather than P waves (PdP), and vice versa. In some areas it cannot be identified at all (Houard and Natal, 1993). One motivation for this study was provided by this surprisingly strong variation of detected seismic reflectors as described by Weber and Davis (a)
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(1990) and Weber (1993), and more recently by Kendall and Shearer (1994). These studies present maps of D" reflectors showing regions where reflected phases (PdP and SdS) have been identified and also regions where no reflection could be detected. The puzzling fact is, however, that the so-called 'bounce points' of seismic rays showing and not showing a reflection are located within the same Fresnel zone; the area that constructively contributes to the reflected phase. Therefore, the size of the Fresnel zone restricts the spatial resolution of identifiable areas with or without a reflector. The second topic of this paper deals with steeply inclined reflectors and was stimulated by Christensen (1987) who suggested from geodynamical modelling that piles of slab material may be deposited at the CMB. Such a feature could imply steep and laterally varying reflectors in the lower mantle. All of the models proposing a reflection from the top of D" have been restricted so far to single 'footprints' on a spherical interface even if adjacent 'footprints' have been on significantly different reflector levels. Here we allow for topography. In the next section we explain why D" reflections can be more easily detected at an epicentral distance larger than 70° and comment on some interesting implications of a phase shift in the seismic wavelet to commonly applied processing
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90
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techniques such as beam-forming and phase picking. Section 3 is dedicated to the application of Kirchhoff synthetics, our major modelling tool, and the analysis of the Fresnel zone. There we address the problem of adjacent bounce points with varying reflectivity. Potential features of D" reflectors such as a steep slope and a laterally varying impedance contrast are described and modelled in Section 4. In the last section we conclude with a short discussion and suggest future work.
2. Wide angle reflections from D" The amplitude and therefore, the possible detection of PdP and SdS is controlled by the respective reflection coefficient of the interface. Lay and co-workers (e.g. Lay and Helmberger,
275
1983) proposed an S-velocity contrast of 2.8% for a reflector at the top of D". Subsequently, Weber and Davis (1990) used P- and S-velocity contrasts of 3% and 2% in their respective D" models P W D K and SWDK. The physical interdependence between the P- and S-velocity contrasts under pressure and temperature conditions prevalent in D" is still an open question but it is most likely that one constrains the other (Li et al., 1991). Fig. 1 shows the amplitude of the reflection coefficient versus incidence angle for PdP and SdS (SH) and a velocity jump ranging from 1 to 3% which covers all existing models. The steep amplitude increase between 70 ° and 75 ° demonstrates that the favourable epicentral distance range lies, depending on the earth model and the focal depth beyond approximately 70 ° and 60 ° for P and S waves, respectively. This explains why almost all observations of PdP and SdS have been made in this distance range. The steep increase of the reflection coefficient near the critical angle indicates the high sensitivity of the amplitude to the angle of incidence and therefore, to any inclination of the reflector. This point will be further examined when different reflector models are described in Section 4. Owing to the existence of inhomogeneous waves beyond the critical angle the reflection coefficient becomes complex and results in a phase advance of the reflected wave with respect to the incident wave. Fig. 2 shows the phase distortion for an example of PdP generated by the Kirchhoff method for a spherical boundary 290 km above the CMB and a P velocity increase of 3%. For longer epicentral distances, depending on the velocity gradient beneath the interface, the diving waves (PdP, PDDP etc.) are superimposed. Fig. 3 shows this part of the travel-time triplication for the model PWDK. We restrict our investigation to reflections only, i.e. we model the seismic wave in an epicentral distance range where the reflected phase (PdP) is dominant. A phase shift such as the phase advance described above has important implications for any method that uses a reference wavelet for the detection and identification of seismic phases. This is obvious for straightforward phase picking where a high amplitude section of a wavelet pop-
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
276
PWDK model, source depth = 100 km
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ping out of the noise is correlated without allowing for the phase shift. But even more adequate techniques like matched filters (Schlittenhardt, 1986, Neuberg and Wahr, 1991) or phase stripping (Kendall and Shearer, 1994) which work properly for a wave with an undisturbed phase spectrum such as PcP or subcritical PdP would suffer from a phase difference between the wavelet being analysed and the reference wavelet unless the phase shift is taken into account. Depending on the wavelength of the seismic signal under consideration this leads to a systematic bias towards earlier or later arrivals. For a shortperiod (1 Hz) PdP wavelet the error can be as large as +0.5 s and for a long-period (10 s) SdS wavelet + 5 s. This leads to an equally systematic bias in the determination of the reflector level of + 15 to + 50 kin. Even though this might not be considered significant in view of other error sources in travel-time determination, the phase advance can affect the determination of amplitudes as well. As PdP and SdS are very weak phases, several authors have applied array techniques to enhance the signal-to-noise ratio. Stacking and beam-forming methods are likewise affected by a phase advanced wavelet if the aperture o f the seismic array or network covers an epicentral distance
range over which the waveform changes significantly. Weber and Davis (1990) use the Gr~ifenberg Array with an aperture of only 1° with respect to their source-receiver constellation and do not face this problem when applying beamforming methods to their data. However, other investigations (e.g. Houard and Nataf, 1993; Shibutani et al., 1994) use networks covering a larger distance range of 8 ° to 10° and possibly destroy their signal by stacking differently phaseshifted seismic wavelets rather than enhance it. An appropriate way would be to stack subsets of seismic traces recorded over a smaller epicentral distance range, and subsequently to account for the phase advance. Although the phase shift seems to be only an obstacle for seismic data processing it can also be used to further constrain the reflecting interface. Rather than using only the amplitude of a wavelet to characterise the boundary, incorporating the phase provides the additional constraint for the determination of both the impedance contrast and the angle of incidence. This is used in a deconvolution scheme described in Pointer and Neuberg (1995) where the incident P wave is phase advanced before it is used as a reference wavelet to retrieve the differential travel-time and the amplitude ratio. 3. Kirchhoff synthetics and the effective Fresnel zone
Kirchhoff synthetics are generated according to Huygens' principle where a wavefield is represented by the superposition of elementary sources located at any surface in the confined volume under consideration. The Kirchhoff method has often been applied to those seismological problems that involve the modelling of seismic waves interacting with an interface of irregular topography. For a full description of the method see, e.g. Scott and Helmberger (1983). We used the formulation corresponding to Neuberg and Wahr (1991) in the frequency domain 1 c R(~b) u(r, to)=~--~Jso ll' itof( ~o)
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J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281
where the scalar function u is a component of displacement, f(o)) the Fourier transform of an arbitrary incident wavelet, l and l' the ray paths through a homogeneous mantle with velocity V, ~b the angle of incidence and R(~b) the complex reflection coefficient. The numerical integration is carried out over an area of S O= 70 ° x 40 ° representing even for very large epicentral distances at least one Fresnel zone at the D" reflector for a spherical earth. For a 1 Hz wavelet and an epicentral distance range between 83 ° and 101 ° the Fresnel zone ranges from, approximately, 600 × 300 to 1250 x 350 km. The epicentral distances for the homogeneous mantle are determined in such a way that the seismic rays through the homogeneous mantle match the angle of incidence of rays propagating through the more realistic P W D K earth model. The complex reflection coefficient R(~b) plays the key role in generating the reflected wavefield. It follows from its amplitude behaviour (Fig. 1) that the contributions to the reflected wavefield beyond the critical angle of incidence are higher than for smaller angles. Hence, the complex reflection coefficient represents a weighting function over the integration area, i.e. the Fresnel zone. In the case where the integration area contains rough topography the concept of a confined Fresnel zone breaks down as the contributions to the reflected wavefield emerge from isolated patches of piecewise smooth reflectors. But even for a completely smooth reflector with the impedance characteristic of D" the part of the Fresnel zone which we refer to as the effective Fresnel zone yielding the major contribution to the seismic reflection is by far smaller than the classical Fresnel zone defined by the limits of constructive interference. Its size is determined by the magnitude of the complex reflection coefficient together with the source-receiver geometry. The resolution of any seismic investigation is restricted by the size of the Fresnel zone even if the seismic ray concept and corresponding 'bounce points' suggest an unlimited one. A smaller, effective Fresnel zone, however, results in a higher resolution and furthermore, in the separation of 'footprints' on different reflector
277
levels situated within one classical Fresnel zone. This is further explored in the presence of lateral inhomogeneities.
4. Potential D" features Three different representations of lateral inhomogeneities in D" are considered in detail: (1) a laterally varying impedance contrast on a constant reflector level, (2) a sharp edge or cliff, and (3) a steeply inclined reflector linking two different reflector levels. When using Kirchhoff synthetics to generate the seismic reflections from these features we need to restrict the epicentral distance to a range where the reflection provides 3pc 104 103
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278
J. Neuberg, T. Pointer~Physics of the Earth and Planetary Interiors 90 (1995) 273-281
the dominant part of the wavefield rather than a larger distance where other seismic phases of the travel-time triplication such as the head and diving wave could be superimposed. For a constant velocity or a negative velocity gradient beneath the interface the reflection remains dominant over the entire distance range. The simplest case of a lateral heterogeneity at the top of D" is a varying density or seismic velocity contrast as it could be produced by a lateral change in chemical composition or temperature. The impedance contrast has been increased from 0 to 3% over a lateral transition distance of 200 km while the reflector level has been kept constant. Fig. 4 shows the reflected wavelets. They exhibit a remarkably strong amplitude variation over a distance which is short compared with the size of a Fresnel zone. This is due to the effective Fresnel zone as discussed above and is also reinforced by the simultaneous change in reflectivity. Seismic noise in real data further disguises weak phases and observations could be made that show a reflection from one area and none from an adjacent one, even though the Fresnel zones are overlapping. In this way a patchy distribution of reflectors and a lack of those within confined areas could be explained by the rapidly increasing reflection coefficient whose effect on the amplitude is even enhanced by a laterally varying impedance contrast. Such a feature is not unlikely for a heterogeneous boundary layer like D". The model of a sharp edge or cliff is from a seismological point of view the extreme case of a varying impedance contrast where the distance of lateral transition is reduced to zero. Such an abrupt change in chemical composition or reflector level (rather than t e m p e r a t u r e change) would produce a similarly sharp variation in the amplitude of the reflected wavefield. The striking point is, however, that such an abrupt lateral change is not necessary to explain the seismic observations. Therefore, one does not need to use a somewhat unrealistic model when a smoother and hence, more realistic lateral transition produces the same effect as demonstrated above. Steeply inclined reflectors a t the base of the mantle could be an indication for deposited slab
material forming the heterogeneous D" layer where the seismic impedance contrast represents an interface between two completely different chemical compositions. The emphasis here is on the orientation of the interface and its interaction with an incident seismic wave. Several models of reflector topography linking two horizontal levels are depicted in Fig. 5. The difference in elevation
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J. Neuberg, T. Pointer/Physics o]"the Earth and Planetary, Interiors 90 (1995) 273-281
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between the two levels and the transition range are the two parameters that control the curvature of the intermediate steeply inclined reflector. Fig. 6 shows the corresponding set of reflections. As the epicentral distance increases and the effective Fresnel zone moves over this feature, a second seismic wavelet with a different phase, amplitude and travel-time emerges. The differences in travel-time and phase correspond to the different reflector levels. The waveform of both wavelets is again determined by the change of reflection coefficient with epicentral distance but is also strongly affected by focusing and defocusing according to the curvature of the reflector linking both levels. The inverse problem to obtain the location of the reflector from seismic traveltimes is a classical migration problem and well established in exploration seismology. However, the data coverage needed to carry out such an inversion cannot be achieved yet in global seismology. So far only horizontal reflectors, or spherical reflectors in the case of a global earth model, have been considered where the calculation of travel-times for a given earth model is straightforward. It is essential for the detection of steeply inclined reflectors however, to allow for the possibility of two wavelets rather than one in a time window which is big enough to cover an entire depth range in D". Weber and K6rnig (1992) searched seismic bulletins for additional seismic arrivals between P and PcP and identified regions of D" where many additional seismic phases have been reported. If some of these anomalous phases were caused by steep reflectors but were interpreted as being caused by spherical interfaces the corresponding reflection amplitudes and levels would naturally show a large scatter.
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280
J. Neuberg, 1". Pointer~Physics of the Earth and Planetary Interiors 90 (1995) 273-281
The increasing number and quality of observations might enable us in future to look for seismic reflectors of any kind in a way similar to the migration problem in exploration seismology, including amplitude and phase information. For now, the approach could be to study D" in an area such as beneath northern Siberia where the data coverage for D" reflections is sufficiently good; start at a well-identified reflector and work through several Fresnel zones towards areas without a reflector while searching for phase changes, amplitude variations and secondary seismic arrivals.
5. Conclusions A phase advance affecting overcritical reflections from the top of D" has certain implications for commonly applied seismological methods. (1) Stacking of seismic traces for the enhancement of the signal-to-noise ratio should only be carried out over a short epicentral distance where the phase advance is negligible. Stacking over a longer range in the presence of a phase shift spoils the signal rather than enhances it. (2) Travel-time determinations made by phase picking or cross-correlation methods need to allow for the phase advance in order to avoid a systematic bias which would otherwise lead to a systematic error in obtaining reflector levels. When applying cross-correlation techniques both the travel-time and the amplitude determination would be affected if the phase shift were ignored. (3) The phase advance also carries valuable information about the complex reflection coefficient and can be used to place an additional constraint on the determination of a reflector model. The amplitude of the reflection coefficient for commonly accepted models of D" acts as a weighting function over the area which constructively contributes to the reflected wave field and reduces it to a smaller effective Fresnel zone. By using Kirchhoff synthetics a set of reflected waves from a laterally varying impedance contrast has been generated. The combination of such a feature and the reduced size of the Fresnel zone
increases the spatial resolution of the reflected wavefield. In this way we explain strong amplitude variations from reflections originating from adjacent areas within one Fresnel zone. Steeply inclined reflectors produce secondary wavelets with strong amplitude variations and phase distortions which may be difficult to identify if a detection procedure is set up for a single horizontal (or spherical) reflector. If such a wavelet is found in seismic data the interpretation as a horizontal reflector will result in a strong amplitude and travel-time scatter.
Acknowledgements The authors are grateful to Michael Weber for helpful discussions, and to Michael Kendall and Peter Shearer for providing a preprint of their paper. Leo Lisapaly is thanked for providing some of the figures and checking the Kirchhoff code thoroughly. Constructive comments by Michael Kendall and Hanneke Paulssen are gratefully acknowledged.
References Bullen, K.E., 1950. An earth model based on a compressibility-pressure hypothesis. Mon. Not. R. Astron. Soc., Geophys. Suppl., 6: 50-59. Christensen, U.R., 1987. Formation and dynamics of a compositional boundary layer at the base of the mantle. Eos, Trans. Am. Geophys. Union, 68: 1488. Doornbos, D.J. and Hilton, T., 1989. Models of the coremantle boundary and the travel times of internally reflected core phases. J. Geophys. Res., 94: 15741-15751. Houard, S. and Nataf, H.-C., 1993. Laterally varying reflector at the top of D" beneath northern Siberia. Geophys. J. Int., 115: 168-182. Kendall J.M. and Shearer, P.M., 1994. Lateral variations in D" thickness from long-period shear wave data. J. Geophys. Res., 99: 11575-11590. Lay, T., 1989. Structure of the core-mantle transition zone: a chemical and thermal boundary layer. Eos, Trans. Am. Geophys. Union, 70: p. 49. Lay, T. and Helmberger, D.V., 1983. A lower mantle S-wave triplication and the shear velocity structure of D". Geophys. J. R. Astron. Soc,, 75: 799-837. Li, X.-D., Giardini, D. and Woodhouse, J.H., 1991. The
J. Neuberg, T. Pointer/Physics of the Earth and Planetary Interiors 90 (1995) 273-281 relative amplitudes of mantle heterogeneity in P velocity, S velocity and density from free-oscillation data. Geophys. J. Int., 105: 649-657. Neuberg, J. and Wahr, J., 1991. Detailed investigation of a spot on the core-mantle boundary using digital PcP data. Phys. Earth Planet. Inter., 68: 132-143. Pointer, T. and Neuberg, J., 1995. Deconvolution of overcritical P-wave reflections from the top of D". Phys. Earth Planet. Inter., 90: 283-301. Schlittenhardt, J., 1986. Investigation of the velocity- and Q-structure of the lowermost mantle using P c P / P amplitude ratios from arrays at distance of 700-84 °. J. Geophys., 60: 1-8. Scott, P. and Helmberger, D.V., 1983. Applications of the
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Kirchhoff-Helmholtz integral to problems in seismology. Geophys J.R. Astron. Soc., 72: 237-254. Shibutani, Tanaka, T., A., Kato, M. and Hirahara, K., 1994. A study of P-wave velocity discontinuity in D" layer with J-array records: preliminary results. J. Geomag. Geoect., 45: 1275-1285. Weber, M., 1993. P and S wave reflections from anomalies in the lowermost mantle. Geophys. J. Int., 115: 83-210. Weber, M. and Davis, J.P., 1990. Evidence of a laterally variable lower mantle structure from P- and S-waves, Geophys. J. Int., 102: 231-255. Weber, M. and K6rnig, M., 1992. A search for anomalies in the lowermost mantle using seismic bulletins. Phys. Earth Planet. Inter., 73: 1-28.