Surface Topography of the Lower Part of 

Columbia Glacier, Alaska, 1974-1981

« ^Us-A .>

 jimp * 

U.S. GEOLOGI 'RV'EY PROFESSIONAL PAPER 1258-E
/-*-

L.

Surface Topography of the 

Lower Part of Columbia Glacier, 

Alaska, 1974-1981

By L A. RASMUSSEN and M. F. MEIER

STUDIES OF COLUMBIA GLACIER, ALASKA

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1258-E

Obtaining surface altitudes on a square grid 

by applying the method of "optimum 

interpolation " to irregularly positioned altitudes 

obtained by photogrammetry.

Revised May 1985

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON 1985
DEPARTMENT OF THE INTERIOR 

DONALD PAUL HODEL, Secretary

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Library of Congress Cataloging in Publication Data

Rasmussen, L. A.

Surface topography of the lower part of Columbia Glacier, Alaska, 1974-81.

(U.S. Geological Survey professional paper; 1258-E)

Bibliography: p.

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

1. Columbia Glacier (Alaska) I. Meier, Mark Frederick, 1925- . II. Title. III. Series: Geological Survey professional

paper;1258E.

GB2427.C64R377 1984 551.3'1'097983 84-600191

For sale by the Distribution Branch, U.S. Geological Survey 

604 South Pickett Street, Alexandria, VA 22304
CONTENTS

	Page

Symbols and abbreviations V

Abstract El

Introduction 1

Acknowledgments 4

Original topographic data 4 

Altitude errors in individual photogrammetric points 

Spatial distribution of individual photogrammetric points 

Statistical properties of the surface topography ^

A preliminary interpolation 13

Correlation calculations and comparisons 13

First approximation to the interpolated altitudes 18

The correlation function 19

The interpolation algorithm 21

Optimum interpolation 21

Some independent test data 22

The results 25

Discussion 27

Application to future work 28

References cited 29

Appendix A: Fourier transform of the correlation function 33

Appendix B: Columbia Glacier surface altitudes 34

ILLUSTRATIONS

FIGURE 1. Index map of Columbia Glacier, Alaska E2

2. Surface topography of lower part of Columbia Glacier on July 27,1974 *>

3. Surface topography of lower part of Columbia Glacier on September 1,1981 (detail of original map) ^

4. Surface topography of lower part of Columbia Glacier on September 1,1981 °

5. Surface topography of lower part of Columbia Glacier altitude decrease between July 27,1974 and September 1,1981 ®

6. Photograph of surface roughness 10

7. Spatial classification of photogrammetric data H

8. Time profiles of surface altitude change I4

9. Altitude-change correlation as a function of distance 16

10. Altitude-change correlation as a function of direction 1^

11. Mathematical functions fit to empirical correlation data 23

12. Spatial distribution of ground-truth-sample points 24

13. Comparison of ground-truth-sample errors with algorithm-estimated errors 25

14. Effect on ground-truth-sample errors of possible systematic error in the photogrammetric data 26

15. Comparison of ground-truth-sample errors and adjusted algorithm-estimated errors 27

16. Fraction of data points used that are from same flight as point being interpolated, versus number of data points used 28

17. Norm field coefficients as functions of time 29

III
IV CONTENTS

TABLES

TABLE 1. Date and coverage of aerial photography °

2. Estimated error distribution of individual photogrammetric point altitudes 

3. Spatial distribution of individual photogrammetric points over the 24 squares shown in figure 4 1 2

4. Estimated average altitude changes from July 27, 1974, until the date of the indicated flight for the lower reach and the

lower part of the central basin * 4

5. Result of interpolating at a point using a regional second-degree polynomial corrected by averaging the residuals of the

three individual photogrammetric points nearest that point ^

6. The coefficients aL and bL used in equation 15 to interpolate the norm field fL(x,y) for flight L from the digitizations of the

two maps /2 (x,y) and /30 (x,y) 19

7. Values of the correlation coefficient calculated from the deviations from the norm fields for the indicated time difference 

(T) intervals and the indicated horizontal distance (d) intervals 

8. Distribution of 58-point surface-survey sample 25
SYMBOLS AND ABBREVIATIONS

SYMBOLS AND ABBREVIATIONS

Symbol Name

A Correlation study point at x=5,250 m, 

y=27,500 m 

a Coefficient for obtaining norm field 

from 1974 and 1981 maps 

a Year.

a Coefficient of correlation function 

B Correlation study point at x=6,500 m, 

y=22,000 m.

b Coefficient for obtaining norm field 

from 1974 and 1981 maps 

18 Coefficient of correlation function 

C Correlation study point at jc=6,250 m, 

y=18,500m.

D Correlation study point at x=8,250 m, 

y- 18,500m.

d Horizontal distance 

E Correlation study point at x=6,750 m, 

y=15,000m.

Ej Error in fitting IPP's with j\x,y) 

EG Interpolation error estimated by 

algorithm 

E'G Actual interpolation error with groundtruth sample 

E"G Version of E'Q adjusted for errors in 

surveying and time discrepancy 

Ej Error in digitizing 1974 and 1981 

maps 

Ep Random error of IPP altitudes 

Er Error in approximating r*(r,d) by 

R(Ttd) 

ET Departure of IPP's from digitization of 

1974 and 1981 maps 

Ew Part of ET due to IPP being on wrong 

side of contour 

E2 Error in approximating IPP's by 

P(x,y) 

Ea As Ey but corrected by "nearest three" 

residual averaging 

e Mathematical constant, 2.71828....

F Correlation study point at jc=8,750 m, 

y=15,000 m 

f(x,y) Norm field 

/ Subscript denoting row of 762.5-meter 

data grid.

IPP Individual photogrammetric point (coordinates determined by vertical aerial photography).

i Base of imaginary numbers.

J Subscript denoting column of 762.5- 

meter data grid.

KQ Modified Bessel function of the second 

kind.

km Kilometer.

L Subscript denoting flight number.

X Fraction of IPP's from same flight as 

point being interpolated 

m Number of IPP's common to 1974 and 

1981 maps 

m Meter.

N Order of linear system for determining 

interpolation weights 

Units 

(where applicable)

Dimensionless

m 

km

m 

m

Dimensionless

Dimensionless 

Dimensionless

Dimensionless

Symbol Name

n Sample size, used generically 

P(x,y) Second-degree polynomial 

p Coefficient of P(x,y). 

$ Fourier transform of correlation function.

0 Coordinate axis of $. 

\l/ Arbitrary rotation angle. 

R(r,d) Correlation function 

r(d,B,r) Correlation between Z(0,0,£+r) and 

Z(d,8,t) 

r*(r,d) Empirical correlation of IPP departures 

from j\x,y) 

r^(d,r) Azimuth-averaged r(d,6,r) 

7^(0,7) Distance-averaged r(d,6,r) 

t Time coordinate 

At Time difference of surface surveyed altitude from some tL 

r Time lag 

6 Azimuth (compass direction). 

UTM Universal Transverse Mercator. 

u Component of surface velocity in xdirection 

V Variance of IPP's about f(x,y) 

v Component of surface velocity in ydirection 

w Linear-combination weights for interpolation algorithm 

x Horizontal coordinate, positive to 

east 

£ Local coordinate within 200-meter digitization square 

y Horizontal coordinate, positive to 

north 

y Coordinate of u-advected IPP 

Z Glacier surface altitude above National

Geodetic Vertical Datum of 1929,

_ herein referred to as sea level 

Z Average of Z at vertices of 200-meter

digitization square 

ZIPP Altitude coordinate of an IPP 

Z(0,0,t+r) Altitude of point used in correlation

study 

ZL(xk ,yk ) IPP altitude for fcth point of flight!/ 

Z*(x,y) Altitude estimated by interpolation

algorithm 

Z'L(x,y) Ground-thruth altitude adjusted to flight 

date £L 

Z(x,y) Altitude at (x,y) by advecting IPP with 

v for r 

AZ Deviation of IPP from norm field, as 

used in weights equation 

AZ* Value of AZ estimated by algorithm 

AZL(x^yk) Deviation of IPP from fL(x,y) 

dZ Systematic error in the photogrammetric data 

f Local coordinate within 200-meter digitization square 

< > Integer function operator; that is, <x> 

is the integer part of x.

Units 

(where applicable)

Dimensionless 

m

Dimensionless 

Dimensionless

Dimensionless 

Dimensionless 

Dimensionless 

a

a 

a

-i

ma 

m2

ma"1

Dimensionless

m

200m

m 

m

m 

m

m 

m

m 

m 

m

m 

200m

Note: Use of names of firms or their products in this report does not imply endorsement.

STUDIES OF COLUMBIA GLACIER, ALASKA

SURFACE TOPOGRAPHY OF THE LOWER PART OF 

COLUMBIA GLACIER, ALASKA, 1974-81

By L. A. RASMUSSEN and M. F. MEIER

ABSTRACT

The method of optimum interpolation is used to get surface altitudes on the nodes of a square grid superimposed over a map of the 

lowest 18 kilometers of Columbia Glacier, a large, grounded, icebergcalving glacier near Valdez, Alaska Vertical aerial photography has 

been obtained about five times a year since July 1976. For each of 29 

flights, the altitude of the glacier surface has been determined photogrammetrically at the locations of prominent surface features. The 

large amount of data, the noisy character of the topographic data due 

to the rough surface, and a modeling need to know precisely how the 

ice surface changes with time require a sophisticted interpolation procedure. The recently developed method of optimum interpolation is 

used here because it minimizes the interpolation error by developing 

and using knowledge of the statistical properties of the data The 

reduction of these surface altitude and velocity data will lead to the 

best data set ever generated for a glacier, and that data will be 

valuable for investigating calving mechanics, for studying glacier 

dynamics, and particularly for using in modeling the drastic retreat of 

Columbia Glacier.

Topographic maps of the glacier surface were obtained from flights 

on July 27, 1974, and September 1, 1981. The first approximation of 

the topography on the date of a particular intervening flight is taken 

to be the linear combinaton of the two maps that best fits the photogrammetric data from that flight. The deviations of the individual 

photogrammetric points about this first approximation are the quantities to which the method of optimum interpolation is actually 

applied in estimating the altitudes of the grid nodes. Between 116 and 

202 photogrammetric points are available from flight to fight.

The statistical properties of the surface topography are described in 

terms of the correlation between the altitude changes at a point and 

the changes at some other point. This correlation has a high degree of 

homogeneity and isotropy, both spatially and temporally, and was 

little affected by the glacier flow. This remarkable regularity permits 

the use of a very simple implementation of the optimum interpolation 

algorithm.

The random error in the photogrammetrically determined altitudes 

is estimated at 3.5 meters by comparing the individual points with 

maps made from the same flights. A ground-truth sample of 58 individual and averaged points was used to test the interpolated results 

for three flights. The asymmetry of these sample deviations about the

first-approximation surfaces suggests that there may be a systematic 

error of about +1.5 meters in the photogrammetrically determined 

altitudes. The standard error in the altitudes interpolated on the grid 

nodes is estimated to be 2.5 meters.

INTRODUCTION

Columbia Glacier is a large (67 km long and 1,100 km2 

in area), calving glacier near Valdez, Alaska (fig. 1). 

Although much of its bed is below sea level, the glacier 

is grounded everywhere except for some short, floating 

segments along its margins where it spills into icedammed, freshwater lakes. The terminus pushes 

against a moraine shoal; the top of the shoal is about 

20 m below sea level, but a short distance upglacier the 

bed is about 400 m below sea level.

Recent studies of calving glaciers in Alaska have 

shown that calving speed is related to water depth at 

the terminus, and that if a glacier retreats off of a shoal 

so that its terminus faces deep water, the retreat will 

become rapid and irreversible (Post, 1975). Columbia 

Glacier clearly has the potential to make a drastic retreat. This situation is of practical as well as academic 

interest, because drastic retreat will cause a large 

increase in iceberg discharge; some icebergs from Columbia Glacier drift into shipping lanes in Valdez Arm 

(KoUmeyer and others, 1977).

To predict whether Columbia Glacier might begin 

rapid retreat and, if so, when and by how much the iceberg discharge might increase, the U.S. Geological 

Survey began an intensive study in 1977 involving 

aerial photography and ground-based and boat-based 

fieldwork (Meier and others, 1978). Results of the fieldwork during the 1977-78 principal data year have been

El
E2 STUDIES OF COLUMBIA GLACIER, ALASKA

147°00'

 61°00

L 

480000mE SOOOOOmE

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

dots at 2-kilometer intervals along the longitudinal coordinate system. The border of the interpolation domain is dotted.
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E3

reported (Mayo and others, 1979; Sikonia and Post, 

1980), and a preliminary prediction was issued in June 

1980 (Meier and others, 1980). Two postulated formulations of the calving relation were investigated (Brown 

and others, 1982; Sikonia, 1982). Detailed calculations 

from three predictive models (Rasmussen and Meier, 

1982; Sikonia, 1982; Bindschadler and Rasmussen, 

1983) soon followed; they all reinforced the initial expectation of substantial retreat during the mid-1980's, 

along with an order-of-magnitude increase of the iceberg 

discharge.

The ground-based and boat-based fieldwork was supplemented by vertical aerial photography done about 

five times a year since July 1976, usually for the 

18-kilometer-long lower part of the glacier only. For 

each flight, photogrammetric analysis has yielded accurate coordinates of points on the glacier surface 

(Meier and others, 1984) and, because identifiable surface features can be followed form one flight to the next, 

average flight-to-flight surface velocities have been 

determined (Fountain, 1982).

This photogrammetric data set is characteristically 

different from data sets obtained by traditional field 

programs, in that the data are numerous (about 7,000 

separate point coordinate determinations), the data are 

irregularly distributed in both time and space, and some 

important information (such as changes in glacier thickness over time) may be immersed in noise owing to the 

spatially irregular glacier surface. The data contain only 

the two vriables, surface altitude and displacement. 

Numerical modeling of changes in flow and shape of a 

glacier requires (1) data interpolated on a grid fixed in 

space, (2) special treatment to obtain the best possible 

estimate of time changes in that field, and (3) knowledge 

of the errors and statistical properties of the data set. 

These modeling requirements, imposed on a nontraditional data set, make it necessary to use a sophisticated 

data-analysis procedure. Use of this procedure may 

have broader application, especially as modern methods 

of data acquisition greatly increase the complexity of 

data analysis.

The central problem in this analysis procedure is the 

need to use all relevant information in the interpolation 

algorithm. Only by doing so can the signal-to-noise ratio 

be maximized. Large-scale swells and swales in the surface topography tend to persist because they are caused 

by the underlying bedrock, and this means that the 

interpolation scheme can be aided if the temporal as 

well as the spatial dimension is included. Thus, generalpurpose contouring programs, which deal with a twodimensional field only, are not sufficient. Constructing 

topographic maps by photogrammetry from each set of 

aerial photographs also ignores the temporal dimension 

and, in addition, would be prohibitively expensive.

The method of interpolation reported here incorporates both temporal and spatial information. It has 

the additional advantage of requiring and producing information on the statistical structure of the field. 

Statistical structure refers to how well the altitude of a 

certain point correlates with the altitudes of neighboring points, distributed in both space and time, and 

whether that correlation depends on location (a test of 

homogeneity) or on direction (a test of isotropy). Defining this statistical correlation function is critical to the 

method of optimum interpolation and also enables 

determination of the interpolation error.

The interpolated surface topography, when combined 

with the surface velocity and other variables on the 

same grid, will constitute an extraordinarily rich data 

set for use in time-dependent, two-dimensional models 

of glacier flow. The data set must be internally consistent with respect both to the continuity equation and to 

the flow law, and it must be faithful, within the bounds 

of observational error, to the field data. Because Columbia Glacier is a calving glacier, the data set will be 

useful for continuing the investigation of the calving 

mechanism as well as for studying glacier dynamics 

generally. Because rapid retreat appears to have begun 

(Meier and others, 1984), the future extension of the 

data set will include the only detailed, high-quality data 

on this process in existence.

The report begins with a description of the photogrammetric data in the section "Original Topographic 

Data": first the inherent altitude error is estimated, and 

then the temporal and spatial distributions of the 

data are described. Because the benefit of low interpolation error is gained at the cost of first determining the 

variable's statistical properties, several preliminary interpolation schemes are introduced in "Statistical Properties of the Surface Topography"; these results are 

then used to estimate the point-to-point correlation of 

altitude changes, between points displaced from one 

another in both space and time, and to determine 

whether the correlation is homogeneous and isotropic. 

Described in "First Approximation to the Interpolated 

Altitudes" are how the 1974 and 1981 topographic 

maps were used to make norm fields for the times of 

intervening flights and how deviations of irregularly 

positioned photogrammetric points were established as 

the variable actually interpolated; also discussed is obtaining a mathematically proper function for representing the correlation. In "The Interpolation Algorithm," 

the derivation of optimum interpolation is briefly 

recapitulated and several of its chief properties are mentioned, the result of applying it to the photogrammetric 

data is compared, for several tens of points, with altitudes surveyed from the glacier surface, and the exact 

specification of the algorithm actually used is given.
E4 STUDIES OF COLUMBIA GLACIER, ALASKA

Finally, in the section titled "Discussion," errors and 

shortcomings of the results are considered, the elusiveness of defining the surface altitude of a severely fractured and irregular ice mass is mentioned, and speculations are offered on extrapolating the data set.

ACKNOWLEDGMENTS

We are particularly grateful to Alan Thorndike for 

valuable counsel on the statistical basis of this work, 

and to him, Craig Lingle, and Uwe Radok for helpful 

reviews of the manuscript.

ORIGINAL TOPOGRAPHIC DATA

Two topographic maps for the lower reach of the 

Columbia Glacier surface (indicated by the box in fig. 1) 

were produced photogrammetrically at 1:50,000 scale 

and 10-meter contour interval for the flights on July 27, 

1974 (fig. 2), and September 1, 1981. For these and the 

intervening 27 flights, the glacier surface altitude was 

determined photogrammetrically at numerous irregularly positioned points (table 1). Except for the September 1, 1981, map, which was produced by Air Photo 

Tech of Anchorage, Alaska, all the processing was performed by the USGS Western Mapping Center at 

Menlo Park, Calif. The coordinates of the individual 

points are compiled in Fountain (1982). The 1974 map 

was smoothed by the photogrammetrist, but the photogrammetrist who made the 1981 map followed contours 

in and out of crevasses. Figure 3 shows a section, at 

1:50,000 scale, of the original September 1, 1981, map.

Both maps were digitized by visually interpolating 

altitudes on a 200-meter square grid (4 mm at 1:50,000 

scale). The uncertainty of how to interpolate between 

contour lines was used, to a modest degree, to attempt 

to make the resulting difference field relatively free of 

spurious small-scale features. The manual contouring of 

the digitized values for the 1981 map is shown here 

(fig, 4) because its smoothness, which is comparable 

with that of the 1974 map, makes 1974 to 1981 changes 

in the glacier surface easier to discern than using the 

original 1981 map. The box in figure 4 indicates the section shown in figure 3, and the difference between the 

1974 and 1981 maps is shown in figure 5.

The digitizations of the two maps are used as the 

bases for the ultimate interpolation to the nodes of the 

square grid. This requires determining the altitudes, at 

the times of each of the two maps, at the x,y location of 

any of the (irregularly positioned) photogrammetric 

points from any of the intervening flights. For interpolating between the nodes of the 200-meter grid, a 

piecewise planar surface is assumed to pass through the 

altitudes at the nodes. Every grid cell is subdivided into 

four triangular regions, each of which has as its vertices

the center of the square and two adjacent vertices of the 

square. The altitude Z at the center of the square is 

taken to be the average of the altitudes at the square's 

four vertices; this Z is also the ordinate at the center of 

the square of the plane best fitting the points at the four 

vertices in the least squares sense. For each triangle, the 

surface is defined to be the plane passing through the 

altitudes at the triangle vertices. That is, if a local coordinate system is chosen so that the coordinates of the 

surface at the vertices of the square are (£,f,Z)=(0,0,Z00), 

(1,0,Z10), (l,l^n), (0,1,Z01), then the surface is interpolated according to

Z= .(i)

The interpolation surface is continuous in ordinate but 

has discontinuity of derivative at the edges of the 

triangles, both within a square and from square to 

square. Although it is not relevant to the present 

analysis, this interpolating surface has a practical 

advantage over other zero-order-continuity interpolating surfaces, such as the bidimensional linear interpolation: the construction of a normal from an external 

point is a direct, linear calculation. The result of this 

interpolation for the July 27, 1974, map is denoted 

f2(x,y), and for the September 1, 1981, map, f^i^y), the 

numbers 2 and 30 corresponding to the numbers of 

aerial photographic flights as listed in table 1.

ALTITUDE ERRORS IN 

INDIVIDUAL PHOTOGRAMMETRIC POINTS

The lower reach is a very rough surface (fig. 3, 6), and 

this roughness may contribute to the lack of perfect 

agreement between the maps and the individual photogrammetric points (IPP's) from the same flights. When 

drawing a contour line on a map, the operator smooths 

out small-scale roughness and attempts to do so without 

introducing bias. When picking IPP's for coordinate 

determination, the operator must see nearby surface 

topography to form a stereo image; thus, isolated peaks 

or depressions are not used. However, the local surface 

seen around an IPP does not necessarily correspond to a 

mean surface over a larger area. Coordinate determination is accurate to within about 2 m, but it is subject to 

human error (R. W. Olsen, 1980, oral commun.). Because 

of the roughness of the actual surface, the representativeness of any particular point is a serious consideration.

For each of the two maps, the distribution of the 

departures of the IPP altitudes from the digitization of 

the map was carefully examined, to estimate how much 

of the discrepancy was due to interpolation error and how
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E5

400000 £

2 3 KILOMETERS

495000raE 405000raE

Topography by U.S.G.S., WMC rrom

photography taken July 27, 1974. 0___1________ 

Modified by Project Office, Glaciology, ' ' " ' 

WRD, Tacoma.

FIGURE 2. July 27, 1974 (flight 2), surface topography, in meters above sea level, with 10-meter contour interval, 

prepared photogrammetrically by the U.S. Geological Survey Western Mapping Center at Menlo Park, Calif. 

Approximate scale is 1:100,000. The border of the interpolation domain is dotted. The field of view of figure 6 

is indicated.
E6 STUDIES OF COLUMBIA GLACIER, ALASKA

TABLE I. Date and coverage of aerial photography

[Decimal year indicates time of flight; t=1978.000 at 0000 hours on January 1, 1978, and increases by 1/365.2422 for each day thereafter. Coverage designations: a, most of glacier; 

b, whole glacier; c, lowest 4 km: d, lower reach; e. icefall reach; f, central basin. IPP=individual photogrammetric point ]

Flight 

Number

1 

4 

5 

7

10 

11 

1 9

1 o

14

15 

i a

i n

I Q

1 Q

Ofi

91

99

f>Q

24 

OK

oc

97

9Q

9Q

Qf>

much was 

results are 

defined to

Date

1957

Tnlir 9Q

1974

- July 27 

1976

- July 24 

- Oct. 1

- Nov. 17

1977

Tan 1Q

-Mar. 7

£\\)L, ^O

- July 7

.n.ug. £tij

- Nov. 8

1978 

- Feb. 28

- April 19

- July 30

- Aug. 26 

- Nov. 8

1979 

-Jan. 6

- April 12

- Aug. 18 

- Oct. 20

1980 

- Feb. 29

- May 12

Tnlir 99

QaT»f 9

- Oct. 30

1981 

-Mar. 7

- June 16

- Sept. 1

Flight Total 

Decimal altitude number of 

year (m) Coverage IPP's

1957.574 9,000 a 266 

1974.568 7,920 b 551

1976.561 5,490 c 137 

.750 5,490 c 148 

.879 5,490 c 73

1977.051 5,490 c 68 

.180 5,490 c 64 

.309 5,490 c 154 

.418. 5,490 c 196 

.514 7,010 d 222 

.659 7,010 d 227 

.854 7,010,8,230 d, f 202

1978.160 6,400 d 202 

.297 7,010, 7,770 d, e, f 193 

.442 7,010, 7,770 d, e, f 272 

.576 7,010, 8,530 d, e, f 247 

.650 7,010 d 193 

.853 5,490 d 193

1979.014 6,100 d 169 

.277 7,010 d 175 

.628 7,010 d 151 

.800 7,010 d 162

1980.162 7,010 d 146 

.361 7,010 d 149 

.556 7,010 d 134 

.671 7,010 d 130 

.830 7,010 d 127

1981.180 7,010 d 125 

.457 7,010 d 123 

.667 7,010 d 123

due to inherent error in the IPP altitudes. The 

summarized in table 2. The total departure is 

be the difference, ET=Zlpp(x,y) fa,y). To inZIPP-Z (ZIPP<£) 

Ew=<0 (Z<ZIPP <Z+10) .. (2) 

ZIPP-(Z+10) (ZIPP >Z+10)

vestigate the discrepancy that may be due to interpolation error, both in the digitization and in the interpolation within the 200-meter grid, only those points on the 

wrong side of a contour are considered. That is, if an IPP 

with altitude ZIPP falls between contours Z and Z+10 m, 

its departure Ew is reckoned as follows, regardless of the 

position of the point relative to the two contours:

This is the extreme interpretation of the topography 

between the contours to give the minimum possible 

\EW\; for example, \ET\ may be as great as 10 m when 

Ew=0. The mean square total departure E\ between 

the IPP altitudes and the digitization of the map is
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E7

4 KILOMETERS

FIGURE 3. Surface topography of lower part of Columbia Glacier, 

section of September 1, 1981 (flight 30), map. Surface topography 

in meters above sea level, with 10-meter contour interval, prepared 

photogrammetrically by Air Photo Tech of Anchorage, Alaska 

Approximate scale is 1:50,000. Shown are original contours (light) 

and smoothed contours (heavy).

assumed to consist of two parts: that due to the two 

sources of interpolation error E2 and that due to the actual departure Ep of the IPP altitudes from the true 

surface altitudes. Assuming these two components to 

be statistically independent,

ET2 =EP2 +E2 .

(3)

If the excess of the total departure above the wrongside departure is assumed to be equally divided between interpolation error and further contribution to

Ep, 2 then

2

(4)

so that, from equations 3 and 4:

(5)

Each map, therefore, affords a separate estimate of Ept 

as shown in table 2. An additional assumption underlying this analysis is that an IPP's map position is correct, and that any error it may have is only in its 

altitude. The sensitivity of the ultimate results to the 

uncertainty of the actual value of Ep will be investigated in a later section.

SPATIAL DISTRIBUTION OF 

INDIVIDUAL PHOTOGRAMMETRIC POINTS

If the IPP's are to be used as the source data for interpolating altitudes on the nodes of the square grid, it is 

important that they give good coverage over the domain of the interpolation. The spatial distribution of the 

IPP's is examined in two ways. Shown in figure 7 is a 

partitioning of the studied region into 24 squares, each 

2 km on a side. Table 3 gives the percentage distribution of the points, square by square, for flights 2 

through 30. Consistent with table 1, the total number of 

points falling within the 24 squares generally declines 

with time after flight 9. The nonuniformity of the distribution over the squares follows a pattern with strong 

persistence; for example, square 21 is relatively densely 

covered, and square 14 is relatively sparsely covered, for 

all 22 full-coverage flights (flights 9 through 30). Fortunately, the increase of surface roughness downglacier is 

accompanied by an increase of IPP density downglacier.

Another indication of the quality of the spatial distribution of the IPP's is how well they sample the 7.1-year 

(1974-81) altitude-change field (fig. 5), which is considered separately in two sections. The dashed line in 

figure 7 approximates the position on both maps of the 

zone of steep slope running the full width of the glacier 

at the constriction. The two sections are the region common to both map digitizations, as subdivided by the 

dashed line; they are here termed the central basin 

(above, 36 km2) and the lower reach (below, 69 km2).

Neglecting possible digitization error here, the true 

average altitude change over the 7.1 years is obtained 

by subtracting the 200-meter digitizations of the two 

maps; for the central basis the altitude fell 12.0 m, for 

the lower reach it fell 27.0 m, and for the combined 

region it fell 21.8 m. What is taken here to be the true 

change is determined by about 900 digitization nodes in 

the central basin and by about 1,700 nodes in the lower 

reach, a much greater density than the IPP's. Table 4 

gives for each full-coverage flight, for each section, the 

IPP-sampled 7.1-year change as a fraction of the true 

7.1-year change; this fraction is used to scale the IPPsampled altitude change from July 27,1974, until a particular flight, in order to estimate the true altitude 

change until that flight. There may be two causes of the 

apparently better sampling in the lower reach: (1) the 

five-thirds greater density of points and (2) the greater 

altitude change there, which must be compared with the 

estimated error in the IPP altitudes.

The time profile of the altitude change for each section is shown in figure 8; for both sections the change 

was much greater over the second half of the 7.1-year interval than over the first half. This must be regarded as 

only a zero-order estimate of the altitude change.

STATISTICAL PROPERTIES OF THE 

SURFACE TOPOGRAPHY

Knowledge of the statistical structure of a field variable is important to the specification of an algorithm for 

interpolating among given values of that variable. The
E8 STUDIES OF COLUMBIA GLACIER, ALASKA

4QQOOOmE

2 3 KILOMETERS Base by Air Photo Tech, Inc. n .

from September 1, 1981 air photography, ____________

modified and smoothed by Project

Office, Glaciology, WRD, Tacoma

FIGURE 4. Smoothed September 1,1981 (flight 30), surface topography, in meters above sea level, with 10-meter 

contour interval. The dashed box indicates the section shown as figure 3. Approximate scale is 1:100,000. 

The border of the interpolation domain is dotted.
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E9

4QQOOOmr 4QCOOOm£ 4QQOOOmp

3 KILOMETERS

405000mE

Base by Air Photo Tech, Inc. 0 1 2 

from September 1, 1981 air photography, ' ' ' ' 

modified and smoothed by Project 

Office, Glaciology, WRD, Tacoma.

FIGURE 5. Altitude decrease between July 27,1974, surface topography (fig. 2) and smoothed September 1,1981, 

surface topography (fig. 4), with 10-meter contour interval. Approximate scale is 1:100,000. The border of 

the interpolation domain is dotted.
E10 STUDIES OF COLUMBIA GLACIER, ALASKA

FIGURE 6. Photograph taken on October 8,1975, showing the roughness of the glacier surface. The major crevasse valleys seen here are about 

20 m deep and are spaced about 50 m apart. The location of this picture is shown on figure 2. U.S. Geological Survey photograph by 

L. R. Mayo.

TABLE 2. Estimated error distribution of individual photogrammetric point altitudes

[Errors: Between all n points and map digitization (EJf, between digitization and only points on wrong side of a contour (Ew), due to digitization interpolation (E^, and between all points

and true surface topography (E ); see eq. 2-5]

July 27, 1974 

(n=216)

September 1, 1981 

<n=120)

Minimum (m) 16

Mean (m) 

Maximum (m) 

Mean square (m ) 

Root-mean-square (m)

-0.9 

30 

16.8 

4.1

-12

-0.1

27 

5.3 

2.3

5.8 

2.4

11.0 

3.3

-9

0.7 

23 

19.4

4.4

-6 

0.4

17 

6.8 

2.6

6.3 

2.5

13.1 

3.6
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 Ell

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FIGURE 7. Spatial classification of individual photogrammetric points (IPP's). The twenty-four 2-kilometer squares are the 

basis for the distribution statistics given in table 3. The six lettered crosses (A, B, C, D, E, F) are the study points for investigating the statistical structure of the surface topography. The small circles are the IPP's from flight 11 (at the beginning of the principal data year on August 29, 1977); the solid circles are odd-numbered points, and the open circles are 

even-numbered points. The dashed line separates the central basin (above) and the lower reach (below). Also shown are some 

of the rows (I) and columns (J) of the 762.5-meter data grid. The dotted border shows the region common to the digitizations of the two maps (figs. 2, 4).
M

h->

to

TABLE 3. Spatial distribution of individual photogrammetric points (IPP's) over the 24 squares shown in figure 4

[The total refers only to those points falling within one or another of the 24 squares; because of rounding, the sum of the percentages may not be 100. The average percentage value is 4.2]

Total 

Flight number of IPP's

2 

3 

4 5 

6 

7 

8 9 

10 11 12 13 14 15 

16 

17 18 19 

20 21 22 

23 24 25 26 27 28 29 30 

205

119130

56

4948

108169187

192

175172

161

159157168166

155157133143

129127119118117115110112

Average number per 

square

8.55.05.42.32.0

2.0

4.57.07.8

8.0

7.37.26.76.66.5

7.0

6.96.56.5

5.5

6.0

5.4

5.35.04.94.94.84.64.7

Square-by-square distribution of total number of IPP's, in percent

1

1

00000

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SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E13

interpolation method adopted here is that known as 

"optimum interpolation" (Gandin, 1963). Because the 

time rate of change of the surface altitude has profound 

glacier-dynamics implications, through the equation of 

continuity, the interpolation is conducted not only 

within the x,y domain but also within the t domain. The 

salient statistical representation to be obtained is the 

correlation function, which states how well the altitude 

at some point correlates with the altitude at some other 

point; the method of optimum interpolation has these 

correlations as its coefficients. A further consideration 

is whether the same function applies regardless of where 

within the region the two points are and regardless of 

the relative orientation of the two points. These two 

properties, respectively, homogeneity and isotropy, are 

usually associated with the x,y domain, but they also 

have analogs in the t domain.

A PRELIMINARY INTERPOLATION

Optimum interpolation cannot be applied until after a 

correlation function is assumed. Because the original 

data occur irregularly in space and time (table 1, fig. 7), 

a preliminary interpolation scheme is necessary for investigating the statistical structure of the surface 

topography, from which the correlation function can be 

deduced. The scheme devised here uses an isotropic 

inverse-square-distance interpolation among the residuals from a second-degree polynomial that best fits, in 

the sense of least squares, all of the IPP's for a particular flight, for either the lower reach or the central 

basin. That is, first the polynomial

(6)

(7)

minimizing

is formed, where the summation is over all re of the 

IPP's for that flight for that region. Then, to interpolate 

the altitude Z*(x0,y0) at some arbitrary point, the residuals at the three nearest points are averaged by weighting each by the reciprocal of the square of the distance 

from (*0,;y0):

L(Z(x,y)-P(x,y)] I [(x-x^+jyl,\l[(x-x/+(y-y/\

where the summations are here over the three points 

nearest (x0,y0).

Two different variations of this scheme were considered. Polynomials of degree one and degree zero were 

also used, and the averaging of residuals was also done 

for all points within each of several fixed radii of influence ranging from 500 to 3,000 m. These variations 

were tested by applying them to those nodes of the 

762.5-meter grid for which values could be determined 

from the digitizations of the two maps. For the "nearest 

three" averaging rule, the second-degree polynomial 

agreed better with the digitized values than the firstdegree, which agreed better than the zero-degree. For 

the second-degree polynomial, the "nearest three" 

averaging rule gave results better than most of, but not 

all of, the several influence radii tested. The use of a 

restrictively small radius of influence along with the 

zero-degree polynomial, on the other hand, often produced very poor results.

Table 5 shows the result of using the "nearest three" 

rule with the second-degree polynominal for each of the 

two maps, for the lower reach and for the central basin. 

Except for the case of the central basin on September 1, 

1981, for which 25 interpolations were made from only 

21 IPP's whose distribution is highly skewed, the 

residual-averaging substantially reduced the error.

CORRELATION CALCULATIONS 

AND COMPARISONS

The statistical structure of the surface topography is 

investigated by using the correlation between the altitude at some point (0,0,t+r) and the altitude at some 

other point (d,6,t):

r(d,6,T)=

f (9)

where (d,6) are polar coordinates in the x,y plane and t is 

time. Time is defined in terms of decimal years, with 

t= 1978.000 at 0000 hours on January 1, 1978, and increases by 1/365.2422 for each day thereafter. The summations are over as many flights as possible, subject to 

the constraints

t> 1977.418 

t+r< 1981.667 (10)

and the bar denotes averaging over those same flights; 

when T=0, the sumations are over all 22 flights (flights 

9 through 30). Equations 6-8 are used to interpolate the 

altitudes at each of the six study points A, B, C, D, E, F 

(fig. 7) and at each of 56 points surrounding each study 

point. The surrounding points consist of all combina-
E14 STUDIES OF COLUMBIA GLACIER, ALASKA

TABLE 4. Estimated average altitude changes from July 27, 1974, until the date of the indicated flight (table 1) for the lower reach and the

lower part of the central basin

[All change values are in meters, and all are negative, indicating a falling surface]

Flight

9 .

10 .

11 .

12 .

13 .

14 .

15

16

17

18

19 .

20

21

22

23 -

24

25 -

26

27 -

28

29 -

30 .

Number of 

sample 

points

Lower

reach

. 135

. 143

. 143

. 129

. 134

. 132

. 119

. 124

. 131

. 128

. __ __ __ _ 114

. 117

. 104

. 114

. 106

. 107

. 100

. 100

. 99

. 99

. 96

. ge

Central

basin

41

53

57

53

47

37

42

40

41

46

42

42

33

35

26

29

26

24

23

24

20

22

Ratio of sampled 

7.1-year change to 

true 7.1-year change

Lower

reach

1.05

1.04

1.03

1.04

1.04

1.03

1.01

1.02

1.02

1.00

1.01

1.01

1.02

1.01

.99

.98

.98

.98

.98

1.00

1.00

.97

Central

basin

1.22

1.16

1.10

1.12

1.08

1.09

1.09

1.11

1.14

1.09

1.07

1.06

1.07

1.05

1.11

1.05

1.11

1.15

1.15

1.23

1.23

1.22

Sampled change 

from July 27, 1974, 

until flight date

Lower

reach

3.4

5.3

8.9

14.5

11.7

11.6

9.7

13.2

15.2

17.4

16.4

15.0

18.5

21.3

21.5

20.5

22.6

25.3

22.7

17.6

17.9

26.3

Central

basin

0.9

0.6

3.2

5.6

1.9

2.3

1.8

5.6

6.5

5.5

6.7

6.0

8.1

10.2

8.3

9.1

9.4

10.8

11.2

9.2

12.8

14.6

Scaled change 

from July 27, 1974, 

until flight date

Lower

reach

3.3

5.1

8.7

14.0

11.2

11.2

9.6

13.0

14.9

17.4

16.3

14.9

18.1

21.1

21.7

20.8

23.1

25.8

23.2

17.5

17.9

27.0

Central

basin

.8

.5

2.9

5.0

1.8

2.1

1.6

5.0

5.7

5.0

6.2

5.6

7.6

9.7

7.5

8.6

8.5

9.4

9.7

7.5

10.4

12.0

Scaled change as 

fraction of total 

7.1 -year change

Lower

reach

.12

.19

.32

.52

.42

.41

.35

.48

.55

.64

.60

.55

.67

.78

.80

.77

.85

.95

.86

.65

.66

1.00

Central

basin

.06

.04

.24

.42

.15

.18

.14

.42

.48

.42

.52

.47

.64

.81

.63

.72

.71

.79

.81

.63

.87

1.00

LU 

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nr -.qn

27 JULY 1974

* I i i - i i I I

\\/^\ .

\ *' S^X,

 \^ .^ 9 »x^ .S \^

~ \ /*\ * *~~"* ~~

\ /*~* \ NcB

 \

\ ^9\

V \ .-_

x~x / \ -

V \

 LR

I I I I l I I

1974.0 1976.0 1978.0 1980.0 1982.0

FIGURE 8. Time profiles of the surface altitude change since July 27,1974, averaged over the lower reach (LR) and over the central basin (CB).
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E15

tions of d=200, 400, ..., 1,400 m and 0=E, NE, N, ... , 

SE. For each study point and for each time lag r, the 

correlation coefficient is averaged over direction for 

each distance

(11)

and is averaged over distance for each direction

(12)

The curves in figure 9 are TB (d,r) versus d for the six 

study points and for T= 1.0, 0.5, 0, 0.5, and 1.0 a 

Similarly, the curves in figure 10 are Td (6,7) versus 6. For 

each study point, when r¥=0, the values of Z(0,0,£+r) are 

interpolated from a hand-drawn curve through the 

altitudes for each of the 22 flights for that study point; 

the values of t used in equation 9 are those flight dates 

that satisfy the associated constraints (relation 10), so 

that between-flight interpolation need not be conducted 

for any of the surrounding points. The flights that can 

be used with T= 1.0, 0.5, 0, 0.5, and 1.0 a are, respectively, 15-30, 13-30, 9-30, 9-27, and 9-25 (table 1).

The question of spatial homogeneity can be examined 

by comparing, in either figure 9 or figure 10, the six 

curves occurring vertically for any one of the five time 

lags T. In the case of perfect homogeneity, the curves for

TABLE 5. Result of interpolating at a point using a regional seconddegree polynomial corrected by averaging the residuals of the 

three individual photogrammetric points (IPP's) nearest that 

point

July 27, 1974 

(flight 2)

September 1,1981 

(flight 30)

Lower 

reach

Central 

basin

Lower 

reach

Central 

basin

Number of IPP's

Number of interpolation 

points (at nodes of the 

762.5-meter data grid) 

E2 , root-mean-square error 

at interpolation points 

using only the second 

degree polynomial fitted 

to IPP's (m)

Ey root-mean-square error 

at interpolation points 

when "nearest three" 

residual averaging used 

with second degree polynomial (m)

176

107

35

25

101

103

21

25

13.3 14.5 12.9 9.8

6.4 11.4 8.1 10.3

the six different study points would be identical. The 

most substantial departure from this condition is that 

the curves for E and F for T= ±0.5 a, and the curves for 

A for all T, are markedly depressed below the average.

The question of spatial isotropy can be examined by 

considering the variation with direction within each of 

the curves of figure 10. In the case of perfect isotropy, 

each curve would be a horizontal line segment; perfect 

spatial homogeneity would further require that for any T 

the curves would all have the same rvalue for all six 

study points.

The apparent departure from perfect isotropy is not 

reduced if the flow of the glacier is considered. The open 

circles in figure 10 represent distance-averaged correlations from equations 9 and 12, but the Z(d,6,t) values 

used in equation 9 are interpolated by equation 8 from 

altitudes at IPP's that have been allowed to move with 

the glacier flow from time t until time t+r. Because the 

y-component of velocity v is much greater than the 

^-component (Fountain, 1982), only the former is used 

in advecting the points; that is,

where

Z(x,y)=Z(x,y),

t+T

y=y+\v[x,y(t)}dt

(13)

(14)

is used instead of Z(x,y) in equation 8. Because y=y 

when 7=0, and because few velocity data are available 

for the central basin, circles do not accompany the 

curves in figure 10 either for 7=0 or for study point A.

The question of temporal isotropy can be examined 

by comparing the curves (for any study point in either 

figure 9 or figure 10) for T= 0.5 a and 7=+0.5 a, and 

for T= 1.0 a and 7=+1.0 a There is a high degree of 

left-right (negative T, positive T) symmetry in both 

figure 9 and figure 10.

Although the ideals of homogeneity and isotropy are 

not unambiguously demonstrated by the curves of 

figures 9 and 10, neither is any other pattern of correlation unambiguously demonstrated by them. Possible 

causes for the discrepancies from the ideals include: 

the n used in equation 9 is determined by the number 

of flights satisfying condition (relation 10), which 

ranges from n=22 for 7=0 down to n=lQ for T= 1.0 

a; the Z(0,0,£+r) values used in equation 9 were interpolated visually from curves drawn by hand through 

the Z(0,0,tL ) values at flight dates tL; correlations at 

study point A are relatively more strongly contaminated by the altitude errors in the IPP's than are 

the study points in the lower reach, where the altitude 

changes are more than twice as great as they are in the 

central basin. The small number of realizations (that is, 

22 or fewer flights, depending on the value of T)
E16 STUDIES OF COLUMBIA GLACIER, ALASKA

0.5

-0.5 L 

+ 1.0

r +0.5

r +0.5

0 

1.0r

r +0.5

0 

+ 1.0r

+0.5

-0.5 

+ 1.0r

+0.5

-0.5L

J____I

0 1000

T=-1.0a

0 1000 0 1000 0 1000

DISTANCE, IN METERS

r=0a r=0.5a

0 1000

r=1.0a

FIGURE 9. The function r6(d,r) versus d for each study point (A,... ,F) and for each of the five indicated time lags r 

(see eq. 9, 11). Although continuous curves are shown, re(d,r) is known only at the eight d-values 0, 200, 400, 

...,1,400 m.
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E17

I r

i.0r

+ 0.5

-0.5 L 

+ 1.0r

r +0.5

0 

+ 1.0

r +0.5

0 L 

+ 1.0

r +0.5

0 L 

1.0 r

+ 0.5

OO

-0.5 

+ 1.0r

+ 0.5 

0

-0.5

i i >

B

ENWSE ENWSE ENWSE ENWSE ENWSE

r=-l.0a r = -o.5a r=Qa r=0.5a r=1.0a

FIGURE 10. The function rJ(6,T) versus 6 for each study point (A, ..., F) and for each of the five indicated time lags T 

(see eq. 9, 12); although continuous curves are shown, rJ(6,T) is known only at the eight 0-values E, NE, N, 

..., SE. The open circles indicate values obtained by using individual photogrammetric points allowed to move 

with the glacier flow (eq. 13, 14).
E18 STUDIES OF COLUMBIA GLACIER, ALASKA

precluded investigation of temporal homogeneity, 

which is usually termed "stationarity." Because no 

other pattern is unambiguously demonstrated by the 

curves of figures 9 and 10, the assumptions of 

homogeneity and isotropy are made, both spatially 

and temporally, including the type of isotropy that, 

surprisingly, permits neglecting the effect of the 

glacier flow when 7=^0.

FIRST APPROXIMATION TO THE 

INTERPOLATED ALTITUDES

Because optimum interpolation can be applied better 

to deviations from a norm than to the raw values of the 

variable themselves (Gandin, 1963, p. 86-92), an interpolation scheme is adopted to provide that norm. The 

correlation function then expresses the correlation of the 

deviations from this norm as a function simply, because 

of the assumptions of homogeneity and isotropy, of the 

horizontal distance d=[(^c1 x^+iy^y^2]1'2 between two 

points Z(xl,yl,t^ and Z(x2,y2,t2) and of their time difference 7=1^ £2 |. As a consequence of adopting this 

interpolation scheme, two decisions concerning the domain of the interpolation are made: (1) altitudes are interpolated only for the times of the flights, although there is 

no theoretical bar to interpolating altitudes at any arbitrary time, and (2) a single procedure is applied to the 

entire combined region of the lower reach and the central 

basin, thereby avoiding the introduction of a discontinuity between these two regions, but at the risk of increasing the interpolation error.

For each flight, the norm field fL(x,y) is obtained by 

taking the altitude change from flight 2 until flight L to 

be a linear function of the change from flight 2 until flight 

30:

or (15)

in which the coefficients are estimated by minimizing

E12 Ef~ mT rj (16)

where the sum is over all mL IPP's for that flight having 

(x,y) within the domains of the digitizations of the two 

maps.

A split sample analysis of the mL points is used to estimate the root-mean-square error in approximating the 

true topography by the norm field fL (x,y). The assumption is made that if the mL points are a good sample of 

the true topography, subject to their inherent altitude 

error, then a randomly chosen large subset of them is 

also a good sample. Two subsets of the points are used, 

the odd-numbered ones and the even-numbered ones, 

taking them in the order in which they appear in Fountain (1982). The ^-distributions of the two subsets are 

shown in figure 7 for flight 11, which is typical of these 

distributions for the other flights. First, (OX,)ODD an<^ 

(6L )ODD are obtained by using only the odd-numbered 

points in equations 15 and 16; then I^^VEN *s computed 

by using (O^DD and (&L )ODD in the application of equation 16 to only the even-numbered points. Second, 

(Ey)ODD is computed by applying equation 16 to only the

odd-numbered points, using faL)EVEN anc^ ^L^EVEN °k~ 

tained by applying equations 15 and 16 to only the evennumbered points. Third, (aj)^^ and (bj)^^ are obtained 

by using all mL points in equations 15 and 16; then 

^ALL *s computed by using those coefficients in the applications of equation 16 to all mL points. Table 6 gives 

mL, all three sets of coefficients, and all three values of 

EF. Although flights 3-8 only partially cover the region 

(table 3), and therefore are not used in constructing the 

correlation function, they are amenable to the application of equations 15 and 16 and are therefore included in 

these results.

Another interpretation of Ef is as an indication of the 

variance of the data's deviations about the norm. 

Although there is a pronounced spatial distribution of 

the total altitude change between the times of the two 

maps (fig. 5), with the magnitude of the change generally decreasing upglacier, a second-degree polynomial 

(eq. 6) very poorly represents the ^-distribution of the 

squares of the deviations from fL(x,y). The goodness of 

fit (r2) of the polynomial is also given in table 6; its low 

values support the reasonableness of the assumption of 

spatial homogeneity of the variance.

Because the IPP altitude errors are assumed to be uncorrelated with the altitudes themselves, the variance V 

of the altitudes about the norm field is estimated as the 

excess of the average E2 over E2 . The split-sample 

values are averaged because they are independent estimates, in that points used to compute E. are separate 

from those used to form the norm-field coefficients, 

whereas (Ef)ALL is computed from the same points used 

to form the coefficients. Taking E2 =12 m2, the result is 

also coincidentally V=12 m2 , compared with 11 m2 if 

the (E)ALL were used. The variance is assumed to be constant in time as well as in space, and the flight-to-flight 

variation of Ef is interpreted to be random-sampling 

variation. If a separate V were to be associated with 

each flight, and if each were to be obtained by subtracting E2 = 12 m2 from Er 2 then V would vary from flight to 

flight between extreme values of zero for flight 10 to
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E19

TABLE 6. The coefficients aL and bL used in equation 15 to interpolate the norm field fL (x,y) for flight L from the digitizations of the two

maps f2 (x, y) and f30 (x, y)

[The coefficients are estimated from all mL points (ALL), from only the odd-numbered points (ODD), and from only the even-numbered points (EVEN). The error is estimated by using (o^, 

6i*ODD wit'1 only the even-numbered points to give (#JEVEN. by using (o^, ^(gyj^ with only the odd-numbered points to give (EJODD, and by using (a^, bj)^^ with all mL points to 

give USJ^LL- The goodness of fit (Fit) is for a second-degree polynomial P2(x,y) fit to the squares of the deviations of all mL points from fL(x,y]

Flight mL ODD EVEN ALL .Ey(m) Fit

aL bL aL bL aL bL EVEN ODD ALL

5 51 1>007 H6 £72

6 48 0.439 -4.2 .798

8

1 AA K.d A d O A *7O

10 196 .342 4.3 .366

1O 1 QO f7f\£i A FJ Ti A

13 181 .744 89 785

1 A 1£Q CO'7 *7C nn A

16 164 .578 2.7 .617

17 1 7O C.CC. O A £ff\ A

19 156 .906 7.0 .691

9ft 1 K.Q £KQ O 7 COX

99 1XO QQ1 O 1 7Q1

OO 1OO 1 A1 O KC 1 AAO

OC 1 *>/» ^TKQ If! Or K

97 1 OO 77i? 1 ^7 rrrro

1.4 0.063 0.7 4.62 3.41 4.03 0.02 

8.5 .487 8.1 4.42 3.81 4.11 .21 

11.0 .767 9.8 4.77 4.62 4.66 .21 

10.3 .906 9.9 7.52 8.67 7.58 .32

7.6 .602 1.2 6.31 8.68 6.80 .34 

1.7 .598 1.7 4.68 5.38 4.70 .27 

3.8 .520 5.0 4.92 5.40 5.04 .19 

4.0 .360 6.2 3.94 4.57 3.95 .12 

4.9 .353 4.5 3.45 3.04 3.24 .04

3.1 .458 3.5 3.60 4.80 4.22 .08 

5.2 .710 5.0 6.57 6.98 6.77 .28 

9.7 .763 9.3 6.30 6.77 6.53 .16 

9.5 .725 8.3 5.82 6.81 6.29 .11 

7.7 .645 7.6 4.51 4.60 4.47 .02

3.4 .594 3.0 4.57 3.73 4.16 .02 

1.5 .635 2.2 3.87 4.02 3.91 .05 

6.8 .908 7.0 5.27 4.63 4.90 .10 

2.7 .801 4.9 5.45 5.43 5.08 .11 

2.3 .650 2.6 5.24 4.60 4.92 .02

-0.2 .601 -1.6 4.10 4.30 4.07 .07 

0.6 .846 1.5 3.90 4.37 4.01 .09 

5.3 1.008 5.4 5.17 5.14 5.15 .08 

1.6 0.808 1.1 4.34 4.18 4.19 .05

0.3 .801 -0.9 5.00 4.12 4.53 .01 

1.9 .906 -0.7 4.38 4.90 4.44 .01

o n nnn in ^01 A nc A to no

28 123 .369 -6.1 .514 -4.0 .425 -5.4 5.89 5.50 5.41 .18

29 116 .349 -8.2 .443 -6.1 .385 -7.5 5.41 4.85 5.09 .12

OA i i O o AO o t &f\e r» t Q*7f* OK 4 AQ O 41 O 7K f\A

34 m2 for flight 12. Part of the variation of Ef may be 

caused by variation in Ep, but estimates of that quantity are unavailable for flights 3 through 29. If a variation of equation 15 were used, one that lacked the constant term bL, then the variance would be 14 m2 and the 

deviations about the norm field would have a nonzero 

mean.

THE CORRELATION FUNCTION

The empirical function r*(r,d) gives the correlation 

between deviations from the norm field fL(x,y) at different points and different times. Following from the 

assumptions of homogeneity and isotropy, the function 

depends only on the horizontal distance d between the 

two points and on the time difference r. 

For T=0, any of the flights can be used with itself to 

estimate the d-dependence r*(0,d)- For r^O, the time 

distribution of the flights (table 1) permits investigating

r*(r,d) for only selected T. Table 7 gives the value of r* obtained, after the fashion of equation 9, by considering all 

the deviations

AZL(^y,)=ZL(^yfe )-/L(^yfe ) (17)

falling within each of seven different r-intervals and 

within each of ten different d-intervals. These are calculated directly from the deviations, without being 

vitiated by the crude interpolation necessary to get the 

Z(0,0,t+r) values between flight dates for computing 

r(d,6,r) at the six study points. 

The empirical r*(r,d) distribution of table 7 has several 

prominent characteristics: (1) it indicates a moderately 

well behaved function of d and T, (2) it approaches unity 

as d and T both go to zero, (3) for all T, it approaches zero 

for large d, and (4) for small d, it remains positive for all 

T. The x,y,t density of the IPP's is high enough that 

several points can easily be obtained for interpolation if
E20 STUDIES OF COLUMBIA GLACIER, ALASKA

TABLE 7. Values of the correlation coefficient calculated from the deviations from the norm fields for the indicated time difference (T) intervals and the indicated horizontal distance (d) intervals

[The value of the correlation coefficient is shown above, and the numbers of irregularly positioned photogrammetric points falling within the particular r-interval and d-interval is shown 

below. The pairs of flights having the particular time difference are also shown. The correlation for r=d=0 is assumed to be unity]

T-interval 

(a)

0 ________

.241 

to .283

to .534

to .796 

dan

to 1.024 

1.136 

to 1.202 

1.295 

to 1.358

Flights

 (9,9), (10,10), ...

(30,30)

 (9, 11), (13, 15), (14, 16), 

(16,18),(19,20),(25,27), 

(28,29)

(21,23),(23,26),(26,28), 

(28,30)

(18,21),(19,22),(22,25), 

(26,29)

 (9, 15), (11,17), (12, 18), 

(14,20),(17,21),(23,28), 

(26,30)

(12 19) (15 21) (17 22)

(19,23),(21,27)

(15,22),(18,23),(19,24), 

(23,29),(24,30)

d-interval (m)

0-40

(1.000) 

1

.700 

168

.409 

108

.167 

68

.304 

55

.417 

61

.481 

34

180-220

8

.717 

168

.401 

66

.206 

113

.293 

117

.624 

142

.478 

129

380-420

.622 

151

.473 

95

.396 

224

.301 

188

.197 

172

.346 

199

.338 

188

580-620

.574 

448

.436 

298

.302

287

.244 

299

.389 

298

.402 

393

.218 

326

780-820

.413 

479

.367 

359

.208 

273

.281 

379

.176 

422

.293 

464

.191 

392

980-1,020

.380 

937

.309 

541

.219 

451

.243 

532

.251 

461

.268 

579

.103 

505

1,480-1,520

.238 

1,060

.196 

660

.173 

546

.071 

641

.204 

701

.135

848

.169 

665

1,980-2,020

.139 

1,109

.049

758

.108 

711

.043

773

.074 

770

-.012 

904

.052 

706

2,480-2,520

.043 

1,300

.045 

760

.014 

739

-.001

844

-.047 

789

-.045 

996

-.023 

884

2,980-3,020

-.115 

1,301

-.003 

850

-.088 

764

-.074 

919

-.075 

822

-.082 

1029

-.030

877

they are required to be within a d and a T substantially 

smaller than the ranges of those variables represented 

by the empirical data. If the correlation function is 

known perfectly (Gandin, 1963, p. 70-72), the interpolation error decreases as more and more points are 

used, albeit more and more slowly. However, the correlation function here is not known perfectly.

Requiring a point to be within d=l km and 7=0.39 a 

still includes a large number of points, and it also restricts the domain over which r*(r,d) must be approximated functionally to the domain over which the empirical values are themselves better behaved. This domain actually used is twice that large (d<2 km and 

7 < 0.78 a), because the method of optimum interpolation 

requires the correlations between the included points 

themselves, as well as between the included points and 

the point where the interpolation is made; for example, 

two of the included points may each be within a kilometer of the point where the interpolation is made, but 

they may be 2 km away from each other.

Because the correlation function must have a positive 

Fourier transform (Gandin, 1963, p. 38-40), the class of 

functions that can be used for approximating the r*(r,d) 

data is severely restricted. Two analytical functions 

that are widely used (Gandin, 1963; Thiebaux, 1975) are 

the damped cosine and the Gaussian

U8)

The empirical data do not exhibit oscillation of sign, 

which would imply using the damped cosine. Another

function having a positive transform (Oberhettinger, 

1973) is

R(r,d)= a

a'+r2

(19)

which approximates the empirical data much better 

than equation 18 does.

The relative suitabilities of function 18 and function 

19 for approximating the r*(r,d) values of table 7 are 

compared in figure 11. In both, the coefficients are 

chosen to minimize

E2=

30

(20)

in which the summation is over all 30 combinations of 

7=0, 0.27, 0.51, and 0.78 a and d=0, 0.2, 0.4, 0.6, 0.8, 

1.0,1.5, and 2.0 km except for the two smallest d-values 

when 7=0. The minimizing coefficients are a=2.79, 

(8=1.11 in function 18 for Er=0.109, and a=0.470, 

0=0.755 in function 19 for Er=0.069. Not only are the 

residuals produced by function 19 smaller than those 

produced by function 18, but they are also more randomly distributed. Function 18 systematically produced negative residuals for small values of either rord 

and positive residuals for large values; this is true to a 

lesser degree with function 19 as well.

Although the assumption of spatial homogeneity permits determining r as a function of only the distance d 

as well as the time difference 7, the Fourier transform
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E21

must be of the same dimensionality as that of the space 

in which the data points exist. The transform

 ^oo ^ oo ^

in which x and y denote the two components of the 

distance d, may be shown to be positive for all finite, 

positive values of the coordinates 0lt 02, and 03 (appendix A).

THE INTERPOLATION ALGORITHM

Because the present work is a part of a larger effort in 

investigating the behavior of Columbia Glacier, and of 

glaciers generally, rather than pursuing questions in 

data analysis itself, no direct comparison has been made 

between various interpolation methods. Optimum interpolation is widely used in the objective analysis of 

meteorological variables. In making an individual interpolation, it tends toward using all of the available data 

and it provides an estimate of the interpolation error. In 

analyzing rainfall patterns in south-central France, 

Creutin and Obled (1982) compared it with several other 

methods, including the highly similar kriging, and 

found it to be superior, overall. Although it is called optimum interpolation, it extrapolates as well; that is, it is 

not restricted to estimating values at points that are in 

any sense between the given points.

OPTIMUM INTERPOLATION

The interpolated altitude at some point for flight L, 

Z*L(x,y), is obtained by adding the estimated deviation 

there, AZ*, to the norm:

Z*L(x,y)=fL{x,y)+AZ*0 (22)

The method of optimum interpolation (Gandin, 1963) is 

used to obtain AZJJ as a linear combinatin of the deviations AZ at N nearby points selected, in general, both 

from flight L and from other flights close in time to 

flight L:

N

AZ*= (23)

The weights w{ are obtained from a system of simultaneous linear equations that minimizes (and, therefore, 

gives the method its name) the error:

M

El = (AZ;-AZ0 (24)

k=i

in which the summation is over a larger number M of 

applications of the method at some particular point 

(x,y), and AZ0 is the true deviation from fL(x,y) there.

If equation 23 is substituted into equation 24, and if 

the squaring is performed, then

-i M

i E (AZO ) M k=i

N 

-2 E

M

M k=1

N

+ E

N

E

ww. M

. (25)

If AZ is assumed to have zero mean over large M, and if 

its variance is taken to be constant, as suggested by 

table 6, then the first term on the right-hand side of 

equation 25 is the variance of AZ, which is denoted V. If 

equation 25 is divided by V it becomes

N N

E*GIV=l-2 wrio+

N

E (26)

where r. is the correlation between AZ. and AZj, and is 

calculated from equation 19 using the distance d and 

time difference r between the two data points having 

those deviations. Minimizing EG with respect to the w. 

is accomplished by requiring

dEffldw^O (27)

for l<i<N, and solving the resulting linear system, in 

which the r.= l,

N

E W=

7=1

(28)

to obtain the wt. If equations 28 are multiplied by wp 

and if the sum is taken over i, then equation 26 may be 

simplified to

N

E (29)

The condition 0 <E2G < V is always fulfilled if the matrix 

of equation 28 is positive definite, which is ensured
E22 STUDIES OF COLUMBIA GLACIER, ALASKA

(Gandin, 1963, p. 39) if the correlation function used to 

form the TV has a positive Fourier transform.

The foregoing development is for the simplified 

special case in which the AZ{. are known exactly. 

Because they are not, being subject to the IPP altitude 

error given by equation 5, the linear system must be 

modified (Gandin, 1963, p. 79) to

N 

£

.7=1

(30)

for determining the MA, which are then used in equation 

23 to get the interpolant AZ* and in equation 29 to get 

the error estimate EG. Using equation 30 is predicated 

on three assumptions regarding the IPP altitude errors: 

(1) they have zero mean, (2) they are not correlated with 

the true altitude values, and (3) they are not correlated 

with each other. The detailed comparison of the IPP's 

with the two maps, to compute the error Ew of equation 

2, revealed no contradiction of these assumptions.

The presence of the error term in equation 30 has two 

important consequences (Gandin, 1963, p. 80-84). First, 

it increases the interpolation error EG. Second, the value 

of Ep that produces the smallest EG is its true value; 

that is, EG will be increased if a value of Ep is used that 

is either larger or smaller than its true value.

If the errors at the individual points are indeed uncorrelated, or poorly correlated, the resulting interpolation 

error EG can be less than Ep. As the points used tend to 

be farther away from the interpolation point, and the 

correlations ri0 tend to zero, the weights also tend to 

zero, and the interpolated value Z*L(x,y) tends to the 

norm fL(x,y), whether or not the error term is present. 

However, if the interpolation is conducted among the 

values of the variables themselves, instead of among the 

deviations from the norm, this does not occur unless the 

algorithm is modified in a way that increases the error 

(Gandin, 1963, p. 89).

SOME INDEPENDENT TEST DATA

An abundance of ground-truth topographic data was 

obtained for Columbia Glacier by standard surveying 

methods (Mayo and others, 1979), and the data are useful for investigating the dependence on N, and for other 

purposes. Unfortunately, the ground-truth altitudes 

Z(x,y,tL+At) are for dates slightly different from the 

flight dates tL, and the interpolation model obtains 

altitudes only for the flight dates. For use in comparing 

the interpolated values against them, the ground-truth 

altitudes are adjusted to the nearest flight dates; it is 

assumed that the time rate of change of the actual 

altitude at some point (x,y) is equal to the average time

rate of change of the norm field there during the time interval between the two flight dates tL and tL' that 

bracket the date tL+At of the point to be adjusted. That 

is, the adjusted value is given by

, (31)

in which L'=L+At/| At|.

Many surface altitude points given by Mayo and 

others (1979) are for times close to one or another of the 

three flights 11,15, and 17. Because of the practical difficulty created by the extreme roughness of the glacier 

surface, the observations are skewed spatially; within 

the combined region, they are concentrated primarily in 

the central basin and secondarily near the centerline in 

the lower reach (fig. 12). To relieve the skewness in the 

sample of observations selected to compare the interpolation algorithm against, central basin points were required to be nearer a flight date than were lower reach 

points. Another device used for improving the spatial 

uniformity of the sample was the averaging, of all three 

space coordinates, of observations lying within about 

100 m of one another. To permit investigation of the effect of the number of points N used in the interpolation 

equations 30, only those ground-truth points were admitted to the sample that had at least 10 points among 

the IPP's within 1.0 km and 0.39 a. Table 8 gives the 

distribution, by Z and by At, of the 58-point sample 

selected; the greatest equation 31 adjustment was 0.8 

m, and the root-mean-square value was 0.3 m.

The three flight-by-flight sections of the sample are 

highly similar in the distribution of their departures 

Z'L(x,y) fL(x,y) from the norm field. The mean departures for the three sections are, respectively, 2.7, 3.7, 

and 2.3 m; the median departures are 3.1, 4.0, and 

 2.1 m; and the departures are nearly uncorrelated 

with the altitude itself, the three r2 values all being less 

than 0.04. The mean for the entire sample is 2.8 m, the 

median is 3.0 m, and 46 of the 58 departures are 

negative. A systematic discrepancy between the 

photogrammetric data and the surface-survey data may 

be the source of the predominant negativeness of the 

departures.

The error estimate E'G obtained by comparing the adjusted ground-truth altitudes Z'L(x,y) and the altitudes 

Z*(x,y) interpolated by the algorithm is given by

(32)

in which the summation is over the 58 points in the 

ground-truth sample. It is compared in figure 13 with
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 

~~ 1.0

E23

r 0.5

r 0.5

0.5

0.5

r=0.27a

r=0.78a

0 1 2 

DISTANCE, IN KILOMETERS

FIGURE 11. Comparison of best fitting correlation function (eq. 18) with a=2.79, #=1.11 

(dashed line), best fitting correlation function (eq. 19) with a=0.470, 0=0.755 (solid line), 

and empirical r*(r,d) from table 7 (solid circles).

EG as estimated by the algorithm (eq. 29). Each is shown 

as a function of N; for any N, the IPP's having the 

highest correlation with a particular point (x,y,t) 

through equation 19 are the ones used in estimating

Z*(x,y,t). Because neither the variance V of the IPP's 

about the norm field nor their random error Ep is known, 

the sensitivity of both EG and E'G to these two quantities 

is also examined. The actual error E'G depends only on
E24 STUDIES OF COLUMBIA GLACIER, ALASKA

30

25

CO 

DC 

LJJ

H 

LJJ

20

15 -

code

0

1

2

3

4

5

6

0

2

flight(s)

11

15

17

11,15

11,17

15,17

11,15,17

4

5

242

6

16

21

4o° 6

10

x, IN KILOMETERS

FIGURE 12. Spatial distribution of individual and averaged points surveyed from

the surface.

the ratio E]JV because only that quantity appears in 

equation 30, from which the interpolating weights wi are 

obtained. The estimated error EG depends on each of 

them independently, on the ratio in equation 30 and on

V alone in equation 29, and is much more sensitive to 

the values of these algorithm parameters than E'G is.

In the special case in which the employed correlation 

function exactly describes the true statistical structure
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E25

TABLE 8. Distribution of58-point surface-survey sample

[The first number in each table entry is the number of single points, and second is the 

number of averaged points. The time difference between the surface survey and the flight 

date is At, in days]

Z-interval 

(m)

Flight 100-199 200-299 300-399 400-499 500-599

11 +4 

-2 

-4 

-5 

-8 

15 -1 

-2 

-3 

-4 

-5 

3,0

2,0 

1,0

2,2 

0,1

6,1

1,0 

2,1

1,0 

1,0

1,2

2,0

4,0

17 +8 

+7 

+6 

+5 

5,5 4,1

1,2

1,1

2,3

of the variable being interpolated, and the observational 

error Ep and the variance V are both perfectly known, 

the interpolation error EG is a nonincreasing function of 

the number N of data points used in equations 23 and 

28. None of these ideals is fulfilled by the Columbia 

Glacier photogrammetric data. Belousov and others 

(1968, p. 92) found that for interpolating the geopotential height of an isobaric surface in the atmosphere, N 

should be 5 or 6 for dense distributions of points and 

might be increased "to 7 or 8, but no more" for sparse 

distributions.

If a systematic error 8Z in the photogrammetric data 

is assumed to contribute to the discrepancy between the 

two sets of data, and if it were to be subtracted out, then 

the interpolated values would be adjusted to 

Z*L(3c,y) 6Z, the estimated error EG would be unaffected, 

and the sample error would become, for some particular 

N, ^

(33)

58

This adjusted error is shown as a function of N in figure 

14 for selected 6Z, using V=Ep2=I2 m2.

The Z'L(x,y) themselves are subject to two sources of 

error, the surveying itself and the equation 31 adjustment to the flight dates. If the mean square error for 

these two sources combined is taken to be 1 m2, then the 

corrected error, for some N and some 6Z, is

(EG)*=(E'G) 2 -I . (34)

This is also shown as a function of N in figure 14 for the 

6Z=0 case.

THE RESULTS

The optimum interpolation results for flights 3-29 appear in appendix B. The interpolated altitude Z?L(&y) is 

given to the nearest tenth of a meter, and the estimated 

error <EG > +1 is given in meters. Although the number 

of tenths of a meter in Z*L(x,y) is not a significant figure, 

dropping it would lead to an unacceptably large rounding error in forming the average time rate of change of 

surface altitude between flights, especially for short interflight intervals. Also given in appendix B are the 

digitization interpolation results for flights 2 and 30, and 

the estimated digitization error Er (eq. 4, table 2) is 

rounded up to the next whole meter: <Ef> +1=3 m.

actual error E G

 estimated error EG

(12,24),(6,12) 

(12,12) 

(24,12),(12,6)

values in parentheses are 

(V,E2P), both in square meters

FIGURE 13. Error EG estimated by the algorithm (dashed line) and 

actual error E'G (solid line) between algorithm results and altitudes 

surveyed from the glacier surface, as functions of the number N of 

individual photogrammetric points (IPP's) used for interpolating at 

any particular point The sensitivity of these two quantities to the 

variance V of the IPP's about the norm field and to the inherent 

IPP altitude error E* is included for various postulated values for 

those two parameters. The actual error E'G depends only on the 

ratio E2 IV because only that quantity appears in equation 30, from 

which the interpolating weights are obtained. The estimated error 

EG depends on each of the two parameters separately, on the ratio 

in equation 30 and on V alone in equation 29. Although continuous 

curves are shown for clarity, the quantities exist only for integer N.
E26 STUDIES OF COLUMBIA GLACIER, ALASKA

Altitudes are interpolated only for the flight dates tL 

and only on those nodes (I,J) of the 762.5-meter data 

grid, on which velocities are given in Fountain (1982), 

that are common to the domain of the digitizations of the 

two maps. The (x,y) coordinates of the IPP's are related 

to these grid coordinates and to the Universal 

Transverse Mercator (UTM) System according to

/ 

J 762.5

65,648-y 

10,458+x

(35)

and

0.9996

UTM Easting-490,000 

UTM Northing-6,750,000

. (36)

Because the photogrammetric coverage of flights 3-8 

was of only the lower part of the lower reach, the correlation study was conducted only for flights 9-30; however, 

the algorithm based on flights 9-30 was also applied to 

flights 3-8, but only in the vicinity of the photogrammetric coverage.

Because it is so restricted spatially and temporally, 

the ground-truth sample probably provides a good estimate of neither the systematic error in the photogrammetric data nor the actual error EG in the interpolation 

algorithm. Moreover, the error EG is only estimated by 

the algorithm; the correlation function, the variance 

about the norm, and the random error are all imperfectly known. Of several possible ways of reconciling 

the apparent difference between EG and E'G, increasing 

the estimate of EG is probably the safest. The evidence 

from the ground-truth sample is too weak to justify 

instituting a systematic 6Z adjustment of the photogrammetric data, and it is too strong to ignore. As the 

uncertainties in the interpolation model tend to cause 

EG to be underestimated, and as the scale of errors does 

not warrant reporting it more precisely than in whole 

meters, simply rounding up each estimate to the next 

greater integer number of meters is consistent with 

both these considerations. The 58-point root-meansquare of these rounded-up values is shown in figure 15 

as the curve labeled < EG > +1; for comparison, it shows 

the EG curve from figure 13 for V=Ep2=l2 m2 , and it 

shows the EG curve from figure 14 for 6Z=0.

All three curves in figure 15 are decreasing functions 

of N for 0<AT<9. So also is the fraction \(N) of the N 

selected interpolating points that are from the same 

flight as the point being interpolated (fig. 16); it is also 

obtained from the ground-truth sample, in which flights 

9-12 were used for interpolating for flight 11, flights 

13-17 for flight 15, and flights 14-19 for flight 17. The 

advantage of a small \(N) is the heightened flight-toco

DC

LUlLU

Uj

N?

^

_ o

Uj

E G (SZ) 

E"G

8Z=0

8Z=1m

<5Z=2m

5

N

10

FIGURE 14. Effect, on actual error E'G between algorithm results 

and altitudes surveyed from the surface, of some postulated values 

of the possible systematic error 5Z in the photogrammetry data 

(solid line). Also shown is the adjusted error E"G obtained by taking 

6Z=0 but assuming a 1-square-meter mean square error for the 

combined effect of surveying error and error in the equation 31 adjustment to the flight dates (dashed line). Although continuous 

curves are shown for clarity, the quantities exist only for integer 

values of the number N of individual photogrammetric points used 

by the algorithm for interpolating at any particular point.

flight influence on the interpolation; if each flight's data 

were interpolated independently, the inferred surface 

topography would have an erratic time variability.

Finally, the algorithm (eq. 22, 23, 30) is used with 

both the variance V and mean-square random error Ep 

taken to be 12 m2 . It is applied to the N< 10 IPP's 

having the N highest correlations with the point being 

interpolated, provided they are all within a distance 

d<1.0 km and time difference r<0.39 a of that point. 

Equation 19, with a=0.470 a and |3=0.755 km, is used 

to approximate those correlations, as well as the correlations between the N points themselves. The estimated 

error EG is obtained from equation 29 and is reported as
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E27

the next greater integer <EG > + 1. If no IPP meets the 

d and T conditions, then the norm fL(x,y) is used as the 

interpolated altitude, and the error is taken to be 

<VF>+l=4m.

The algorithm-estimated error EG, through equations 

19 and 29, is a decreasing function of the (x,y,t) density 

of the IPP's. The root-mean-square of the rounded value 

<EG > + 1, when averaged over the 154 interpolated 

points, rises from about 2.5 m for the early, 190-IPP 

flights to about 3.1 m for the late, 120-IPP flights. It 

also rises, secondarily, during periods of long interflight 

time intervals and falls during periods of short ones. 

The time-density dependence is weaker than the spacedensity dependence; when the interflight interval is 

large, there are still many r=0 IPP's from the flight of 

the date for which points are interpolated.

DISCUSSION

No analysis of the interpolated topography is contained in this report. The altitudes are given to one more 

decimal place (tenths of a meter) than is significant as 

altitude explicitly. The additional decimal place is included because the data implicitly contain information 

on the altitude change rate, which has profound 

dynamic significance. Hounding the interpolated altitudes to the nearest whole meter would seriously degrade this implicit information.

Minor details in each of the two maps may be overrepresented in the interpolations. The actual surface 

topography Z(x,y,t) may have short-lived, perhaps fastmoving, small-scale features. If so, one of these features 

present at the time of one or another of the maps will be 

projected through the time interval, in the same location but with diminishing intensity, to the time of the 

other map. The norm fields fL(x,y), which constitute the 

basis of the interpolation, are linear combinations of the 

two maps, but the pattern of map-to-map altitude 

changes (fig. 5) exhibits little congruence between the 

minor details of the two maps. For any particular flight 

date, this effect is neutralized, partially, by the interpolation among the IPP deviations from the norm field. 

Because the results include values interpolated only on 

the 762.5-meter grid, this overemphasis is tolerable; 

however, if maps were to be prepared for the intermediate flight dates, or if the interpolation were conducted on a fine grid, the fictional replication of these 

ephemeral minor details would not be tolerable. The 

number of points interpolated on the grid nodes is 

roughly the same as the number of IPP's.

The correlation function R(r,d) does not fit the empirical r*(r,d) exactly, and the ideals of homogeneity and 

isotropy are fulfilled only approximately. These imperfections affect the results, ultimately, as irregularities

CO

DC in 

K ui

* 3

of 

o

DC 

DC 

111

I I I I T

I 1

5

N

10

FIGURE 15. Error EQ estimated by the algorithm using V=E2 = 

12 m2 from figure 13, the error E"G from figure 14, and the error 

(EQ ) + \ formed as the root-mean-square of the algorithm-estimated 

errors for individual points being rounded up to the next whole 

number of meters. Although continuous curves are shown for clarity, the quantities exist for only integer values of the number N of 

individual photogrammetric points used by the algorithm for interpolating at any particular point.

in the weights wt. Generally, unless the data density is 

very low, slight irregularities in the weights produce 

only a slight increase in the actual interpolation error 

(Gandin, 1963, p. 67). The good fit of the norm fields to 

the IPP's, without any interpolation among the 

residuals, would itself yield glaciologically usable data; 

the error for the ground-truth sample was E'G =4.15 m 

when only the norm field (Af=0) was used.

Probably the most surprising result is that the radial 

symmetry of the empirical correlation function is independent of the glacier flow (eq. 13, 14; fig. 10); that is, 

for finite time lag T, the correlation is still isotropic. 

Perhaps the high ratio of seasonal variation to longterm trend (fig. 8) masks the effect of kinematic waves
E28 STUDIES OF COLUMBIA GLACIER, ALASKA

1.0

A (N) 0.5

I I I

5

N

10

FIGURE 16-Fraction \(N) of the number N of individual photogrammetric points used for interpolating that are from the same flight as 

the point being interpolated. These results are from the 58-point 

surfat mdividual "d averaSed P<^ts surveyed from the glacier

on the correlation statistics, or perhaps none were present. The space-time variation of surface velocity (Meier 

and others, 1984) has this same synchronous behavior 

over the same domain, except for very near the calving 

terminus.

APPLICATION TO FUTURE WORK

A few conclusions can be drawn regarding design of 

data-collection programs. If flight-to-flight average 

velocities are required, the flight frequency is bounded 

by the persistence of identifiable natural features, such 

as crevasse patterns or other surface figuring. Because 

the interpolation error rises gently as the IPP density 

declines, the volume of photogrammetric processing for 

any particular flight could probably be reduced, so long 

as the coverage is complete.

Although altitudes are interpolated here only for the 

dates of the photo flights, it is not because IPP's must 

be available for the same time as the time coordinate of 

the point being interpolated; obtaining the correlation 

values for equation 30 from equation 19 does not depend on any of the time lags being zero. Instead, it is 

because the flight dates themselves give a reasonably 

nch time distributon of values, and because the method 

of equations 15 and 16 can be used only for the flight 

dates. However, the time variation (fig. 17) of the coefficients (dubj)^ of table 6 is reasonably weU behaved, so 

that norm fields for times between the flight dates could 

be obtained by performing (yet another) preliminary 

interpolation to get a(t), b(t) for some arbitrary time.

Moreover, if the photogrammetric processing of 

flights later than September 1,1981, is used for producing only IPP's instead of a later map, equations 15 and 

16 could obtain norm fields for those flight dates as extrapolation outside the time interval of the two existing 

maps. As in interpolating between flight dates, the 

algorithm (eq. 22, 23, 30) would then proceed as usual, 

once a norm field is obtained. The suitability of such extrapolated norm fields would not have to be established 

independently; as long as the equation 16 error E2f remained reasonably low, the norm field could be confidently accepted as the basis of the interpolation.

Good ground truth is essential to determining absolute altitude. However, if there were a systematic error, constant in x and y, and if the bed topography were 

inferred from the same data set, then the dynamic 

variables-velocity, glacier thickness, and surface 

slope-would be unaffected by this error. For a calving 

glacier, though, accurate absolute altitudes would be 

necessary to represent the terminus physics properly.

For use in dynamic models, the altitude errors are 

probably less significant than the spatial and temporal 

variations in the crevasse pattern and in the density of 

the snow and firn layers. In determining the normal 

stress at a point on the glacier bed, some account must 

be taken of the compactness and density of the overlying material. Perhaps a mass-equivalent topography 

could be employed, so that the thickness times a constant modeling density would equal the actual 

thickness-compactness-density integral. Whereas this 

topography might properly represent the distribution of 

the normal stress, along with its gradient, it probably 

would not properly represent the distribution of the 

shear stress. However, if this concept is furthered, it 

may be profitable to consider a "dynamically equivalent 

topography" to which the flow law, including an explicit 

representation of longitudinal stress, could be applied 

exactly by using local derivatives, thus avoiding the 

need for a slope-averaging artifice. It is not obvious how 

to do slope averaging in a domain with two horizontal 

space dimensions, but the exact application of such a 

flow law has been successful in time-dependent, twodimensional modeling (Rasmussen and Campbell, 1973) 

of a synthetic data set.

The altitudes interpolated here are accompanied by 

an error estimate. If fields of other variables (velocity, 

bed topography, mass balance, and thickness change) 

were similarly prepared, it may be possible to construct 

a data set that is consistent through the dynamic equations, for some reasonable choice of values for the flowlaw parameters, and that is faithful, within the error 

bounds, to the field data Two partial solutions of this 

problem have already been achieved: Sikonia (1982, 

written commun.) obtained a two-dimensional data set 

that is consistent through the continuity equation, and 

Bindschadler and Rasmussen (1983) obtained a onedimensional data set that is consistent through both the 

continuity equation and the flow law.
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E29

1.0 -

0.8

0.6 -

0.4 -

0.2 -10

1978.0 1979.0 1980.0 1981.0

FIGURE 17. Linear combination coefficients (a^bj)^^ from table 6 for forming the norm fields fL(x,y). They are connected by smooth curves 

(solid line for a, dashed line for b) to indicate that the coefficients may be well enough behaved in time to permit interpolating between 

flight dates, first for the norm field f(x,y,t) and then for surface altitudes

REFERENCES CITED

Belousov, S. L., Gandin, L. S., and Mashkovich, S. A., 1968, Computer processing of meteorological data: Leningrad, Hydrometeorological Publishing House. [English translation, Israel Program for Scientific Translations, Jerusalem, 1971.]

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.

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.

Creutin, J. D., and Obled, C., 1982, Objective analysis and mapping 

techniques for rainfall fields: an objective comparison: Water Resources Research, v. 18, no. 2, p. 413-431.

Dwight, H. B., 1961, Tables of integrals and other mathematical 

data: New York, Macmillan, 336 p.

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

and velocity: data set (1957-1981): U.S. Geological Survey OpenFile Report 82-756,225 p.

Gandin, L. S., 1963, The objective analysis of meteorological fields: 

Leningrad, Hydrometeorological Publishing House. [English 

translation, Israel Program for Scientific Translations, Jerusalem, 1965.]

Kollmeyer, R. C., Motherway, D. L., Robe, R. Q., Plate, 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 yean 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 OpenFile Report 78-264,78 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 disintergration of the lower reach of the 

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, 1984, Photogrammetric determination of surface 

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

Alaska: U.S. Geological Survey Professional Paper 1258-F (in

Oberhettinger, Fritz, 1973, Fourier transforms of distributions and 

their inverses: New York, Academic Press, 167 p.

Post, Austin, 1975, Preliminary hydrography and historic terminal 

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

Hydrologic Investigations Adas 599,3 sheets.

Rasmussen, L. A., and Campbell, W. J., 1973, Comparison of three 

contemporary flow laws in a three-dimensional, time-dependent 

glacier model: Journal of Glaciology, v. 12, no. 66, p. 361-373.

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

of the drastic retreat of Columbia Glacier, Alaska- U.S. Geological Survey Professional Paper 1258-A, 23 p.

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

to Columbia Glacier, Alaska: U.S. Geological Survey Professional 

Paperl258-B,74p.

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.

Thiebaux, H. J., 1975, Experiments with correlation representations for objective analysis: Monthly Weather Review, v. 103, 

p. 617-627.

APPENDIXES A, B

SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E33

The Fourier transform,

oo oo oo

 oo oo ^

APPENDIX A: FOURIER TRANSFORM 

OF THE CORRELATION FUNCTION

4>(02,03)=402

02+x2+y2

(A-l)

is required to be positive for all values ^ > 0 for 7= 1, 2, 3. 

Because, by using Euler's formula, the exponential may 

be written as

sn

[cos (02*+03y)+; sin (02ac+03y)] , (A-2)

and because the coordinates T, x, and y are independent 

of each other, equation A-l may be written

where

cos 0J7+J sin 0

dr

(A-3)

(A-4)

and

 oo oo

cos (02oc+03y)+i sin

dxdy. (A-5) |32 +*2 +y2

In equation A-4, because the denominator is an even 

function of T and sin 0jT is an odd function of r, the imaginary part vanishes; and, because cos 0^ is also an 

even function of T, the real part is (Dwight, 1961, formula 859.001)

cos 0:7 , _

C?+T2

(A-6)

Similarly, in equation A-5, the imaginary part 

vanishes, and the real part may be written as

cos (02oc+03y)

dxdy (A-7)

0 0

and depends only on the quantity 0|3=(02+03). which is 

generalized by setting 02=02(3 cos * and 03=02>3 sin *:

o o

cos cos sn ,, (A g

If the substitutions are made,

cos

 sn

sn

cos

(A-9)

from which jc 2+y2=j:2+y2 and dxdy=dxdy, equation 

A-8 becomes (Dwight, 1961, formulas 120.01, 859.041):

' dydx

cos(023Jc)

cos(023Jc)

tan"

(A-10)

where J^ is the zero-order modified Bessel function of 

the second kind, which is a positive function ranging 

from +°° for zero argument toward zero for large 

argument. 

Finally, combining equations A-3, A-6, and A-10,

$(01 ,02,03)=27r2a02<T*iQ' KQ(0 V0f+03r) , (A-l 1) 

which is positive for all values of the 0^..
E34 STUDIES OF COLUMBIA GLACIER, ALASKA 

APPENDIX B: COLUMBIA GLACIER SURFACE ALTITUDES

Given in the following grids are results for flights 2 

through 30 (table 1). For flights 3 through 29 the altitudes are the results of optimum interpolation, but for 

flights 2 and 30 they are from the digitizations of the 

topographic maps for the dates of those two flights. The 

first figure given is altitude, in meters above sea level

(Z* in the case of interpolated altitudes); the figure after 

the solidus (/) is the estimated error, in meters 

( <EG +1 > in the case of interpolated altitudes). The column numbers (/) of the 762.5-meter data grid are given 

across the top of the array of altitudes, and the row 

numbers (/) are given at the left.
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E35

	Flight 2, July 27, 1974 (1974.568)

	19 20 21 22 23 24 25 26 2? 

4? 532.2/3 527.4/3 529.3/3 551.5/3 569.8/3 565.6/3 561.7/3 0/0 0/0

48 531.0/3 521.9/3 525.0/3 540.2/3 556.4/3 543.5/3 533.5/3 0/0 0/0

49 529.8/3 512.1/3 511.4/3 525.9/3 526.3/3 524.2/3 527.6/3 0/0 0/0

50 515.5/3 492.6/3 493-5/3 499.9/3 513.6/3 527.1/3 532.7/3 0/0 0/0

51 503.3/3 494.4/3 486.0/3 496.3/3 531.1/3 538.3/3 537.2/3 0/0 0/0

52 481.0/3 485.9/3 480.1/3 483.7/3 495.7/3 522.6/3 533.8/3 0/0 0/0

53 483.6/3 473-5/3 463.6/3 437.1/3 404.3/3 0/0 0/0 0/0 0/0

54 0/0 377.0/3 404.3/3 412.6/3 394.5/3 383.6/3 0/0 0/0 0/0

55 343.1/3 358.9/3 388.2/3 406.8/3 401.7/3 379.0/3 0/0 0/0 0/0

56 341.0/3 360.0/3 376.6/3 385.7/3 374.3/3 343.6/3 0/0 0/0 0/0

57 330.4/3 341.2/3 350.2/3 323-9/3 308.2/3 299-0/3 285-0/3 0/0 0/0

58 322.6/3 334.0/3 333-8/3 322.6/3 300.3/3 296.6/3 292-0/3 0/0 0/0

59 299-7/3 309-3/3 302.5/3 297-0/3 288.0/3 288.2/3 271-7/3 244.6/3 209-0/3

60 278.0/3 288.5/3 293-3/3 271.4/3 270.6/3 282.6/3 256.8/3 249.1/3 231.2/3

61 0/0 256.8/3 267.9/3 258.5/3 271.0/3 273-3/3 246.6/3 234.7/3 0/0

62 0/0 0/0 255.5/3 258.0/3 254.9/3 250.2/3 240.9/3 224.1/3 0/0

63 0/0 210.5/3 235.7/3 249.5/3 237-6/3 236.4/3 232.2/3 216.4/3 188.4/3

64 0/0 202.9/3 218.3/3 209.6/3 212.6/3 226.6/3 215.9/3 197.1/3 173-0/3

65 0/0 0/0 192.3/3 192.8/3 204.5/3 216.8/3 206.2/3 192.9/3 169-9/3

66 0/0 0/0 176.6/3 185-8/3 192.7/3 183.6/3 184.8/3 171.2/3 154.3/3

67 0/0 0/0 139-3/3 181.1/3 178.2/3 167.4/3 170.9/3 165.0/3 122.5/3

68 0/0 0/0 0/0 135-7/3 146.8/3 140.2/3 142.5/3 132.9/3 91.8/3

69 0/0 0/0 0/0 136.7/3 133-5/3 0/0 0/0 0/0 0/0
E36 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 3, July 24, 1976 (1976.561)

19 20 21 22 23 2*1- 25 26 27

61 0/0 253.7/3 263.4/2 254.1/2 267.9/3 270.2/4 243.2/4 232.4/4 0/0

62 0/0 0/0 254.7/2 255,5/2 253.2/2 248.6/2 238.3/3 222.2/4 0/0

63 0/0 207.0/2 234.9/2 248.4/2 233-7/2 232.2/2 229.9/2 216.4/3 186.0/4

64 0/0 204.2/2 217.2/2 205.5/2 209.4/2 224.4/2 215.3/2 198.2/2 172.3/3

65 0/0 0/0 190.3/2 188.7/2 199.4/2 213-3/2 202.4/2 192.6/2 168.2/3

66 0/0 0/0 173-9/2 179.4/2 184.3/2 177.9/2 178.7/2 169.9/2 149.9/4

67 0/0 0/0 128.8/2 172.8/2 168.4/2 161.8/2 159-9/2 159-5/2 118.3/4

68 0/0 0/0 0/0 122.4/2 137-9/2 131-8/2 131-7/2 123-3/2 81.1/4

69 0/0 0/0 0/0 119-1/3 114.2/3 0/0 0/0 0/0 0/0
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E37

Flight 4, October 1,1976 (1976.750)

19 20 21 22 23 24 25 26 27

61 0/0 248.1/4 257.8/2 248.8/2 262.1/3 263.9/3 237.5/4 227.8/4 0/0

62 0/0 0/0 249.1/2 249.7/2 246.8/2 242.4/2 232.3/3 217.2/3 0/0

63 0/0 202.8/2 229.4/2 242.4/2 228.4/2 226.3/2 223.5/2 210.8/2 180.1/4

64 0/0 201.5/2 211.6/2 198.5/2 205.3/2 219.5/2 209.7/2 192.4/2 166.8/3

65 0/0 0/0 184.4/2 182.0/2 192.9/2 206.7/2 196.5/2 186.3/2 161.8/3

66 0/0 0/0 169.0/2 170.4/2 176.0/2 172.8/2 172.5/2 164.6/2 142.8/4

67 0/0 0/0 124.1/2 164.8/2 156.3/2 156.2/2 153.7/2 154.4/2 112.5/4

68 0/0 0/0 0/0 113.8/2 126.5/2 126.1/2 126.6/2 115.9/2 71.2/4

69 0/0 0/0 0/0 108.0/2 97.7/2 0/0 0/0 0/0 0/0
E38 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 5, November 17, 1976 (1976.879)

19 20 21 22 23 24 25 26 27

64 0/0 199.2/3 207.5/3 194.0/3 203.9/2 217.2/2 205.7/3 188.4/3 163.1/4

65 0/0 0/0 180.1/3 178.1/3 189.3/2 202.0/2 193.0/3 182.0/3 157.5/4

66 0/0 0/0 166.4/2 166.5/2 170.2/2 169.4/2 168.7/3 160.2/3 138.2/4

67 0/0 0/0 122.2/2 159.9/2 150.4/2 153.4/2 150.0/3 151.0/3 108.7/4

68 0/0 0/0 0/0 109.6/2 122.5/2 122.8/2 124.1/2 111.7/3 65.3/4

69 0/0 0/0 0/0 101.7/2 89.0/2 0/0 0/0 0/0 0/0
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E39 

Flight 6, January 19, 1977 (1977.051)

19 20 21 22 23 24 25 26 2?

64 0/0 197.2/3 208.3/3 195.3/3 202.6/2 216.6/3 203.9/3 188.2/3 163.2/4

65 0/0 0/0 181.1/3 178.1/3 189.1/2 200.9/3 192.7/3 182.4/3 157.1/3

66 0/0 0/0 166.9/3 169.5/3 172.6/2 168.9/3 168.4/3 159.9/3 138.8/3

67 0/0 0/0 124.3/2 163.3/2 153.1/2 153.0/2 152.6/3 150.9/3 107.5/3

68 0/0 0/0 0/0 113.9/2 126.3/2 125.6/2 127.4/2 116.2/3 67.7/4

69 0/0 0/0 0/0 109.2/2 98.9/2 0/0 0/0 0/0 0/0
E40 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 7, March 7,1977 (1977.180)

19 20 21 22 23 24 25 26 27

64 0/0 197.9/3 207.7/3 193.8/3 202.4/2 216.4/2 203-7/3 189.3/3 164.5/3

65 0/0 0/0 180.2/3 177.0/3 189.0/2 201.6/2 193.9/3 182.3/3 156.7/3

66 0/0 0/0 167.7/3 168.4/3 172.5/2 170.9/3 169.5/3 158.3/3 138.2/3

67 0/0 0/0 124.7/2 163.4/2 154.3/2 154.8/2 153-7/3 150.3/3 106.5/3

68 0/0 0/0 0/0 113.2/2 129-1/2 122.5/2 124.7/2 113-2/3 65.8/3

69 0/0 0/0 0/0 117-9/2 102.8/2 0/0 0/0 0/0 0/0
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E41

Flight 8, April 23,1977 (1977.309)

19 20 21 22 23 24 25 26 27

64 0/0 203.1/3 213.9/3 198.9/3 205.9/2 220.5/2 208.9/3 195.6/3 170.7/3

65 0/0 0/0 185.2/3 182.5/3 193.4/2 207.8/2 199.0/3 188.4/3 162.1/3

66 0/0 0/0 172.9/2 173.0/2 179.3/2 176.7/2 174.5/3 164.1/3 143.9/3

67 0/0 0/0 130.7/2 169.1/2 162.0/2 160.5/2 160.1/3 155.9/3 111.6/3

68 0/0 0/0 0/0 119.8/2 136.2/2 127.3/2 127.8/2 118.3/3 71.6/3

69 0/0 0/0 0/0 126.5/2 111.8/2 0/0 0/0 0/0 0/0
E 42 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 9, June 2, 1977 (1977.418)

	19 20 21 22 23 24 25 26 27

47 536.0/3 530.3/2 530.5/2 554.0/2 572.9/2 568.0/3 565.0/4 0/0 G/0

48 533.5/3 525.0/3 525.0/2 541.1/2 559.5/2 544.7/3 535.4/3 0/0 0/0

49 531.7/2 513.2/2 510.5/2 526.6/2 528.1/3 527.4/3 531.0/3 0/0 0/0

50 514.7/2 490.8/3 490.5/3 499-9/3 515.0/3 529.8/3 535.9/3 0/0 0/0

51 500.6/3 491.7/3 482.8/3 494.3/2 532.8/3 540.7/3 538.7/3 0/0 0/0

52 477.2/3 484.2/2 477.4/2 480.9/3 494.6/2 524.5/3 537.6/3 0/0 0/0

53 484.9/2 473.8/2 462.9/2 434.5/2 406.3/2 0/0 0/0 0/0 0/0

54 0/0 379.2/2 405.4/2 410.6/2 394.1/3 381.3/3 0/0 0/0 0/0

55 340.4/3 359.4/3 390.1/2 406.9/2 398.8/2 375.2/3 0/0 0/0 0/0

56 335.9/3 357.1/3 376.6/2 384.2/3 373.4/3 344.4/3 0/0 0/0 0/0

57 327.4/2 340.9/3 350.9/2 322.0/3 306.2/3 298.3/3 283.2/3 0/0 0/0

58 325.1/2 337.1/3 334.1/2 319-7/2 298.4/2 296.4/3 289.2/3 0/0 0/0

59 300.0/3 310.7/3 302.0/2 294.6/2 284.4/3 286.6/3 270.6/3 246.1/4 210.1/4

60 279.8/3 289.7/3 291.0/3 268.1/2 266.5/3 278.8/3 253.6/2 251.0/2 233.3/4

61 0/0 258.1/3 265.7/3 256.2/2 267.5/2 270.8/3 242.9/3 235.9/3 0/0

62 0/0 0/0 255.7/2 256.7/2 253.4/2 247.6/2 237-7/3 225.2/3 0/0

63 0/0 212.0/2 237-8/2 247.8/2 234.4/2 233-0/2 229.4/2 218.7/3 190.5/3

64 0/0 207.3/2 220.1/2 205.1/3 210.2/2 225.2/2 214.2/2 201.3/3 176.5/3

65 0/0 0/0 191.6/2 188.7/2 199-1/2 213-7/2 204.4/2 194.5/3 167.8/3

66 0/0 0/0 178.1/2 179.2/2 187.0/2 182.0/2 179-8/2 169-8/2 150.3/2

67 0/0 0/0 138.7/2 175.7/2 170.2/2 166.2/2 166.8/2 161.7/2 117-2/2

68 0/0 0/0 0/0 124.8/2 142.0/2 133-9/2 134.0/2 126.0/3 79-3/3

69 0/0 0/0 0/0 134.1/2 122.3/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E43

	Flight 10, July 7, 1977 (1977.514)

	19 20 21 22 23 24 25 26 27

47 534.9/3 529.1/2 529.3/2 552.9/2 571.5/2 566.4/3 563.2/4 0/0 0/0

48 531.7/3 523.8/2 523.7/2 539.6/2 557.7/2 542.8/2 533.5/3 0/0 0/0

49 530.7/2 512.2/2 509.2/2 525.0/2 526.0/2 525.6/3 529.4/3 0/0 0/0

50 513.3/2 489.2/3 488.7/2 498.0/3 512.8/3 528.0/2 534.1/2 0/0 0/0

51 499.1/2 490.2/2 481.1/3 492.7/2 530.6/2 539.1/3 537.1/3 0/0 0/0

52 475.7/2 482.6/2 475.5/2 479.2/2 493-5/2 523.0/3 536.1/3 0/0 0/0

53 483.4/2 472.5/2 461.3/2 433.3/2 404.7/2 0/0 0/0 0/0 0/0

54 0/0 377-7/2 403.5/2 408.8/2 393.3/2 379.9/3 0/0 0/0 0/0

55 339.3/3 358.7/2 389-2/2 405.5/2 397-7/2 373-9/3 0/0 0/0 0/0

56 334.4/3 355.6/3 375.6/2 382.3/2 372.7/3 342.9/3 0/0 0/0 0/0

57 325.6/2 338.7/3 349.8/2 320.2/2 304.4/2 296.8/3 282.0/3 0/0 0/0

58 324.3/2 336.1/3 332.3/2 317.5/2 296.5/2 294.9/3 287.7/2 0/0 0/0

59 298.9/3 309.4/3 300.2/2 292.5/2 282.3/2 284.9/3 269.2/3 244.7/3 208.6/4

60 278.5/3 288.6/3 289-5/2 266.5/2 264.7/2 277-0/3 251-9/2 249-5/2 231.8/4

61 0/0 256.9/3 264.2/3 254.9/2 265.4/2 269-4/3 241.6/3 234.5/2 0/0

62 0/0 0/0 254.2/2 255-4/2 252.0/2 246.1/2 236.7/2 223-9/3 0/0

63 0/0 210.7/2 236.5/2 246.3/2 232.8/2 231-5/2 228.4/2 217-8/3 189-4/3

64 0/0 205.8/2 218.8/2 203-7/2 208.1/2 223-3/2 212.5/2 200.3/2 175-5/3

65 0/0 0/0 190.4/2 187-2/2 196.9/2 211.7/2 202.4/2 193-3/2 166.8/2

66 0/0 0/0 176.8/2 177-7/2 185-0/2 180.2/2 178.2/2 168.7/2 149-2/2

67 0/0 0/0 139-0/2 173-3/2 169-1/2 164.6/2 165-8/2 160.6/2 115-7/2

68 0/0 0/0 0/0 124.5/2 139-6/2 133-1/2 132-3/2 125-7/3 78.0/3

69 0/0 0/0 0/0 130.7/2 120.7/2 0/0 0/0 0/0 0/0
E 44 STUDIES OF COLUMBIA GLACIER, ALASKA

Flight 11, August 29, 1977 (1977.659)

	19 20 21 22 23 24 25 26 27

47 533.2/3 527.2/2 528.0/2 551.3/2 569.4/3 563.9/3 561.8/4 0/0 0/0

48 529.3/3 521.7/2 522.1/2 537.5/2 555.0/2 539.8/2 531.5/3 0/0 0/0

49 528.9/2 510.4/2 507.8/2 522.2/2 523.0/2 523.0/3 527.3/3 0/0 0/0

50 511.5/2 486.9/2 486.7/3 495.3/3 509.6/3 524.8/2 531.7/2 0/0 0/0

51 497.1/2 487.6/2 478.9/2 489.8/2 527.9/2 536.3/3 534.6/3 0/0 0/0

52 473.4/2 480.1/2 472.8/2 476.5/2 490.1/2 520.0/3 534.2/3 0/0 0/0

53 481.0/2 469.3/2 458.3/2 431.1/2 403.6/2 0/0 0/0 0/0 0/0

54 0/0 375.5/2 400.8/2 406.5/2 391.6/2 377.0/3 0/0 0/0 0/0

55 336.9/3 356.3/3 386.4/2 402.7/2 395.4/2 371.2/3 0/0 0/0 0/0

56 331.2/3 352.4/3 372.6/2 378.7/2 369.8/3 340.0/3 0/0 0/0 0/0

57 322.5/2 336.4/3 346.9/2 316.8/2 300.8/2 293.6/3 278.7/3 0/0 0/0

58 321.8/2 332.0/3 327.2/2 313.6/2 293.0/2 291.7/3 284.3/2 0/0 0/0

59 295.9/3 306.6/3 296.1/2 287.7/2 278.6/2 281.6/3 265.9/3 241.9/3 205.6/4

60 275.4/3 285.8/3 285.9/3 262.2/2 261.3/2 272.7/3 248.6/2 246.1/2 229.0/4

61 0/0 253.2/3 260.8/3 251.8/2 261.6/2 265.1/3 238.5/3 231.1/3 0/0

62 0/0 0/0 250.6/2 252.3/2 247.8/2 242.5/2 233.6/2 221.0/3 0/0

63 0/0 208.2/2 233.2/2 242.2/2 229.1/2 228.0/2 224.8/2 214.6/3 185.8/3

64 0/0 202.9/2 214.8/2 199.4/2 204.6/2 218.3/2 209.3/2 196.7/3 172.2/3

65 0/0 0/0 186.5/2 182.5/2 190.4/2 205.4/2 197.7/2 189.5/2 164.1/2

66 0/0 0/0 173.7/2 172.0/2 178.0/2 175.4/2 173.0/2 165.6/2 146.0/2

67 0/0 0/0 136.4/2 166.6/2 161.5/2 158.5/2 160.1/2 156.9/2 113.3/2

68 0/0 0/0 0/0 119.7/2 132.8/2 128.2/2 128.3/2 121.9/2 73.5/3

69 0/0 0/0 0/0 114.0/2 104.3/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974^81 E 45

Flight 12, November 8,1977 (1977.854)

	19 20 21 22 23 24 25 26 2?

47 531.7/3 525.4/2 526.9/3 549.9/3 567.5/3 561.9/4 562.2/4 0/0 0/0

48 527.1/3 519.4/2 520.9/2 535.5/2 552.1/2 536.7/2 530.6/3 0/0 0/0

49 528.0/2 508.8/2 506.2/2 519.6/2 520.4/3 521.3/3 525.4/3 0/0 0/0

50 510.0/2 485.6/3 485.8/3 493.2/3 507.1/3 522.9/2 530.4/3 0/0 0/0

51 495.8/3 486.1/2 477.7/2 487.0/3 525.6/2 533.4/3 532.3/3 0/0 0/0

52 471.4/2 478.0/2 470.4/3 474.7/2 486.3/2 517.2/3 533.7/3 0/0 0/0

53 479.5/2 464.0/3 455.0/2 428.7/2 401.9/2 0/0 0/0 0/0 0/0

54 0/0 374.1/2 398.8/2 406.3/2 391.3/2 374.6/3 0/0 0/0 0/0

55 335.3/3 357.1/3 385.8/2 401.9/2 393.3/2 368.0/3 0/0 0/0 0/0

56 329.4/3 352.5/3 369.9/2 376.3/2 367.7/3 337.2/3 0/0 0/0 0/0

57 320.1/2 334.5/3 343.7/2 314.9/2 297.2/3 289.0/3 275.2/3 0/0 0/0

58 318.1/2 327.3/3 320.4/2 309.7/2 290.5/2 288.3/3 280.2/2 0/0 0/0

59 291.3/3 301.6/3 291.8/2 281.9/2 274.2/2 277.6/3 262.6/3 238.8/4 202.3/4

60 271.3/3 281.3/3 281.5/3 257.5/2 256.6/2 267.7/3 244.9/3 241.9/3 226.0/4

61 0/0 247.9/3 256.7/3 247.3/2 256.5/2 259.2/3 234.2/3 226.5/3 0/0

62 0/0 0/0 245.0/2 248.1/3 242.5/2 237.4/2 229.6/3 216.3/3 0/0

63 0/0 204.9/2 229.2/2 238.1/2 223.5/2 222.0/2 218.8/2 209.6/3 180.6/3

64 0/0 200.9/2 210.6/2 193-7/2 199.8/2 211.9/2 203.6/2 191.5/3 167.6/3

65 0/0 0/0 182.1/2 174.9/2 182.9/2 195.2/2 191.7/2 183.6/2 160.7/3

66 0/0 0/0 169.0/2 161.9/2 166.6/2 166.6/2 166.8/2 160.3/2 142.7/3

67 0/0 0/0 129.0/2 154.1/2 147.3/2 148.5/2 150.1/2 151.6/2 110.8/2

68 0/0 0/0 0/0 108.4/2 120.3/2 119.6/2 119.4/2 115.1/3 66.7/3

69 0/0 0/0 0/0 100.6/2 86.7/2 0/0 0/0 0/0 0/0
E 46 STUDIES OF COLUMBIA GLACIER, ALASKA

Flight 13, February 28, 1978 (1978.160)

	19 20 21 22 23 24 25 26 27

47 534.6/3 528.5/2 529.4/2 551.5/2 569.9/3 565.1/4 566.7/4 0/0 0/0

48 529.8/3 522.6/2 524.3/2 538.4/2 554.7/2 539.7/3 534.3/4 0/0 0/0

49 531.6/2 511.4/2 508.8/2 521.6/2 523.5/3 524.2/3 528.4/4 0/0 0/0

50 513-4/2 489.0/2 489.0/2 495.4/3 510.4/3 526.4/3 534.1/3 0/0 0/0

51 499.5/2 491.3/2 482.6/3 491.3/3 529.9/3 537.5/3 535.8/3 0/0 0/0

52 474.9/2 483.1/3 477-3/2 479.3/2 491.5/3 522.2/3 538.1/4 0/0 0/0

53 484.1/2 468.3/2 459.6/2 433.5/2 405.2/2 0/0 0/0 0/0 0/0

54 0/0 379.2/2 404.2/2 412.1/2 396.8/2 379.2/3 0/0 0/0 0/0

55 342.5/3 364.5/3 391.5/2 408.7/2 399.0/3 372.7/3 0/0 0/0 0/0

56 336.1/3 358.7/3 376.4/3 383.1/2 373.0/3 341.7/3 0/0 0/0 0/0

57 325.1/2 340.7/3 348.1/2 321.1/2 301.5/2 292.3/3 279-3/3 0/0 0/0

58 319-5/2 331.2/3 325.0/2 314.1/3 295.2/2 291.1/3 283.5/2 0/0 0/0

59 292.2/3 304.0/3 294.7/2 283.0/2 276.3/2 280.2/3 266.1/2 242.0/3 205.6/4

60 273.0/3 282.6/3 283.6/2 258.2/2 258.4/2 270.1/3 248.9/3 245.0/2 229.4/4

61 0/0 249.5/3 258.8/3 249.3/2 258.5/2 260.5/3 236.9/3 228.6/3 0/0

62 0/0 0/0 246.5/3 248.8/2 243.0/2 237-3/3 231.9/2 218.7/3 0/0

63 0/0 207.5/2 230.7/2 241.0/2 225.0/2 223.1/2 219.7/2 210.9/3 182.6/3

64 0/0 204.8/2 214.4/2 196.4/2 200.2/2 212.3/2 203.6/2 192.2/3 169.9/2

65 0/0 0/0 185.1/2 177.2/2 183.5/2 196.2/2 191.3/3 183.7/2 164.4/3

66 0/0 0/0 170.0/2 162.8/2 167.3/2 166.0/2 165.6/2 163.1/2 146.1/2

67 0/0 0/0 128.7/2 154.6/2 149.2/2 148.8/2 148.6/2 154.1/2 115.8/2

68 0/0 0/0 0/0 108.2/2 125.3/3 118.6/2 117.8/2 118.7/2 70.7/3

69 0/0 0/0 0/0 104.1/2 95.6/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 47

	Flight 14, April 19, 1978 (1978.297)

	19 20 21 22 23 24 25 26 2? 

4? 534.4/3 528.3/2 528.7/2 551.1/2 569.5/3 565.0/4 565.9/4 0/0 0/0

48 529.2/3 522.6/2 523.8/2 538.0/2 554.7/2 539.7/3 534.0/4 0/0 0/0

49 530.9/2 510.4/2 507.9/2 521.3/2 522.9/3 523.0/3 527.7/4 0/0 0/0

50 512.7/2 488.1/2 488.7/2 494.2/3 509.4/3 525.5/3 533-6/4 0/0 0/0

51 499.3/2 490.8/2 481.9/3 491.3/3 530.2/3 537.4/3 535.8/3 0/0 0/0

52 474.3/2 483.2/3 477.7/2 479.0/2 491.2/3 523.9/3 538.4/3 0/0 0/0

53 484.5/2 467.9/2 459.6/3 432.8/2 403.5/2 0/0 0/0 0/0 0/0

54 0/0 378.0/2 403.6/2 411.7/2 397.1/2 379-3/3 0/0 0/0 0/0

55 342.1/3 362.8/2 390.9/2 408.3/2 397.5/3 372.7/3 0/0 0/0 0/0

56 336.0/3 358.8/2 376.5/3 382.3/2 372.4/2 341.4/3 0/0 0/0 0/0

57 324.8/2 340.3/3 348.2/2 321.0/2 300.7/2 292.2/3 279.5/3 0/0 0/0

58 318.7/2 330.3/3 327.1/2 314.2/2 294.5/2 290.7/3 283.6/2 0/0 0/0

59 291.0/3 302.3/3 293.9/2 282.7/2 275.6/2 278.9/3 265.8/2 241.9/3 205.5/4

60 272.8/3 282.6/3 283.0/2 257.1/2 257.3/2 270.7/2 248.8/3 245.4/2 229.4/4

61 0/0 249.2/3 258.3/3 248.9/2 258.3/2 260.4/2 236.7/3 228.2/3 0/0

62 0/0 0/0 245.0/3 247.8/2 242.2/2 236.1/3 231.1/2 218.1/3 0/0

63 0/0 205.8/2 229.0/2 241.3/2 224.4/2 222.4/2 218.7/2 210.2/3 182.3/3

64 0/0 204.7/2 214.2/2 196.2/2 199.0/2 213.4/2 202.2/2 191.3/3 169.6/2

65 0/0 0/0 184.6/2 177.5/2 184.6/2 198.3/2 191.3/3 182.5/2 163.7/3

66 0/0 0/0 168.1/2 164.7/2 169.6/2 166.8/2 167.0/2 163.5/2 146.5/2

67 0/0 0/0 128.2/2 157.6/2 151.5/2 151.1/2 149.6/2 153.4/2 115.2/2

68 0/0 0/0 0/0 109.1/2 128.7/2 119.4/2 119.7/2 119.4/2 71.1/3

69 0/0 0/0 0/0 109.0/2 100.6/2 0/0 0/0 0/0 0/0
E 48 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 15, June 11, 1978 (1978.442)

	19 20 21 22 23 24 25 26 2?

47 534.6/3 528.6/2 529.1/2 551.8/2 569.9/3 565.4/4 565.6/4 0/0 0/0

48 529.5/3 522.3/2 524.0/2 538.7/2 555.6/2 540.3/3 534.0/4 0/0 0/0

49 531.2/2 510.8/2 508.4/2 522.3/2 523.8/3 523.0/3 527.8/4 0/0 0/0

50 513.3/2 489.1/2 490.0/2 494.7/2 509.7/3 525.5/3 533-7/4 0/0 0/0

51 500.2/2 491.2/2 481.4/2 492.3/3 530.1/3 538.3/3 536.5/3 0/0 0/0

52 475.8/2 484.1/3 478.4/2 478.4/2 492.1/2 526.0/2 538.8/3 0/0 0/0

53 486.1/2 470.7/2 461.1/3 432.8/2 403.8/2 0/0 0/0 0/0 0/0

54 0/0 376.6/2 403.5/2 410.9/2 397.9/2 380.7/3 0/0 0/0 0/0

55 341.4/3 361.5/2 389.1/2 407.4/2 395.7/3 373.2/3 0/0 0/0 0/0

56 337.3/3 359.7/3 377.5/2 381.7/2 372.4/2 342.6/3 0/0 0/0 0/0

57 325.8/2 340.5/3 349.0/2 320.6/2 300.9/2 294.0/3 281.1/3 0/0 0/0

58 319.5/2 330.6/3 329.2/2 315.4/2 294.9/2 291.1/3 285.6/2 0/0 0/0

59 291.4/3 302.2/3 294.1/2 283.8/2 276.1/2 279.6/3 267.3/2 242.8/3 206.4/4

60 273.9/3 284.3/3 283.8/2 258.7/2 258.5/2 272.9/2 250.5/3 246.8/2 230.2/4

61 0/0 250.1/2 259.4/3 250.3/2 261.3/2 262.0/2 238.2/2 229.5/2 0/0

62 0/0 0/0 245.8/3 248.7/2 243.6/2 237.5/3 232.1/2 218.8/3 0/0

63 0/0 205.5/2 228.2/2 242.5/2 226.1/2 223.9/2 219.7/2 211.3/3 183.6/3

64 0/0 204.8/2 214.7/2 197.6/2 200.8/2 215.8/2 203.4/2 191.3/3 170.7/2

65 0/0 0/0 185.0/2 179.4/2 187.6/2 201.7/2 193.3/2 182.8/2 164.8/3

66 0/0 0/0 168.3/2 168.3/2 175.1/2 170.5/2 170.5/2 164.7/2 147.2/2

67 0/0 0/0 130.2/2 163.3/2 157.2/2 154.9/2 154.0/2 154.5/2 116.3/2

68 0/0 0/0 0/0 112.8/2 132.3/2 122.8/2 121.8/2 121.7/2 73-3/2

69 0/0 0/0 0/0 113.9/2 105.4/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-^81 E 49

	Flight 16, July 30, 1978 (1978.576)

	19 20 21 22 23 24 25 26 2?

47 530.7/3 524.9/2 525.6/2 548.7/2 566.2/3 562.1/4 561.3/4 0/0 0/0

48 526.0/3 518.8/2 520.2/2 534.9/2 551.7/2 536.7/3 530.1/4 0/0 0/0

49 527.9/2 507.8/3 505.4/2 519-1/2 519.5/2 518.0/3 523.6/4 0/0 0/0

50 510.6/2 485.8/2 487.0/2 490.7/2 504.9/3 520.7/3 529.4/4 0/0 0/0

51 497-7/2 487.3/2 477.7/2 488.3/2 524.5/3 534.4/3 533.5/3 0/0 0/0

52 472.8/2 480.7/3 474.0/2 474.1/2 486.7/2 523.5/2 535.8/3 0/0 0/0

53 483.6/2 468.3/2 456.6/3 428.9/2 399.6/2 0/0 0/0 0/0 0/0

54 0/0 371.3/2 398.8/2 405.9/2 392.2/2 377.2/3 0/0 0/0 0/0

55 336.4/3 355.3/2 383.3/2 402.0/2 391.6/2 369.9/3 0/0 0/0 0/0

56 334.6/3 355.3/3 373.2/2 376.8/2 366.5/2 338.0/3 0/0 0/0 0/0

57 322.7/2 337.0/3 343.9/2 315.8/2 297.1/2 291.0/3 277.8/3 0/0 0/0

58 316.0/2 326.7/3 326.2/2 312.3/2 290.6/2 287.0/3 282.6/2 0/0 0/0

59 288.6/3 298.7/3 290.3/2 281.2/2 272.9/2 276.0/3 263.9/2 239.2/3 202.8/4

60 270.8/3 281.5/3 280.1/2 255.2/2 256.1/2 270.2/2 247.5/3 243.5/2 226.5/4

61 0/0 246.8/2 255.6/3 247.7/2 258.0/2 259.0/2 234.6/2 225.9/2 0/0

62 0/0 0/0 241.9/3 245.5/2 240.8/2 233-7/3 227-9/3 215.2/3 0/0

63 0/0 201.9/2 223-5/2 238.8/2 223-2/2 221.1/2 216.7/3 207.6/3 180.2/3

64 0/0 201.2/2 210.8/2 194.3/2 198.8/2 212.8/2 200.4/2 187.7/3 167.1/2

65 0/0 0/0 181.2/2 176.5/2 184.8/2 199.4/2 190.7/2 179.0/2 160.5/2

66 0/0 0/0 165.3/2 165.0/2 172.7/2 167-3/2 167.6/2 161.1/2 143.5/2

67 0/0 0/0 127.4/2 164.0/2 156.1/2 152.8/2 151.9/2 151.4/2 111.6/2

68 0/0 0/0 0/0 111.9/2 129.2/2 121.4/2 123.8/2 119-3/2 70.8/2

69 0/0 0/0 0/0 114.7/2 104.6/2 0/0 0/0 0/0 0/0
E 50 STUDIES OF COLUMBIA GLACIER, ALASKA

Flight 17, August 26, 1978 (1978.650)

	19 20 21 22 23 24 25 26 27

47 529.8/3 523.8/2 524.9/2 547.8/2 565.0/3 560.9/4 560.3/4 0/0 0/0

48 525.1/3 517.8/2 519.1/2 533.5/2 550.2/2 535.1/3 528.9/4 0/0 0/0

49 527.1/3 507.5/3 504.4/2 517.2/2 517.6/2 516.5/3 522.4/4 0/0 0/0

50 510.4/2 485.5/2 486.1/2 489.2/2 503.1/3 519.0/3 528.0/4 0/0 0/0

51 496.8/2 485.9/2 476.5/2 486.8/2 523.2/3 532.7/2 532.2/3 0/0 0/0

52 471.5/2 479.1/3 472.1/2 472.1/3 484.8/2 522.1/2 534.8/3 0/0 0/0

53 482.4/2 466.5/2 455.2/3 427.6/2 398.2/2 0/0 0/0 0/0 0/0

54 0/0 368.6/2 396.8/2 403.7/2 389.5/2 375.5/3 0/0 0/0 0/0

55 334.4/3 352.6/2 380.9/2 399.5/2 389-3/2 368.1/3 0/0 0/0 0/0

56 333.2/3 353.5/3 370.1/2 374.6/2 364.1/2 335.9/3 0/0 0/0 0/0

57 321.4/2 335.1/3 341.6/2 313-3/2 295.1/2 288.9/3 275.9/3 0/0 0/0

58 314.4/2 325.4/3 324.3/2 310.7/2 288.8/2 285.3/3 280.6/2 0/0 0/0

59 287.3/3 297.2/3 289.4/2 279.7/2 271.3/2 273.6/3 262.2/3 237.6/3 201.2/4-

60 269.6/3 280..7/3 278.5/3 253.2/2 254.3/2 268.4/2 246.1/3 241.9/2 225.1/4

61 0/0 245.4/2 254.1/3 246.1/2 256.2/2 257.2/2 232.8/2 223.9/2 0/0

62 0/0 0/0 240.3/3 244.0/2 239.3/2 231.6/2 225.7/3 213.0/3 0/0

63 0/0 200.4/2 221.7/2 237.1/2 221.4/2 219.3/2 214.6/3 206.3/3 178.1/3

64 0/0 199.8/2 208.4/2 192.6/2 197.6/2 211.3/2 199.0/2 185.6/3 165.3/2

65 0/0 0/0 179.7/2 174.6/2 182.9/2 197.0/2 188.7/2 177.1/2 158.7/2

66 0/0 0/0 163.9/2 162.5/2 170.1/2 165.6/2 165.7/2 159.1/2 141.2/2

67 0/0 0/0 126.0/2 161.7/2 154.0/2 150.3/2 149.6/2 149.7/2 109.6/2

68 0/0 0/0 0/0 108.9/2 126.1/2 119.7/2 122.1/2 117.2/2 68.3/2

69 0/0 0/0 0/0 113.6/2 100.6/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 51

	Flight 18, November 8, 1978 (1978.853)

	19 20 21 22 23 24 25 26 2?

47 533.3/2 525.9/2 527.4/3 550.6/3 567-6/4 562.7/4 563.7/4 0/0 0/0

48 527.1/3 520.5/2 521.7/2 535.2/2 550.8/3 536.1/3 531.4/4 0/0 0/0

49 529.4/3 510.0/3 505.3/2 518.9/2 518.7/2 518.0/3 524.5/4 0/0 0/0

50 513.2/2 487.4/2 486.5/2 489.6/2 504.0/2 519.8/3 530.0/4 0/0 0/0

51 498.5/2 485.9/2 476.8/2 486.2/2 523-7/3 533.1/2 533.5/4 0/0 0/0

52 471.7/2 478.3/3 470.7/2 472.2/3 484.5/2 521.6/3 537.7/3 0/0 0/0

53 482.8/3 463-7/2 453.8/3 426.7/2 399.5/2 0/0 0/0 0/0 0/0

54 0/0 367.3/3 394.5/2 402.3/2 387.2/3 374.1/3 0/0 0/0 0/0

55 332.6/3 352.4/2 379-8/2 399.0/2 386.9/2 365.5/3 0/0 0/0 0/0

56 332.3/3 351.6/3 367-3/2 371.0/2 361.8/2 333-2/3 0/0 0/0 0/0

57 320.8/2 334.3/3 340.4/3 311-2/2 293.4/2 286.6/3 273-1/3 0/0 0/0

58 314.0/3 324.7/3 321.6/2 308.8/2 287.7/2 284.6/3 277.3/2 0/0 0/0

59 286.6/3 299.9/3 290.4/3 278.8/2 270.6/2 271.8/3 260.6/3 237.5/3 201.0/4

60 269-6/3 280.6/3 277.0/3 250.9/2 252.2/2 265.9/3 244.7/3 240.6/2 225.2/4

61 0/0 244.3/3 252.4/3 243.7/2 253.5/2 255.3/2 230.4/2 222.1/3 0/0

62 0/0 0/0 239.1/3 243.8/2 237.5/2 228.9/2 221.8/3 210.2/3 0/0

63 0/0 201.7/2 221.4/2 235.0/2 219.2/2 216.2/2 211.8/3 203-6/3 175.3/3

64 0/0 201.9/2 207.3/2 188.1/2 196.1/2 209-0/2 197.5/2 183.8/3 163.1/2

65 0/0 0/0 177.3/2 171.2/2 179.4/2 193.0/2 186.1/2 175.2/2 156.1/3

66 0/0 0/0 163.8/2 156.1/2 162.7/2 163.3/2 161.5/3 155.7/2 137.4/3

67 0/0 0/0 124.2/2 153.3/2 145.4/2 145.1/2 143.0/2 147.4/2 106.8/3

68 0/0 0/0 0/0 103.7/2 118.4/2 113-9/2 115.0/3 111.6/2 62.5/3

69 0/0 0/0 0/0 111.3/2 88.5/2 0/0 0/0 0/0 0/0
E 52 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 19, January 6, 1979 (1979.014)

	19 20 21 22 23 24 25 26 27 

4? 531.1/2 524.1/2 526.1/3 548.7/3 566.0/4 561.8/4 562.1/4 0/0 0/0

48 525.9/3 518.5/2 520.3/2 534.5/2 549-9/3 535.5/3 530.2/4 0/0 0/0

49 527.8/3 508.0/3 503.8/2 518.1/2 518.4/2 517.3/3 523.6/4 0/0 0/0

50 511.4/3 485.8/2 484.4/2 489.0/2 504.1/3 519.5/3 528.9/4 0/0 0/0

51 497.2/2 484.9/2 476.2/2 485.8/3 523.5/2 532.2/3 533.3/4 0/0 0/0

52 471.2/3 478.1/3 469.7/2 473.5/3 486.6/3 521.5/3 537.6/3 0/0 0/0

53 482.8/3 465.6/3 455.6/2 427.5/2 399.1/3 0/0 0/0 0/0 0/0

54 0/0 370.2/3 395.8/3 402.8/2 387.6/3 374.6/3 0/0 0/0 0/0

55 334.2/3 353.6/2 380.8/2 399-9/2 389-1/2 366.9/2 0/0 0/0 0/0

56 333-4/3 352.2/3 365-7/2 372.2/2 363-2/3 333-1/3 0/0 0/0 0/0

57 321.0/2 334.6/3 340.7/3 311-4/2 293-2/3 286.3/3 273-3/3 0/0 0/0

58 314.1/3 325.5/3 321.4/2 308.6/3 287.5/2 284.3/3 277-1/2 0/0 0/0

59 288.0/3 301.0/3 291-3/3 279-2/2 271-1/2 271-6/3 259-6/3 236.7/3 200.7/4

60 269.9/3 281.1/3 278.3/3 251.0/2 251.6/2 265.7/3 244.6/3 240.3/2 224.6/4

61 0/0 245.1/3 253-5/3 241.8/2 253-7/2 256.2/2 230.5/3 222.0/3 0/0

62 0/0 0/0 240.8/3 243-9/2 237-9/2 229-8/2 221.5/3 210.5/3 0/0

63 0/0 202.8/2 223-9/2 235.7/2 219-8/2 216.7/2 212.1/3 203-4/3 175.6/3

64 0/0 202.1/3 208.8/2 189-3/2 196.6/2 209-2/2 198.4/2 184.1/3 163-6/2

65 0/0 0/0 179-3/2 172-3/2 180.3/2 193-2/2 186.9/2 176.2/2 156.8/3

66 0/0 0/0 165.1/2 158.5/2 164.8/2 164.1/3 162.0/3 156.1/2 138.1/3

67 0/0 0/0 126.0/2 154.0/2 147-6/2 145-3/2 143-1/2 147-5/3 107.1/3

68 0/0 0/0 0/0 107-4/2 120.1/2 114.4/2 116.1/3 113-8/2 65.0/3

69 0/0 0/0 0/0 115.9/2 93-2/2 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 53 

	Flight 20, April 12,1979 (1979.277)

	19 20 21 22 23 24 25 26 27

47 529.3/2 523.2/2 524.7/3 547.3/2 565.3/3 561.1/4 560.6/4 0/0 0/0

48 524.7/3 517.1/3 519.7/2 534.2/2 550.1/3 536.7/4 530.7/4 0/0 0/0

49 525.9/3 506.2/3 503.3/3 518.4/3 519.2/3 518.3/3 523.6/4 0/0 0/0

50 510.1/3 484.7/2 483.6/2 489.9/2 505.2/3 520.3/3 529.3/4 0/0 0/0

51 496.6/2 485.0/3 477.0/2 486.5/3 525.2/3 533.2/3 533.3/4 0/0 0/0

52 472.7/3 478.5/3 470.6/3 474.2/3 487.5/3 520.4/3 536.3/3 0/0 0/0

53 483.7/3 467.6/3 457.3/2 428.6/3 399.5/3 0/0 0/0 0/0 0/0

54 0/0 372.4/3 399.3/3 405.1/2 389.1/3 375.8/2 0/0 0/0 0/0

55 334.8/3 354.1/2 384.1/2 401.3/2 393.1/2 369.3/3 0/0 0/0 0/0

56 333.2/3 351.6/3 365.7/2 373.7/2 364.8/3 334.1/3 0/0 0/0 0/0

57 321.0/3 334.7/3 341.7/3 312.7/3 293.9/3 285.1/3 273.4/3 0/0 0/0

58 314.0/3 325.7/3 322.2/3 308.3/3 287.8/2 284.4/3 277.4/2 0/0 0/0

59 289.1/3 301.3/3 291.4/3 279.5/2 271.1/3 271.3/3 258.4/3 235.9/3 200.7/4

60 269.8/3 280.6/3 278.4/3 250.4/2 251.5/2 264.7/3 243.8/3 240.2/3 224.4/4

61 0/0 245.6/3 253.3/3 241.4/2 253.9/2 256.6/2 230.4/3 222.8/3 0/0

62 0/0 0/0 241.3/3 244.4/2 238.8/2 232.2/2 222.8/3 211.0/3 0/0

63 0/0 202.4/2 225.4/3 236.4/2 220.5/2 218.1/3 213.7/3 204.1/3 176.6/3

64 0/0 200.2/3 208.9/3 191.6/2 197.1/2 210.8/2 199.9/3 184.4/3 163.6/3

65 0/0 0/0 179.0/3 174.1/3 182.4/3 196.7/3 189.3/3 177.4/3 157.2/3

66 0/0 0/0 165.0/2 162.3/2 169.4/2 166.8/3 164.4/3 156.0/2 137.6/3

67 0/0 0/0 128.0/2 158.9/2 154.1/3 150.5/3 146.5/2 147.3/3 106.5/3

68 0/0 0/0 0/0 115.3/2 128.3/2 119.0/2 116.9/3 114.0/3 67.4/3

69 0/0 0/0 0/0 123.6/2 104.2/2 0/0 0/0 0/0 0/0
E 54 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 21, August 18, 1979 (1979.628)

	19 20 21 22 23 24 25 26 27 

4? 526.5/2 519.6/2 521.1/3 543.6/3 561.9/4 557.4/4 556.6/4 0/0 0/0

48 522.9/3 515.0/3 516.7/2 530.8/2 546.4/3 533.8/4 526.8/4 0/0 0/0

49 522.8/3 504.5/3 501.5/3 515.0/3 515.5/3 516.0/4 520.5/4 0/0 0/0

50 508.0/3 483.2/3 481.9/2 487.9/2 502.8/4 518.1/4 525.5/4 0/0 0/0

51 494.3/2 483.2/3 474.1/2 483.6/3 522.2/3 530.3/3 530.1/4 0/0 0/0

52 469.7/3 475.3/2 467.0/3 470.5/3 483.0/3 516.2/3 532.8/3 0/0 0/0

53 479.0/3 461.4/3 451.1/3 422.6/3 393-3/2 0/0 0/0 0/0 0/0

54 0/0 367.1/2 392.8/2 399-3/2 383.6/2 371.7/2 0/0 0/0 0/0

55 332.8/3 351.4/2 379.5/2 395.8/2 388.1/3 366.3/3 0/0 0/0 0/0

56 330.8/3 349.0/3 362.4/2 367.4/2 359.4/2 329.9/3 0/0 0/0 0/0

57 318.2/3 332.1/3 340.1/2 309.6/3 291.3/3 280.7/3 268.8/3 0/0 0/0

58 310.7/3 323.8/3 319.7/3 304.4/2 284.4/2 281.0/3 274.4/2 0/0 0/0

59 285.6/3 297.2/3 287.4/2 276.7/3 270.0/3 269.2/3 254.4/3 232.3/3 196.9/4

60 265.7/4 276.2/3 274.9/3 249.4/2 250.1/2 260.8/3 239.7/3 236.8/2 221.0/4

61 0/0 241.5/3 249.4/3 238.6/2 252.0/2 254.1/2 227.7/3 220.2/3 0/0

62 0/0 0/0 237.6/3 241.5/2 236.8/2 231.6/2 221.4/3 207.5/3 0/0

63 0/0 196.8/2 221.6/3 233.1/2 217.4/2 216.4/2 212.8/3 200.0/3 171.5/3

64 0/0 194.2/3 204.6/2 189.8/2 194.7/2 208.0/3 198.5/3 182.3/3 159-3/3

65 0/0 0/0 174.3/2 171.1/3 179.9/2 194.8/3 187.4/3 175.8/3 153.5/3

66 0/0 0/0 159.6/2 159.2/2 167.4/3 164.4/3 163.4/2 153-6/3 133-5/3

67 0/0 0/0 123-5/2 158.0/2 155.4/3 150.2/3 145.8/2 144.7/3 101.5/3

68 0/0 0/0 0/0 113.4/2 129.0/2 119-7/2 114.6/2 109-0/3 64.4/3

69 0/0 0/0 0/0 118.7/2 104.3/2 0/0 0/0 0/0 0/0
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-^81 E 55

	Flight 22, October 20, 1979 (1979.800)

	19 20 21 22 23 24 25 26 27

47 527.7/3 519.9/2 521.1/3 544.0/3 562.8/4 557.9/4 558.5/4 0/0 0/0

48 522.3/3 515.3/3 517.3/2 531.2/3 546.8/4 534.4/4 528.0/4 0/0 0/0

49 523.0/3 504.8/3 502.1/3 515.1/3 515.9/3 518.1/4 521.4/4 0/0 0/0

50 507.9/3 483.5/3 481.8/2 487.5/2 503.1/4 518.8/4 526.4/4 0/0 0/0

51 493.9/2 481.7/3 473.2/2 482.4/3 522.3/4 530.3/3 529.9/4 0/0 0/0

52 467.9/3 473.6/2 465.4/3 468.6/3 479.4/3 514.2/3 534.0/3 0/0 0/0

53 477.7/3 455.6/3 447.0/3 420.7/3 393.1/2 0/0 0/0 0/0 0/0

54 0/0 363.9/2 390.2/2 397.2/2 381.5/2 369.3/2 0/0 0/0 0/0

55 330.5/3 349.6/3 377.0/3 393.7/2 385.3/3 363.3/2 0/0 0/0 0/0

56 329.1/3 346.6/3 360.4/2 364.8/2 357.3/2 328.9/3 0/0 0/0 0/0

57 316.2/3 329.9/3 337.5/2 307.8/3 289.8/3 278.1/3 265.1/3 0/0 0/0

58 309.1/3 322.4/3 318.2/3 301.8/2 283.1/2 279.5/3 271.5/2 0/0 0/0

59 282.6/3 295.7/3 286.1/3 273.7/3 267.6/3 266.9/3 252.5/3 231.3/3 195.3/4

60 263.7/4 274.0/3 272.6/3 247.2/2 247.9/2 257.8/3 237.7/3 235.2/2 220.2/4

61 0/0 238.6/3 246.2/3 236.3/2 249.4/2 251.1/3 225.7/3 218.6/3 0/0

62 0/0 070 233.4/3 239.0/2 234.4/2 229.1/2 218.4/3 204.7/3 0/0

63 0/0 194.8/2 219.0/3 230.1/2 213.8/2 213.2/2 209.8/3 197.1/3 167.7/3

64 0/0 194.0/3 201.7/2 185.1/3 192.3/2 204.8/3 195.4/3 180.1/3 156.3/3

65 0/0 0/0 170.6/2 167.0/3 175.4/2 191.2/3 184.7/2 172.9/3 149.7/3

66 0/0 0/0 156.4/2 153.1/2 160.2/3 160.5/3 159.8/2 150.8/3 130.0/2

67 0/0 0/0 118.3/2 149.9/2 147.0/3 144.0/3 141.7/3 141.9/3 99.1/3

68 0/0 0/0 0/0 104.9/2 121.5/2 115.0/2 106.6/2 101.8/3 57.1/3

69 0/0 0/0 0/0 110.8/3 91.6/2 0/0 0/0 0/0 0/0
E 56 STUDIES OF COLUMBIA GLACIER, ALASKA

Flight 23, February 29, 1980 (1980.162)

	19 20 21 22 23 24 25 26 27

47 530.4/3 522.5/2 524.3/3 546.9/3 566.0/4 560.0/4 561.5/4 0/0 0/0

48 523.6/3 516.5/3 519-2/3 534.4/3 551.2/3 537.0/4 530.6/4 0/0 0/0

49 524.3/3 505.6/3 503-3/3 516.9/3 518.8/3 520.6/4 523-8/4 0/0 0/0

50 509.1/3 484.6/3 482.5/2 489.0/2 504.7/4 520.9/4 528.9/4 0/0 0/0

51 495.7/2 483.4/3 475.0/3 484.2/3 525.2/4 532.5/3 531-9/4 0/0 0/0

52 468.5/3 473-9/3 466.9/2 470.9/3 481.7/3 516.3/3 537-3/3 0/0 0/0

53 478.4/3 454.0/3 447.9/3 423.6/2 396.9/2 0/0 0/0 0/0 0/0

54 0/0 363.7/3 390.8/3 399.8/3 382.2/2 369-8/2 0/0 0/0 0/0

55 330.2/3 348.9/3 376.9/3 395.9/3 386.9/3 363-9/2 0/0 0/0 0/0

56 328.3/2 346.1/2 361.0/2 366.0/2 360.2/2 330.6/3 0/0 0/0 0/0

57 316.0/3 328.1/3 334.7/2 305.8/2 289.0/3 278.6/3 264.3/3 0/0 0/0

58 310.0/3 322.8/3 318.8/3 302.9/3 284.1/2 280.3/3 272.3/2 0/0 0/0

59 282.1/3 297.5/3 290.1/3 273-8/2 267-2/3 267-7/3 254.9/3 233-1/3 196.4/4

60 264.0/4 274.0/3 274.4/3 247.6/2 246.7/2 259-7/2 240.0/3 236.2/2 221.6/4

61 0/0 239.4/3 245.5/3 234.1/2 248.9/2 251.4/3 226.2/3 220.2/3 0/0

62 0/0 0/0 230.7/3 236.0/2 232.9/2 227.5/2 217.9/3 206.5/3 0/0

63 0/0 196.8/2 218.7/3 228.3/2 212.5/2 211.6/2 208.1/3 198.4/3 169-3/3

64 0/0 196.4/3 202.2/3 182.1/2 190.0/2 202.9/3 193-9/3 180.0/3 157-5/3

65 0/0 0/0 170.6/2 165.3/2 173-6/2 189-3/3 182.7/3 170.7/2 149.1/3

66 0/0 0/0 157.0/2 151.4/3 157.3/2 159-6/3 157-0/2 149-6/3 130.4/3

67 0/0 0/0 116.2/2 146.0/3 142.6/3 141.7/3 139-1/2 142.0/3 100.5/3

68 0/0 0/0 0/0 103-3/2 120.7/3 112.1/3 102.0/2 101.5/3 58.5/3

69 0/0 0/0 0/0 110.6/2 87-3/3 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 5?

	Flight 24, May 12, 1980 (1980.361)

	19 20 ,21 22 23 24 25 26 2?

47 528.2/3 521.4/2 523.1/3 546.2/3 564.5/4 557.9/4 558.3/4 0/0 0/0

48 522.6/3 514.8/3 517.6/3 534.5/3 551.4/3 535.7/4 527.9/4 0/0 0/0

49 522.1/3 503.6/3 501.6/2 515.1/2 517.9/3 518.1/4 521.3/4 0/0 0/0

50 506.9/2 483.7/3 481.9/2 487.8/2 503.1/4 518.9/4 526.4/4 0/0 0/0

51 494.4/2 483.0/2 474.4/3 484.8/3 524.5/3 531.9/3 530.4/4 0/0 0/0

52 467.7/3 473.4/3 466.7/2 472.1/3 483.1/3 516.0/3 533.9/3 0/0 0/0

53 477.4/3 455.0/3 448.2/3 423.6/3 395.7/3 0/0 0/0 0/0 0/0

54 0/0 361.9/3 390.3/3 400.0/3 381.1/2 369.6/2 0/0 0/0 0/0

55 328.6/3 346.5/2 375.7/3 395.4/2 387.4/2 363.4/2 0/0 0/0 0/0

56 326.2/2 344.9/2 361.6/2 369.0/2 361.8/2 329.7/3 0/0 0/0 0/0

57 315.3/3 326.2/3 332.4/2 302.7/2 286.1/2 278.0/3 264.6/3 0/0 0/0

58 309.3/3 322.1/3 319.7/2 303.4/3 282.8/2 280.1/3 273.0/2 0/0 0/0

59 282.0/3 296.7/3 290.6/3 274.2/2 267.2/2 268.0/2 254.9/3 232.3/3 195.6/4

60 263.2/4 272.9/3 275.3/3 247.7/2 247.1/2 261.7/2 239.9/3 235.2/2 220.1/4

61 0/0 240.0/3 245.9/3 233.9/2 249.0/2 253.1/3 226.8/3 219.6/3 0/0

62 0/0 0/0 231.6/3 234.6/2 232.8/2 227.8/2 219.1/3 207.5/3 0/0

63 0/0 197.6/2 219.1/3 230.9/2 214.0/2 213.0/2 209.8/3 199.7/3 170.8/3

64 0/0 194.8/3 201.8/3 183.9/2 190.7/2 203.9/2 194.4/3 180.1/3 157.7/3

65 0/0 0/0 170.8/2 166.0/2 176.6/2 191.7/3 183.2/3 171.0/2 149-9/3

66 0/0 0/0 158.3/2 155.7/3 162.3/2 161.8/2 157.9/2 149.9/3 131.9/3

67 0/0 0/0 116.6/2 151.0/2 148.8/3 146.1/2 142.0/2 143.3/3 102.4/3

68 0/0 0/0 0/0 109.1/2 126.1/2 117.6/3 108.1/2 105.7/2 65.3/3

69 0/0 0/0 0/0 115.9/2 96.2/3 0/0 0/0 0/0 0/0
E 58 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 25, July 22, 1980 (1980.556)

	19 20 21 22 23 24 25 26 27

47 527.2/3 520.8/2 522.2/3 545.4/3 563.1/4 556.0/4 556.3/4 0/0 0/0

48 521.7/3 514.4/3 517.0/3 533.9/2 550.3/3 534.2/4 525.9/4 0/0 0/0

49 520.7/3 502.4/3 500.5/2 513.2/2 516.7/3 516.1/4 519.4/4 0/0 0/0

50 505.3/2 482.0/3 481.9/2 486.4/2 501.4/4 517.0/4 524.4/4 0/0 0/0

51 492.7/2 481.5/2 473.9/3 483.8/3 523.6/3 530.3/3 527.9/4 0/0 0/0

52 465.8/3 472.3/3 465.5/3 471.6/3 483.0/3 514.2/3 530.9/3 0/0 0/0

53 476.5/3 454.0/3 446.8/3 422.7/3 393.4/3 0/0 0/0 0/0 0/0

54 0/0 360.6/3 388.4/3 398.6/2 379.2/2 367.4/2 0/0 0/0 0/0

55 324.5/3 342.4/2 373.3/2 393.7/2 384.6/2 360.5/2 0/0 0/0 0/0

56 322.4/2 342.4/2 359.8/2 367.6/3 359.4/2 327.0/2 0/0 0/0 0/0

57 312.2/3 323.8/3 329.9/2 302.0/2 282.5/2 275.8/3 262.5/3 0/0 0/0

58 306.8/3 319.3/3 316.8/2 302.1/3 280.1/2 277.5/2 270.1/3 0/0 0/0

59 279.4/3 294.2/3 289.9/3 272.1/3 265.5/2 265.1/2 252.0/3 229.7/3 193.7/4

60 261.0/4 270.8/2 273.6/3 246.7/2 245.9/2 259.2/2 236.8/3 231.8/2 217.6/3

61 0/0 238.0/2 244.8/3 232.5/2 247.6/2 251.2/3 224.5/3 216.5/3 0/0

62 0/0 0/0 230.7/3 232.5/2 230.8/2 225.7/2 216.8/3 204.8/3 0/0

63 0/0 196.1/2 217.6/3 228.4/2 213.0/2 211.0/2 207.1/3 196.8/3 168.3/3

64 0/0 191.8/3 197.7/2 181.4/2 189.9/2 202.8/2 192.7/3 177.8/3 155.0/3

65 0/0 0/0 166.7/2 162.9/2 175.0/2 190.4/3 182.0/3 168.4/2 146.5/3

66 0/0 0/0 155.9/2 154.4/3 163.2/2 162.7/2 157.1/2 147-7/3 128.7/3

67 0/0 0/0 113.4/2 152.2/2 150.7/3 148.1/3 141.1/2 140.3/3 98.8/3

68 0/0 0/0 0/0 111.2/2 126.2/2 119.3/2 109.4/2 102.5/2 64.3/3

69 0/0 0/0 0/0 115.4/3 95.2/3 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974^81 E 59

	Flight 26, September 2, 1980 (1980.671)

	19 20 21 22 23 24 25 26 27

47 527.1/3 520.5/2 521.7/3 544.9/3 562.4/4 555.0/4 556.0/4 0/0 0/0

48 520.9/3 514.1/3 516.9/3 532.8/2 549.3/3 533.1/4 525.3/4 0/0 0/0

49 520.1/3 501.7/3 499.9/2 512.8/3 516.1/3 515.4/4 518.6/4 0/0 0/0

50 504.3/2 481.0/3 481.5/2 485.4/2 500.4/4 516.0/4 523.7/4 0/0 0/0

51 491.2/2 480.1/2 472.9/2 482.5/3 522.7/3 529.3/3 526.1/4 0/0 0/0

52 464.1/3 471-2/3 464.1/3 470.2/3 481.2/3 512.3/3 529.0/4 0/0 0/0

53 475.1/3 451.8/3 444.7/3 421.2/3 393.8/3 0/0 0/0 0/0 0/0

54 0/0 357.6/3 386.5/3 397.2/2 377.8/2 365.2/2 0/0 0/0 0/0

55 321.1/3 340.2/2 372.2/2 392.3/2 381.8/2 357.9/2 0/0 0/0 0/0

56 319.9/2 339.7/2 357.6/2 365.1/3 356.2/2 323-9/2 0/0 0/0 0/0

57 309.5/3 321.8/3 327.3/2 300.9/2 282.2/2 273.1/3 259.9/3 0/0 0/0

58 304.1/4 315.9/3 314.3/2 299.9/2 277.6/2 274.9/2 266.5/3 0/0 0/0

59 276.4/3 292.0/3 287.3/3 269.9/3 263.2/2 262.5/2 249.4/3 227.7/3 191.8/4

60 259.0/4 268.8/2 271.7/3 244.6/2 243.5/2 255.8/3 233-5/3 229-3/2 215.9/3

61 0/0 235..4/2 242.9/3 231.5/2 245.5/2 248.2/2 221.6/3 213-8/3 0/0

62 0/0 0/0 229-0/3 230.8/2 228.6/2 223-2/2 214.2/3 201.8/3 0/0

63 0/0 193.7/2 215.8/3 226.4/2 210.8/2 208.2/2 204.0/3 193-8/3 165.2/3

64 0/0 190.0/3 194.0/2 177-8/2 187-8/2 200.2/2 190.4/3 174.9/3 152.1/3

65 0/0 0/0 162.6/2 159-3/2 171.7/2 187.4/3 179.6/3 165.0/2 143.3/3

66 0/0 0/0 152.6/2 150.9/3 159-4/2 160.5/2 154.6/2 144.2/3 124.3/3

67 0/0 0/0 110.6/2 148.9/2 147.2/2 145.9/3 137.7/2 136.9/3 95.2/3

68 0/0 0/0 0/0 108.1/3 123-1/2 117.3/2 106.6/2 97.6/2 59.4/3

69 0/0 0/0 0/0 112.0/3 89.6/3 0/0 0/0 0/0 0/0
E 60 STUDIES OF COLUMBIA GLACIER, ALASKA

Flight 27, October 30, 1980 (1980.830)

	19 20 21 22 23 24 25 26 27

47 527.2/3 520.8/2 521.7/3 544.8/3 562.0/4 555.2/4 555.4/4 0/0 0/0

48 521.9/3 514.8/3 517.4/3 532.9/2 549.1/3 533-3/4 525.2/4 0/0 0/0

49 520.3/3 501.5/3 499-9/2 512.9/3 516.8/3 515.3/4 518.6/4 0/0 0/0

50 504.6/2 481.3/3 481.8/2 485.9/2 501.1/4 516.3/4 523-7/4 0/0 0/0

51 490.8/3 481.0/2 474.0/2 483.0/3 523-3/4 529.6/4 525.3/4 0/0 0/0

52 465.0/3 472.5/3 466.3/3 472.3/3 484.0/3 512.9/3 526.1/4 0/0 0/0

53 476.0/3 454.5/3 447.5/3 423.7/3 395.8/3 0/0 0/0 0/0 0/0

54 0/0 358.8/3 389.2/3 399.5/3 380.1/2 366.6/2 0/0 0/0 0/0

55 322.2/3 341.3/2 373.8/2 394.4/3 384.1/2 359.2/2 0/0 0/0 0/0

56 321.3/2 341.3/3 359-2/3 367-0/3 357-8/3 324.3/2 0/0 0/0 0/0

57 310.6/3 323.1/3 329.5/2 302.9/3 286.0/2 276.1/3 261.4/3 0/0 0/0

58 304.5/4 316.5/3 316.2/2 302.5/2 279.1/2 275.7/3 267.5/3 0/0 0/0

59 278.3/4 293.6/3 289.2/3 272.8/3 265.1/2 264.0/3 250.3/3 228.6/4 193.0/4

60 260.7/4 270.9/2 274.1/3 247.0/2 246.6/3 257.5/3 234.9/3 230.4/2 216.5/4

61 0/0 237*1/3 246.0/3 234.8/2 248.3/2 250.5/2 223.1/3 214.4/3 0/0

62 0/0 0/0 232.5/3 233.8/2 231-3/2 225.6/2 216.3/3 202.7/3 0/0

63 0/0 194.9/2 217.9/3 228.8/2 213.1/2 211.1/3 205.8/3 194.4/3 166.3/3

64 0/0 190.5/3 195.3/3 181.7/2 189.6/2 202.0/2 192.7/3 176.1/3 153.5/3

65 0/0 0/0 165.2/3 162.0/2 174.8/2 191-1/3 183-1/3 167-6/2 145.8/3

66 0/0 0/0 155.0/2 153-2/3 163-2/2 163-5/2 158.7/2 146.6/3 126.8/3

67 0/0 0/0 112.9/2 152.5/2 151-3/3 147-3/3 141.3/3 139-5/3 98.6/3

68 0/0 0/0 0/0 110.4/3 127-2/3 121.5/2 112.3/3 100.5/2 62.9/3

69 0/0 0/0 0/0 116.6/3 96.1/3 0/0 0/0 0/0 0/0
SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 61

	Flight 28, March 7, 1981 (1981.180)

	19 20 21 22 23 24 25 26 27

47 523.7/3 516.9/2 521.1/3 543.4/3 560.6/4 555.6/4 553-7/4 0/0 0/0

48 520.9/3 512.5/3 517-7/3 533-9/3 548.6/3 532.8/4 524.5/4 0/0 0/0

49 519-8/4 501.3/3 500.6/3 516.9/3 518.1/3 514.9/4 518.3/4 0/0 0/0

50 505.0/3 481.7/3 481.6/3 488.6/3 502.8/4 516.8/4 523-3/4 0/0 0/0

51 490.7/3 483.8/2 475.7/2 484.8/3 522.6/4 530.0/4 526.4/4 0/0 0/0

52 467.7/3 476.2/3 469.6/3 474.0/3 486.6/3 514.4/3 525.1/4 0/0 0/0

53 476.5/3 462.3/3 453.2/3 428.1/3 398.4/3 0/0 0/0 0/0 0/0

54 0/0 365.0/3 394.9/3 403.9/3 384.3/3 371.0/2 0/0 0/0 0/0

55 326.1/3 345.0/3 378.9/3 398.8/3 389.4/2 363-7/3 0/0 0/0 0/0

56 324.3/2 345.0/3 364.6/3 373-7/3 360.8/3 327-3/3 0/0 0/0 0/0

57 315.2/3 327.3/3 333.8/2 306.5/3 290.5/3 280.8/3 265.9/3 0/0 0/0

58 309.3/4 320.9/3 320.4/2 307.8/2 284.9/2 280.1/3 272.9/3 0/0 0/0

59 284.2/4 297.9/3 292.7/2 280.6/3 270.1/2 269.5/3 255.3/4 231.6/4 196.2/4

60 264.4/4 274.6/3 279.5/3 253.1/3 252.2/3 264.1/3 239.3/3 233.8/3 218.7/4

61 0/0 240.0/3 251.4/3 240.8/3 254.5/2 255.0/2 227.7/3 218.3/3 0/0

62 0/0 0/0 237.7/3 239.0/3 236.6/2 231.1/3 221.6/3 207.8/4 0/0

63 0/0 197.3/2 221.0/3 233.4/2 217.8/2 216.1/3 211.4/3 198.7/3 172.2/3

64 0/0 190.4/3 198.2/3 186.7/2 192.4/2 208.3/2 198.6/3 181.6/3 158.5/3

65 0/0 0/0 171.9/3 168.3/2 182.5/3 199.9/3 191.0/3 175.8/2 152.8/3

66 0/0 0/0 162.0/2 162.4/3 175.1/3 171.7/3 169.0/3 154.4/3 134.4/3

67 0/0 0/0 120.8/2 161.0/3 162.8/2 155.0/3 152.0/3 146.5/3 104.0/3

68 0/0 0/0 0/0 120.1/3 136.4/3 129.4/3 121.6/3 109.1/3 72.0/3

69 0/0 0/0 0/0 127.9/2 114.7/3 0/0 0/0 0/0 0/0
E 62 STUDIES OF COLUMBIA GLACIER, ALASKA

	Flight 29, June 16, 1981 (1981.457)

	19 20 21 22 23 24 25 26 27

47 519.0/3 510.7/2 517.0/3 540.9/3 558.6/4 553-9/4 551.8/4 0/0 0/0

48 518.1/3 508.6/3 516.2/3 531.9/3 546.6/4 531.3/4 522.7/4 -0/0 0/0

49 518.1/4 500.1/3 500.3/3 516.7/3 516.5/3 513.2/4 516.5/4 0/0 0/0

50 503.2/3 480.6/3 480.8/2 488.4/3 502.0/4 515.2/4 521.6/4 0/0 0/0

51 489.6/3 483.2/3 474.5/2 483.8/3 519-8/4 527.4/4 524.8/4 0/0 0/0

52 467.2/3 475.6/3 468.4/3 471.3/3 480.9/3 509-3/3 523.3/4 0/0 0/0

53 473.4/3 462.2/3 453.4/3 427.0/3 395.6/3 0/0 0/0 0/0 0/0

54 0/0 365-1/3 395.0/3 403.2/3 385.1/3 371.3/2 0/0 0/0 0/0

55 326.8/3 345.6/3 378.4/3 398.0/3 390.0/2 363.3/3 0/0 0/0 0/0

56 324.9/2 346.0/2 363.2/3 372.5/3 360.8/3 325.9/3 0/0 0/0 0/0

57 315.8/3 328.5/2 332.9/3 304.9/3 289-3/3 280.6/3 266.0/3 0/0 0/0

58 308.5/4 320.7/3 318.6/2 306.6/2 284.9/2 281.2/3 273.0/3 0/0 0/0

59 283.2/4 294.9/3 289.6/2 280.6/3 270.1/2 270.8/3 255.6/3 230.5/4 194.9/4

60 262.7/4 272.1/3 277.1/3 254.3/3 252.6/2 266.2/2 241.1/3 233.8/3 217.7/4

61 0/0 237.8/3 250.8/3 242.3/2 255.3/2 256.2/3 228.3/3 218.6/3 0/0

62 0/0 0/0 236.6/3 240.0/3 238.0/2 231.5/3 221.3/3 207.5/3 0/0

63 0/0 195.6/2 219.4/3 232.9/2 218.8/2 217.0/2 212.0/3 198.6/3 171.6/3

64 0/0 188.9/3 197.6/3 187.4/2 193.3/2 208.6/2 198.1/3 179.6/3 156.0/3

65 0/0 0/0 171.8/3 168.2/2 182.4/2 199.0/2 188.3/3 171.4/3 149.4/3

66 0/0 0/0 160.8/2 160.8/2 172.7/2 168.6/3 165.8/3 150.2/3 131.2/3

67 0/0 0/0 119.7/2 159-7/3 160.0/2 151.8/2 150.1/2 144.4/3 101.4/3

68 0/0 0/0 0/0 115.8/2 133.8/3 124.3/3 118.6/3 108.8/3 72.5/3

69 0/0 0/0 0/0 124.9/3 114.1/3 0/0 0/0 0/0 0/0
	SURFACE TOPOGRAPHY, LOWER PART OF COLUMBIA GLACIER, ALASKA, 1974-81 E 63

	Flight 30, September 1, 1981 (1981.667)

	19 20 21 22 23 24 25 26 27

47 524.2/3 517.0/3 518.8/3 541.0/3 559.0/3 553.5/3 554.3/3 0/0 0/0

48 517.9/3 511.4/3 515.1/3 528.5/3 544.1/3 530.1/3 523-7/3 0/0 0/0

49 518.0/3 500.4/3 500.1/3 513-9/3 514.5/3 513-8/3 517-1/3 0/0 0/0

50 502.5/3 479-3/3 480.5/3 486.6/3 500.9/3 514.5/3 522.2/3 0/0 0/0

51 488.7/3 478.1/3 471.2/3 479-6/3 517-6/3 524.7/3 523-4/3 0/0 0/0

52 463-1/3 468.9/3 461.3/3 463-5/3 473-4/3 506.0/3 524.8/3 0/0 0/0

53 468.0/3 448.3/3 441.6/3 416.4/3 389-6/3 0/0 0/0 0/0 0/0

54 0/0 354.3/3 383-3/3 393-3/3 376.3/3 363-6/3 0/0 0/0 0/0

55 319-4/3 339.6/3 369-5/3 387-3/3 377.1/3 355.3/3 0/0 0/0 0/0

56 317.4/3 337.9/3 354.8/3 361.2/3 351.3/3 321.9/3 0/0 0/0 0/0

57 307-6/3 320.8/3 329-3/3 305.2/3 287.4/3 275.1/3 260.8/3 0/0 0/0

58 301.7/3 311.4/3 309.2/3 301.2/3 280.8/3 275.9/3 266.6/3 0/0 0/0

59 275.5/3 288.0/3 281.8/3 272.0/3 265.4/3 265.7/3 250.0/3 226.5/3 190.8/3

60 257.3/3 266.9/3 269.8/3 247.5/3 247.6/3 256.6/3 233.6/3 228.4/3 214.4/3

61 0/0 232.5/3 243.5/3 235.7/3 247.7/3 248.1/3 221.7/3 213.0/3 0/0

62 0/0 0/0 229.0/3 234.9/3 230.8/3 224.8/3 214.9/3 200.8/3 0/0

63 0/0 187.6/3 211.1/3 224.3/3 211.6/3 210.1/3 205.7/3 193-3/3 163-4/3

64 0/0 186.0/3 191-9/3 179-0/3 189.8/3 203-3/3 192.3/3 174.2/3 149-9/3

65 0/0 0/0 163-9/3 163-2/3 174.7/3 190.0/3 182.2/3 166.8/3 143-6/3

66 0/0 0/0 151.4/3 151.6/3 158.0/3 160.5/3 158.2/3 145-0/3 124.8/3

67 0/0 0/0 112.7/3 145.9/3 142.5/3 143-8/3 141.0/3 137-4/3 95-8/3

68 0/0 0/0 0/0 101.9/3 116.4/3 111.0/3 106.1/3 95-6/3 52.8/3

69 0/0 0/0 0/0 104.5/3 84.7/3 0/0 0/0 0/0 0/0