Three-dimensional, prestack, plane wave migration of teleseismic P-to-S converted phases: 1. Theory

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2112, doi: /2001jb000216, 2003 Three-dimensional, prestack, plane wave migration of teleseismic P-to-S converted phases: 1. Theory Christian Poppeliers 1 and Gary L. Pavlis Department of Geological Sciences, Indiana University, Bloomington, Indiana, USA Received 1 August 2000; revised 17 July 2002; accepted 25 November 2002; published 20 February [1] We present the theoretical foundations for a prestack migration technique to image teleseismic P-to-S converted phases. The method builds on teleseismic P wave deconvolution, pseudostation stacking [Neal and Pavlis, 1999] and on the idea of using a plane wave decomposition for imaging as introduced by Treitel et al. [1982]. Deconvolution operators are constructed by pseudostation stacking of the array aligned to the incident P wave arrival times to produce a space-variable deconvolution operator. The resulting data are then muted to remove the deconvolved direct P wave pulse and pseudostation stacked over a grid of feasible slowness vectors. The pseudostation stack interpolates the wave field onto a regular grid along Earth s surface producing a series (one per slowness vector) of uniformly sampled three-dimensional data cubes (two space variables and time). The plane wave components can be propagated downward using a form of approximate ray tracing with a three-dimensional Earth model. This yields a series of distorted cubes topologically equivalent to the original uniformly sampled data cubes. These data volumes are summed as a weighted stack with the weights derived from an integration formula for inverse scattering based on the generalized Radon transform. This allows an image of the subsurface to be constructed on an event by event basis beneath the array. We apply this technique to data from the Lodore array that was deployed in northwestern Colorado. The results suggest the presence of a major lithospheric-scale discontinuity defined by a south dipping boundary. INDEX TERMS: 7260 Seismology: Theory and modeling; 7205 Seismology: Continental crust (1242); 7294 Seismology: Instruments and techniques; KEYWORDS: imaging, migration, inverse scattering, converted waves, Born approximation, plane wave Citation: Poppeliers, C., and G. L. Pavlis, Three-dimensional, prestack, plane wave migration of teleseismic P-to-S converted phases: 1. Theory, J. Geophys. Res., 108(B2), 2112, doi: /2001jb000216, Introduction [2] It has become apparent in the past decade that the coda of the teleseismic P wave contains a wealth of data on structures in the lower crust and upper mantle. It is now clear that the region below the crust-mantle boundary has complex geologic structures that scatter teleseismic P waves to yield scattered P waves and P-to-S converted phases that form the P coda. Since pioneering work by Vinnik [1977] it has become apparent that the P-to-S converted phases, which we shall henceforth refer to as P d S phases, are a valuable tool for illuminating structures in the upper mantle. The main reason for this is that P d S phases arrive with a lag relative to the P arrival time that can be mapped directly to depth and they are polarized approximately orthogonal to the incident P wave. These properties seem to make 1 Now at Department of Earth Science, Center for Computational Geophysics, Rice University, Houston, Texas, USA. Copyright 2003 by the American Geophysical Union /03/2001JB000216$09.00 estimation of P d S phases relatively robust compared to other components of the wave field. [3] The sophistication of P d S imaging has been evolving rapidly since the emergence of portable, broadband arrays over the past decade. The earliest approaches built on the single station methods of Vinnik [1977] and Langston [1977]. That is, with widely spaced stations, which were the norm of the time, the only strategy for imaging structure below a station was to combine the data from many events to produce a one-dimensional (1-D) model for the region below the station (e.g., Langston [1979, 1989], Owens et al. [1987], or Gurrola et al. [1995]). With the emergence of broadband arrays, many people have recognized that the expanded data should provide more detailed information about the subsurface than could be achieved with the single-station methods. The earliest efforts used simple ray back projection methods combined with stacking of multiple events [Dueker and Sheehan, 1997, 1998; Sheehan et al., 1995; Bostock, 1996, 1997, 1998; Kosarev et al., 1999]. Recently, several groups have attempted the application of various forms of migration to P d S wave fields. Shearer et al. [1999] experimented with using Kirchoff migration to deal with a similar problem to ESE 18-1

2 ESE 18-2 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION look at S to P scattering in SS precursors. Ryberg and Weber [2000] demonstrate the application of Kirchoff depth migration to synthetic data generated by finite difference techniques demonstrating that the concept of migration of P d S wave fields is theoretically sound. Levander et al. [2000] experimented with Kirchoff migration applied to data from the linear MOMA array. Finally, Bostock and colleagues [Bostock and Rondenay, 1999; Bostock et al., 2001; Rondenay et al., 2001; Shragge et al., 2001] have recently developed an inverse scattering formulation for direct imaging of broadband array data based on the formulation of Beylkin and Burridge [1990]. The technique we introduce here as has strong similarities to that of Bostock et al. [2001] in that both use an inversion based on what has come to be called the generalized Radon transform [Miller et al., 1987]. The major difference is in the fundamental approach. Bostock et al. [2001] use a pure inverse scattering approach which fundamentally requires all the data to be present simultaneously. The problem is tractable because the generalized Radon transform provides an exact inverse in the form of an integral equation that eliminates the need for a matrix inversion on a massive scale. We approach the problem from a perspective more in line with seismic reflection processing. The key concepts in all reflection processing is reducing noise, which can be loosely defined as components of the wave field that don t match your imaging condition, by focused stacking. Here we do this by building on a concept called pseudostation stacking introduced by Neal and Pavlis [1999]. Pseudostation stacking was originally conceived as a way to interpolate P d S wave fields onto a uniform grid of image points. The key insight for this paper, however, came from Treitel et al. s [1982] use of the t p transform in the migration of standard seismic reflection data. This gave us the idea to utilize plane wave decompositions as the foundation for imaging P d S wave fields. [4] The goal of this first paper in a two-part series is to introduce the theoretical aspects of our 3-D imaging technique. Additionally, we demonstrate that the concepts introduced here are sound by applying a 2-D implementation of our algorithm on some simple simulation models. As a final step, we apply the technique in 2-D to data from single event recorded by the Lodore array in northwestern Colorado. This single event illuminates a lithospheric scale discontinuity that we interpret as correlative with the Cheyenne belt. Our results suggest that a lithospheric scale discontinuity separates the Archean age Wyoming Province from the Proterozoic age Colorado Plateau. The second paper of this series focuses on the practical aspects of applying our imaging technique to image Earth structure, namely, stacking multiple, migration P d S images and a more detailed investigation of the imaged Cheyenne Belt structure. 2. Fundamentals [5] The methodology described in this paper is founded on the distorted wave Born approximation [Taylor, 1972; Devaney and Oristaglio, 1984]. The Born approximation assumes scattering is induced by small enough perturbations in the background medium velocity that the wave field can be written as a simple linear combination of two terms: u impulse ðr; s; t Þ ¼ G 0 ðr; s; tþþu sc ðr; s; tþ; ð1þ where r and s are source and receiver locations, respectively, t is time, and G 0 (r, s, t) is the incident wave field in the reference medium that propagates from r to s [Miller et al., 1987]; u sc is the scattered wave field. We use the impulse subscript on u in equation (1) to emphasize that our starting point is Green s function for the medium. A fundamental assumption of all teleseismic body wave imaging, which this paper shares, is that the observed teleseismic wave field can be expressed as ur; ð s; t Þ ¼ s 0 ðþ t * u impulse ðr; s; tþ ¼ s 0 ðþ t * ½G 0 ðr; s; tþ þ u sc ðr; s; x; tþš; ð2þ where s 0 is the quantity commonly called the source time function and the asterisk denotes convolution in time. The solution of the forward problem for computing u sc for an elastic medium is given by Wu and Aki [1985], Beylkin and Burridge [1990], and Bostock et al. [2001]. 3. Time Domain Deconvolution and Wave Field Interpolation [6] For imaging P d S modes a key assumption is that the longitudinal component (the vertical is often used by some people as an approximation, but in our implementation we have used the longitudinal component) is dominated by s 0 and the transverse components (2 degrees of freedom) are related by a simple convolution that can be written in matrix form as h ¼ Vi; where h is the recorded radial or transverse trace, V is a n m matrix with the rows of the matrix being filled with the discretely sampled longitudinal seismogram [e.g., Gurrola et al., 1995, equation (3)], and i is the time series of P d S arrivals that we seek to estimate that is commonly called a receiver function. [7] We use the multichannel, generalized inverse form of equation (3) introduced by Neal and Pavlis [1999, 2001]. This solution utilizes two extensions to previous P d S wave field deconvolution procedures. First, we use the pseudostation stacking method described by Neal and Pavlis [1999]. This approach interpolates the wave field onto a uniform grid of surface image points, x i (i =1,2,..., N g, where N g is the number of grid points), using a weighted slant stack of N s recording stations, ûx ð i ; tþ ¼ XNs j¼1 w ij u j t p x 0 j where here and in related equations that follow we use u with various appendages to denote a set of threecomponent seismograms. In equation (4), û is the pseudostation stack, and u j is the original seismic data recorded at station j whose location in space is x j 0. The seismograms, u j, are time shifted by a plane wave lag, t p x j 0, based on the slowness vector p determined by conventional array processing. Note that in practice we have found the plane wave approximation used in equation ð3þ ð4þ

3 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION ESE 18-3 (3) does not correctly account for time shifts due to near receiver structure and static corrections are required to properly align the stack with most real data. In practice this means we align all signals with first arrival times. The weights, w ij, are defined by a spatial averaging function. Following Neal and Pavlis [1999], we use a Gaussian function w ij ¼ w x i x 0 j ¼ exp kx! i x 0 j k2 2L 2 ; ð5þ where L is an aperture parameter whose scale depends on the experimental geometry and the local scattering properties of the lithosphere. [8] Neal and Pavlis [2001] show that pseudostation stacking in combination with time domain deconvolution can be viewed as a three-dimensional (two space dimensions and time) inversion operator that deconvolves the incident wave field from finite, irregularly sampled data. They also argue that an important property of pseudostation stacking in deconvolution is that it is a compromise between two existing end-members. That is, standard, single-station deconvolution methods use point samples of the wave field. The other end-member is that suggested by Li and Nabelek [1999]. They design one deconvolution operator for the entire array using a simple stack of all data. Pseudostation stacking is a compromise that effectively smooths the incident wave field over the scale length defined by L. This is a useful generalization as Neal [2001] showed that the appropriate scale for L depends on the scattering properties of the Earth beneath the array. In any case, the use of pseudostation stacking in deconvolution achieves two useful things simultaneously: (1) it provides a mechanism to interpolate the wave field from an irregular to a regular grid; and (2) it produces a localized average of the wave field around each image point that provides an improved estimate of the incident wave field in the presence of scattering. 4. Plane Wave Depth Migration 4.1. Wave Field Separation [9] A reliable deconvolution is a fundamental first step in imaging of P-to-S conversions. Not only does the quality of the deconvolution control the effective signalto-noise ratio of the data [Poppeliers, 2001; Poppeliers and Pavlis, 2003] being analyzed, but the process is also fundamental to wave field separation. Wave field separation is problematic in imaging P d S modes because the signals of interest, in general, are buried underneath the incident wave field. The reason that P d S imaging is tractable results from two fundamental properties of P d S modes: (1) S modes always travel slower than the incident P wave causing conversions to arrive with a predictable time delay relative to P, and (2) S modes are polarized in a direction orthogonal to P in an isotropic media. We now show the first property can be exploited to achieve wave field separation of the deconvolved data through muting. The second aids the separation process when ray coordinates are used. [10] Pseudostation stacking allows us to produce an estimate of the deconvolved vector wave field at an arbitrary image point, u d, through the relation û d x 0 j ; t ¼ ¼ ¼ V g i ðx i ; p 0 Þûx ð i ; tþ X V g Ns i j¼1 w ij u j t p 0 x 0 j V g i G 0 þ V g u sc ; ð6þ where V i g is the generalized inverse operator at pseudostation imaging point i constructed from the pseudostation stack at x i from Toeplitz equation (3). We use p 0 for the slowness vector to emphasize it is defined by the incident wave field. In the section that follows p becomes a variable. The second equality in equation (6) follows by substitution from equation (4) which defines the pseudostation stack. [11] If the distorted wave Born approximation is valid, we can substitute equation (2) in equation (6) to produce û d ðx i ; t Þ¼ðV g i s 0 " X Ns Þ * w ij G 0 x 0 j ; s; t j¼1 þ XNs j¼1 # w ij u sc x 0 j ; p 0 x 0 j ; where s 0 is the discrete form of s 0 (t) that is required since we express the deconvolution operator in matrix form. Conceptually, equation (7) is equation (2) with a deconvolved source time function, V i g s 0. The left term in the brackets is the incident wave field passed through pseudostation stacking and the right term is the scattered wave field passed through the same operator. [12] A simple method for wave field separation of P d S modes follows directly from equation (7). The initial pulse on the deconvolved seismograms is polarized in the longitudinal direction and represents the deconvolved incident wave field that we identify with G 0 in equation (7). Consequently, we need only define a front-end mute function 8 < 1 t > t; Mt ðþ¼ : 0 otherwise: with t = t t P defined as a delay time relative to the P wave arrival time, t P. Wave field separation is achieved if t is a long enough time to effectively eliminate the term involving G 0 in equation (7) so we can write Mt ðþû d ðx i ; tþ Mt ðþ " X Ns rt ðþ * j¼1 ð7þ ð8þ # w ij u sc x 0 j ; t p 0 x 0 j : ð9þ This says that the muted, deconvolved, pseudostation stacked traces (left-hand side of equation (9)) can be viewed as a stack of the impulse response of the scattered wave field convolved with r(t), where rt ðþ¼v g s 0 : ð10þ

4 ESE 18-4 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION We use the symbol r(t) to clarify that this function is a row of the resolution matrix defined by R = V g V which is familiar to many readers from geophysical inverse theory. [13] It is important to recognize that there are at least three fundamental limitations that follow from equation (9). First, the result is only as good as the deconvolution operator V g. Sidelobes in r(t) and random noise amplified by the deconvolution will introduce noise into the result (Poppeliers [2001] or Poppeliers and Pavlis [2003]). Second, the matrix used to construct V g uses longitudinal component traces that are commonly affected by a variety of P multiples that are difficult to remove by the deconvolution methods described above. Finally, there are sourceside complications that currently limit the data that can be utilized in P d S imaging. The most common example is intermediate depth events where the pp phase arrives within the time window of interest. In the language of equation (7) the approximation fails because the incident wave field has a component at large t that cannot be muted. The way to deal with this problem at present is to use only data from deep events where pp is completely outside the analysis time window or to use shallow events where depth phases can be absorbed into an equivalent source time function Plane Wave Decomposition [14] The approach we use is based on a 3-D plane wave decomposition and a comparatively simple imaging condition that builds on work by Treitel et al. [1982]. The fundamental concept is to decompose the deconvolved wave field estimates into a complete suite of plane wave components as in the well know t p transform and then recombine them in space with the proper imaging condition (Figure 1). [15] Analytically, we express this as û d ðx i ; p; tþ ¼ XNs a ij MðtÞV g u j j¼1 t dp x 0 j ; ð11þ where dp = p p 0 is the slowness vector relative to the incident wave field. It defines a plane wave shift of the muted, deconvolved seismograms (MV g u j ) that can be used to point each pseudostation array through a full range of feasible propagation velocities and azimuths. [16] We use the symbol a ij instead of w ij in equation (11) to emphasize that these are not necessarily the same weights as those used to design the deconvolution operator in equation (6). We can expect that the optimal choices for a ij and w ij are completely different because the objectives are different: deconvolution requires the best possible estimate of the incident wave field, while in equation (11) our primary objective is to interpolate the wave field onto a regular grid in space-time and slowness. In all the actual calculations presented in this paper the a ij and w ij weights are computed with equation (5) using the same aperture parameter L. This is almost certainly not an optimal choice, but full appraisal of the optimal choice of the a ij relative to w ij is a subject for future work. [17] With these caveats, there are two secondary issues related to equation (11) the reader should recognize. First, this approach assumes that the scattered wave field can be locally treated as a plane wave. This approach is analogous to concepts used by Hill [1990, 2001] in his Gaussian beam migration operator. Like the Gaussian beam approach this provides a useful way to control the smoothing of the migrated image in a known way. Second, equation (11) clarifies a very pragmatic reason why the mutes, M, are necessary. Without mutes, the large deconvolved pulse at zero time lag, which is commonly an order of magnitude larger than deconvolved P d s modes, would splash across the stacked section defined by equation (11) and dominate the result. We stress the fact that applications of mutes is a nonlinear processing step and they must be applied after deconvolution Imaging Condition [18] The methodology we introduce here builds on two quite different papers on migration: (1) Treitel et al. [1982] describe an imaging condition in 2-D for a seismic section decomposed into plane wave components, and (2) Miller et al. [1987] treated the migration problem as an inverse scattering problem and developed an analytic solution that has an implicit plane waveform in three dimensions. In particular, they find hfðx 0 Þi ¼ 1 Z cos 3 aðr; x 0 ; sþ p 2 c 3 0ð x 0ÞAðr; x 0 ; sþ u scðr; s; t ¼ t 0 Þd 2 xðr; x 0 ; sþ; ð12þ where F is the scattering potential that we wish to estimate from the scattered wave field, u sc. The fraction involving the velocity, c 0, at the image point x 0, the amplitude term, A, and the scattering angle, a, is a weighting term that scales how different points in the scattered wave field are stacked at a particular image point. (Details of the geometry of the terms in this equation are shown in Figure 6 of Miller et al. [1987] which will not be repeated here for brevity s sake.) Because we are attempting to recover only P d S scattering potential we can use a modified version of the acoustic approximation of Miller et al. [1987]. It should be noted, Figure 1. (opposite) The concepts of plane wave migration. (a) A scatterer that sheds P d S modes in a host of directions (dashed lines) that are not parallel to the incident wave field (thick, solid line). If we record the P d S modes at a series of stations at the surface, the nonincident P d S modes will be bright when the array is aiming in a direction parallel to the ray parameter of the nonincident P d S mode. This is because of the coherent pseudostation stack for that particular ray parameter. We define symbols that enter in equations (12) and (17). (b) A different perspective. A collection of pseudostation stacked seismograms on a regular grid can be beam formed along a given slowness vector p that is related to a downward projection along parallel ray paths to a given depth referred to here as an image surface. (c) Extension of this concept to three dimensions. We can define a time to depth transformation operator that maps the data space onto physical space using a ray-based back projection. For velocity models that vary only in the vertical direction, constant depth surfaces are surfaces of constant delay time, t, defined in equation (15). In three dimensions the image surface is warped into an irregular surface as illustrated here.

5 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION ESE 18-5 however, that a more complex elastic wave formulation derived by Beylkin and Burridge [1990] and implemented by Bostock and Rondenay [1999] and Bostock et al. [2001] can, in principle, recover elastic wave properties of the medium by including additional scattered wave field components. We take a different approach and treat the problem strictly as imaging aimed at recovering the P d S scattering potential. [19] The above inversion formula assumes that ray theory can be used to describe the propagation of the P d S wave field in the medium. We further assume that by some other methodology (e.g., teleseismic P and S wave tomography) we have a reliable three-dimensional P and S wave velocity model which we will write as v p (x) and v s (x), respectively. If we let i S (x i, p, z) denote the ray path computed in v s for slowness vector p from the pseudostation position x i to a

6 ESE 18-6 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION given depth z, then the propagation time for the S phase to a given scattering point at depth z can be computed from the path integral Z t s ðx i ; p; zþ ¼ S i ðx i;p;zþ 1 v s ds; ð13þ t s is not the same as the delay time, t, used in equation (11) because t is defined relative to the incident P wave arrival time. For this we need to also compute the similar travel time Z t p ðx i ; p 0 ; zþ ¼ P i ðx i;p;zþ 1 v p ds; and we can then compute a delay time as ð14þ tðx i ; p; zþ ¼ t s ðx i ; p; zþ t p ðx i ; p; zþ: ð15þ [20] Equation (15) is the basic imaging condition for this method. Although it is equivalent to the point source geometry shown in Figure 1a, it is better understood as a plane wave imaging condition for transforming a full gather of time domain data at constant p to depth. Figure 1c illustrates what this mapping operation does. It takes an initial cube of data (fixing p on the left hand side of equation (11) yields a function of x i and time lag, t) and maps the result onto a distorted cube in space. In the case of a 1-D velocity model this process can be visualized by taking a deck of playing cards and skewing them to have nonvertical sides. Each card would represent one time step in the raw data. The line formed by the edge of the deck of cards is what we call S i. In a three-dimensional velocity model the process can be visualized as allowing the playing cards to separate and become wrinkled into nonplanar surfaces. Finally, we note that equation (15) is equally appropriate for a spherical Earth. The only additional problem is that ray paths have to be computed in spherical geometry so the distorted cube in Figure 1c is warped into the shape of a spherical shell Final Stack [21] Given the conversion from delay time to depth with equation (15) the final image is constructed as a weighted stack ^fðx 0 Þ¼ Z q2 ðp;zzþ f2 ðp;zþ q 1 ðp;zþ f 1 ðp;zþ W½qðp; zþ;fðp; zþš^u t d½ x 0;qðp; zþ; fðp; zþšdfdq; ð16þ where f and q are standard azimuthal and zonal angles in spherical coordinates. They are direct functions of the ray geometry used in the imaging condition (equation (15)) and hence depend explicitly on the slowness vector, p, and depth. û t d is the data defined by equation (11) mapped into the image space, x 0 as described above. In addition, we added the superscript t to u d to emphasize that we use the data rotated into a direction orthogonal to the computed surface emergence direction of the ray corresponding to an t S phase with slowness vector p. (u d remains vector valued, but is reduced from three to two components.) Figure 1 illustrates the rational for this rotation. It focuses the stack on converted S modes. [22] Equation (16) describes the summation of energy along isochron surfaces defined by the difference in P and S travel times. Note, this is quite similar to diffraction stack migration, except we have included a weighting term W which is often neglected in diffraction stack migration schemes. In Appendix A we show how to modify Miller et al. s [1987] inversion formula to derive the following formula for the function W in equation (16): Wðqðp; x 0 Þ; fðp; x 0 ÞÞ ¼ kr stðr; x 0 Þþr p tðx 0 ; sþk 3 8p 2 : ð17þ A The length of r s t and r p t scale with the velocity model s slowness at the image point for S and P, respectively and have the geometry shown in Figure 1. In Figure 1a the vector x is a unit vector pointing in the direction defined by the vector r s t + r p t. Since it defines a direction, x directly maps onto the angles in spherical coordinates given in equation (16). A is an amplitude term that corrects for geometric spreading. [23] Miller et al. [1987] and Bostock et al. [2001] utilize a change of variables to recast this integral into a form involving experimental coordinates (e.g., midpoint coordinates in a common offset gather). The plane wave formulation make this unnecessary because the domain of integration in equation (16) maps directly to spherical coordinates. The range of solid angles covered varies spatially, however, due to depth dependent ray bending and spatial variation in effective coverage that strongly limits the range of integration on the edges of the array. (We discuss pragmatic consequences of this in section 7.) [24] The existing theoretical foundation of inverse scattering can be used to argue that equation (17) is the correct weighting function. Equation (17) is needed in order to preserve amplitudes on the migrated image. That is, our objective is to produce an image scaled to the P-to-S scattering potential at each image point, and equation (17) is the correct form within the limits of the various approximations made here. Given the station density of existing teleseismic data, however, we have found the results depend weakly on the details of how this weighting function is defined. To test the robustness of our theoretical framework we experimented with end-members of a weighting function that varied from W = constant to W defined as a linear weighting function peaked at scattering angle near 180 (the opposite of equation (17) which has a minimum at 180). The results differed very little. The likely reason is that the impulse response of the effective migration operator [Schneider, 1999] depends only weakly on the form of the weighting function. 5. Tests With Synthetic Data [25] Simulations are a necessary tool to validate our theoretical approach and study potential limitations. We first examine a pure diffraction hyperbola modeled by a point scatter in a homogeneous medium. In our test case, the source time function is a simple second derivative Gaussian

7 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION ESE 18-7 Figure 2. A 2-D point scatterer simulation. (left) Raw, unstacked time record section of a pure diffraction hyperbola produced by a point scatterer in a constant velocity medium (velocity of 5 km/s). The trace separation is 1 km, and the illuminating wave field is vertically incident. (middle) Raw data after the pseudostation stacking operation. Note how the pseudostation stacking operation enhances the flat portions of the diffraction hyperbola but produces a stair-step effect on the dipping portions. (right) Plot produced by applying our migration operator (in two dimensions) to this synthetic data set. Applying our plane wave migration scheme to this data set collapses the diffraction hyperbola. Notice, however, that the point diffractor is smoothed in the horizontal direction. This is due to the pseudostation stacking operation incorporated into the migration scheme. The width of the pseudostation stacking window is indicated at the top. function of specified frequency. Hence this test is focused exclusively on the migration process and does not account for the interaction of migration and deconvolution that is present in real data. By applying our plane wave migration in 2-D we produce the results shown in shown in Figure 2. The plane wave migration operator is applied in two dimensions by limiting the azimuthal angle (f in equation (16)) to one value, in this case f = 0. Experience in reflection seismology imaging suggest that the results obtained from a 2-D implementation are no different than a full 3-D implementation if no out-of-plane arrivals are present. Note that we successfully recover the point diffractors. The point diffractors, however, are smeared horizontally, but this is expected when one considers that pseudostation stacking is incorporated into the migration algorithm. If we consider equation (11), then we can see that the final image will be smoothed in the horizontal direction. If we use a pseudostation stacking window with aperture L, the final diffraction point will be smoothed horizontally over a scale length L. While the inclusion of the pseudostation stacking operator may seem to have undesirable effects on the final image because it limits horizontal resolution, this is tempered by the fact that the same property reduces background noise by lateral smoothing. Furthermore, it is intuitively clear that it is not reasonable to expect that it is possible to obtain a horizontal resolution of a migrated image that is better than the nominal station spacing of the array [Poppeliers, 2001]. Thus, by setting the pseudostation stacking window L in plane wave migration to the nominal station spacing and using all of the array data, we are not significantly reducing the resolution of the migrated image. [26] Figure 3 is a related test with a horizontal and dipping layer in a homogeneous medium. Although the flat layer is recovered with minimal distortion, we see migration artifacts for the dipping layer that take two main forms. The first artifact is an overall decrease in the dominant frequency content of the imaged structure, which is likely due to

8 ESE 18-8 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION Figure 3. Dipping layer simulation for an array with a uniform station spacing. (left) A dipping layer with a moveout slowness of 0.07 s/km after pseudostation stacking of a synthetic array of uniform station spacing. The recording stations are spaced 4 km apart, the pseudostation spacing is 4 km, and the velocity of the model is a constant 6 km/s. The horizontal and vertical scales are approximately equal. Notice the distortion of the dipping layer introduced by the pseudostation stacking operation. (middle) Result of applying the plane wave migration. The flat layer is recovered with very little distortion or vertical smearing. Note that the dipping layer is no longer shingled. The plane wave migration operator reduces the frequency content of the imaged layers, but high-frequency relics are introduced due to spatial aliasing. (right) Application of a low-pass filter, eliminating these high-frequency relics. stacking the plane wave components. The second artifact is in the form of high-frequency arrivals after the main arrival. However, the artifacts are not severe enough to hide the underlying structure and the true dip is correctly recovered. Additional tests like this show that as the dip of a layer increases, the degree of migration artifacts also increase. This indicates to us that these artifacts are a result of spatial aliasing and imply that there is a limit to the dip which we can coherently resolve with our migration scheme. [27] An important phenomenon is also observed in Figures 2 and 3 that provides some insight into why this technique can work even in the presence of relatively wide station spacing. The pseudostation stacking operation sums data in localized zones with a simple plane wave correction defined by a specified slowness vector. Therefore the pseudostation stacking operation will enhance signals across the pseudostation array aperture (L in equation (5)) that have a dip consistent with the specified slowness. However, when stacking at a slowness inconsistent with the actual dip the pseudostation stacking operation will produce a stair-step, or shingled, type pattern seen in the unmigrated panels of Figures 2, 3 4. Thus, even though the pseudostation stacking method breaks up a dipping layer when stacking on the incident wave field s slowness, p 0,the same feature will be enhanced when the array is pointed with the correct slowness for that feature. The final stack enhances the dipping layers when they are present and cancels the components of the wave field that shingle. [28] As a final test, we simulate the same Earth model as in Figure 3. However, we use a nonuniform station spacing (Figure 4). This simulates a common geometry of broadband arrays deployed to date. When the recording array has a nonuniform and highly irregular station spacing, our migration scheme produces more high-frequency migration artifacts. Because the plane wave migration scheme utilizes pseudostation stacking to interpolate the wave field between real recording stations, a sparse station spacing will result in horizontal spatial aliasing of smaller horizontal wave numbers. This phenomenon is well documented in the seismic reflection literature, where it is shown that the ability to image steeply dipping layers without spatial aliasing is a function of station density [e.g., Yilmaz, 1987, pp , ]. For plane wave migration of P ds images, we see that we are presented with the same fundamental limitation; the ability to effectively image dipping layers is a function of the frequency content of the wave field, the magnitude of dip, the uniformity of station spacing, and the array density. [29] The high-frequency relics in the migrated image are due to spatial aliasing introduced by the final stacking operation defined by equation (16). Therefore, by applying a low-pass filter to the final, migrated image we can reduce the spatial frequency content of the image to a value appropriate for the dip of the structure we are imaging and the frequency content of the original wave field. Thus we can eliminate the high-frequency relics based on the

9 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION ESE 18-9 Figure 4. Dipping layer simulation with nonuniform station spacing. The actual stations locations are indicated by the dots. The input model is identical to that in Figure 3. (left) Time record section after pseudostation stacking and (middle) migrated image. (right) A low-pass-filtered version of Figure 4 (middle) applied to remove aliasing artifacts. assumption that the frequency content of the relics simply do not exist in the original wave field. Empirically, we find that if we limit the spatial wavelength of the migrated image to greater than half the magnitude of the spatial smoothing window, then we can mitigate the effects of the highfrequency relics, as shown in Figures 3 and Applications to the Lodore Array [30] To demonstrate the utility of our migration scheme on real data, we focus our investigation on the Cheyenne belt in northwestern Colorado (Figure 5). The Cheyenne belt has been interpreted as a suture between the Archean rocks of the Wyoming province to the north and the Ma eugeoclinal rocks to the south [Karlstrom and Houston, 1984; Bickford, 1988; Karlstrom and Humphreys, 1998; Smithson and Boyd, 1998; Lester and Farmer, 1998]. Current models suggest that the suture is a thrust zone with the younger rocks to the south accreted onto the Archean Wyoming craton. While geologic investigations of the Cheyenne belt are necessarily limited to the surface expression of the suture, broad-scale geophysical data indicates that the suture may extend for significant depths into the lithosphere. [eg. Henstock et al., 1998; Lowry and Smith, 1995; Lerner-Lam et al., 1998]. However, because broad-scale geophysical studies are typically not designed to obtain a resolution fine enough to directly image individual fault zones, little is known concerning the structure of the suture at depth. Likewise, targeted, high-resolution geophysical studies such as seismic reflection profiles [e.g., Knapp et al., 1996] are often not capable of interrogating the mantle to lithospheric depths due to the large artificial sources required. Therefore applying plane wave migration to receiver function images of this area is well suited to provide high-resolution images of this Proterozoic suture. [31] For the preliminary results presented in this paper, we applied our migration algorithm to a single, deep (209 km) South American subduction zone event (m b = 5.2) recorded by the Lodore array in northwestern Colorado. The approximate distance of the event is 50 with a back azimuth of 132. Note that we only apply plane wave migration to a single event in order to test the algorithm and to demonstrate its effects on single events. Imaging with multiple events presents a new challenge that is the primary topic of Poppeliers and Pavlis [2003]. [32] The Lodore array is located directly over the inferred location of the Cheyenne belt because it was deployed to image this feature. For this preliminary work, we reduced the problem to two dimensions to make the problem computationally tractable. The two-dimensional sections shown in this paper strike roughly perpendicular to the Cheyenne belt. In this way, we minimize out-of-plane arrivals and are able to produce a reasonable approximation of an infinitely long 2-D model. [33] In Figure 6, we compare images which have been produced by (1) pseudostation stacking and deconvolution and (2) plane wave migration. We argue that the Moho is visible in both images, but with a different appearance. The most striking difference is a change in frequency content of the images. As we showed with the synthetic data (Figures 3 and 4), plane wave migration tends to lower the frequency content. Although we could have applied filtering to the images to equalize the frequency content, we chose to display the images after processing without any postprocessing filtering, as this allows us to make direct compar-

10 ESE POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION Figure 5. The Lodore array in northwestern Colorado. The array stations are shown by the inverted triangles, and the inferred location of the Cheyenne belt is shown as the heavy, dashed line. The grid of pseudostation used to produce the 2-D P ds images shown in Figure 6 are shown by the dots (adapted from Neal and Pavlis [1999]). isons of the algorithms. Another apparent difference in the two images in Figure 6 is that what we interpret as the Moho appears to be a slightly different depths in each image. This is likely due to two factors working in tandem. First, the tendency of plane wave migration to lower the overall frequency content of the final image has slightly reduced the vertical resolution of the imaged Moho. Second, plane wave migration may be inducing a slight phase shift to the data in the final image which gives an appearance that the Moho reflector has shifted in depth. The phase shift could be induced by an incorrect velocity model and would be more apparent for single-event images. By stacking multiple plane wave migrated images, the phase shift would cancel. [34] The Moho appears to truncate at the surface location of the Cheyenne belt. The truncation of the Moho suggests a steeply dipping discontinuity (labeled as CB in Figure 6) which corresponds well to the south dipping suture noted by Lester and Farmer [1998] and Smithson and Boyd [1998] and may mark a sharp transition from the Archean lithosphere of the Wyoming Province to the north and the Proterozoic lithosphere of the Colorado Plateau to the south. The exact nature of this transition is not known; however, our results imply that the Moho is displaced, suggesting that the suture extends to lithospheric depths. The image we see in Figure 6 may be the expression of a Proterozoic aged lithospheric fault system separating the Archean Wyoming Province from the Proterozoic Colorado Plateau. This implies that the lithosphere on either side of the suture has distinctly different physical properties, and can be imaged seismically. Indeed, this interpretation is supported by estimates of effective elastic thickness [Lowry and Smith, 1995] and seismic velocity studies [Henstock et al., 1998; Lerner-Lam et al., 1998]. In all these studies, it was found that a significant change in lithospheric properties is present across the Cheyenne belt. 7. Discussion and Conclusions [35] The spatial resolution of migrated seismic images has been studied extensively [e.g., Beylkin et al., 1985; Cohen et al., 1986; Bleisten, 1987; Chen and Schuster, 1999]. The general consensus is that the horizontal and vertical resolution is a function of the array aperture, the depth of the scatterer, and the spectral bandwidth of the illuminating wave field. While stacking multiple events will tend to broaden the spectrum of the illuminating wave field, the depth of a particular point scatterer and the array aperture are fixed. Therefore there are limits to the depth at which a given array can accurately see. Chen and Schuster [1999] indicate that the horizontal resolution of a point scatterer is linearly proportional to the depth of the point scatterer, scaled by a constant. However, this relationship was developed for seismic reflection methods, in

11 POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION ESE Figure 6. The effect of migration compared to pseudostation stacking. These results were produced from a single event recorded by the Lodore array. (top) Passed through a 2-D implementation of the plane wave migration procedure. (bottom) Pseudostation stacked data back projected along the incident ray pay. Note that the image produced by the plane wave migration operator has a lower overall frequency content (no filters have been applied to this image after the migration), which is what we observe in the synthetic tests. A and A 0 correspond to the locations shown on Figure 5, and CB is the location of the Cheyenne belt at the surface. The discontinuity of the Moho and a deeper (70 km) structure suggest that a lithospheric scale discontinuity exists between the Archean lithosphere to the north and the Proterozoic lithosphere to the south. In both sections the width of the pseudostation stacking window is 15 km, and the pseudostation spacing is 4 km. which a spherical wave reflects off of a point scatterer at depth. However, it illustrates the point that any migration scheme will have depth resolution issues which are based on array aperture. [36] The technique we describe here has coverage issues that limit resolution in different ways than those encountered in seismic reflection. These coverage issues are similar to those encountered in body wave travel time

12 ESE POPPELIERS AND PAVLIS: PLANE WAVE MIGRATION approach of Bostock et al. [2001] has great promise, but because it considers all data simultaneously, it cannot readily compensate for variations in source quality. As Poppeliers and Pavlis [2003] show, our approach is amenable to developing a weighted stack that can balance the results for source variation and uncontrolled variation in source locations. In any case, the reader must recognize that direct imaging of this kind of data is at its infancy and a large number of important details remain to be worked out before we can develop this technology to a level approaching that of modern reflection seismic imaging. Figure 7. Resolution limits of migration. A simple geometric example shows that the depth to which we can accurately migrate a continuous horizon is a function of the array aperture and the range through which we sweep the ray parameter. In the case of a vertically incident wave field, an array aperture of 130 km, and 30 > d >30, the deepest that a flat, continuous horizon is completely migrated is 120 km (corresponding to layer B). When we attempt to image deeper layers, our migration becomes less effective; in other words, layer C is less migrated than layer B because fewer plane wave components are summed to produce the result. tomography with arrays like this. That is, one can see that if a P d S phase originates from a location outside the array, then our migration scheme will correctly place this P d S arrival outside the volume directly beneath the array. From a relatively simple geometric argument (Figure 7), we can see that if the ray parameter is shifted by ±30 and our array aperture is 140 km, then the deepest that a flat conversion horizon will be reconstructed from the full range of slowness vectors is 120 km. This is less than one array aperture. Deeper targets and areas outside the central zone will of necessity be subject to more artifacts from incomplete coverage. This is an important point to remember when interpreting an image such as that presented in Figure 6. Notice that the image becomes less coherent at depths >80 km. While the inaccurate velocity model we used for the migration is a likely contributor to this incoherence, the incomplete ray coverage at this depth (see Figure 7) may also be important. [37] An important distinction between our approach to migration of P d S phases and others that have been tried recently is that our approach is directly comparable to a prestack migration method in reflection seismology. This is important because, as we emphasize in part II of this paper, natural sources are widely variable in signal strength due to large variations in source size. Backprojection methods [Sheehan et al., 1995; Dueker and Sheehan, 1997, 1998; Kosarev et al., 1999] are not complete migration methods and cannot discriminate edges correctly or remove the effects of diffractions. They are loosely equivalent to CMP stacking, and would require poststack class migration to remove diffraction artifacts. The inverse scattering Appendix A: Weighting Function for PdS Inversion [38] We begin with equation (17) of Miller et al. [1987] f ðx 0 Þ ¼ 1 Z 8p 2 Z d 2 x d 3 xd 00 ½x ðx 0 xþšf ðxþ; ða1þ where f(x 0 ) is the scattering potential function in the Earth at x as recorded by a receiver at position x 0 and x is a unit vector pointing in the direction perpendicular to a given isochron surface. For the P d S imaging problem it has the geometric relationship shown in Figure 1. Note that equation (A1) includes a delta function in the integrand. The delta function in the volume integral in equation (A1) collapses the spatial integration to the set of points satisfying the travel time relation shown by equation (15). This can be thought of as an isochron surface defined by the velocity of P and S phases. [39] Miller et al. [1987] give a lengthy mathematical argument to relate equation (A1) to the classical inverse Radon transform. The key argument is that a tangent to the isochron surface through an image point can be viewed as the plane of integration in the forward Radon transform. Their derivation is based on the acoustic scattering case. In the acoustic case the normal to the isochron surface is defined by the bisection of the lines formed by incident and scattered rays at each image point. (The geometry is shown in Figure 6 of Miller et al. [1987].) In the P to S conversion case the comparable relation is kbk¼kr s tðr; xþþr p tðx; sþk; ða2þ where r s t(r, x) and r p t(x, s) are vectors defined by the ray geometry at the subsurface image point x. The geometry is illustrated in Figure 1a which should be compared directly with Figure 6 of Miller et al. [1987]. b is the P d S equivalent of the comparable quantity defined in equation (23) of Miller et al. [1987]. With this association the remainder of their argument can be followed to yield the following inversion formula that is the direct analog of equation (27) of Miller et al. [1987]: Z ˆfðx 0 Þ¼ d 2 xðr; x 0 ; sþ kr stðr; x 0 Þþr p tðx 0 ; sþk 3 8p 2 u sc ðr; s; t¼t 0 Þ; A ða3þ where the t = t 0 condition means we extract the sample that satisfies the travel time imaging condition for a P to S