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Appendix A Seismic Reflection Migration
The pointwise migration method (Clarke and Echelmeyer, in press) assumes that a single layer of homogeneous material (i.e. temperatre icw with a constant P-wave velocity) overlies bedrock, that the geophones used in the analysis are placed in a horizontal linear array and that the reflection points are in a vertical plane passing through the array. No corrections for surface statics were made in this analysis, because the vertical offset for all geophones and shots was less than 10m and often less than 1 m.
Two pairs of data are needed to calculate the depth, slope and horizontal position of the reflector: the first arrival times at any two geophones (typically the first and the last of the spread) and the distance to those geophones from the shot along the surface. If discrepancies arise between several reflections, priority is based upon the clarity and strength of the reflected waves.
The most important consideration when choosing arrival times is obtaining the correct move-out, which is the slope of a theoretical line that best fits the first arrival times of the reflected-wave packet on the geophone traces. The reduction of seismic data was conducted independently by two of the authors and checked for comatency. If discrepancies in arrival-time picking were greater than 1 ms and could not be resolved, the wave was discarded. The estimated error in choosing arrival times for clear reflections is therefore ± 1.0 ms. Errors in determining arrival times from the seismogram will result in incorrect move-outs that are either parallel but shifted (with end points shifted by the same amount) or skewed (with end points shifted by different amounts) to the correct line. Parallel move-outs that are 1 ms in error result in less than 2% errors in both depth and horizontal location of the reflector. Skew move-outs that are off by 1 ms often produce much larger errors in both depth (5–150%) and horizontal location (100–4000%). Because the error due to skewed move-out can be so large, they often can be easily detected and corrected. We believe that the reduced seismic data presented in this paper are free from these larger errors, giving a random-error estimate of approximately ±30 m (2%) at the deepest point (1477m). If several clear reflections overlap when plotted, this random error can be reduced to the scatter of the reflections’ position.
Another potential source of error in seismic results is choice of compressional wave velocity. The literature is noticeably lacking in consensus for a P-wave speed in temperate ice, perhaps because P-wave speed is very sensitive to temperature variations between 0° and −1°C. Röthlisberger (1972) concluded that results from various authors of 3600–3620 m s−1 and 3670–3700 m s−1 have equally valid data supporting them. We used the mean of these suggested values, 3650m s−1, consistently throughout the calculations. Our measurements on Taku Glacier show that the average direct P-wave speed, determined from the move-outs of ten seismo-grams, was 3700 ms−1 with a standard deviation of 46 m s−1. The relationship between direct-wave speed at the surface and bulk ice speed varies with surface location and probably changes with season; for example, the direct-wave speed could be slower than the bulk-ice speed due to crevassing or faster due to a lower average temperature. Increasing the speed used in the calculations to 3700 m s −1 increases the thickness estimate by approximately 21m (1.4%) at the deepest point. Therefore, we conservatively estimate the maximum systematic error at ±2%, or approximately 30m at the deepest point of Taku Glacier. Propagation of maximum random and systematic errors results in error bounds of ±43m.
Appendix B APPLICATION OF ICE-DEFORMATION THEORY
Measured surface velocity on Taku Glacier can be used to ice thickness following techniques developed by Nye (1965) for ice flow down a uniform parabolic channel and, when combined with a measured thickness, estimate basal motion. The center-line surface velocity due to ice deformation is given by
where n is the flow-law exponent (n= 3), A is the flow-law parameter (5.3 × 10−24Pa−3s−1 from Paterson (1981, p. 39)), ρ is the average density of ice, g is the acceleration of gravity, α is the surface slope averaged over 6–10 times the glacier thickness, H is glacier thickness at the center line and f is a shape factor which accounts for the drag of the valley walls, Equation (1) assumes a basal shear stress calculated by τ
b = fρgH sin α. Inverting for thickness from Equation (1) is usually an iterative process because f depends on H.
Fig. 9. Shape factor, f, as a function of W, the ratio of half-width to thickness. A polynomial (solid line) was fitted through elliptical shape-factor values (circles) given by Nye (1965). This function is used to eliminate the need to iterate solutions for thickness in Equation(1).
We presenr a simple method to solve for thickness directly. A polynomial, f(W), was fitted through the values of the shape factor given by Nye (1965) as a function of W (Fig. 9), where W is the ratio of glacier half-width, w½, to center-line thickness, H.
Normalizing Equation (1) by ud
evaluated for W = 1 (i.e. Η = w½
and f = 0.445) = yields a normalized velocity, U, which is independent of actual glacier thickness, width and surface dope
U from Equation (3) is shown in Figure 10 as a function of W. Grouping the Constant terms of Equation (1) together, the value of u
d(W = 1) for a given surface slope and width can be calculated as follows
where w is the glacier width in meters and ud
is given in m a−1.
To solve for thickness H, evaluate the lefthand side of Equation (3) by setting u
d(W) equal to measured surface speed and then finding u
d(W = 1) from Equation (4). W can then be estimated from Figure 10, or by solving Equation (3), and then the center-line depth from H = . Estimates based upon these calculations are very sensitive to surface slope and can easily be in error by ±100 m.
For example, consider a glacier that has a width of 4.5 km, a surface slope of 1.0° averaged over 10 km, and a measured surface velocity of 220 m year−1. From Equation (4), u
d(W = 1) = 691 m a–1. The dimensionless velocity ratio, U, is 220/691 = 0.318. From Figure 10, W= 1.67 and thus H = 2250m/1.67 = 1350m.
If the estimated thickness is greater than the measured thickness, this difference may indicate basal motion, due to the ice sliding over bedrock Of a layer of deforming subglacial till. The ratio of basal motion to measured surface motion can be estimated by first determining the contribution of deformation to surface speed, ud
, using the measured thickness for H in Equation (1). The remainder of measured surface speed and this deformation speed is an estimate of basal motion.
Fig. 10. Normalized velocity U vs W. U is the ratio of the measured surface velocity to the deformational velocity with W=1 from Equation (4). Center-line thickness can be determined froom W using this figure.