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## Images in: Chapter 9

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### Seismic data

Images - Chapter 9 Examples of primaries, receiver ghosts (receiver-side reverberations), free-surface multiples (source-side reverberations), and internal multiples in OBS data. These events can be decomposed into (a) upgoing and downgoing events (U/P), or (b) into P-wave and S-wave arrivals (P/S).

Seismic data recorded on geophones deployed on land, on the sea floor, or in boreholes contain upgoing and downgoing P- and S-wave arrivals. Some events are primaries, whereas other events are multiples. Most current imaging tools in seismic data processing require that multiples or at least downgoing components of the recorded data must be attenuated. The imaging tools also require that P-wave and S-wave arrivals be separated and imaged separately. To fulfill these requirements, petroleum seismologists have developed various techniques for splitting recorded seismic data into upgoing and downgoing P- and S-wave arrivals, into total upgoing and downgoing wavefields, and into primaries and multiples.

Wave-equation-based wavefield decomposition plays an important role in the preprocessing of seismic data. Wavefield decomposition by means of the wave equation can be roughly divided into two categories. In the first are methods that separate the elastic wavefield into upgoing and downgoing P- and S-waves (for short, denoted by P/S decomposition), or separate the wavefield into total upgoing and downgoing waves (for short, denoted by U/D decomposition). In the second are methods that attempt to eliminate free-surface related multiples (and possibly internal multiples). Whereas P/S and U/D decomposition typically act on one shot gather at a time, full surface-related multiple removal requires a considerably larger data volume. The two categories of preprocessing can be run independently of each other.

### Amplitude spectra

Images - Chapter 9 Amplitude spectra of (a) hydrophone (pressure), and (b) geophone (vertical component of the particle velocity) for a water depth of 25 m.

(a) shows the amplitude spectrum of the ghost on hydrophone data for a water depth of z=25 m. There is no information about the seismic signal at the notch positions. To solve this notch problem, Barr and Sanders (1989) introduced paired pressure and particle-velocity detectors (dual sensors) in the water-bottom cable, leading to improved resolution by proper combination of the hydrophone and geophone signals in processing. This improved resolution occurs because the vertically oriented geophones have notches at frequencies different from those of hydrophones so that the sum of hydrophone and scaled geophone measurements has no receiver ghost.

(b) shows the amplitude spectrum of the ghost on geophone data for the water depth of z=25 m. Thus, since the geophone recording has no zero amplitude in its spectrum where the hydrophone recording has notches with zero amplitude in its spectrum, the geophone provides the missing information about the seismic signal.

### U/D decomposition

Images - Chapter 9 (a) Model of a water layer of thickness 150 m above a half-space. All events below the sea floor are downgoing and thus absent in the upgoing component following from the U/D decomposition just below the sea floor. The events are labeled by Dnnn,'' where D'' means that the event is downgoing below the sea floor and nnn'' indicates the traveltime in ms. (b) Modeled data of pressure, the vertical component of the particle velocity scaled by the P-wave impedance of the sea floor, and the upgoing component of the vertical traction below the sea floor.

### U/D decomposition

Images - Chapter 9 (a) Two-layer model above a half-space. The events are labeled by Dnnn'' and Unnn,'' where D'' and U'' mean that the events are downgoing or upgoing, respectively, below the sea floor, and nnn'' indicates the traveltime in ms. (b) Modeled data of pressure, the vertical component of the particle velocity scaled by the P-wave impedance of the sea floor, and the upgoing component of the vertical traction below the sea floor. The last trace shows the effect of free-surface multiple elimination.

The second model here includes layer of thickness 125 m below the water layer. The P-wave velocity in this layer is 2000 m/s. The reflection coefficient of the sea floor is again RPP=0.45, and the reflection coefficient of the interface below is 0.16. For this model the zero-offset pressure and the vertical component of the particle velocity scaled by the sea floor P-wave impedance are displayed in Figure 9-4b. Observe that this simple model gives a quite complicated seismic response. To analyze the seismograms, we have in (a) sketched the events up to 625 ms. D and U denote downgoing and upgoing events, respectively, just below the sea floor. Computing the upgoing component S3(U) eliminates all downgoing events just below the sea floor. The upgoing primary, U225, with a traveltime of 225 ms is clearly present. The water-layer multiples D300, D500, D700, etc., are clearly eliminated. The internal multiple U350 arriving at 350 ms is hardly visible because it has reflected twice at the lower interface with a small reflection coefficient. Even if the model is simple, it has several noteworthy characteristics. Receiver ghosts and reverberations are downgoing events just below the sea floor and therefore attenuated. However, the free surface produces multiples that are upgoing events below the sea floor. Such free-surface-related events like U425, U550, and U625 are sometimes called source-side multiples. They remain as part of the upgoing wavefield S3(U). In Chapter 10 we will derive algorithms that eliminate all free-surface-related multiples. Free-surface multiples are events that have at least one reflection at the free surface. The last seismogram in (b) shows the response of the model when all free surface related multiples are absent. Compared to the upgoing field S3(U) just below the sea floor the free-surface demultipled seismogram is almost free of multiples. Free-surface multiple attenuation is therefore a better demultiple tool than U/D decomposition below the sea floor.

A second notable characteristic of the model is that the upgoing primary, U225, in (b) is weaker on the pressure recording than on the particle-velocity recording scaled by P-wave impedance.
Recall that this particular scaling of the particle velocity makes the downgoing events just below the sea floor equal in amplitude between the pressure and particle-velocity seismograms. Stated differently, relative to this primary amplitude, the downgoing multiples are stronger on the hydrophone than on the geophone. To see why this is so, we make the following observations. The pressure is recorded from a hydrophone placed just above the sea floor. Since the vertical component of the particle-velocity field is continuous at the sea floor, we may, for the present analysis, without loss of generality, assume that the vertically oriented geophone is also located just above the sea floor. Consider a downgoing multiple of unit amplitude hitting the sensors. Since the sensors are sitting infinitesimally above the interface when they measure the downgoing event, they will at the same time measure an upward-reflected event with amplitude strength equal to the reflection coefficient RPP of the sea floor. Since the hydrophone sensor is isotropic (insensitive to the wave's direction), it measures upgoing and downgoing events without distinction---that is, the amplitude sum, 1+RPP. On the contrary, the geophone is sensitive to orientation and measures upgoing and downgoing events with an opposite sign. In this particular case the geophone measures the amplitude 1-RPP. With RPP>0, it follows that the downgoing multiple is stronger on the pressure seismogram than on the vertical component of the particle velocity seismogram.

### U/D decomposition (Cont'd)

Images - Chapter 9

### Snapshots of wave propagation

Images - Chapter 9 (a) Snapshots of wave propagation of normal stress (negative of pressure in fluid) and the vertical component of the particle velocity in a model that consists of a homogeneous solid half-space overlain by a water layer. Observe that the downgoing wave of the normal stress field has a polarity opposite to that of the vertical component of the particle velocity, whereas the upgoing wave has the same polarity for the two wavefields. The top half-space is water (P-wave velocity is 1.5 km/s, density is 1.0 g/cc), and the bottom half-space is a solid (P-wave velocity is 2.0 km/s, S-wave velocity is 1.0 km/s, and density is 2.0 g/cc). Indicates upgoing waves.

### Snapshots of wave propagation (Cont'd)

Images - Chapter 9 (b) U/D decomposition of the snapshots in (a) using the small-angle approximation of slowness scalars. Notice the scalars used here are based only on the elastic parameters of the first layer, which is filled with water. D indicates downgoing waves, {\it U} indicates upgoing waves, RD indicates residues of downgoing waves, and {\it RU} indicates residues of upgoing waves.

### Cartesian versus the spherical coordinates

Images - Chapter 9 Cartesian (k1, k2. k3) versus the spherical (ω/VP, θ, φ) coordinates in the wavenumber space.

### Pressure P recorded just above the sea floor

Images - Chapter 9 Pressure P recorded just above the sea floor (a), and vertical-traction components computed just below the sea floor: (b) total upgoing waves S3(U); (c) upgoing P-waves S3(UP); (d) upgoing S-waves S3(US); (e) total P-waves S3(P); (f) total S-waves S3(S).

### Vertical component of the particle velocity

Images - Chapter 9 Vertical component of the particle velocity, Ν3, recorded just below the sea floor (a), and its components computed just below the sea floor: (b) total upgoing waves Ν3(U), (c) upgoing P-waves Ν3(UP), (d) upgoing S-waves Ν3(US), (e) total P-waves Ν3(P), and (f) total S-waves Ν3(S).

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