# Model experiments

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

We shall analyze the problem of near-surface model estimation using two sets of synthetic data. The near-surface model for the first data set comprises a single layer with an undulating refractor and a flat topography. The near-surface model for the second data set comprises multiple layers below an irregular topography. We shall examine the extent of resolving the long- and short-wavelength anomalies by way of the refraction and residual statics correction methods described in residual statics corrections and refraction statics corrections.

Figure 3.4-14a shows an earth model that comprises a simple subsurface velocity-depth model and a near-surface model that comprises a single layer with an undulating refractor but flat surface topography. The refractor geometry has wavelength variations that range from short wavelengths that are less than a cable length to long wavelengths greater than a cable length. The weathering velocity is 1200 m/s and the refractor velocity is 2000 m/s.

Figure 3.4-14b shows the zero-offset section derived from the earth model in Figure 3.4-14a. Note the traveltime distortions associated with the flat reflectors. Also, in this section we see the multiples from the refractor and the peglegs from reflector 1. The zero-offset section is appropriately aligned with respect to the velocity-depth model in the lateral direction. The objective in this model experiment is, following the application of refraction and residual statics corrections, to obtain a stack response similar to this zero-offset section.

A two-way acoustic wave equation was used to model a total of 154 shot records along the line. Both shot and receiver group intervals are 50 m, and the number of channels is 97, including the zero-offset trace. The split-spread recording geometry has a maximum offset of 2350 m. Selected shot records shown in Figure 3.4-15 from the synthetic data set associated with the earth model exhibit traveltime distortions on the reflection events caused by the undulating refractor. Note also the distinct refracted arrivals and the ground-roll energy.

Figure 3.4-16a shows the CMP-stacked section with no statics corrections. Compare with the zero-offset section (Figure 3.4-14b) and note some differences. As a result of velocity discrimination, multiples have been attenuated to some extent by CMP stacking. Both the CMP-stacked and zero-offset sections have the imprint of the near-surface effects on reflection times. Note that CMP stacking has given rise to the spurious structural discontinuities on reflections below 1 s between CMP 100-200. Following residual statics corrections (Figure 3.4-16b), these short-wavelength anomalies appear to have been resolved. The moderate-to-long wavelength anomalies expressed by the reflection traveltime undulations, however, have remained in the section. These anomalies have been resolved by refraction statics corrections as shown in Figure 3.4-17a using, in this case, the generalized linear inverse (GLI) method to solve equation (52). Nevertheless, some residual anomalies still remain. After residual statics corrections based on the solution to equation (25), the remaining long-wavelength anomalies are untouched, while the residual short-wavelength anomalies have been further resolved. Unfortunately, not all of the traveltime distortions due to the near-surface layer (the undulating refractor R in Figure 3.4-14a) have been eliminated. Note, for instance, the slight undulations in Figure 3.4-17b on events at 0.5 and 1 s which correspond to horizons 1 and 2 in Figure 3.4-14a. Note also in Figure 3.4-17b the distorted structural high represented by the reflection between 1-1.5 s, which corresponds to horizon 3 in Figure 3.4-14a, and the sagging reflection below 1.5 s, which corresponds to horizon 4 in Figure 3.4-14a.

The results of the GLI statics estimates are summarized in Figure 3.4-18. For the variable-thickness estimate (equation 52), the weathering velocity was set to 1200 m/s — the correct velocity for the nearsurface layer. Frame 1 shows the estimated GLI parameters — the intercept time anomalies (equations 53a, 53b), as a function of the shot-receiver station number. Frame 2 shows the pick fold, namely the number of picks in each shot (denoted by ×) and receiver (denoted by the vertical bars) gather. Note the tapering of the pick fold at both ends of the line.

A quantitative measure of the accuracy of the GLI solution to refraction statics is the sum of the differences between the observed picks tij and the modeled traveltimes ${\displaystyle t'_{ij}}$ (equation 52) over each shot and receiver gather. These residual time differences are plotted in frame 3 of Figure 3.4-18. Large residuals often are related to bad picks. Nevertheless, even with good picks, there may be large residuals attributable to inappropriateness of the model assumed for the near-surface.

Figure 3.4-18 also shows the estimated weathering thicknesses at all shot-receiver stations (frame 4). Finally, the computed statics and the near-surface model are shown in frames 5 and 6, respectively.

Uncertainty in the assumed value for weathering velocity is an important practical consideration in refraction statics. Figure 3.4-19 shows results of GLI statics solution using two different weathering velocities. Compare with the result using the correct weathering velocity (Figure 3.4-17a) and note that the GLI solution tolerates reasonable departures from the correct weathering velocity.

The ability of statics solutions to resolve the effect of the near-surface layer with the undulating refractor shown in Figure 3.4-14a is further tested by applying the generalized reciprocal method (GRM) described by equations (50a, 50b). Compare the results of refraction statics corrections using the GLI method (Figure 3.4-17) and the GRM method (Figure 3.4-20), and note that, in this case, the differences are marginal. Nevertheless, it appears that neither of the statics solutions appear to have resolved the time anomalies caused by the undulating near-surface layer, completely. Assumptions made about the near-surface model always limit the resolving power of all the statics corrections methods.

Although in most cases the near-surface often consists of a low-velocity weathering layer, in some exploration basins, a single-layer near-surface model may not be adequate. We shall examine the response of statics solutions to a multilayer near-surface model shown in Figure 3.4-21a. The zero-offset section at z = 0 is free of the near-surface effects (Figure 3.4-21b) since the nearsurface anomalies are above z = 0 in the model (Figure 3.4-21a). Following the application of refraction and residual statics corrections, ideally, we should like to obtain a stack response similar to this zero-offset section.

The near-surface consists of layers with irregular geometry and an irregular topography. A two-way acoustic wave equation was used to model a total of 154 shot records along the line. Both shot and receiver group intervals are 50 m, and the number of channels is 97, including that associated with the zero-offset trace. The split spread recording geometry has a maximum offset of 2350 m. Selected shot records from the synthetic data set associated with the earth model (Figure 3.4-22) exhibit traveltime distortions on reflection events resulting from the complex near-surface model. Note also the distinct refracted arrivals and the ground-roll energy.

Figure 3.4-23a shows the CMP-stacked section with no statics corrections. Compare with the zero-offset section (Figure 3.4-21b) and note the significant differences. Note the severe time anomalies caused by the complexity of the near-surface layer. Some short-wavelength anomalies have been resolved by residual statics corrections (Figure 3.4-23b). Nevertheless, this stacked section is far from implying the simple subsurface structure (Figure 3.4-21a).

By using the near-surface model (Figure 3.4-21a), statics at all shot-receiver stations were calculated by hand and corrections were applied to the data. The resulting stacked section is shown in Figure 3.4-24a. Note that most of the long-wavelength anomalies have been removed. Remaining disortions on reflection times, particularly betwen CMP 300-400, imply that correcting for the near-surface effects by statics shifts applied to CMP traces is a simplistic approach given the complexity of the near-surface model. Residual statics corrections do not help in removing the time anomalies that are beyond the limit of statics corrections (Figure 3.4-24b).

Now assume a near-surface model that comprises a single layer with constant velocity (1400 m/s). The top and base of this layer are defined by the elevation curve and the flat datum at z = 0, respectively. Then, compute the elevation statics at each shot-receiver station using the thickness of the constant-velocity layer, and apply them to the CMP traces. Figure 3.4-25 shows the CMP stack with elevation statics and the subsequent residual statics corrections. Compare with Figure 3.4-23 and note that much of the time anomalies are due to elevation differences along the line.

Elevation statics corrections are basically a simple alternative to statics corrections based on an estimate of a near-surface model. Using the GLI solution based on equation (52), inversion of the refracted arrivals yields the stacked section in Figure 3.4-26a. Here, the near-surface model was assumed to consist of a single layer, as in the case of elevation statics corrections, but with varying refractor geometry. Note the improvement on reflection times on the stack associated with the GLI solution. Clearly, a more complicated, multi-layered near-surface model can, in principle, be estimated from inversion of refracted arrivals. However, the more complicated the model, the more parameters need to be specified. This in turn will require a more complicated inversion scheme. Generally, in practice, one should model the near-surface simply. If traveltime distortions are not resolved adequately by a simple near-surface model, it often means that the problem is not solvable by statics methods. Specifically, the near-surface corrections should not be done using vertical time shifts applied to CMP traces. Under those circumstances, very little can be achieved by residual statics corrections (Figure 3.4-26b). Instead, the problem should be characterized as dynamic and be solved by [[introduction to earth modeling in depth|earth modeling in depth.

The results of the GLI statics estimates are summarized in Figure 3.4-27. For the variable-thickness single-layer near-surface, the weathering velocity was assumed to be 1400 m/s. Frame 1 shows the estimated GLI parameters — the intercept time anomalies (equation 53a,52) over each shot and receiver gather is shown in frame 3. Large residuals, in this case, are attributable to the inappropriateness of the model assumed for the nearsurface. Figure 3.4-27 also shows the estimated weathering thicknesses at all shot-receiver stations (frame 4). Finally, the computed statics and the near-surface model are shown in frames 5 and 6, respectively.

For comparison, the GRM statics solution for the multilayered near-surface model of Figure 3.4-21a is shown in Figure 3.4-28. While both the GLI (Figure 3.4-26a) and GRM (Figure 3.4-28a) solutions are comparable, it appears that neither of the statics solutions appears to have resolved the time anomalies caused by the complex near-surface layer, completely. Again, assumptions made about the near-surface model always limit the resolving power of all the statics methods.

## Equations

 ${\displaystyle t'_{ij}=s_{j}+r_{i}+G_{k}+M_{k}x_{ij}^{2}}$ (25)

 ${\displaystyle t_{+}=t_{ABCD_{2}}+t_{D_{1}EFG}-t_{ABFG}-{\frac {D_{1}D_{2}}{v_{b}}}}$ (50a)

 ${\displaystyle t_{-}=t_{ABCD_{2}}-t_{D_{1}EFG}+t_{ABFG}.}$ (50b)

 ${\displaystyle t'_{ij}=T_{j}+T_{i}+s_{b}x_{ij}}$ (52)

 ${\displaystyle T_{j}={\frac {z_{j}{\sqrt {v_{b}^{2}-v_{w}^{2}}}}{v_{b}v_{w}}}}$ (53a)

 ${\displaystyle T_{i}={\frac {z_{i}{\sqrt {v_{b}^{2}-v_{w}^{2}}}}{v_{b}v_{w}}}}$ (53b)

 ${\displaystyle t'_{ij}=T_{j}+T_{i}+s_{b}x_{ij}}$ (C-54)