Refraction interferometry for residual statics solutions Chao Zhang* , Jie Zhang, University of Science and Technology of China (USTC); Zhikun Sun, GeoTomo LLC Summary Refraction traveltimes have long been applied for deriving long-wavelength statics solutions. They are also applied for deriving residual statics, but it requires providing with substantially accurate traveltime picks for the calculation. In this study, we present a residual statics method that applies interferometric theory to produce four stacked super-virtual refraction gathers with significantly improved signal-to-noise ratio. They include forward and backward super-virtual refraction gathers for receivers and shots. Picking first arrivals on these four gathers followed by applying a set of equations is able to derive reliable residual statics solutions. This approach can help dealing with noisy data and also avoid using traveltime picks from shot gathers. We demonstrate the approach by applying to synthetic data as well as real data. Introduction Refraction methods have become a standard statics approach for obtaining high-quality seismic sections in land seismic data processing. In seismic industry, current long-wavelength statics approaches include delay-time method (Gardner, 1939), generalized linear inversion (Hampson and Russell, 1984), traveltime tomography (Zhang and Toksoz, 1998), and waveform inversion (Sheng et al., 2006). For residual statics solutions, there are also many methods such as reflection stack-power maximization method (Ronen and Claerbout, 1985), refraction waveform residual statics (Hatherly et al., 1994), and refraction traveltime residual statics (Zhu and Luo, 2004). The accuracy of the refraction static correction largely depends on the quality of first arrival traveltimes. However, in some areas, refraction traveltimes cannot be accurately picked at far offset. To partially overcome this problem, the theory of refraction interferometry was developed to enhance the signal-to-noise ratio of head-wave arrivals (Dong et al., 2006; Bharadwaj and Schuster, 2010). Dong et al. (2006) demonstrated the method using land data over a salt dome in central Utah and later Nichols et al. (2010) demonstrated its effectiveness over a hydro-geophysical research site in Idaho. Mallinson et al. (2011) presented an extension of refraction interferometry which creates virtual far-offset refraction arrivals by the combination of both correlation and convolution of traces with one another to create what is denoted as super-virtual refraction traces. They presented a workflow for super-virtual refraction interferometry (SVI) and applied the method to both synthetic and field data, later Bharadwaj et al. (2012) presented the rigorous theory of SVI.

In this study, we borrow the concept of super-virtual refraction interferometry but for calculating refraction residual statics. Details are discussed in the following. Interferometric refraction residual statics Figure 1a shows a schematic illustration of refraction raypath in a simple layer model. The traveltime difference between two adjacent receivers (R1 and R2) decomposes on: 1) the horizontal segment; 2) the difference of two upcoming raypaths to R1 and R2 respectively. Then, we set up Equation 1 that includes residual statics.

Figure 1: (a) Schematic illustration of refraction raypath in 2D layer model. (b) Sketch for receiver-pair obtaining the signal from both left and right sources, black line represents the refraction raypath from the left shot and blue line denotes the signal from the right shot. Assuming long-wavelength statics has been corrected, the following differential equations are set:

( ) ( ) 12 12 1 2 - 1 - res res T d s=Δ ×

( ) ( ) 23 23 2 3 - 2 - res res T d s=Δ × (1)

… … ( ) ( ) , 1 , 1 1 - - k k k k kres k res k T d s+ ++ =Δ ×

where res(k+1), res(k) are the receiver residual statics at the location k+1 and k, respectively; , 1 ( 1) ( )k kT t k t k+Δ = + − represents their traveltime differences, dk,k+1 denotes the distance interval between two adjacent receivers, and sk is the slowness along refraction path.

Low velocity High velocity

s1 s2 sk

d12        d23                … …                  dk,k+1    

R1 R2 R3 R(k) R(k+1) Source

(b)

(a)

s1 s2

d12                     d12 

res(2) res(2)

res(1) res(1)

Low velocity High velocity

Right source R1 R2 Left source

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Refraction interferometry for residual statics solutions

We find that the slowness can be represented by traveltime differences. Taking the receivers R1 and R2 as an example, they receive signal from both left and right shots, as shown in Figure 1b. Assuming upcoming raypaths from left and right to the same receiver equal, we can establish Equation 2 and Equation 3, respectively.

( ) ( ) 12 1 122 1res res d s T− + × =Δ (2)

( ) ( ) 12 2 211 2res res d s T− + × =Δ (3)

Where 12TΔ and 21TΔ denote the traveltime differences between R1 and R2 from left shots and right shots, respectively. Combining Equation 2 and Equation 3, we then obtain Equation 4, which eliminates the residual statics.

( )1 2 12 12 21s s d T T+ × = Δ + Δ (4)

In a similar way, we apply this process to other receiver pairs, and obtain the following equations:

( )1 2 12 21 12/s s T T d+ = Δ + Δ

( )2 3 23 32 23/s s T T d+ = Δ + Δ (5)

… … ( )1 , 1 1, , 1/k k k k k k k ks s T T d+ + + ++ = Δ +Δ

If we assume that s1 is equal to s2, then we infer all slowness values by the recursive formula, namely, s1, s2, ... sk+1. Assuming the first statics zero, we can derive statics for the remaining receivers (shots) iteratively following Equation 1 if we can obtain △T. We could obtain traveltime differences using traveltime picks between any two adjacent traces in the same shot gather. Nevertheless, there is a better and more efficient way to do that. Instead, we apply super-virtual interferometry method to calculate the quantity. Super-virtual refractions are generated in two steps: the first step involves correlation of the data to generate traces with virtual head wave arrivals, and the second step involves convolution of the data with the virtual traces to create traces with super-virtual head wave arrivals. Theoretically, correlation of adjacent receiver pair from left to right in sequence for any post-critical source position should lead to a virtual trace with the same virtual refraction traveltime, so stacking traces over all post-critical source positions will enhance the SNR of the virtual refraction by N , similar to that in except the virtual refraction traces are convolved with the actual refraction traces and stacked for different geophone positions to give the forward super-virtual traces with a SNR enhanced by N , as seen in Figure 2a. Here N is the number of post-critical shot positions that coincide with receiver locations. Another super-virtual traces can

also be generated backward as shown in Figure 2b. Including another two super-virtual gathers for shots, there should be four stacked super-virtual gathers totally. In each gather, the maximum peak time of a trace is the traveltime difference that we need to calculate residual statics.

(a) (b)

Figure 2: Schematic illustration of (a) forward super-virtual traces for receivers, (b) backward super-virtual traces for receivers. Numerical example We shall use a 2D synthetic dataset to demonstrate how to calculate interferometry residual statics. In Figure 3a, it shows a nice free shot gather, and in Figure 3b the same gather is applied with surface consistent statics at far offset with arbitrary statics between -20 ms and 20 ms.

(a) (b)

(c) (d)

Figure 3: (a) Original synthetic data. (b) Data applied with arbitrary statics. (c) Data with random noise (S/N=2). (d) Early arrival after applying inside and outside muting.

R1 R2 R3 R4 … R(k)

Right shot

R1 R2 R3 R4 … R(k)

Left shot

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Refraction interferometry for residual statics solutions

Then random noise created with signal-to-noise ratio of 2 is further added to data as shown in Figure 3c. The influence of noise on the far offset refractions is significant. It is difficult to pick the first arrivals accurately. We further roughly mute the data and keep the early arrivals as shown in Figure 3d, and the same process is applied to all shot gathers. The output shall be used for creating super-virtual refraction gathers. Following the approach that we describe above, Figure 4 presents two interferometry gathers for receivers and two for shots. Because of stacking power, these gathers show good quality of signals. Picking the first arrival traveltimes in these gathers, we obtain forward and backward traveltime differences for both receivers and shots.

(a)

(b)

(c)

(d)

Figure 4: (a) Forward super-virtual refraction gathers for receivers. (b) Backward super-virtual refraction gathers for receivers. (c) Forward super-virtual refraction gathers for shots. (d) Backward super-virtual refraction gathers for shots.

After picking, we calculate slowness along the refractor following Equation 4 and apply the picks and slowness in Equation 1 to infer residual statics for all receivers and shots in a surface consistent manner. Figure 5a shows the true residual statics applied to data, and Figure 5b presents interferometric residual statics. In Figure 5c, their differences are plotted as well, and it shows that the differences are less than 2 ms.

(a)

(b)

9000 9500 10000 10500 11000-2

-1

0

1

2

X (m)

Diff

eren

ce (m

s)

shotsreceivers

(c)

Figure 5: (a) True residual statics for receivers (blue), and shots (red). (b) Interferometric residual statics for receivers (blue), and shots (red). (c) Differences between true and resolved statics values for receivers (blue), and shots (red). Field data We demonstrate the statics solutions using a real dataset from China. A total number of 243 shots were recorded with the shot interval of 40 m and receiver interval of 20 m. The number of channels is 610. Figure 6 displays a shot gather and it clearly shows that the far offset traces suffer from noise so that picking the first break is very difficult. We also select a good shot gather that the variations in the first arrivals indicate strong statics effects (Figure 7a).

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Refraction interferometry for residual statics solutions

Figure 6: A shot gather with low SNR at far-offset We apply the refraction interferometry approach to process the data and pick traveltime differences on the super-virtual refraction gathers. Figure 7b presents the same shot gather after the residual statics is applied.

(a)

(b)

Figure 7: A shot gather (a) before and (b) after interferometric residual static correction applied. Figure 8 shows stacked sections obtained from data without residual static correction (Figure 8a), and data with interferometric residual static correction (Figure 8b). The

comparison clearly indicates the impact of the residual statics solution which improves the event continuity significantly.

(a)

(b)

Figure 8: The stacked sections (a) without residual static correction, (b) with interferometric residual static correction applied. Conclusions We developed a residual statics approach that applies refraction interferometry to help the calculation of statics. The approach can handle very noisy data in which the first picks are hard to pick. Tests with synthetics and real data suggest that the method is robust in dealing with noise and it can help improve stack quality significantly. The method is refraction based, thus, it is valid only when the near surface refractions are reliable. If the near surface area is too complex such that wave propagation is beyond refractions, then the method may fail. Acknowledgements We appreciate the support from GeoTomo, allowing us to use TomoPlus software package to perform this study.

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http://dx.doi.org/10.1190/segam2014-1267.1 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2014 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

Bharadwaj, P., and G. T. Schuster, 2010, Extending the aperture and increasing the signal-to-noise ratio of refraction surveys with super-virtual interferometry: Presented at the AGU Annual Meeting.

Bharadwaj, P., G. T. Schuster, I. Mallinson, and W. Dai, 2012, Theory of supervirtual refraction interferometry: Geophysical Journal International, 188, no. 1, 263–273, http://dx.doi.org/10.1111/j.1365-246X.2011.05253.x.

Dong, S., J. Sheng, and G. T. Schuster, 2006, Theory and practice of refraction interferometry: 76th Annual International Meeting, SEG, Expanded Abstracts, 3021–3025.

Gardner, L. W., 1939, An areal plan of mapping subsurface structure by refraction shooting: Geophysics, 4, 247–259, http://dx.doi.org/10.1190/1.1440501.

Hampson, D., and B. Russell, 1984, First-break interpretation using generalized linear inversion: 54th Annual International Meeting, SEG, Expanded Abstracts, 532–534.

Hatherly, P. J., M. Urosevic , A. Lambourne, and B. J. Evans, 1994, A simple approach to calculating refraction statics corrections: Geophysics, 59, 156–160, http://dx.doi.org/10.1190/1.1443527.

Mallinson, I., P. Bharadwaj, G. T. Schuster, and H. Jakubowicz, 2011, Enhanced refractor imaging by super-virtual inteferometry: The Leading Edge, 30, 546–550, http://dx.doi.org/10.1190/1.3589113.

Nichols, J., D. Mikesell, and K. V. Wijk, 2010, Application of the virtual refraction to near-surface characterization at the Boise Hydrogeophysical Research Site: Geophysical Prospecting, 58, 1011–1022.

Ronen, J., and J. F. Claerbout, 1985, Surface-consistent residual statics estimation by stack-power maximization: Geophysics, 50, 2759–2767, http://dx.doi.org/10.1190/1.1441896.

Sheng, J., A. Leeds, M. Buddensiek, and G. T. Schuster, 2006, Early arrival waveform tomography on near-surface refraction data: Geophysics, 71, no. 4, U47–U57, http://dx.doi.org/10.1190/1.2210969.

Zhang, J., and M. N. Toksöz, 1998, Nonlinear refraction traveltime tomography: Geophysics, 63, 1726–1737, http://dx.doi.org/10.1190/1.1444468.

Zhu, W. H., and Y. Luo, 2004, Refraction residual statics using far offset data: Presented at the Geo2004 Conference.

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