A raypath-consistent receiver correction in PS converted wave processing through seismic interferometry: New application for tropical zones

The estimation of static corrections is an issue still unsolved for PS converted wave processing. Due to the PS converted wave usually arriving at the surface at non-zero angles, the surface consistent approach is no longer valid, and corrections become non-stationary, i.e. the correction is not static. Seismic interferometry is used in receiver gathers transformed to the radial domain to estimate functions that contain the delay caused by the weathered layer, considering the emergence angle of the PS converted wave. Inverse ﬁlters, derived from these functions, are applied by convolution to the raw traces to supply traces corrected for weathering layer effects. Seismic interferometry was satisfactorily tested in two synthetic models and then applied to a 2C seismic line from the Llanos Basin (Colombia). This is the ﬁrst application of the technique in Colombia, initially developed for permafrost zones, with different assumptions and surface complexity; and it resulted in an improved PS converted wave image.


Introduction
Static corrections, which can reach up to 100 ms for receivers, are an inherent problem of PS converted wave (PS-CW) processing that has not been completely solved. The difficulties related to receiver corrections include the masking of the first arrivals of the refracted PS-CW by surface waves, and the low energy of the refracted PS-CW, which prevents the use of refraction methods. Due to several factors such as water level, permafrost, and calcareous soils, among others, the V p /V s relation in the weathered layer can be non-uniform and as high as 10, that makes the P wave statics less useful for the correction of PS-CW statics.
There are two main approaches to the calculation of PS-CW statics. The first one requires a known shear velocity model, usually obtained by direct measurements (borehole data, refraction, Rayleigh wave inversion). The second one assumes a surface consistent model, as in the statics correction methods available for acoustic wave processing (Guevara et al., 2015). However, the surface consistent assumption is not always valid, and, in the case of PS-CW, it causes error during processing. Seismic interferometry, an adaptation of optical interferometry (Wapenaar et al., 2010a;Wapenaar et al., 2010b), creates new seismic data from recorded wave fields through the correlation of seismic traces. Seismic interferometry has been used to find and apply receiver corrections in zones with complex weathering (Bakulin & Calvert, 2006). To avoid the surface consistent assumptions, an adaption of seismic interferometry, referred as raypath interferometry uses constrained common emerging angle gathers instead of common converted point gathers -CCP gather for correcting the PS-CW image (Henley, 2012). Henley (2012) developed raypath interferometry for zones with permafrost layer (high velocity). In this research, seismic interferometry was applied to a PS-CW dataset acquired in the Llanos Basin, Colombia, where different conditions are present. For example, the near-surface has a low-velocity layer composed of clay matrix and sand channels, typical of tropical soils. As a matter of experience, the available static methods do not properly correct the distortion caused by the weathered layer in PS-CW images in the Llanos basin.

Materials and methods
The stability and robustness of raypath interferometry was validated by testing two models that simulated the geological features of the Llanos Basin, e.g.: a weathered layer with water level at depths between 20 m and 50 m, sand channels with low seismic velocity in a clay matrix (Guayabo Formation) and different values of thickness and depth for the weathered layer. From now on, the weathered layer will be called LVL because it is generally a Low-Velocity Layer. As shown in Fig. 1 A, the first model contains an LVL with a thickness between 180 m to 250 m, and two flat reflectors at depths of 400 m and 700 m. The LVL of Model 2, which is shown on Fig. 1 B, has a water level at 50 m of depth with a thickness between 110 m to 160 m and lateral velocity variations represented by sand channels, whose shear velocity and acoustic velocity are 400 m/s and 900 m/s, respectively. There are two reflectors cut by a fault in the middle of the section below the LVL. Fig. 1 A and 1 B indicate the velocity values V s (x, z) and V p (x, z) for both models.
Both synthetic models were simulated, based on a finite difference modeling algorithm, with a split-spread pattern with 480 channels, a maximum offset of 2 400 m, a receiver interval of 5 m, and a source interval of 20 m. A 35-Hz Ricker impulse was used as wavelet with a sample interval of 2 ms. Raypath interferometry was then applied to a two-component seismic line located at Quifa Project and acquired by Pacific E&P Corporation. The Quifa area of the Llanos Basin features Late Eocene to Oligocene formations where heavy oil deposits have been found, and the presence of hydrocarbons is associated with channel deposits. The provided PS-CW image was compared with a previous image obtained (Buitrago, 2016) through a conventional PS-CW processing sequence (Lu & Hall, 2003).

Theoretical framework
A static correction is a vertical time shift applied to a seismic trace to correct the LVL effects associated with its changes in composition, elevation, thickness and lateral facies (Sheriff, 1991). The surface consistent correction procedure assumes that the raypaths through the LVL layer are near vertical, an approximation that is only valid when the velocity of the LVL layer is much smaller than the velocity of the underlying layer. This condition is unusual in the Llanos Basin, where the path followed by PS converted waves across the LVL can reach emergence angles higher than twenty degrees, due to smooth velocity variations. In this case, the surface consistent approximation is no longer valid, and the receiver correction becomes non-stationary, i.e. a static correction cannot correct distortions along the whole PS-CW trace (Cova et al., 2017). The raypath consistent assumption, which replaces the surface consistent constraint, is implemented through the radial trace transform (Claerbout, 1975), which extracts the plane wave components of the wavefield and provides common ray parameter gathers. Common shot gathers S(x, t ) are mapped from the offset-time domain (x, t ) to common emergence angle gathers S (v, t ) in the radial domain (θ, t ) through the  Fig. 2 A illustrates the raypaths for a common shot gather and Fig. 2 B shows the raypaths for a common ray parameter gather, where rays emerge at the same angle. Fig. 2 C illustrates the process of extracting radial traces from a shot gather where the amplitude of each sample is represented by a color. The process includes a) straight lines with different slopes or velocities are traced from the origin, b) the travel times and amplitudes that each line intercepts constitute the new trace in the radial domain with a constant velocity established by the slope. The mapping is well-behaved everywhere except at the origin, but the discrete sampling is not the same in both domains, and interpolation is required from one domain to the other to avoid aliasing. Fig. 3 shows the reflection seismogram recorded on an LVL that overlays the second layer. To remove the travel time across the LVL, the g trace is shifted by time T(ACB) and the resulting trace is kinematically equivalent to the h trace recorded as if a virtual source and receiver were redatumed to the base of the LVL. Similarly, this process is obtained by means of cross-correlation, a key concept for interferometry. The cross-correlation of the f trace with the g trace is equal to the h trace. For raypath interferometry, the uncorrected trace represents the g trace. The trace f is an operator obtained by the cross-correlation of the g trace with a pilot trace. Applying the operator by cross-correlation (or its time-reverse version by convolution), it corrects the g trace.

PS converted wave processing
The input for the receiver correction of the LVL effects for PS-CW is the NMO corrected set of common receivers (CR) gathers with source statics provided previously by acoustic wave processing. The PS-CW processing sequence applied follows the guidelines proposed by Lu & Hall (2003), but the receiver statics module is replaced by the interferometry approach, whose workflow is depicted in Fig. 4. On the other hand, the PS-CW image obtained by Buitrago (2016) followed a modified Lu & Hall (2003) sequence. The traces of CR gathers (including a deconvolution and band pass and f-k filters to attenuate the coherent noise) are sorted by ascending offsets and then transformed to the radial domain. Later, the radial traces are sorted by ray parameter and receiver to create common angle (CA) gathers, where each ray parameter represents an exit angle. The traces of CA gathers contain the seismic response associated with rays emerging with the same angle and transforming the LVL distortion into a stationary one. Now, the cross-correlation is used to identify the effects caused by LVL. The pilot traces that compose the reference wavefield are generated by a partial running stack of the original traces over an aperture to create a smoothed model of the section. Subsequently, all the traces are correlated with their corresponding pilot traces to capture the delay times and Additionally, inverse filters derived from final correlation functions are applied to each trace in the original common angle gathers by convolution to remove the delays and phase distortions caused by LVL. Each inverse filter corrects a single trace in a CA gather, which produces a significant improvement in the continuity of reflectors. During the interferometric process, the entire correlation function is used because it includes the differences in phase and amplitude associated with the travel through the LVL. Finally, the CA gather is sorted and transformed to the x − t domain by the Inverse Radial Transform in order to produce a CR gather. The final PS-CW stacked section is obtained by stacking all the resulting CR gathers.

Results
Stacked PS-CW sections of Models 1 and 2 obtained using interferometry were compared with stacked PS-CW sections with a simple correction based on the Vp/Vs average ratio for the weathered layer. Initially, long-period statics were observed for Model 1 due to the thickness and lateral velocity variations in the weathered layer. Moreover, short-period statics were added intentionally during processing. Fig. 5 A depicts the result after a correction based on P wave statics multiplied by a Vp/Vs ratio. Fig. 5 B shows the stacked PS-CW image of Model 1, after raypath interferometry, where the removal of short-period and long-period statics can be observed.
The initially corrected PS-CW stacked section of Model 2 shows mild undulations that are still affecting the reflectors (Fig. 6 A). These undulations are related to changes in depth and thickness, a lateral variation of velocity produced by some channels, and the water table in the weathered layer. Fig. 6 B shows the stacked section corrected by interferometry, where the geometry of the reflectors and the fault plane is improved. A depth migration was necessary to locate both reflectors in the right position and attenuate the fault shadow effects. Fig. 7 A shows the P wave stacked section of the real data obtained using a conventional process of static correction where weathering effects are partially corrected. Some residual effects are observed in the Carbonera Formation reflectors (indicated by the arrows), and in the lack of continuity of some reflectors in the same formation. Fig. 7 B shows the PS-CW stacked section without any correction (neither source nor receiver) and indicates large weathered level effects (more than 100 ms).  As illustrated in Fig. 8 A, the PS-CW stacked image obtained by the conventional static correction, a modification of Lu & Hall (2003) sequence that uses the P wave receiver statics at the receiver statics calculation onset, is still affected by some weathered level effects. This is caused using P wave statics that transfer the deficiencies of the P wave static correction process to the PS-CW image. This is indicated by the presence of the same statics of the P wave stacked image. Finally, the results of the interferometric application are shown in Fig. 8 B, where the residual effects have been removed and the continuity of the reflectors is improved.   Lu and Hall (2003) obtained by Buitrago (2016). B) Result of the Interferometry used to solve receiver statics for the PS-CW image.

Discussion
Interferometry is a good option to obtain a high-quality PS converted wave image for some problematic zones with gas-bearing layers or highly complex LVL, where it may be difficult to obtain a P wave image.
There are two weaknesses related to raypath interferometry. Firstly, the amplitude and phase changes produced by the application of the inverse filters to the raw seismic traces may make them inappropriate for seismic inversion procedures or amplitude attribute analysis. Cova  transformation instead of the radial trace transformation, but they found a difficulty with the potential loss of data fidelity. Accordingly, other techniques that do not modify the amplitude and phase spectrum of the original data should be researched.
Secondly, interferometry, as used in this research, is not currently recommended for zones with faults and structural complexity (e.g. pinch-outs). The picking of the guide horizons becomes problematic in that case.