Inverse-Ray Imaging of a 3D Homogeneous Layer Based on Common-Shot Gathers of Oblique Profiles

Imaging of 3D structural interfaces through reflected rays shooting from common-shot gathers is presented in this paper. First, by fitting the reflected arrivals picked from common-shot gathers, we calculate apparent dips and the shortest distances between sources and reflectors along two profiles. Then, based on the geometry of the profiles and a planar reflector, a unit normal vector of the reflector is determined from the apparent dips and the azimuths of two oblique profiles. A special case, when apparent dips are zero along two parallel profiles, for determining the reflector normal is also investigated. We propose three criteria to ensure the selected travel-times along two profiles resulting from the same planar reflector. These are that firstly, the same shortest distance from sources to the reflector is utilized; and secondly, we want to ensure the same normal of the reflector, and finally, the same ray distance. between profiles or if at least two profiles are within a Fresnel zone. Although this technique deals with a single-layered problem currently, it is fundamentally important when we extend it to image inhomogeneous multi-layered media.

fixed velocity of the upper layer, an exact velocity analysis of the lowest layer was also presented recently (Wang 2005b).
The aim of this study is to fully use the existing 2D MCS profiles for constructing 3D structures since a 3D seismic survey is expensive. In this paper, we develop a prestack inverseray imaging of a 3D homogeneous layer based on inverse rays propagating along MCS profiles. Although the target for prestack imaging of a 3D homogeneous layer is similar to Pivot's work (1997), this technique is more efficient for handling MCS data recorded along oblique profiles. Furthermore, by considering the travel-time gradients of MCS data, this method can be extended for imaging 3D inhomogeneous multi-layered media.
In the following, we determine the side effect of inverse rays perpendicular to the profiles from at least two common-shot gathers recorded along oblique profiles. Based on the ray geometry of apparent dips along oblique profiles, a unit normal vector of a planar reflector and the reflection points are obtained (Chang 2004). We further propose three criteria for selecting the proper travel-time sets from all common-shot gathers. Finally, the prestack inverse rays are applied to image the bathymetry of a forearc basin, in the southernmost Ryukyu subduction zone where MCS lines are dense, and a complex model of SEG/EAEG over-thrust.

THEORY
Inverse rays propagating in 3D can be determined based on the ray geometry from at least two non-parallel profiles. By considering the least-square travel-time fitting of a commonshot gather along a profile, Wang (2002) derived the velocity (V) of a homogeneous layer, the apparent dip (α π i − /2) of a planar reflector, and the shortest distance (h i ) between the source ( S i ) and the reflector (Fig. 1). Therefore, a unit normal vector of a 3D planar reflector can be calculated from apparent dips of the reflector based on two common-shot gathers along two oblique profiles (Appendix A). However, the question arises as to how we can properly select two common-shot gathers in order to calculate an accurate normal of a planar reflector. In this paper, we present three criteria for selection of two common-shot gathers to determine the reflector normal and the reflection points.

Criteria for Incorporating Travel-Time Sets
For determining a reflector normal, travel-time sets of two common-shot gathers along two profiles are selected from reflected arrivals of all 2D MCS profiles. We propose the following criteria to ensure the selected travel-times result from the same planar reflector.

The Shortest Distance between the Source and the Reflector
Let the travel-time sets be selected from two common-shot gathers along oblique profiles (or the apparent dips are zero along parallel profiles) for which the shortest distances (h i or h j ) between the sources ( S i or S j ) and the reflector can be determined from fitting of the travel-time sets along profiles (Wang 2002). In particular, a reflection point (S j * ) of a ray emitting from the source ( S j ) and reflected back to the same source ( Fig. 1) can be written as: If the travel-time sets are associated with the same reflector, then projection of the vector S S i j * onto the reflector normal should be equal to the shortest distance h i , i.e., To substitute equation (1) into equation (2), we derive the first criterion of properly incorporating the travel-time sets for which the shortest distances (h i and h j ) between the sources ( S i and S j ) and the unit normal (N) can be determined. Fig. 1. Geometry of reflected rays shooting from sources ( S i and S j ) at the surface (XY plane), reflecting at a planar reflector ( S i * , A ik , S j * , and A jk ), and arriving at receivers ( R ik and R jk ) along two oblique profiles ( X i and X j ). h i and h j are the shortest distances from sources S i and S j to the reflector, respectively. Direction angles of S S i i * (or S S j j * ) with respect to the profile and Z axis are denoted by α i (or α j ) and φ , respectively. Projection of S S i i is along the dip line of the reflector.

The normal Vector of the Reflector
If an apparent dip along a profile is not zero (or α π i ≠ /2 in Figs. 2a, b), then, by considering the side effect perpendicular to the profile, the reflector is a tangent plane of a circular cone rotated about the profile and with a vertex: at the surface ( Fig. 2a, b). The vector B S i i * and S S i j * * lie on the reflector and can be written as: with the aid of equations (1) and (4). Therefore, the cross product of two vector B S i i should be checked to see whether it is parallel to the reflector normal (N). Alternatively, if an apparent dip along a profile is zero (or α π j = /2), then the reflector is a tangent plane of a circular cylinder rotated about the profile and with a radius of h j (Figs. 2b,c). In this case, the vectorX j along the profile lies also at the reflector. Therefore, the cross product of vectorsX j and S S i j * * should be parallel to the reflector normal. Special cases of the apparent dips about zero along two parallel profiles (Fig. A2) and along two oblique profiles (Fig. 2c) should also be investigated. According to equations (A4) and (A7), the true dip of the reflector for the later case is zero whereas the reflector of the former case is not necessarily horizontal. It can be also readily shown that the criteria introduced in Section 2.1 apply for both special cases.

Computation of Reflection Points
If a unit normal vector of a reflector (N), computed from equation (A4) or (A7) by considering two common-shot gathers along two profiles satisfies the criteria introduced in Section 2.1, we can determine the reflection points on the planar reflector. Let reflected rays be shot from a common source ( S i ), be reflected at reflection points ( A ij ), and arrive at multichannel receivers ( R ij ). According to the ray geometry in Fig. 3, the shortest distances between the receivers and the reflector are: Planar reflectors imaged from tangent planes of (a) two circularly conical envelopes when α α π i j , / ≠ 2, (b) a circularly conical envelope when α π i ≠ /2 and a circularly cylindrical envelope when α π j = /2, and (c) two circularly cylindrical envelopes when α α π i j , / = 2 along oblique profiles X i and X j . B i and B j are vertices of circularly conical envelopes along profiles X i and X j , respectively. Vectors where projection of the source ( S i ) and the receivers ( R ij ) along the reflector normal and onto the reflector are: Finally, we can also check whether the ray distance and the layer velocity are consistent with the reflected travel-times: 3. RESULTS

Hoping Basin
A forearc basin, Hoping Basin, is located near the Ryukyu Arc, Luzon Arc and Yaeyama Accretionary Prism in the southernmost Ryukyu subduction zone where the plate motion and earthquakes are active (Fig. 4). Subduction and arc-continent collision in the Ryukyu forearc region have resulted in an old Suao Basin beneath the Hoping Basin (Lallemand et al. 1997) and warping of the arc basement east of the Hoping Basin (Wang et al. 2004). Furthermore, on 31 March 2002, an earthquake with a magnitude of 6.8 and a focal depth of 9.6 km beneath the Hoping Basin resulted in strong damage in Taipei at an epicentral distance of about 100 km. Based on migration velocity analysis and horizon picks over MCS stacked sections, four major sedimentary layers of the Hoping Basin are imaged through 3D inverse reflected rays (Wang 2005a). In this paper, we use reflected arrivals of the common-shot gathers along seven MCS profiles in the Hoping Basin and apply the inverse rays to image the bathymetry of the basin.
Reflected rays are traced on the high-resolution bathymetry of the Hoping Basin (Fig. 5a). For inverse-ray imaging along seven MCS profiles, there are 450 shots with a shot interval of 1.25 km and a receiver interval of 0.5 km that are less dense than the real data set. According to the smoothness and continuity of the travel-time curves resulting from the same planar reflector, any two travel-time sets of common-shot gathers along oblique profiles are selected for determining the reflector normal from equation (A4). Criteria of the shortest distance between the source and the reflector, the reflector normal and the ray distance introduced in Sections 2.1 and 2.2 are considered to delete the improper selection of the travel-time sets.
Since the basin bathymetry (width, length and depth are about 67 km, 78 km and 5 km as shown in Fig. 5) is relatively flat, the out-of-plane reflections are limited. This implies that 2D inverse rays along MCS profiles are sufficient to image most of the bathymetry below the profiles. However, due to the sparse spacing of 10 -35 km, proper inverse-ray imaging cannot be achieved when the variation of the bathymetry exceeds 300 m between neighboring profiles (elliptical areas in Fig. 5b).

SEG/EAEG Over-Thrust Model
In quest of oil and gas embedded in the complex faults and folds, a 3D over-thrust model with eleven layers developed by SEG and EAEG has been intensively investigated for seismic modeling and inversion. In this study, we only image the fourth layer of the over-thrust model from the top (Fig. 6) since the complex faults exist in the central region of the layer. Reflected rays on the fourth layer of SEG/EAEG over-thrust model are traced along orthogonal profiles with profile spacing of 1 km and 4 km, respectively, in the complex and flat regions (black dots in Fig. 7c). Along a profile, the shot interval, near offset, receiver interval and channel number are set to 1.25 km, 0.21 km, 0.1 km and 48, respectively. A lot of reflection points in the central region of the thrust faults are not located along profiles since out-of-plane reflections are occurred due to the complexity of SEG/EAEG over-thrust model. Fig. 4. Seismic lines and bathymetry (Liu et al. 1998) in the southernmost Ryukyu subduction zone. White frame indicates the study area of the Hoping Basin. Reflected interfaces of the over-thrust model (Fig. 7a) are imaged by incorporating traveltime sets across profiles. However, due to incorrect selection of travel-time sets across profiles in the complex structure of the over-thrust model, the depth errors of inverse-ray imaging are greater than 0.2 km in the central fault zone (red crosses and circles in Fig. 7b). These large errors can be readily identified and thus can be disregarded when the imaging density is less than 200 points km 2 − (Fig. 7c) and/or the uncertainty of the imaging depth is greater than 0.2 km (Fig. 7d). Although inverse-ray imaging based on a 2D acquisition is insufficient to determine the whole interface of the model, it provides a general trend of the interface depth for which 3D imaging from other techniques is difficult based on the 2D data set. If the profile spacing is a few times denser than the current spacing, the full coverage and the accurate depth of the imaging interfaces can be expected.

CONCLUSION
Inverse-ray imaging of a planar reflector is presented based on reflected travel-times of common-shot gathers along two oblique profiles. We first assume all travel-times resulting from the same planar reflector, and then use any two travel-time sets along two profiles to determine the reflector normal. Several criteria based on the shortest distance between the source and the reflector, the same normal vector of the reflector, and the same ray distance are subsequently considered to delete incorrect results.
Applications of this method to a relatively flat basin and a complex over-thrust demonstrate the capability of inverse rays for imaging pseudo-3D reflectors between sparse profiles. Once travel times have been recorded along at least two profiles within a Fresnel zone, this method can resolve the structural interfaces for which 3D imaging from other techniques is difficult based on 2D data sets. Therefore, the inverse-ray method developed in this paper can image a pseudo-3D structure even when profile spacing (> 1 km) is far beyond the grid spac-  ing required in the classical approaches such as migration and tomography. Although the prestack inverse rays presented in this paper are less practical due to their limitations in determining of a homogeneous layer, they will be extended to image 3D inhomogeneous multi-layered media. For example, sedimentary and crustal structures can be investigated based on common-shot gathers from MCS acquisition and the common-receiver gathers from ocean-bottom seismometers (OBS) survey. The existing MCS/OBS profiles off eastern Taiwan provide good data sets for imaging seismogenic structures and for evaluating earthquake hazard in the southernmost Ryukyu subduction zone.
angles of the dip line (θ and θ i ) with respect to the X and X i axes are related to each other by considering the profile azimuth ( θ i * ) as shown in Fig. A1, we observe that: To substitute the equation (A3) into equation (A2), we obtain sin cos sin sin sin( ) sin sin cos cos Therefore, two profiles used for determining the reflector normal should not be parallel to each other. Equations (A1) and (A4) also indicate that X and Y components of the unit normal vector of the reflector can be determined from equation (A4)  = ] is when their apparent dips are zero (α π i = /2) so that equation (A4) is no longer applicable. For determining the unit normal vector of a reflector in this special case (Fig. A2), a unit vector perpendicular to the profiles is set:ˆs inˆcos* * X e e i x i y which is along the direction of the dip line of the reflector. By considering the right triangle ∆S EF i as shown in Fig. A2, the true dip (φ ) of the reflector can be determined from: Therefore, when the apparent dips along two parallel profiles are zero, the unit normal vector of the reflector is:ˆs Fig. A2. A planar reflector imaged from a tangent plane of two circularly cylindrical envelopes when α α π i j , / = 2 along two parallel profiles. X ⊥ is a unit vector perpendicular to the profiles. Fig. A1. Two oblique profiles ( X i and X j ) and the dip line of a planar reflector projected onto the surface. Azimuths of the profile X i (or X j ) and the dip line are denoted by θ i * (or θ j * ) and θ , respectively. θ i (or θ j ) is the angle between the dip line and the profile X i (or X j ).