Comparison among Interferometric, Reduced-time and Kirchhoff Migration of CDP Data

One of the difficulties in seeing beneath salt is that the migration velocity in the salt and above is not well known. This can lead to defocusing of migration images beneath the salt. In this paper we show that reduced-time migration (RTM) and interferometric migration (IM) can mitigate this problem. Reduced-time migration shifts the data with the time difference between the calculated arrival time τ ref sg and natural arrival time τ̃ ref sg of a reference reflection, where s and g denote the source and receiver locations on the surface and τ ref sg is calculated by raytracing through the salt velocity model. We use the term natural and calculated to represent, respectively, the arrival times that are velocity-independent (times directly picked from the data without knowledge of the velocity model) and velocity-dependent (times calculated by raytracing through a given velocity model). The benefit of RTM is a significant reduction of defocusing errors caused by errors in the migration velocity. Interferometric migration on the other hand requires 1) extrapolation of the surface data below salt using the natural arrival times τ̃ ref sg of the subsalt reference reflector, and 2) migration of the extrapolated data below the salt. The benefit with IM is that no salt velocity model is needed so the model-based defocusing errors are, in theory, eliminated. To reduce computational time and artifacts we implement IM with a semi-natural Green’s function. We use the term semi − natural because the Green’s function is constructed by a combination of model-based (calculated) and picked (natural) traveltimes. Since no explicit data extrapolation is needed, IM with the semi-natural Green’s function approach is more cost efficient than the standard IM. We also tested both RTM and IM with semi-natural Green’s functions on both synthetic and field CDP data sets. Results show that they both can remove the significant kinematic distortions due to the overburden without knowledge of the overburden velocity, and 1 appear to overcome some distortion problems in the migration images obtained by Kirchhoff migration with residual statics. INTRODUCTION Similar to static problems due to topography and near-surface velocity variations, the arrival time distortions caused by an uncertain overburden velocity, such as complex basalt layers or salt domes, can severely degrade the quality of the migration image for the deep structures. Static corrections are sometimes not enough to solve this problem, and so datuming techniques are sometimes used before migration to correct the arrival time distortion before migration. However, most conventional datuming techniques (Berryhill, 1979, 1986; Yilmaz and Lucas, 1986; Bevc, 1995; Schneider et al., 1995) require a detailed knowledge of the velocity model above the datum horizon. Recently, several natural or velocity-independent datuming techniques were developed. Kelamis et al. (1999) presented a wave equation datuming scheme based on the principles of Common Focusing Point (CFP) technology (Berkhout, 1997a, 1997b). The velocity-depth model is expressed in terms of model-independent propagation operators which can be updated in an iterative way (Bolte and Verschuur, 1998). The virtual source imaging methods proposed by Calvert et al. (2004) and Bakulin and Calvert (2004) use the VSP traces as natural Green’s functions to extrapolate the surface data below salt without needing to know the salt or overburden model. However, one of the earliest natural redatuming methods is daylight imaging proposed by Claerbout (1968) and Rickett and Claerbout (1999). Here, traces are correlated and summed to transform free-surface multiples into primaries, where the sources are redatumed from depth to be on the free surface. An extension of daylight imaging is interferometric migration (IM) by Schuster (2001), Sheng (2001), Yu and Schuster (2001, 2004), and Schuster et al. (2004a, 2004b). Interferometric imaging performs both extrapolation and migration using the crosscorrelated data. The extrapolation part in daylight imaging or interferometric migration is similar to the virtual source approach of Calvert et al. (2004) and Bakulin and Calvert (2004), except that IM typically uses single arrivals for the time shifts while virtual source imaging uses a train of arrivals followed by deconvolution (Schuster and Zhou, 2005). A problem with interferometric migration is that it can be computationally expensive. A related interferometric method is that of migration by semi-natural Green’s functions introduced by Schuster (2003). In this method, the imaging condition for migrating multiples is constructed by a combination of traveltimes computed from the model and traveltimes picked from data. Under the stationary phase approximation and Fermat’s principle, the multiples are migrated without the expensive crosscorrelation of the data. One of the most recent methods for natural data imaging is that by Sheley and Schuster (2003), who suggested a simple way to mitigate migration defocusing prob2 lems caused by an incorrect migration velocity and statics: i.e., time advance the data by the arrival time of the direct wave. No data extrapolation is needed. Migration of reduced− time crosswell data resulted in more sharply focused and more accurately located interfaces between the wells compared to standard migration for 5-10 percent errors in the migration velocity. In this paper we show how the interferometric migration with semi-natural Green’s functions and RTM methods can be used to improve the quality of migration images obtained from CDP data. Instead of reducing CDP data by the direct-wave traveltimes, we reduce them by the adjusted traveltimes of a reference reflector. RTM and IM with semi-natural Green’s functions are tested on synthetic and field CDP data sets. Results show that the migration images below the reference layer from RTM and IM are more coherent than those from standard migration when there is an uncertain overburden velocity model above the reference layer. The first part of this paper describes the theory of RTM and IM for CDP data. This is followed by a section which presents test results for both synthetic and field data. THEORY Reduced-time Migration We now discuss the theory for reduced-time migration of CDP data. Figure 1 illustrates a reference reflector located just below the salt, where the geometries of the reflector and the salt body are not well known. The goal is to image the oil reflector despite significant errors in the velocity model of the salt. It is assumed that the migration velocity is well known below the reference reflector. At high frequencies, the primary reflections from the oil reflector and the associated reflections from a nearby reference reflector are given, respectively, by d(g, s) = r(xo)W (ω) e sxoxog, (1) d(g, s) = r(xo)W (ω) e sxoxog, (2) where the source is at s, geophone is at g, W (ω) is the spectrum of a source wavelet, the reflection coefficient and geometrical spreading effects are represented by r(xo) and r(x′o), and the specular reflection points along the oil and reference interfaces are given by xo and x ′ o, respectively. Also, τ̃ab represents the natural traveltime between a source at a and a geophone at b where the tilde denotes a picked or natural traveltime. Since both RTM and IM proposed in this paper need to identify the reflection event associated with a reference reflector, we limit the natural traveltime τ̃ab to the arrival time associated with the strongest energy (one of the local minimum arrival times when there are multi-pathing events between the source a and geophone b). 3 The reduced-time idea (Sheley and Schuster, 2003) is to shift the data by the time of the natural reference reflection time: φ(g, s) = d(g, s)oile−iωτ̃ ref sg , (3) where τ̃ ref sg = τ̃sx′o + τ̃x′og is the reference reflection time. These natural reference traveltimes can be picked from the data (typically picked from common offset gathers) and inserted into equation 3. Alternatively, the reference reflections in the data can be windowed and correlated with the original traces to get a good approximation to φ(g, s). For imaging beneath the reference reflector, the reduced data are migrated using the following formula: m(x) = ∑ ω ∑ s ∑ g φ(g, s)e−iω(τsx+τxg−τ ref sg , (4) where x is the trial image point beneath the reference reflector and τ ref sg is the calculated, not natural, reference reflection time. This calculated time τ ref sg is obtained based on the migration velocity above the reference layer and is used to roughly cancel the natural time shift τ̃ ref sg in equation 3. If there are no velocity errors between the migration velocity model and the true model, then the calculated time shift will exactly cancel the natural time shift in equation 4. However, if there are migration velocity errors, then the τ̃ ref sg − τ ref sg term will mitigate defocusing due to migration velocity errors. To justify this statement, substitute equations 1 and 3 into equation 4 to get: m(x) = ∑ ω ∑ s ∑ g d(g, s)oile−iω(τsx+τxg− 2ref timing error { }} { [τ ref sg − τ̃ ref sg ] , = ∑ ω ∑ s ∑ g r(x)W (ω)e−iω( 2oil timing error { }} { τsx + τxg − τ̃sxo − τ̃xog − 2ref timing error { }} { [τ ref sg − τ̃ ref sg ] , (5) where 2 = τ ref sg − τ̃ ref sg is the timing error between the computed and natural times for the reference reflection if x ≈ xo. Similarly, 2 is the timing error for the oil reflection, and according to our assumption about an erroneous salt model, all of the timing errors are confined to the miscalculations of traveltimes for waves propagating in the salt. Therefore, if the trial image point is at the specular reflection point x = xo, and the oil reflector is sufficiently near the reference reflector so that both the reference and 4 oil ray segments in the salt nearly coincide, then their timing errors are approximately the same: 2 ≈ 2. At x = xo, equation 5 becomes m(xo) = ∑ ω ∑ s ∑ g r(xo)W (ω)e −iω(2oil−2ref , = r(xo) ∑ ω ∑ s ∑ g W (ω), (6) which gives coherent summation over all frequencies for x = xo. The timing errors due to an incorrect migration velocity in the salt+overburden are canceled out by the natural and model-based time shifting of the data. This self-cancellation of timing errors due to an incorrect reference model is a robust feature not enjoyed by the interferometric migration method. Since we do not know the exact position of the reference layer, we approximate the location of the reference layer with the information picked from the standard Kirchhoff migration image. Although an error in the overburden velocity will result in an error in the positioning of the reference layer, we will empirically demonstrate in the next section that RTM and IM can accurately focus the reflection images under a reference layer even with a rough estimate of the reference layer’s position. In addition, we also assume that xo is located in the mid-point position defined by the shot and geophone locations. This assumption is valid for a flat reference layer in a layered overburden velocity model. For a complex earth model, this assumption results in timing errors which increase in proportion to the model complexity. One possible remedy is to use estimates of the reference layer dip and make the appropriate adjustments to the up and downgoing reference reflection times. If the xo point is much deeper than the reference reflector, then the 2 oil might not be a good approximation to 2 because there will be too large of a difference between the segment lengths of the oil and reference rays in the salt. One possible remedy is to estimate a scaling factor α = ray/ray so that ατ̃ ref is used as the time shift in equation 3. As shown in Figure 1, ray = syo + gy′ o and ray ref = sx′o + x′og are the segment lengths of the oil and reference reflection rays in the overburden structure, respectively. To approximate the segment lengths of the reflection rays, straight raypaths between shot/geophone position and the reflection point xo or xo are assumed in this paper. In summary, the implementation of RTM includes three steps: 1. Pick the reference reflection times τ̃ ref sg . 2. Create the shifted data with equation 3. 3. Migrate the shifted data with the reduced-time migration imaging condition in equation 4. Here, τ ref sg is the calculated reference reflection time with the migration velocity model. We assume that the reflection point xo is located at the 5 reference layer with a horizontal position coincided with the mid-point position defined by the source s and geophone g locations. Interferometric Migration with Semi-Natural Green’s Function The virtual source (Calvert et al., 2004) and interferometric migration (Schuster et al, 2004a, 2004b) approaches use the natural events (such as the direct arrivals) instead of the velocity model to extrapolate surface data below salt. A third approach related to IM is that of migration by semi-natural Green’ functions (Schuster, 2003). The natural downward continuation can be expressed by inserting natural picked traveltimes into the prestack Kirchhoff datuming algorithm: d(y′,y)oil = ∑ s ∑ g d(g, s)e −iω(τ̃ sy +τ̃ gy′ , (7) where s, g, y, y′ represents the shot/geophone positions at the surface and datum, respectively; τ̃ ref sy and τ̃ ref gy′ denote the natural traveltimes associated with a salt model as shown in Figure 2. Note, equation 7 differs from standard datuming methods (e.g. Bevc, 1995) because it does not require a velocity model. The amplitude term describing the geometrical spreading and obliquity is omitted for simplicity because the goal of IM is to remove the kinematic effects of the salt. Migration of the redatumed data in equation 7 yields the interferometric migration formula: m(x) = ∑ ω ∑ y ∑ y′ d(y′,y)oile−iω(τyx+τxy′ , = ∑ ω ∑ s ∑ g ∑ y ∑ y′ d(g, s)e −iω(τ̃ sy +τyx+τ̃ gy′ yx, (8) where x is the trial image point beneath the reference reflector. According to the stationary phase theory, the dominant contributions in the summations ∑ y and ∑ y′ are from the specular rays connecting the shot s, geophone g and the image point x (Jiang et al., 2005). This can be proved by substituting equation 1 into equation 8: m(x) = r(xo) ∑ ω W (ω) ∑ s ∑ g [∑ y e iω ( specular { }} { (τ̃ ref syo + τyoxo)− diffraction { }} { (τ̃ ref sy + τyx) ) ∑ y′ e iω ( specular { }} { (τ̃ ref gy′o + τy′oxo)− diffraction { }} { (τ̃ ref gy′ + τy′x) )] , (9) where the diffraction times are always greater or equal to the specular times when x → xo. The stationary phase approximation says that the dominant contributions to the 6 summations ∑ y and ∑ y′ are when the exponential arguments become stationary, i.e., the diffraction points y and y′ coincide with yo and yo, respectively (see Figure 2). In this case, the exponential arguments in equation 9 become zero so that the summation over s, g and ω will be perfectly coherent estimate of r(xo) weighted by the wavelet. To reduce the computational expense and correlation artifacts in equation 8 we replace the diffraction times in equation 8 with the specular times (Schuster, 2003): m(x) ≈ ∑ ω ∑ s ∑ g d(g, s)e −iω(τ̃ syo +τyox+τ̃ gy′oyox, (10) where the subscript o denotes the positions on the specular ray. The specular times τ̃ ref syo and τ̃ ref gy′o can be obtained by applying Fermat’s principle to the picked times τ̃ ref sy , τ̃ ref gy′ and model-based times τyx and τy′x: τ̃ ref syo + τyox = miny[τ̃ ref sy + τyx], and τ̃ ref gy′o + τxy′o = miny′ [τ̃ ref gy′ + τy′x], (11) where miny denotes the operation for finding the minimum traveltime τ̃ ref sy +τyx along the different y positions in the reference layer. In this paper, we assume single-pathing in the structure above the reference layer and limit the search area to be adjacent to the line defined by a straight line connecting the source s and the trial image point x. This assumption might limit the use of semi-natural Green’s functions for subsalt imaging with significant multi-pathing. In the future, we might be able to overcome this limitation by finding several local minimum traveltimes along the reference layer. In summary, the implementation of IM with semi-natural Green’s function includes four steps: 1. Pick the reference reflection time τ̃ ref sy and τ̃ ref yg . 2. Use the sub-overburden velocity and ray tracing to compute τyx for y at the reference interface and x below overburden. 3. Use Fermat’s principle in equation 11 to get τ̃sy0 + τy0x 4. Migrate with equation 11. NUMERICAL EXAMPLE The RTM method and IM method with semi-natural Green’s functions are tested on a synthetic and a field marine data set in this section. The resulting images are compared to the standard Kirchhoff migration method.