Reflection seismic data from block F3 in the Dutch North Sea exhibit many large-amplitude reflections at shallow horizons typically categorized as bright spots. In most cases, these bright reflections show a significant “flatness” that contrasts with local structural trends. Although flat spots in thick reservoirs are often easily identified, others within thin beds or near reservoir edges can be difficult to identify and are poorly understood. Many of the shallow large-amplitude reflections in this block are dominated by flat spots. We investigated the tuning effects that such flat spots cause as they interacted with reflections from the top of the reservoir. We first studied the zero-offset “wedge-model” tuning effects of the flat spot with overlying bright spots, dim spots, or polarity reversals. We then expanded that model to examine prestack tuning effects, as well as the results from inclusion of postcritical flat spot reflections in the final stack. We observed that under certain conditions, the reflections could appear to be somewhat flattened bright spots; those conditions might be met frequently in practice, and they should be considered in routine interpretation. In the North Sea case, we concluded that this tuning effect was the primary cause of the brightness and flatness of these reflections.
Observations of flat bright spots
Direct hydrocarbon indicators (DHIs) have long been used successfully in exploration projects (Brown, 2012). Bright spots, for example, demonstrate an increase in (negative) reflection coefficients as the water sand beneath a higher impedance cap rock transitions to a gas sand or oil sand. These changes can be recognized easily and are often exploited because of their prominence in a seismic section. Flat spots that represent reflections from the hydrocarbon-water contact are also easy to recognize because of their unconformable flatness, and they are always positive in sign. Dim spots result from a cap rock that is of lower impedance than the underlying water sand and hydrocarbon sand. Brown (2012) emphasizes that dim spots may represent overlooked exploration targets because they are often difficult to identify. This difficulty can arise because of the dim spots’ low amplitude and because tracking of prominent (positive) reflections may follow a flat spot event rather than the dimmed reservoir top. Polarity reversals exist where the cap rock is of higher acoustic impedance than the hydrocarbon sand, but of lower impedance than the water sand. These are presumably difficult to recognize for the same reasons as dim spots.
We present an interesting case study from offshore Netherlands block F3, which leads us to more general observations. The area exhibits significant bright reflections that are very flat in nature, contrasting with the structural trend of the surrounding rocks. The data set used for this work (provided by dGB Earth Sciences for use with the OpendTect software suite) includes poststack 3D seismic data with limited well-log data from four wells. This data set also demonstrates excellent examples of gas chimneys, DHIs, and stratigraphic features. Of interest in our study are small shallow uneconomic reservoirs that exhibit very bright and flat reflections. Figure 1 shows one example of these reservoirs. Throughout this paper, we will focus on this specific reservoir, considering it to be representative of other nearby reservoirs that exhibit the same phenomenon. Note the brightness of the reservoir reflections marking the top of the reservoir. It is this bright nature of these reflections that has led to their categorization as bright spots in the literature (Schroot and Schuttenhelm, 2003).
To estimate the departure of this reflection from the structural trend, we compared it with immediately overlying layers. The green line in Figure 1 shows a tracked horizon indicative of the structural trend. A phantom horizon was created from the green horizon by shifting it downward an appropriate time. This phantom horizon is displayed in blue below the tracked horizon. Although the phantom horizon is conformable with the water-sand reflections along the flanks, it is not conformable with the bright reflections associated with the gas reservoir at the crest. This observation suggests that the “red” reflections (negative, indicating an acoustic impedance decrease) are not simple bright spots, but they are influenced by the underlying flat spot. The flatness and the brightness of the reservoir reflections need to be explained if we are to properly interpret similar features elsewhere.
Basic techniques used in this study
To examine these flat and bright features, we created synthetic data from different models. We examined tuning effects for normal incidence on thin hydrocarbon beds. Then, we examined nonnormal incidence prestack tuning and the effect of postcritical phase shifts on stacking.
The shallow layers in block F3 consist of alternating beds of shale and sand. Tuning effects are often observed in thin beds (Ricker, 1953; Widess, 1973; Kallweit and Wood, 1982; Chopra et al., 2006). The reservoirs under discussion exhibit features that suggested to us that tuning plays an important role in the generation of these flat reflections. Most tuning models consider reflections from the top and bottom of sand layers. Here, we expect this tuning to arise from the reflections from the top of the reservoir and the flat spot reflection at the gas/water contact (GWC). First, we examine this issue from a normal incidence assumption, as is done in most thin-bed studies.
We also examine the effect of amplitude variations with offset or angle (AVO), and we include postcritical reflections in our models. The effect of postcritical reflections is usually not considered for stacking purposes, but these (uneconomic) reservoirs lie at very shallow depths (), and it is very likely that postcritical reflections have been recorded. These postcritical reflections may or may not have been muted prior to stacking. Postcritical reflections involve a phase shift, and if these reflections are included in the stacking process, the wave shape and amplitude of the final stacked event will change. In addition, these rocks are highly unconsolidated and the elastic properties of the rocks will be strongly influenced by the nature of the pore fluid (e.g., Hilterman, 2001). Therefore, we expect to observe a large velocity contrast at the GWC. This, in turn, will result in a small critical angle for flat spot reflections.
The work involved four basic steps. First, we undertook simple rock-physics modeling to estimate several unknown formation properties needed for analysis. Then, we conducted forward seismic normal-incidence modeling to study possible tuning effects. The third step involved AVO analysis and the effect of postcritical reflections. Finally, we investigated the AVO influence on the overall stack with and without normal moveout (NMO) stretch and muting. We conclude that thin-bed tuning effects of DHIs (particularly dim spots and polarity reversals) with their underlying flat spots may result in flat bright reflections that one may interpret as bright spots.
In this section, we first provide the methodology used in our study, and then we apply it to a model designed to resemble the lithology present in North Sea block F3.
Tuning effects for normal incidence
Ricker (1953), Widess (1973), Kallweit and Wood (1982), and Chopra et al. (2006) study tuning effects in detail and establish tuning thicknesses and resolvable limits. Widess (1973) concludes that for bed thicknesses thinner than half of the seismic wavelength (, where is the dominant wavelength), the reflections from the top and bottom of the layer interfere in ways that change the shape and amplitude of the wavelet. As the bed thins to one-fourth of the wavelength (), the amplitude of the tuned wavelet grows and reaches a maximum through the constructive interference of the side and main lobes of the two reflections. This thickness is called the tuning thickness. When the bed thickness reaches one-eighth of the wavelength (), the composite wavelet resembles a derivative of the original waveform and no change in trough-to-peak time will be observed. The amplitude then decreases toward zero as the bed continues to thin (Widess, 1973; Kallweit and Wood, 1982). Widess (1973) points out that a thin-bed thickness should be at least one eighth of the dominant wavelength to be delineated. However, in the presence of noise, the resolution is usually taken to be (Chopra et al., 2006).
Most of these studies used a wedge model similar to that shown in Figure 2, with a seismic section (Figure 2a) and tuning curve (Figure 2b). In this case, a 50-Hz (dominant frequency) Ricker wavelet was convolved with opposite but equal reflection coefficients at the upper and lower interfaces. Some published examples (e.g., Robertson and Nogami, 1984) used reflection coefficients that are identical rather than opposite in polarity.
In our case, however, we are interested in a different sort of wedge model, shown in Figure 3. This model is more appropriate for a wedge of hydrocarbon sand between overlying shale and underlying water sand. Many characteristics of the conventional wedge model are seen here. In contrast to a conventional wedge model, the polarity of the overlying shale-sand interface may differ from the polarity of underlying flat spot. The difference in polarity occurs when the impedance of the gas sand is lower than the impedance of the overlying shale. Detailed physical properties of the rock matrix, pore-fluid, and composite rocks used in modeling are given in Tables 1–23.
Tuning effects for amplitude variation with offset
Rutherford and Williams (1989) group AVO responses into three classes (I–III) based on normal incident reflection coefficient and AVO behavior, whereas Castagna et al. (1998) add an additional class IV. Some other authors have identified additional “classes,” but most authors refer only to these four. Another important type of AVO behavior is exhibited by all flat spots (see the following paragraph), but it is generally not given its own classification. Figure 4 shows the four classic AVO classifications plus the AVO behavior of flat spots.
Most bright spots exhibit class III AVO behavior; that is, they exhibit a negative reflection at zero offset () that increases (negative) with increasing offset. These are the most-often cited DHIs, and they are often easily recognized.
For most dim spots, class I (or II) AVO characteristics are observed. They exhibit a positive and decreasing positive amplitude with increasing offset, perhaps eventually becoming negative. Because the dim spot classification is based on stacked images, the stack must still be positive even if the amplitudes at far offsets are negative.
All flat spots exhibit similar behavior, in which the reflection is always positive and its amplitude increases with offset. This is because the effect of fluid substitution in a given reservoir rock will always result in low acoustic impedance (hydrocarbon zone) over high acoustic impedance (water zone). Because the shear modulus is unchanged, the negligible change in shear velocity across the interface is due only to the density effect. In most classification systems, this flat spot behavior is not recognized as a distinct class.
Because the amplitudes of the offset traces contribute to the amplitude of the final stack, it is important to consider those amplitudes in interpreting the final stack. This is often ignored when the stack is treated, as it often is, as an acoustic or zero-offset section. In this study, we examine this effect and its significance for tuning.
Normal moveout correction, stretch, and muting
In addition to AVO, the final stack can be impacted by the changes in wavelet shape with offset that result from stretching and phase shifts. Stretching can occur as a result of NMO correction, and phase shifts occur naturally in postcritical reflections.
The amount of extra traveltime () due to NMO for reflections observed at nonzero offset can be readily observed and computed (Buchholtz, 1972). Because conventional NMO correction uses different values of at different two-way reflection times, the result is a distortion of the wavelet. This distortion is more pronounced for early times and long offsets, and includes a reduction in high-frequency content (Shatilo and Aminzadeh, 2000). Buchholtz (1972) points out that the most severe stretching of the wavelet occurs at the intersections of reflection hyperbolas.
The usual solution for the NMO stretch problem is to discard or mute the severely stretched part of the traces, dependent on time and offset (Buchholtz, 1972). Usually a stretch limit of 50% (occasionally up to 100%) is taken to determine the muting zone of the CMP gather.
The tuning models described above are applied to the case of the flat bright events in North Sea block F3. In this section, we review the rock-physics components of the model.
The location of the (uneconomic) gas reservoir of interest is inline 210–250, crossline 1050–1200 at a depth of 520–560 ms in the survey from block F3 of the Dutch North Sea. None of the four wells drilled in the block penetrated this reservoir, but a nearby well (F03-4) provides data for the water-saturated equivalent sand as well as for the overlying shale. Figure 5a shows the well location and identifies the water-saturated sand with respect to the reservoir. This well has only gamma-ray (GR) and sonic logs available at this depth. The sonic velocities for the overlying shale and water sand were obtained from the well logs (Figure 5b). The fluid and grain properties were assumed from standard fluid and mineral properties. All of these properties are listed in Table 1. The sonic log indicates that the overlying shale has a lower compressional velocity than the water sand. Neglecting the density differences across the interfaces, the sonic measurements imply that the gas zone should not exhibit a bright spot, but it should display a polarity reversal or a dim spot. In the following section, we will see that simple fluid substitution further suggests that the gas zone should exhibit a polarity reversal (positive reflection over the water sand and negative reflection over the gas sand), rather than a dim spot.
Rock-physics modeling results
To overcome the limited log data available, we used Gardner et al.’s (1974) law to estimate density and Greenberg and Castagna’s (1992) model for the shear velocities of the water sand and the shale. We then used the Gassmann (1951) equation for fluid substitution in the sand, assuming 80% gas saturation and normally pressured gas typical of this depth.
The results from Gassmann fluid substitution are shown in Figure 6. The formation is very sensitive to pore fluid, and replacement of water with gas decreases the impedance dramatically. The results of this rock-physics modeling are tabulated in Table 2. Having estimated the rock properties, forward modeling studies were conducted to study the possible tuning effects and are presented in the following sections.
Tuning effect analysis at normal incidence (zero-offset)
To generalize beyond our specific case, the normal-incidence tuning effect was analyzed with three different models using only one variable: the acoustic impedance of the overlying shale. That is, using the properties observed for the water sand in block F3 and our calculated properties for the gas sand, we varied the properties of the overlying shale to model a (1) bright spot over a flat spot, (2) polarity reversal over a flat spot, and (3) dim spot over a flat spot. Details are provided in Table 3.
A geologic wedge model was used for tuning analysis, with an upper interface dipping at . A zero-offset synthetic seismic section was generated by convolving a 50-Hz Ricker wavelet (approximating the spectrum of the original seismic data) with the geologic model (Figure 7). Figure 8 shows the synthetic seismograms generated for all three cases. The red lines follow the negative (trough) reflections in the region where the tuning effect has distorted the wavelets, whereas the blue lines show the actual boundary of the wedge model. Note that in some cases, we show red lines away from the tuning effect to emphasize possible confusion from poorly tracked events.
Tuning effect on zero-offset sections
The top model in Figure 8 shows the synthetic seismograms for the polarity-reversal case. In this model, the shale impedance is less than that of water sand but greater than that of gas sand (Table 3). This model best represents the F3 reservoir as estimated from the log data. Note that in this model, the tuning effect near the pinchout results in a nearly flat reflection event, changing the apparent dip to approximately (from ). In the pinchout zone, the negative side lobe (indicated by an arrow in Figure 8) of the flat spot appears just above the reservoir top. Interpretation of stacked sections from such a geologic feature may be difficult to perform correctly because the polarity reversal is disguised as a flat, bright event.
The synthetic seismograms for the bright spot model are shown in the middle of Figure 8. In the bright spot model, the shale impedance is greater than that of the water sand and the gas sand (Table 3). This model shows negligible flattening due to tuning, and interpretation of such a structure is probably straightforward.
The results from dim spot modeling are remarkable, as shown in the bottom of Figure 8. (In the dim spot model, the impedance of the shale is less than that of the gas sand and the water sand.) Because dim spot amplitudes are low, the tuning effect at pinchout may not be noticeable. Going updip from the flank of the reservoir, the thin gas zone results in constructive interference between the positive main lobe of the dim spot (the reservoir top) and the positive main lobe of the flat spot (the GWC), and of their respective sidelobes, resulting in slightly enhanced reflections. Then, as the gas zone reaches one-fourth of the wavelength in thickness (the tuning thickness in a conventional wedge model), this constructive interference is reduced and destructive interference between sidelobes and mainlobes results in smaller amplitudes, reaching a minimum at one-fourth of the wavelength. As the thickness continues to increase to one-half of the wavelength, there is a slight constructive interference between the sidelobes of the two main reflections, resulting in a slightly enhanced negative reflection (the red line in Figure 8). It is possible that one’s eye, or an automatic tracking system seeking a bright spot, would continue to follow the negative sidelobe of the flat spot. The resulting seismic section shows a strong negative flat reflection over a strong positive flat reflection; this response is similar to that observed in F3 (Figure 1). These reflections exhibit a dip of flat to rather than the model dip of . This is not necessarily a result of tuning, but it is an effect of the low-amplitude dim spot itself.
We show that tuning effects can have significant effects on a true zero-offset seismic section, in which a flat spot terminates against a dipping interface. In the next section, we analyze and discuss amplitude versus offset and the importance of varied stacking angle ranges on the tuning effect.
Amplitude variation with offset and stacking up to and beyond critical offset
Amplitude variation with offset analysis
In contrast to the four conventional AVO classes, the negligible shear velocity contrast across any hydrocarbon-water contact makes its AVO response distinctive and unique. Specifically, the reflection coefficient for any flat spot is always positive and increases with offset, as ias shown in Figure 4.
Figure 9 shows the AVO curves out to large angles for all four DHIs. To have confidence at large angles, it is necessary to solve the full Zoeppritz (1919) solution (we used code provided by the Consortium for Research in Elastic Wave Exploration Seismology [CREWES], which solves the equations as written by Aki and Richards, 1980). For reflections that occur at an interface at which the velocity increases, a critical angle will be encountered, beyond which the reflections undergo strong phase rotation. The flat spots and dim spots will always exhibit supercritical phase rotation (see the flat spot and dim spot curves in Figure 9b). Because the velocity increase across a flat spot can be significant, the critical angle may occur at surprisingly small angles, such as the 52° shown in our example. This may or may not be within the range of recorded data. Because postcritical reflections always undergo phase rotation, if stacking involves these postcritical reflections, the stacked output will exhibit large-amplitude nonzero-phase wavelets. This effect may compound the similar wavelet distortion caused by tuning.
Figure 10 shows the result of convolving the AVO response in Figure 9 with a 50-Hz Ricker wavelet. This includes phase shifts for the flat spot and dim spot that extend beyond critical. The seismograms on the far-right in Figure 9 show the result of stacking these events over different angle ranges as indicated.
As the stacking ranges increase, all hydrocarbon indicators other than the flat spot show an increase in brightness of the stacked trace without any change in wave shape. (The dim spot model does not include many traces beyond critical offset in the final stack, and the effect of its phase rotation is minimal.) The flat spot, on the other hand, shows not only an increase in amplitude but also a change in shape. For example, a zero-phase Ricker wavelet when stacked over 0°–70° approaches the appearance of a 90° wavelet. The earliest part of this is a “trough” easily misinterpreted as a negative reflection coefficient. Hence, flat spot events stacked beyond critical offset can generate spurious bright reflections that might be categorized as bright spots by the interpreters.
Muting based only on the NMO stretch may not remove the postcritical reflections and could mislead interpreters. Because muting is usually based on distortion caused by NMO, and not by angle, it is possible that some flat spots are stacked beyond critical angle.
Synthetic stacked seismograms in the wedge model
As we have seen, a zero-phase Ricker wavelet, when stacked over a range of 0°–70°, undergoes a significant phase rotation and might appear as a negative reflection event. Here, we investigate the effect of stacking over varying angle ranges within the wedge model.
To see the combined effect from tuning (due to a dipping interface encountering a flat GWC) and stacking with AVO and postcritical reflections, we ran the wedge model described earlier, but this time using wide-angle stacks rather than zero-offset reflections. Three different models (polarity reversal, bright spot, and dim spot) were prepared for 0°–30°, 0°–60°, and 0°–70° stacks, respectively, recalling that the critical angle for the flat spot in this model is 52°. The synthetic seismograms in Figure 11 show that the combined effect of tuning and supercritical stacking enhances the flatness of those spurious bright reflections. Table 4 summarizes the dip (in milliseconds/trace) of reflections affected by tuning effects and stacking ranges compared with the original geologic model dip, for these three different models.
The modeling results summarized in Table 4 show a significant decrease in interpreted dip angle for all models. This suggests that observed reflection events from the top of the reservoir can appear as a “flat” event, discordant with the structural trend because of tuning. The effect is quite striking if the top of the reservoir exhibits a polarity reversal or a dim spot over the gas zone.
Even the bright spot model shows that stacking beyond the critical angle can result in some apparently flatter events. The strong reflection from the reservoir top probably helps to avoid improper interpretation in most cases.
The maximum flattening effect was observed for the dim spot model. In this model, the top of the reservoir is nearly invisible, and this probably has the most significant implication for interpretation. These flat reflections in the case of a dim spot, marked with the red line in Figure 8, were caused by the tuning effect of the side lobes and are exaggerated when the nonzero-phase wavelet is stacked beyond the critical angle. Although the sonic log from a nearby well suggests that the reservoirs in the F3 block most likely exhibit polarity reversal, the dim spot model seems to best match the F3 seismic data.
For our particular data set, we know neither the muting criterion used in the original processing nor how much stretch was caused by NMO correction, so the previous discussion and analysis were restricted to stretch-free modeling. In the next section, NMO stretch and muting will be considered in our analysis.
Normal moveout stretch and muting analysis
To estimate possible NMO stretch for different mutes, we extend our modeling to include the effects of NMO stretch on stacking.
To convert angles of incidence to offset distances, we created a true-depth model that is intended to resemble the area of block F3, based on the well-log data. This simple subsurface model used homogeneous flat layers extending from the surface to the base of the gas reservoir (Figure 12). We used the reservoir and shale parameters for the polarity-reversal case. In this model, we found the “critical offset” for the flat spot reflection (the surface offset distance to the critical angle of 52°) to be approximately 1300 m.
Normal moveout stretch and muting analysis
The “percent of changing frequency” criterion (Yilmaz, 1987), which qualifies the NMO stretching (frequency distortion in which events are shifted to lower frequencies) by the percent of changing frequency, is adopted here. A stretch limit of 50% changing frequency is taken to determine the muting zone and yields muting at 1100 m offset for the time of the flat spot reflection. This implies that postcritical reflections from the GWC (occurring at approximately 1300 m offset for our subsurface model) would have been automatically muted in our example. The details are provided in Figure 13, which shows the synthetic CMP gather before NMO correction and after NMO correction, with and without muting of events beyond critical offset.
In addition, the reflection hyperbolas’ intersection of the seafloor reflection with the flat spot reflection can be observed at approximately 1200 m. Because the maximum NMO stretch usually occurs when some of the reflection hyperbolas intersect one another (Buchholtz, 1972; Shatilo and Aminzadeh, 2000; Zhang et al., 2013), the traces around that offset are often muted; in our case, this offset happens to be close to the critical offset for the flat spot reflections. We conclude that the postcritical seismic data from flat spot reflections in F3 were most likely muted before stacking and excluded from the stacked output.
Results and comparison with original seismic data
Based on the analyses described above, reflections from GWC are likely to have been muted beyond the 1100-m offset, corresponding to an angle of incidence of 47°. Figure 14a shows the synthetic seismograms that result from stacking over an angle range of 0°–47° for the polarity-reversal case previously shown for different angle ranges. These perhaps best represent the example used from North Sea block F3. The synthetic seismograms demonstrate the flat, bright reflections caused by tuning and stacking. We think that this explains the observations in the actual stacked seismic data, as shown in Figure 14b.
Results and discussion
Bright reflections in block F3 of the Dutch North Sea were analyzed for possible sources of their flatness. Tuning and postcritical stacking were evaluated as possible reasons. For this purpose, zero-offset and wide-angle stacked sections were prepared by forward modeling. NMO stretch and muting were investigated to evaluate the possibility that stacking extended beyond the critical offset for our data set.
The tuning effect results showed a significant decrease in the reflection dip for the polarity-reversal and the dim spot cases. The polarity-reversal case is suggested for F3 based on well logs and the strength of the reflections, but the dim spot example also fits the data. In both cases, the synthetic seismograms show that the reflections appear to become very flat for thicknesses under one fourth of the wavelength. For the dim spot case, the modeling results are particularly striking because the dim spot event is of very low amplitude and the bed boundary could be misidentified from the (enhanced) sidelobe above the GWC. The result for the dim spot is a strong negative flat reflection over a strong positive flat-spot reflection; this response is similar to that observed in F3. If an interpreter or tracking system is seeking a bright spot, they might think that they have found it in these large negative amplitudes.
If postcritical events were included in the stacking output for our data set, additional distortion to the event could have resulted. Flat spot events stacked beyond critical offset can generate spurious bright reflections that might be categorized as bright spots. The two phenomena (tuning effect and supercritical stacking) could act together, strongly modifying the final results in an actual reservoir. This effect is even more striking if the top of the reservoir exhibits a polarity reversal over the gas zone. For our particular data set, the stretching and muting are unknown, but our simple models suggest that postcritical seismic data were excluded in the stacking output.
Based on these analyses, we conclude that the tuning effects and postcritical stacking can make bright reflections in F3 flatter and brighter. Postcritical stacking likely did not occur, and the tuning effect is presumed to be the main source of the bright, flat events in the F3 data. The tuning effects can be significant for both dim spots or polarity reversals, making these reflections appear as bright spots. Care should always be taken when interpreting stacked data, with the recognition that it is not the same as zero-offset data.
The stacking of flat spot reflections beyond the critical angle can boost their amplitudes significantly while accompanied by a significant phase shift in the stacked output. Tuning effects can also change the amplitudes and apparent polarity and phase in the cases we examine. Individually, these effects can result in fairly flat, bright negative events overlying strong positive reflections. The effect can be strong enough that it can even make a dim spot appear as a (flat) bright spot.
Postcritical seismic data of flat reflections were most likely excluded from the stacked output of our data set, based on our NMO stretch and muting analysis. We conclude that the tuning effect is the key reason for the flatness of the bright reflections at shallow depths in block F3 of the Dutch North Sea. We further conclude that these bright reflections are not typical bright spots but appear as such because of the tuning effect.
In addition, although for our particular data set, postcritical offset data were probably muted based on traditional criteria, we recommend that care should be taken while dealing with reflection data containing a wide range of incidence angles, where those criteria may not be routinely applied (e.g., crosswell-seismic data). Muting applied solely on the basis of NMO stretch might include postcritical reflections, and the stacked output will be significantly altered.
We thank dGB Earth Sciences for providing the data of block F3 and the OpendTect software used here for analysis and display. We also want to thank CREWES at the University of Calgary, who created the MATLAB codes that we used for amplitude variation with offset modeling. We appreciate the comments of several reviewers, leading to improvements in the manuscript.
Qiang Guo received a B.S. (2012) in geophysics from China University of Geosciences, Wuhan, China, and an M.S. (2014) in geophysics from Michigan Technological University. He is working with Xiamen Construction and Development Corporation as an engineering geophysicist. His current research interests include seismic data processing and interpretation, rock-physics modeling, and refraction seismology.
Nayyer Islam received a B.S. in geological engineering from the University of Engineering and Technology, Lahore, Pakistan, and he joined that institution as a lecturer in 2008. He served at that position until coming to Michigan Technological University, where he received an M.S. (2011) and a Ph.D. (2014). Currently, he is working with BP America as a petrophysicist. His research interests include reflection seismology, rock-physics modeling, and petrophysics (i.e., seismic petrophysics).
Wayne D. Pennington received degrees in geology and geophysics from Princeton University, Cornell University, and the University of Wisconsin-Madison. He is currently the dean of the College of Engineering at Michigan Technological University, where he has worked since 1994. Prior to that, he was at Marathon Oil Company’s Petroleum Technology Center in Littleton, Colorado, following several years on the faculty at the University of Texas at Austin. He has served as a first vice-president of SEG, as the president of the American Geosciences Institute, and as a Jefferson Science Fellow at the U.S. Agency for International Development.