1. Introduction

Amplitude vs offset (AVO) analysis of seismic data is a powerful indicator of changes in physical properties with depth (see Reference Sheriff and GeldartSheriff and Geldart, 1995). The porosity, saturation (see Reference Nolan and EchelmeyerNolan and Echelmeyer, 1999) and the type of pore fluid are the primary controlling agents in the strength of the AVO effect. I report on a modification of the AVO technique that simplifies it and makes it more robust under some conditions. I calculate layer properties by measuring the incidence angle at which the reflection phase undergoes a 180° shift.

The acoustic impedance of a water-saturated till is close to that of ice. A working hypothesis of active ice-stream flow is that subglacial till is mobilized, resulting in a meters-thick dilatant till layer that is pervasively sheared to produce fast flow (Reference Alley, Blankenship, Bentley and RooneyAlley and others, 1987a, Reference Alley, Blankenship, Rooney and Bentleyb; Reference Tulaczyk, Scherer and ClarkTulaczyk and others, 2001b). Such a layer would have significantly lower seismic velocities than a similar, non-dilatant, non-deforming layer (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others, 1986). Distinguishing between these subglacial conditions is key to a fuller understanding of the ice-stream systems.

Measuring these basal conditions is possible, but difficult, with traditional seismic techniques that determine the compressional- and shear-wave seismic velocities (α and fi, respectively) directly in the target layer by detecting a reflection from beneath the layer. Because of the thinness of the layer (on the order of a half-wavelength in the case of the deforming-till hypothesis), determining the seismic velocities of the layer directly by measuring the move-out of the layer-base reflection is difficult. Direct measurement of the layer velocities has been shown to be successful in one case (Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others, 1987), though generating the source shear waves is difficult. Another technique has been to measure the amplitude ratio between the primary and first multiple reflection and invert for the acoustic impedance of the layer (Reference SmithSmith, 1997). Both techniques require excellent signal-to-noise ratio and high frequencies, which are not always possible. Exploiting the phase reversal vs offset (when there is such a reversal) is a more robust technique.

This layer is thicker than about 5 m (a quarter-wavelength of our dominant frequency of 80 Hz). Reflections from layers thinner than a quarter-wavelength would tend to average the reflectivity behavior of the upper and lower boundaries (see Reference Nolan and EchelmeyerNolan and Echelmeyer (1999) for a more detailed discussion).

Reflection seismic data were collected along a profile that crossed the southern margin of one of the tributaries that feed Ice Stream C (Fig. 1). This tributary (hereafter C1) has sharp lateral boundaries (as identified by velocity data (Reference Anandakrishnan, Blankenship, Alley and StoffaAnandakrishnan and others, 1998; Reference JoughinJoughin and others, 1999) and satellite imagery (Reference Hodge and DoppelhammerHodge and Doppelhammer, 1996)) but a relatively diffuse up-glacier boundary. The satellite imagery shows that distinct flowlines form downstream of a region of rough surface topography that is correlated to the edge of the sedimentary basin (Reference BellBell and others, 1998). The velocity data (Reference JoughinJoughin and others, 1999) show that the relatively higher flow speeds that are associated with ice streaming extend farther into the interior of the ice sheet. Nevertheless, there is a change in the flow behavior of the ice as it crosses the up-glacier edge of the sedimentary basin: a flattening of the surface slope and reduction in driving stress (Reference Bindschadler, Bamber, Anandakrishnan, Alley and BindschadlerBindschadler and others, 2001), the formation of distinct surface flow stripes (Reference Hodge and DoppelhammerHodge and Doppelhammer, 1996), a sharpening of the lateral boundary, and a narrowing of the main flow zone, that suggest a distinct change that is associated with and possibly due to the presence of subglacial sediments. For this reason, I refer to this zone as the “onset” of this tributary of Ice Stream C (Reference Bindschadler, Bamber, Anandakrishnan, Alley and BindschadlerBindschadler and others, 2001).

Fig. 1. Location map of part of the seismic line, including flow speeds. the boundaries of the ice stream are picked from satellite imagery.

The seismic program consisted of collection of multifold reflection data using a towed snow streamer and gimballed geophones (Reference Sen, Stoffa, Dalziel, Blankenship, Smith and AnandakrishnanSen and others, 1998). The travel-time data have been analyzed and inverted for subglacial sedimentary structure (Reference Anandakrishnan, Blankenship, Alley and StoffaAnandakrishnan and others, 1998) and for internal-layer character (Reference BurkettBurkett, 2000). Here I examine the AVO effect of the reflection data in which the compressional-wave reflectivity rp(i) varies with i, the angle of incidence (defined as the angle of the seismic ray path relative to the reflection interface normal).

2. Reflectivity Theory

The amplitude and phase of seismic reflections off interfaces between materials with distinct acoustic parameters (e.g. acoustic impedance, compressional-wave (P-wave) velocity σ, and shear-wave (S-wave) velocity f ) can be strongly variable as a function of the ray paths angle of incidence. The normal-incidence P-wave reflectivity for a ray incident fro mt he i = 1 to the i = 2layeris

where zi = Piσ_i is the acoustic impedance in the ith layer. For this experiment, ice density is P ₁ = 920 ± 2 0 kg m_{– 3} and the P-wave velocity is α ₁ = 3831 ±15 ms–₁ (Reference Anandakrishnan, Blankenship, Alley and StoffaAnandakrishnan and others, 1998). The normal-incidence reflection is positive or “normal” for rp0 > 0, and negative or reversed for rp0 ≤ 0. The normal-incidence phase is reversed only if z2<z ₁. The range of densities and P-wave velocities in the subglacial sediments, for which z2 ≤ z ₁, is small. For a realistic range of sediment densities (1900–2200 kg m–₃), the velocities for which the normal-incidence reflection would be negative are σ2 ≤ 1860 ms–1 to σ2 ≤ 1600 ms–1, respectively. Though the density of these subglacial sediments has not been measured, farther downstream the density is p = 1950 kg m–₃, calculated from the reported density of solids of 2640 kg m–₃ (Reference Tulaczyk, Kamb and EngelhardtTulaczyk and others, 2001a) and a porosity of 40 %. Thus the lower limit of 1900 kg m–₃ and upper limit of 2200kgm_{– 3} , corresponding to a porosity of «25%, are reasonable.

The incidence-angle dependence of the reflection coefficient (calculated from the so-called Zoeppritz equations) is tabulated in the “scattering matrix” (Reference Aki and RichardsAki and Richards, 1980), of which the r_p(i) reflectivity is a complicated function of P- and S-wave velocities and densities in both layers. The AVO variation is particularly strong for reflection horizons that separate regions with large differences in Poisson’s ratio σ (Reference ShueyShuey, 1985). For a downward increase in Poisson’s ratio, as would be the case for ice over unconsolidated sediments, r_p increases with increasing angle. Further, if the normal-incidence reflectivity is negative, then the reflection phase will change as the angle of incidence crosses the angle of intromission (the angle at which the reflection is extinguished). It is this “crossing angle” i_c that I measure and use to determine subglacial sediment properties.

Although I lack absolute reflection amplitudes, the AVO effect can be used to constrain the conditions at the bed of the ice. I n particular, the reflection from the base of the ice stream has a distinct change in phase from negative at near-offsets to positive at intermediate offsets if these conditions are met:

1. 1. r_p0 ≤ 0 because the acoustic impedance of the subglacial sediments is less than that of the ice; and

2. 2. r_p(i) > 0 for some i, with a crossing angle i _c ; this is most likely due to an increase in Poisson’s ratio from the ice (σ =0.33) to the sediments (σ ∼ 0.45).

The crossing angle increases with increasing shear-wave velocity. For an assumed sediment P-wave velocity of α ₂ = 1.7 km s–1, the crossing angle ranges from i_c = 11° for f 2 = 0.1 km s–₁ to i_c = 25° for f 2 = 0.7 kms–₁ (Fig. 2) (this is the likely range of S-wave velocities in sediments (Reference HamiltonHamilton, 1976)).

Fig. 2. Plot of reflection amplitude vs angle of incidence for ice over dilatant till: σ1= 3.83km s-1, β1 1.95kms–¹,ρ₁ 920 kg m–3,p2 = 2.1kg m–3.for the upper set of curves, σ2 1.7 km s–¹ and for the lower set, σ₂ = 1.6 km s^–1. the shear-wave velocity f2 ranges from 0.1 km s^–1 (upper curve) to 0.7 km s–¹ (lower curve) in both sets of curves. the zero-crossing and i_c increases over that range.

3. Fieldwork and Processing

The analysis was conducted on shot gathers because of the low common-midpoint multiplicity (CMP fold); the trace spacing in all the figures is equal to the receiver spacing of 25 m. We justify a posteriori the use of shot gathers rather than CMP gathers in the analysis. The region of interest for this study is of angles of incidence i ≤ 15°. For those angles, the actual spread of reflection points at the base of the ice is 475 m at the maximum ice thickness of h = 1800 m, and shorter yet in regions of thinner ice. I assume that bed conditions do not change with position at that scale.

3.2. AVO

These results can be further strengthened by examining the shear-wave velocity in the sediments. As the seismic streamer used in the experiment did not allow direct collection of shear-wave data, we extract that information from the crossing-angle data. Figure 3 is a typical ice-stream example demonstrating the crossing-angle behavior, with traces at near offsets having negative phase, and traces at far offsets having positive phase; I have annotated the figure to show the change in phase across the crossing angle. Contrast this to Figure 4, a shot record from the ice sheet where the reflections are all the same phase. The shot records from the ice sheet all show r_p0 > 0 and no crossing angle. The poor-quality shots at the margin of the ice stream are because of the noise from buried crevasses. I n general, the crossing angle could not be localized to better than 3–4 traces (approximately ±1.5°). The crossing-angle data (Fig. 5) show considerable scatter, but none are 4 13°. The crossing angles near the shear margin are slightly higher than those in the interior, but that difference is only barely significant, and I do not insist upon it. All the shots within the ice stream consistently show a phase reversal.

Fig. 3. Plot of a shot record from the ice stream at location km 59.1. each trace is separated by 25 m; the trace label is approximate angle of incidence at the base of the ice sheet. the reflection energy at low angles of incidence has a negative “first break”−a trough. the first break at large angles is positive − a filled-in peak. the angle separating the two is the crossing angle. the uncertainty in choosing the crossing angle is 3–4 traces or ±1.5°.

Fig. 4. Plot of a shot record from the ice sheet at location km 38.1, over the region of ice-sheet sediments. each trace is separated by 25 m: the trace label is approximate angle of incidence at the base of the ice sheet.

Fig. 5. plot of the crossing angle _ic along the seismic line. the approximate region of the ice stream (beyond km 56) is indicated. the shear margin is characterized by buried crevasses and a rapid change in ice-flow velocity.

Several approximations were made in the analysis, but introduce inconsequential errors. We calculate the ice thickness using a four-layer firn velocity model assuming constant thickness along our line, and a constant P-wave velocity of σ1 = 3831ms–1 below the firn. For simplicity I have replaced the firn with a layer of ice with equivalent vertical travel time, which introduces a small horizontal position error in the source and receiver. The maximum bed slope along our line is approximately 2° in the ice stream and 4° in the shear margin, and I can correct for much of that in our angle-of-incidence calculations. The cross-line slopes are not well known, but the airborne radar surveys suggest that they are less steep than the in-line slopes (personal communication from D. D. Blankenship and D. L. Morse, 2001) . I estimate the error in our angle of incidence arising from these approximations as Δ i = ± 2°.

In the ice stream, the data show that the reflection changes phase at a mean crossing angle i_c = 9.1° ±2.3; added to the error in angle-of-incidence determination, I estimate an error Δ i _c = ± 3°. The observed i_c = 9° suggests that the P-wave velocity σ ₂ ∼ 1.7 kms–₁ (see Fig. 2). A higher σ₂ would make the normal-incidence phase positive (E_{p 0} > 0), which is not observed; alternatively, a lower σ₂ would push the zero-crossing angle i_c to the right for all f2, above what is observed (Fig. 2). For an assumed sediment P-wave velocity of σ₂ = 1.7 kms–₁, this crossing angle suggests an extremely low S-wave velocity of f2 = 0.1 km s–₁ ± 0.2. The value of f2 ∼ 0.1 kms–₁ is fully consistent with that measured by Reference Blankenship, Bentley, Rooney and AlleyBlankenship and others (1986, Reference Blankenship, Bentley, Rooney and Alley1987) at UpB on Ice Stream B. Our observed P- and S-wave velocities are difficult to explain in any way except through a high-water-pressure, poorly consolidated, soft layer beneath Ice Stream C.

4. Conclusion

The normal-incidence phase and AVO behavior of the seismic reflection off the base of the ice in the onset region of Ice Stream C is consistent with a subglacial layer with low P-wave velocity σ₂ = 1.7 km s–₁ and extremely low S-wave velocity β2=0.1kms–₁ .

We suggest that this layer is a dilatant and presumably deforming till layer that underlies the ice stream. This dilatancy is well established within a few tens of km of the onset of the ice stream, as evidenced by the consistency of the AVO effect at all the shots (300 m shot spacing) in the ice stream. This behavior is also consistent with the boundaries of the sedimentary basin determined by refraction shooting (Reference Anandakrishnan, Blankenship, Alley and StoffaAnandakrishnan and others, 1998). In addition, these reflections are of distinctly different character from reflections off non-ice-stream subglacial sediments and off crystalline basement.We suggest that subglacial till deformation is an important mechanism for the initiation of rapid ice-stream flow.