DETERMINATION OF VOLUMES
Glacier volumes were found by determining the area
between each set of bedrock and surface contours, a pair of which define
a volume element (fig. 7). Areas were measured with a planimeter or a
digitizer to within 3 percent accuracy and were completed for the length
of the glacier. Volumes CI is the contour interval.
The contour interval used was 200 feet on Mount Rainier and 100 feet on Mounts Hood and Shasta and the Three Sisters. Glacier volumes were calculated by drainage area and altitude. Values for each drainage area, by mountain, are listed in tables 2 through 5. The percentage of the total ice and snow volume measured by radar varied for each mountain as follows: 62 on Mount Rainier, 83 on Mount Hood, 53 on the Three Sisters, and 19 percent on Mount Shasta. The error in calculating volume varied with each glacier, depending on the number of measurement points per glacier. Error in snow patch and ice boundary corrections was estimated at 5 percent, and error from the original topographic maps was less than 5 percent. A conservative estimate for error in the volume of measured glaciers is ±20 percent.
Several different approaches have been made for estimating the volume or average thickness of unmeasured glaciers. For instance, Post and others (1971, p. 4) related average thickness to area size classes, using several measured glaciers for calibration. Kotliakov (1980) also incorporated the type of glacier in a similar scheme. Brückl (1973), Müller (1976, p. 12), Shih and others (1981, p. 194), Zhuravlev (1980), and Macheret and Zhuravlev (1982, p. 310) utilized an empirical relation of the form
where is
the mean glacier thickness, m are coefficients derived from regression analysis, A is
the glacier area, and m ~0.5.Paterson (1970, p. 43) proposed that the shear stress on the bed be treated as a constant. For a simple, infinitely wide glacier with laminar flow, the shear stress on the bed τ is
where ρ is the ice density,
where α is the surface slope and
Because glacier characteristics are related to latitude and climatic conditions, it was necessary to develop a volume estimation method for use in the Cascade Range. The volumes of unmeasured glaciers were estimated from statistical analysis of characteristics for all measured glaciers except the Whitney, which was examined during later study (Kennard, 1983). The variables required for these volume estimations can be determined from topographic maps and aerial photographs. Paterson's assumption of a constant basal shear stress, which is equivalent to the assumption that glacier flow can be treated by plasticity theory (Nye, 1951, p. 554), was tested by comparing shear stress (eq. 5) with other glacier characteristics. It was found that the larger glaciers had shear stress values in the expected range of 1<τ<2 bars. However, the smaller glaciers had lower values of shear stress, ranging down almost to zero. It would appear that some glaciers reach a critical basal shear stress, and for these glaciers, the flow is sufficiently fast to adjust the longitudinal profile to a dynamic equilibrium so that the product of thickness and surface slope is related to that stress. Other glaciers are too small to reach that critical shear stress, and their profiles are determined less by dynamic considerations than by local variations in snow drifting and melting. In an analysis of the measured glaciers (table 1), it was found that most glaciers having a length greater than 8,500 ft obtained a critical shear stress, and most glaciers having a length less than this did not. It was also found that an estimated basal shear stress τ* in pounds per square foot for the larger glaciers could be calculated by the empirical relation TABLE 1.—
where Σ V = Ah, the estimated volume V* in cubic
feet can be calculated by using the estimated shear stress τ*
according to
where area and slope were measured at 1,000-foot intervals and then summed. Small glaciers, those that do not obtain the critical shear stress, were generally those less than 8,500 ft in length (Kennard, 1983). The empirical relationship is
where When the estimation methods were developed they were tested by application to glaciers with measured volumes. The standard deviations of errors of estimated and measured volumes were as follows: 5 percent for large glaciers (with volumes found by using equations 6 and 7), 16 percent for small glaciers (using equation 8), and 13 percent for the groups together. Together with a ±20 percent uncertainty in finding measured glacier volumes, the error in using the estimation method is assumed to be about ±25 percent.
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