Normal modes in seismic data â Revisited

GEOPHYSICS, VOL. 77, NO. 4 (JULY-AUGUST 2012); P. W27–W40, 22 FIGS., 4 TABLES. 10.1190/GEO2011-0094.1

Normal modes in seismic data — Revisited

M. Landrø1 and P. Hatchell2

and presentation is given in the excellent text book written by Ewing et al. (1957). They study the seismic response from a source in the water layer, embedded in a layered half-space. More recent books discussing this are written by Frisk (1994) and Jensen et al. (2000). In modern university courses, and especially for those specializing in petroleum geophysics, there is hardly any focus on normal modes. Because the mathematical derivation is fairly long and tedious, there is a tendency to skip this in university teaching. The same statement is to some extent also valid for Lamb’s problem (Lamb, 1932). Seventy years ago, there were several papers focusing on deriving equations and approximations to describe the complex seismic signal measured at a hydrophone far away from the source. In underwater acoustics however, mode theory is used extensively (Frisk, 1994; Jensen, 2000). In Ewing et al. (1957), one example of field data is presented in Figures 4-12. Another, more recent example can be found in Greene and Richardson (1988), where they analyze data from the Beaufort Sea. The focus of this paper is threefold: First, to honor the basic and important work done by the scientists mentioned above; second, to derive a new approximation for the minimums of the normal mode group velocities; and third, to briefly demonstrate that some of the characteristic features related to normal modes are actually observed in field data. During the last decade, the discussion on how much seismic activity harm fish and fishing activity has been more and more in focus (Amundsen and Landrø, 2011). At long distances away from the source, the major part of the wave field is actually composed of normal modes, and hence, we think it is timely to revisit this old theory again. Various types of seismic modeling, including the use of normal mode theory, may play an important role as a tool to determine safety zones for seismic surveys in sensitive areas. In addition to the environmental aspects of normal modes, it is interesting to explore whether they can be exploited in seismic analysis. Hatchell et al. (2007) present one example where the focus is on monitoring velocity changes in the water layer. Another recent proposal is presented by Landrø and Amundsen (2011), where they suggest using the refracted wave to monitor changes in the layer below the water layer.

ABSTRACT At long distances from a seismic shot, the recorded signal is dominated by reflections and refractions within the water layer. This guided wave signal is complex and often is referred to as normal or harmonic modes. From the period equation, we derive a new approximate expression for the local minima in group velocity versus frequency. We use two data sets as examples: one old experiment where the seismic signal is recorded at approximately 13 km offset and another example using life of field seismic data from the Valhall Field. We identify four and five normal modes for the two examples, respectively. A fair fit is observed between the estimated and modeled normal mode curves. Based on the period equation for normal modes, we derive a simple, approximate equation that relates the traveltime difference between various modes directly to the velocity of the second layer. Using this technique for offsets ranging from 6 to 10 km (in step of 1 km), we find consistent velocity values for the second layer. We think that this method can be extended to estimate shallow lateral velocity variations if the method is applied for the whole field. We find that the simple equations and approximations used here offer a nice tool for initial investigations and understanding of normal modes, although a multilayered method is needed for detailed analysis. A comparison of three vintages of estimated normal mode curves for the Valhall field example representing seabed locations shifted by 1 km indicates that minor shifts in group velocity minima for the various modes are detectable.

INTRODUCTION Almost 70 years have elapsed since Pekeris (1948) and Press and Ewing (1950) presented their work on normal (or harmonic) modes related to seismic wave propagation. A comprehensive overview

Manuscript received by the Editor 9 March 2011; revised manuscript received 16 January 2012; published online 27 June 2012. 1 NTNU, Department of Petroleum Engineering and Applied Geophysics, Trondheim, Norway. E-mail: [email protected]. 2 Shell Internation Exploration and Production Inc., Houston Texas, USA. E-mail: [email protected]. © 2012 Society of Exploration Geophysicists. All rights reserved. W27

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Landrø and Hatchell

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NORMAL MODES A comprehensive description of normal modes generated within a layered half-space is given in the well-known textbook written by Ewing et al. (1957). Figure 1 is modified from their book and shows a liquid half-space over an infinite liquid layer. Based on seismic theory, it is possible to derive approximations for the recorded wave field at a large horizontal distance from the source, and the detailed derivation of this can be found in their excellent book. In the analysis of the seismic data recorded at a long distance from a seismic vessel, we will use the period equation (equation 4–78 in Ewing’s book)

qffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 c2 ρ2 α21 − 1 tan kH − 1 ¼ − qffiffiffiffiffiffiffiffiffiffiffi2ffi ρ1 1 − c 2 α21

(1)

α2

Here, c is the frequency-dependent phase velocity of the normal modes, k is the wavenumber, H is the water depth, α1 and α2 denote the P-wave velocity of the water layer and the first layer below the seabed, and ρ1 and ρ2 denote densities for the corresponding layers, respectively. From this equation, it is possible to determine the phase velocity (c) as a function of frequency for each mode. Because the left side of equation 1 is periodic, we will get multiple solutions, leading to the harmonic modes. Once the phase velocity for each mode is determined, it is possible to estimate the group velocity by taking

the derivative of the frequency with respect to the wavenumber k. An example of phase velocities and corresponding group velocities is shown in Figure 2. In Ewing’s book, there is no explicit formula given for the group velocity. It is shown in Appendix A that the group velocity (U) can be derived directly from equation 1

α2 κ 2 1 1 ; U ¼c− κ21 α21 ρ2 1 2 c 1 þ ρ1 Hk κ2 þ κ3 α2 cos kHκ1

(2)

2 2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where κ 1 ¼ ðc2 ∕α21 Þ − 1 and κ 2 ¼ 1 − ðc2 ∕α22 Þ. This equation can also be found in Frisk (1994, equation 5.165, page 154) using slightly different notation. From equation 2, we see that the group velocity approaches α1 when c → α1 and α2 when c → α2 , as expected. From Figure 2, we see that the phase velocity for each mode starts at the velocity for the second layer and asymptotically approaches the water velocity for higher frequencies. The asymptotic behavior is observed for the group velocity, apart from the fact that the group velocity reaches a local minimum for a given frequency for each mode. We will derive approximations for these frequencies in this paper. If the second layer is a solid layer, the shear velocity will enter into this period equation, as derived by Ewing et al. (1957, equation 4.154)

ffi sffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 c2 ρ2 β42 α21 − 1 c2 c2 q ffiffiffiffiffiffiffiffiffiffiffi ffi 1 − tan kH − 1 ¼ 4 1 − α21 ρ1 c4 1 − c22 α22 β22 α2

H α1 , ρ1 α 2 , ρ2

Figure 1. A point source in a liquid layer over a liquid half-space. Paths of multiple reflected and refracted waves which may interfere and build up the oscillatory refraction wave. This figure has been modified from Figure 4-1 and 4-7 in Ewing et al. (1957).

c2 2 − 2− 2 : β2

(3)

Comparing this equation with equation F-1 in Yilmaz (2001), we think there is a misprint in equations F-2b and F-2c. If the two terms under the square root in these equations are interchanged, the correspondence with equation 3 is retained. For the field data used in this study, the shear-wave velocity for the second layer is assumed to be fairly low, typically of the order of 2–300 m∕s. This means that we can assume β2 ∕c ≪ 1, and hence

sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 tan kH −1 α21 qffiffiffiffiffiffiffiffiffiffiffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 ρ2 α21 − 1 β22 c2 β 3 ≈ − qffiffiffiffiffiffiffiffiffiffiffi2ffi 1 − 4 2 − i4 1 − 2 32 : ρ1 1 − c 2 c α2 c

(4)

α2

Figure 2. Phase (solid lines) and group (dashed lines) velocities estimated from equation 1 for the four first mode defined by the period equation 1. The sum of multiple reflections and refractions within the water layer as shown in Figure 1 may be represented as multiple normal modes, each with a unique frequency-dependent phase velocity and a corresponding group velocity. Water velocity is 1485 m∕s, velocity of second layer is 1700 m∕s. The density ratio is 1.8, and the water layer thickness is 75 m. The velocities are normalized with respect to the water velocity.

This means that the dispersion relation has an imaginary part, corresponding to an imaginary part of the wavenumber, which again means attenuation. For a shear-wave velocity equal to 300 m∕s; the imaginary part of equation 4 is equal to 0.01. We will therefore neglect this term. Furthermore, the real part correction term in equation 4 is of the order of 12%, and therefore we will use equation 1 in the further analysis. For higher values of shear-wave velocity in the seabed, the full period equation 3 has to be used to determine the group velocity for each mode. A numerical example showing the effect of a nonzero shear-wave velocity is shown in Figure 3. We clearly see that the effect of introducing the shear velocity is to


Normal modes in seismic data

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push the local minimums for the group velocity toward higher frequencies. If we increase the shear-wave velocity of the second layer above the P-wave water velocity, we might get more than one local minimum for mode numbers larger than 1. See, for instance, Figure 4-18 in Ewing et al. (1957).

EXPLICIT AND APPROXIMATE EXPRESSIONS FOR GROUP VELOCITY Equation 2 gives an exact expression for the group velocity, and in Appendix B we derive approximate values for the frequency at which the local minimums in the group velocity occur

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 α2 α4 ð1 þ γ 3 þ 23 α22 þ 19 α24 γ −3 Þ 1 1 rffiffiffiffiffiffiffiffiffiffiffiffi ≈ α2 ρ1 2πH 1 − α12 2 2 2 α1 2 1 α2 −2 3 − 3 þ × γ γ 3 9α21 α22 ρ2 α1

f min n

Figure 3. The effect of introducing a solid water bottom with shearwave velocity: Solid lines represent a shear-wave velocity of zero, and dashed lines represent a shear-wave velocity of 500 m∕s in the second layer. The P-wave velocity in the second layer is 1700 m∕s, and the density contrast 1.8. The imaginary term in equation 4 is neglected.

(5)

where γ is given by equation B-14. From Figure 2, we observe that the minimum group velocity decreases with increasing mode number. In Appendix C, we derive the following expression for the envelope of the minimum of the group velocity

U≈

α21 : c

(6)

Figure 4 shows a comparison between exact and approximate envelope curves for the minimum group velocity for water depths of 75 and 500 m, respectively. We see that the approximation (equation 6) is more accurate for higher frequencies, and that all curves α2 approach the theoretical value of U∞ ¼ α12 in the high frequency limit. In Appendix C, it is also shown that the minimum group velocity for each mode is approximately equal to

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u α22 u −1 α2 u U n ≈ u1 þ 1 2 ; α2 t πρ α2 3 1 þ ð2n − 1Þ 2ρ1 α22 α21

(7)

Figure 4. Comparison of estimated envelope curves for the minimum group velocity (dashed lines) and exact curves (solid lines) for water depths of 75 and 500 m, respectively. Notice that all curves approach the theoretical value of α1 ∕α2 ¼ 1485∕1725 ¼ 0.86 in the high frequency limit.

2 1

where n is the mode number. For high mode numbers (n), we see that the asymptotic value is U ∞ ¼ α21 ∕α2 , again as expected. Figure 5 shows that the accuracy of equation 7 increases with n. For the example shown in Figure 5, the error for n ¼ 1 is 4% and 0.8% for n ¼ 10. If we let α2 ∕α1 ¼ 1 þ ε, and assume that ε ≪ 1, it is shown in Appendix C that the relative increase (ε) in P-wave velocity between the layer below the seabed and the water layer velocity, is approximately equal to

εn ≈

−1 ΔT n πρ 1 1þ 1 1− 2 Tn 2ρ2 ð2n − 1Þ3

(8)

where ΔT n is the two-way traveltime difference between the first mode and mode n (this means that equation 8 is valid only for n > 1). If we assume that this relative velocity increase is not varying with n, we find that the relative time difference is

Figure 5. Comparison of estimated relative group velocity minima using equation 8 (squares) and exact minima (circles). The relative errors are most significant for low mode numbers (n). Parameters used: α1 ¼ 1485 m∕s, α2 ¼ 21725 m∕s, H ¼ 75 m, ρ1 ¼ 1000 kg∕m3 ; and ρ2 ¼ 1800 kg∕m3 .



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ΔT n πρ −1 1 ≈ε 1þ 1 1− : 2 Tn 2ρ2 ð2n − 1Þ3

(9)

This equation offers a direct relation between observed traveltime differences between higher order modes and the first one and the relative velocity increase from the water layer into the first layer

below seabed. As an example we have computed the “Lyman series”, that is those mode combinations involving the first mode and the three others (Figure 6). The results are summarized in Table 1, where we observe a reasonable agreement between the estimates of the velocity for the second layer. As expected, the velocity estimate based on the two first modes is somewhat lower than the proceeding values. These estimates are somewhat lower compared to those estimated directly from the refracted wave, where we find values ranging from 1705 m∕s to 1778 m∕s. The estimates presented in Table 1 are far from accurate, and it is hard to assess uncertainties to the estimates. However, it demonstrates that there is a potential for measuring the group velocity minimums for each mode and use it to estimate the velocity ratio at the seabed. According to equation 9, the density ratio will also influence the estimate of the seabed velocity contrast. In our calculations here, we have assumed a density contrast of 1.8 at the seabed. Figure 7 shows the asymptotic behavior of the relative timeshift versus mode number, for two density ratios. In the asymptotic limit, we find that

ΔT ∞ πρ1 ε∞ ≈ 1þ : T∞ 2ρ2 Figure 6. Frequency analysis of one single trace from the field data set. The trace has been filtered with 48 different narrow band-pass filters with gradually increasing center frequency. The bandwith of each filter is only a few Hz. We identify four distinct harmonic modes from this figure. Identifying minimum traveltimes for the first four modes observed in the field data (shown by dashed lines): T1 ¼ 3.38 s, T2 ¼ 3.70 s, T3 ¼ 3.83 s, and T4 ¼ 3.87 s. Note that the time scale is arbitrary, because there is no control on when the source actually was fired.

Table 1. Estimated values for epsilon based on time shifts between various modes compared to the first, extracted from Figure 6. Mode transition 1–2 1–3 1–4

Time difference (s)

ε

α2 ðm∕sÞ

0.32 0.45 0.49

0.12 0.14 0.14

1670 1690 1690

Figure 7. Relative traveltime difference between mode n and the first mode, assuming a constant velocity ratio, for two density ratios: Black line: 1.8 and red line: 1.5. Parameters used: α1 ¼ 1485 m∕s, α2 ¼ 1725 m∕s, and H ¼ 75 m.

(10)

This means that the relative velocity increase is directly related to the observed traveltime shift and the density ratio.

VARYING VELOCITY, WATER DEPTH, AND DENSITY So far, we have found that it is possible to identify the group velocity for several normal modes directly from a single trace frequency analysis. To assess how useful such an analysis might be for long offset seismic data, we will study the sensitivity versus variation in some of the key parameters in the period equation (equation 1 or 4). Figure 8 shows that the velocity of the second layer (the layer below seabed) has a strong influence on the group velocity behavior for all modes. We see that the effect of increasing the velocity in the second layer is to shift the group velocity minimums toward lower frequencies and to decrease the actual value of each minimum. Furthermore we see that the effect increases with increasing frequency. The effect of changing the density ratio between the second layer and the water layer, is less pronounced, as shown in Figure 9. Also here, we see that the effect of increasing the density in layer two is to push the group velocity minimums toward lower frequencies and to decrease the actual value of each group velocity minimum. The effect of changing the water depth is significant, as shown in Figure 10. For large water depths, the normal modes are squeezed toward lower frequencies. However, the actual group velocity minimum value is independent of the water depth. This means that changing the water depth is equivalent to squeezing the group velocity curves. This effect is also observed directly from equation 1 where the kH-term only appears on the left side. Increasing the water depth is equivalent to reducing the wavenumber k, and hence also the frequency. From equation 7 we observe the same feature: The minimum group velocity is independent of the water depth, and decreases with increasing mode number (n). In real sediments where the velocity is not constant with depth, guided modes in deeper water will have lower frequencies that see deeper into the earth and break this symmetry.



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For deepwater environments, the normal modes move to low frequencies that are often below the source frequencies in use for reflection seismic acquisition and so these effects are rarely seen on oil-gas exploration surveys. For example, equation 4–80 in Ewing et al. (1957) shows that the cutoff frequency of the nth normal mode occurs at frequency

f cutoff ¼

ð2n − 1Þα1 α2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4H α22 − α21

(11)

For a deepwater environment where H ¼ 1000 m, and the two velocities are 1500 and 2000 m∕s, this gives a cutoff value of 0.56 Hz. Hatchell and Mehta (2010) analyzed deepwater passive acoustic waves using interferometry of Ocean Bottom Node (OBN) recordings made during a survey acquired in 1000 m water depth. The main arrivals they saw were from normal modes that are the ambient noise in deepwater environments. Estimations of the phase velocity dispersion curves (Figure 11) show three normal modes that are well predicted by the period equation (equation 1).

Figure 8. The effect of changing the velocity on the normalized (with respect to the water velocity) group velocity for various harmonic modes. Solid lines show the four first harmonic modes (black, green, red, and blue) for a velocity of 1700 m∕s in layer two, and dashed lines show corresponding modes for a velocity of 1800 m∕s in layer two. Notice that the effect increases with increasing mode number. Density ratio: 1.8.

DESCRIPTION OF THE EKOFISK DATA SET The field data were recorded as a part of an experimental study of various types of noise that may influence seismic data quality. The data acquisition was performed in the Ekofisk area in the southern part of the North Sea. The water depth in this area varies between 70 and 75 m. In February 1989, M/V Rigmaster recorded seismic noise from another seismic vessel acquiring seismic data. The Rigmaster source array was not active. The data were recorded on a 240 channel system, with 12.5 m between the hydrophone groups and a total streamer length of 3000 m. The cable depth was 9 m. A sample rate of 2 ms was used, and the data were recorded with a 0–154 Hz (72 dB∕oct) filter. The data were resampled from 2 to 4 ms after applying a 5.3 (18 dB∕oct) to 90 Hz (72 dB∕oct) DFS V instrument filter. One hundred thirteen records of 7 s were acquired. The positions and heading of each of the two vessels are shown in Figure 12. Figure 13 shows mid traces of all the 113 records. The noise bursts from the other vessel is slightly shifted (by half a second or so) from one record to the next. This is probably due to slightly different timing between each shot on the shooting vessel and the timing between each record on the recording vessel. A comparison of a near, mid and far offset trace is shown in Figure 14. On all traces we notice the refracted wave (harmonic oscillations between 2 and 3 s) followed by the water wave (including all water layer reverberations and the normal modes). Unfortunately, this data set is not available in digital format any more, and therefore a more comprehensive analysis involving for instance, an inversion method using a multilayered water and subsurface model is not possible.

Figure 9. The effect of changing the density ratio between layer two and the water layer. Solid lines show density ratio of 1.8 and dashed lines show density ratio of 2.2. Velocity of layer two is 1700 m∕s.

ESTIMATING DISPERSION CURVES FROM SEISMIC DATA The most common way to estimate dispersion curves from seismic single sensor data is to apply multiple filter analysis to the data (Dziewonski et al., 1969). Hatchell et al. (2007), used the τ-p method proposed by McMechan and Yedlin (1981) to identify the normal modes, and Hatchell and Mehta (2010) used the method proposed by Park (1998). However, for the Ekofisk data set, the streamer is not inline oriented with respect to the shot (Figure 12), and therefore, we use a slightly different method using only one

Figure 10. The effect of changing the water layer thickness. Solid lines show the four first harmonic modes for a water depth of 75 m (assuming a velocity of 1700 m∕s in the second layer), and dashed lines show corresponding modes for a water layer thickness of 300 m. Notice that the major effect is to move all modes to a significantly lower frequency. Density ratio: 1.8.


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single trace as input to identify the normal modes. Figure 15 shows a time-frequency analysis plot, where we have used a narrow sliding band-pass filter (2.5 Hz wide, with a Hanning taper window) on the midstreamer trace for record two. By using this simple technique the various modes separate quite well, and we may interpret four modes within the frequency range between 0 and 120 Hz. The red solid lines in the figure shows an attempt to model the phase velocity traveltime (upper curves) and the corresponding group velocity traveltime for the first four modes. In the modeling, we have assumed that the distance between source and receiver is 13 km, and

that the velocity of the second layer is 1700 m∕s. The group velocity curves fit reasonably well, however, we observe that the actual location of the minimums is displaced toward higher frequencies for higher modes. If we assume that the “effective” velocity of the second layer decreases with frequency (similar to what we discussed for the refraction wave), we are able to achieve a better fit between modeled and observed minimums for the group velocity. We find that a choice of 1800, 1690, 1650, and 1630 m∕s (for the first four modes) gives a good fit between observed and modeled group velocity minimums. However, the modeled traveltimes do not fit very accurately by using these values. This might be due to the fact that we do not actually know the source-receiver distance exactly. The distance was obtained from the observer log, and is not very precise. Another possible explanation for the deviation between modeled and observed group velocities is the effect of introducing the shear velocity into the period equation (see equation 4). The effect of taking the shear-wave velocity into account is also to shift the group velocity minimums toward higher frequencies. Despite these discrepancies between modeled and observed modes, we conclude that we may interpret four modes based on simple analysis of a single trace from this data set. There is no doubt that to increase the accuracy and usefulness of such an analysis our ad hoc and trial and

Figure 11. Comparison of measured phase velocity dispersion and the period equation in 1000 m water depth (Hatchell and Mehta, 2010).

Figure 12. Position and heading of the recording vessel (Rig Master) and the shooting vessel (Pacific Horizon) during the acquisition of the seismic data.

Figure 13. Single trace plot (midstreamer trace) of all 113 records.

Figure 14. Selected traces (near, mid and far) from record 2. Notice the refracted wave (harmonic oscillations) arriving ahead of the water wave.


Normal modes in seismic data error based method should be replaced by a multilayer inversion technique, possibly taking both velocity variations in the water layer as well as a layered earth model including nonzero shear-wave velocities into account.

USING VALHALL LIFE OF FIELD SEISMIC DATA The Valhall Field is one of several chalk fields in the southern part of the Norwegian part of North Sea, close to the border between Norway and Denmark. The field was discovered in 1975, and production started in 1982. In 2003, 120 km of seismic cables were trenched into the seafloor covering an area of 45 km2 above the field. In total, 2400 receiver 4C (3 geophones and one hydrophone) were deployed (Barkved et al., 2003). A separate shooting vessel

Figure 15. Frequency analysis of the mid streamer trace for record two. The trace has been filtered with 48 different narrow band-pass filters with gradually increasing center frequency. The bandwidth of each filter is only a few Hz. Notice four distinct harmonic modes. Also observe the weaker refraction signal observed between 2 and 3 s. Red solid lines show estimated traveltimes for the group velocity assuming a water velocity of 1485 and 1700 m∕s for the second layer. The corresponding traveltime for the phase velocities are shown by dashed red lines.

Figure 16. Constant receiver gather (hydrophone component) from Valhall, after LMO (using 1500 m∕s) and adjustment to 4 s twoway traveltime. Typical velocity for the shallowest is estimated at approximately 1750 m∕s.

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covering a regular grid of 50 × 50 m at the surface was used. The water depth at Valhall is approximately 70 m. More than 11 life of field seismic (LOFS) surveys have been acquired so far. The acquisition pattern used for these LOFS surveys enables us to use and analyze data with offsets up to 10 km. When analyzing these data with respect to normal modes, we applied a linear moveout correction using a water velocity of 1500 m∕s, and adding 4 s to the data, as shown in Figure 16. A similar analysis as presented in Figure 6, is presented in Figure 17, for an offset of 10 km. After selecting the 10 km offset trace shown in Figure 16, a 10-Hz wide sliding frequency window was used, followed by a windowed (length of 50 ms) rms-calculation in the time domain. We observe the same trend as shown in Figure 6, and the five observed events are interpreted as normal modes, corresponding to the characteristic group velocity minimums. The fifth mode was judged to be slightly weaker and less precisely defined than the first four, and was therefore not used for further analysis. Again we used equation 8 to estimate ε-values for mode transitions between one and four. The nice feature of equation 8 is that there is a first order relation between observed traveltime shifts between normal modes and the velocity of the second layer α2. Equation 8 is also dependent on the density ratio between the second layer and the water layer, but this may be regarded as a second order effect. In our analysis, we used a value of 1.6 for this ratio, based on well log information from the Valhall field. In equation 8, we need to know the traveltime of each mode (T n ¼ Uxn ). For simplicity, we assumed that Un ≈ α1 ¼ 1470 m∕s. The actual group velocity value is somewhat less than this value, however, the inaccuracy introduced by this assumption is judged to be minor. From Figure 17, we extracted traveltime differences between the various modes by simple picking. The pick was done by choosing a traveltime in the middle of the black maximum shown in Figure 17. Estimated epsilon-values and corresponding velocities for layer two are listed in Table 2. We observe that the estimates are fairly consistent using the first four modes. The average velocity of the second layer is around 1670 m∕s. However, the uncertainty associated with this number is significant, due to the width (100 ms) of the normal mode curves in Figure 17. The thin

Figure 17. Time-frequency plot for a source-receiver offset of 10 km. A 10-Hz wide sliding frequency filter has been used for the same seismic trace, after linear moveout correction using 1500 m∕s, and shifting to 4 s traveltime. Five modes may be interpreted. The solid curves represent modeled group velocities for various choices of water depth and seabed (second layer) velocity: red: 80 m and 1700 m∕s; cyan: 70 m and 1700 m∕s. The velocity of the water layer is 1470 m∕s and the density ratio between the second and first layer is 1.6.



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solid color-coded lines in Figure 17 show traveltime curves corresponding to group velocities for the first five modes (using equation 2) for four combinations of water depth and velocity for the second layer. Based on this type of trial and error based “inversion” method it is hard to determine the optimal parameter choice. However, it seems like a choice close to 1700 m∕s and an effective water depth of 80 m gives a reasonable fit to the data for most offsets. This is also reasonably close to estimates based on equation 8, presented in Table 2. It should be emphasized that there are several uncertainties and weaknesses associated with this simple type of analysis, which will be addressed in the discussion section. Using a shorter offset, 6 km, we obtain similar results, as shown in Figure 18. This is as expected, because the local minimums for the group velocity (equation 5) is independent of source-receiver distance. Table 3 shows the same analysis as presented above for five different offsets, ranging from 6 to 10 km. A remarkable consistency in the velocity estimates for layer two is observed, all velocity values ranging from 1676 to 1725 m∕s. Furthermore, we observe that there is a slight trend of reduced velocities with increasing offset. Although this trend is within the anticipated uncertainty of the method, we think it indicates that there is a potential for estimating velocity variations over the whole field based on this technique. Relative measurements

Table 2. Estimated values for epsilon based on time shifts between various modes compared to the first, extracted from Figure 15, assuming a density ratio of 1.6 between the second layer and water layer. Mode transition 1–2 1–3 1–4

Time difference (s)

ε

α2 ðm∕sÞ

0.24 0.30 0.36

0.135 0.133 0.144

1668 1665 1682

Figure 18. Time-frequency plot for a source-receiver offset of 6 km. A 10-Hz wide sliding frequency filter has been used for the same seismic trace, after linear moveout correction using 1500 m∕s, and shifting to 4 s traveltime. Five modes may be interpreted. The solid curves represent modeled group velocities for various choices of water depth and seabed (second layer) velocity: red: 80 m and 1700 m∕s; cyan: 70 m and 1700 m∕s. The velocity of the water layer is 1470 m∕s and the density ratio between the second and first layer is 1.6.

of traveltime shifts are more precise than one measurement, and by using much more data than shown in this example, more precise maps of shallow velocity variations might be produced. A comparison between Figures 17 and 18 shows a clear shift toward higher frequencies for the local minimums of the group velocity. Such a shift is normally indicating a change in water depth, because to the first order, the traveltime shifts are influenced by changes in the velocity contrast at the seabed, while a horizontal shift in such plots are dominated by changes in water depth. In this case, we know that the seabed subsidence at the field has been up to 6 m, so it is likely that the actual change in water depth between the 6 and 10 km offset data is less than 6 m. For higher order modes (n > 1), equation B-19 for the frequency minimums is simplified to equation 11. From equation 11, we clearly see that the frequency stretch we observe in the Valhall data between 6 and 10 km might be caused by a change in water depth or a change in velocity for the second layer. A simple modeling exercise using equation 11 shows that the minimum frequency is sensitive to water depth and velocity changes. Figure 19 shows that, for instance, a combined reduction in water depth of 5 m and a reduction in velocity for layer two of 20 m∕s results in frequency stretch of roughly 10 Hz.

Table 3. Estimated velocities for layer two m∕s for offsets ranging from 6 to 10 km, using equation 9. Density ratio between second layer and water layer is 1.6. Notice a small decrease in velocity with offset, and a slight increase in velocity from lower to higher modes. Mode

6 km

7 km

8 km

9 km

10 km

Average

1–2 1–3 1–4 Average

1676 1709 1725 1703

1694 1711 1714 1706

1687 1698 1706 1697

1692 1680 1680 1677

1668 1665 1682 1672

1683 1692 1701 1692

Figure 19. Minimum frequency for mode four using equation 13 as a function of water depth and velocity for the second layer. The water velocity is 1470 m∕s. Notice that a change in water depth and a change in velocity both influence the frequency. The colorbar is in Hz, and the arrow illustrates that a combined reduction in water depth of 5 m and a corresponding velocity reduction of 20 m∕s leads to an increase in the minimum frequency of roughly 10 Hz.



RELATIVE LATERAL VELOCITY VARIATIONS An alternative way of analyzing the LOFS data is to keep the offset fixed and move from one receiver station to the next, for instance, by 1 km or denser. In this way, we can compare relative changes based on the estimated dispersion curves. Figure 20 gives an illustration of this procedure: We observe a minor decrease in the vertical shift between for instance modes one and four when as the lateral position is changed from 0 to 1 and 2 km. The red solid dispersion curves in Figure 20 are identical, and we can also spot a minor horizontal shift between modes one and four as the lateral displacement increases. An alternative and probably more accurate way of analyzing such data could be to implement a semblancelike analysis for each mode, for instance, where the velocity of the second layer is used as the semblance-parameter to determine. In this way, an average velocity could be determined for each receiver location for a constant offset, and, in principle, a velocity map could be derived. Other methods for more accurate velocity analysis based on the estimated dispersion curves, could be to crosscorrelate the dispersion signals for each mode, and estimate relative timeshifts combined with equation 8 to estimate relative velocity changes. Once the velocity attribute is derived based on vertical shifts of the estimated dispersion curves, it could also be interesting to estimate the horizontal shift as an extra seismic attribute. Given that the velocity attribute has been reasonably well determined it should be possible to estimate this second attribute, which could be denoted

W35

as effective water depth. Both these attributes are limited by the fact that they are estimated using a two-layer model as the basis. Hale (2009) suggested an interesting crosscorrelation method designed for estimation of horizontal and vertical timeshifts from time-lapse seismic data. We think that it is possible to adapt this method to determine horizontal and vertical displacements between two dispersion curves, for instance, like the one displayed in top and middle of Figure 20.

COMPARISON WITH VELOCITY FIELD FROM SHALLOW SEISMIC We have now demonstrated that we can estimate relative stable velocity values for the second layer from measuring time shifts between various normal modes. The limitation is that we have used a simple two-layer model, and it could therefore be interesting to compare this simple model with other data. One such “independent” data set is stacking velocities from a recent shallow seismic survey that was acquired over the field in 2009. The velocities we picked are not exactly from the same location as the LOFS data, however, we think it is still useful to compare the velocity profile from the shallow seismic data with our two-layer velocity profile. Other issues that might complicate this comparison is that the normal mode analysis will measure horizontal velocities, while the shallow seismic survey will see velocities that are closer to vertical. The LOFS survey used in our examples was acquired in a relative cold season, and we estimated the water velocity at this time of the year to be close to 1470 m∕s. The value from the shallow seismic is higher, around 1490 m∕s, and we think that most of this difference is due to seasonal changes (Hatchell et al., 2007). Table 4 shows stacking velocities and corresponding interval velocities (based on Dix equation) from this survey. We notice that the water velocity continues below the seabed (which is at approximately 103 ms in this case). The water velocity continues approximately 21 m below the seabed (28 ms in TWT). This observation is the background for using an “effective water depth” that is actually larger than the actual water depth: Because the P-wave velocity remains equal to the water velocity for the first 10–20 m into the sedimentary layers, it might be reasonable to use a water depth that is 10 m larger than the actual value. A comparison of the two velocity models is shown in Figure 21. The two models differs, however, an inspection by eye tells us that the two-layer model is in relatively good agreement with the multilayer model based on the shallow seismic. Table 4. Example of stacking velocities for a shallow seismic survey at the Valhall field. Interval velocities are estimated using Dix equation. TWT (ms)

Figure 20. Time-frequency plots for 6 km offset for three laterally displaced receiver locations: 0 km (top), 1 km (middle), and 2 km (bottom). The red solid line shows the group velocity versus frequency for a water depth of 80 m, water velocity of 1470 m∕s and a velocity of the second layer of 1700 m∕s. The density ratio is 1.6. These red curves are identical for the three plots. We notice a minor reduction in the vertical spread between the modes as well as a horizontal shift toward higher frequency as the lateral displacement increases from 0 to 2 km.

0 105 133 169 292 354 498

Velocity (m∕s)

Interval velocity (m∕s)

Estimated depth (m)

1493 1493 1518 1564 1659 1705 1784

1493 1493 1608 1723 1781 1907 1964

78 99 128 159 268 327



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THE REFRACTED WAVE

DISCUSSION

At long distances from a point source, the refracted wave arrives prior to the water wave, because the velocity of the second layer is larger than the water velocity. As discussed by Ewing et al. (1957), this wave is close to monochromatic for a two-layer model. For a multilayered earth model, refracted waves propagating with various velocities might occur, and this assumption of a monochromatic refraction wave will be violated. In such a case, a full waveform inversion technique is probably needed to resolve and obtain quantitative information about the subsurface. However, for the two-layer model, we assume that the refracted wave has a phase velocity close to α2 , we see from equation 1 that the right side goes to infinity, which means that

The purpose of this paper is to revisit and to explore to what extent the simple two-layer normal mode theory can be used for practical purposes and initial analysis of (ultra) far offset seismic data. First of all, it is clear that we observe events in the time-frequency domain that corresponds fairly well with the normal mode theory. By analyzing the refracted wave using the simple two-layer model with constant water velocity and constant seabed velocity, we find seabed velocities varying between 1700 to 1780 m∕s, for the Ekofisk data set. Using the traveltime differences between the four modes identified on a frequency-traveltime plot of the water wave, results in a seabed velocity approximately equal to 1690 m∕s. The discrepancy between these two estimates is smaller than the expected uncertainty. A more accurate estimation requires a more comprehensive analysis of the data. For the Ekofisk data set, we lack control of the precise distance between the source and the receiver, and this leaves some extra uncertainty associated with this data set. From the Valhall LOFS data, we observe five modes for offsets between 6 and 10 km. We find that a linear moveout filter enhances and eases the identification of the normal modes (Figure 15). The weakest point in our analysis of this data is the assumption of a twolayer model. We know that there is a gradual increase in P-wave velocity with depth, and for the Valhall case the velocity increase below the seabed is very weak. As suggested by the estimated velocities from a shallow seismic survey at Valhall in 2009, a layered model consisting of 5 to 6 layers covering the first 300 m below seabed should be closer to reality. However, like most inversion procedures, we will then face the traditional problem of nonuniqueness; the more layers we introduce the higher will the nonuniqueness of the inversion be. Another weakness of our analysis is that we assume isotropic rocks. It is expected that anisotropy will influence our analysis, because the normal modes will probably be dominated by near to horizontal wave propagation. The shear-wave velocity for the first layer below seabed at the Valhall field is known to be relatively small, less than 200–300 m∕s, however, this will also influence the dispersion curves, as shown in Figure 3. In addition to this, the relatively wide maximum for the normal modes (up to 100 ms measured in two-way traveltime, see Figure 17) introduces

π kn H ≈ ð2n − 1Þ rffiffiffiffi : α2 2 α22 − 1

(12)

1

The typical frequencies for the refraction modes are approximately given as (letting f ¼ kα2 ∕ð2πÞ

fn ¼

ð2n − 1Þα1 α2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 4H α22 − α21

(13)

which is the same as the cutoff frequency given in equation 11 (or equations 4–80 in Ewing et al., 1957). This means that we assume that the frequency minimums are close to the cutoff frequencies. In our case, for the Ekofisk example, using α2 ¼ 1725 m∕s we get f 1 ¼ 9.8 Hz, f 2 ¼ 29.2 Hz, and f3 ¼ 48.8 Hz, which correspond nicely with the frequency peaks shown in Figure 22. If we estimate the four peaks directly from Figure 19, we get f 1 ¼ 9.0 Hz, f 2 ¼ 28.0 Hz and f 3 ¼ 49.4 Hz, and f 4 ¼ 70.5 Hz, corresponding to a monotonic decreasing estimate for α2 : 1778, 1752, 1716, and 1705 m∕s. This monotonic decrease may be related to a velocity increase with depth below the sea bottom, and that lower frequencies “see” deeper into the subsurface. This also means that our two-layer model is not sufficient to analyze our data in more detail. In other words, the refracted wave is actually composed of several refracted waves, and the softer the seabed is the more pronounced will deeper refractions be.

Figure 21. Estimated velocity versus depth based on normal modes (black solid line) compared to velocity profile (interval velocities) from shallow seismic survey.

Figure 22. Frequency spectrum of the mid trace from Figure 14, analyzing only the refracted wavetrain. Time window used for frequency analysis is between 2.0 and 2.7 s.


Normal modes in seismic data inaccuracies related to precise velocity estimates based on this method. The influence of a layered water layer has not been addressed, and we expect that such velocity variations will alter and maybe improve the results presented here. Typical velocity variation versus depth in this area of the North Sea is of the order of 20–25 m∕s, or 1.5%. A significant uncertainty related to our estimates of the velocity for layer two for the Valhall case is coupled to the picking of the traveltime shifts. Based on eye inspection, a reasonable uncertainty for this picking procedure is around 40 ms. From equation 8, we find that this uncertainty corresponds to an uncertainty of 40 m∕s in the estimated layer velocity. A more sophisticated method for traveltime picking, for instance, to pick the time corresponding to the maximum peak for all dispersion curves (see Figure 16) will probably reduce this uncertainty. Especially when we compare dispersion curves for slightly different locations and keep the offset constant, we think this uncertainty might be reduced significantly. In such a case, a crosscorrelation technique between dispersion curves from different locations might offer a precise tool for estimating shallow velocity variations. Unfortunately, our method is not sufficiently accurate to embark on a more detailed study to quantify and determine parameters from a multilayer model. The filtering technique shown in Figures 6 and 15 demonstrates the problem: It is hard to determine the time and frequency of the local minimums for these curves, and therefore we judge that a more detailed analysis is not fruitful on this data set. Maybe other methods that reveal the normal modes clearer and crisper than what is shown in Figures 6 and 17 can be used as a basis for such an analysis, but this is speculation. Despite this, we think that this paper demonstrates that there are possibilities for using the normal mode theory to understand the seismic signal measured at very long distances from the seismic source. To what extent and how useful this can be for exploration geophysics remains to be seen. As the amount of seabed data increases, the availability of long offset (more than 6 km) data will increase, because the shooting pattern for such surveys often covers an area bigger than 6 × 6 km.

CONCLUSIONS By using new approximations based on the theory developed for normal modes, we find that it is possible to relate (with some inaccuracies) observed traveltime differences corresponding to various normal modes directly to the velocity contrast at the seabed. These time shifts are also sensitive to the seabed density contrast. The influence of the seabed shear-wave velocity is low, given that the S-wave velocity is low. This method is only valid for the acoustic case, and for a two-layer model: water overlaying an infinite solid half-space. A more realistic analysis will require an inversion based normal mode method including several layers in the water layer and for the subsurface below the seabed. From a seismic data set where the source-receiver distance is 13 km, we identify four normal modes. It is not straightforward to fit the observed group velocity versus frequency behavior for these four modes using a constant value for the seabed P-wave velocity contrast. Frequency analysis of the refracted wave shows four distinct maxima, corresponding to four values for the seabed P-wave velocity, gradually decreasing from approximately 1780 to 1700 m∕s. We interpret this as a corresponding velocity increase

W37

with depth, and assume that the lower frequencies see deeper into the formation below the seabed. Using LOFS data from the Valhall field we identify five normal modes after application of a linear moveout filter. A trial and error based fitting method results in a seabed P-wave velocity of approximately 1700 m∕s, which is in reasonable agreement with velocities obtained from a shallow seismic data set in the same area. We derive an approximate relation between traveltime shifts between the minimums for various normal modes for the group velocity, and the seabed velocity contrast. This equation shows that the P-wave velocity of the layer below seabed is approximately 1690 m∕s. This value is slightly lower than that estimated from the refracted wave only.

ACKNOWLEDGMENTS PGS is acknowledged for permission to use data from the seismic noise project acquired by SERES in 1989. The data were acquired by M/V Rigmaster in February 1989, and analyzed by Martin Landrø and Svein Vaage in 1989. We want to thank the crew on M/V Rigmaster and the following companies for sponsoring the SERES noise project: Norsk Hydro, Statoil, Elf Aquitaine, Fina Exploration, Nopec and CGG. Martin Landrø acknowledges the financial support from the Norwegian Research Council to the ROSE consortium at NTNU. Four reviewers are acknowledged for numerous suggestions and critical comments that improved the paper significantly. The Valhall Licence Partners, BP Norge AS and Hess Norge AS is acknowledged for permission to use their field data. Olav Barkved is acknowledged for many discussions and for providing data from the Valhall field. Finally, we want to thank Børge Arntsen and Egil Tjåland for several discussions and comments to the manuscript.

APPENDIX A EXPLICIT EXPRESSION FOR THE GROUP VELOCITY The group velocity (U ¼ dω∕dk) can be determined either by using finite difference expressions for cðfÞ estimated from equation 1, or by deriving an analytical expression for the group velocity. An alternative derivation can be found in Frisk (1994, p. 152–154). The group velocity is given as (equations 4–94 in Ewing et al., 1957)

U ¼cþk

dc . dk

(A-1)

The left side of the period equation (equation A-1) can be written as a function of the two independent variables k and cand the right side is a function of only one independent variable cthat is

fðc; kÞ ¼ gðcÞ;

(A-2)

where

sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 fðc; kÞ ¼ tan kH −1 α21 and


(A-3)


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ffi qffiffiffiffiffiffiffiffiffiffiffi c2 ρ2 α21 − 1 gðcÞ ¼ − qffiffiffiffiffiffiffiffiffiffiffi2ffi . ρ1 1 − c 2

(A-4)

ffi qffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffi c2 − 1 2 2 α1 c ρ ffi; tan kH − 1 ¼ − 2 qffiffiffiffiffiffiffiffiffiffiffi ρ1 1 − c22 α21

α2

Taking partial derivatives on both sides of equation A-2 yields

(A-5)

This means that ∂f ∂k

dc ¼ ∂g ∂f . dk ∂c − ∂c

(A-6)

qffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffiffi2ffi c2 − 1 and κ 2 ¼ 1 − αc2 , α2 1

2

∂f Hκ1 ¼ ; ∂k cos2 kHκ1

(A-7)

∂f kHc ¼ ; ∂c α21 κ1 cos2 kHκ 1

(A-8)

∂g ρc ¼− 1 ∂c ρ2

κ2 α21 κ1

þ ακ2 1κ

2 2

κ22

(B-1)

α2

∂f ∂f ∂g dc þ dk ¼ dc. ∂c ∂k ∂c

Replacing the square roots by κ 1 ¼ we find

equations for the phase and group velocities of normal modes as a function of frequency. From the period equation

.

(A-9)

Inserting equations A-7, A-8, and A-9 into equation A-6 and then into equation A-1 yields the following expression for the group velocity

α2 κ 2 1 1 . U ¼c− κ 21 α21 1 2 2 c 1 þ ρ1ρHk þ kHκ cos 1 κ2 κ 3 α2

(A-10)

2 2

Another version of this equation (using slightly different notation) can be found in Frisk (1994, equation 5.165). From this equation, we observe that when the phase velocity c approaches the velocity of the second layer α2, then κ 2 → 0, and the last term in equation A-10 vanishes, and the group velocity is equal to the velocity of the second layer. The second layer also will vanish when c → α1 , and then the group velocity will be equal to the velocity of the water layer, as expected. However, for phase velocities between α1 and α2 , the last term in equation A-10 will be nonzero and give a group velocity varying between the upper α2 and lower bound α1 .

APPENDIX B APPROXIMATE FREQUENCY FOR MINIMUM GROUP VELOCITY In contrast to the group velocity estimation presented in Appendix A, the following approximation for the minimum group velocity has not been presented previously, as far as we are aware. For practical purposes, it is often convenient to derive approximate

we see that in the limit c → α2 will the right side of equation B-1 be significantly larger than 1. For x ≫ 1, we have

tan

−1

π x ¼ − tan 2

−1

1 π 1 ≈ − . x 2 x

(B-2)

Applying the inverse tangential to the period equation and using the approximation in B-2, we obtain

kHκ 1 ≈

π ρ1 κ 2 þ ; 2 ρ2 κ 1

(B-3)

qffiffiffiffiffiffiffiffiffiffiffi ffi qffiffiffiffiffiffiffiffiffiffiffi2ffi 2 where κ 1 ¼ αc2 − 1 and κ 2 ¼ 1 − αc2 . Including higher order 1 2 modes (n) in this equation and using k ¼ 2πf∕c; we find

c 1 ρ1 κ 2 fn ≈ þ ð2n − 1Þ. H 4κ1 2πρ2 κ 21

(B-4)

Of particular interest is to find an approximation for the minima in the group velocity. From Figure 2, we observe that these minima occur fairly close to the region where the phase velocity is significantly larger than α1 . We will exploit this asymmetrical behavior of the group velocity curves to derive approximations for the frequencies corresponding to the minima in the group velocity. Furthermore, we observe that this assumption is more valid for higher mode numbers (the group velocity minimum is closer to the lower cutoff frequency). To find an approximation for the minimal group velocity for each mode, we will approximate equation A-10 for the two end states, c → α1 and c → α2

α21 κ21 U1 ≈ c − ; ρ2 2 c 1 þ ρ1 Hkκ kHκ cos 1 2

U2 ≈ c −

ρ1Hkα22 κ32 cρ2 cos 2 kHκ1

.

(B-5)

(B-6)

Now, we will assume that the minima for the group velocity occur when the upper and lower approximations are equal (U1 ¼ U 2 ). This assumption is based on the following observation: If the two approximations are valid beyond the group velocity minimum (from each side), then they have to cross exactly at the minimum. If the two approximations are not valid all the way to the minimum (which is the case in most circumstances), this assumption will be less and less valid, and the error will increase as the deviation between the approximations in equations B-5 and B-6 and the exact group velocity curve increases. This assumption leads to



kH ≈

ρ2 α21 κ21 − α22 κ22 . ρ1 α22 κ32

(B-7)

Notice that the mode number n is included in this equation, which means that we get one solution for each mode. From the definition of x, we see that the phase velocity is given as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 þ x2 c ¼ α1 t α2 1 þ x2 α12

Replacing the wavenumber k by frequency we find that

f min ≈

cρ2 α21 κ 21 − α22 κ 22 ; 2πHρ1 α22 κ 32

(B-8)

which relates the frequency for minima in the group velocity to the phase velocity c uniquely. Combining equations B-4 and B-8 now yields

c 1 ρ κ cρ2 α21 κ21 − α22 κ22 þ 1 22 ð2n − 1Þ ¼ . (B-9) H 4κ 1 2πρ2 κ1 2πHρ1 α22 κ32 Introducing x ¼ κ 1 ∕κ 2 into B-9 and rearranging, we find

x4 − x2

α22 πρ α2 ρ2 α2 − ð2n − 1Þ 1 22 x − ð2n − 1Þ 12 22 ¼ 0. 2 α1 2ρ2 α1 ρ2 α1 (B-10)

Equation B-10 is a depressed quartic equation, and a general solution to this type of equation was derived by Lodovico Ferrari (see Carpenter, 1966). We will, however, simplify equation B-10 one step further, and assume that the last term is negligible compared to the other terms. Assuming that x > 1, and that the density contrast in the seabed is large, we find that the ratio between the last two terms in equation B-10 is

πρ2 x ∼ 6: 2ρ1

(B-11)

This number is obviously not significantly larger than one, however we will use this assumption to simplify equation B-10. In cases where the ratio given in B-11 is closer to one, one has to solve the full quartic equation given in B-10. After neglecting the last term in equation B-10 we find the following cubic equation for x

x3 − x

α22 πρ α2 − ð2n − 1Þ 1 22 ¼ 0. 2 α1 2ρ1 α1

(B-12)

This equation has three solutions, one real and two imaginary. Selecting the real solution and after some Taylor expansion we find that

1 1 3 πρ α2 3 α2 2ρ2 α21 xn ≈ ð2n − 1Þ 1 22 þ 22 ; (B-13) 2 2ρ2 α1 3α1 ð2n − 1Þπρ1 α2 which is an approximate solution for x at the minimum for the group velocity. If we introduce the parameter

γ ¼ ð2n − 1Þ

πρ1 α22 ; 2ρ2 α21

(B-14)

equation B-13 simplifies to 1

xn ¼ γ 3 þ

α22 −1 γ 3. 3α21

(B-15)

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(B-16)

2

and that

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α22 − α21 : κ2 ¼ α22 þ x2 α21

(B-17)

Inserting this into equation B-8, we find an explicit approximate expression for the frequency where the group velocity has a minimum

f min n

pffiffiffiffiffiffiffiffiffiffiffiffiffi ρ2 α1 1 þ x2 α21 2 rffiffiffiffiffiffiffiffiffiffiffiffi 2 x − 1 ; ≈ α2 α2 ρ1 2πH 1 − α12

(B-18)

1

where x is given by equation B-15. Inserting for x in equation B-18 gives the following approximation for the frequencies corresponding to the minimum group velocity for each mode n

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 α2 α4 1 þ γ 3 þ 23 α22 þ 19 α24 γ −3 1 1 rffiffiffiffiffiffiffiffiffiffiffiffi ≈ α2 ρ1 2πH 1 − α12 2 2 2 α1 2 1 α2 −2 3 − 3 þ γ γ × ; 3 9α21 α22 ρ2 α1

f min n

(B-19)

where γ is dependent on the mode number n as shown in equation 2 B-14. For n larger than one, the γ −3 -terms may be neglected. If we test equation on the examples shown in Figure 9, we estimate the four first frequencies to be 11.7, 31.7, 52.8, and 73.2 Hz, respectively. For comparison, the exact frequency minima are 15.2, 33.9, 53.5, and 73.2 Hz, respectively. As expected, the accuracy of equation B-19 increases with increasing mode number, because γ increases with n, and hence the approximations made in the derivation of the equation is less accurate when n is small.

APPENDIX C APPROXIMATION FOR THE ENVELOPE OF THE GROUP VELOCITY MINIMA Both from synthetic modeling and field data, the minima for the group velocities are a prominent feature, as shown in Figures 2 and 15. It would be useful to derive an approximation for the envelope of these minima. The frequency for each minimum is given by equation B-19, and the group velocity is given by equation A-10. If we assume that the cosine-squared term in equation A-10 is small (this is a valid assumption if the mode number is high, and less valid for smaller mode numbers), we obtain a simple equation for the envelope curve defined by the minimum group velocity for each mode



W40

U ≈c−

α21 κ 21 α21 ¼ . c c

(C-1)

Here, c is given by equations B-16. As the mode number approaches infinity, we see that c ¼ α2 , leading to the asymptotic minimum group velocity

U∞ ¼

α21 ; α2

(C-2)

which is a surprisingly simple result. We notice that this asymptotic value is independent of water depth and density ratio, only the velocity contrast at the seabed. Figure 4 shows a comparison between the exact envelope curve and equation C-1 for water depths of 75 and 500 m, respectively. We notice that all curves in Figure 4 approach the asymptotic value given by equation C-2 in the high frequency limit. In the numerical example shown in Figure 8, we use α1 ¼ 1485 m∕s, α2 ¼ 1725 m∕s leading to U ∞ ∕α1 ¼ 0.86. The density contrast is equal to 1.8. As expected, the accuracy of equation C-1 is worst for low frequencies and shallow water depths. The relative error for the first mode at 75 m depth is approximately 6%. One way to improve the accuracy of the approximate envelope curve given by equation C-2 is to solve the full quartic equation B-10 and keep the cosine-square term in equation A-10. Inserting for c from B-16 into equation C-1 yields

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x2 α21 ∕α22 . U ≈ α1 1 þ x2

(C-3)

Rearranging this equation and approximating x with only the first term from equation B-15, we find that

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u α22 u −1 α2 u U n ≈ u1 þ 1 2 . α2 t πρ α2 3 1 þ ð2n − 1Þ 2ρ1 α22 α21

(C-4)

2 1

Equation C-4 is similar to the energy levels in quantum mechanics, and we see that as the mode number approaches infinity, the group velocity approaches the asymptotic value given by equation C-2. If we use equation C-4 to compute, for instance, the difference in squared minimum group velocity between mode n and mode 1, we find (let y ¼ α2 ∕α1 ¼ 1 þ ε, and assume that ε ≪ 1)

U 21 − U 2n ≈

2 α21 2 πρ1 2 −3 2 ð y − 1Þ 1 þ y ð1 − ð2n − 1Þ−3 Þ. 2ρ2 y2 (C-5)

The group velocity difference is now related to the traveltime (corresponding to the minimum for each mode) in the following way

ΔU ΔT ¼− . U T

(C-6)

Combining equations C-5 and C-6, we find the following approximation for ε

ε≈

−1 U 2 ΔT πρ1 1 . 1 þ 1 − 2 2ρ2 α21 T ð2n − 1Þ3

(C-7)

A reasonable approximation for U is U ≈ α21 ∕α2 ⇒ that U∕α1 ≈ 1 þ ε, which means that

ε≈

−1 ΔT πρ 1 : 1þ 1 1− 2 T 2ρ2 ð2n − 1Þ3

(C-8)

REFERENCES Amundsen, L., and M. Landrø, 2011, Seismic sources, Part IX, seismic surveys and fish: GeoExpro, 8, no. 4, 64–68. Barkved, O., K. Buer, K. B. Halleland, R. Kjeldstadli, T. Kleppan, and T. Kristiansen, 2003, 4D seismic response of primary and production and waste injection at the Valhall field: 65th Annual International Conference and Exhibition, EAGE, Extended Abstracts, A22. Carpenter, W., 1966, On the solution of the real quartic: Mathematics magazine, 39, 28–30, doi: 10.2307/2688990. Dziewonski, A. S., S. Bloch, and M. A. Landisman, 1969, A technique for the analysis of transient seismic singals: Bulletin of the Seismological Society of America, 59, 427–444. Ewing, W. M., W. S. Jardetsky, and F. Press, 1957, Elastic waves in layered media: McGraw-Hill. Frisk, G. V., 1994, Ocean and seabed acoustics, a theory of wave propagation: Prentice Hall. Greene, C. R., and W. J. Richardson, 1988, Characteristics of marine seismic survey sounds in the beaufort sea: Journal of the Acoustical Society of America, 83, 2246–2254, doi: 10.1121/1.396354. Hale, D., 2009, A method for estimating apparent displacement vectors from time-lapse seismic images: Geophysics, 74, 99–107, doi: 10.1190/1 .3184015. Hatchell, P., and K. Mehta, 2010, Ocean bottom seismic (OBS) timing drift correction using passive seismic data: 80th Annual International Meeting, SEG, Expanded Abstracts, 29, 2054–2058. Hatchell, P. J., P. B. Wills, and M. Landrø, 2007, Analysis of guided waves recorded on permanent ocean bottom cables: 69th Annual International Conference and Exhibition, EAGE, Extended Abstracts, P063. Jensen, F. B., W. A. Kuperman, M. B. Porter, and H. Smith, 2000, Computational ocean acoustics: Springer. Lamb, H., 1932, (first published in 1879), Hydrodynamics: Cambridge University Press. Landrø, M., 2007, Attenuation of seismic water-column noise, tested on seismic data from the grane field: Geophysics, 72, no. 4, V87–V95, doi: 10.1190/1.2740020. Landrø, M., and L. Amundsen, 2011, Using water layer normal modes to detect shallow gas and CO2 leakage: EAGE workshop on permanent reservoir monitoring. Landrø, M., and S. Vaage, 1989, Seismic interference noise recorded: by M/ V Rigmaster, February 1989, Seres Internal Report. McMechan, G. A., and M. J. Yedlin, 1981, Analysis of dispersive waves by wave field transromation: Geophysics, 46, 869–874, doi: 10.1190/1 .1441225. Park, C. B., R. D. Miller, and J. Xia, 1998, Imaging dispersion curves of surface waves on multichannel record: 68th Annual International Meeting, SEG, Expanded Abstracts, 1377–1380. Pekeris, C. L., 1948, Theory of propagation of explosive sound in shallow water: Geological Society of America Memoir, 27, 1–117. Press, F., and M. Ewing, 1950, Propagation of explosive sound in a liquid layer overlying a semi-infinite elastic solid: Geophysics, 15, 426–446, doi: 10.1190/1.1437607. Yilmaz, O., 2001, Seismic data analysis, investigations in geophysics, No. 10, SEG.


Normal modes in seismic data â Revisited

Normal modes in seismic data â Revisited

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normal modes ArXiv version

Quasi-Normal Modes in Topologically Massive Gravity

Split fermion quasi-normal modes

Normal Modes UX Maturity Model

Elliptical normal modes and - Michigan Technological University

Magnetic Normal Modes of Bi-Component Permalloy

stress-induced seismic anisotropy revisited - Semantic Scholar

Holographic Preheating: Quasi-Normal Modes and Holographic ...

Wormhole Perturbation and Its Quasi Normal Modes

Computing Linear and Nonlinear Normal Modes

Quasi-Normal Modes and Gravitational Wave Astronomy

Normal Modes and Essential Dynamics - Semantic Scholar

Data Semantics Revisited - CiteSeerX

Nonlinear Normal Modes in a System with ... - Cornell University

Torsional Network Model: Normal Modes in ... - Semantic Scholar

Global Nonlinear Normal Modes in the Fullerene Molecule C60 arXiv

Polytropic equilibrium and normal modes in cold atomic traps

Dynamics and instantaneous normal modes in a liquid with density ...

Rossby Normal Modes in Basins with Barriers - Woods Hole ...

Perturbations and quasi-normal modes of black holes in Einstein ...

Molecular vibrations and normal modes in L-prolyl ... - Semantic Scholar

Normal modes of relativistic systems in post-Newtonian approximation

Cytoskeleton influence on normal and tangent fluctuation modes in the ...

Normal modes in seismic data â Revisited