1

Imaging teleseismic P to S scattered waves using the Kirchhoff integral

Alan Levander1 and Fenglin Niu Department of Earth Science, Rice University

William W. Symes Department of Computational and Applied Mathematics, Rice University

Abstract. The development of high quality portable broadband seismographs and subsequent deployment of large numbers of these instruments in temporary target-oriented arrays or in fixed geometry and mobile arrays as planned for the USArray component of Earthscope presents a number of imaging opportunities using scattered waves. Such imaging, analogous to pre- and post-stack depth migration imaging developed for petroleum exploration [e.g., Biondi and Bevc, 2005, this volume], offers structural seismologists interested in crust and upper mantle structures the opportunity to develop impedance-like images of high resolution relative to regional or global tomograms. In this paper we describe one such means of imaging with scattered P to S-waves making using of the Rayleigh-Sommerfeld diffraction integral, discuss its strengths, limitations, and aspects of the required array geometry, present synthetics tests to demonstrate the utility of the algorithm and to explore some of the issues associated with scattered wave imaging, and apply the imaging system to several datasets.

One dataset is from a cratonic region, the Kaapvaal craton, in which we have clearly imaged the 410-km and 660-km discontinuities, and have some energy arriving at the 520-km discontinuity, as well as at the expected depth of the base of the craton. The second dataset is from a Cenozoic orogenic belt, the southern Rocky Mountains, where we identify the crust-mantle boundary, and a Proterozoic subduction slab in the uppermost mantle that can be interpreted as the southern edge of the Wyoming craton. Speaking broadly, the migrated images provide relatively high quality images of the structures expected from the classical seismic Earth models: the Moho and the transition zone discontinuities. In addition the images show details of smaller scale features that are either unidentified, or are poorly resolved in tomograms made from the same datasets.

Converted wave imaging and traveltime tomography complement one another: The tomograms provide the laterally variable velocity structures needed for image focusing. The converted wave images provide higher resolution images of both lateral and vertical impedance changes, removing some of the smearing inherent in tomographic methods. 1 Corresponding author: alan@rice.edu

2

1. Introduction

The deployment of relatively large numbers of portable broadband seismographs recording unaliased teleseismic wavefields permits analysis of scattered waves from the receiver array side of the upper mantle using depropagation and focusing algorithms. These methods, known collectively in the petroleum and active source seismic communities as depth migration, have been applied in a number of variations to teleseismic datasets using temporary broadband deployment datasets. Ryberg and Weber [2000] compare reflection processing and receiver function processing methods, including post-stack Kirchhoff depth migration using synthetic data. Sheehan et al. [2000] present a diffraction stack depth migration of receiver functions from the Snake River Plain Array. Bostock and Rondenay [1999] describe a diffraction stack integral to migrate elastic wavefields and apply the method to 2-D P-SV finite-difference synthetic seismograms. This group followed with a series of papers describing development and application of generalized inverse radon transform methods to seismic imaging with scattered P and S waves, followed by applications to synthetic data and the Cascadia dataset [Bostock et al., 2001; Shragge et al., 2001; Rondenay et al., 2001]. Poppeliers and Pavlis [2003a and 2003b] employ a plane wave receiver function migration approach subsequent to resolving the teleseismic wavefield into plane wave components. Levander [2003] describes a Kirchhoff integral pre-stack depth migration of receiver functions using a 1-D velocity model. Wilson and Aster [2003] show examples of pre-stack Kirchhoff depth migrations from the Rio Grande Rift-Colorado plateau region. Here we elaborate on the imaging theory utilizing Kirchhoff methods, present several synthetic examples to illustrate resolution, image fidelity, and aspects of velocity analysis, and then show two field examples. The first field example is from the southern African craton, the second is from the western U.S. orogenic plateau.

2. Methodology

2.1 Theory Pre- and post-stack imaging of surface or borehole recorded acoustic wave signals with the Kirchhoff integral is well established in the petroleum industry [e.g., Schneider, 1977; Wiggins, 1984; Wiggins and Levander, 1984; Biondi and Bevc, 2005]. Lafond [1991] and Lafond and Levander [1993] developed equations for pre-stack Kirchhoff depth migration in heterogeneous media using asymptotic approximations to the wave equation. Keho [1986] developed algorithms for elastic wave migration and material properties inversion of surface and vertical seismic profile data from the elastic wave representation theorem. Here we confine our attention to the acoustic problem. Starting from the Helmholtz equations one can apply Green’s theorem to the integral solution for the Green’s function to obtain (see appendix A1):

!

U(r,") =1

4#dS

0$ [G(r0,r,")

%U(r0,")

%n&U(r

0,")

%G(r0,r,")

%n] (1)

where

3

U(r0, ω) is a scalar field recorded along a surface So, S0 is a surface encompassing the volume of interest,

G(r0,r,ω) is the Green’s function for a medium with equivalent velocity structure, r0 is location along the surface, r is location anywhere in the propagation space,

n is the normal direction to the surface, positive outward, and ω is angular frequency. For a planar surface, the first and second terms of the integral are equivalent. For a variable height surface this is approximately true, provided the wavelength of the topography is large compared to the wavelength of the incident field [e.g., Wiggins, 1984; Hill and Wuenschel, 1985]. This is known as Kirchhoff’s approximation [Beckman and Spizzichino, 1963]. For migration we make use of this approximation and an asymptotic solution to the 3-D Green’s function (see appendix A1):

!

G (r,r' ,") = A(r,r' ) exp(i"# (r,r' )) (2) where τ is the travel time between r’ and r, and satisfies the eikonal equation, and A is the amplitude between r’ and r and satisfies the transport equation. We can write (1) as

!

U (r,") =#1

2$dS0% U (r0,")[

&A

&n+ i"A

&'

&n] exp(i"' ) (3)

Assuming that amplitude fluctuations are small relative to travel-time fluctuations, a far-field approximation, and taking the gradient of the travel-time field with respect to the observation surface gives

!

U (r,") =i"

2#dS0$ U (r0,")

A(r0,r) cos%

c(r0)exp(i"& ) (4)

where θ is the angle between the normal of the integration surface and the wave vector, and c(r0) is the velocity at the surface. This is a form of the Rayleigh-Sommerfeld diffraction integral, which propagates a 3-D scalar wavefield recorded along the surface, S(r0), to anywhere, r, in a medium characterized by a traveltime field τ(r0,r) and an amplitude field A(r0,r) provided that the far-field and Kirchhoff approximations are valid. Image formation requires an imaging condition as well as the (de)propagation operator described above. To form an image, the (de)propagation time, τ, is equivalent to the physical propagation time of the experiment, t; that is, seismic events are back extrapolated to the points at which they were scattered in the subsurface. An image of scattering locations is then given by

4

!

I (r) =1

4" 2d#

$%

%

& (i#) dS0$L

L

& exp($i#(t $' ))U (r0,#)A(r,r0) cos(

c(r0) t='

(5)

where L represents the half aperture of the observation array. We assume that no waves are incoming from outside of the aperture anywhere on S, and the field U and its derivatives are zero outside of the aperture. As in diffraction theory, this violates a principle of complex analysis [e.g., Goodman, 1996], but in practice works well. To adapt this formulation to teleseismic imaging we make a number of additional assumptions to retain the scalar wave formulation, and that are needed by the experiment and target geometries. The first of these is that the receiver function [e.g., Langston, 1979; Ammon, 1991] is an accurate approximation to the converted S wave field on the array side of a P-wave teleseismic travel path, in which case a slight modification to (5) can be used for imaging the converted S-wave field. The imaging condition is then:

!

0 = " p (r,re ) +" s (r0,r) # te (r0 ,re ) (6) where te(r0,re) is the recording time at the receiver referenced to the earthquake origin time at re. The effects of two-dimensional scattering across an array from waves not oriented orthogonally to the structural dip of a 2-dimensional geologic structure are accounted for by correcting the propagation phase in the out of plane direction [Bostock et al., 2001]. Taking into account the amplitude of the incoming P-wave field at the scattering point, AP(re,r), and defining SPS(r) as a pseudo-scattering coefficient consisting of the ratio of converted S to incident P:

!

SPS(r) = A

S(r) /A

P(r)

the image from a single earthquake recorded by the array is given as

!

I (r) = SPS (r) =1

4" 2d#

$%

%

& (i#) dS0$L

L

& exp($i#(t $ (' P +' S $ py ( y ))

RF (r0 ,#)AS (r,r0) cos) (r0)

AP (r,re )* (r0) t= te

(8)

where RF is the receiver function recorded at r0, SPS is the scattering coefficient for P to S conversion at the point r, AP(re,r) is the P wave amplitude at r from a source at re, AS(r,r0) is the S wave spreading, and β(r0) is the shear wave velocity at the surface. The correction for out of plane propagation with respect to a reference coordinate, y’, using the ray parameter in the coordinate orthogonal to the image, py, is included in the phase term [Bostock et al., 2001].

5

In practice the travel times and amplitudes of the P and S fields are calculated using an upwind finite-difference solver to account for heterogeneity in the imaging region [van Trier and Symes, 1991; Araya and Symes, 1996], with travel times and amplitudes for the P wave incident at the bottom of the imaging region calculated using an average one-dimensional model [Shearer, 1999]. The receiver functions are imaged one by one. Post-migration processing includes dip limiting the partial images, and editing unreliable partial images. The final image is made by registering the image from each earthquake onto a common coordinate system and summing images from all earthquakes.

2.2 Limitations The fidelity of the receiver function as a representation of the converted S field has been called into question [Pavlis, unpublished] as a result of incomplete deconvolution of the source signature, and waves not following the simple ray geometries inherent to the method. To improve the isolation of S, Vinnik [1977] suggests rotating the data into longitudinal, transverse, and orthogonal components to prior to deconvolution, now known as the LQR method. Reading et al. [2003] developed a rotation taking into account free surface effects as means to further improve the isolation of the S field. Park and Levin [2000] use a wavelet decomposition to construct receiver functions that appears to stabilize the process in a more robust manner than simple spectral division with whitening or time domain prediction error filtering. The Park-Levin method also provides reliability estimates of the receiver functions. Other complications are associated with multiple P to P and P to S reflections in the crust, and P to surface waves scattering at the free surface or at the basement contact to sedimentary basins [Dueker and Sheehan, 1997; Levander and Hill, 1985]. Bostock et al. [2001] and Wilson and Aster [2003] make use of the multiple reflections to enhance image quality of the Moho multiple reflections. Multiples and coherent surface-scattered energy still remain the most obvious source of noise on the common-conversion-point (CCP) stack and depth migrated converted S wave sections. Experiment and target geometries pose several problems. At present most broadband experiments are either 2-D (i.e., quasi-linear) arrays or coarse 3-D arrays (i.e., areal arrays). Coarse arrays lead to spatial aliasing of the signals of interest, and short arrays restrict the depth to which reliable images can be made. The first Fresnel zone of the wavefield at the surface must be well sampled in a Fourier sense for image reconstruction (Figure1), providing guidelines on both spatial sampling and array length. If we assume a 2-D array oriented approximately along the x-axis with a 2-D (x-z) scattering structure beneath it, we can develop what is termed a 2.5-D approximation to account for 3-D wave propagation [e.g., Bleistein, 1986] by integrating (4) along the y-axis and applying the stationary phase approximation. This provides addition phase and amplitude weighting terms in the imaging integral including the out of plane wavefront curvature.

6

3. Applications

3.1 Synthetic Examples First we examine the fidelity of the imaging system and the effects of unrecognized velocity perturbations of intermediate scale using synthetic seismograms. We assume that the one-dimensional P and S structure are reasonably well known, but that unrecognized low or high velocity perturbations distort the wavefield. For calculating the ray geometries we use the exact array dimensions and station distribution as the Kaapvaal Seismic Experiment of the Kaapvaal Project [Carlson et al., 1996], described below, and chose one of the earthquake locations from the experiment. The synthetic seismograms have been calculated by Kirchhoff forward modeling, through a modified IASP91 [Kennett and Engdahl, 1991] velocity model (Figures 2a and 2b). A Ricker wavelet source pulse with a 0.056-0.22Hz bandpass is used and correct P to S scattering coefficients are calculated. The first synthetic example, shown in Figure 2, was designed to test the imaging system resolution at various depths. The migration model a slightly modified IASP91 model, is described further in the next section. Figures 2a and 2b are point response functions constructed for dips up to 70o and 40o from the P-incidence direction for point scatterers at (x,z)=(450, 325) km, (450, 400) km, and (450, 660) km. Figures 2c-2e show the horizontal point spread functions for the 70o and 40o images. The half-width at half maximum in the images is approximately 20-30 km; the 40o image significantly reduces the sidelobe energy at shallower depths where spatial aliasing becomes a problem (see Figure 1). Figure 3 shows the fidelity of the imaging system for the two different maximum dips used in depth migration due to random noise. A random noise field has been bandpassed at 0.033-0.15 Hz, corresponding approximately to the bandpass of the field seismograms described in the next section. The noise images show that no energy collects at rapid variations in the velocity field, such as the mantle transition zone discontinuities, and that no prominent dipping events are introduced as a result of irregularities in array geometry, but that subsurface illumination is indeed uneven as a result of irregularity and relative sparseness of the receiver array (Δx~35 km, with dropouts producing array gaps of ~100 km). Note that there is a “crease’ in the images at (x,z)=(+200, >400) km, a result of the uneven surface coverage. Note also the increase in apparent lateral continuity as the dip bandwidth of the image is decreased. The second model is designed to examine the effects of unrecognized velocity fluctuations on the image, by adding a 2-D fluctuation consisting of a 400 km wide, 200 km deep anomaly centered at (x,z)=(450, 300) km superimposed on the model of the preceding example (Figures 4a and 4b). The anomaly has a maximum of +3.0% in P-velocity and +5.0% in S-velocity. Three P to S converting surfaces were superimposed on the velocity model: 1) A conversion corresponding to a –2.5% S velocity perturbation was generated along a cosine-shaped interface at depths from 225 to 325 km to simulate a hypothetical base of continental lithosphere. 2) A conversion corresponding to a +3.6% velocity contrast was generated at 400-km depth. 3) A conversion corresponding to a +6.1% velocity contrast was generated at the 660-km discontinuity. Figures 4c and 4d show the synthetic data depth migrated images for maximum dips 40o. Illumination along the conversion interfaces is relatively uniform over large

7

distances, and the negative polarity event is well imaged and detectable until it merges with the multiple reflections at the right hand side of the figure. Migrating with the correct velocity model produces the image in Figure 4c, whereas migrating with the exact 1-D model but without including the 2-D perturbations leads to the image in Figure 4d. Significant distortions in the depth of the 410- and 660-km discontinuities are evident in the migration with the 1-D velocity model. The 410- and 660-km discontinuities are imaged approximately 5 and 10 km, respectively, shallower than their true depths. Lastly, we examine signal detection in the presence of random noise added to the synthetic seismograms prior to imaging. Figure 4e shows a 40o image from the same synthetic example but to which 0.033-0.16 Hz bandpass random noise has been added. The noise has a maximum amplitude corresponding to the maximum amplitude of the negative polarity event at 225-325 km depth. The negative polarity event is still well imaged and continuous across the section in the presence of the random noise. Note that the crease in the section at (x,z)~(200, > 400) km is evident in this section as well, but recall that this image is made for a single synthetic source.

3.2 Field Data Example: Kaapvaal Craton The South Africa Seismic Experiment of the Kaapvaal Project deployed 55 PASSCAL broadband seismometers at 82 sites in South Africa, Zimbabwe and Botswana in 1997-1999 (Figure 5). The seismic array is approximately linear and extends ~2000 km southwest to northeast across the Archean Kaapvaal and Tanzania cratons and the Limpopo mobile belt and the adjacent Proterozoic Namaqua-Natal mobile belts. We present a number of images made from 9 earthquakes whose back azimuths lie within 30o the array axis. We made receiver functions from the vertical and radial component data using the LQR method [Vinnik, 1977]. The receiver functions have been bandpass filtered at 0.033-0.15 Hz. The high cut frequency was chosen to avoid spatial aliasing due to large stations spacing (average Δx~35km, refer to Figure 1). The large array aperture and station spacing make the Kaapvaal seismic array useful for imaging the mantle above and through the transition zone as we have shown in the synthetic tests. The data were migrated using a variation of the 1-D IASP91 reference velocity model. The first modification was to add a 40 km thick continental crust. Crustal velocities were taken from the crustal model of Christensen and Mooney [1995] for cratons (Figure 6). A second modification was to reduce the compressional velocities in the mantle between the crust and the 410-km discontinuity to account for a depleted cratonic root (Figure 6) [Lee, 2003; Niu et al., 2004]. Iron depletion of the upper mantle during craton formation not only increases the Mg# of the rocks forming the upper mantle, but also affects the Vp/Vs ratio largely by reducing compressional velocities relative to S for a fertile mantle. Niu et al. [2004] determined that a 2.5% reduction of compressional velocity in the upper mantle was adequate to substantially reduce the travel time residuals from a regional earthquake recorded across the array, but that shear velocities did not require appreciable adjustment. The model shown in Figure 6, which we refer to informally as the IASP91 Craton model, has compressional velocities reduced by 2.5% from the base of the crust to the top of the transition zone, and shear velocities increased by 0.5% in the same region relative to IASP91.

8

Figure 7a shows the 45o dip image focused with the 1-D IASP91 Craton velocity model. Migrations using the IASP91 model without adjustment placed the top of the transition zone at z < 390 km similar to the depth determined by Gao et al. [2002]. The array station spacing is too great to permit producing a continuous image of the Moho (see Figure 1). The first laterally continuous events are the crustal multiples appearing at depths of ~150-225 km, a result expected from the synthetic tests shown in Figure 4. The top of the transition zone appears at ~400 km depth, and the bottom of the transition zone at about ~660 km. A zone of relatively weak events appears between the top of the transition zone and the end of the crustal multiples at ~225 km. Additional laterally discontinuous events appear within the transition zone. Figure 8b compares the depth migrated image with CCP gather images of the same data using a 4th-root stacking method [Niu et al., 2004]. The 400- and 660-km discontinuities in the two images are in good agreement, although the peaks in the CCP stacks are considerably sharper, a consequence of the correlation properties of the 4th-root stack. Note that the Moho, crustal multiples, and transition zone discontinuities are almost the only appreciable energy on the CCP stacks. Figure 8 shows P and S wave tomography images from James et al. [2001] that were used to create the two-dimensional migration velocity model. The velocity fluctuations were added to the modified IASP91 Craton velocity model; although we note that the fluctuations are relatively small for both P and S. Figure 9 shows a 45o dip image focused with the 2D migration model. The transition zone discontinuities are imaged more coherently, with the top of the transition zone at 411.5+/- 10.6 km from X=0 to X=1800 km, and the bottom at 648.5+/- 14.6 km. This corresponds reasonably well to previous estimates [Niu et al., 2003], and gives a transition zone thickness of 237.0+/-18.0 km, about 5-6 km less than global averages. Above the transition zone energy is now coherent over distances of hundreds of kilometers across most of the section under the Kaapvaal craton. Both positive and negative amplitude events are prominent beneath the multiple train and the 410 discontinuity.

3.3 Field Data Example: Cheyenne Belt, Western United States Our second field data example is from a Precambrian boundary separating Archean and Proterozoic terranes in the orogenic plateau of the western United States (Figure 10). The teleseismic data were recorded along a ~250 km long array crossing the Cheyenne Belt, one of the principal targets of the CD-ROM experiment [Karlstrom and CD-ROM Working Group, 2002]. The Cheyenne Belt is a profound boundary separating a series of Proterozoic island arc terranes that began accreting to the Wyoming craton at ~1.8 Ga. The surface expression of the boundary is in places as narrow as 100m. Some 150 kilometers to the west previous active source seismic data [Henstock et al., 1998; Snelson et al., 1998; Gorman et al., 2002] have suggested that this boundary is only a few hundred kilometers wide at depths of ~100-150 km with highly modified North American mantle lying south of the Cheyenne Belt. Henstock et al. [1998] inferred the presence of an upper mantle low velocity zone with a thin lid just beneath the Moho. To the north, beneath the Wyoming craton, these studies show a mantle with cratonic velocities, although the depth of the high velocities was only constrained to ~100 km. S wave analysis using partitioned waveform inversion by van der Lee and Nolet [1997] suggests that the Wyoming craton does not extend to what are considered normal cratonic depths (i.e., > 200 km).

9

P and S tomography as well as the receiver functions were provided by Yuan and Dueker [2005] and Zurek and Dueker [2005]. Station spacing along this array is Δx~15km, allowing us to approximately double the bandwidth of the previous example (0.033 < f < 0.33 Hz; see Figure 1), although the shorter array permits imaging only to about the depth of the beginning of the crustal multiples. In this example the velocity fluctuations in P and S are at least five times larger than in the previous example (compare Figures 8 and 11). The migration velocity model was constructed starting with Yuan and Dueker’s [2005] 1-D reference model for the P and S tomography, replacing the crust and uppermost mantle with velocities determined from CD-ROM refraction experiment [Levander et al., 2005], adding the IASP91 model for the deeper mantle, and adding the 2-D fluctuations from the tomography. The 2-D pre-stack depth migration of 7 earthquakes has a number of interesting features (Figure 12). The Moho is well imaged across the ~250 km profile, and is in fairly good agreement with that determined from refraction, although the Moho depth from the refraction data is somewhat shallow and smooth compared to the migrated teleseismic image, likely the result of 1) the layer based model parameterization used for the refraction analysis [Zelt and Smith, 1992], 2) illumination of the Moho from above in the refraction experiment and from below in the teleseismic experiment, and 3) the boundary being complex rather than a simple first order discontinuity [Keller et al., 2005]. The Moho shows lateral variability, particularly in the distance range of 100-125 km, where the reflection and tomography interpretations have placed the crustal suture between the Proterozoic terranes and the Archean terranes. In this region the wide-angle active source data show complicated PmP reflections, and the 2D prestack depth migration indicates a weakly converting Moho. The zone of multiples appears at the bottom of the depth section at z~175 km. Between the Moho and the crustal multiples are an arcuate, concave up positive event in the uppermost mantle in the south, and a series of north dipping negative and positive polarity events in the northern half of the section. The arcuate positive event in the south corresponds to a low velocity upper mantle anomaly in the tomography images. To the north the shallowest positive polarity event we interpret as a boundary between the southern edge of the Cheyenne belt mantle, and a piece of island arc mantle subducted beneath the northern edge of the Archean Wyoming province, an interpretation made by Zurek and Dueker [2005]. A similar interpretation was made from a CCP image along the Deep Probe teleseismic profile across the Cheyenne Belt to the west by Crosswhite and Humphreys [2003]. We interpret the subparallel negative polarity event beneath this as the base of the stalled subducting plate, in agreement with the tomography images of Yuan and Dueker [2005]. The negative polarity results from higher velocity in the slab fragment than the mantle immediately beneath and to the south of it. We suggest that this negative polarity event marks the southernmost extent of the modern Wyoming craton.

4. Discussion

A number of aspects of the 2-D pre-stack depth migrated images are worth further discussion. The Kaapvaal experiment tomography produced relatively small P and S velocity fluctuations (~±0.5% and

10

~±1.0%, respectively). The migration image shows a series of weak events in the 225-400 km depth range that correspond roughly to the base of the tectosphere as interpreted by James et al. [2001]. The two transition zone discontinuities are very well imaged. Between the transition-zone discontinuities a number of laterally variable events may correspond to the 520 discontinuity, although some of these events are clearly noise or artifacts. The images made from the Cheyenne Belt CD-ROM array data are in reasonable agreement with both the tomography images and surface-source reflection and refraction images, the combined experiments have identified what we interpret to be largely Paleoproterozoic subduction-accretion structures. The smaller station spacing than in the Kaapvaal Seismic Experiment has allowed imaging with higher frequencies and has produced a better resolved image of a complex uppermost mantle. The outlines of the large tomography anomalies (±2.5% in P and ±5.0% in S) correspond well to converting surfaces in the depth migration images given the vertical smearing inherent in teleseismic tomography. In developing the images from the field data we have not accounted for anisotropy in the velocity field. Although adding another level of complication, this can be done by incorporating anisotropic eikonal solvers in calculating the Green’s functions in the imaging operator [e.g., Qian and Symes, 2002]. Anisotropy is small under the Kaapvaal array, with S-wave splitting less than 0.6 s [Silver et al., 2001]; whereas under the Cheyenne Belt CD-ROM array shear wave splits are significantly larger, 1.0-2.0 s and appear to be dependent on azimuth of approach [Dueker, personal communication].

5. Conclusions

Direct coherent imaging techniques have seen widespread application in many branches of applied physics and medical imaging [Blackledge, 1989], and now in application to teleseismic P to S conversion data provide significantly greater resolution of mantle impedance boundaries than is available from tomography methods. The tomography methods produce a necessary element for imaging with converted waves, as large scale velocity perturbation structure is essential for proper image focusing. Ultimately some of the uncertainty in vertical positioning of tomography velocity anomalies may be resolved using focusing criterion similar to those developed for active source pre-stack imaging [e.g., Lafond and Levander, 1993; Morozov and Levander, 2002]. Extension of this method to data from dense two-dimensional arrays to obtain images of three-dimensional structures is relatively straightforward. With the development of dense portable array broadband seismology, it is possible to image the structural geology of the lower crust and upper mantle, as well as make detailed examinations of lateral variations in the transition zone phase change boundaries resulting from thermal effects and volatiles. As Figure 1 shows, the constraints imposed on this type of imaging largely arise from the instrument resource base, as array aperture and dense spatial sampling make competing and equally important demands on experiment design for direct imaging of the Earth.

11

Acknowledgments. AL and FN would like to thank David James and other members of the Kaapvaal Seismic Experiment group and Ken Dueker, Brian Zurek, and Huaiyu Yuan at the University of Wyoming for providing seismic data, receiver functions, and tomography models used in the depth migrations. This research was supported in part by NSF grants EAR-0225670 to AL and EAR-0346203 to AL and WWS, and by Rice University. Lastly AL gratefully acknowledges a Green Scholarship for sabbatical leave at the Scripps Institution of Oceanography in 1999-2000, where this work was begun. Many people in the Scripps community made helpful suggestions, particularly Peter Shearer, Frank Vernon, and Gabi Laske.

References Ammon, C.J., The isolation of receiver effects from teleseismic P waveforms, Bull. Seismol. Soc. Am., 81,

2504-2510, 1991. Araya, K., and W.W. Symes, 2D and 2.5D Kirchhoff inversion using upwind finite difference amplitudes. Society

of Exploration Geophysicists, 66th annual international meeting (expanded abstracts), 503-506, 1996. Beckman, P., and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, International

Series of Monographs in Electromagnetic Waves, 4, Pergamon Press, Inc., 1963. Biondi, B., and D. Bevc, Subsurface imaging of complex structures by reflection seismic data, in Array Analysis of

Broadband Seismograms, edited by A. Levander and G. Nolet, Washington, D.C., American Geophysical Union, 2005.

Blackledge, J.M., Quantitative Coherent Imaging: Theory Methods, and Application, 362pp, Academic Press, 1989.

Bleistein, N., Two-and-one-half dimensional in-plane wave propagation, Geophysical Prospecting, 34, 686-703, 1986.

Bostock, M. G., and S. Rondenay, Migration of scattered teleseismic body waves, Geophys. J. Int., 137, 732-746, 1999.

Bostock, M.G., S. Rondenay, and D.S. Shragge, Multiparameter two-dimensional version of scattered teleseismic body waves, 1. Theory for oblique incidence, J. Geophys. Res., 106, 30771-30782, 2001.

Carlson, R.W., T.L. Grove, M.J. de Wit, and J.J. Gurney, Program to study the crust and mantle of the Archean craton in southern Africa, EOS Trans. AGU 77, 273-277, 1996.

Christensen, N.I., and W.D. Mooney, Seismic velocity structure and composition of the continental crust: A global view, J. Geophys. Res., 100, 9761-9788, 1995.

Crosswhite, J.A., and E.D. Humphreys, Imaging the mountainless root of the 1.8 Ga Cheyenne Belt suture and clues to its tectonic stability, Geology, 31, 669-672, 2003.

Dueker,K., and A. Sheehan, Mantle discontinuity structure beneath the Colorado Rocky Mountains Region, J. Geophys. Res., 103, 7153-7169, 1997.

Dueker, K., X. Yuan, and B. Zurek, Thick-structured Proterozoic lithosphere of the Rocky Mountain region, GSA Today, 11, 4-9, 2001.

Gao, S.S, P.G. Silver, K.H. Liu, and Kaapvaal Seismic Group, Mantle discontinuities beneath southern Africa, Geophys. Res. Lett., 29, 10.1029/2001GL013834, 2002.

Goodman, J.W., Introduction to Fourier Optics, MacGraw-Hill, New York, 1996. Gorman, A.R., R.M. Clowes, R. M. Ellis, T. J. Henstock, G.D. Spence, G. R. Keller, A. R. Levander, C. M.

Snelson, M. J. A. Burianyk, E. R. Kanasewich, I. Asudeh, Z. Hajnal and K. C. Miller, Deep Probe – Imaging the roots of western North America, Canadian Journal of Earth Sciences, 39, 375-298, 2002.

Henstock, T.J., and the Deep Probe Working Group, Probing the Archean and Proterozoic lithosphere of western North America, GSA Today, 8, 1-5 & 16-17, 1988.

Hill, N.R., and P.C. Wuenschel, Numerical modeling of refraction arrivals in complex areas, Geophysics, 50, 90-98, 1985.

James, D.E., M.J. Fouch, J.C. VanDecar, S. van der Lee, and Kaapvaal Seismic Group, Tectospheric structure beneath Southern Africa, Geophys. Res. Lett., 28, 2485-2488, 2001.

12

Karlstrom, K., and CD-ROM Working Group, Structure and evolution of the lithosphere beneath the Rocky Mountains. GSA Today, 4-10, 2002.

Keho, T.H., The Vertical Seismic Profile: Imaging in Heterogeneous Media. Earth, Atmospheric, and Planetary Sciences. Cambridge, MA, MIT: 304, 1986.

Keller, G.R., K.E. Karlstrom, M.L. Williams, K.C. Miller, C. Andronicos, A. Levander, C. Snelson, and C. Prodehl, The dynamic nature of the continental crust-mantle boundary: Crustal evolution in the Southern Rocky Mountain region as an example, in The Rocky Mountains: An Evolving Lithosphere, Geophysical Monograph Series 154, American Geophysical Union, Washington, D.C., 403-420.

Kennett, B.L.N., and E. R. Engdahl, Traveltimes for global earthquake location and phase identification, Geophys. J. Int.,105, 429-465, 1991.

Lafond, C.F., A unified approach to complex seismic imaging problems, Dept. of Geology and Geophysics, Houston, TX, Rice University: 133, 1991.

Lafond, C.F., and A. Levander, Migration moveout analysis and depth focusing, Geophysics, 58, 91-100, 1993. Langston, C.A., Structure under Mt. Ranier, Washington, inferred from teleseismic body waves. J. Geophys. Res.,

84, 4749-4762, 1979. Lee, C.-T. A., Compositional variation of density and seismic velocities in natural peridotites at STP conditions:

Implications for seismic imaging of compositional heterogeneities in the upper mantle, J. Geophys. Res., 108, 2441 doi:10.1029/2003JB002413, 2003.

Levander, A., USArray design implications for wavefield imaging in the lithosphere and upper mantle, The Leading Edge, 22, 250-255, 2003.

Levander, A.R., and N. R. Hill, P-SV resonances in irregular low-velocity surface layers, Bull. Seismol. Soc. Am., 75, 847-864, 1985.

Levander, A., C.A. Zelt, and M.B. Magnani, Crust and upper mantle velocity structure of the Southern Rocky Mountains from the Jemez Lineament to the Cheyenne Belt, in K. Karlstrom and G.R. Keller, eds., The Rocky Mountain Region: An Evolving Lithosphere, Geophysical Monograph Series 154, American Geophysical Union, Washington, D.C., 293-308.

Morozov I.B., and A. Levander, Depth image focusing in travel-time map based wide-angle migration, Geophysics, 67, 1903-1912, 2002,

Morozova, E., X. Wan, K.R. Chamberlain, S.B. Smithson, R. Johnson, and K.E. Karlstrom, Inter-wedging nature of the Cheyenne belt--Archean-Proterozoic suture defined by seismic reflection data, in The Rocky Mountain Region: An Evolving Lithosphere, Geophysical Monograph Series 154, American Geophysical Union, Washington, D.C., 217-226.

Niu, F., A. Levander, C.M. Cooper, C-T.A. Lee, A. Lenardic, and D.E. James, Seismic constraints on the depth and composition of the mantle keel beneath the Kaapvaal Craton, Earth Planet. Sci. Lett. 224, 337-346, 2004.

Park, J., and V. Levin, Receiver functions from multiple-taper spectral correlation estimates, Bull. Seismol. Soc. Am., 90, 1507-1520, 2000.

Poppeliers, C., and G.L. Pavlis, Three-dimensional, prestack, plane wave migration of teleseismic P to S converted phases I: Theory, J. Geophys. Res., 108, doi:10.1029/2001JB000216, 2003.

Poppeliers, C., and G.L. Pavlis, Three-dimensional, prestack, plane wave migration of teleseismic P-to-S converted phases: 2. Stacking multiple events. J. Geophys. Res., 108, 10.1029/2001JB001583, 2003.

Qian, J., and W.W. Symes, Finite-difference quasi-P traveltimes for anisotropic media, Geophysics 67, 147-155, 2002.

Reading, A., B. Kennett, and M. Sambridge, Improved inversion for seismic structure using transformed , S-wavevector receiver functions: Removing the effect of the free surface, Geophys. Res. Lett., 30, 1981, doi:10.1029/2003GL018090, 2003.

Rondenay, S., M.G. Bostock, and J. Shragge, Multiparameter two-dimensional version of scattered teleseismic body waves, 3. Application to the Cascadia 1993 data set, J. Geophys. Res., 106, 30,795-30,807, 2001.

Ryberg, T., and M. Weber, Receiver function arrays: a reflection seismic approach, Geophys. J. Int., 141, 1-11, 2000.

Schneider, W.A., Integral formulation for migration in two and three dimensions, Geophysics, 43, 49-76, 1977. Shragge, J., M.G. Bostock, and S. Rondenay. Multiparameter-two dimensional inversion of scattered teleseismic

body waves, 2. Numerical examples, J. Geophys. Res., 106, 30783-30793, 2001.

13

Silver, P.G., S.S. Gao, K.H. Liu, and the Kaapvaal Seismic Group, Mantle deformation beneath southern Africa, Geophys. Res. Lett., 28, 24385-432, 2001.

Shearer, P.M., Introduction to Seismology, Cambridge University Press, Cambridge, U.K., 1999. Sheehan, A. F., P. M. Shearer, H.J. Gilbert, K.G. Dueker, Seismic migration processing of P-SV converted phases

for mantle discontinuity structure beneath the Snake River Plain, Western United States, J. Geophys. Res., 105, 19055-19065, 2000.

Snelson, C.M., T.J. Henstock, G.R. Keller, K.M. Miller, and A. Levander, Crustal and uppermost mantle structure along the Deep Probe seismic profile, Rocky Mountain Geology, 33, 181-198, 1998.

van der Lee, S., and G. Nolet, Upper mantle S velocity structure of North America, J. Geophys. Res., 102, 22,815-22,838, 1997.

van Trier, J., and W.W. Symes, Upwind finite-difference calculation of traveltimes, Geophysics, 56, 812-821, 1991.

Vinnik, L.P., Detection of waves converted from P to SV in the mantle, Phys. Earth Planet. Inter., 15, 39-45, 1977. Wiggins, J.W., Kirchhoff integral extrapolation and migration of nonplanar data, Geophysics, 49, 1239-1248, 1984. Wiggins, J.W., and A. Levander, Migration of multiple offset synthetic vertical seismic profile data in complex

structures, in Vertical Seismic Profiles, M. Simaan editor, JAI Press, Greenwich, CT, 269-289, 1984. Wilson, D., and R. Aster, Imaging crust and upper mantle seismic structure in the southwestern United States using

teleseismic receiver functions, The Leading Edge, 22, 232-237, 2003. Yuan, H., and K. Dueker, Upper mantle tomographic Vp and Vs images of the Rocky Mountains in Wyoming,

Colorado, and New Mexico: Evidence for a thick heterogeneous chemical lithosphere, in The Rocky Mountain Region: An Evolving Lithosphere, Geophysical Monograph Series 154, American Geophysical Union, Washington, D.C., 329-346.

Zelt, C.A., and R.B. Smith, Seismic traveltime inversion for 2-D crustal velocity structure, Geophys. J. Int., 108, 16-34, 1992.

Zurek, B., and K. Dueker, Lithospheric stratigraphy beneath the Southern Rocky Mountains, in The Rocky Mountain Region: An Evolving Lithosphere, Geophysical Monograph Series 154, American Geophysical Union, Washington, D.C., 317-328.

A.Levander and F. Niu, Department of Earth Science, Rice University, 6100 Main Street, Houston, TX 77005, USA

(alan@rice.edu, niu@rice.edu) W.W. Symes, Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, TX

77005, USA (symes@caam.rice.edu)

(Received XXXXX XX, 2004; revised XXXX XX, 2004; accepted XXXX XX, 2005) Copyright 2005 by the American Geophysical Union

14

Appendix A1 We start with two scalar wave equations, one a homogenous equation and the other defining the Green’s function:

!

"2U (r, t) #

1

c2

$ 2

$t 2U (r, t) = 0

and (A1)

!

"2G (r,r' , t) #

1

c2

$ 2

$t 2G (r,r' , t) = #4%& (t)& (r # r' )

for an impulsive source at location r’ at time t=0. Fourier transforming over time twice gives homogeneous and inhomogenous Helmholtz equations:

!

"2U (r,#) + k

2U (r,#) = 0

and (A2)

!

"2G (r, # r ,$) + k

2G (r, # r ,$) = %4&' (r % # r )

with k, the wavenumber, defined as ω/c. Green’s theorem relates the values of two twice differentiable fields, U and G, at any point in a volume specified by r to the values of the their normal derivatives at the surface of the volume specified by r0:

!

dV (r) U (r,")#2G (r,r' ,") $G (r,r' ,")#2

U (r,")[ ]%%%

!

= dS0 (r0)"" U (r0,#)$G (r0,r' ,#)

$n%G (r0,r' ,#)

$U (r0,#)

$n

&

' ( )

* +

(A3) Substituting the Helmholtz equations into Green’s integral and using the sifting property of the three dimensional delta function gives

!

U (r' ,") =#1

4$dS0 (r0)%% [U (r0,")

&G (r0,r' ,")

&n

!

"G (r0,r' ,#)$U (r0,#)

$n] (A4)

15

This integral has a simple interpretation: Given measurements or estimates of two fields U and G, and their normal derivatives on a closed surface S defined by r0, we can predict the value of the field U anywhere within the volume. The integral has been widely used in diffraction theory and forms the basis of a class of seismic migration operators. For the migration example we measure U or its derivative along S, and choose a Green’s function that we can calculate that either makes one term vanish or allows us to simply combine the two terms in the integral. Note that the Green’s function is an auxiliary field, chosen to simplify the integral, and does satisfy the same boundary conditions as the field we observe. Note also that we have as yet said nothing about the structure of the velocity field, c(r). In active source seismics, U is either the acoustic pressure in marine surveys, or a scalar representing particle velocity from a vertical component geophone. For our purposes here we assume that it is a scalar representing SV wave amplitude resulting from conversion from an incident P wave on scattering elements in the imaging volume. As a simple example of a commonly used Green’s function, design a Green’s function that vanishes on S, i.e., G(r0)=0. Let c be a constant, let S be defined as z0=0, and V(r) enclose z > 0. The 3-dimensional frequency domain freespace Green’s function is

!

GFS(r0,r,") =

exp(±i" r0 # r /c)

r0 # r (A5)

A Green’s function that vanishes on z0=0 is

!

G (r0 ,r,r' ,") =exp(±i" r0 # r /c)

r0 # r#exp(±i" r0 # r' /c)

r0 # r'

(A6) where (A7) The second term in the Green’s function is referred to as the image source. Evaluating the normal derivative of the Green’s function at z=0 gives

!

"G

"n= 2

"GFS

"nz=0

(A8)

16

In global seismology a constant velocity Green’s function is inappropriate, due to the large velocity gradients in the Earth. Asymptotic approximations to the Green’s function are used instead (Equation 2). The time domain representation of the free space Green’s function is

!

GFS(r,r' , t) =

" (t # r # r' /c)

r # r' (A9)

which can be replaced with the asymptotic form

!

GFS(r,r' , t) " A(r,r' )# (t $% (r,r' ))

equivalent to the Fourier representation (A10)

!

GFS(r,r' ,") = A(r,r' ) exp(i"# (r,r' ))

where τ is a solution of the eikonal equation

!

1

c2(r)

= "# (r,r' )2 (A11)

and A(r,r’) is a solution of the corresponding transport equation, and is equivalent to the geometrical spreading. Figure Captions Figure 1. Fresnel radius vs period for teleseismic body waves used for receiver function imaging. The figure shows the Fresnel radius at the Earth's surface for the PREM velocity model for imaging targets at various depths. The dashed black lines are the average station intervals for the Kaapvaal Seismic Experiment (Δx~35km-100km) and the CD-ROM experiment (Δx~15km). The intersections of the horizontal solid lines (twice the station interval) and the Fresnel curves indicate the minimum periods for which the arrays will record unaliased signals for horizontally traveling waves (dip angle = 90o). For migration imaging, the first Fresnel zone must be recorded completely and spatially unaliased, putting constraints on both array length and station spacing (dip limiting the images relaxes the spatial sampling constraint, but degrades resolution). For targets at transition zone depths the sampling criteria is met for periods greater than about 3s. For targets at 100km depth the Kaapvaal data are aliased at ~6s. For shallower targets the Kaapvaal array is aliased at all periods. The CDROM array with 15km spacing has unaliased signals at ~3s for depths greater than 50 km.

17

Figure 2. Response tests for point scatterers at shallow mantle (z=300km), and transition zone depths (z=400km and 660km) for maximum migration dips of (a) 70o and (b) 40o. (c-e) The horizontal point spread functions for the three anomalies show that the lateral resolution is about 20-30km for either of the migration dips, but that sidelobe energy at shallower levels (z=300km) is significantly reduced at the lower dip. The horizontal point spread functions show that the amplitude of the sidelobes is significantly smaller than the peaks. Figure 3. Random noise tests for migrations dips of (a) 70o and (b) 40o. This test is designed to identify any imaging system artifacts resulting from the structure of the migration velocity model (1-D models in Figures 4 and 6), or arising from irregularities in the recording geometry. A significant crease in the migrations is evident in both images at (x,z)~(200, > 400) km. Figure 4. (a) Two-dimensional P velocity field used for depth migrations in subsequent panels. Scattering surfaces are shown as dashed lines, with a negative polarity conversion at the “base of lithosphere”, and positive conversion at the transition zone discontinuities. (b) Velocity perturbation added to modified IASPI 91 model (see Figure 6) to examine the effects of velocity uncertainty on imaging. The maximum perturbation in the center of the anomaly is 3.0% in P and 5% in S (not shown), with linear gradients to its edges. (c) Pre-stack migration of synthetic receiver functions with multiples using the correct velocity model shown in Figure 4a and retaining dips to ±40o of the incident P wave. The “base of lithosphere” and the transition zone discontinuities are well imaged. (d) Pre-stack migration of synthetic receiver functions with multiples using the 1-D velocity model shown in Figure 6. The “base of lithosphere” is almost unaffected by the velocity anomaly, whereas the 400- and 660-km discontinuities are incorrectly imaged too shallowly by about 5 and 10 km, respectively. (e) Pre-stack image to which bandpass filtered random noise was added to the synthetic seismograms. The amplitude of the noise was chosen to match that of the “base of the lithosphere” event. In the presence of modest amplitude random noise the small amplitude event is still well imaged. Figure 5. Map of southern Africa showing the Kaapvaal Seismic Experiment array and locations of earthquakes used in this study (inset). Figure 6. Left: Modified IASP91 Craton velocity model. Vp was reduced 2.5% and Vs increased by 0.5% from the IASP91 model between the base of the average cratonic crust of Christensen and Mooney [1995] to the top of the transition zone to account for the reduced Mg# of depleted cratonic lithosphere rocks [Lee, 2003]. Right: The cratonic lithosphere has a significantly lower Vp/Vs ratio than the IASP91 model. Figure 7. 1D Migration (a) and 4th-root CCP stack (b). Both the migration and the CCP stack were focused using the IASP91 Craton velocity model shown in Figure 6. The location of the transition zone discontinuities, as well as crustal multiples are virtually identical in the two displays. The sharpness of the CCP stack results from the 4th root stacking technique which uses correlation properties of the wavefield. Figure 8. P and S tomography models for the Kaapvaal craton [James et al., 2001]. Note the relatively small fluctuations in both P and S, and the lack of correlation of the anomalies with the transition zone discontinuities. James et al. [2001] interpret the bottom of the high velocity anomalies beneath the Kaapvaal craton at z~300-350km as the base of the Kaapvaal craton.

18

Figure 9. (a) Pre-stack depth migration of Kaapvaal seismic data using the two-dimensional velocity fluctuations of Figure 8 superimposed on the IASP91 Craton model (Figure 6). Dips to ±45o from the angle of the incident P wave have been retained. The transition zone discontinuities are better imaged in the 2-D migration than in the 1-D migration of Figure 7. (b) Interpreted depth migration superimposed on P tomography model of Figure 8. (c) Interpreted depth migration superimposed on S tomography model of Figure 8. The 2-D migration produces a clearer image in the zone between the base of the crustal multiples through the transition zone. Figure 10. Map of the CD-ROM experiment area. CD-ROM investigations included deep crustal reflection and refraction profiles, and two teleseismic array deployments. We show results from pre-stack depth migration of the northern array, crossing the Cheyenne belt, a Paleoproterozoic suture separating the Archean Wyoming province from the Proterozoic Yavapai province. Thin lines centered on each teleseismic array indicate the back-azimuths of earthquakes used for receiver function imaging. Figure 11. Left: Compressional and shear velocity perturbations in imaging region as determined from teleseismic tomography [Yuan and Dueker, 2005]. Right: One-dimensional model developed by combining CD-ROM refraction and teleseismic tomography results with the IASP91 model. Figure 12. (a) Pre-stack depth migration of CD-ROM receiver functions using the two-dimensional velocity fluctuations and 1D model of Figure 11. Dips to ±45o of the incident P wave have been retained. Superimposed on the image are the interpretation of a coincident deep crustal reflection profile [Morozova et al., 2005], and velocity contours to the Moho from the coincident refraction profile [Levander et al., 2005]. Positive polarity events are the Moho (black dashed line in 12b), an upper mantle arcuate event at (x,z)~(>175, 75-100) km and an upper mantle north dipping event at (x,z)~(25-75, 60-100) km. A prominent upper mantle north dipping negative polarity event appears at (x,z)~(> 75, 60-175) km. (b) P tomography model of Yuan and Dueker [2005] superimposed on the image: The arcuate event corresponds to the base of a low velocity anomaly in the upper mantle under northern Colorado (blue dashed line). The north dipping events correspond to the top and bottom of a subducted Paleoproterozoic slab that stalled after partial subduction (red and blue dashed lines, see Yuan and Dueker, 2005). We suggest that the north dipping negative event (red dashed line) now marks the northern edge of the Wyoming craton.

0

200

400

600

800

Dep

th (

km)

0 200 400 600 800 1000 1200 1400 1600

b) 40 Migrationo

0

200

400

600

800

Dep

th (

km)

0 200 400 600 800 1000 1200 1400 1600Distance (km)

a) 70 Migrationo

Figure 2ab

Point Spread / 70o (Dashed) / 40o (Solid)

0 200 400 600 800 1000 1200 1400 1600 -5

0

5

10

0 200 400 600 800 1000 1200 1400 1600 -10

010203040

0 200 400 600 800 1000 1200 1400 1600 -6 -4 -2

02

Z=300 km (-2.5%)

Z=410 km (3.60%)

Z=660 km (6.14%)

Distance (km)

Figure 2cde: Levander et al.

c)

d)

e)

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600Distance (km)a) 70 dip

o

Figure 3ab

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600Distance (km)b) 40 dipo

Levander and Niu: Figure 4ab

a)

b)

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600Distance (km)

0 1 2 3

dVp/Vp %

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600Distance (km)

11 10 9 8 7 6Velocity (km/s)

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600b) 1-D Velocity Model

Levander and Niu: Figure 4abc

c) 2-D Velocity Model w Random Noise0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600

0

200

400

600

800

Dep

th (

km)

-400 -200 0 200 400 600 800 1000 1200 1400 1600

a) 2-D Velocity Model

M

M

M

16o 20o 24o 28o 32o 36o

-32o

-28o

-24o

-20o

-16o

NAMIBIA

ZIMBABWE

R.S.A.

Harare

Capetown

Kimberley

Bushveld Complex

Mag

ondi Belt

Okwa Belt

Khe

iss B

elt

Cape Fold Belt

Kaapvaal Craton

Limpopo Belt

3

62

415

BOTSWANA

B

B'

Namaqua-Natal Belt

0

100

200

300

400

500

600

700

800

900

0

100

200

300

400

500

600

700

800

9003 4 5 6 7 8 9 10 11 12

3 4 5 6 7 8 9 10 11 12

km/s

IASP91 Craton S

Dep

th (

km)

0

100

200

300

400

500

600

700

800

9001.70 1.75 1.80 1.85 1.90

1.70 1.75 1.80 1.85 1.90

IASP91Craton

IASP91CM (1995)CratonicCrust

Vp/Vs

IASP91Craton P

IASP91

Levander & NiuFigure 5

km/sLevander & NiuFigure 6

0

200

400

600

800

Dep

th (

km)

0 200 400 600 800 1000 1200 1400 1600 1800

Distance (km)

-5.0 -2.5 0.0 2.5 5.0Amplitude

0

200

400

600

800

Dep

th (

km)

0 200 400 600 800 1000 1200 1400 1600 1800

Kaapvaal CratonSW NE

Kaapvaal CratonSW NE

0

200

400

600

800

Dep

th (

km)

dVp/Vp

-0.5 0.0 0.5dVp/Vp%

0 200 400 600 800 1000 1200 1400 1600 1800

0

200

400

600

800

Dep

th (

km)

180016001400120010008006004002000

dVs/Vs

0.0dVs/Vs%

-1.0

SW Kaapvaal Craton NE

Figure 8: Levander & Niu

SW Kaapvaal Craton NE

Kaapvaal CratonSW NE

-800

-600

-400

-200

0

Dep

th (

km)

Figure 9ab: Levander & Niu

-800

-600

-400

-200

0

Dep

th (

km)

0 200 400 600 800 1000 1200 1400 1600 1800

Kaapvaal Craton

Multiples

MultiplesMultiples

Multiples

410

660

252

-110

254

-106

256

-104

258

-102

34 34

36 36

38 38

40 40

42 42

Colorado

New Mexico

Wyoming

Cheyenne

Belt

Jemez

Lineament

Teleseismic StationRefraction

Reflection

Proterozoic

Proterozoic

Archean

Shotpoint

Figure 10: Levander and Niu

3

4

5

6

7

8

9

10

km

/s

Vs

V

p

Ref

ract

ion

Tom

ogra

phy

bac

kgro

und

1D V

eloc

ity M

odel

Figure 11 :Levander and Niu

-200

-150

-100-5

00

Depth (km)

-300

-250

-200

-150

-100

-50

050

100

150

200

250

300

Dis

tanc

e (k

m)

-3-2

-10

12

3dV

p/V

p %

-200

-150

-100-5

00

Depth (km)

-300

-250

-200

-150

-100

-50

050

100

150

200

250

300

Dis

tanc

e (k

m)

dVs/

Vs %

-6 -

4 -

2 0

2

4 6

8

50

100

150

200

250

300

350

400

Sout

h

2

D T

omog

raph

y V

eloc

ities

Nor

th

CB

South North

Proterozoic Archean

Figure 12a: Levander and Niu

-200

-150

-100

-50

0

Dep

th (k

m)

050100150200

Distance (km)

4.8

6

6.2

6.8

7

Moho

6.4

6.6

WY

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

1.77 Ga gabbro/granite

Punderthrust

slab

Pgm

Pgm

Prb

Crustal Multiples

Moho

CB

South North

Proterozoic Archean

Figure 12b: Levander and Niu

-200

-150

-100

-50

0

Dep

th (k

m)

050100150200

Distance (km)

4.8

6

6.2

6.8

7

Moho

6.4

6.6

WY

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

SSSS

1.77 Ga gabbro/granite

Punderthrust

slab

Pgm

Pgm

Prb

Crustal Multiples

Moho