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Front Cover: U.S. Geological Survey photograph No. 81R3-083 by Austin Post, September 26, 1981.
Bed Topography Inferred from 

Airborne Radio-Echo Sounding of 

Columbia Glacier, Alaska

ByC S. BROWN, L A. RASMUSSEN, and M. F. MEIER

STUDIES OF COLUMBIA GLACIER, ALASKA

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1258-G

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1986
DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Library of Congress Cataloging-in-Publication Data

Brown, C S.

Bed topography inferred from airborne radio-echo sounding of Columbia Glacier, Alaska.

(Studies of Columbia Glacier, Alaska) (Geological Survey professional paper ; 1258-G)

Bibliography: p. G24

Supt. of Docs, no.: I 19.16:1258-G

1. Columbia Glacier (Alaska) 2. Echo sounding. 3. Radar in glaciology. I. Rasmussen, L.A. II. Meier, Mark

Frederick, 1925- . III. Title. IV. Series. V. Series: Geological Survey professional paper ; 1258. 

GB2427.C64B76 1986 551.3'12'097983 85-600300 ____ _____________

For sale by the Books and Open-File Reports Section, U.S. Geological Survey, 

Federal Center, Box 25425, Denver, CO 80225
CONTENTS

Page

Abstract ------------------------------------------------- Gl

Introduction --------------------------------------------- 1

Radio-echo sounding development ------------------------- 4

The airborne radio-echo sounding system 

Development of the system ---------

Airplane positioning ---------------

Data analysis -------------------------

Signal extraction ------------------

Internal consistency ---------------

Error in echo arrival times ---------

Page

Interpretation of airborne radio-echo sounding data --------- G9

Refraction and reflection represented in two

dimensions -------------------------------------- 10

Refraction and reflection represented in three

dimensions -------------------------------------- 16

Columbia Glacier data ------------------------------------ 17

Discussion ----------------------------------------------- 23

References cited ------------------------------------------ 24

Appendix A: Echo arrival times --------------------------- 25

ILLUSTRATIONS

	Page

FIGURE 1. Index map of Columbia Glacier, Alaska ---------------------------------------------------------------------------- G2

2. Vertical aerial photograph of the terminus of Columbia Glacier, December 14, 1984 ---------------------------------- 3

3. Oblique aerial photograph showing the roughness of the Columbia Glacier surface near the terminus ------------------ 4

4. Map of flight paths and sounding positions from which the Columbia Glacier bed topography was inferred ------------- 6

5. Photographs of two reconstructed airborne radio-echo profiles across the lower part of Columbia Glacier --------------- 7

6. Graph of frequency distribution of altitude of the airplane above the glacier surface ---------------------------------- 9

7. Diagram showing a ray refracted according to Snell's law ----------------------------------------------------------- 10

8. Map of surface topography of lower part of Columbia Glacier on August 26, 1978 ------------------------------------- 11

9 17. Diagrams showing:

9. Refraction according to Snell's law at the air-ice boundary for n = 1.78 in the ice ------------------------------ 12

10. Reflection loci for two soundings from above the glacier surface and for one from the surface itself ------------- 12

11. Reflection loci from several positions along a flight path ----------------------------------------------------- 13

12. Effect of airplane altitude on the radius of curvature of the reflection locus ----------------------------------- 13

13. Effect of the locus slope on the radius of curvature of the reflection locus ------------------------------------- 14

14. Effects of errors in echo arrival time and airplane altitude --------------------------------------------------- 14

15. Hypothetical bed profile and profiles inferred from echoes received at different altitudes ----------------------- 15

16. Arrival-time hyperbolas caused by hypothetical persistently reflecting points on the glacier bed ---------------- 16

17. Snell's law in three dimensions ---------------------------------------------------------------------------- 16

18. Map of inferred bed topography ----------------------------------------------------------------------------------- 18

19. Diagram of vertical cross section showing glacier surface and reflection lobes for 31 positions along flight path

N2500 ------------------------------------------------------------------------------------------------------- 19

20-22. Maps showing:

20. Sources of reflection lobes forming envelope of lobes --------------------------------------------------------- 20

21. Adjustment of envelope-method bed according to directly measured bathymetry ------------------------------- 21

22. Nadir-method results and surface soundings ---------------------------------------------------------------- 22

TABLE

TABLE 1. Summary of errors in inferring bed profile of figure 15, using each of the two methods, on data from each of three 

indicated altitudes -------------------------------------------------------------------------------------

Page

G16

in
IV

Symbol 

C

H

km 

m 

mm 

MHz

n

q

R

(IS

t

SYMBOLS AND ABBREVIATIONS

SYMBOLS AND ABBREVIATIONS

Name Unit

Speed of light in air --------------------- 300 m/jis

Airplane altitude above glacier surface - - - m

Kilometer.

Meter.

Millimeter.

Megahertz.

Index of refraction of ice ---------------- dimensionless

Length of ice-segment of ray ------------- m

Radius of curvature of reflection locus - - - - m

Length of air-segment of ray ------------ m

Second.

Microsecond.

Time ----------------------------------- $

Arrival time for echo from glacier bed - - - - (is 

Arrival time for echo from glacier

surface ------------------------------- (is

Horizontal distance from airplane position

to refraction point -------------------- m

Horizontal coordinate, positive to east - - - - m

Horizontal coordinate, positive to north - - m 

Vertical coordinate, positive upward ----- m

Coefficients of plane locally approximating 

glacier surface ----------------------- dimensionless

Inclination of ice-ray from the vertical - - - degrees 

Inclination of air-ray from the vertical - - - degrees
STUDIES OF COLUMBIA GLACIER, ALASKA

BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO 

SOUNDING OF COLUMBIA GLACIER, ALASKA

By C S. BROWN, L.A. RASMUSSEN, and M.F. MEIER

ABSTRACT

The first airborne radio-echo sounding of a temperate glacier was 

performed in 1978 at Columbia Glacier, a large (1,100-squarekilometer), grounded, iceberg-calving glacier 38 kilometers west of 

Valdez, Alaska. The sounding system used a low frequency (about 

1.5 megahertz) to overcome scatterings from water-filled voids in the 

ice, a short pulse, and an untuned receiver. Transverse and longitudinal profiles were flown over the lower 7 kilometers of the glacier. 

The received signal, the horizontal position of the airplane, and its 

altitude above the glacier surface were recorded by an FM tape 

recorder. For the data analysis, pictures of received energy from each 

flight profile were reconstructed from the taped data using an oscilloscope. Use of intersecting profiles allowed an internal consistency 

check to determine whether the correct bed reflection had been chosen. A three-dimensional geometric method of determining the envelope of the reflection lobes was developed for interpreting the data, 

instead of the differential method used by previous investigators. 

This analysis provided bedrock altitude determinations at every 

node of a 200-meter square grid. The probable error in the inferred 

bed altitudes was estimated to be 30 meters; the greatest depth was 

370 meters below sea level.

INTRODUCTION

Columbia Glacier (fig. 1) is a large, grounded, calving glacier near Valdez, Alaska, the terminus of the 

trans-Alaska pipeline. In 1978 it was 66.6 km long and 

1,100 km2 in area. Many large tributaries coalesce to 

form the lower trunk of the glacier, which fills a deep 

fiord and ends in a grounded, calving terminus that 

stands up to 90 m above sea level. From the time of the 

first recorded observation in 1794 (Vancouver, 1798) 

until 1978, the terminus had been relatively stable, 

ending partly on Heather Island (fig. 2) and partly on a 

submerged moraine shoal in water less than 100 m 

deep. Upglacier from the terminus, however, the bed is 

nearly 400 m below sea level.

Nearly all grounded, iceberg-calving glaciers in 

Alaska, such as Columbia Glacier, have experienced 

large-scale, asynchronous advances and retreats; Columbia Glacier, however, was the only glacier that extended in 1978 to its neoglacial moraine. The largescale advances and retreats of calving glaciers are due 

not solely to climatic variations; water depth at the 

terminus is a critical factor. When a glacier retreats 

into a deep basin or fiord, instability results and the 

calving flux increases (Post, 1975; W.O. Field, oral 

commun., 1980; Brown and others, 1982).

In 1974, Columbia Glacier appeared to be entering a 

period of retreat (Post, 1975; Sikonia and Post, 1980). 

Since 1977, the U.S. Geological Survey has documented an accelerated rate of thinning in the terminus 

region and a trend of increasingly larger embayments 

forming, as an even greater portion of the terminus 

retreats from the crest of the shoal each summer and 

fall. Once the glacier has backed off the moraine shoal 

into the deep water upglacier, rapid and irreversible 

retreat is expected to begin. In October of 1984, all of 

the terminus was 1 2 km behind the shoal in water 

averaging 120 m deep; all evidence indicates that the 

large-scale recession has begun.

A direct consequence of rapid retreat is an increasing 

flux of icebergs. Even prior to 1974, small icebergs from 

the glacier drifted toward and occasionally into the 

shipping lanes in Valdez Arm (Kollmeyer and others, 

1977); these shipping lanes are used by tankers transporting oil from the Alaska pipeline terminal at 

Valdez.

To determine when this rapid retreat might begin, 

and by how much the iceberg discharge might be increased, an intensive study was begun by the Geological Survey in 1977 (Meier and others, 1978). A prediction was made (Meier and others, 1980) that the glacier 

would begin an irreversible retreat within a few years. 

Details of each phase of this study are being published 

as a series of scientific papers, of which this is one. The 

other papers discuss the relationship of calving speed to 

water depth and other variables (Brown and others, 

1982); surface topography (Rasmussen and Meier,

Gl
G2

61°30'-

61°00f -

147°15'

STUDIES OF COLUMBIA GLACIER, ALASKA

147°00' 146°45' 146°30'

FIGURE 1. Index map of Columbia Glacier, Alaska. Arrows indicate direction of ice flow. Main ice stream is indicated by longer 

arrows and dots at 2-kilometer intervals along the longitudinal coordinate system. Boxed area is region with inferred bed 

topography from the airborne radio-echo sounding.
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G3

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FIGURE 2. Vertical aerial photograph of the terminus of Columbia Glacier, December 14, 1984. Also indicated is the position of the 

terminus on August 26, 1978, and the position of the moraine shoal.

1985); mass balance observations; field measurements 

of velocity, thickness, and thickness change (Mayo and 

others, 1979); photogrammetric determination of surface altitude, terminus position, and ice velocity (Fountain, 1982; Meier and others, 1985); the development of 

a data set that satisfies the continuity equation; the 

adjustment of velocity data to obey continuity (Rasmussen, 1985); a continuity model of the terminus retreat and the rate of iceberg discharge (Rasmussen and 

Meier, 1982); finite-element (Sikonia, 1982) and finitedifference (Bindschadler and Rasmussen, 1983) models 

of the flow of the lower glacier, which also predict iceberg flux; and several additional topics.

Glacier thickness is a principal variable in the equation of continuity, in the flow law, and in calving relationships; thus, it is crucial in any dynamic glacier 

model, such as those developed for predicting the behavior of Columbia Glacier. Thickness cannot easily be 

measured directly; drilling in the lower reach of 

Columbia Glacier is expensive, relatively slow, and 

prohibitively dangerous (fig. 3). Thickness, however, 

can be determined indirectly from the rather easily 

measured surface topography and an estimated bed topography. Currently, polar ice bodies are sounded by 

radio-echo methods. When it became necessary to know 

the thickness of Columbia Glacier, efforts to sound this
G4 STUDIES OF COLUMBIA GLACIER, ALASKA

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FIGURE 3. Oblique aerial photograph showing the roughness of the Columbia Glacier surface. View is to the west, about 2 kilometers above 

the terminus. Average relief from summits to valleys is about 25 meters; local relief may range up to 40-50 meters. USGS photograph 

by M.F. Meier, August 14, 1984.

large, temperate glacier were begun. This was a 

formidable task, as radio-echo sounding techniques of 

temperate (nonpolar) ice bodies are relatively new and 

are still very much in the developmental stage. This 

paper outlines the development of radio-echo sounding 

methods for temperate glaciers, describes the collection 

and analyses of data in this first airborne radio-echo 

sounding of a temperate glacier, and explains how the 

bed topography, and thus the glacier thickness, were 

inferred from these data.

RADIO-ECHO SOUNDING DEVELOPMENT

Measurements of glacier thickness by radio methods 

were first attempted with some success in 1927 and 

1928 (Evans, 1963c). Since the first reported radio

sounding of polar ice in Antarctica in 1957 (Evans, 

1963a), sounding of polar ice bodies by radio-echo 

methods has continued in Greenland and Antarctica 

(see, for instance, Evans, 1963b, 1967; Swithinbank, 

1968; Robin and others, 1970; Morgan and Budd, 1975) 

and elsewhere. Recently, much emphasis has been 

placed on the interpretation of these returns (see, for 

instance, Harrison, 1970; Oswald, 1975; and Robin, 

1975).

Research in applying radio-echo sounding methods 

to temperate glaciers first had to address the problems 

of sounding ice at the melting temperature, ice that is 

less homogeneous than polar ice. Smith and Evans 

(1972) determined that the difficulty of sounding temperate glaciers was due not to the high temperature, 

impurity content, or ice fabric but to the masking of the
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G5

true bottom return by the diffuse return from the 

englacial scatterers, or inhomogeneities. Shortly thereafter, Watts and England (1976) correctly attributed 

these inhomogeneities to water-filled voids in the ice, 

determined that frequencies less than 10 MHz were 

needed to overcome the scatterings, and suggested that 

a short-pulse radar with an untuned receiver could be 

used at these low frequencies. This system differs from 

radars typically used for polar ice, which employ a modulated carrier signal and a tuned radio receiver. 

Vickers and Bollen (1974) introduced the technology of 

resistively loaded antennas, from which there is practically no reflection from the ends of the arms, to the 

radio-echo sounding of temperate glaciers. With this 

system, they successfully sounded depths of more than 

200 m on South Cascade Glacier, a temperate glacier in 

the North Cascade Mountains of Washington, using 

5 MHz frequency and then applied the system to 

Columbia Glacier (Vickers and Bollen, 1974). Since 

then, scientists at the University of Iceland developed 

a profiling system based on these principles, employing 

frequencies of 2 to 10 MHz, and used it successfully on 

the Vatnajokull and Myrdalsjokull icecaps (Ferrari 

and others, 1976; Bjornsson and others, 1977; 

Sverrison and others, 1978; Bishop and others, 1979; 

Bjornsson, 1982). The system was towed behind a motorized sled and was the first successful attempt at continuous profiling on the surface of a temperate glacier.

THE AIRBORNE RADIO-ECHO 

SOUNDING SYSTEM

DEVELOPMENT OF THE SYSTEM

Sounding work was begun on Columbia Glacier during the summer of 1974 (Vickers and Bollen, 1974), 

when approximately 70 surface radar and gravimetric 

spot measurements were made, all in the lower 30 km 

of the main trunk glacier. For the purposes of numerical modeling, however, data from a much denser network of points were necessary. A continuous profiling 

system could not be operated on the glacier surface 

because of the many locations rendered inaccessible by 

the highly crevassed surface (fig. 3).

On the basis of successful sounding of South Cascade 

and Columbia Glaciers with a surface system, Watts 

and Wright (1981) designed an airborne radio-echo 

sounding system for use at Columbia Glacier, and used 

this system during August and September 1978. This 

was the first airborne radio-echo sounding of a temperate glacier. Owing to the severe time constraint under 

which the system was developed and flown, many problems involving data acquisition and analysis remained 

unsolved.

Data were obtained from 12 east-west and 7 northsouth profiles covering the lower 7 km of the glacier 

(fig. 4), which is the most important region for studying 

the glacier flow. The airborne radar system was carried 

in a Fairchild-Pilatus Porter1, a short-takeoff-andlanding airplane sufficiently large and powerful to 

carry all the equipment and personnel required and 

capable of flying at speeds low enough for radar work, 

around 100 knots (kn; 185 kilometers per hour (km/ 

hr)).

Watts and Wright (1981) modified their surface 

monopulse generator for the airborne work; they were 

able to use transmitting and receiving antennas similar in principle to those used in the surface system but 

modified because they had to be fed in the center, yet 

could be attached to the airplane only at the ends. Because of these modifications and the proximity of the 

metal airplane, the emitted wave train contained about 

six cycles instead of the more desirable single sinusoid 

(Churchill and Wright, 1978).

An oscilloscope on board the airplane sampled the 

received signal and transformed it to an audio frequency that was recorded on a seven-track FM tape 

recorder. The oscilloscope also provided an output, 

recorded on another channel, that gave an absolute 

time correlation between echoes and their delay from 

the initial transmitted pulse. Photographs (fig. 5) were 

then taken of an oscilloscope composite reconstruction 

made by playing back the magnetic tapes. Time delay 

from the original transmitted pulse is plotted vertically, and the horizontal position represents movement 

above the surface; brightness was modulated in proportion to received (echo) voltage.

AIRPLANE POSITIONING

The horizontal position of the airplane was recorded 

in the local x,y coordinate system devised for Columbia 

Glacier which is related to the geodetic UTM (Universal Transverse Mercator) system according to

'x\ i / UTM Easting 490 km 

0.9996 \UTM Northing 6,750 km

(1)

The x,y position was determined by an automated navigation system using two microwave transponders located at known points off the edge of the glacier. It 

measured the times of the responses from the transponders, and a small computer on the airplane then calculated the position. The distances to the transponders, 

and thus the x,y position, were recorded every 0.5 s.

'The use of brand names in this report is for identification purposes only 

and does not imply endorsement by the U.S. Geological Survey.
G6 STUDIES OF COLUMBIA GLACIER, ALASKA

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FIGURE 4. Flight paths and sounding positions from which 

the Columbia Glacier bed topography was inferred. Dots 

represent the sounding positions; their location corresponds 

to the point where the echo arrival times fy were read off the 

reconstructed profiles at approximately 5-millimeter intervals. If the maximum value of tb did not occur at one of these

roughly equally spaced positions, its position was also included. Note the close proximity of the Boom Boom, Barefoot, and Easy ridges; contours are given in meters above sea 

level. The location of the region depicted is indicated by the 

box in figure 1.
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G7

Flightline Surface return

Bed return

B

Surface return

Flightline

Bed return

FIGURE 5. Reconstructed airborne radio-echo profiles across the lower part of Columbia Glacier: (A) profile at y = 14.4 kilometers, illustrating a reasonably good return; (B) profile at y = 12.9 kilometers, illustrating a poor return and the difficulties in determining t^. Each 

profile is shown twice, once with and once without the surface return and the selected bed return identified.
G8 STUDIES OF COLUMBIA GLACIER, ALASKA

Twelve east-west profiles and seven north-south profiles were flown (fig. 4). The east-west profiles are 

termed the "North family," owing to their northward 

progression at 500-m increments from their baseline, 

and the north-south profiles are similarly termed the 

"West family."

The airplane's radar altimeter reading was also 

recorded every 0.5 s, giving H, the altitude in meters 

above the glacier surface. This was the only altitude 

recorded on the tape. The altitude at which the airplane was to fly each profile was predetermined and 

logged prior to flight; this was the barometric altitude 

to the nearest 100 ft (30 m). The horizontal distance 

along the flight path from the beginning of each profile 

was also recorded for use in locating the photographed 

arrival times in real space.

DATA ANALYSIS 

SIGNAL EXTRACTION

The value of tb (x, y, z), the round-trip signal time to 

the bed from a particular airplane position, was measured manually from a photograph of an oscilloscope 

reconstruction of the profile. Figure 5 shows the reconstruction of both a good profile and a poor one; most 

were somewhere in between in quality. Obviously 

there were several difficulties to overcome before tb 

could be measured. First was the problem of correlating 

the several bands (cycles) of the reflection with those of 

the transmitted pulse. Not only were there multiple 

bands, but each usually was not continuous across the 

profile. Sophisticated off-line signal processing could be 

used to compress these multiple bands, making it 

easier to distinguish the bottom echo. There also was 

the problem of the valley wall echoes masking returns 

from the glacier bed, particularly on the west edge of 

the glacier against the Boom Boom and Barefoot ridges 

(fig. 4). These valley wall returns likewise could have 

been removed by signal processing or eliminated at the 

outset by flying much closer to the glacier surface.

Owing to the electronic saturation of the airborne 

receiving system by the transmitted pulse, no echoes 

could be detected during the first several microseconds. 

Thus, variations in ts were recorded but the absolute 

arrival time ts of the echo from the surface was not 

recorded and the t -datum line, commonly referred to as 

the flightline, does not appear on the reconstructed profile pictures (the lines at the very top of the pictures are 

the tail end of the recovery from the saturation of the 

electronics). This flightline is necessary to measuring 

tb , and reconstructing it was difficult. To approximate 

its position on the picture, the surface reflection corresponding to the selected bed return first had to be determined. In many cases there was not a one-to-one

correspondence between the several surface bands and 

the several bands from the bed return. Usually the 

brightest and widest band was chosen; from the internal consistency check (below) it could be determined 

whether the corresponding bed reflection had been chosen correctly. Two points then were plotted, one at each 

end of the profile above the selected surface band at a 

distance corresponding to the average altitude of the 

airplane above the surface. This distance was determined by knowing the number of microseconds per division on the oscilloscope and by knowing the signal 

speed in air. A line drawn between these points was 

used as the £-datum line. The error in using this tdatum line was determined to be well within the 30-m 

error in airplane altitude H.

INTERNAL CONSISTENCY

The final phase of the data reduction was checking 

for internal consistency by comparing the two tb values 

at each intersection of two profiles. Each tb value was 

adjusted to account for possible differences in altitude 

at which the two profiles were flown. This was done by 

reducing each tb by tH, the amount of time for the airwave to travel the distance H:

t'=tb -tH =tb -ZH/c , (2)

in which c=300 m/jxs is the signal speed in air. The 

difference |^\r-^l between the reduced tb values for the 

east-west profile and the north-south profile was calculated for each of the 80 profile intersections. Although 

this formulation embodies the strict assumption that 

the echo comes from the nadir, the difference \t^ ty\ is 

little affected if the echo comes from elsewhere, so long 

as the echoes sensed on the two profiles come from 

approximately the same location. That is, the difference between the two nadir distances to the glacier 

surface is used to approximate the difference between 

the two slant distances actually traversed by the signal. Because the difference in H was generally small 

compared with H (fig. 6), this weaker assumption 

seems reasonable. As the error in measuring tb on the 

photograph was a maximum of 0.45 jxs, this value was 

used as the maximum allowable difference value of 

l^v"~^wl- Sixty-three percent of the differences were actually less than 0.20 jxs. One difference was 0.46 jxs, for 

which there was no obvious explanation.

Matching t^ and t^ at each intersection provided a 

check for consistency between profiles. To check for 

consistency within a single profile, the band chosen as 

representing the bed reflection on the photograph from 

which one tb was obtained had to be the same band used 

to measure the tb values of the neighboring intersection 

points on the profile. At 5 (out of 138) profile intersec-
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G9

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AIRPLANE ALTITUDE, H, 

ABOVE GLACIER SURFACE, IN METERS

FIGURE 6. Frequency distribution of altitude H of the airplane 

above the glacier surface. Shown are the number f(H) of the 678 

airplane positions within each of the indicated 20-meter-wide altitude intervals.

tions, it was necessary to choose a band different from 

the neighboring intersection. There is no definitive explanation for this; numerous factors undetected in the 

oscilloscope plot affect the reflected waveform. Once 

again, signal processing would probably yield more 

consistent results by virtue of putting almost all the 

energy into a single band.

Apparent internal consistency would result if every 

tb in every profile were misinterpreted in the same way. 

The good agreement between the tb values and both the 

directly measured bathymetry and the radar soundings 

from the surface makes this unlikely.

ERROR IN ECHO ARRIVAL TIMES

Although there were difficulties involving both the 

data acquisition and the data analysis, there were only 

two major sources of error. The first was in the altitude 

used to position the t-datum line on the photographs of 

the reconstructed profiles. This altitude was the barometric altitude rounded to the nearest 100 ft (30 m) at 

which it was predetermined the airplane should fly. It 

was referred to as the "average profile altitude," measured above mean sea level. It is assumed that the 

instrumental and reading error in altitude values during flight is of the order of 10 m, to which must be added 

the effect due to atmospheric pressure fluctuations, 

perhaps another 10m. Therefore the aggregate error in

flying the predetermined altitude is assumed to be 

about 30 m. Thus, at any given point on a profile and, 

therefore, in the t -datum line, there was an error of 

±30 m in the altitude value used. This corresponds to 

±0.20 |xs on the photograph.

The second major source of error was in measuring tb 

on the photographs. The bands chosen as representing 

the bed reflections on the profiles averaged 1 mm in 

width. Thus, this was the average amount by which the 

distance from the t -datum line to the chosen reflection 

could vary. By using the predetermined number of microseconds per division on the oscilloscope and the reduction factor between the oscilloscope screen and the 

photograph, the conversion factor on the photograph 

was calculated to be 0.30 (xs/mm (microseconds per millimeter). Thus the 1 mm variation in measuring tb corresponds to 0.30 (xs. The possible inaccuracy in measuring tb on the photograph was less than 0.3 mm (0.09 jxs) 

and is considered to be negligible in relation to the 

other, much larger errors.

Instrumental errors, including those in the 

radio-echo sounder, oscilloscope, and the transponders 

used to determine the x,y position of the airplane, were 

deemed to be insignificant compared with the other 

errors. Thus, if the two sources of error are considered 

to be independent, the error in tb is [(0.20 (xs) 2 

+ (0.30 jxs)2] 1/2 =0.36 (xs. This error results directly in 

error in glacier thickness; however, as will be shown, 

the two are not strictly proportional. If the echo is assumed to come from the nadir, then 0.36 (xs corresponds 

to 30 m.

INTERPRETATION OF 

AIRBORNE RADIO-ECHO SOUNDING DATA

The interpretation of the radio-echo sounding data is 

based on simple, known geometric principles, which 

are briefly reviewed here. Although they are ultimately applied three-dimensionally, they are initially 

described in two dimensions for ease of presentation. 

The simplicity of the mathematical model follows from 

several assumptions concerning the physics of the signal propagation and of the composition of the glacier, 

as well as recognition of the low quality of the original 

data.

Signal strength is assumed to be unreliably known, 

so that echo arrival time is the only variable used in 

inferring the distance traveled by the signal. The signal is assumed to be omnidirectional, so that no direction is ruled out as a candidate from which an echo may 

be received. Because of the uncertainty with which arrival times tb were read from the reconstructed profiles, 

a simple geometric method of determining the envelope 

of the reflection lobes was selected for interpreting the
G10 STUDIES OF COLUMBIA GLACIER, ALASKA

data instead of the differential method used by Harrison (1970), which is based on the rate of change of the 

signal arrival time as the airplane moves along its 

flight path. The signal is assumed to travel in air at the 

rate of c =300 m/fxs (meters per microsecond); within 

the glacier it is V/i times this rate, where rc = 1.78 is 

taken to be the index of refraction of ice.

Although the glacier surface is highly irregular and 

fractured (fig. 3), the mathematical model assumes a 

much smoother, average surface at which refraction 

follows Snell's law (fig. 7):

sin (j>= sin 6 . 

n

(3)

In fact, it assumes a piecewise planar surface between 

given altitudes on the nodes of a 200-m square grid. 

These were obtained by applying the method of optimum interpolation to photogrammetrically determined altitudes of irregularly positioned points appearing on a sequence of vertical aerial photographs

H

Air

Ice

FIGURE 7. A ray refracted according to Snell's law (equation 3).

taken on August 26, 1978, from about 6,800 m above 

the glacier surface (Rasmussen and Meier, 1985). This 

surface topography is represented in figure 8 by a manual contouring of the altitudes on the 200-m square 

grid. Because the soundings were taken in late August 

over the part of Columbia Glacier having surface altitudes less than 300 m above sea level, no correction 

such as that used by Harrison (1970) was applied to 

the passage of the signal through a firn layer, which 

would have an n-value smaller than that for ice. Another assumption that is made is that the arrival times 

that are used are from echoes occurring only at the 

glacier bed, rather than from englacial bodies of liquid 

phase water.

Change of airplane position during the period between signal transmission and reception is neglected. 

The 100-kn airplane speed (Watts and Wright, 1981) 

produces only 1 mm of displacement in 20 (as, a time 

that exceeds all signal arrival times obtained over Columbia Glacier. The 1 s consumed by the sampling oscilloscope in constructing a complete sweep (R.D. 

Watts, written commun., 1985) corresponds to a displacement of about 50 m.

REFRACTION AND REFLECTION 

REPRESENTED IN TWO DIMENSIONS

The dependence of the ice angle cj> on the air angle 6 

that arises from using n = 1.78 in equation 3 is shown in 

figure 9. This relationship gives a maximum value of 

(j>=34.18° when 6=90°, and the slope dfyldft decreases 

continuously from V* at 0=0°, to 0 at 0-90°.

The distance equation for an echo with arrival time 

tb is

ctb =2 (r + nq) (4)

in which r is the length of the air leg and q is the length 

of the ice leg (fig. 7). It is easily shown that the ray that 

obeys Snell's law is the one that minimizes the travel 

time between (x,z) and (Q,H).

A parametric representation of the x,z locus is easily 

obtained from equations 3 and 4 in terms of 6:

x =

z =

 . ^- sin 0 cos0 2

H

cos 0 2

(5)

This formulation is valid only for an idealized glacier 

surface lacking curvature. Although it is not possible 

here to give a closed-form expression z=f(x) for the 

locus, equations 5 may easily be transformed so that
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING Gil

11 -

10

789 

x, IN KILOMETERS

10 11 12

FIGURE 8. Surface topography of lower part of Columbia Glacier on August 26,1978. The location of this region is indicated 

by the box in figure 1. The contour interval on the glacier is 10 meters, with occasional 5-meter dashed contours. Datum 

is sea level.
G12 STUDIES OF COLUMBIA GLACIER, ALASKA

0 30 60 90 

AIR ANGLE, 6, IN DEGREES

FIGURE 9. Refraction according to Snell's law at the air-ice 

boundary for n = 1.78 in the ice: (A ) Specimen refracted rays. The 

ice ray and the air ray are both identified by the incidence angle 

6 of the air ray, as measured from the vertical. (JB) Ice angle <}> as 

a function of the air angle 6. The slope d<b/dQ decreases continuously from l/n at 6=0 to 0 at 6=90° (where <J>=34.18°).

the parameter is r or u or the locus slope dz/dx. Half of 

each of the x,z loci for three different airplane altitudes 

H=Q, 200, and 800 m is shown in figure 10; all three 

loci are scaled to have the same 2 = 393 m depth (corresponding to tb = 10 jxs when #=800 m, which is typical of the Columbia Glacier data) at *=0. The H=0 

locus is constructed on the assumption that the antennas are in dielectric contact with the ice and, therefore, 

that no refraction occurs.

The x,z curve is strictly just a locus, in the sense that 

any point on it could be the source of the echo having 

the arrival time that determines the curve. If several 

loci were available from closely spaced airplane positions, then a glacier bed could be inferred from their 

envelope. No part of the bed can be above the envelope 

because, for one or more of the observation positions, an 

echo would have been received with an earlier arrival 

time than the one from which the locus was constructed. Several such loci, and a bed profile that is 

consistent with them but that is not uniquely deter1000

o

u- 500 

oc

-500

H is airplane altitude above 

glacier surf ace, in meters

Air

Ice

0 500 1000 

HORIZONTAL DISTANCE, x, IN METERS

FIGURE 10. Reflection loci for two soundings from above the glacier 

surface and for one from the surface itself. For each locus (solid 

curve), a few generating rays (dashed line) are shown. For the two 

soundings from above the surface, the rays refract at the surface 

according to Snell's law (eq. 3, fig. 9). A horizontal glacier surface 

is assumed; the complete reflection locus in three dimensions is 

actually the body of revolution obtained by rotating the locus about 

the 2-axis.

mined by them, are shown in figure 11. Any other 

curve that was tangent to each of the several loci, and 

that nowhere rose above their envelope, might also be 

the bed profile.

Glaciological and geological knowledge must influence the bed inference, for radio-echo sounding can 

only impose some constraints on the bed topography, 

not determine it exactly. For example, the envelope 

itself is consistent with all the data; however, on those 

grounds, the locus-intersection cusps would not be interpreted as bed features. A dense array of loci, from a 

continuum of observation points along the flight path, 

would eliminate these cusps, but it would still yield 

only an upper-bound constraint on the actual bed topography.

The locus (eqs. 5) may be differentiated to give its 

slope, a convenient expression for which is

? , , . _ , 2 -2 n\ -i-=tan <}>=sin 0 (n* snr 0) (6)

The maximum value dz/dx =0.679 occurs at the maximum value <J>=34.18°. A second differentiation yields
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G13

BEDROCK

FIGURE 11. Reflection loci from several positions along a flight path. A bed profile that is consistent with these hypothetical loci, but that

is not uniquely determined by them, is also shown.

the radius of curvature, which may be written in the 

form

-3/2

^ \nH l-(Ai 2 -!) m -z\ . (7) 

dx \ ) | \dx/

At x 0, where dzldx =0, the radius is R =nH z. This 

is significant because the radius increases at the same 

rate as the depth increases, but it increases n times 

faster than the airplane altitude increases. For a surface sounding (H=Q), the radius of curvature at x 0 

equals the depth. The ratio of these two, R(Q,z)/ 

R(H,z), as a function of the ratio of the airplane altitude to the depth, is shown in figure 12. The radius of 

curvature may also be considered a function of the 

slope at some particular depth. The ratio R(H,z,Q)/ 

R(H,z,\dzldx\) as a function of the slope is shown in 

figure 13 for selected values of the ratio of the airplane 

altitude to the depth. These two diagrams demonstrate 

that as airplane altitude increases, the radius of curvature of the locus increases very rapidly and so does 

the smallest scale of bed features that can be inferred 

from the data.

Errors in the calculation of the bed altitude can be 

related to errors in the original data. It is easily seen 

from equations 5 that at x=0 and error of 8z=8.43 m

.0001 .001 .01 1 10 100 1000 10.000

H/lzl

FIGURE 12. Effect of airplane altitude on the radius of curvature of 

the reflection locus. For both the surface sounding and the sounding from airplane altitude H, the radius of curvature R is for the 

nadir, at depth z, where the slope of the locus is dzldx, =0 (eq. 7). A 

horizontal glacier surface is assumed.
H is airplane altitude above 

glacier surface, in meters

STUDIES OF COLUMBIA GLACIER, ALASKA

0.4 0.6

idz/dxl

FIGURE 13. Effect of the locus slope on the radius of curvature of 

the reflection locus (R ). Two reflection loci at depth z are both 

from airplane altitude H but from different airplane positions, 

so that one has locus slope dz/dx and the other has zero slope. 

The radius-of-curvature ratio is given as a function of slope for 

selected H-io-\z\ ratios, including the special H=0 case of surface soundings (eq. 7). A horizontal glacier surface is assumed.

H is airplane altitude above 

glacier surface, in meters

500 1000 

HORIZONTAL DISTANCE, x. IN METERS

FIGURE 14. Effects of errors in echo arrival time and airplane altitude. For the three reflection loci shown in figure 10, the underestimate 5z of the glacier thickness is shown as a function of horizontal distance, for errors in arrival time 5£6 = -0.1 microseconds and 

in airplane altitude §H = +15 meters. Only part of the H =0 curve 

is shown; this error increases very rapidly, because the locus from 

a surface sounding becomes so steep as the locus approaches the 

glacier surface (fig. 10).

may be produced by an error of either §tb = 0.1 JJLS or 

of 8# = 15 m, regardless of the depth z. Elsewhere it is 

more complicated, as §tb produces a greater change 

than the comparable §H does; the influence of §H is 

subdued by a cos 6-effect, whereas §tb acts directly 

along the ray path. The sensitivity of the loci of figure 

10 to errors of §tb = -0.1 JJLS and S# = 15 m is illustrated 

in figure 14. Whereas a small §tb for a surface sounding 

produces a large 82 at a large horizontal distance, the 

envelope of a profile of surface soundings (fig. 11) would 

be no more affected by that §tb than a profile of aerial 

soundings.

Inferring the bed topography from radio-echo sounding data is an inverse problem, many characteristics of 

which are readily revealed by considering the forward 

problem. A hypothetical bed topography, having no 

variation in the other horizontal direction and underlying a horizontal glacier surface, is shown in figure 15. 

It also shows the envelope that would be obtained from 

loci constructed from arrival times that would be received from that bed along flight paths at #=200 and 

800 m above the glacier surface and at the surface 

itself. A slight idealization is invoked at each end of the

bed profile (points A and M), where it is assumed that 

a high curvature feature exists having the capability of 

reflecting back along the path from which it came a ray 

that might have come from any direction; otherwise, an 

echo would not be received at sounding positions near 

that same end, for the ray could not be normal to the 

bed at its endpoints. Were this not assumed, no echoes 

would be received toward the ends of the sounding profiles.

As would be expected from equation 7, the greater 

the altitude from which the sounding is done, the less 

detail is obtained in the envelope and the greater the 

error is. In fact, the 800-m profile is controlled by just 

short stretches of each of the five bed prominences (A, 

C, E, G, M); they control only part of the 200-m profile, 

which also receives echoes from bed points between the 

five prominences, and they control even less of the profile of soundings taken from the glacier surface. The 

range of control of each of these five stretches is indicated by the refracted ray making up the beginning 

and the end of each range.

The bed profile in figure 15 was composed so as to 

emphasize several different effects (eq. 7) on the curva-
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G15

oc

UJ

LLI

O

D

800

- 600 

LLI

o

400

CO

E 200 

O

o

DC -200

-400

-600

Flight path, H=800

Envelope method 

H=200

Hypothetical bed

500 1000 1500 2000 

HORIZONTAL DISTANCE, x, IN METERS

2500 3000

FIGURE 15. Hypothetical bed profile and profiles inferred from 

echoes received at different altitudes. Over those segments where 

an inferred profile is not indicated, it is understood to be coincident 

with the bed. The ranges of control of each of the five prominences,

which collectively control the entire record of soundings from 

800 meters, are indicated by the refracted rays making up the 

beginning and end of each range. The discussion in the text is 

keyed to the lettered features.

ture of the reflection locus. Because of the effect of 

airplane altitude, the depression at B is detected in the 

200-m data but not in the 800-m data. Because of the 

effect of depth below the surface, the depression at B is 

detected in the 200-m data but the one at F, which has 

the same curvature, is not; the same is true of the 

depressions at D and I with respect to the data from 

soundings at the surface. The depression at D which 

is at the same depth as the one at B but has greater 

curvature is not detected in the 200-m data, whereas 

the one at B is. The depression at L has the same depth 

and the same curvature as the one at B, but it is inclined from the horizontal; thus, the one at B is detected in the 200-m data but the one at L is not.

A separate effect occurs at H, J, and M, where the bed 

slope exceeds the maximum value of 0.679 that a locus 

from an aerial sounding can achieve. Therefore, these 

steep stretches could not be detected by any aerial data. 

The separation of the surface-sounding envelope from 

the bed at H and J is not caused by the steepness of the 

bed slope, but rather by the curvature of the depression 

at I.

Shown also in figure 15 is the bed profile that would 

be obtained from the 800-m data were the echo assumed to come always from the nadir. If the apropriate 

linear transformation (ct=8QQn -z) is made to the vertical scale, then the 800-m nadir curve becomes the 

tb (x) curve of the original data. The curves arising from 

making the same assumption with the 200-m and surface data are omitted for reasons of graphical clarity, 

but all six different bed inferences are briefly summarized in table 1.

Persistently reflecting points on the glacier bed generate hyperbolic segments (Harrison, 1970) on the signal profile tb (x\ whether soundings are taken from the 

glacier surface or from above it. Such a hyperbolic segment is the nadir-method curve from Jt=2,140 to 

2,900m of figure 15, which is the same as the tb (x) 

profile if the appropriate linear transformation is made 

to the vertical scale. Five such hyperbolas are shown in 

figure 16 for persistently reflecting points at depths of 

200 and 500 m as detected from the glacier surface and 

from altitudes of 200 and 800 m above it. The slope 

dtb /dx is zero at x =0, where the sounding is taken from 

directly above the reflecting point. It may be shown 

that the limiting slope, at x=°°, is dtb /dx 2/c=6.7 JJLS/ 

km for surface soundings, and is dtb ldx =2n/c = 11.9 JJLS/ 

km for soundings taken from above the surface.
G16 STUDIES OF COLUMBIA GLACIER, ALASKA

TABLE 1. Summary of errors in inferring bed profile of figure 15, using each of the two methods, on data from each of three indicated altitudes

[m, meters]

Root-mean-square error - - -

Location, x ---------------

(m) 

(m)

(m)

H=o

13 

44

2,140

Envelope method

#=200 m

33

Qfi

9 ^, 120 i.&\j

#=800 m

67

1 CO

2,150

H=Q

34 

90

2,000

Nadir method

H=200 m

57 

19Q

2,050

#=800 m

90 

186

2,240

REFRACTION AND REFLECTION 

REPRESENTED IN THREE DIMENSIONS

The two-dimensional representation of the refraction-reflection geometry is easily extended to three dimensions. For the case of a horizontal glacier surface, 

the complete locus is the body of revolution formed by 

rotating the x,z locus (fig. 10) about the z-axis. If the 

surface is not horizontal, but is planar, the rotation is 

about the normal to the plane.

Snell's law (eq. 3) is easily expressed in three dimensions. The point Oc^i^i) on the air ray and the point 

(x2 ,y2 ,Z2) on the ice ray are each a unit distance from the 

origin, which is where the refraction occurs (fig. 17).

- H is airplane altitude above

glacier surface m meters

i I i I I i L l i

0 500 1000 1500 

HORIZONTAL DISTANCE, x, IN METERS

FIGURE 16. Arrival-time hyperbolas caused by hypothetical persistently reflecting points on the glacier bed. Each t^x) hyperbola is 

identified by the depth z of the point below the horizontal glacier 

surface and by the altitude H above it from which the point is 

detected. The horizontal distance from the point is x.

The plane ooc + ^y+yz 0 has its coefficients scaled as 

to magnitude and sign so that, respectively, 

a2 +|32+72=l and ax^fiy^yz^cos 9>0. Then,

(8)

in which

k=cos 0 (n2 sin2 0) =(l-n2 sin2 <f>) -n cos <f> 

and

-cos <$>=ax2+fiy2+yz2 .

Normal 

to plane

Air ray

Normal 

to plane

FIGURE 17. Snell's law in three dimensions. The angles 0 and <j> obey 

equation 3. The points (^1,^7,21) and (x2 ,y2,Z2^ each a unit distance 

from the refraction point, are related by equation 8.
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G17

Either the 6-form or the 4>-fbrm of k is used, according 

to whether (x2,y2,z2) is found from (xi,y^,z{) or the other 

way around. The plane determined by the two rays is 

normal to the refracting plane. The signal's travel time 

between (x^y^Zi) and (x2,y2 ,z2) is IGSS when passing 

through the origin than it would be were it to pass 

through any other point in the plane. In the analysis of 

the Columbia Glacier data, a plane was used to approximate the surface topography lying between an airplane position and a location where the altitude of the 

reflection locus is desired.

If the glacier bed is idealized as a differentiate, 

mathematical surface, then a normal exists at every 

point on the bed. A ray that strikes the bed at some 

point by following some path other than along the normal will be reflected, but it will not return along the 

same path. Because the airplane's motion is negligible 

compared with the speed at which the signal travels, 

only a ray that travels along a normal to the bed will 

produce an echo that reaches the airplane. Equivalently, an echo will be received only from points where 

the three-dimensional reflection locus is tangent to the 

bed. For any particular point on the bed, then, an echo 

can be received at only one particular airplane position 

at any particular altitude. Because the ray path must 

be normal to the bed, its direction cosines are uniquely 

determined by the direction cosines of the bed at that 

point. This unique ray path through the ice is refracted 

at the glacier surface, according to Snell's law, into a 

unique ray path through the air.

It is also assumed that the laws of geometric optics 

apply; that is, the wavelength of the signal is much 

smaller than the roughness elements on the bed. 

Small-scale roughness, which could not be analyzed, 

could cause an unknown amount of diffraction or other 

effects that might complicate the analysis.

COLUMBIA GLACIER DATA

The echo arrival times tb for the 678 airplane positions (fig. 4) are tabulated in appendix A. The arrival 

times for the airplane-bed-airplane round trip are in 

microseconds. A frequency distribution of the altitude 

H of the airplane above the glacier surface appears as 

figure 6. Arrival times were read at about every 5 mm 

along the reconstructed profile; if the maximum value 

of tb for the east-west flight paths did not occur at one 

of these equally spaced positions, an extra point at the 

maximum tb location was also included because it gives 

the greatest depth to the bed.

The reflection lobes implied by the set of coordinates 

(x, y, z, tb ) for each of the 678 airplane positions were 

constructed by using a plane to approximate the glacier 

surface locally. At each node of a 200-m square grid,

the altitudes of all lobes reaching that node were compared, and the deepest one was taken to be the envelope 

altitude there. The bed topography was inferred by contouring the unadjusted envelope altitudes (fig. 18) except near the terminus, where directly measured 

bathymetry was used.

A vertical cross section (fig. 19) shows the relationship between the reflection lobes and their envelope for 

one flight path. The section, along y = 15.4 km, was 

chosen because the contours are nearly north-south in 

direction, so that there is little transverse bed slope. It 

demonstrates that the normality condition is satisfied 

in the x ,z plane, because where the refracted rays reach 

the envelope they are normal to it. An indication that 

the normality condition is satisfied in the x ,y plane is 

provided by figure 20, which shows at each grid node 

which one of the reflection lobes forms the envelope 

there; the straight line joining the center of the reflection lobe and the node is generally normal to the contour there. The fact that the envelope is reached by 

almost all of the reflection lobes is evidence of the high 

internal consistency of the data.

The envelope near the terminus is made up of lobes 

from airplane positions that are about a kilometer upglacier (figs. 4, 20). Because the echoes having the arrival times that determine these lobes are likely from 

bed locations near the flight path, the envelope near 

the terminus is artificially shallow. This is confirmed 

by the direct bathymetric measurements, some of 

which were made since the terminus retreated behind 

its August 26, 1978, position (Post, 1975; Post, written 

commun., 1979-1984). The inferred bed topography 

(fig. 18) is taken to be the directly measured bathymetry where it exists and in a slender zone upglacier from there it is taken to be an interpolation 

between it and the envelope of the reflection lobes. The 

adjustment made to the envelope (fig. 21) is greatest 

near the terminus, where the directly measured 

bathymetry exists and where the airborne coverage is 

poor, and fades to zero at the upglacier margin of the 

interpolation zone.

The difference between the nadir method and the 

envelope method is shown in figure 22, which also 

shows the difference between the envelope method and 

the reflection lobes from some surface soundings. 

Where the envelope directly beneath an airplane position is horizontal, the nadir method and the envelope 

method give the same result; if the bed has any inclination, the nadir method gives a shallower estimate of the 

bed altitude. Whereas the envelope-method bed is contoured from values on the 200-m square grid, the difference between the nadir method and the envelope 

method exists only directly beneath an airplane position; therefore, the contouring of the difference is less
G18 STUDIES OF COLUMBIA GLACIER, ALASKA 

"I I I I I \ T

10

7 8 9 10 

x, IN KILOMETERS

1 1 12
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G19

1200

Airplane positions

1000

CO

cc

LJJ

H 

LU

LU

LJJ

LU 

CO

<

LLI

800

600

400

200

Air rays

LU -200

o

m

Reflection lobes 

A

VERTICAL EXAGGERATION 2:1

-400

u" 1000

§ 800

p 600

< 4 00

200

0

-200

-400

Air rays

Ice rays

B Envelope of lobes

7 8 

X. IN KILOMETERS

10 11

FIGURE 19. Vertical cross section along y = 15.4 kilometer showing 

glacier surface and reflection lobes for 31 positions along flight 

path N2500. For each airplane position are shown (A ) the vertical 

cross section through its reflection lobe as well as the air rays from 

the intersections of the lobe and the glacier surface, and (B ) the air 

ray refracted at the glacier surface into the ice ray that reaches 

envelope of lobes. In (B ) are shown the envelope and those rays

reaching it at integral multiples of 200 meters along the x-axis; 

also shown are some rays at intermediate points on the envelope 

from airplane positions for which the lobe does not form the envelope at any 200-meter multiple. For clarity, the lobes from other 

flight paths are not shown; the actual envelope is slightly deeper, 

by as much as 5 meters at x =5.8, 6.4, and 7.8 kilometers.

FIGURE 18. Inferred bed topography. Contour interval is 100 meters 

on exposed bedrock and 20 meters elsewhere. Terminus position 

(heavy solid curve) is taken from August 26, 1978, vertical aerial 

photography. Bed topography in front of terminus and in densely

ruled region behind terminus is taken from direct bathymetric 

soundings (Austin Post, written commun., 1979-1984). Bed topography in sparsely ruled region is taken from an extrapolation of the 

adjustment (fig. 21) in the densely ruled region".
G20 STUDIES OF COLUMBIA GLACIER, ALASKA

789 

x, IN KILOMETERS

10 11 12
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G21

CO

oc

LU

15

14

13

12

11

10

789 

x, IN KILOMETERS

10 11 12

FIGURE 21. Adjustment of envelope-method bed according to directly measured bathymetry (Austin Post, written commun., 

1979-1984), a negative value indicating lowering of the inferred 

altitude. Contour interval is 20 meters. The adjustment in the 

sparsely ruled region is an extrapolation of the adjustment in the

densely ruled region, where direct bathymetry was taken. The 

amount of the adjustment is not defined all the way down to the 

August 26, 1978, terminus position because the envelope method 

could not be applied that far away from the airplane positions 

(dots) from which echo arrival times are available.

FIGURE 20. Sources of reflection lobes forming envelope of lobes. For 

each node of the 200-meter square grid is shown the airplane position (small circle) whose reflection lobe forms the envelope at that 

node. At some intermediate points are shown the airplane position 

whose lobe forms the envelope at that point but does not form the

envelope at any grid node. For a few airplane positions, the echo 

arrival time presumably was underestimated, so that the resulting 

lobe was too shallow to reach the envelope. The bed contours shown 

are those obtained from the envelope method without modification 

according to the directly measured bathymetry.
G22 STUDIES OF COLUMBIA GLACIER, ALASKA

CO

cc

LU

21

20

19

18

17

16

2 15 

X

14

13

12

11

10

789 

x, IN KILOMETERS

10 1 1 12
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G23

accurate than the contouring of the envelope-method 

bed. The maximum difference between the envelopemethod bed and a surface sounding, however, does not 

occur directly beneath the position from which the 

sounding was taken unless the envelope-method bed is 

horizontal there. The location and the amount of the 

algebraically maximum height of the envelope-method 

bed above the sphere, which is the reflection lobe of the 

surface sounding, are shown in figure 22.

DISCUSSION

If there is a strong law that says that every echo must 

come from someplace and that the bed there must be 

normal to the refracted ray, then there is probably also 

a weak law that says that an echo comes from only one 

place. Were the weak law valid, then the regions where 

it is likely that the bed altitude is overestimated are 

those regions where the envelope at many grid nodes is 

formed by a single reflection lobe; according to this 

weak law, the bed coincides with the envelope at only 

one point within such a region and is below it elsewhere. Several candidates for such regions are revealed by the source map (fig. 20). The most conspicuous occurrence of this is the sequence of closed 

depressions along the bottom of the valley, where the 

curvature of the bed likely exceeds that of the reflection 

lobes. Each of the dominating lobes here is produced by 

the maximum tb of an east-west profile. Were the data 

free of error, the true bed altitude would be given at one 

point per reflection lobe and the altitude of the bed 

would be overestimated elsewhere. The resulting surface will be much smoother than the actual bed topography. When the airplane is 800 m above ice that is 

500 m thick, and the bed slope does not exceed 0.2, the 

radius of curvature of the reflection lobe is about 

2,000 m (eq. 7; figs. 12, 13).

In addition, data error must be considered. For one 

thing, it may be the reason some reflection lobes apparently do not reach the envelope (fig. 20). It would be 

possible to adjust the tb to make such a lobe reach the 

envelope which is the ultimate test of internal consistency of the data but this would not modify the envelope, which is what the bed topography is inferred from.

The effect of errors in tb or H can be determined from 

equations 5, as illustrated in figure 14. The estimated 

0.36 (JLS error in tb , assuming the airplane altitude to be 

exactly 800 m, corresponds to an error of about 30 m for 

the point on the lobe directly beneath the airplane, and 

an error of about 35 m at a horizontal distance of 1 km. 

Conversely, if tb is correct, the estimated 30-m error in 

airplane altitude corresponds to an error of about 17 m 

directly beneath and about 12 m one kilometer away.

A biased error distribution in the inferred bed topography results. Bed prominences control the data and 

cause overestimation of the bed altitude elsewhere, 

whereas the data error may induce an error of either 

sign.

This project was not only the first airborne radioecho sounding of a temperate glacier, but it also was 

performed under a severe time constraint. If time had 

allowed testing of the entire system including the 

data analysis prior to the actual sounding of Columbia Glacier, many of the problems encountered would 

have been recognized and solved. With this in mind, 

there are a few fundamental suggestions to those interested in work on airborne radio-echo sounding of temperate glaciers.

First, in order to infer a glacier bed topography from 

airborne radio-echo sounding, it is necessary to have 

the location of the initial pulse (the flightline on the 

reconstructed photograph), tb , and knowledge of the 

glacier surface topography. The last can be obtained by 

recording ts , by having a known surface topography, or 

by recording both the barometric and radar altitudes. 

Because of the possible error due to changing atmospheric conditions, the barometric altimeter needs to be 

carefully calibrated before and after each flight.

Second, when attempting to sound a deep, narrow, 

temperate, valley glacier, the gross geometry of the 

wave the radius of curvature of the reflection lobe 

increases at the rate of 1.78 times H necessitates flying as close to the glacier surface as possible. Thus, the 

dead time of the system (a minimum of 3.5 (JLS in this 

case) must be reduced.

Finally, careful consideration needs to be given to 

the data acquisition System and to the data reduction. 

It is necessary to think through completely what data 

are desired and how they will be analyzed, so it can be

FIGURE 22. Nadir-method results and surface soundings. Solid contours (100-meter interval) show exposed bedrock and bed topography inferred by the envelope method; they agree with figure 18 

except in its ruled regions, where they differ by the amount shown 

in figure 21. Dashed contours (20-meter interval) show height of

nadir-method bed above envelope-method bed. Spot values show 

the difference, in meters, between the envelope-method bed and 

surface soundings (Mayo and others, 1979), a positive value indicating that the envelope method gave a higher altitude.
G24 STUDIES OF COLUMBIA GLACIER, ALASKA

determined what variables should be measured and in 

what format the measurements should be recorded. 

The system should also allow easy recovery of the raw 

recorded data.

REFERENCES CITED

Bindschadler, R.A., and Rasmussen, L.A., 1983, Finite-difference 

model predictions of the drastic retreat of Columbia Glacier, 

Alaska: U.S. Geological Survey Professional Paper 1258-D, 

17 p.

Bishop, J.F., Gumming, A.D.G., Ferrari, R.L., Miller, K.J., and 

Owen, G., 1979, The 1977 Cambridge University expedition to 

Vatnajokull, Iceland: Cambridge, University of Cambridge, 

Dept. of Engineering, Special Report 6.

Bjornsson, Helgi, 1982, Drainage basins on Vatnajokull mapped by 

radio echo soundings: Nordic Hydrology, v. 13, no. 4, p. 213-232.

Bjornsson, Helgi, Ferrari, R.L., Miller, K.J., and Owen, G., 1977, A 

1976 radio echo sounding expedition to the Vatnajokull Ice Cap, 

Iceland: The Polar Record, v. 18, no. 115, pp. 375-377.

Brown, C.S., Meier, M.F., and Post, Austin, 1982, Calving speed of 

Alaska tidewater glaciers, with application to Columbia Glacier: 

U.S. Geological Survey Professional Paper 1258-C, 13 p.

Churchill, R.J., and Wright, D.L., 1978, An airborne radio-echo 

sounder for measurements in temperate glaciers: Colorado State 

University Department of Electrical Engineering, Final Report, 

63 p.

Evans, S., 1963a, Radio techniques for the measurement of ice thickness: The Polar Record, v. 11, no. 73, p. 406-410.

 1963b, International co-operative field experiments in glacier 

sounding: Greenland, 1963: The Polar Record, v. 11, no. 75, 

p. 725-726.

 1963c, Correspondence: Radio techniques for the measurement of ice thickness: The Polar Record, v. 11, no. 75, p. 795. 

-1967, Progress report on radio echo sounding: The Polar

Record, v. 13, no. 85, p. 413-420.

Ferrari, R.L., Miller, K.J., and Owen, G., 1976, The 1976 CambridgeReykjavik Universities expedition to Vatnajokull, Iceland: Cambridge, University of Cambridge, Dept. of Engineering, Special 

Report 5, 62 pp.

Fountain, A.G., 1982, Columbia Glacier photogrammetric altitude 

and velocity: Data set (1957-1981): U.S. Geological Survey 

Open-File Report 82-756, 226 p.

Harrison, C.H., 1970, Reconstruction of subglacial relief from radio 

echo sounding records: Geophysics, v. 35, no. 6, p. 1099-1115.

Kollmeyer, R.C., Motherway, D.L., Robe, R.Q., Platz, B.W., and 

Shah, A.M., 1977, A design feasibility study for the containment 

of icebergs within the waters of Columbia Bay, Alaska: U.S. 

Coast Guard Final Report 128, 135 p.

Mayo, L.R., Trabant, D.C., March, Rod, and Haeberli, Wilfried, 1979, 

Columbia Glacier stake location, mass balance, glacier surface 

altitude, and ice radar data 1978 measurement year: U.S. Geological Survey Open-File Report 79-1168, 72 p.

Meier, M.F., Post, Austin, Brown, C.S., Frank, David, Hodge, S.M., 

Mayo, L.R., Rasmussen, L.A., Senear, E.A., Sikonia, W.G., 

Trabant, D.C., and Watts, R.D., 1978, Columbia Glacier progress 

report December 1977: U.S. Geological Survey Open-File Report 78-264, 56 p.

Meier, M.F., Rasmussen, L.A., Post, Austin, Brown, C.S., Sikonia, 

W.G., Bindschadler, R.A., Mayo, L.R., and Trabant, D.C., 1980,

Predicted timing of the disintegration of the lower reach of 

Columbia Glacier, Alaska: U.S. Geological Survey Open-File Report 80-582, 47 p.

Meier, M.F., Rasmussen, L.A., Krimmel, R.M., Olsen, R.W., and 

Frank, David, 1985, Photogrammetric determination of surface 

altitude, terminus position, and ice velocity of Columbia Glacier, 

Alaska: U.S. Geological Survey Professional Paper 1258-F, 41 p.

Morgan, V.I., and Budd, W.F., 1975, Radio-echo sounding of the 

Lambert Glacier Basin: Journal of Glaciology, v. 15, no. 73, 

p. 103-111.

Oswald, G.K.A., 1975, Investigations of sub-ice bedrock characteristics by radio-echo sounding: Journal of Glaciology, v. 15, no. 73, 

p. 75-87.

Post, Austin, 1975, Preliminary hydrography and historic terminal 

changes of Columbia Glacier, Alaska: U.S. Geological Survey 

Hydrologic Investigations Atlas 559, 3 sheets.

Rasmussen, L.A., 1985, Adjusting two-dimensional velocity data to 

obey continuity: Journal of Glaciology, v. 31, no. 108, p. 115-119.

Rasmussen, L.A., and Meier, M.F., 1982, Continuity equation model 

of the predicted drastic retreat of Columbia Glacier, Alaska: U.S. 

Geological Survey Professional Paper 1258-A, 23 p.

 1985, Surface topography of the lower part of Columbia 

Glacier, Alaska, 1974-1981: U.S. Geological Survey Professional Paper 1258-E, 63 p.

Robin, G. de Q., 1975, Radio-echo sounding: Glaciological interpretations and applications: Journal of Glaciology, v. 15, no. 73, p. 49 

64.

Robin, G. de Q., Swithinbank, C.W.M., and Smith, B.M.E., 1970, 

Radio echo exploration of the Antarctic ice sheet: International 

Symposium on Antarctic Glaciological Exploration, Hanover, 

N.H., 1968, Proceedings, p. 97-115.

Sikonia, W.G., 1982, Finite-element glacier dynamics model applied 

to Columbia Glacier, Alaska: U.S. Geological Survey Professional Paper 1258-B, 74 p.

Sikonia, W.G., and Post, Austin, 1980, Columbia Glacier, Alaska: 

Recent ice loss and its relationship to seasonal terminal embayments, thinning, and glacier flow: U.S. Geological Survey Hydrologic Investigations Atlas 619, 3 sheets.

Smith, B.M.E., and Evans, S., 1972, Radio echo sounding: Absorption 

and scattering by water inclusion and ice lenses, Journal of 

Glaciology, v. 11, no. 61, p. 133-146.

Sverrison, Marteinn, Johannesson, /Evar, and Bjornsson, Helgi, 

1978, A radio-echo equipment for depth sounding of temperate 

glaciers: Reykjavik, Science Institute, University of Iceland, Report RH-78-16, 15 p.

Swithinbank, C.W.M., 1968, Radio echo sounding of Antarctic 

glaciers from light aircraft: IUGG/IASH General Assembly of 

Bern, 1967: Reports and discussions, p. 405 414.

Vancouver, George, 1798, A voyage of discovery to the North Pacific 

Ocean and round the world in the years 1790 1795: London, 

printed for G.G. and J. Robinson, Paternoster-Row, and 

J. Edwards, Pall-Mall, v. 3.

Vickers, R.S., and Bollen, Robert, 1974, An experiment in the radio 

echo sounding of temperate glaciers: Menlo Park, Calif., 

Stanford Research Institute, Final Report, 16 pp.

Watts, R.D., and England, A.W., 1976, Radio-echo sounding of temperate glaciers: Ice properties and sounder design criteria: Journal of Glaciology, v. 17, no. 75, p. 39-48.

Watts, R.D., and Wright, D.L., 1981, Systems for measuring thickness of temperate and polar ice from the ground or from the air: 

Journal of Glaciology, v. 27, no. 97, pp. 459-469.
BED TOPOGRAPHY INFERRED FROM AIRBORNE RADIO-ECHO SOUNDING G25

APPENDIX A: ECHO ARRIVAL TIMES

The following tables give the echo arrival times at 

each airplane position along the 12 east-west and 7 

north-south profiles. The numbers preceded by "N" or 

"W" indicate flight paths shown in figure 4. The arrival 

times (t) are the roundtrip airplane-bed-airplane signal 

times, in microseconds; x and y are the horizontal coordinates of the airplane in the local Columbia Glacier 

coordinate system (eq. 1), and z is the vertical coordinate, in meters above sea level.

x y z t

N500

6649 13398 877 9.55 

6986 13403 874 9.38 

7419 13389 906 8.92 

7904 13403 919 8.54 

8424 13428 926 7.93

HI 000

6371 13873 1021 9.87 

6854 13862 1020 10.33 

7327 13870 1031 10.01 

7791 13877 1032 9.40 

8252 13882 1028 8.78 

8719 13889 1022 8.26 

9216 13903 1027 7.82 

9692 13909 1060 7.11 

10168 13915 1092 7.15

M1500

6187 14395 924 7.94 

6706 14380 900 9.05 

7220 14391 903 9.59 

7735 14402 905 8.91 

8249 14412 908 8.15 

8733 14443 917 7.53 

9235 14437 911 7.37 

9737 14431 906 7.22 

10243 11125 907 6.65

N2000

6555 14845 1034 9.57 

7010 14872 10U5 11.04 

7339 14884 1043 11.31 

7844 14883 1035 10.44 

8350 14879 1027 9.73 

8854 14887 1020 9.27 

9358 14906 1016 8.54 

9824 14904 1017 7.87 

10285 14899 1019 7.33

N2500

6347 15U17 939 8.61 

6864 15399 936 10.40 

7207 15406 940 10.98 

7721 15116 946 10.20 

8236 15127 952 9.13 

8761 15121 951 8.86 

9302 15113 917 8.20 

9801 15107 937 7.18 

10330 15120 918 6.85

x y z t

5967 13386 888 7.64 

6308 13392 882 8.83 

6786 13400 876 9.80 

7127 13398 884 9.17 

7574 13387 914 8.80 

8077 13111 921 8.38 

8597 13136 928 7.62

5728 13874 996 8.82 

6018 13881 1022 9.31 

6533 13868 1020 10.13 

7012 13865 1021 10.27 

7181 13873 1035 9.83 

7915 13879 1031 9.18 

8105 13881 1026 8.60 

8889 13896 1018 8.10 

9371 13905 1038 7.69 

9850 13911 1070 7.31 

10326 13916 1103 7.06

5509 14379 870 6.73 

5847 14391 905 7.28 

6360 14388 914 8.30 

6877 14384 901 9.41 

7392 11395 901 9.31 

7906 11105 906 8.66 

8411 14123 911 7.92 

8900 11141 915 7.11 

9402 11135 910 7.35 

9901 11129 905 7.07 

10119 11121 918 6.38

5919 11809 1018 8.21 

6252 14827 1026 8.72 

6707 14851 1037 10.01 

7126 14876 1014 11.10 

7507 11885 1011 11.18 

8012 11881 1032 10.11 

8518 11878 1021 9.58 

9022 11893 1019 9.08 

9517 11908 1016 8.28 

9978 11902 1018 7.68 

10182 11897 1026 7.17

5657 15152 916 6.61 

6002 15135 913 7.77 

6520 15108 937 9.15 

7035 15103 938 10.91 

7378 15110 912 10.80 

7893 15120 918 9.91 

8107 15130 951 9.23 

8913 15120 950 8.67 

9175 15109 915 7.95 

9977 15112 911 7.26 

10507 15125 951 6.66

x y z t

6137 13389 885 8.35 

6179 13395 880 9.22 

6820 13101 874 9.70 

7269 13394 895 9.01 

7736 13395 917 8.68 

8250 13119 921 8.18 

8770 13111 930 7.20

5887 13881 1019 9.08 

6210 13877 1022 9.60 

6691 13861 1020 10.30 

7169 13867 1028 10.16 

7638 13875 1031 9.62 

8098 13880 1029 8.97 

8559 13886 1025 8.13 

9057 13901 1017 7.95 

9533 13907 1019 7.56 

10009 13913 1081 7.21

5678 14385 887 6.99 

6015 14397 923 7.60 

6531 14381 904 8.67 

7019 11387 902 9.77 

7563 11398 905 9.11 

8078 11109 907 8.10 

8571 11135 914 7.71 

9067 14139 913 7.38 

9569 11133 908 7.31 

10071 11127 903 6.88 

10595 11122 928 6.08

6101 11818 1022 8.10 

6101 11836 1030 9.12 

6858 11863 1011 10.53 

7175 11878 1011 11.39 

7675 11881 1038 10.81 

8181 11880 1029 9.90 

8687 11880 1021 9.13 

9190 11900 1018 8.83 

9671 11906 1017 8.07 

10131 11901 1018 7.50 

10708 11891 1037 7.03

5829 15113 911 7.19 

6171 15126 911 8.15 

6692 15100 936 9.76 

7095 15101 939 11.05 

7550 15113 911 10.52 

8061 15123 950 9.65 

8581 15128 953 9.01 

9122 15117 918 8.15 

9636 15107 910 7.71 

10151 15116 911 7.05 

10683 15129 955 6.17

x y z t

N3000

5852 15825 1041 8.03 

6375 15854 1039 9.56 

6901 15886 1034 10.70 

7341 15896 1027 11.32 

7682 15895 1020 10.76 

8136 15893 1010 10.28 

8540 15880 1005 9.56 

8933 15895 1005 8.89 

9327 15931 1008 8.71 

9721 15967 1010 8.49

M3500

5559 16427 967 7.75 

6097 16412 968 8.51 

6629 16399 968 9.17 

7138 16401 961 10.67 

7477 16403 956 11.22 

8014 16409 957 10.46 

8552 16414 958 9.20 

9092 16406 951 8.65 

9588 16405 950 8.44 

10076 16406 951 7.77 

10565 16406 951 6.70

N4000

5374 16851 1052 7.73 

5841 16844 1049 8.33 

6309 16845 1043 9.06 

6778 16845 1036 10.01 

7247 16846 1030 11.25 

7585 16851 1024 12.20 

8033 16864 1018 11.06 

8462 16876 1014 9.90 

8891 16888 1010 9.17 

9411 16891 1008 8.67 

9954 16893 1008 8.23 

10497 16895 1007 7.78 

11040 16897 1006 7.20

H4500

5447 17418 996 7.83 

5986 17398 998 8.31 

6517 17404 1009 9.26 

7044 17421 1016 10.41 

7179 17452 1012 11.79 

7916 17476 1009 11.15 

8460 17446 1005 9.74 

9003 17422 1003 8.41 

9544 17450 1009 8.07 

10076 17457 1021 7.36

N5000

5274 17865 1015 7.67 

5730 17878 1020 8.32 

6187 17892 1025 9.21 

6643 17905 1030 10.08 

7128 17857 1035 11.04 

7458 17868 1035 11.95 

7955 17892 1034 11.21 

8452 17916 1033 10.33 

8884 17908 1028 9.33 

9366 17907 1019 8.04 

9902 17916 1005 6.46

x y z t

5222 15839 998 5.56 

5537 15832 1019 6.61 

6024 15833 1042 8.68 

6550 15865 1037 9.96 

7076 15897 1033 11.03 

7379 15896 1026 11.23 

7833 15894 1016 10.58 

8278 15891 1008 10.07 

8671 15874 1004 9.31 

9065 15907 1006 8.78 

9459 15943 1009 8.70 

9868 15975 1011 8.28

4841 16449 969 5.66 

5201 16437 967 6.49 

5738 16422 968 8.09 

6276 16406 969 8.70 

6799 16400 965 9-55 

7308 16402 959 11.13 

7655 16405 956 11.04 

8193 16410 957 10.06 

8732 16413 956 8.91 

9262 16405 950 8.59 

9750 16405 950 8.29 

10239 16406 951 7.44

4755 16880 1049 6.59 

5064 16865 1051 7.29 

5528 16844 1053 7.93 

5997 16844 1047 8.55 

6466 16845 1041 9.36 

6931 16846 1034 10.38 

7404 16846 1028 11.73 

7747 16856 1021 11.84 

8176 16868 1017 10.64 

8605 16880 1013 9.61 

9049 16890 1009 8.99 

9592 16892 1008 8.52 

10135 16894 1008 8.09 

10678 16896 1007 7.60

4720 17469 1002 7.27 

5083 17444 999 7-98 

5628 17406 995 7.84 

6163 17400 1002 8.62 

6693 17407 1013 9.60 

7218 17434 1014 10.90 

7566 17459 1011 11.74 

8098 17466 1007 10.73 

8641 17136 1001 9.22 

9181 17131 1005 8.25 

9725 17159 1012 7-91 

10250 17152 1032 6.91

4629 17872 1004 5.64 

4953 17868 1010 6.60 

5426 17870 1017 7.89 

5882 17883 1022 8.59 

6339 17896 1027 9-51 

6805 17890 1032 10.35 

7292 17860 1035 11.55 

7624 17876 1034 11.72 

8120 17900 1034 10.92 

8600 17915 1032 10.02 

9026 17904 1027 8.93 

9545 17910 1014 7.57 

10080 17919 1000 6.10

x y z t

5379 15836 1008 6.05 

5695 15828 1030 7.28 

6200 15843 1040 9.14 

6725 15876 1036 10.31 

7227 15896 1030 11.23 

7530 15895 1023 10.96 

7984 15894 1013 10.44 

8409 15885 1006 9.82 

8802 15883 1005 9.08 

9196 15919 1007 8.73 

9590 15955 1010 8.64

5021 16442 966 6.06 

5380 16432 967 7.06 

5918 16417 968 8.31 

6455 16401 969 8.90 

6968 16401 963 10.09 

7392 16402 958 11.29 

7834 16407 957 10.79 

8373 16412 958 9.61 

8912 16410 954 8.73 

9425 16405 950 8.53 

9913 16406 951 8.06 

10402 16406 951 7.08

4909 16873 1050 7-00 

5219 16858 1051 7.52 

5684 16844 1051 8.13 

6153 16845 1045 8.79 

6622 16845 1038 9.67 

7090 16846 1032 10.79 

7558 16850 1025 12.25 

7890 16860 1020 11.46 

8319 16872 1016 10.25 

8748 16884 1011 9.37 

9230 16891 1009 8.83 

9773 16893 1008 8.37 

10316 16894 1007 7.94 

10859 16896 1007 7.41

4901 17457 1000 7-76 

5265 17431 998 7.96 

5810 17395 995 8.03 

6340 17402 1006 8.94 

6870 17109 1017 9.97 

7392 17446 1013 11.51 

7740 17472 1010 11.49 

8279 17456 1006 10.25 

8822 17426 1003 8.75 

9364 17441 1007 8.17 

9902 17463 1016 7.68

4791 17870 1007 6.01 

5115 17866 1013 7.19 

5578 17874 1019 8.10 

6035 17887 1024 8.90 

6491 17900 1028 9.80 

6966 17873 1033 10.66 

7416 17866 1035 12.00 

7789 17884 1034 11.48 

8286 17908 1033 10.63 

8742 17911 1030 9.69 

9188 17904 1023 8.50 

9723 17913 1009 6.95
G26 STUDIES OF COLUMBIA GLACIER, ALASKA

x y z t

N5500

5580 18413 1039 8.12

6095 18429 1042 9.03

6631 18420 1041 9-93

7188 18386 1038 11.82

7560 18363 1035 11.92

8086 18368 1038 10.85

8610 18376 1041 9.62

9134 18386 1043 8.52

9658 18398 1045 6.36

H6000

6337 18890 1041 9.61

6862 18889 1030 10.97

7264 18887 1022 11.98

7742 18854 1025 11.35

8216 18856 1023 10.24

8671 18874 1020 8.77

9126 18891 1016 7.86

W500

9505 14054 1032 7.71

9503 14466 1031 8.17

9505 14876 1032 8.39

9517 15281 1037 8.50

9529 15685 1040 8.65

9542 16081 1034 8.94

9549 16479 1031 9.00

9533 16889 1038 8.76

9534 17291 1040 8.35

9568 17680 1034 8.18

W1000

8993 1^508 1064 8.50

9012 14905 1064 9.39

9042 15303 1058 9-30

9039 15707 1051 9.16

9019 16115 1043 8.98

9022 16515 1042 9.31

9038 16911 1046 9.24

9044 17310 1049 8.65

9029 17714 1050 8.92

9010 18111 1054 9.16

8988 18500 1059 8.85

8989 18913 1064 8.40

9008 19346 1069 8.23

9021 19768 1073 8.16

9033 20186 1078 8.12

W2000

8000 13952 1078 9.45

8003 14349 1043 9.38

8014 14758 1043 10.01

8024 15166 1043 10.27

8004 15574 1037 10.37

7988 15984 1033 10.63

7992 16410 1038 11.01

7996 16832 1043 11.30

7999 17229 1047 11.31

7994 17632 1051 11.24

7962 18056 1054 11.36

7943 18484 1059 11.38

7971 18922 1067 11.17

7990 19360 1073 10.93

8002 19799 1078 10.71

x y z t

4816 18404 1008 5.38

5198 18409 1024 6.63

5752 18418 1040 8.47

6267 18434 1043 9.28

6817 18409 1040 10.47

7346 18377 1036 12.41

7737 18363 1036 11.64

8261 18371 1039 10.31

8785 18379 1042 9.32

9309 18390 1044 7.94

5639 18895 1054 8.46

5987 18891 1048 9.12

6512 18890 1037 9.99

7037 18889 1026 11.64

7387 18884 1021 11.85

7913 18844 1026 11.05

8368 18862 1022 9.70

8823 18880 1019 8.45

9281 18897 1015 7-38

9281 18897 1015 7.38

9510 13778 1042 7.32

9504 14191 1032 7.89

9502 14603 1031 8.21

9509 15011 1033 8.44

9521 15417 1039 8.53

9533 15817 1038 8.74

9546 16213 1033 9.00

9544 16616 1033 8.95

9527 17025 1040 8.63

9545 17421 1038 8.26

9579 17809 1032 7.87

8981 13978 1056 8.22

8987 14243 1060 8.21

8996 14640 1066 8.88

9022 15038 1062 9.41

9052 15435 1056 9.25

9032 15843 1048 9.11

9013 16251 1040 9.04

9028 16647 1043 9.32

9044 17043 1047 9.05

9039 17445 1049 8.65

9024 17849 1051 9-08

9003 18240 1055 9.11

8980 18629 1060 8.68

8995 19058 1066 8.32

9013 19489 1071 8.20

9025 19907 1075 8.14

7997 13427 1138 9.91

7998 13690 1108 9.63

8000 14083 1063 9.39

8007 14485 1043 9.51

8017 14894 1043 10.19

8018 15302 1041 10.30

7997 15709 1035 10.44

7989 16126 1034 10.75

7994 16553 1039 11.14

7997 16965 1044 11.33

8000 17361 1048 11.28

7983 17774 1052 11.26

7952 18198 1055 11.42

7952 18630 1061 11.32

7981 19068 1069 11.09

7994 19506 1075 10.85

8007 19945 1079 10.64

x y z t

5007 18406 1016 5.91

5390 18411 1032 7.51

5924 18424 1041 8.77

6446 18432 1042 9.55

7003 18398 1039 11.13

7374 18375 1036 12.33

7911 18366 1037 11.36

8435 18373 1040 9.92

8959 18383 1043 8.98

9483 18394 1044 7.22

5812 18892 1051 8.85

6162 18891 1044 9.35

6687 18890 1033 10.44

7212 18888 1023 11.92

7565 18869 1022 11.62

8065 18850 1025 10.69

8519 18868 1021 9.18

8974 18886 1017 8.15

9515 13639 1052 7.11

9506 13917 1033 7.52

9503 14328 1031 8.05

9501 14740 1030 8.28

9513 15146 1035 8.47

9525 15552 1041 8.58

9538 15949 1036 8.84

9550 16345 1031 9.02

9538 16752 1036 8.87

9523 17161 1042 8.49

9557 17550 1036 8.25

8984 14111 1058 8.20

8990 14376 1062 8.30

9002 14773 1066 9.22

9032 15170 1060 9.36

9046 15571 1053 9.20

9026 15979 1046 9.04

9017 16384 1041 9.22

9033 16779 1045 9-30

9049 17175 1048 8.81

9034 17579 1050 8.75

9018 17981 1052 9.15

8995 18370 1057 9.01

8982 18769 1062 8.52

9002 19202 1067 8.27

9017 19628 1072 8.18

9029 20046 1076 8.13

7998 13558 1123 9.76

7999 13821 1093 9.53

8001 14215 1048 9.36

8010 14622 1043 9.76

8020 15030 1043 10.24

8011 15438 1039 10.33

7991 15845 1033 10.52

7991 16268 1036 10.88

7995 16695 1041 11.24

7998 17097 1045 11.33

8001 17493 1050 11.25

7973 17915 1053 11.30

7941 18339 1057 11.12

7962 18776 1064 11.25

7986 19214 1071 11.01

7998 19652 1076 10.78

x y z t

W2500

7521 14069 1032 9.83

7515 14478 1033 10.20

7507 14883 1035 11.14

7490 15283 1033 11 -29

7472 15684 1030 11.09

7514 16093 1027 11.19

7556 16504 1026 11.81

7555 16916 1030 12.27

7553 17328 1035 12.00

7532 17742 1037 11.86

7510 18157 1040 12.02

7514 18579 1040 11.98

7525 19003 1040 11.73

7541 19418 1040 11.54

7559 19831 1040 11.42

W3000

6993 13781 1070 10.61

6995 14209 1071 10.61

7004 14648 1074 11.00

7005 15046 1073 11. 13

7003 15426 1071 11.69

6993 15820 1068 11.20

6976 16223 1066 10.80

6979 16633 1063 10.72

6990 17044 1061 10.64

6995 17461 1056 10.57

6996 17885 1047 10.84

6994 18305 1041 11.06

6988 18716 1044 11.40

6982 19126 1046 11.77

6975 19537 1049 12.10

W3500

6519 13446 1047 10.46

6555 13974 1032 10.10

6574 14512 1028 9.57

6558 15062 1030 9.56

6542 15611 1031 9.78

6525 16147 1035 9.63

6507 16683 1038 9.41

6540 17229 1035 9.44

6581 17777 1031 9.87

6581 18303 1035 9.75

6573 18831 1039 10.12

6558 19381 1039 10.61

6543 19931 1040 10.94

W4000

6155 13590 1048 9.75

6165 14153 1042 9.12

6169 14713 1039 8.61

6139 15251 1038 8.66

6108 15789 1036 8.83

6042 16328 1033 8.94

5987 16846 1037 8.47

5934 17362 1041 8.48

5895 17888 1046 8.78

5907 18449 1058 8.88

5910 18981 1064 9.11

5911 19506 1069 9.41

5915 20061 1074 9.71

x y z t

7511 13499 1030 9.58

7516 13784 1031 9.74

7521 14208 1032 9.88

7513 14613 103^ 10.58

7501 15017 103t 11.26

7484 15417 1032 11.24

7483 15820 1029 11.05

7530 16230 1027 11.36

7556 16641 1027 12.06

7555 17053 1032 12.19

7546 17466 1036 11.93

7525 17881 1038 11.88

7507 18297 1040 12.06

7518 18720 1040 11.90

7529 19143 1040 11.66

7547 19556 1040 11.49

7564 19968 1040 11.40

6997 13222 1070 10.42

6995 13501 1070 10.68

6992 13920 1070 10.57

6998 14356 1072 10.71

7007 14792 1075 11.14

7005 15173 1073 11.58

7002 15553 1070 11.57

6987 15955 1068 11.03

6973 16358 1065 10.73

6983 16770 1063 10.72

6993 17181 1060 10.58

6995 17602 1053 10.66

6996 18026 1044 10.92

6992 18442 1042 11.14

6986 18852 1044 11.54

6980 19263 1047 11.88

6973 19673 1050 12.20

6494 12762 1113 9.17

6506 13104 1080 9.81

6528 13619 1035 10.42

6568 14151 1030 9.87

6569 14696 1029 9.54

6553 15245 1030 9.59

6536 15791 1032 9.91

6519 16326 1036 9.47

6513 16864 1038 9.42

6554 17412 1034 9.49

6585 17955 1031 10.00

6579 18477 1036 9.74

6568 19014 1039 10.30

6553 19564 1040 10.74

6165 12854 1066 8.97

6160 13222 1057 9.12

6156 13776 1044 9.75

6170 14342 1041 8.74

6159 14892 1038 8.61

6128 15431 1037 8.71

6096 15969 1036 8.89

6022 16503 1034 8.87

5969 17018 1038 8.40

5917 17534 1042 8.60

5899 18075 1050 8.81

5909 18631 1061 8.94

5910 19156 1066 9.21

5911 19685 1071 9.51

x y z t

7514 13642 1031 9.67

7518 13926 1031 9.79

7518 14343 1033 9.97

7510 14748 1034 10.92

7496 15150 1033 11.30

7478 15550 1031 11.16

7499 15956 1028 11.08

7546 16367 1026 11.57

7556 16778 1029 12.24

7555 17190 1034 12.09

7539 17604 1037 11.88

7517 18019 1039 11.95

7511 18438 1040 12.04

7521 18862 1040 11.81

7535 19281 1040 11.60

7553 19693 1040 11.45

6996 13361 1070 10.63

6994 13641 1070 10.66

6992 14063 1070 10.57

7001 14502 1073 10.85

7006 14919 1074 11.28

7004 15300 1072 11.69

6998 15686 1069 11.40

6982 16089 1067 10.91

6976 16495 1064 10.71

6986 16907 1062 10.69

6995 17320 1059 10.54

6996 17744 1050 10.75

6996 18168 1040 10.99

6990 18579 1043 11.26

6984 18989 1045 11.66

6977 19400 1048 11.99

6500 12933 1097 9.40

6513 13275 1063 10.21

6541 13797 1033 10.30

6579 14329 1028 9.65

6563 14879 1029 9.54

6547 15428 1031 9.66

6531 15969 1033 9.84

6513 16504 1037 9.41

6527 17046 1037 9.43

6567 17594 1032 9.64

6583 18129 1033 9.90

6577 18652 1038 9.92

6563 19198 1039 10.46

6548 19748 1040 10.84

6163 13038 1061 8.77

6158 13406 1052 9.56

6161 13965 1043 9.54

6175 14530 1039 8.63

6149 15072 1038 8.63

6118 15610 1037 8.76

6069 16149 1035 8.93

6004 16675 1035 8.65

5952 17190 1039 8.41

5899 17706 1044 8.71

5903 18262 105t 8.84

5909 18806 1062 9.02

5910 19331 1067 9.31

5913 19873 1072 9.61