## Oblique incidence

Obliquely incident waves provide important information on P-wave velocities in hydrate deposits through two observable properties: traveltimes and amplitudes. Wide-angle traveltimes provide the long-wavelength velocity structure of the sediment column through normal-moveout and stacking velocity analysis (e.g.,

(Ecker et al., 2000; Hyndman and Spence, 1992)), forward modeling, or inversion ((Katzman et al., 1994; Tinivella and Accaino, 2000)). The accuracy of velocities determined by traveltime analysis increases with the maximum incidence angle. In deep water, for example, long-offset ocean-bottom seismic data (e.g., (Katzman et al., 1994; Korenaga et al., 1997)) generally provides better velocity determinations than data acquired on a streamer of limited length.

Partitioning of the amplitude of a P-wave that impinges obliquely on a boundary is described by a complex set of equations, the Zoeppritz equations (Zoeppritz, 1919). Assuming an incoming P-wave, two important phenomena occur: (1) The slope of R as function of incidence angle depends on the shear modulus and hence Vs (e.g. Ostrander, 1984). (2) At oblique angles, some amplitude is turned into a S-wave (P-to-S converted wave, PS-wave), due to the horizontal traction imposed by the incident P-wave. Figure 1 shows amplitude partitioning for different contrasts of seismic properties at layer boundaries. Fig. 1: Amplitude as a function of incidence angle for an incident P-wave, reflected P-wave and reflected P-to-S converted wave. Absolute values normalized to the amplitude of the incoming P-wave. Elastic parameters are realistic for unconsolidated marine sediments close to the seafloor: Upper layer: Vp: 1600 m/s; Vs=200 m/s (left), 400 m/s (right), density=1700 kg/m3. Lower layer: : Vp: 1600 m/s; Vs=400 m/s (left), 200 m/s (right), density=1700 kg/m3. Note the differences in the P-reflection response depending on whether Vs increases or decreases. Reflected P-amplitudes increase until all P-energy is reflected at the critical angle, 62.7° for this velocity contrast. The P-to-S arrival has a negative polarity for a Vs decrease (where the polarity of the arrival for a Vs increase is defined as positive).

Fig. 1: Amplitude as a function of incidence angle for an incident P-wave, reflected P-wave and reflected P-to-S converted wave. Absolute values normalized to the amplitude of the incoming P-wave. Elastic parameters are realistic for unconsolidated marine sediments close to the seafloor: Upper layer: Vp: 1600 m/s; Vs=200 m/s (left), 400 m/s (right), density=1700 kg/m3. Lower layer: : Vp: 1600 m/s; Vs=400 m/s (left), 200 m/s (right), density=1700 kg/m3. Note the differences in the P-reflection response depending on whether Vs increases or decreases. Reflected P-amplitudes increase until all P-energy is reflected at the critical angle, 62.7° for this velocity contrast. The P-to-S arrival has a negative polarity for a Vs decrease (where the polarity of the arrival for a Vs increase is defined as positive).

Seismic rays that hit a layer boundary at an oblique angle follow Snell's law of optics. The incidence angles of incoming, reflected and transmitted P-and S-waves are inversely related to the layer velocities. Hence

Figure 2 shows schematic raypaths across a layer boundary and the annotation used in this chapter.

PS-r

vp=ie

Vs=2C

PS-r

vp=ie

Vs=2C PS-t

PS-t

Fig. 2: Ray paths according to Snell's law, for an incoming P-wave (P-i). Elastic model as in Fig. 1, with a Vs increase. P-r: reflected P-wave (usually labeled PP); PS-r: reflected P-to-S converted wave (usually referred to as "PS-wave"); P-t: transmitted P-wave; PS-t: transmitted P-to-S converted wave.

Amplitude analysis provides information that cannot be gleaned from traveltime analysis alone. For example, amplitude-versus-offset modeling is widely used to estimate Vs contrasts from P-wave data. Waveform inversion can provide accurate models of short-wavelength P-velocity variations, as discussed in a later section (Korenaga et al., 1997; Minshull et al., 1994; Pecher et al., 1996a; Singh et al., 1993).