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dissertation entitled

REMOTE SENSING 0F WHOLE-FIELD FLOW RATES
OF GLACIERS AND OTHER LARGE BODIES
USING INTERFEROMETRIC METHODS

presented by

EDGAR GEORGE CONLEY

has been accepted towards fulfillment
of the requirements for

Ph. D. degreein APPLIED MECHANICS

,/%mi6&m/

ajol‘ professor
GARY LEE CLOUD

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DePa

REMOTE SENSING OF WHOLE—FIELD FLOW RATES
OF GLACIERS AND OTHER LARGE BODIES
USING INTERFEROMETRIC METHODS
BY

Edgar George Conley

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Metallurgy, Mechanics and Materials Science

1986

 

 

 

 

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ABSTRACT
REMOTE SENSING 0F WHOLE—FIELD FLOW RATES
OF GLACIERS AND OTHER LARGE BODIES
USING INTERFEROMETRIC METHODS
By
Edgar George Conley

The flow of glacier ice is mapped using noncoherent (white)
light speckle interferometry. Time-lapse, high resolution,
double exposure of a sun—illuminated glacier surface yields
photographic transparencies which diffract coherent light in
the manner of Young's experiment. In Young’s experiment, a
pair of knife—edge slits diffract light such that the
resulting interference fringe patterns are indicative of the
spacing between the slits. Such slits may be generated
photographically simply by imaging shadows appearing on a
non-coherently illuminated, optically rough surface. Fringe
patterns arising from the double exposed glacier surface
image yield information about ice displacement taking place
during the exposure interval. The method is non-contacting,

non-invasive and whole-field.

This thesis describes experimental application of the white
light speckle method to the surfaces of both the Nisqually
Glacier, WA and to the Ptarmigan Glacier, Juneau, AK.
Noncoherent imaging and coherent optical processing theory
are presented along with experimental and data analysis

methods.

 

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Edgar George Conley

Limitations on applicability, resolution and accuracy are
imposed by weather, atmospheric turbidity and camera motion
occurring between exposures. Weather problems can be
minimized by proper choice of field season. Methods to
’freeze' atmospheric turbulence and to gauge the motion of

the camera are proposed.

The results from the field seasons are consistent with the
flow-rates collected by conventional means at the Nisqually,
whereas poor weather ruined results at the Ptarmigan. On
the limited areas of overlap for which comparison can be
made, this method yields surface flow rates of 0.6 meters
per day while surveying methods yield about 0.4 meters per

day.

Good agreement with the results of traditional measurement
techniques Suggests other applications of the white light

speckle method. These include, determination of river and
sea-ice flow, mine wall strains, volcanic slope stability,

and possibly, tectonic plate strains.

Ga]

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ACKNOWLEDGEMENT

The greatest debt owed by an author is to the many
scientists who took the time and effort to write down their
knowledge. I thank them. The greatest personal debt owed
by this author is to my advisor and confidant, Professor
Gary Cloud. I wish to express my appreciation for his many

years of unfailing assistance and encouragement.

I am also grateful to advisory committee members Professors
T. H. Edwards, J. F. Martin and H. F. Bennett whose care and

concern, on my behalf, was obvious.
Finally, I acknowledge the patience and dedication of Mr.

Eric Burgoon who, during the early stages of the research,

offered invaluable assistance in both field and laboratory.

iv

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TABLE OF CONTENTS

1. INTRODUCTION

I. THE GEOPHYSICAL MEASUREMENT PROBLEM ..................... 1
II. HISTORICAL BACKGROUND-SPECKLE INTERFEROMETRY ........... 3
A. Young's experiment .................................... 3
B. The laser speckle phenomenon .......................... 4
C. Laser speckle applied to displacement measurement ..... 6
D. White light speckle method ............................ 8
E. Merging problem and solution .......................... 9
III. APPLICATION TO GLACIERS .............................. 10
A. Glacier study ........................................ 10
B. Historical background-glacier flow rate measurement..13
IV. SUMMARY ............................................... 14
V. DISSERTATION ORGANIZATION .............................. 14

2. THEORY OF IMAGING AND FILTERING

I. IMAGING ................................................ 16
A. Coherent illumination ................................ 17
B. Noncoherent illumination ............................. 30
C. The effect of aberrations ............................ 37

II. FILTERING ............................................. 39

III. SUMMARY OF IMAGING AND FILTERING RESULTS ............. 51

IV. SOME PRACTICAL ASPECTS ................................ 59
A. Sensitivity .......................................... 59

a. Large shear ......................................... 60

b. Small shear ......................................... 62

B. Fringe pattern measurement ........................... 68
C. Other limitations .................................... 71
V. WHOLE FIELD FILTERING .................................. 71
VI. SUMMARY ............................................... 72

3. METHOD AND RESULTS

I. FIRST YEAR EXPERIMENTS ................................. 73
A. Preliminary development tests ........................ 73
B. Field tests-Nisqually Glacier, Washington ............ 86
C. Fringe pattern analysis .............................. 90
D. Preliminary conclusions .............................. 98

II. YEAR-TWO EXPERIMENTS ................................. 102
A. Preliminary development tests ....................... 102
B. Field tests - Nisqually Glacier, Washington ......... 107
C. Field tests - Ptarmigan Glacier, Alaska ............. 110
D. Fringe pattern analysis ............................. 112

III. SUMMARY - EXPERIMENTAL RESULTS ...................... 117

v

 

4. DISCUSSION AND CONCLUSIONS

I. APPARENT SYSTEM RESOLUTION ............................ 123
II. ERROR ANALYSIS ....................................... 136
A. Fringe pattern measurement .......................... 136
B. Rigid body motion determination ..................... 137
C. Ice motion projection onto glacier surface .......... 139
D. Accumulated errors .................................. 140
III. COMPARISON OF RESULTS WITH ICE FLOW THEORY .......... 142
IV. FURTHER STUDY ........................................ 148
A. Increasing precision ................................ 148
a. Rigid body motion determination .................... 148
b. Interferometric determination of rigid body motion.150
B. Other applications of the white light method ........ 150
APPENDICES

1. PHOTO PROCESSING DETAILS
2. COMPUTER CODE - FORTRAN V

vi

 

 

 

LIST OF TABLES

3.1 ICE DISPLACEMENT COMPONENTS FOR 24 HOUR

INTERVAL - NISQUALLY ................................. 122
4.1 SPECTRAL RESPONSE OF IMAGING SYSTEMS FOR
VARIOUS CONDITIONS ................................... 125
vii

 

 

CHAPTER 1

Figure
Figure
Figure

1.1
1.2
1.3

CHAPTER 2

Figure
Figure
Figure

Figure
Figure

Figure
Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

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LIST OF FIGURES

Young's experiment ............................ 5
Speckle pattern micrograph (from Cloud [63])..7
Young's fringes generated from specklegram....7
Imaging geometry ............................. 21
Reference sphere aperture approximation ...... 21
Geometric interpretation of aperture
autocorrelation .............................. 36
Autocorrelation for circular aperture ........ 36
Autocorrelation for circular aperture with

small de-focus ............................... 4O
Filtering geometry ........................... 42

Ideal rectangular frequency function, its
auto—correlation and squared

autocorrelation .............................. 48
Squared autocorrelation of aperture function

- light intensity distribution on transform

plane ........................................ 57
Cosine-squared (Young’s) fringes for moderate
object displacement between exposures ........ 57

Product of Figures 2.7a and 2.7b - transform
plane intensity for fine object structure
recorded through circular aperture for

moderate object displacement ................. 57
Narrow-band object spatial frequency
distribution ................................. 58

Product of a, b and d above - transform

plane intensity for narrow-band object

spatial frequencies recorded through

circular aperture ............................ 58
Transform plane fringes for object

displacement which is of the same order as
speckle cell size ............................ 61
Transform plane fringes for object

displacement which is less than speckle cell

size ......................................... 61
Airy disk intensity distribution - point
spread function for circular aperture ........ 65

viii

 

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F

REFEREE.

Fi

Fi

 

Figure 2.10 Rayleigh criterion for minimum resolvable
separation between object points (referred
to film plane) ............................... 65
Figure 2.11 Typical non-ideal point spread function for
which the Rayleigh criterion will be invalid.67
Figure 2.12 Fringe pattern shift with non-unity
visibility for moderate object displacement..67

Figure 2.13 Whole-field filtering scheme ................. 70
CHAPTER 3
Figure 3.1 Communication Arts Building - artificial
field displacement experiment subject ........ 75
Figure 3.2 Whole—field fringes from Communication Arts
Building image ............................... 75

Figure 3.3 Fringe bands extending from upper left to
lower right enclosed in halo from artificial
field displacement tests - subject, asphalt

walkway ...................................... 8O
' Figure 3.4 Gravel parking lot image ..................... 80
Figure 3.5 Diffraction halo and fringe bands running
from lower left to upper right - subject,
graveled parking lot ......................... 80
Figure 3.6 Nisqually area map — the symbols . and‘
indicate the year-two camera locations ....... 85
Figure 3.7 Print of double—exposure 24 hr plate,
Nisqually, 1981 (year-one) ................... 91
Figure Fringe patterns from Nisqually icefall ....... 92

3.8
Figure 3.9 Fringe patterns from Nisqually down-glacier
(bands run from upper left to lower right)...92

Figure 3.10 Computer projection algorithm ................ 96
Figure 3.11 Final results from yearvone .................. 97
Figure 3.12 Euler angle algorithm ....................... 101
Figure 3.13 Specialized camera from year-two ............ 106
Figure 3.14 Camera sites relative to glacier surface....106
Figure 3.15 Prints from 24hr stereo plates, year-two....109
Figure 3.16 Fringe patterns from rock strata, Ptarmigan

glacier valley wall (fringe bands extend from
upper left to lower right) .................. 109
Figure 3.17 Ice displacement results from one of the
year—two stereo plates showing 24 hour ice
displacement with: no rigid body correction,
raw rigid body correction and forced, rigid
body correction ............................. 118
Figure 3.18 24 hour Nisqually ice displacement for years
—one (marked with I) and -two (marked Oand
A: ) field seasons with one comparison point

from Hodge [28] ............................. 119
Figure 3.19 Repeat of Figure 3.18 with results (markedCD)
from single plate (whose mate was lost) ..... 120

ix

 

CHAPTER 4

Figure 4.1 Modulation transfer function for
atmospherically degraded images - abstracted

from von der Luhe [57] ...................... 131
Figure 4.2 Hufnagel’s atmospheric turbulence model -
abstracted from Korff [56] .................. 131

 

 

 

CHAPTER 1

INTRODUCTION

I. THE GEOPHYSICAL MEASUREMENT PROBLEM

Fundamental principles of mechanics have, in the past three
decades, been applied to the study of motion of tectonic
plates, to the stability of volcanic slopes, to the flow of
ice and it's interaction with surroundings, and to other
naturally occurring systems. Theories of rheology, mass
transport, and heat transfer are being related to knowledge
gained through observation. Physical principles are
insufficient to render a tractable set of equations:
Phenomenological relations are required. Since metrology is
the principal means of gathering the required facts,
accurate experimental determination of boundary conditions
is essential. Such information is needed to guide the
thinking of theoreticians, as well as to check the

predictions of their models.

Major components of the boundary set of geophysical
processes are surface velocity and strain rate maps.
Geophysical systems, however, pose a singular problem in
measurement. Very long time scales and distances (i.e. gage
lengths) are inherent in traditional methods. Measuring

techniques, in general, locate objects or distinctive

 

 

 

 

features in the the field of interest relative to one
another or to some fixed point by surveying, photogrammetry,
taping, etc. Large systems require extensive grids of
either natural or artificial markers, while instruments and
methods must yield displacements of these features accurate
to better than one percent. The North American continent,
for example, drifts west about .1 meter per year [1] and
measures roughly 5000 kilometers from coast to coast.
Determination of annual continental strain rates requires a
technique accurate to about 1 part in 50 million. Such
accuracy exceeds the limits of standard measurement

strategy.

Accuracy is also eroded by certain common analysis
techniques. Average velocity of a glacier surface feature,
for example, is calculated by dividing the distance the
feature has traveled by the elapsed time. This value is
assigned to the point mid-way between the feature's initial
and final positions. Rasmussen [2] has pointed out that,
"The average velocity does not occur at the mid—
point...un1ess the actual velocity distribution belongs to a
restricted class of mathematical functions". Errors

amounting to several percent can result.

In addition to accuracy limitations, mechanicians also note
the conflict between Eulerian (spatial) and Lagrangian

(material) coordinates systems inherent with long time

 

 

 

 

scales and gage lengths. An Eulerian system should be used
since strain is a function of both time and space. Long
time scales, required to obtain meaningful readings, result
in the gross movement of the gage site which obscures the
Eulerian character of the flow. The Lagrangian view, which
devotes attention to a single particle flowing in the system
of interest, is most often used in the analysis of fluid
flow. Long gage lengths, which cannot account for strain
gradients within a gage site, obscure Lagrangian depiction.
Thus, efforts devoted to field strain rate measurement of
many important geophysical processes were frustrated, and
new methods were called for. Noted glaciologist W. S. B.
Patterson [34], in 1969, wrote, "...now the prime need is

for more, or rather better, data."

II. HISTORICAL BACKGROUND - SPECKLE INTERFEROMETRY

Experimentalists required a simple technique having short
gage lengths and time scales which yields whole-field maps
of surface strain rates and velocities. Noncoherent (white)
light speckle interferometry is such a technique. This
optical method is an offshoot of the new, but well
understood, methods of holographic interferometry. Five
important developments are relative to the maturation and

application of the method:

 

 

 

A. Young’s experiment

In 1889, English physician Thomas Young discovered that a
pair of closely spaced knife-edge slits, when illuminated by
nearly coherent light, gives fringes which are indicative of
the spacing and orientation of the slits as in Figure 1.1.
The distance between the slits is approximated by the

relation

for d >> A [1.1)

where d is the distance between the slits, A is the
wavelength of the slit illuminating coherent light, L is the
distance between the plane of the slits and the observation
screen (on which the fringe patterns are viewed) and S is
the separation between adjacent fringe axes. (Derivation of

this formula is given in Chapter 3.)

B. The laser speckle phenomenon

The speckle effect, visible when coherent (laser) light
impinges on a surface, was considered a nuisance during the
first experiments in holographic interferometry. Coherent
light, reflected from a surface with inherent natural
roughness, recombines at the eye to form bright ’speckle

cells’, covering about 5% [9] of a dark background. This

 

illuminating
beam

wavelength: x

vounc’s EXPERIMENT .

   
  

FRINGE
INTENSITY

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observation
screen

on;

Figure 1.1 Young's experiment

 

m

random interference, which causes optical 'noise' in
holography is, nevertheless, uniquely characteristic of the
illuminated surface; if the surface moves, so does the
intact pattern of speckles. High resolution, double
photographic exposures of such a surface, taken before and
after small, uniform in-plane displacement of the surface,
renders rows of equally sheared speckle patterns appearing
in the processed image. When viewed under the microscope,
such an image has a distinctive 'corn-row' pinhole
appearance as shown in Figure 1.2 (reproduced from Cloud
[63]), first noticed by Burch and Tokarski [3]. As reported
in their landmark paper, these workers photographically
produced a multiple pinhole image structure which acted as a
diffraction grating. Such a grating diffracts coherent
light in the same manner as Young’s optical slits. Referred
to the film plane, the object displacement between exposures

is thus given by eqn. (1.1}.

C. The laser speckle method applied to displacement and

strain measurement

Merging the ideas in the last two paragraphs, Archbold,
Burch and Ennos [4], applied the laser speckle method to
surface displacement analysis. If photographic resolution
is sufficiently high, if the speckle cells are ’contrasty’,
and if displacement is neither too large nor too small,

Young’s fringes result if the double exposed, processed

 

Figure 1.2 Speckle pattern micrograph (from Cloud [63])

Young's Fringes

 

 

Figure 1.3 Young’s Fringes generated from specklegram

 

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photograph (specklegram) is interrogated with a narrow beam
of coherent light as in Young's experiment, Figure 1.3.
(Range of useful sensitivity is discussed in the following
chapter.) Furthermore, the fringe patterns will be
indicative of the displacement vector, which is
perpendicular to the fringe axis, in the plane of the film,
according to eqn. {1.1}. In a practical application of
Archbold's experiment, Cloud [5] has successfully gaged the
rotation of an aluminum disk using the laser speckle

technique.
D. The white light speckle method

In a critical development, Boone and DeBacker [11], in 1975,
applied the speckle method to a specially prepared
retroreflective surface which was illuminated by noncoherent
(white) light. Extending this idea, Boone and DeBacker also
reported the requisite speckle structure also appeared in
the photographic image when the light source is placed to
emphasize natural surface roughness. In their experiments,
a concrete beam (presumably not polished) was illuminated
with an ordinary 250w lamp placed close to the plane of the
surface. Thus, interference, as in laser speckle, was not
necessary for the production of speckle cells. Geometric
shadows appearing on the object surface suffice.

Illumination with a (relatively expensive) coherent source

 

 

and special surface preparation was avoided, paving the way

for many new applications.

Other investigators quickly followed, reporting novel
applications and extensions of the new method. In 1975,
Burch and Forno [12] reported success using the same ideas
and 'tuned’ apertures. Special masks fitted to the
recording camera lens selected specific frequency bands and
preferred displacement directions, thereby filtering
unwanted frequencies and noise from the optical signal
before photographic recording. Maddux [14], in 1975,
reported on an automated digital data acquisition system
which logs fringe pattern information using diode arrays and
computing machines. In a biomedical application, Cloud [15]
applied the white light method to strains in skin in vivo.
Except for the work of this laboratory [59] [60] [61] [62],
little practical application of the method outside of the

laboratory has been reported.

E. Merging problem and solution

Applications of optical metrology offered in the literature
as illustrations of a new theory or technique are usually
experiments using engineering materials which measure less
than 1 meter in length. This is understandable since these
objects are easily handled in machine shop and laboratory.

Hence mechanicians, expecially experimentalists, can become

 

 

 

 

10

'stuck' thinking small. Geologists, on the other hand,
accustomed to working with bodies many kilometers in size
would not be given to inquire of the latest techniques to
detect displacements on the order of the wavelength of
visible light in 'typical' engineering materials such as

steel, aluminum or composites.

In 1978, Cloud [5], [15] noted that the texture of ice
(appearing in a photograph) of an Alaskan glacier seemed to
have the prescribed roughness which made Boone and
DeBacker’s experiment successful. Glaciers are heavily
marked by surface undulations, crevasses, suncups, windborne
dust, moraine (till) and other materials. Cloud speculated
that such large and small scale random patterns could be
utilized to generate the speckle effect and proposed to the
National Science Foundation [16] a study of the feasibility
of applying the white light speckle method to glaciers and
to other dynamic geological systems. For the first time,
powerful, new interferometric techniques were brought to

bear upon large, naturally occurring surfaces.

 

 

11

III. APPLICATION TO GLACIERS

A. Glacier study

Among the reasons to study glaciers is that glaciers move.
Movement, amounting to roughly 1 meter per day [17], is
relatively fast compared with other geophysical processes
and so provides ’quick' verifications of new methods in
large scale metrology and flow theories. Glaciers thus
provide a convenient proving ground for basic concepts
regarding metamorphic flow of rock within the earth.
Justification lies partly in the fact that rock, as well as
ice, is a polycrystalline substance and should deform in
similar ways [18]. Theories of plate tectonics, according
to Weertman [l9], face crucial tests in the interiors of
glaciers. Glacier flow, on a smaller scale, mimics the
behavior of large ice sheets. An understanding of these
bodies is important since large scale changes in polar ice,
for example, affect world climate and sea level. Thus,
recent theories of glacier mechanics may have application

beyond the immediate goal.

In addition to yielding understanding of basic geology, the
study of glacier flow deserves special attention. Glaciers
are an important shaper of topography in the form of erosion
and deposition [40]. As world population expands, and more

demand is placed upon natural resourCes, geologists will

 

 

 

12

require detailed histories of the forces which shape the
surface of the planet in order to exploit these resources.
Glacier regions, for example, form important hydrological
power sources in the Netherlands [21] and elsewhere in
Europe [22]. Understanding of glacial mass balance is
necessary to provide for the increased electrical demands of

these areas.

Glaciers also provide a sensitive climatological indicator
[23]. Now, glaciers cover about 10% of the earth's surface.
Siberia, Europe and most of North America were cloaked
during the last ice age [26] but despite much work,
dramatic, sudden growth of ice sheets remains unexplained.
Understanding the growth of large ice masses requires
sophisticated appreciation of long and short term dynamical
processes. Detailed knowledge of glacier dynamics is also
critical to dating layers of ice extracted from boreholes.
Pollen, oxygen, radioactivity content and other substances
trapped in the ice offer clues to past climates and to
predictions of possible future glacier behavior. The
'greenhouse theory', for example, may be substantiated by
determining the carbon dioxide content of air bubbles

trapped in ancient glacier strata.

 

 

13

B. Historical background-glacier flow rate measurement

As in other geophysical disciplines, the early 1950's saw
the concepts of mechanics and physics applied to the glacier
flow problem (e.g. [24], [25]). Accurate knowledge of flow
rates was required and geologists turned to orthodox
methods. The classic method of glacier surface strain
measurement is to set stakes in the ice and periodically
measure the distance between the stakes with tape,
theodolite, photogrammetry or, more recently, electronic
distance measuring devices. Strains are calculated by
dividing the change of distance by the original distance
between stakes. While this method is simple and
straightforward, its precision is impaired by several
factors. Physical access to the stake site is necessary; an
obvious drawback during certain seasons on glaciers with
steep icefalls. Stakes tip, melt into or out of the the ice
(c.f. [43]), or become covered with snow since measurement
over long time periods (perhaps years, e.g. Hodge [28]) may
be required. Whole field flow or strainrate determination
requires a grid of stakes covering much of the surface.

Such grids are impractical to put in place and maintain
except on a very small scale. Because of these pitfalls the
staking method is used principally to determine centerline
flow rates [28] or shear flow across selected cross sections

[29].

 

14

Warner [30] and Warner and Cloud [31] have used a
substantially different approach. A classic strain rosette,
using wires frozen into the glacier surface yielded
strainrates for a temperate valley glacier. Short gage
lengths and time scales resolved the conflict between
Lagrangian and Eulerian coordinate systems but the method is

tedious and provides information for just one point.

IV. SUMMARY

The white light speckle method provides field workers with a
simple technique having infinitely small gage lengths and
relatively short time scales which yields full field maps of

glacier surface flow rates. The acquisition of data

requires only a perch from which to obtain high resolution
photographs of the straining surface, lighting conditions
emphasizing natural surface roughness and two clear periods
of weather providing optical access to the subject. Glacier
surface displacement rates are subsequently determined in

the laboratory through coherent optical processing and

comparison with topological maps.

V. ORGANIZATION OF DISSERTATION

The following chapter details the theory of data recording
(imaging) and processing (filtering) supporting the

interferometric speckle technique. Relevant practical

 

 

15

aspects of applying the noncoherent method to displacement

analysis are discussed.

Chapter 3 details the feasibility of applying the technique
to the surface of two temperate valley glaciers: Nisqually
Glacier, Mt. Rainier National Park, WA, and the Ptarmigan
Glacier, Juneau Ice Field, Juneau, AK, USA. Surface flow
rates and geometry of each body are relatively well

documented, allowing limited comparison of the results of

the new method to the results of established techniques.

This chapter recounts experimental procedures and elaborates
results of field seasons 1981 and 1982. Choice of
instruments and methods for the second field season were
largely determined by the experience of the first year.

Preserving chronological order, preliminary conclusions,

based on first year experiments are given in this section.

Discussion and conclusions, error analysis, recommendations
for further study and adaptation of the optical method to

other measurement problems are given in the final chapter.

 

CHAPTER 2
THEORY OF IMAGING AND FILTERING

P.M. Duffieux, in 1946 [50], was first to suggest that

Fourier methods be applied to optics. Other workers have

extended this notion until, at this time, there is little

distinction between the mathematics of optics and

communication theory. Following these pioneers, this paper

examines the imaging and filtering processes using frequency

analysis.

I. IMAGING

Recent texts (c.f. [46], [47]) describe the noncoherent—

light imaging process as an extension of the coherent-light

process. This is the method used below. Comparisons are

made, where informative, between the coherent and
noncoherent cases. One-dimensional equations are used
because of their relative simplicity and facility. Little
generality is lost by employing one-dimensional analysis.
In some circumstances, however, appeal is made to intuition

by describing the equations in terms of their multi-

dimensional effects.

Wave energy which is capable of interfering with itself is

considered to be spatially coherent. In the first case

17

discussed below, the object surface is illuminated with

coherent light. Such illumination, when reflected or

scattered by an optically rough surface, interferes

(recombines) at the eye to form a random pattern of bright

points known as speckle cells. In the second case discussed

below, the object surface is considered to be illuminated
with noncoherent light. Noncoherent illumination is not
capable of producing interference effects. The mathematics

in this chapter provides theoretical basis for this

behavior.

The restricted (Fraunhoffer) case is assumed throughout,

where object and image points are located at very small

angles from the optical axis.

A. Coherent illumination

Consider the case where an object surface is illuminated
with spatially coherent light and, given the amplitude of
the reflected light as a function of object plane

coordinates, the corresponding amplitude distribution of the

image is to be determined. Following Born and Wolf [20],

assumptions are as follows:

major
Coherent illumination of the object surface.
Small angles with respect to the optical axis. This
is an important restriction; point sources produce spherical

wavefronts. If a sufficiently small spherical portion is

 

18

viewed, however, the wavefront can be considered plane.

Such a condition meets the so-called Fraunhoffer diffraction
restrictions. Fraunhoffer conditions have wide application
in applied optics, greatly simplifying the mathematics.

Unity magnification.

For the real images considered, the subscripts
indicating object and image points indicate object-image
inversion. That is, left on the object is right on the
image.

Unity index of refraction

Isoplanatism, defined as the existence of a region
sufficiently small such that a displacement in the object
plane results in no change of shape, orientation or
distortion of the image. In this region the point spread
function (that is, the image of a point source) is constant

as the source point explores the object surface.

Integration of complex quantities often results in a
constant multiplier. Such constants, indicating phase
change of a wave propagating through a lens, losses due to
reflection or absorbtion or diminished intensity of
spherically radiating waves, for example, are usually

neglected in this type of work.

The observable or detectable quantity is intensity, defined

in terms of the complex amplitude as

 

19

me = [U(x>12 = Hos) U*<x>.

Recalling the properties of complex functions, squaring the
complex amplitude obliterates phase information; it is
unnecessary to carry such arithmetic baggage through the
integration process. Thus, the expression, "...ignoring
nonessential phase terms (or constants)...", is often used
following an integration. The consequence is less

cumbersome, and perfectly useful expressions.

Consider the case where light energy is transmitted through
an aperture from object to image planes as in Figure 2.1.
The nomenclature is as follows:

xo , x1 and a are coordinates in the object, image and

aperture planes respectively.

Uo(xo) is the complex amplitude of the light

disturbance on the object plane. In this development, the
notion of complex amplitude will be used. Eliminating the
time dependence of the electric field vector, complex
amplitude is a function of position only, thus reducing and
simplifying the analysis with so-called 'scalar diffraction
theory'.

K(xo;xl) is the transmission function (also called the

transmission factor or the amplitude impulse response).
This function is defined as the complex amplitude of the

disturbance on the image plane due to a disturbance in the

 

20

object plane which is of unit amplitude and zero phase and
which extends over a unit length of the object line (in the
current one-dimensional system). The general notational

form, K(xo;x1), indicates a nonlinear phase change occurring

in the process of transmitting wave energy from object to
image. 'Nonlinear phase change’ means that the phase of a

wave, propagating for example, from xo to x1, does not

depend simply on the distance between the two points.
Below, K is given a more restrictive, linear form.

Ui(x1) is the complex amplitude of the light

disturbance in the image plane.

Beginning the analysis, consider the contribution of light

amplitude in the image plane at location xl from a light
disturbance originating at X0 in the object plane. The
contribution is dUi(xl) = Uo(xg K(xo;x1) dxo. The total

is

amplitude at point xl

Ui(x1) = [+go(xo) K(xo;xl) dxo, {2.1}

which is recognized as the convolution of the input

amplitude with the amplitude impulse response.

OBJECT
PLANE

 

21

APERTURE
PLANE

Figure 2.1 Imaging geometry

   
 

REFERENCE
SPHERE

IMAGE
PLANE

 

 

 

Figure 2.2 Reference sphere for circular aperture

 

22

The integral extends formally to infinity but actually
extends only to the limit of the object. In the remainder
of this chapter, all integrals will extend to i w, except
where indicated. This is done, it shall be seen, to

maintain the formalism of the Fourier integral.

Now the task is to generate an expression for K, the

amplitude impulse response. Consider a unit source placed

I
in the object plane at x0, that is,

Uo(xo)=6(xo- xé),

where 6 is the Dirac delta function. Substituting this into
eqn. {2.1) and noting the sifting properties of the delta

function gives

.Ui(x = K(x;;x {2.2}

1) 1)'
That is, the transmission function now represents the image

plane disturbance due to a point source.

Now take a large sphere (the reference sphere) centered

I
about a point, x1, on the image plane as shown in Figure

I I
2.2. For the time being, let the reference axis be xoxl.
The classic assumption here is that such a large spherical

surface (compared with the aperture), at very small angles

 

23

is a good approximation to the aperture plane. (This is the
Fraunhoffer condition.) Let there be a point disturbance on
the object plane and let the resulting disturbance, H(a), on
the reference sphere (the sphere is represented by a circle

in the vertical one dimensional analysis) be written as

ZfllR
H(a) = an)?“ A) (2.3)

where R is the focal length of the imaging lens placed in
the aperture and x is the wavelength of the illuminating
light. The phase of G is the phase aberration of the system
while the amplitude of G is the amplitude distortion of the
image forming wave. G is the so—called 'pupil function’ and
is taken to be constant inside the aperture and zero

outside. Mathematically this is expressed as

Though it is filled with a lens, the aperture is assumed
’clear' for the following reason. It is assumed that thin
lenses act as light benders only. They simply affect the
phase of the wave energy passing through them. As long as
the energy is directed to the necessary place (i.e. the
focal plane), phase information, for the present purpose, is

'nonessential'. In the absence of such a lens the focal

 

 

24

plane would simply be displaced to another location (i.e. at
infinity).

I

The disturbance at point x1 in the neighborhood of x1 on the

image plane is the sum of the contributions from each point
on the reference circle which approximately fills the
aperture. Using the Huygens—Fresnel principle wherein each
point Q on the wavefront is considered to be the source of a
new wavelet, the total disturbance on the image plane is
written

saxfl)
Ui(xl) = I H(0) s A dd, {2.4}

where s is the distance from the aperture point Q to the

image plane point x1. This simplified version of the

integral theorem of Kirchhoff is the well-known Fraunhoffer
equation; it is obtained when very small field angles are
assumed. The integrand phase factor exp(2nis/A) and the
factor l/s, (the latter diminishes amplitude with distance)
occur during the propagation of a spherical wave. In Born
and Wolf's rigorous derivation, the denominator s is left in
the equation until it is shown to contribute only a
nonessential constant. This is true only of the denominator
since 5, in the exponent, causes the exponential to vary
rapidly over the imaging plane. Note that when the
Fraunhoffer equation is again invoked in the filtering

section below, the integrand denominator is ignored from the

 

25

onset. In any event, the Fraunhoffer equation states that
the amplitude in the receiving plane is found simply by
integrating the phase factor over the aperture (or, in the
filtering stage, the integration is carried out over the

area subtended by the beam).

Now substituting eqn. {2.3) into eqn. {2.4} and again

ignoring constants gives

ex (_2wiR) ex (Zris)
Ui(xl)= [mm—£2 A 1—5-9 A (10. (2.5)

 

Based on the geometry of Figure 2.2, it is evident that as
the triangle OPQ becomes horizontally stretched, the
distance (R - s) will become a very good approximation to

q . x1, where q is the unit vector directed from O to Q, the

point of disturbance on the reference sphere. (See Born and
Wolf [20] for a mathematically rigorous derivation.) Thus,

for small angles, R - s 2 q - x Rewriting and noting that

1.
in the one dimensional system, the magnitude of q in the

direction of x is thus, s x R - g (x

9.
1 R’ 1)'
The lateral distance from the center of the reference sphere

to the neighboring image point may be represented by the

I I
distance x1- x1, which is approximately equal to x1- x0

 

 

26

(again for small angles), and thus, 3 x R - % (xl- xé).
Substituting this expression for s into eqn. [2.5) gives
2wiR 2;;[R - g(x - x')]
Ui(xl) - I G(a) eXP(' A ) ex?‘ A R 1 ° ’ do
_U_ - I
R [R — R(x1 x°)]

Where R >> a and R >> (x1- x0), which must hold for the
assumed small angles, the integrand denominator reduces to
the constant R2 (which can thus be ignored) and this gives,
. I
2n1(x1- xo)o

Ui(xl) = I C(a) e “R dd. {2.6}

Consider again the transmission function K. K accounts for
amplitude and phase variation as light energy propagates
through the optical system. If this function is
sufficiently well behaved over a small region of space, it
is said to be space (or shift)-invariant. That is, a small
shift in object plane coordinate will result in no

distortion of the image. Throughout this small region, K is
linear depending only on the distance (xl— xé) between the
object and image points. In this case, the transmission
function may be written K(x;;xl) = K(xl- xé), where K is now

defined for a very small ’isoplanatic' region of the system.

 

27

Finally, comparing eqn. {2.6} with eqn. {2.2} and dropping

the primes it is seen that

2£i(x1- xo)a
K(xl- x0) = I C(a) exp[- AR ] do. (2.7}

Thus, the amplitude impulse response of the optical system
is dependent on the pupil function which is, itself,

independent of the object point.

Now an arbitrary transmission function, K(x), may be

expressed in terms of its Fourier Integral:
K(x) = I k(u) exp(—2nixu)du, {2.8a}

where x is the space variable and u is the frequency

variable.

Eqn. (2.8a) can be written in shorthand notation as

K(x) = FT'l(k(u)}.

Since K(x) and k(u) constitute a transform pair,

k(u) = I K(x) exp(2niux) dx = FT{K(x)}, (2.8b}
where k(u) is the frequency response. The amplitude impulse
response and the frequency response are Fourier transforms

of one another.

 

28

Note that if the variables are changed, letting

u = a/AR, {2.9}

then du = do, ignoring constants. Also, if x = (x1- x0),

then eqn. {2.7} and eqn. (2.8a) are similar. Moreover,

k(a/AR) = G(a), {2.10)

i.e. the frequency response is simply a scaled version of

the pupil function.

Recalling a is the coordinate in the aperture plane, it is
seen that the maximum spatial frequency transmitted by a

circular aperture of radius a, for example, is

a
umax— AR“ {2.10a}

Therefore, spatial frequencies appearing on the object plane

greater than umax are 'clipped’ in a manner similar to the
action of an electronic low-pass filter and do not appear in

the image.

In optical work, k(u) is known as the imaging system

frequency response. The modulus of the complex function k

is called the modulation transfer function (MTF).

 

29

Physically, the MTF plays the part of reducing the image

contrast of the higher frequency components.

Note that eqn. {2.1) may be written in a simpler manner by
using the convolution theorem (c.f. [46]) for Fourier
transforms. The theorem states that the Fourier transform
of the convolution of two functions is equal to the product
of their individual transforms. Specifically for the

present case and, using the shorthand notation,

FT{Ui(xl)} = FT(J Uo(xo) K(xl- x0) dxo} = uo(u) k(u).

where, uo(u) = FT{UO(xo)} and, ui(u) = FT{Ui(xl)).
Thus, ui(u) = uo(u) k(u). (2.11}

This simple relation implies that the image is a

superposition of space-harmonic components uo, and that each
component is weighted by the frequency response function

k(u).

For the present purposes there is no longer a need to
consider imaging with coherent illumination. The theory is

sufficiently developed to extend to the noncoherent case and

to make comparisons where necessary.

 

30

B. Noncoherent illumination

Consider the case where an object surface is illuminated by
light having a narrow temporal-frequency spectrum but which
is spatially noncoherent. As before, the observed intensity
is the quantity of interest. Again following Born and Wolf

[20], and using the same assumptions as before, let Io(xo)
be the light intensity occurring over a length dxo on the

object plane. The contribution this source makes to the

intensity at a point x1 on the image plane is

2
dIi(x1) = Io(xo)[K(xl- xo)] dxo, {2.12}

where K is again the transmission function (impulse
response) of the imaging system, defined in the previous
section. It can be shown (c.f. [46] or [47]) that the
noncoherent transmission function is directly proportional
to the square of the modulus of the coherent transmission

function. Thus the total intensity at point xl on the image

plane is
2
Ii(xl) = I Io(xo)[K(xl— xo)] dxo. {2.13}

Certain aspects of coherent and noncoherent imaging can now

be compared. Unlike the coherent impulse response, which is

31

in general complex valued, the noncoherent impulse response
is real and non—negative. With coherent illumination,
therefore, interference can occur, and the resulting image
intensity will exhibit zeros where the corresponding object
points are not zero. This is the speckle effect and
explains why images of noncoherently lit objects do not
suffer from zeros arising from interference; speckle
patterns on such images are simply the result of zeros (i.e.
shadows) of sufficient size appearing on the object surface.
Eqn. (2.13) is identifiable as the convolution of the object
intensity distribution with the squared modulus of the
transmission function. However, it may be expressed in a
much simpler manner. Start by taking the Fourier transform

of each factor:
FT(Ii(xl)) = ¢i(u), {2.14}

FT(Io(xo)} = ¢o(u), (2.14b}

FT{[K(x1- x0)]2} = exp (-27riuxo) FT{[K(x1)]2)

= exp (~2wiuxo) L(u),

32

where, in the last line, a shift in the space coordinate
simply results in a linear phase shift in frequency space.

Ignoring such a phase shift, the equality can be written,
2 2
FT{[K(x1- x0] } = FT{[K(xl] } = L(u) {2.14c}

As in the coherent case, the Fourier transform of the
impulse response is known as the frequency response

function, L(u), of the noncoherent imaging system.

Using eqn. {2.14c} and again using the convolution theorem,

eqn. {2.13} may be written simply as

¢i(u) = ¢o(u) L(u), {2.15}

and it is seen that the spatial frequency spectrum of the
image is, as with coherent illumination, the product of the
object spectrum and the frequency response function.

(Compare to eqn. {2.ll}).

Once again the task is to determine the form of the

. 2 . .
noncoherent transmission function, [K(xl)] . Later, it Wlll

be shown that FT{[K(x1)]2} is equal to the complex

autocorrelation of k(u) where FT{K(xl)} = k(u).

Let P(x) and Q(x) be arbitrary complex functions and let

33

FT{P(x)} = p(u), and FT{Q(x)} = q(u). As a corollary of the

convolution theorem it can be stated that
FT{P(X)Q(X)) = I p(fl) q(u - fl) dfi = p(u) * q(u). {2.16}

where the asterisk indicates convolution and fl is the
shifting variable of integration. In words, the Fourier
transform of a product of two functions is equal to the

convolution of their individual transforms.

Continuing, let the conjugate of the complex function
*
Q(x) = P (x). Then from the properties of complex
*- * .
functions, FT{P (x)} = p (-u). Thus, uSing {2.16},

FT{P(x) P*(x)} = p(u) * p*<-u>.

Also, using the definition of convolution and using p as the

dummy variable,
7? 7k *
N®*p(m)=IMMp(4u-M)W=JPW)pw-UNM

The right hand term in the above expression is the formal
definition of complex autocorrelation of p(u). In this

thesis complex autocorrelation will be abbreviated,

al—
p(u) O P (u)-

- <1 ~~.—-_:.,.:_.2 :1 l _ ,, .l. ”a. 23.45;th ,7. "_'.‘,EZ'. f
M —

34

Autocorrelation is often encountered in optics work, and

indeed occurs in the following section on filtering.

Summarizing the results, and using shorthand notation, from

eqn. {2.l4c},
2 *
FT{[K(X)] } = L(u) = k(u) o k (u). {2.17}

Repeating eqn. {2.9}, u = fiR’ recall A is the wavelength of

the illuminating source. In the case of noncoherent
illumination, A can be considered to be the average
wavelength if the bandwidth AA, centered about A, meets the
restriction, A >> AA. Since this is true (c.f. [20] or
[46]) in the visible light range, in the remainder of this
(imaging) section A indicates the average illuminating
wavelength. Thus from eqn. {2.10}, eqn. {2.17} can be

Written as,
I “‘ _— G a C) G 0' {2 18}

where G is again the zero-one pupil function defined on the
seventh page of this chapter. In words, the noncoherent
frequency response function is given by the complex

autocorrelation of the aperture stop.

35

For a circular aperture of radius a, the autocorrelation of
the pupil function can be visualized by the area common to
two identical displaced circles as in Figure 2.3a. The
normalized frequency response is plotted in Figure 2.3b
where it is evident that the higher frequencies are imaged
at reduced contrast. Khetan and Chiang [13] perform this
integration (autocorrelation) and give the imaging system

normalized frequency response function as

L(u) = [cos'1<p) - p(l - p>1/21, (2.18a)

where p = uAR/2a, is the normalized position in the
frequency domain. It is evident from Figure 2.3a that when
the circles are displaced by 2a the frequency response
function is zero. Thus, object spatial frequencies greater

than

u = Zé {2.18b}

' ' the maximum
cannot appear in the image. Note that umax’

spatial frequency that appears in the image for coherent
illumination (see eqn. {2.lOa}), is exactly 1/2 of that for

the noncoherent case.

autocorrelation

 

 

— 1
I»! '1
2.">
trequency
response

Figure 2.3a Top - Geometric interpretation of aperture
function
Figure 2.3b Bottom - Autocorrelation for circular

aperture

 

37

Noncoherent imaging is summarized by substituting eqn.
{2.18} into eqn. {2.15} and then taking the inverse Fourier

transform of each side,
I (x > = FT‘lm <g new) 0 c*<a>1} {2 19)
i l 0 AR ' '

For a lossless system (i.e. no absorbtion or reflection),
eqn. {2.19} gives the noncoherent-illumination image

intensity.
C. The effect of aberrations

Goodman [47] has shown that, in general, minimum attenuation
will occur with no aberrations and that their presence will
decrease the image intensity to varying degrees. (For a
complete review, see Born and Wolf [20] or Welford [37].)
In the brief discussion on aberrations which follows, the
effect of defocus is presented. Defocus is the principal,
correctable problem of a quality camera system typically
used for speckle interferometry. For well corrected
systems, aberrations such as chromatic, spherical and
astigmatic, for example, are presumed less significant and
certainly less amenable to correcting from the view of an

interferometrician.

 

38

In the presence of aberrations, the pupil function will take

the form

G'<a) = 6(0) epr-Zf-i- WM].

where W(a) is the wavefront aberration. The phase advance,
W(o), expresses the deviation from the ideal spherical
wavefront shape launched from the aperture plane. As
before, the noncoherent transfer function will be the
complex autocorrelation of the pupil function but the
computation will not be straightforward (compared, for
example, with the geometrical interpretation shown in Figure

2.3a).

Born and Wolf's [20] 'displacement theorem' (see also
Welford [37]) establishes that a small change, 2, along the
optical axis from the ideal focal length of the receiving
plane results in no change in the intensity distribution.

From this and geometry they develop the simple relation,
0 2
We) = Imp 2,

as the phase change of the wave being received at the new,

defocused location.

 

 

39

Thus, if the amplitude is constant over the aperture plane

(which is the usual assumption), the pupil function is
6'0» = 6(0) ex [ii-(9)22]
P A R '

Born and Wolf [20] have performed the autocorrelation of the
circular pupil function of a system suffering small amounts
of defocus. Figure 2.4, showing image plane intensity

distribution, is abstracted from their results. The number

next to each curve is the value of the parameter,
raZ
111—2) (R) Z,

where z is the amount of defocus. For values of m greater
than one, it can be seen that the frequency response
function rapidly deteriorates at higher frequencies. For
typical camera settings used in the year-two study, this

equation implies 2 must be held to less that 0.4mm.
II. FILTERING

Speckle interferometry delineates displacements from a
double-exposed photographic image of the displaced object.
Assume the first image is recorded before displacement d, in
the one—dimensional object space, and the second image is

recorded after displacement. Also, assuming unit

 

response

Irequency

40

 

 

 

 

a 1

Figure 2.4 Autocorrelation for circular aperture with

small defocus

41

magnification and photographic resolution is sufficient to
record the finest object detail, the total exposure is

E(x) = [I(x) + I(x + d)]t, where I(x) is the image intensity
(the subscripts from the previous section have been dropped)

and t is the exposure time, assumed equal for each exposure.

Following Burch and Tokarski [3] and occasionally Khetan and
Chiang [13], the amplitude transmission for a linearly

developed negative film is

A(x) -= b — Vct[I(x) + I(x + an {2.20},

with b and c constants.

In the filtering process, the processed negative
(specklegram) is interrogated by a narrow beam of coherent
light as in Figure 2.5, detailed from Figure l.3. If d is
not too small or too large (as explained at the end of this
section), Young’s fringe patterns will appear on the
observation screen. Such a screen may be called the

transform plane, according to the following theory.

Pertaining to Figure 2.5, the nomenclature is as follOWS:

x is a point inside the illuminated area of the
specklegram,

s is a distance on the transform plane measured from

the center of the illuminating beam,

42

llanslorm GEOMETRY
plane

specklegram
plane

  

°:m

 

 

 

 

 

Young’s fringes

Diifraction halo on
transtorm plane L

  

\.
«‘ i
\.

 
  

wavelenglh \

Figure 2.5 Filtering geometry

43

r is the distance between x and s, and

D locates the centerline of the illuminating beam.

From geometry, r = [L2 + (S - (X - 9))211/2-

When L >> s and L >> x then a good approximation is,

~ 5 + 25D (x — D) sx
r ~ L + 2L + 2L - L , {2.21)

using the first two terms of the binomial theorem.

Referring to Figure 2.5, the complex amplitude of the
diffraction spectrum in the far-field region is given once

again by the Fraunhoffer equation (c.f. [20] or [48]):

Ut(s) = c LMX) aim-27% r) dx.

Here, Ut(s) is the complex amplitude in the transform plane,

A(x) is the amplitude transmittance of the specklegram, C is
a constant indicating nonessential phase terms, and the
distance r is explained above. Cast in this form, it can be
seen that the distribution of the diffracted light in the

transform plane is directly proportional to the Fourier

44

transform of the input signal amplitude. Since

[Ut(s)]2 = It(s), the transform plane intensity is

It(s) = [c I A(x) exp(-;§l r) dx]2, {2.22)

where the integral formally extends to infinity but actually

includes only the interrogating beam width.

Consider again eqn. {2.21). Note that the first two terms
are not functions of x; they contribute only to a phase
change in the wavefront intercepting the transform plane,
and may, therefore, be ignored. Also, for small angles,

except near the center of the illuminating beam,

wléfl

x éLD 2<< ABS(-§%), thus, r z ~
Conceivably, for very small values of 5, this inequality
would not be true. As will be seen below, however, the
transform plane intensity expression contains a singularity
at s = O and is not valid at this point. The above
approximation for r holds, therefore, because very small
values of s are excluded.

It is convenient to change variables at this point. Let

45

w = — . {2.223}

Now, using both the approximation for r and eqn. {2.22a},

eqn. {2.22} reduces to,

It(w) = [I A(x) exp(2niwx) dx]2, {2.23a}

ignoring constants.

Using the Fourier transform shorthand notation, defined in

this (filtering) section for an arbitrary function a, as

a(w) = FT{A(X)) = I A(x) exp(2wiwx) dx {2.23b}
and

A(x) = FT-l{a(w)} = J a(w) exp(-2wixw) dw, {2.23c},
and substituting eqn. {2.20} into eqn. {2.23a} gives
It(w) = [FT{b — ct(I(x) + I(x + d))}]2. {2.23d}

Consider the term I(x + d). As in the imaging process, a
shift in cartesian space results only in a phase shift in
frequency space, with the frequency function remaining

unchanged (c.f. eqn. {2.14c}). That is,

46

FT{I(x + d)} = exp(2niwd) FT{I(x)}.

Thus, eqn. {2.23d} becomes
It(w) = {b6(w) - ct[FT{I(x)}][l + exp(2wiwd)])2, {2.24}

where the Fourier transform of a constant b is the scaled

Dirac delta function.

Ignoring the 6 function term, which represents the bright
spot on the transform plane where w = s = 0, gives the

transform plane intensity as
2 . 2
It(w) = [FT{I(x)}] [l + exp(2w1wd)] . {2.25}

Before considering eqn. {2.25} quantitatively, a brief,
qualitative discussion of each factor may prove helpful. It
will be found that the factor [FT{I(x)}]2, represents a
circular area of random patches surrounding a central bright
spot on the transform plane. Such an area is called the

diffraction halo; it’s size will be shown to depend on the

spatial frequency of specklegram area subtended by the

. 2
interrogating beam. The factor [1 + exp(2niwd)] , acts to

modulate the diffraction halo rendering a set of Young's

47

interference fringes which are indicative of the

displacement vector d.

Consider the factor [FT{I(x))]2. Since amplitude squared is

intensity, and using the convolution theorem,

[FT{I(X)}]2 = [FT{[U(X)]2}]2 = [u(w) o u*(w)]2, {2.26}

where FT[U(x)] = u(w). In words, the autocorrelation of the
geometric aperture is the spatial frequency content of the

specklegram image.

To get an idea of the transform plane intensity (that is,
the diameter of the diffraction halo), a spatial frequency
function, u(w), must be assumed. Burch and Tokarski [3],
assumed that an ideal specklegram diffuser would contain
spatial frequencies up to a certain maximum. They reasoned
that the frequency content would be relatively constant near

zero and fall off rapidly near wmax' Their normalized,

ideal diffuser thus takes the shape (in frequency space) of
a rectangle function, symmetric about the origin as shown in

Figure 2.6.

Complex autocorrelation of the rectangular frequency
distribution gives the triangular region so labelled in

Figure 2.6. Light intensity distribution is the square of

48

 

 

 

 

 

x
\
~l‘
0x , *Ԥ\
/ ° K
\ \
\O
.9
Q\
Wmax wmax

Figure 2.6 Ideal rectangular frequency function, its auto-

correlation and squared autocorrelation

49

the autocorrelation and is also plotted in Figure 2.6.
Clearly, the diffraction halo intensity drops off rapidly
with increasing frequency. Indeed, most of the light energy

is concentrated in the middle of the diffraction pattern
between the limits i Wmax’
It may be seen in Figure 2.6 that the extent of the
diffraction halo in the transform plane is proportional to
twice the maximum spatial frequency contained in the

specklegram image. That is,

s
max
wmax= 2AL ' {2'26b}

 

Use of this formula will be made in the following chapter
when the spatial frequency of specklegram images are

inferred from the size of the transform plane diffraction

pattern.

Further explanation of the extent of the halo is given by
Huygens. A beam of light impinging on a series of optical
slits is deviated from its path at an angle proportional to
Thus, the finer the

the spatial frequency of the slits.

pitch of the specklegram image, the greater the angle, and

the larger the halo diameter.

50

Consider the second factor of eqn. (2.25},

[l + exp(2niwd)]2. Making the Euler substitution,
exp(2niwd) = cos(2nwd) + i sin(2nwd),

and squaring, it is found that

[1 + exp(2riwd)]2= 2[1 + cos(2nwd)]

= 4 cosz(nwd), {2.27}
using the trigonometric identity for cosine squared.
Eqn. {2.27} has the usual zeros when,

wd = (n + 1/2), with n = i 0, 1, 2,... {2.28}

Young's fringes, appearing in the diffraction halo, will
have minima where

s = (n + 1/2) AL, with n = i o, 1, 2,... {2.29}

The spacing or separation, S, between adjacent fringe ‘axes’

is typically

S‘=s]n+l-S.ln=

{[(n + 1) + 1/21-[n + 1/21>% .

 

51

which reduces to S = A5 . Restating, the film plane

displacement occurring between exposures is

I»
ml."

which is exactly equation {1.1}, mentioned in Chapter 1 in
relation to Young's experiment. This remarkably simple

result is critical to the success of speckle interferometric

methods.
Substituting eqn. {2.27} into eqn. {2.25}, and ignoring
constants, gives finally the transform plane intensity as a

function of image intensity and speckle shear (film—plane

displacement):
I _ 2 2
t(w) — [FT{I(x)}] cos (nwd). {2.30}

III. SUMMARY OF IMAGING AND FILTERING RESULTS

From the first section (Chapter 2.I.B), repeating the

noncoherent case, image intensity is

I(x) = FT'lloo(§-R)[c(a) o G*(a)]), {2.19}
n

 

52

where the subscripts on I and x, indicating image plane
coordinates, have been dropped. Now, the object
illuminating, noncoherent, average wavelength An, is
distinguished from the coherent, filtering beam wavelength,

A .
c

Writing this inverse transform out longhand (i.e. using the

formal definition, eqn. (2.8a}),

I(x) = I @o(§nR) [C(U) O G*(a)] exp[-2ni(§nR)x]da. {2.19a}

Using the definition for the Fourier transform in this
section (eqn. {2.23b}), and substituting eqn. {2.19a} into

eqn. {2.30} gives

It(W) = [I( J{ ¢o(§nR) [0(0) 0 0*(0)] eXP[-2Wi(§nR)X] d0}

exp[2nixw] dx)]2cosz(nwd). {2.31}

Taking the transform of each side of eqn. {2.19} gives
g — g * Th' ex ression is
n n
substituted into eqn. (2.31}. Then, repeating eqn. (2.22a}

(w = L), collecting exponents, and rearranging the order

s
A
c

 

 

53

of integration in eqn. {2.31}, the transform plane light

intensity distribution is

s
It(A L)
C

-2s 22
= [I éi {J exp[-21rix(>‘nR - ACL)]dX) do] cos (nwd). (2.32}

But, from the orthogonality properties of exponential
functions (i.e. the transform of a constant is a delta

function), the inner term is
I exp[-2«ix(§ R - f L)] dx — s<§ R - f L). {2.33}
n c n c

Substituting eqn. (2.33} back into eqn. (2.32}, and using

the sifting properties of the delta function,
E _ 2 1 E 2 2
t A L) ‘ [I ¢i(AnR) 6(AnR ' ACL) do] °°s (”Wd)

= [¢i(§ L)]2 0052(nwd). {2.34}
' c

Note that,

§ _ s *

54

Substituting eqn. {2.35} back into eqn. {2.34), gives

s ' s * 2 2 sad
It AcL) = (<D°(’\cL) [C(s) o G (s)]} cos ( "cL . {2.36}

In words, the halo intensity on the transform plane is the
object plane frequency distribution times the
autocorrelation of the pupil function. The halo itself is

modulated by the ‘cosine-squared‘ Young's fringe patterns.

Equation {2.36} can be characterized by considering two

extremes. Consider the two limiting cases where the object
contains many spatial frequencies (wide—band), or where the
object contains relatively few spatial frequencies (narrow-

band).

First consider the wide-band case. Recalling Io = FT{QO},
if Io contains many frequencies, then @o will be nearly

constant over a wide range, and will appear, in frequency
space, as the rectangle function in Figure 2.6. (Here, it
is assumed that the optical system is capable of imaging
fine object structure; the case where system resolution is
inadequate, that is, where object structure is very fine, is
discussed in the section on sensitivity, below.) Thus from
eqn. {2.36}, and ignoring this constant, the diffraction

pattern intensity will be given by

 

 

 

55

«sd

I (E [6(5) 0 c*(s)]2cos2(TL).
C

t ACL) = (2.37}

which is simply the squared autocorrelation of the pupil
function modulated by the cosine-squared term. Ignoring the
last term, eqn. {2.37} is the halo intensity distribution
for very fine object structure derived by Chiang [44], and

Meynart [45], who each used the statistical optics approach.

If the imaging aperture is circular, the autocorrelation is

given by eqn. (2.18a), thus eqn. {2.37} becomes

-1 s ~ g 2 1/2 2 2 gsd
{cos (8 ) ~ 5 [l -(S ) ] } cos (ACL)- (2.37a}

The autocorrelation of a circular pupil is shown in Figure
2.3b while the squared autocorrelation is plotted in Figure
2.7a, in transform plane coordinates. To review, Figure
2.3b is the frequency response (MTF) of the imaging lens
while Figure 2.7a is the transform plane light intensity
distribution. For a wide band of object spatial frequencies
with moderate object displacement between exposures, the
cosine-squared fringes are plotted in Figure 2.7b. (Tick
marks on the horizontal axis are located at half-multiples
of in). The product of curves shown in Figures 2.7a and

2.7b represents eqn. {2.37a}, and is plotted in Figure 2.7c.

56

(Figures are on following page)

Figure 2.7a Top - Squared autocorrelation of aperture
function - light intensity distribution on

transform plane

Figure 2.7b Middle - Cosine-squared (Young's) fringes for

moderate object displacement between exposures

Figure 2.7c Bottom — Product of Figures 2.7a and 2.7b -
transform plane intensity for fine object
structure recorded through circular aperture

for moderate object displacement

 

57

    
 

halo lnlonuty

Figure 2.7a

Figure 2 . 7b 1‘

'mn

       

   

lnlonllly

Figure 2.70

Figl

 

Fig

 

 

58

M

I — o.= \‘LI.

 

 

Figure 2.7d Narrow-band object spatial frequency

distribution

  

L— .a—4____. _.
l

Ll,

 

I .__.

Figure 2.7e Product of 2.7a, b and d above - transform
plane intensity for narrow-band object spatial

frequencies recorded through circular aperture

 

 

 

 

As 51

aCl'I

vall
frel
Fig
pro

and

 

 

59

Assume instead, that Io varies slowly with period (z l/Wa)
across the object surface. @o will then exhibit a fairly

sharp peak centered about the average spatial frequency

value, Sa= Acha. Thus, with narrow-band object spatial
frequency content, the resulting Q0 may be represented by

Figure 2.7d, plotted in transform plane coordinates. The
product of the functions represented by Figures 2.7d, 2.7a
and 2.7b, according to eqn. {2.36}, is plotted in Figure
2.7e. Thus Figure 2.7e gives the normalized transform plane
intensity for a relatively narrow band of spatial
frequencies appearing on the object surface, which have been

recorded through a circular aperture.

V. SOME PRACTICAL ASPECTS

A. Sensitivity

Sensitivity of the speckle method depends on relative
speckle size. If the specklegram consists of rather coarse
speckle cells, or, looking at it another way, if film plane
displacement is about the same size as the speckle cells,
few fringes will appear in the diffraction halo, rendering
fringe measurement difficult. If, on the other hand, the
image has very fine structure (relative to the object
displacement), sensitivity will be limited exclusively by

system resolution. Each possibility is discussed below.

 

 

 

 

Sens
mett
smal
sma]

case

Sub.
rel
tha
int
fig
2.7
2.7
the
two

int

 

 

60

a. Speckle shear approximately equal to speckle cell size

Sensitivity of an interferometric displacement measuring
method is a function of the recorded speckle size. The
smallest detectable displacement is fixed by the size of the
smallest individual speckle (shadow) cells. Consider the
case where the smallest speckle cell is about the same size

as the speckle shear, that is, when l/wmaxz d. Upon

filtering such a specklegram, the distance between adjacent
fringes is given by eqn. {1.1} as S = AL/d. From eqn.

{2.26b}, the maximum halo radius is found to be

Substituting eqn. {1.1} into eqn. {2.22a}, and using the
relationship between speckle size and shear, it is seen

that, Smax= 2S. The resulting diffraction halo light

intensity distribution is illustrated in Figure 2.8a. This
figure, according to eqn. {2.37a}, is a product of curves
2.7a and 2.7b, where the cosine-squared extrema in Figure
2.7b have simply become more coarse (moved outward). Note
that there will appear but two distinct minima; the outer

two will likely be obscured by rapidly diminishing halo

intensity.

 

 

 

61

 

 

 

Figure 2.8a Transform plane fringes for object displacement

which is of the same order as speckle size

 

Figure 2.8b Transform plane fringes for object displacement

which is less than speckle cell size

 

 

 

As

ext

1111

noi

em

C at

 

Fol

Th:

1iI

C0]

 

1111-

re

ml

 

 

62

As object displacement, d, becomes smaller yet, fringe
extrema move farther outward. The practical resolution
limit of the white-light technique, then, depends on system
noise and on the actual shape of the halo intensity
envelope. A reasonable guess as to when the fringe minima

can no longer be distinguished would be at l/2wmaxz d.

Following the same reasbning as above, this is where

The transform plane light intensity distribution for this
limiting situation is shown in Figure 2.8b. Object
displacement smaller than 1/2 of the speckle cell size
would certainly obscure extrema location, reducing

confidence in primary data.

b. Speckle shear much greater than speckle cell size

If the object structure is very fine, recorded speckle size
is governed by either aperture size or system resolution.
Aperture size controls the spatial frequency appearing in
the image as explained in the imaging section above.
Clearly, only those cells which are larger than a certain
minimum will be imaged. Thus, for a given system, the
recorded speckle size is inversely proportional to the F—

number (= F# = focal length/aperture diameter). Chiang [6]

 

 

ha:

CO?

CO]

 

wi

 

re

de

 

 

63

has demonstrated this effect showing smaller halo diameters
corresponding to increased imaging F# with all other

conditions equal.

Resolution, that is, the ability to resolve two nearby
points, is theoretically a function of F# as prescribed by
eqn. {2.18b}. At the glacier imaging F#’s, (8 and 16) for
example, the maximum object spatial frequency admitted by

the imaging lens was

2a

l l ., -
“max: AR = AF# = 6xlOE-4x8 ” 208 hues/mm

l ~ .
and, 6xlOE-4xl6~ 104 lines/mm,

for the year~one and -two camera systems respectively. It
will be seen in the next chapter that the practical
resolution limit is, of course, less than these values. In
a real imaging system, therefore, resolution is not
dependent solely upon aperture size as given by eqn.
{2.18b}. There is further 'spreading’ of an object point
into an image patch as energy is diffracted traversing

through the system.

Resolution is quantitatively evaluated in terms of the point
spread function (PSF), alluded to above. The Fraunhoffer

(far—field) diffraction pattern of a point source, imaged

 

 

 

th:

an:

in'

p1.

th

is

i1

Af

SL‘

ar

be

t}

If

C(

 

 

 

 

64

through an aberration-free circular pupil, is shown by Born
and Wolf [20] to be the familiar Airy disk. Airy disk
intensity is a decreasing Bessel function. Thus, image

plane intensity is given by
I(s')=<—1:‘—HJ V" )2, {2.38}

2na ,

AR 5 '. s' is the radial distance from

7.".
where v = = AF#S
the center of the pattern, a is the radius of the pupil, R
is the focal length of the imaging lens, A is the

illuminating wavelength and J1 is the Bessel function.

After Welford [37], this normalized intensity distribution

is plotted in Figure 2.9. If two object point sources are

 

sufficiently close, their respective Airy patterns will add
and, at some minimum separation distance, their images will
become indistinguishable. Lord Rayleigh, in 1879, proposed
that two nearby points of equal intensity, are just
resolvable if the principal intensity maximum of one
coincides with the first intensity minimum of the other.
The Bessel function zero (i.e. the first dark ring of the
Airy disk) occurs when v z 3.83. According to Rayleigh's
criteria, the minimum resolvable distance between object

points (with unit magnification) is therefore

3’. .~, 33 3.83 z .6lAF#. {2.39}
mln 7r

 

65

 

 

1
2
2J1“)
v
>.
t:
m
2!
m
p-
.2.
A—
é 4

  

 

  
  
 

l

l l

l l
L— aos —>]

Figure 2.9 Top - Airy disk intensity distribution - point
spread function for circular aperture
Figure 2.10 Bottom — Rayleigh criterion for minimum
resolvable separation between object

points (referred to film plane)

 

The

66

The additive intensity of two spectral components which are

just resolved are shown in Figure 2.10.

Note that the Airy pattern is simply the result of
diffraction. It is the PSF of a substantially perfect
system. In the presence of aberrations the PSF changes
shape in a complicated way and may deviate to a multi—peaked
curve as shown, for example, in Figure 2.11. Quantitative
effects of specific or combinations of aberrations on PSF
are not readily found. In fact, such tests are difficult to
perform on specific instruments. Welford [37], however,
generalizes that minor aberrations lower the central

intensity, and more light appears in the dark rings, while

 

the half-width of the central maximum does not change.
Thus, with increasing aberrations, the image of a point
source simply appears as a larger, less well-defined patch,

reducing resolution.

Maximum film plane displacement sensitivity, on the other
hand, is exceeded when speckle correlation is lost, that is,
when fringe patterns are no longer discernible in the
filtering process. For coherent imaging, Archbold and Ennos
[8] have taken this value to be about 5-10 speckle
diameters. Cloud's experiments [5], using different optics,
confirm this estimate. Corresponding sensitivity estimates
for the white light case, however, are not found in the

literature. Upon filtering, white-light generated speckle

 

67

intensity

1

 

 

V——>

 

Figure 2.11 Typical non-ideal point spread function for

which the Rayleigh criterion will be invalid

1

 

 

 

3—0
Figure 2.12 Fringe pattern shift with non-unity visibility

for moderate object displacement

 

 

Fi

w}

ti

 

 

68

patterns produce greater optical ’noise' than the coherent
counterpart. This is due, in effect, to less contrasty
'pinhole' edges of the diffracting speckle cells. It is
expected that the maximum range of displacement sensitivity
(referred to the average speckle diameter) is less then that
for coherent imaging. Rigorous experiments on the upper
bound are not often reported since, in this range, other
displacement measuring methods such as moire and

photogrammetry are more commonly used.

B. Fringe pattern measurement

The envelope which modulates Young’s fringes (shown in
Figure 2.7a) shifts maxima toward the center of the
interrogating beam. This effect is seen in Figure 2.7c,
where the actual maxima, for example, are located at the
tick marks in/2, but the apparent maxima are somewhat closer
to the origin. Therefore, it is better to compute
displacements from minima, but, as Kauffman [51] shows, this

will only be exact when such minima have zero intensity.

In practice, fringe patterns will be superimposed with ’dc’
noise resulting in a typical transform plane intensity
distribution shown in Figure 2.12. This figure was
generated using the (apparently empirical) formula given by

Kaufmann [51] which accounts for such dc offset:

 

 

 

He

av

f1

1c

 

 

69

 

 

 

§__ -1 s s s 2 l/2 2
qu L) = (cos (5 >-[s 1<1 - [S l) >
c max max max
1—_v 2_v 2 amt.
[1+v + 1+v cos (ACL)]. {2.40)
Here
V =(Imax_ Imin)/(Imax+ Imin)’ {2'41}

is Michelson's formula for fringe visibility, and Imax and

1min are the maximum and minimum intensity values of the

cosine squared fringes. Note that eqn. {2.40) is simply
eqn. {2.37a) with visibility incorporated into the second

factor.

The parameters used to generate Figure 2.12 are the same as
those used for Figure 2.7c (except for visibility), thus
allowing comparison. For a typical fringe visibility factor
of 0.4, Figure 2.12 shows a slight shift of fringe minima
away from the origin. Kauffman [51] has shown that such a
fringe shift is dependent upon recorded spatial frequency;
lower frequencies result in higher errors. For the present
study, Kauffman’s analysis predicts 10 to 15% error for
measurement based on fringe minima. Unless the diffraction
halo is removed prior to determination of fringe spacing, it
is more accurate therefore, to calculate displacements from

minima.

 

SPAAL

n
FILTER

:3 D!
LASER

70

"our ‘— r. L,
5"“ 5'6"“ or mmsrom
LENS

A:
W:

NOTE: ISOPLANATIC Fl OIN
SYSTEM PICTURE!) Tm m

COLLIMATOR

Figure 2.13 Whole-field filtering scheme

“YEA
“WORM

 

C. Other limitations

Problems associated with practical application of the white
light speckle technique have been described by many authors.
'Archbold et al [4] note that object surface tilt, out of the
plane of the film, can result in defocus. The tilt problem
is aggravated by large ratios of focal length to object
distance where the subject is close to the camera and the
depth—of-field limitations of the imaging system is
exceeded. Chiang [7] writes that surface tilt can also
result in apparent ficticious in-plane displacement. With
large extended objects located far from the camera, the tilt

problem is considered inconsequential.

 

V. WHOLE-FIELD FILTERING

Optical Fourier processing (c.f. [9], [10]), is well suited
to delineating components of homogeneous spatial frequency
recorded on photographic transparencies. Figure 2.13,
reproduced from Cloud [64], depicts a specklegram placed in
the input plane of such an optical filtering system. An
advantage is that sensitivity can be varied after the data
are recorded. Such a feature is especially valuable in
experimental work where strain magnitudes are not known
beforehand. If the displacement across the image field is
not homogeneous, however, whole-field fringes can be viewed

by placing an aperture in the transform plane. Fringes thus

72

appearing superimposed on the image, represent displacement
contours in the direction of the aperture from the optical
axis. Froely et al [20], Asundi and Chiang [6] and other
experimenters, have suggested, however, that the point-by-
point method yields higher fringe visibility and accuracy
where object surface displacement, referred to the film

plane, is not uniform. Experimental verification of these

limitations are explored in Chapter 3.

VI. SUMMARY

Recording and analysis of optical data have been outlined.
The result is a remarkably simple method to sense small
displacements of remote objects. Straight forward
application of this method to a non-laboratory object is

found in the next chapter.

 

CHAPTER 3
METHOD AND RESULTS
I. FIRST YEAR EXPERIMENTS
A. Preliminary development tests

Choice of photographic equipment for the first year study
was dictated by availability and portability. A Linhoff
View camera and Schneider-Kreuznach Xenar coated lenses with
focal lengths 500mm, 360mm and 300mm were selected. Quality
optics and rugged construction combined with a variety of
accessories made this instrument a good choice in the event

of potentially adverse and otherwise unknown field

conditions.

Use of the 4x5 format is common in experimental mechanics

practice. This format is available in film and glass plate

coated with a variety of scientific emulsions.
Laboratories, including those at Michigan State University,

are geared to processing and handling these films; the 4x5

was the format of choice.

Initial photography experiments utilized large objects under

natural illumination. All were found near the Engineering

Building and all were imaged on several different emulsions

 

74

in both film and plate. Experiments were conducted in order
to establish the necessary mix of F#, focusing technique,
exposure time, processing method, and other variables which

would give contrast and resolution required for fringe

generation.

The Communication Arts Building, located about 250m south of
the experimental mechanics laboratory window, was the first
object photographed, Figure 3.1. Recently constructed of
red brick and illuminated by an early evening sun, the north
face appeared to afford the necessary contrast. Trained on
the center of the ground glass placed in the image plane, a

60x microscope was used to magnify and focus the image.

It was noted that, under magnification, outdoor images
'wiggled' perceptibly during this, and subsequent
experiments. It was difficult to ascertain whether such
quiver was caused principally by camera motion or by
refraction gradients of the turbulent media through which

the images were shot. High summer afternoon ambient

temperatures seemed, however, to exacerbate the problem.

After careful focus, Kodak SO—253 High Speed Holographic
film was exposed for 1/155 at F#8, using a 360mm lens. Film
specifications, developing solutions and other processing

details can be found in Appendix I.

 

 

Figure 3.1 Communication Arts Building - artificial field

displacement experiment subject

 

Figure 3.2 Whole-field fringes from Communication Arts

Building image

 

 

76

Pointwise filtering (as shown in Figure 1.3) of the
processed image produced a diffraction halo about 34mm in
diameter. Other parameters of the filtering exercise were:
filtering (interrogating) beam wavelength, A = 6.3xlOE—4mm,

and filtering (transform) distance, L = 1000mm.

Recall from the previous chapter that the maximum spatial
frequency content of the specklegram is given by the size of
the diffraction halo. Assuming Burch and Tokarski’s [3]

ideal rectangular diffuser, the maximum halo radius is 2wmax

diffraction units as seen in Figure 2.7a. According to eqn.
(2.26b},

2w =s /AL=17———-mm
max max (6.3x10E-4 mm/cycle)(1000mm)'

Wmaxz 13.5 cycles (or lines)/mm.

Several fringe orders were apparent within the halo. Based
on the fringe minima separation of 5.2mm, accounting for
'fringe shifting' equal to roughly 10% and using eqn. (1.1],

the pitch of the specklegram ‘grating' was

6.3xlOE-4xlOOO/5.2 z .l2mm.

This value agrees well with the measured pitch of the

brickwork, calculated as follows. Each course of red brick

 

77

rises roughly 9cm. Based on this pitch, specklegram spatial
frequency is
l c cle

_——X__. x demagnification.
90mm

Taking
demagnification = object distance/image distance

where the lens law dictates that image distance 2 focal
length for remote objects. Specklegram spatial frequency is

then

l x 02%% z 8 lines/mm or 1/8 2 0.13mm/line.

Henceforth, fringe spacings were taken from minima and

corrected for shifting.

A double exposure of the Communication Arts Bldg., with the
camera back rotated about its normal approximately 20min. of
are between exposures, resulted in very small, but nearly
uniform, displacement of the image in the horizontal
direction. Such motion resulted from the position of the
building image on the film with respect to the center of
rotation of the camera back. The building facade appeared

at the 12 o’clock position and near the edge of the film.

 

 

78

Thus, very Small rotations of the camera back appeared as
nearly homogeneous building image displacement. Pointwise
filtering did not yield Young's fringes indicative of such
displacement. This was not surprising since microscopic
examination of the specklegram showed that horizontal
spatial frequency components (i.e. vertical lines of mortar)
exhibit much less contrast than vertical components.
Whole—field Fourier processing, however, resulted in good
visibility of fringe patterns indicating horizontal image
shear, up to order 2, as shown in Figure 3.2. It was
evident that uniform displacement is more easily discerned
using the whole—field method. Suitability of the whole-
field method, in the case of non-uniform displacements, was

determined in subsequent experiments.

The asphalt surface of a partially shaded walkway lying
about 10m beneath a third floor laboratory window served as
the next subject. Young’s fringes, Figure 3.3, appeared
when the artificial field displacement experiment described
above was repeated using the same film (500mm lens, F#8,
l/30s each exposure). Pointwise filtering did not produce
fringe patterns at every point on the image because of
depth of field limitations; nevertheless, this experiment
indicated that the camera/film system was capable of
producing the desired image structure without direct

sunlight illumination.

 

 

 

79

The final development subject was a partially graveled
parking lot located west of the Engineering Building, Figure
3.4. Random patterns such as tire tracks and grading lines
found here proved to be a good photographic approximation to
a glacier surface. Artificial field displacement
experiments were repeated with an evening sun angle of
approximately 60 degrees from the zenith. Photographed from
a perch atop the Engineering Bldg., about 15m high, the
surface extended from 100 to 200m distant. A point about
150m from the camera site was precisely imaged near the
center of the ground glass via microscope. Depth-of-field
was adequate as the image later proved to be focused from
top to bottom of the plate. It became standard practice to
focus on points lying near the top—to—bottom middle of the
field. Final film processing resulted in rows of classic
speckle pairs which, when scanned pointwise from the center
outward, produced increasingly fine, rotating Young’s
fringes. This behavior was exactly as expected. Fringes
generated from image points which lie near the center of
rotation suffer relatively small displacement; according to
eqn. {1.1}, fringes were correspondingly coarse here. The
fringe bands became finer as the radial distance from plate
center increased. The two parking lot images, slightly
rotated with respect to one another, resulted in a series of
Young's fringes whose axes were perpendicular to the local
displacement vector. As the image was scanned along a

chord, the relative angle of displacement changed. This was

 

 

 

 

 

Figure 3.3 Fringe bands (extending from upper left to lower
right) enclosed in halo - from.artificial field

displacement tests — subject, asphalt walkway

 

Figure 3.4 Gravelled parking lot image

 

Figure 3.5 Diffraction halo and fringes (extending from

lower left to upper right)-subject, parking lot

 

 

 

 

 

81

indicated by rotating fringe axes. Figure 3.5 is
representative of the fringe patterns produced across the

central portion of the image.

Performing a calculation similar to that above, the maximum
spatial frequency of the parking lot image was inferred from
the size of the diffraction halo during pointwise filtering.
The spatial frequencies of the images were about 18.3
lines/mm @ F#22 and 25.4 lines/mm @ F#ll, (using Kodak High
Speed Holographic type 80-253 film). The higher spatial
frequency content of this image (compared to that of the
brick facade of the Communication Arts Bldg.) was attributed
to the oblique viewing angle as well as to the fine

structure of the coarse sand and pea-graveled surface.

 

In contrast to the results of the pointwise method, whole-
field, Fourier optical processing of the same parking lot
specklegram yielded fringes of near zero visibility. In
this whole-field process, it was noted that fringe
visibility was directly related to the uniformity of the
spatial frequency of the input signal. It was also noted
that ice displacement is not, in general, uniform across the
large areas of glacier surface which would be recorded in
one frame. In the event of nonhomogeneous field
displacement, pointwise filtering, as in Young's experiment,
"is more tedious than the whole-field approach...[but]

displacements thus obtained are more accurate and fringe

 

 

82

quality is much better", Asundi [6]. Henceforth, study was
devoted to the creation of photographic images having
sufficient diffraction efficiency to produce quality Young's

fringes.

Artificial field displacement experiments were repeated
using glass photoplates. Focusing was somewhat more
reliable using plates in place of film. Misfocus, resulting
from both film warping and imprecise adjustment, was reduced
using Kodak High—Speed Holographic type 131-02 glass plates
and an additional refinement in technique. A rough image
was focused in the normal view camera manner on ground
glass, the glass having the same dimensions and index of
refraction as the photographic plate. A microscope, mounted
on the camera back, was focused on the ground glass at a
point near the center of the image. The microscope was
thereby positioned in proper relation to where the emulsion
plane would be located during actual photography. The
ground glass was then carefully removed from the camera,
leaving the microscope focused near the image plane. While
viewing through the microscope, the image was then refocused
on the image plane by adjusting the camera back/microscope
assembly. This technique is sensitive because the depth of
field of the microscope is very small. The error which was
introduced by focusing initially through the ground glass,
amounted to less than 0.3mm. For the purposes of the

present work, this level of defocus was negligible. This

 

83

conclusion is based on Figure 2.4, where the camera system
frequency response (for defocus equal to 0.3mm) lies between
the curves marked 0 and l. Microscope focusing became

standard practice.

Experiments to estimate the practical resolution limit of
the lenses were performed using a standard optical bar chart
and a sensitive microscope trained on the image plane. Bar
charts contain arrays of lines spaced at uniformly
increasing spatial frequency. After positioning the chart
in an Engineering Building hallway at an appropriate
distance from the camera (58m, in this case), a point near
the image center was magnfied through a microscope and the
pitch at which the lines become indistinguishable was
estimated. This value was taken as the practical resolution
limit of the imaging lens. At F#8 and, at this object
distance, the maximum distinguishable spatial frequency of
the imaging lens was about 1.1 lines/mm (28 lines/in) at the
object; at F#ll the value was about 0.94 lines/mm (24

lines/in).

Final experiments in preparation for glacier field work
included a crude attempt at aperture filtering. An opaque
circular disk was cut from black electrical tape and placed
at the center of the 500mm lens. Such a mask, sized about
one-fourth of the lens aperture, acts as a high pass filter.

With this mask in place, standard artificial field

 

 

 

 

 

 

84

displacement tests, processing and pointwise interrogation,
resulted in reduced fringe quality. Evidently, lower
spatial frequencies than those admitted by the filter were
required for good fringe quality and energy content in the
range of frequencies eliminated by a yet smaller disk was
small enough to have negligible effect. It was concluded
that images having good diffraction efficiency are formed
from low-frequency components. Aperture filtering

experiments were discontinued.

Two other photographic emulsions were also tested. Kodak
Minicard SO-424 Scientific Film was used in the parking lot
tests with marginal results. The visibility of fringe
patterns thus produced were decidedly less than those
produced by the holographic film. Tests with Kodak SO-649F
Spectroscopic plate produced moderately visible fringes, but
this emulsion proved to be relatively slow in white light.
Exposure time just over one second exceeded the timer range
on the available shutters, possibly necessitating manual
shutter operation. With the field season impending,
efforts were concentrated on developing a dependable field

technique using glass plate, while testing other media when

the opportunity arose.

 

 

 

 

85

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Figure 3.6 Nisqually area map - the symbols .and‘

indicate the year-two camera locations

 

 

 

 

86

B. Field tests - Nisqually Glacier, Washington

Artificial field displacement experiments were repeated at
the Nisqually Glacier, Mt. Rainier National Park, WA, USA,
Figure 3.6. Changes in lighting, subject size, camera
positioning and other factors made recalibration necessary.
After several days of trial, cursory examination indicated
fringe patterns approaching the quality of those produced
during the initial tests were generated. Images recorded
from several perspectives, all within 1km of the glacier
snout, were successful. A small (low) sun angle on the
south-flowing glacier surface surface yielded the best
fringe quality; hence, it was decided to initiate time-

interval, double exposure experiments during morning.

With a diffraction halo radius measuring roughly 10mm, the
maximum spatial frequency of the specklegrams recorded
during glacier-site artificial displacement tests was about

7.9 lines/mm.

Site accessibility compromised camera location. A site
located too far from the equipment vehicle would require a
pre-dawn hike up unfamiliar terrain; locating the camera too
close to public parking would tempt curious, if well-
meaning, tourists. Fearing damage to fragile one-inch-thick
topsoil, Park Service personnel did not allow camping near

potential camera sites; equipment remained unattended for

 

 

87

overnight periods. The camera was eventually located on the
side of a hill, partially hidden by a clump af trees, about

50m up from a relatively seldom traveled path.

The Nisqually icefall constitutes the steepest portion of
the glacier surface. Theory predicted relatively rapid ice
flow rates in this area. It was decided that the icefall
offered the best possibility from which to record
discernable ice displacements during convenient time
intervals. ,As explained in Chapter 2, sensitivity of the
interferometric method is fixed by the spatial frequency of
the image: smaller speckle cells result in higher
sensitivity. Based on earlier glacier-site artificial
displacement experiments, the smallest resolvable ice
displacement was 1/7.9 z 0.13mm, referred to the film plane.
Accounting for magnification and perspective, the estimated
icefall velocities of very roughly 1 meter/day would render

speckle shear within the discernable range.

 

Image speckle pattern is a function of both surface
roughness and illumination position. A 24 hour time-
exposure interval resulted in nearly identical sun angles
thereby increasing precision. Thus, the image was centered
near the base of the icefall and a twenty-four hour time-

lapse interval was chosen as the starting point for field

experiments.

 

 

 

 

 

88

With favorable weather, two twenty-four hour, double
exposure experiments were conducted at, or near, the
hillside site. High Speed Holographic emulsion, 500mm lens,
F#8, 1/85 each exposure was used, with the camera mounted on
a sturdy, commercially available tripod. Both the film, in
the first trial, and the photoplate in the second trial,
suffered from excess camera motion incurred between (or
during) exposures. Excessive motion appeared in the image
as rigid body displacement and may be attended by loss of
speckle correlation. A double exposure having virtually
zero time interval was recorded to determine, qualitatively,
camera motion caused by shutter operation. Microscopic
examination of the resulting short-duration image did not
reveal rigid body displacement. The conclusion was that
tourists, animals (many of which appeared to be very tame),
weather, natural settling of the tripod mount or perhaps
seismic activity related to nearby, bestirred Mt. St. Helens
could have caused the shift during the 24 hour interval. A
site which assured sturdy, undisturbed camera support was

required.

The glacier surface proved to be bright enough to image on
the relatively slow Spectroscopic emulsion. Contrary to
expectations, the shutter timing mechanism, supplied with
the camera had sufficient range to accomodate this
photographic medium. Thus, some timeelapse experiments were

performed using this plate.

 

 

 

 

89

Interposing the artificial displacement tests and double
exposure tests were serial, single exposures recorded on a
daily basis. In these experiments, the image was recorded
alternately on forward and reverse emulsion. Such an
arrangement allows the processed photoplates to be
superimposed upon one-another as right and left hand images.
In this manner, and after the manner of Adams [32], the
intent was to place fixed objects, such as rock outcrops
appearing in the image, in exact registration, thereby
subtracting rigid body motion. Three degrees of freedom
complicated plate positioning. Efforts were subsequently

concentrated on developing the double exposure procedure.

With intent to isolate the motion problem, double exposure
photography was conducted at other sites. Images having
time intervals of four and twenty-four hours did not yield
Young's fringes but indicated an important cause of camera
movement. A four-hour double-exposure, using High Speed
Holographic film, 500mm lens, F#22, 1/505 each exposure,
showed no ice or rigid body displacement. An interval of
fourty—eight hours, on the other hand, resulted in complete
loss of speckle correlation owing to excessive camera
motion. Yet longer time intervals exacerbated the motion
problem, suggesting that gradual settling of the tripod over
time was a major contributor. Earthen settling rates tend
to decrease over time. The conclusion was that long-term

sites would tend to offer increased camera stability; a

 

 

90

carefully selected camera site was to remain unmoved during

subsequent field work.

Various means of supporting the camera were not effective
until, finally, the instrument was mounted on a large rock.
Figure 3.7 was reproduced from the resulting twenty-four
hour, double exposed photoplate (Spectroscopic 649-F plate,
500mm lens, F#8, ls each exposure). Upon interrogation,
rows of speckle pairs produced Young's fringes across much
of the glacier surface image. Two such fringe patterns
appear in Figure 3.8 and Figure 3.9, recorded with the
original transparency unmoved. Figure 3.8 was produced as a
narrow laser beam was diffracted by a portion of icefall
image while Figure 3.9 was similarly produced from a point
at a lower elevation, about mid-glacier. The slight
rotation of fringe axes was expected, attributed to a change
in perspective. Appearance of Young’s fringe patterns,
indicative of ice displacement, demonstrated data gathering
practicability of the white light technique and culminated

the 1981 field season.

C. Fringe pattern analysis

The maximum spatial frequency of the glacier specklegram,
corresponding to the 10.5mm diffraction halo radius, was
about 8.3 cycles/mm. Typical speckle cells, viewed under

the microscope and measured with a finely graduated scale,

 

 

 

91

 

Figure 3.7 Print of double-exposure 24hr plate - Nisqually

 

 

Figure 3.8

Figure 3.9

 

 

 

Top - Fringe bands from Nisqually Icefall
Bottom - Fringe patterns from Nisqually down-
glacier (bands extend from upper left to lower

right)

 

 

 

 

 

 

93

exhibited roughly the same size (i.e. 1/8.3 z .12mm).
Note that the intensity of these fringe patterns roughly
correspond to the intensity distribution plotted in Figure
2.8a for which speckle cell size and shear were assumed
equal. It was possible to obtain useful fringe axis

separation and orientation measurements as described below.

Young's fringe patterns, gleaned from the double exposed
glacier surface image, were transformed into ice
displacement rates in order to compare the data with
information gathered by conventional methods. Comparison
was made to determine technique deficiencies and prepare a
list of recommendations to be implemented during subsequent

study.

Data analysis, that is transformation of fringe pattern
parameters into surface velocity maps, required assumptions
regarding optical magnification, camera motion and
perspective. The process evolved into the steps enumerated

below:

(1) Young's fringes were logged from several points of
the double exposed, twenty-four hour image. Fringe
separation and angle of the fringe axis with respect to
vertical were measured. Trees visible in the image
foreground of Figure 3.7 served as vertical reference. The

position of image points relative to prominent features such

 

 

 

 

 

94

as rock cleavers, cirques, and silhouettes appearing on both
the image and topological map were noted. In this manner
the glacier surface topological map location corresponding
to the particular image point was determine fairly
accurately (within about 40m). At each map location, the
slope (i.e. surface normal) was then approximated using map
contour lines. Nine points, resulting in the best 'spread'
across the glacier surface, were selected. Employing eqn.
{1.1), a gross displacement vector was calculated at each

point, in image plane coordinates.

(2) Microscopic examination of the double exposed
image revealed displacement of a fixed object; in this case,
a large boulder resting on a bare rock surface. The rigid
body displacement vector of this point, again determined in
image plane coordinates, was measured with a finely

graduated scale and protractor.

(3) Assuming uniform image plane camera motion, a
specially written computer routine vectorially subtracted
rigid body displacement from the gross displacement found in
step (1), the algorithm again operating in image plane
coordinates. The difference is net ice displacement in

image plane coordinates.

(4) Net ice displacement was projected onto the

glacier using surface normals found in step (1) and the

 

 

 

 

 

 

 

 

95

magnification, defined earlier. Figure 3.10 depicts the

computer projection algorithm.

Nisqually Glacier ice displacement for selected surface
points over a twenty-four hour period in late August, 1981,
Figure 3.11, resulted. Two additional assumptions were
required in order to arrive at this particular ice velocity
map. First, since interferometry does not reveal a sign, it
was assumed that ice flows (generally) downhill. The second
assumption, that of correct rigid body displacement
direction, was less clear intuitively. Another displacement
map could be generated, for example, using the same data but
assuming the opposite displacement direction. Figure 3.11
is simply the most reasonable since the alternative

computation gave ice displacements of two to three times

 

greater magnitude.

The ice displacement magnitudes, extracted by the white
light technique, corresponds to those reported by Hodge
[28], who used conventional methods. The earlier research
team was prevented by severe weather, Hodge writes, from
studying flow in the higher elevations. In areas for which
comparisons can be made (roughly points 7 and 8 in Figure
3.11), Hodge's two year study gives centerline flow rates of

0.4 meters/day while the white light technique yields 0.6

meters/day.

 

 

 

 

 

 

96

   
 

camera
site

 

net image-Plane
displacement

Figure 3.10 Computer projection algorithm

 

97

 

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Figure 3.11 Final results from year-one

 

 

 

 

98

D. Year-one preliminary conclusions

These encouraging results lead to the conclusion that the
white light technique was feasible in application to glacier
study. Preliminary conclusions, based on first year
results, especially those affecting development activity the

following field season, are enumerated below:

(1) Double exposed images were more practicable than
single, serial exposures because of the difficulty in

superimposing images in perfect registration.

(2) Surveying methods to locate features found on the
photographic image were needed. Location of many
significant features would increase the precision with which
fringe generating image points were located on topological
maps. Also, to increase precision, a vertical reference

indicator should be included in the photographic image.

(3) Being of the same order of magnitude as net ice
displacement, apparent image plane rigid body displacement
was a major factor affecting precision. Three steps were

suggested as partial remedies:

(a) Apparent movement suffered by all image points was

to be reduced with the use of sturdier camera mounting.

 

 

 

 

 

 

 

 

 

99

(b) Camera motion, however small, is likely during
time-lapse photography. Quantification of this residual

motion is discussed below.

Displacement in a plane (in this case, the apparent motion
of 'fixed' objects in the film plane) can be resolved into
two components, translation and rotation. Referring to

Figure 3.12, planar motion of any given point, say point M,

can be thought of as a simple translation parallel to some
I
reference point translation, P to P , followed by a finite

rotation about that same reference point, PI. The rotation,
through a so-called Euler angle (angle 0), suffered by any
line in the plane is the same rotation suffered by any other
line. The difference in the motion of any two points is
accounted for by the difference in their respective position
vectors relative to the translated reference point. In the

diagram, relative position vectors are designated Rz’ where

the variable 2 is the label of the point whose rigid body
displacement is to be determined. Three pieces of
displacement information about two points in this plane are
needed to completely specify translation and Euler angle
parameters. They are the two components of the rigid body
displacement vector of one ’fixed' reference point appearing
in the double exposed specklegram, point P, and the
direction cosine (coso) of the displacement vector of

reference point Q. Coordinates of other points relative to

 

 

 

100

reference point P is sufficient, then, to delineate rigid
body displacement of all points lying in the image plane. It
was decided that two reference points, such as P and Q in
Figure 3.12, be incorporated into the image during the

second field season.

(c) As a final check of the rigid body displacement
vector estimates (performed in step (2) of the prior
section), it was proposed that two independent data sets be
acquired simultaneously using a stereo camera system.
Separate ice velocity maps, constructed as described above,

were to be compared, establishing confidence levels.

(4) Figure 3.11 shows displacement rates of points
widely scattered about the glacier surface. Greater detail,
perhaps revealing icefall margin shear rates, would be
realized by either photographing from a closer vantage point
or using telephoto lenses. A perch close to the most active
part of the surface was not accessible to the field crew on
a daily basis; use of long focal length lenses was the

alternative.

(5) Assuming decreasing earth settling rates, camera
stability should increase with site installation time.
During subsequent field work, camera locations would remain

fixed for the duration of the field work stay.

 

 

 

 

 

 

 

101

 
  
 
    
    
   
 
  

rigid body

displacement

gross ice

d isplacem
net ice
.\

displacement

    

Euler angle

 

reference point
translation

Figure 3.12 Euler angle algorithm

   

 

 

 

 

 

102

II. YEAR-TWO EXPERIMENTS

A. Preliminary development tests

Technique development centered on implementing year-one
conclusions. Equipment was assembled and methods were
tested in a manner similar to that of the previous season.
Details of initial development tests are not repeated in
this section unless differences between this and the

previous work were significant.

Alterations in the program were primarily accountable to
camera hardware changes. Identical cameras were assembled
from military surplus 916mm, F#5.6-16, uncoated aerial
photography lenses and rotating plate holders. Lenses and
telescoping camera backs were mounted on either end of a
short length of 6in., Schedule 80 PVC tubing, Figure 3.13.
Camera-back mounted microscopes (again, 60x) were used to
precisely adjust focusing micrometer screws. Machinists
levels, also mounted on the camera backs, were used to

orient the photoplate into the 'vertical' position.

In their original form the military surplus lenses were not
suitable for the intended purpose. Very stiff springs
powered the internal shutters; triggering the shutter
visibly jarred the camera. Experiments showed the timing

mechanism was not repeatable in any range; the exposure

 

 

 

 

 

 

 

103

interval being very roughly dependent on the extent of
spring windup. After the shutters were modified to operate
manually with remote cable release, however, exposure
intervals in ranges over one second proved repeatable.
Exposure times in this range necessitated the use of the

relatively slow Kodak 649-F Spectroscopic emulsion.

Initial photography experiments were, once again, carried
out using the gravel parking lot west of the Engineering
Bldg. Artificial field displacement tests were repeated
with similar results: Young's fringe patterns, though
noisy, were produced from the gravel surface image (using

Spectroscopic 649-F plate, F#16, 25 each exposure).

The very broad spectral sensitivity of Kodak 649-F
Spectroscopic emulsion diminished fringe quality by imaging
chromatically aberrant frequencies. The effect known as
dispersion converges different colors at different focusing
distances. Focusing on one color defocuses other
wavelengths. The unfocused energy remains in the optical
system and serves to diminish the image contrast which is
necessary for good diffraction efficiency. Such a reduction
is manifested in the filtering process by diminished halo
size. Therefore, lens modifications, narrowing input
spatial frequency content (band pass filters) or limiting
the radiation wavelength to the image, were tested. It was

found that colored cellophane lens filters or spatial

 

 

 

 

 

 

 

104

frequency masks (as in year-one) necessitated long exposure
times with no appreciable gain in fringe quality.
Refraction gradients in air resulting from atmospheric
turbulence, random low level oscillations of the earth and
other ’background noise', either mechanical or
electromagnetic in nature, reduce the sharpness of any
photographic image. Long exposure times, necessitated by
the use of colored filters or masks, exacerbate these
problems. Thus, lens modification experiments were
abandoned. Focusing in the middle of the range between the

'sharp blue' and 'sharp red' positions was the compromise.

Resolution tests as described in section A of this chapter,
were conducted with the year—two equipment and with the
target optical bar chart placed at 130m from the camera,
again in an indoor hallway. At the aperture setting used
for all of the year-two experiments, F#l6, it was possible
to distinguish lines having spatial frequency up to about

0.51 lines/mm (13 1ines/in)at the object.

After perfecting focusing technique using Spectroscopic
film, artificial field displacement tests imaged a
relatively broad frequency spectrum of about 19.8 lines/mm,

inferred from the diffraction halo size, using eqn. {2.26b}.

Kodak Minicard SO-423 film, being relatively slow in white

light, seemed to be suitable for use in the manually

 

 

 

 

 

 

 

 

 

105

operated shutter cameras. Several single and double
exposure experiments resulted in apparent misfocus. This
indicated that either the emulsion had inadequate resolving
power to image the necessary structure or film warping was
causing misfocus. Field work was to be conducted using

Spectroscopic plate.

Feasibility study funding was earmarked for application of
the white light technique to two separate glaciers. The
Nisqually glacier was chosen as the first to be photographed
with the newly assembled, specialized equipment for several
reasons. An existing body of knowledge regarding ice flow
behavior was available for comparison purposes, the site is
accessible, the climate relatively benign (in late August)
and the field team was familiar with the area. Workers were
to perfect data acquisition methods at the Nisqually and
then travel to the Ptarmigan glacier, near Juneau, Alaska.
The mile—high icefield abutting the Alaskan coast was to
provide a more rigorous testing ground. Unexpectedly well
studied, the Ptarmigan exists in a severe, changeable and

remote environment.

Satisfied with initial test results and armed with 13 boxes
of equipment, tools, supplies and luggage, the expedition

set out first for Mt. Rainier National Park, WA.

 

 

 

 

 

 

106

 

Figure 3.13 Specialized camera from year—two

 

Figure 3.14 Camera sites relative to glacier surface

 

 

 

 

 

 

 

 

 

 

107

B. Field tests - Nisqually Glacier, Washington

Camera sites were established in the east wall of the
Nisqually glacier valley about 600m apart, Figure 3.14.

Each site was somewhat more remote than those of the
previous field season, requiring longer hikes to set-up and
access. Camera assemblies were enclosed in small tents with
lenses directed toward the icefall area through the open
front flap. Sturdy, lag-bolted, wooden stands, custom
fitted at each site, held the camera bodies elevated at an
angle of roughly 15 degrees from horizontal. At this angle,

the lens rested about 0.3m above the hillside surface.

After delays caused by weather and recalibration of exposure

 

and processing procedures, artificial field displacement
tests, using Spectroscopic 649-F emulsion, proved
successful. Fringe visibility produced from some images was
adequate, the failures underscoring the necessity of precise

focusing. .

‘Simultaneous, time-lapse, double exposure experiments were
initiated several times during the following week by
operators communicating by radio. Environmental factors
such as weather, influx of volcanic haze and dew on the
lenses often delayed the second of the double exposures.

With patience and luck, time-lapse experiments of 4, 24 and

 

 

 

 

 

108

48 hour intervals were completed (using the 649-F film,

F#l6, 23 each exposure).

There was no apparent motion on the 4 hour, double-exposure
plate; no fringe patterns were produced. This result was
similar to results of earlier 4—hour experiments conducted
during year-one field study. Similar also to year~one
results, the 48 hour test images exhibited excessive camera

motion which obliterated speckle correlation.

Twice, simultaneous, 24 hour tests were conducted under good
conditions. A bookkeeping error left one plate uselessly
overexposed while the other three, upon cursory inspection,
showed good exposure and focus. Two plates, recorded
simultaneously from different sites, are reproduced in
Figure 3.15. Careful inspection reveals the slight change
of perspective between the two. In the field, fringes
appeared to be marginally visible. It was assumed, however,

that in the laboratory, visibility would be much improved.

Theodolite surveying was conducted during the double
exposure experiments. Distinctive features found on the
glacier surface images were located with respect to the
camera sites and to existing surveying markers. These
markers were precisely located during a previous surveying
expedition [28] and appear on official USGS maps of the

area.

 

.._.. . a. Imam...

 

bar (I

- 23!.

 

 

109

 

 

Figure 3.16 Fringe patterns from rock strata, Ptarmigan
Glacier valley wall (fringe bands extend from

upper left to lower right)

 

 

 

 

 

 

 

 

llO

Satisfied with the three twenty~four hour plates, and with
the field work still on schedule, the research team
considered taking a chance on the weather and moving the
expedition to the Ptarmigan glacier near Juneau, AK.
Weather in southeast Alaska is indeed unpredictable.
Ascending rapidly up the slopes of the Alaskan icefield,
moisture is squeezed from the prevailing sea-borne winds
over the cities from which the weather reports had been
obtained} In the summer of 1982, however, reports of rainy,
overcast periods had been surprisingly few. Deciding to
continue according to the original research plan, the field

team pressed ahead with experiments in Alaska.

C. Field tests—Ptarmigan glacier, Alaska

Experiments on the Ptarmigan were delayed. Luck had
apparently worn thin as poor weather kept the field team’s
helicopter grounded for several days. Finally, the overcast
lifted long enough to allow transport. Once on the glacier
feeding icefield, two camera sites were established on the
east Ptarmigan glacier valley wall. Camera stands and tents
were well anchored. Environmental changes again forced
recalibration of experimental procedures. Cooler ambient
temperatures changed exposure times and unheated buildings
required development of new cold—processing methods.

Artificial field displacement tests were not immediately

 

 

 

 

 

 

111

successful, again because of poor weather (i.e. near white-

out conditions).

Close examination of the first artificial field displacement
plates, shot through the mist, revealed unexpected results.
Fringe patterns were produced from a rock strata image, the
strata being located across the glacier valley from the
camera sites. The vertical, ice-carved wall did not appear
to have great contrast compared with the adjacent glacier
surface. Nevertheless, noisy fringes, as shown in Figure

3.16, were generated across much of the rock strata image.

A sudden, mid-morning clearing allowed successful completion

of artificial displacement tests. A 24-hour, time-lapse

 

experiment was quickly initiated. Benign weather
transformed, within hours, to 60 knot winds and white-out
conditions. Westerly wind-driven hail pummeled the aluminum
clad encampment huts for the next two days. As workers
ventured out to check the status of the equipment, one
camera was found being dragged about by a shredded,
billowing tent. With moisture covering the lens of the
other camera and no break forecast, it was decided to abort
the experiment. Despite the effort, no ice flow data was

collected during the Alaska sojourn.

After packing the photographic equipment and making ready

for the end-of—season closing of the icefield encampment,

 

 

 

 

112

members of the research team waited for two days until the
white-out lifted sufficiently to allow the helicopter pilot
to locate the camp. Giving the field team a final
impression of its fickleness, the weather rapidly closed in
again, stranding the team for an additional overnight. Late
the next afternoon, all fled the icefield just as the long

winter set in, ending the 1982 field season.

D. Fringe pattern analysis

Detection and analysis of Young's fringe data of the second
field season differed from methods of the first year because
of substantial equipment changes, mentioned above. Young’s

fringe patterns were transformed into ice surface velocity

 

maps in order to compare the new results to those of the
previous year, and to flow rate information gathered by
conventional methods. This comparison was made in order to
compile a list of recommendations aimed at increasing both
precision and facility and to determine the merit of full-

scale development of the new optical technique.

Photoplates, double exposed over 24 hour intervals at the
Nisqually glacier, were analyzed in the steps enumerated
below. General similarity exists between the analysis
methods used following the two field seasons. Differences

are presented in detail.

 

 

 

 

 

 

113

(1) Laboratory detection of Young’s fringe patterns,
generated from each of the three successful plates, was more
difficult than anticipated. Diffraction efficiency of the
Spectroscopic emulsion, when used in the specialized
cameras, was poor. In visual comparison to plates of the
previous year, less contrast was evident. Duplication of
the recent plates on more 'contrasty' holographic emulsion,
by a method identical to contact printing, resulted in
improved fringe visibility when the reproduced film was
interrogated. Compressing many gray tones into few, this
process left the duplicates overexposed in some areas and
underexposed in others. Using the common darkroom practice
of 'dodging’, however, it was possible to partially
compensate for mis-exposure on both the duplicate and the

original.

When this duplication process did not increase fringe
visibility (on areas of glacier surface image thought
necessary for good coverage), mechanical oscillation of the
duplicate image, during the interrogation process, was often

helpful.

In-plane vibration of the 4x5 film, at 50-60Hz and at an
amplitude of about 5mm, spatially averaged optical noise.
This process is somewhat akin to autocorrelation of temporal
data to filter low level, periodic signals from noise.

Temporal autocorrelation makes use of the random nature of

 

 

 

 

 

 

 

 

114

noise; after sufficiently long time period, this
mathematical process ensures that random signals cancel one
another, whereupon the previously camouflaged period of the
desired signal becomes evident. The space oscillation
process, on the other hand, does not reduce 'noise'. Since
speckle cell pitch does not vary significantly over the 5mm
range of motion and 'noise' does vary, the visibility of the
once-masked signal is enhanced. The fundamental difference
in the two signal enhancement techniques lies in the one-
sided nature of light as opposed, for example, to the
positive and negative displacements of seismic signals;
there is no such thing as a light diode. With oscillation,
many stubborn image points finally yielded Young's fringe

patterns of adequate visibility.

 

Maximum spatial frequency of the artificial field
displacement tests glacier images were again inferred from
the size of the diffraction halo. Based on the 12mm halo
diameter (using eqn. {2.26b)), the finest spectral
components were about about 4.8 lines/mm. The 24 hour, time
lapse images exhibited virtually the same maximum frequency.
Reasons for obtaining relatively low resolution in outdoor

conditions are discussed in Chapter 4.

With fringe bands visible, information required to determine
net ice displacement was gleaned from each photoplate in

much the same fashion as in year-one:

 

 

 

 

 

 

 

 

115

(1) Fringe separation and fringe axis data were logged
at several points of the glacier surface image. Fringe axis
minima spacing within the 24-hour-test halo varied over the
image from roughly 5 to 10mm. Location of the fringe
generating points, in image plane coordinates, was also
recorded at this time. The vertical reference direction was
taken as the top edge of the photoplate, which had been
leveled it may be recalled, before exposure. Using data
gathered by theodolite, points on the image Where fringe
patterns were logged were located on the topological map of
the glacier surface. Surface normals were estimated at nine
of the points, giving the best coverage across the glacier
area of interest. Using eqn. (1.1}, a gross displacement
vector was calculated corresponding to each of the nine

points.

(2) In a manner similar to that of year-one analysis,
rigid body motion was detected by microscopic examination of
the processed photoplates. Two 'rigid body' points,
appearing in each image, were examined for spurious
movement. Displacement vectors at each point were estimated
with a finely graduated scale and protractor. Finally,
location, in image plane coordinates, of each ’rigid body'

point was noted.

(3) A specially written computer routine (revised from

year-one) vectorially subtracted rigid body motion from

 

 

 

 

 

116

gross displacement at each generating point on the image.
The Euler angle algorithm, described above, calculates a
unique rigid body displacement vector for each image point.
The appropriately scaled rigid body displacement was then
subtracted from gross displacement at nine of the points
logged in step (1), giving net ice displacement for nine

points on the glacier surface image.

(4) Also in a manner similar to year—one analysis, each net
ice displacement vector was scaled according to

magnification and then projected onto the glacier surface.

(5) The stereo camera system allowed direct comparison
of results when the two resulting velocity maps, from data
acquired simultaneously, were superimposed. The merged
velocity maps did not appear to coincide. If errors were
due to mis-estimates of rigid body motion, than small,
appropriate adjustments of the original measurements would
bring about convergence. In an iterative (and admittedly,
partly intuitive) process, small, incremental changes in
both magnitude and direction of rigid body vectors were made
on each plate. (The computer code for this exercise appears
in Appendix 2.) Alternately repeating steps 3 through 5 on

each plate brought about rough correspondence.

Figure 3.17 shows this ’boot-strap’ process graphically for

one of the 24 hour plates. Consider the nine sets of three

 

 

 

 

 

 

 

 

.17

 

 

 

 

117

vectors originating from common points on Figure 3.17. Each
vector set is shown in proper relation to the others in
glacier surface coordinates. Within a set, each vector
indicates,
(l), the ice flow field resulting from raw data (gross
ice displacement with no rigid body correction, marked N),
(2), raw data with rigid body correction (gross ice
displacement with rigid body correction, marked R), and
(3), raw data with a forced rigid body correction
(marked F).
The vector drawn from the tip of N to the tip of F is the
unique, effective, forced rigid body correction for each
glacier surface point. Corresponding diagrams gleaned from

the other, simultaneously recorded 24 hour plate are

 

similar.

III. SUMMARY - EXPERIMENTAL RESULTS

Figure 3.18 is a plan view of the Nisqually Glacier surface
showing the location and relative magnitude of 'forced
solution’ ice surface displacement during a 24-hour period
in late August, 1982. Nine flow vectors were gleaned from
each of two, simultaneous, double—exposed, high resolution
photoplates. For comparison, the results of year one and
the one nearly-overlapping point from Hodge [28] are

included. Results from each study are in good agreement.

 

 

 

 

 

 

 

 

 

Figure 3 . 17

118

KEY

N -- NO RIGID BODY
CORRECTION

R -- RAW RB CORRECTION

-FORCED RB CORRECTION

R SCALE:
1 M/ D

Ice displacement results from one of the year-
two stereo plates showing 24hr ice displacement
no rigid body correction, raw rigid body

with:

correction and forced rigid body correction

 

 

 

 

 

 

 

 

119

.\ II
\

.\ @5064 ll

. .\\ waflcg:< ilIIIlIIIIWlIlI“ .BA'l

 

CID". ONO.

9 melon/an

 

I?"

Figure 3.18 24hour Nisqually ice displacement for years

-one (marked I ) and -two (markedo andA)

field seasons with one comparison point from

Hodge [28]

 

 

 

 

 

 

 

 

 

 

 

120

wancm2<

o_moas

 

MHodge)
I

Figure 3.19 Repeat of Figure 3.18 with results (marked CD)

from single plate (whose mate was lost)

 

 

 

 

 

121

The third photoplate (whose mate was lost), was analyzed in
the same manner as above, stopping short of the step 5
comparison/compromise. General agreement with the 'forced',
simultaneous solutions is satisfactory, as seen in Figure

3.19.

Tabulated ice displacement components for 24 hour intervals,
gathered from each of four photoplates, is given in Table

3.1 which follows.

 

 

 

 

122

TABLE 3.1

ICE DISPLACEMENT COMPONENTS FOR 24HR INTERVAL-NISQUALLY

in meters/day (rounded off)-refer to Figure 3.19

symbol II Z: <> C)
stereo—pair
point year—one year-two year-two year-two
no. X,Y,Z X,Y,Z X,Y,Z X,Y,Z
magnitude magnitude magnitude magnitude

1 0.6,—2.1,-1.2 0.2,-1.3,-0.5 0.3,-0.8,-0.4 0.6,-1.4,-0.8
2.5 1.4 0.6 .

2 0.2,-2.6,-l.5 0 .3,-l.l,-0.5 0.l,-0.4,-0.4 0.4,-1.2,-0.8
3.0 1.2 0.6 .

l 5

3 0.1,-1,-0.6 -0.l,-0.7,-0.3 -0.2,-0.7,-0.4 0.6,-l,-0.7
1.3 0.8 0.5 1.3

,-O.6

4 0.l,-0.7,-0.5 0.l,-0.4,-0.2 —0.3,-0.4,-0.1 5,-1.1
0.9 0.5 . 1.3

0 5

5 0.2,-O.6,-0.5 0.0,-0.8,-0.4 -0.2,—0.4,-0.2 0.4,-1,-0.6
0.8 .9 0.5 1.2

6 -.2,-1.0,-.4 - 2,-.4,— 2 -.l,- 9,— 4 4,-l.0,-.5
1.1 0.5 .8 1.2

7 2,-.6,- 2 - 2,-.5,- 2 - l,-.9,- 4 4,- 9,-.5
0.6 0.6 1.0 l l

8 0,-.5,— 3 -.l,-.3,-.l O,—.7,-.3 4,-.5,-.5
0.6 0.3 0.8 0.8

9 .l,-.9,-.4 -.l,-1.0,—.2 .4,-.5,-.4 3,-.9,-
0.9 1.1 0.7 1.1

 

 

 

 

 

 

CHAPTER 4

DISCUSSION AND CONCLUSIONS

I. IMAGING SYSTEM RESOLUTION

Sensitivity of the speckle-interferometric method is fixed

by the minimum recorded speckle size. Limiting the spatial

frequency of the specklegram image simply limits the minimum

displacement that can be detected. In the case of pointwise

 

filtering, however, truncated specklegram frequencies
contribute also to a 'fringe shift’ as noted in Chapter 2.
Fringe axis shift partially masks the actual location of the
cosine—squared extrema, resulting in measurement errors.
Thus it is important to identify the causes and extent of

frequency truncation.

To this end, consider the apparent resolution of each camera
system used in this study. Each design appears to image
different maximum frequencies for different conditions. It
will be helpful to compare the imaging characteristics of
each instrument. A common parameter describing the
performance of astronomical telescopes, 'angular
resolution', will be adopted. This parameter is often
expressed in arc-seconds (1 degree = 3600 arc-seconds).
Angular resolution of the cameras used during each of the

two field seasons, under various imaging conditions, is

 

 

 

 

 

124

given below. Resolution values are given by theory or
inferred from either bar chart tests or from diffraction
halo size as described in the following paragraph. The

results are compiled in Table 4.1.

For a circular aperture system, recall that the theoretical
maximum image plane spatial frequency is, repeating eqn.

{2.18b}, umaxs 2a/AR. Here, a is aperture radius, A is the

average noncoherent wavelength of the illumination, taken

here as 6x10-4mm, and R is the focal length of the imaging

 

lens. But, since 2a/R = 1/F#, then umax= l/AF#. Thus, the
maximum theoretical angular resolution, 7, is

u

max

1/
1 z arctan("‘§———) (4,1)

AF# 6F#
R ) = arctan [(10E14)(R)]' (4.2}

 

= arctan(

Equation {4.2} is used to calculate the maximum theoretical
angular resolution entered on the first line of Table 4.1.
In order to calculate the values entered on the second line,
eqn. {4.1} is combined with the values of the maximum
spatial frequencies transmitted by the camera lenses during
optical bar chart tests. Equation 4.1 is combined with the
maximum spatial frequencies inferred from halo diameter (as

reported in Chapter 3) to calculate the remaining two lines.

 

 

 

 

 

 

 

125

TABLE 4.1

SPECTRAL RESPONSE OF IMAGING SYSTEMS FOR VARIOUS CONDITIONS
in arc-seconds; emulsion and test parameters noted

Theoretical maximum -
using eqn. {4.2}

Practical limit - (4.1};
bar chart test results

Artificial field displacement (l3 @ F#8)
tests of parking lot surface,

using (4.1); inferred
from diffraction halo size

Artificial field displacement

tests on glacier surface,
using eqn. (4.1}
(inferred from halo size)

24 hour time lapse test
on glacier surface,
using eqn. {4.1}
(inferred from halo size)

 

YEAR-ONE
500mm lens

2 @ F#8

3 @ F#8

l6 @ F#ll, l/8s
22 @ F#22, 1s
each exposure
Holo-film
80-253

52 @ F#l6
l/SOS each exp

50 @ F#8
ls each exp
Spectro
649-F

916mm lens

2.2 @ F#16

*see text

YEAR-TWO

 

3.1 @ F#l6

ll @ F#l6

25 each exp

Spectro-
plate
649-F

47 @ F#l6
2s each exp
Spectro
649-F

47 @ F#l6
25 each exp
Spectro
649—F

 

 

 

 

 

 

 

126

Note that all year—two experiments were carried out at F#l6,
the glacier imaging setting. Imaging at many different
settings is evident, on the other hand, during year-one
experiments where a variety of emulsions were used. Despite
this diversity, it is possible to compare the performance of
each camera type at constant F#. This means that the
performance of the year—one instrument was inferred from the
results of the artificial displacement tests in which the
graveled parking lot surface was used as the object. It can
be seen that the angular resolution for this set of
experiments became finer as the aperture increased two F-
stops in diameter from F#22 to F#ll. This behavior was
anticipated. Furthermore, an additional increase in
resolution would be expected if the same scene had been
recorded at F#8. Unfortunately, no such experiment
occurred; the results of such a test may be estimated,
however. It is reasonable to assume that if two F-stops
reduced the angular spread by 6 arc-seconds then one
additional F-stop would reduce it linearly by an additional
3 arc-seconds. Thus, it can be assumed that, had the
specklegram been recorded at F#8, the resulting angular
resolution would prove to be about 13 arc-seconds. This

estimated value is denoted in Table 4.1 with an asterisk.

Holographic and Spectroscopic emulsions were both used in
the photography experiments summarized above; these coatings

have comparable resolution but spectral sensitivity of the

 

 

 

 

 

 

 

127

Spectroscopic emulsion is somewhat more broad. Recall that
duplication of the Spectroscopic plate onto more contrasty
Holographic emulsion was required during year-two
experiments in order to increase diffraction efficiency.
Spectroscopic emulsion was used during the first-year,
however, with good results upon filtering the primary plate.
The difference is the coated lenses used in the year-one
experiments. Stray reflected light reduces image contrast.
This is especially true of highly corrected instruments
having a large number of air-glass surfaces; lens coating
substantially reduces internal reflections. It may be
concluded that coated lenses render images having higher
diffraction efficiency, especially when the image is

recorded on emulsion having broad spectral response.

Summarizing Table 4.1, it is seen that the resolution limit
of the year-one and year-two camera lenses is approximately
equal for equal conditions regardless of emulsion. The
resolution limit, as determined from the indoor bar-chart
tests should be taken as a relative measure between the
year-one and -two systems since the eye perceives greater
detail than that which could actually be imaged by the lens.
The three arc-second limit, nevertheless, fits nicely
between the theoretical value and the values determined from

artificial displacement tests.

 

 

 

 

 

 

128

There is noted, however, a significant resolution reduction
when shooting at equal distances outdoors rather than
indoors. Recall optical bar-chart tests were conducted
indoors at a distance of 130m for the year—two camera, while
artificial displacement tests, using the same instrument
outdoors, were conducted with about the same camera-to-
object distance (about 150m). It was assumed that virtually
all spatial frequencies appeared across the surfaces of both
the parking lot and glacier-with proper lighting.

Evidently, refraction gradients in the media through which
the images were collected affects the spectral components in
such a way as to act as a low pass filter. This effect is

known in astronomy as the 'seeing’ limit.

In order to determine this upper frequency bound, some
background is necessary. In a landmark work related to
stellar signal distortion, Fried [54], explains that the
incoming wave front is 'tilted' by turbulent media through
which the front must pass (i.e. tilt is a function of time).
"Tilt displaces the image but does not reduce its sharpness.
If a very short exposure image is recorded, the image...is
insensitive to tilt, which can be a substantial part of the
total distortion". An isoplanatic column of air through
which two nearby beams travel to the imaging lens varies
with Space and time. Refraction characteristics of such a
column are considered constant for a 'short’ time interval.

Short time exposures for astronomical imaging range up to 20

 

 

 

 

129

milliseconds, with most workers using exposure intervals of
less than 10 milliseconds [53]. The minimum glacier
exposure interval was equal to the larger value, but most
photographs were recorded at much longer exposure times.
Wave-front tilt, evidently, caused spurious, high-temporal
frequency image oscillations. These were first noted during
outdoor artificial field displacement experiments. Images
collected outdoors are assumed to be distorted by wave-front
tilt and the spatial frequency band was clipped by an amount

proportional to the level of turbulence.

Astronomers characterize this complication by multiplying
the telescope modulation transfer function (MTF) by a
'structure' function which describes the wave-front
perturbation statistics in the entrance pupil. Some
researchers [54] feel that such a procedure is an
oversimplification, that is, lens and atmospheric effects
cannot be separated in the manner such theory suggests.
Alternative viable theories, hOWever, are simply
unavailable. For the present, workers must cast about
within the sea of speculation to find a suitable multiplying
function. Such a technique is used below. In applying this
theory to the present work, it is assumed that a 'suitable'
frequency—dependent ratio term simply diminished image plane
intensity of the higher spatial frequencies (i.e., the term

multiplies eqn. (2.15)).

 

 

 

 

 

 

 

 

130

Von der Luhe [57] uses the results of Korff [55] to describe
the wavefront statistics and then computes the ratio of the
squared-MTF's, with and without atmospheric turbidity. The
analysis is centered about a very narrow optical frequency
range of wavelength 4.4x10E-4mm. Some of von der Luhe’s
theoretical results are shown (dashed) in Figure 4.1. The
horizontal axis is the non-dimensionalized, relative spatial

frequency, w/w , where w is the maximum theoretical
max max

limit admitted by the aperture (given by eqn. {2.18b}). The
numbers next to each curve are the values of the parameter,

a = ro/D, where D, is the aperture diameter, and r0 is an

atmospheric turbulence parameter, the latter first suggested
by Fried [54]. In order to make use of von der Luhe’s

results, ro must be determined. Unfortunately, very little

is found in the literature related to daytime ’seeing'
extended objects through a finite atmospheric layer of
nearly constant altitude. Astronomers feel that most of the
stellar signal distortion occurs during its passage through
the lowest level of atmospheric layers [27] so a stellar
model may adequately account for the perturbations incurred
by terrestrial waves recorded in the present study. Viewing
angle inclination and station altitude [57] also affect
'seeing’. Thus, a stellar imaging model that accounts for
inclination is sought. Hufnagel’s turbulence model [56],
abstracted in Figure 4.2, is one such prototype. At an

altitude of 10m (glacier images were shot low over both the

 

 

 

 

 

 

 

131

 

 

 

 

Relative Freq uency

Figure 4.1 Modulation transfer function for atmospherically

. degraded images-abstracted from von der Luhe[58]

    

INCLINATION:

-1
~° lo I-
'0

\'6/5(MEtens-MICRONs-s/sl

 

'2
‘0 A A J .
‘00 ‘01 102 103 to“

ALTITU DE (METERS)

Figure 4.2 Hufnagel’s atmospheric turbulence model-

abstracted from Korf [56]

 

 

 

.r
o
O

 

 

 

 

132

glacier valley and over the glacier surface), for
A = 0.5x10E—6m (= 0.5 microns), and an inclination of 1
deg., Hufnagel’s model gives,

roA exp(-6/5) — .02.
Thus, roz 9mm. The parameter a - ro/D, for

D = aperture diameter = focal length/ F-number, is z.15.
Interpolating the appropriate curve of Figure 4.1, it is
seen that fringe intensity is reduced to .5 of its zero
frequency value at a relative frequency of .06. For the
year—two study, this means that the angular resolution is
about l/.06 z 17 times coarser than the theoretical minimum
of 2.2 arc-seconds. Thus atmospheric cut-off is predicted
to be about 37 arc-seconds. The same analysis predicts
significant clipping begins at angular resolution less than
40 arc-seconds for the year-one camera. Discrepancies may
be the result of von der Luhe's single frequency analysis as
opposed to the relatively broad-band, visible light domain
used in the present study. Nevertheless, correspondence
between these numbers and the last entries in Table 4.1 is

very good.

Von der Luhe's work is cited because it is recent (1984) and
because he included experimental results which were garnered
from a large number of solar observations at the Mt. Palomar
Observatory. It is worth noting that the theoretical

attenuation, plotted (solid) for different a’s, seem to

 

 

 

 

 

 

 

 

133

collapse, upon experiment, into the narrow range shown
dashed in Figure 4.1. This behavior is due, von der Luhe
explains, to experimental error. Using von der Luhe's
experimental results and again assuming cut-off is where
intensity falls to .5 of maximum, predicted angular
resolution is about 30 arc-seconds for each camera system.
Again, this value corresponds reasonably well with the

resolving power demonstrated at the glacier site.

Discrepancies between these results and those of von der
Luhe's could result from mis-focus. It is shown (in Chapter
2.I.C) that the glacier instruments, for example, require
focus adjustment accurate to about .5mm. Thermal expansion
or other dimensional changes beyond the control of the
operators may have changed the focal setting and thus
exceeded this bound. The relatively low and narrow
frequency range imaged at the glacier sites tends to
minimize the effects of defocus. Thus, one of the
advantages of atmospheric clipping is less stringent

focusing requirements.

Off—axis contrast losses may have a somewhat stronger effect
than mis-focus. Image points located far from the optical
axis have been gathered from object points which lie outside
the central isoplanatic patch; the MTF depends on field
angle. Since the point spread function is not radially

symmetric (as is usually the case except on the axis) the

 

 

 

 

 

 

 

134

MTF will also vary with azimuth of the object lines defining

it [37]. This is the sum effect of aberrations.

It is difficult to quantify this behavior in the glacier
imaging cameras. When scanned along a radius, the
specklegram images produced fairly constant halo diameter,
indicating constant spatial frequency. The problem may
simply be one of measurement; a few mm change in the
location of an ill-defined boundary is not easily discerned.
Real lenses, however, (see any technical performance report
on quality photography equipment) show a contrast loss in
the range of 20 to 30% from center to edge for most F#‘s.
Since the glacier flow data was gathered from points far
from center of the specklegram image, some additional losses
can be expected (even though such tests of popular camera
equipment are conducted at wider angles). It is concluded
that data gathered from the image center would generally
exhibit higher spatial frequency and thus be in closer

correspondence with von der Luhe's results.

Finally, it is noted that the results of the present study
are in the range of Worden's assertion [53] that a roughly
50-fold reduction in angular resolution is expected from the

theoretical limit when imaging starlight through the

atmosphere.

 

 

 

 

 

 

 

 

135

With so many parameters available, it might be argued, a
moderate correspondence with all the data was inevitable.
Struggling against the paucity of facts related to
terrestrial imaging the agreements are, nonetheless,

pleasing.

For completeness, another technique allowing measurement of
motion which would otherwise be too small is noted here.

The method consists of displacing the camera a known amount
between exposures, in the plane of the film. Thus, object
motion is added to camera motion in the specklegram record.
During analysis, the prescribed camera motion is then
subtracted from the gross image displacement in the manner
given in Chapter 3. Rendering object displacement (referred

to the film plane) in an acceptable range by intentional

 

movement of the camera has been successful in the laboratory
[13]. It is anticipated that this technique can be adapted
to other geophysical problems and to other large dynamical

systems.

While atmospheric effects may limit resolution even in good
weather, sufficient information can be recorded (on
relatively inexpensive film) to determine displacements if

the time interval between double exposures is properly

selected.

 

 

 

 

136

II. ERROR ANALYSIS

Capturing fringe-generating glacier images was the objective
during the year-one field season. Accordingly, parameters
such as vertical reference direction, rigid body
displacement reference point location, topological map
location of glacier surface features, and other data were
not precisely determined during the first Nisqually visit.
Flow rate error estimates for the year-one experiments were
difficult to generate. Such estimates are given below for
the year-two study, however, with a closing note comparing

the results from the two field seasons.

Errors are introduced at three stages in the interferometric
measurement process. Each, in turn, is discussed below; the

accumulated errors are estimated in the final section.
A. Fringe pattern measurement

In an effort to increase speed and accuracy of an otherwise
tedious exercise, electronic devices such as diode arrays
and computing machines have been recently adapted to the
fringe measurement task. An important parameter in the
automated process is fringe visibility which, in this case,
is easily quantified. (Visibility is defined and discussed
with respect to the pointwise filtering method in Chapter

2.IV.B.) The manual (visual) method used in the present

 

 

 

 

 

 

 

137

study, however, does not include quantification of this
parameter. In order to determine the effects of fringe
visibility the value must be estimated; 0.4 seemed
reasonable. In the range of visibility from 0.3 to 0.5, and
with three (or perhaps more) fringes in the diffraction
halo, Kaufmann's analysis [51] predicts the distance between
apparent fringe minima will measure 10% to 15% greater than
the true value. Accordingly, the resulting uncertainty in
the measurement of fringe minima separation distance is
about 5%. Because, according to eqn. {1.1}, film plane
displacement is linear with fringe separation, the gross ice
displacement estimates are also 15% uncertain owing to

poorly specified visibility.

Interferometry reveals not only magnitude but also direction
of the displacement occurring between exposures. It has
been found that such direction (fringe axis orientation)
measurements, with practice and patience, were repeatable to

within :2 deg.

B. Rigid body motion determination

Relative motion of the camera with respect to the object
surface is viewed as the single most important factor
affecting accuracy. As indicated in the previous chapter

(3.VI.C), such motion is detected with photogrammetry

 

 

 

 

 

 

 

 

 

138

methods. Thus, interferometric measurements were degraded

to the lower confidence level of the photogrammetry process.

Microscope measurements of the displacement of 'fixed'
points in the image yield the so-called 'rigid body
displacement' vectors. Magnitude estimates of these vectors
by different members of the research group are repeatable to
i20%, while orientation estimates vary in the range of i5
deg. These potential errors are not as important, however,
as are the differences between the initial rigid-body motion
estimates and the final values arising from the forced
simultaneous solution. Recall from the previous chapter
that associated with each point on the photographic image

there is: (i) a unique gross displacement inferred from

 

fringe patterns; (ii) a unique rigid body displacement
(calculated from Euler angle parameters and original rigid
body estimates); and, (iii) a unique rigid body displacement
vector that corresponds to the final forced solution. As
seen in Figure 3.17 (Chapter 3.II.D), it is clear that final
rigid body motion vectors do not, on the whole, vary largely
from the original estimates. Indeed, the differences in
both magnitude and direction are within the previously

stated range of i20% and i5 degrees.

Means to improve precision by delineating the rigid body
motion interferometrically in subsequent field tests are

proposed in the final section.

 

 

 

 

 

139

C. The projection of net ice displacement onto the glacier

surface

Here, the critical value is the glacier surface location
corresponding to the image point at which fringe patterns
were detected. Optical magnification and geometric
projection of net ice displacement onto the glacier surface
is strongly dependent on these coordinates. During year—two
field work, careful surveying located such glacier surface
points to within a few tens of meters. Magnification was

easy to calculate within a fraction of a percent.

The larger problem was to estimate the surface normals on
what is a highly convoluted surface. Determination of an
'average' slope turned out to be fairly straight—forward
since lasers used in the pointwise filtering process produce
2-3mm diameter beams which subtend a relatively large area
of glacier. This area corresponds to an ellipse, when
projected onto the glacier surface, which is roughly the
size of a football field. Since spatial frequencies are
averaged over precisely the same elliptical area, 'average’

surface slope estimates are justified.

Topological maps show Mt. Rainier’s south flank varies, on
average, between 27 and 38 degrees from horizontal,
gradually decreasing toward the Nisqually snout. Over a

distance of 40 to 50 meters (the maximum surveying error),

 

 

 

 

140

the slope does not change more than 3 degrees; average slope
values are, therefore, accurate to at least i3 degrees, with

the same confidence in average surface normals.

Though topological maps were compiled from eight—year-old
data, general features such as surface slope and icefall

location, do not change dramatically over time. For the

present purposes then, use of 'old' maps contributed

negligible error.

The computer algorithm (shown in Figure 3.10, and documented
in Appendix 11) projects net ice displacement whose
magnitude is proportional to the cosine of the surface
normal. Using the given error limit, the ratio of possible
cosines gives the maximum ice displacement magnitude error.
Near the icefall the surface normal is about 52 degrees from
horizontal.. With i3 degrees confidence in the glacier

surface slope values, the ice displacement may vary about,

_ cos§522 ~ +
[1 cos(52i3)] x 100 ~ _7%.

Thus, near the maximum slope, ice displacement rate errors
are no more than i7% in magnitude. A similar calculation
for down-glacier points, where the slope is about 27

degrees, gives about 110% maximum error.

 

 

 

 

 

 

 

141

D. Accumulated errors

Vector magnitude errors incurred in each of the analysis
stages are multiplicative; their handling is straight-
forward. Angular measurement uncertainties, however, are
coupled to vector magnitudes because of the projection of
the image vector domain (a circle) onto the glacier surface
ice displacement range (an ellipse). For the relatively
small angle measurement errors of the present case, such
coupling can be ignored. Angular uncertainties are thus
additive. The possible range of azimuth for any of the ice

displacement rate vectors shown in Figure 3.18 is thus,

i(2 + 5) = i7 degrees.

Using the possible measurement errors characterized in the
three previous sections, ice flow rate magnitudes may vary

up to

-30%
+35%

1 - (1i.05)(li.20)(1i.07) z
from their given value for up-glacier regions. For dOWn-

glacier areas, the possible error is,

-32%

l - (1i.05)(1i.20)(1i.10) z +39%

 

 

 

 

 

 

 

 

 

 

142

These discrepancies are roughly in the range of seasonal
variations reported by Hodge [28] who, using conventional
methods at the same glacier, measured flow rates varying

about 25%.

Though the above analysis addresses year—two results only,
good comparison between the flow data from the two field

seasons indicates that year-one errors are of similar size.

III. COMPARISON OF RESULTS WITH ICE FLOW THEORY

At the current level of development, the theory of ice flow
is not adequate to describe the behavior of real ice in
either ice sheets or glaciers. Nye [33], Patterson [34]
and, more recently, Hutter [35] are among many who offer
versions of ice rheology in naturally occurring bodies.
With time, succeeding theories have understandably become
more complex and more in need of field data to fix boundary
conditions and to establish phenomenological relationships.
The boundary-value problem, coupled with temperature and
velocities defies exact solution. Also, it is difficult to
identify trends from numerical results, which are inherently
over-specific. The problem is classic: there are simply

too many unknowns to allow a general analytical solution.

As an illustration, consider the ice flow formula developed

by Patterson [34]. Using Glen’s [25] flow law (i.e.

 

 

 

 

 

 

143

stressrate = A x strainraten, where A and n are constants),
Patterson attempts to describe the simplest case of down-
slope, laminar velocity for a deforming infinite ice sheet

by the following formula:

hn+1

u - U = 2[Aoexp(-Q/RT)]{pg sin<a>ln n+1

s b {4.3}
The variables, with comments on their range of values and
possible relationship to the behavior of real glaciers are
described below:

U and U

s b are the longitudinal velocity components of

the surface and base, respectively. These are assumed to be

the only non-zero components. Note that eqn. {4.3) gives

 

differential rates only; it contains no information on
absolute surface velocity. In real glaciers, basal sliding
varies over an enormous range [36], depending on water
storage and efflux, bed obstacle size and spatial frequency,
and whether the creep mechanism or the melt-regelation
process is dominant. Accounting for bed topography requires
detailed seismic, gravity or sonar studies. Such
painstaking work is rarely done on a sufficiently large
extent of glacier to be useful.

A0 is given as a constant but, in fact, depends on the

grain size, ice impurity and crystal orientation [18].

 

 

 

144

Q is the activation energy for polycrystalline ice.
Actually Q is temperature dependent, doubling in value from

—250.C to +100.C, according to Barnes [38]. Barnes reports

that Q seems to be a function of the extent of the liquid

phase at the grain boundaries.

The temperature T is a function of both position and

time in temperate glaciers where the effects of strain

heating are significant [35].
R is the universal gas constant.

The bracketed term in eqn. {4.3} is actually the

viscosity A in Glen's flow law. Ice responds to deviatoric

stress by becoming softer; the bottom layers slide over one-

another more easily than those above, resulting in high

velocity gradients near the sole. Little has been reported

 

regarding the non-Newtonian properties of glacier ice
exhibiting strain-rate dependent viscosity.

Ice density, p, for glacier work, is usually taken as

.9 times the specific gravity of pure ice. Values from

borehole measurements [34] vary, however, from 0.85 to 0.91
g is gravitational acceleration.
a is the angle of the mean slope upon which the ice

slab model slides. This value varies, obviously, along the

course of a real glacier.

The exponent, n, is usually taken as 3 for glacier

work, but Raymond [36] has surveyed experimental results and

 

 

 

 

 

145

found that this number varies from 1.3 to 4.2, with a mean

of about 3.2, and that it is a strong function of stress.
Finally, h is slab thickness, assumed to be constant

Thickness

for the above model, but not for the real case.

of glacier ice is roughly proportional to the inverse of the
slope; mountain flanks supporting glaciers often vary 10 to

20 degrees from head to terminus.

 

Not mentioned above, in relation to the slab model, are

other factors affecting flow:

The effects of valley walls which, in the case of the

Nisqually icefall, are presumably significant since this

body is relatively narrow. Rock side-boundaries may retard

flow by surface friction or accelerate flow by providing a

liquid-ice interface through heat conduction from the

surface [34]. Such a melting condition obviously

contributes to a transverse as well as a downward (toward

the sole) velocity component.
Real glacier flow is seldom planar, due in part to the

so-called ’submergence’ velocity. In the accumulation area

(e.g. above 2200m elevation at Nisqually) ice flow has a

downward velocity component estimated to be 5-10% [34] of

the gross surface displacement vector magnitude. Emergence

velocity in the down-glacier ablation zone is the reverse of

this trend. Emergence is physically understandable; if it

did not occur the surface would become ever steeper.

 

 

 

 

 

 

 

 

 

 

 

146

Average Nisqually surface velocity varies as much as
40% on a monthly basis, as reported by Hodge [28]. Other
glaciers exhibit similar but more extreme behavior. A
theory has been offered by Raymond [36] which correlates
temporal variations of velocity with stress. Observations
by Meier [49], furthermore, have indicated that internal
deformation seems to be controlled by an ’effective' slope.
In Meier's model, effective slope is a function of bed
topography, bed slope and surface slope, the latter changing

slowly with time.

Quantitative comparison of the results of the current study
with ice flow theory is fruitless; by observing the general
characteristics of the above model, however, some of the
trends observed in the velocity map, Figure 3.18, can be
explained. Points 1 and 2 of the year-one results are
located near the steepest part of the Nisqually icefall.
Eqn. {4.3} indicates velocity has a strong dependence on
surface slope; the relatively high rates shown here are

expected.

With other factors constant, general theory predicts that

down-slope flow rates gradually increase in the accumulation
area and gradually decrease in the ablation area. Emergence
velocity contributes to this tendency and is seen to be zero

near the equilibrium line which separates the accumulation

 

 

 

 

 

 

147

and ablation areas. This trend is generally borne out along

midvglacier as in Figure 3.18.

Little general theory has been offered regarding ice
behavior in zones of confluence. The Nisqually and its
major tributary, the Wilson glacier, meet just below the
rock formation known as the Wilson Cleaver. Here, a
distinct, westward flow trend which is not directly down
hill is noted in Figure 3.18. This is possibly the result
of the convergence of differential stress zones. The
greater accumulation area of the Nisqually, in this maritime
region, could account for higher stresses in this body
compared with those occurring at the same elevations in the
Wilson. The meeting of these two systems is accompanied by
a strain—rate component corresponding to the direction of

decreasing stress gradient.

The glacier flow results from the two field seasons are
observed to be self-consistent. They compare well with the
limited available field data collected by conventional

means, and they seem reasonable.

 

 

 

 

 

 

148

IV. FURTHER STUDY

A. Increasing precision

Researchers endeavor to raise the density of information
that is photographically recorded. Generally, this means
increased resolution, but successful efforts to image yet
finer detail must be accompanied by equally well-described
camera motion. Precise determination of camera motion and
techniques to make use of high object-spatial frequencies

follows.

a. Interferometric determination of rigid body motion

Camera motion is probable during time-lapse photography;
reduction of such motion would be effected by steady camera
mounting. Metal stakes, driven into the earth would provide
substantially improved support. Two means to delineate

residual motion are suggested.

Appropriately patterned stationary grids placed in the
viewfield would serve to generate Young's fringes during the
interrogation process. Two focused grids, localized in the
photographic image, would provide sufficient information to
determine translation vectors and Euler angle values; camera
motion, in the plane of the film, would be completely

specified. The photogrammetry exercise of step 2 (in

 

 

 

 

 

149

Chapter 3.1.C) would thus be replaced by the more accurate
interferometric method. Grids may be any contrasting
pattern provided the spatial frequency is high enough to
register small relative camera movement but not so high as
to be affected by atmosphere/lens attenuation. Determining
useable grid patterns and placement would require but a
simple series of on-site experiments carried out
concurrently with artificial displacement tests. A
completely interferometric method would raise precision by

at least an order of magnitude.

Apparent camera motion would be effectively reduced to zero
if two sequentially recorded photoplates (each single—
exposed) were superimposed in exact registration. As

explained in Chapter 3, such a configuration is allowed by

 

recording alternate images on reversed emulsion (i.e.
through the glass). Superimposing photoplates in perfect
registration requires patience, intuition, and perhaps,
specialized three-degree—of—freedom mechanical positioning
equipment. Success has been reported by Adams [32], using
laboratory subjects; it should be adaptable to larger
objects. 'Perfect registration is and ideal condition;
remnant camera motion could be ascertained by a hybrid of

the above two methods.

Another problem, which should be remedied by steady camera

mounting, is worth noting here. Dew collecting on the

 

 

 

 

 

150

lenses often delayed glacier time lapse experiments.

Muirden [58], writing for amateur astronomers, suggests that
a long black-lined tube extending beyond the telescope lens
will help prevent dew from forming. With sturdy mounting,
and well-defined camera motion, the effects of wind on a
large tube extending several diameters from the lens would
not be of concern. In addition, if the camera mount is
sufficiently stout, it should be possible to simply remove

the dew mechanically.

b. Short exposure methods

A series of short exposures, optically processed, allowed

Layberie [39] to resolve dual stars which were far closer

 

than the normal atmospheric seeing limit. Since this 1970
milestone, astronomers have used short exposures times to
'freeze' atmospheric refraction. The ensemble of a large
number of photographs yields detail approaching the
telescope diffraction limit. Unfortunately for the present
work, Layberie's technique yields displacements, not
displacement rates. Very short exposure times,
nevertheless, ought to freeze the effects of atmospheric
turbulence in terrestrial images as well. Upon filtering
such a short—exposure specklegram, refraction gradients will
be manifest by spurious fringe axis separation. At the same
time, the diffraction halo will appear larger, indicating

increased resolution. The trade-off is classic: higher

 

 

 

 

 

 

151

resolution for more noise. A series of on-site, variable—
exposure-time experiments would provide an indication of the
necessary sacrifices in attaining higher resolution.
Addressing once again the desirable characteristics of
photographic emulsion, it seems clear that narrow band, high
response is indicated. Photographic media must image a
narrow frequency band to counteract the contrast reducing
effects of both dispersion and uncoated lenses (if used) and

to facilitate short exposure intervals.

It is expected that statistical treatment of large numbers
of high resolution images, each having well defined camera
motion, will yield accurate flow rate maps. The large
number of images required for accuracy will not limit data
processing.- Comparing many images is now possible with
computer-based digital imaging techniques. Further, with
the anticipated accuracy of such systems, optical processing

may not, ultimately, be required.
B. OTHER APPLICATIONS OF THE WHITE LIGHT SPECKLE METHOD

TIROS-l, launched in 1960, was the first in a series of
weather satellites used to map conditions on earth. Crude
vidicon images provided the first, long-term, birds-eye
views of large scale cloud patterns. Many geodetic
platforms followed TIROS-l into orbit. ERTS images [21],

for example, were used to monitor and predict glacial mass

 

 

 

 

 

152

balance of hydropower regions in Europe. LANDSAT [22]
images were photogrammically analyzed in an attempt to
determine ice flow through Canada's Baffin Bay connecting
the Arctic and Atlantic oceans. Judging from available
reports, however, interferometric methods have not been
applied to geophysical mechanisms using platforms in space.
One reason is resolution: At the present stage of
development, electronic imaging does not preserve the
requisite structure; interferometric techniques require
primary 'analog' character. This limitation is expected to
diminish as digital information packing becomes more dense

and the information itself becomes more easily manipulated.

For the current level of electronic imaging sophistication,

 

however, the space shuttle provides the means to record and
recover primary photographic data. High-resolution
photographs of glacier or sea ice, or of large land masses,
are achievable; the above-detailed experiments have
demonstrated that Young's fringe patterns, indicative of
displacement and strain, can be generated from such images.
Superposition of single-exposure photoplates would be the
required technique in this case where the camera cannot
remain stationary during the time-lapse interval. There are
many additional benefits of a vantage point from space: The
required photographic equipment is small, reliable,
relatively inexpensive and has been proven easy to handle in

weightless conditions. 'Overhead' is a much desired

 

 

 

 

153

perspective from which to record geological processes
occurring on the earth's surface. Reliance on topological
map data is reduced. It is clear that space platforms can
greatly expand the potential of the noncoherent light

method.

 

 

 

 

 

 

 

APPENDICES

 

 

APPENDIX I

PHOTO PROCESSING DETAILS

Kodak Developing solution, Fixing time
Emulsion time (min.) @ 200 (min)
Minicard II D-19, 8 10

Film SO-424

Fast Holographic
Film SO-253 HRP (2:1), 2 5
Plate 131-02

Spectroscopic
Plate 649-F D-19, 7 10
(cold processing @ 15C) D—19, 15 20

 

APPENDIX 2

COMPUTER CODE - FORTRAN V

 

cccccccccccPROGRAM GLA27 cccccccccccccccccc
DIMENSION P(l3),D(3),WKAREA(68),A(3,3),S(3,1),Al(2)
+,W(9,5),CHARA(9,5),CHAR(13)
CHARACTER*1 0,NUM
CHARACTER*5 PTNO
CHARACTER*4 Fl
CHARACTER*8 FILE,CHARA,GAP,GAQ
CHARACTER*12 ZZ1,ZZ2,ZZ3,ZZ4,ZZ5,GFIL
CHARACTER*20 LAO
CHARACTER*44 CHAR,LA1,LA2,LA4,SCALE,LA5
OPEN(7,FILE='OUTPUT')
PRINT '("PROGRAM GLACIER27")'
PRINT ’("ENTER DATA FILE NUMBER”)’
READ(1,'(A4)’)F1
NFLAG=0
PRINT '("IF YOU WISH NUMERICAL RESULTS ENTER L")'
PRINT ’("IF NOT, ENTER 0")'
READ(1,*)NFLAG
FILE='DATA'//Fl
OPEN(5,FILE=FILE)
READ(5,*)P1X,P1Y,TMAG,GP,Q1X,Q1Y,GQ
CCCCCCCC INITIALIZE GRAPHICS CCCCCCCCCCCC
GMA1=30.
CALL INITT(480)
GFIL='G GLA27'//Fl
CALL OPENTK(GFIL,IER)
CALL DWINDO(-400.,388.,000.,600.)
CCCCCCCCCCCC DRAW POINTS AT CAMERA SITES CCCCCCCCC
CALL MOVEA(0.,1070.)
CALL CHARTK('*’,.4)
CALL MOVEA(220.,1600. )
CALL CHARTK(’*’,.4)
SIGN=1
TMAG=SIGN*TMAG
P2X=TMAG*COS(GP)+P1X
P2Y=TMAG*SIN(GP)+P1Y
T=TAN(GQ)
R=SQRT((Q1X—PlX)**2+(Q1Y-P1Y)**2)
TEMP1=1+T**2
TEMP2=2*(—P2X—(T**2)*QlX+T*Q1Y-T*P2Y
TEMP3=(T*Q1X)**2+2*T*Q1X*P2Y
+-2*T*Q1X*Q1Y—2*Q1Y*P2Y
++P2X**2+Q1Y**2+P2Y**2—R**2

I—‘KD

DISC=(TEM2**2)-4*TEMP1*TEMP3
IF(DISC.LT.O)THEN

SIGN=-l

PRINT '("DOES NOT CONVERGE-INITIAL DISPLACEMENT ")'
PRINT ’("DIRECTION CHANGED IN SIGN. ")'
GO TO 1

ENDIF

DISC=SQRT(DISC)
Q2Xl=(-TEMP2+DISC)/(2*TEMP1)
Q2X2=(—TEMP2-DISC)/(2*TEMP1)
Q2Yl+T*(Q2xl-Q1X)+Q1Y
Q2Y2=T*(Q2X2-QlX)+QlY

RX=QlX-P1X

RY=QlY-P1Y

AlX-Q2Xl-P2X

A1Y-Q2Yl-P2Y

A2X=Q2X2-P2X

A2Y-Q2Y2-P2Y

RMS=RX**2+RY**2

CCCCCCCCCCCCALCULATE ROTATIONS CCCCCCCCCCC
A1(1)=ASIN((RX*A1Y-RY*A1X)/RMS)
A1(2)=ASIN((RX*A2Y-RY*A2X)/RMS)

CCCCCCCC SELECT SMALLEST ALPHA-PLATE ROT ANGLECCCCCCC
ALPHA=A1(1)
TRANSQ=SQRT((Q2X1-Q1X)**2+(Q2Yl—Q1Y)**2)
ALPHALT=A1(2)
TRANSQALT=SQRT((Q2X2-QlX)**2+(Q2Y2-Q1Y)**2)
IF(ABS(A1(1)).GT.ABS(A1(2)))THEN
ALPHA=A1(2)
TRANSQ=SQRT((Q2X2-QlX)**2+(Q2Y2—Q1Y)**2)
ALPHALT=A1(1)
TRANSQALT=SQRT((Q2Xl-QlX)**2+(Q2Y1-Q1Y)**2)
ENDIF

CCCCCCCCCC CALCULATE TRANSLATIONS CCCCCCCCC
TX=TMAG*COS(GP)

TY=TMAG*SIN(GP)

CCCCCCCCC OPERATE ON EACH POINT CCCCCC
DO 2 I-1,9
READ(5,*)(P(J),J=1,13)

DX—P(7)*COS(+P(8))
DY=P(7)*SIN(+P(8))

CCCCCCCCCCCCALCULATE ROTATION ABOUT POINTCCCC
V1X=P(12)—P1X
V1Y=P(l3)-P1Y
R2X=V1x*COS(-ALPHA)+V1Y*SIN(-ALPHA)+P2X
R2Y=-V1X*SIN(—ALPHA)+V1Y*COS(-ALPHA)+P2Y

CCCCCC CALCULATE RIGID BODY DISPLACEMENT OF POINT cccccc
RBX=R2X-P(12)

RBY=R2Y-P(13)

CCCCCCCC CALCULATE NET ICE DIP. REL TO FILM PLANE IMAGECCCC
XNET=DX-RBX
YNET=DY-RBY
FDISNET=SQRT(XNET**2+YNET**2)

cccccccccc ACCOUNT FOR OPTICAL MAGNIFICATION CCC

 

 

0D=SQRT((P(1)-P(4)**2+(P(2)—P(5)**2+(P(3)-P(6))**2)
OM=OD/91.7

XNET=XNET*OM

YNET-YNET*OM

CCCCCCC CALCULATE NET ICE DISP. MAG AND ANGLE REL TO FILMCCC
PHI=ATAN2(YNET,XNET)
DNET-SQRT(XNET**2+YNET**2)

CCCCCCC CALCULATE VALUES FOR LEQTZF CCCCCCCC
SI=-ATAN2((P(4)-P(1)),(P(5)-P(2)))
D(l)=DNET*COS(SI)*COS(PHI)
D(2)-DNET*SIN(SI)*COS(PHI)
D(3)=DNET*SIN(PHI)

A(1,1)=P(0)

A(l,2)=P(lO)

A(1,3)=PP(ll)
A(Z.1)=D(3)*(P(5)-P(2))-D(2)*(P(6)-P(3))
A(2.2)=-(D(3)*(P(4)-P(l))=D(1)*(P(6)-P(3)))
A(2.3)=D(2)*(P(4)-P(l))-D(1)*(P(5)-P(2))
(3.1)=D(1)

A(3,2)=D(2)

A(3,3)=D(3)

S(l,1)=0

S(2,1)=0

S(2,l)=DNET**2

M=l

N=3

IDGT=1

IA=3

CCCCCCCC CALL IMSLS SUBROUTINE CCCCCCCCCCC
CALL LEQT2F(A,M,N,IA,S,IDGT,WKAREA,IER)

CCCCCCC DRAW GLACIER DISPLACEMENT VECTOR CCCCCC
X=P(4)*.1
Y=P*5)*.l
CALL MOVEA(X,Y)

O=’0'

CALL MOVEA(X,Y)
VLEN=SQRT(S(1,1)**2+S(2,1)**2+S(3,l)**2)*GMA1
ANGLE=ATAN2(S(2,I),S(l,l))

CALL DRAWR(VLEN*COS(ANGLE),VLEN*SIN(ANGLE))

CCCCCC WRITE POINT NUMBER NEAR ORIGIN CCCCCCC
CALL MOVEA(X,Y)

CALL MOVER(0.,7.)
WRITE(NUM,’(Il)')I
CALL CHARTK(NUM,.7)

CCCCCC WRITE OUTPUT INTO CHAR ARRAY CCCCCC
W(I,1)=FDISNET*10
W(I,2)=S(l,l)

W(I,3)=S(2,1)

W(I,4)=S(3,l)
W(I,5)=SQRT(S(1,1)**2+S(2,l)**28(3,l)**2)
WRITE(CHARA(I,1),'(F7.6)’)W(I,l)
WRITE(PTNO,'(Il)')I

DO 3 =2,5
WRITE(CHARA(I,J),'(F7.l)')W(I,J)

 

 

 

 

 

3 CONTINUE
CHAR(I)= —PTNO//CHARA(I, 1)//CHARA(I, 2)//CHARA(I, 3)
+//CHARA(I, 4)//CHARA((I, 5)
2 CONTINUE
CCCCCCCCC WRITE DATA FILE NAME CCCCCCCCCCCCCC
LAO='DATA FILE NAME:’// F1
CALL MOVEA( 10 ,500. )
CALL CHARTK(LAO,1.5)
CCCCCC DRAW SCALE CCCCCCCCCCCCCCCCCCC
CALL MOVEA(60.,400)
SCALE='SCALE-l METER/DAY:’
CALL CHARTK(SCALE<.9)
CALL MOVER(10.,000. )
CALL DRAWR(GMAI,0.)
CCCCCCC WRITE CHAR ARRAY ON SCREEN CCCCC
CCCCCCC LABEL OUTPUT CCCCCCCCCCCCCCC
LAl=’ROT OF PLATE (deg)&TRANS OF PLATE(Cm)'
WRITE(ZZl,’(F8.4)')ALPHA*57.2
WRITE(ZZ2,'(F7.4)')TMAG
WRITE(ZZ3, '(F7 4) )TRANSQ
CHAR(12)- -221// Pt? //zzz// PtQ //zz3
CALL MOVEA(OS. ,375)
CALL CHARTK(LAI, .950)
CCCCCCCCCC WRITE CURRENT VARIABLES CCCCCCCCCCCCCCC
WRITE(GAP,'(F7.4)')GP
WRITE(GAQ,'(F7.4)')GQ
CALL MOVEA(0.,325.)
LA5-'GammaP='//GAP//' GammaQ’//GAQ//'(radians)’
CALL CHARTK(LA5,.9)
CALL MOVEA(OO.,350)
CCCCCCCCCC WRITE ALTERNATE ALPHA AND TRANSLATECCCCCCCC
CALL MOVEA(0.,300.)
LA4='ALTERNATE ALPHA & TRANS PARAMETERS'
CALL CHARTK(LA4,.9)
WRITE(ZZA,’(F7.4)’)ALPHALT*57.2
WRITE(ZZS,'(F9.5)')TRANSQALT
CHAR(13)=ZZA//' PtQ’//ZZS
CALL MOVEA(00.,275.)
CALL CHARTK(CHAR(I3),.9)
IF(NFLAG.EQ.O)GO TO 6
CALL MOVEA(-90.,230.)
KA2=’POINT D(10*MM) SX SY SZ S(M/D)'
CALL CHARTK(LA2,.900)
DO 5 K-1,9
CALL MOVEA(-60.,(225.-20*K))
CALL CHARTK(CHAR(K),.750)
CONTINUE
CONTINUE
CALL MOVEA(-300.,700.)
CALL ANMODE
PRINT ’("TICKLE VARIABLES")'
PRINT '("TO END ENTER 9")'
WRITE( 1,103)TMAG
WRITE( 1,104)GP

mm

 

 

 

 

 

> 103
104
105
10

WRITE( 1, 105)GQ
TEMP10=TMAG

TEMP11=GP

TEMP12=GQ

PRINT '("ENTER NEW TMAG")'
READ(1,*)TMAG

IF (TMAG.EQ.9)GO T0 10
IF(TMAG.EQ.0)TMAG-TEMP10
PRINT '("ENTER NEW GP")’
READ(1,*)GP

IF(GP.EQ.0) GP=TEMP11’ PRINT '("ENTER NEW GQ”)’
READ(1,*)GQ
IF(GQ.EQ.O)GQ=TEMP12

CALL RECOVR

REWIND(5)

READ(5,*)DUMMY

CALL NEWPAG

GO TO 9

FOMRAT('TMAG-',F9.5)
FORMAT('GAMMAP=’,F9.5)
FORMAT('GAMMAGQ=',F9.5)

CALL ANMODE

CLOSE(5)

CLOSE(7)

CALL CLOSTK(IER)

END

 

 

 

 

 

 

 

 

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1. LePichon, X., "Sea Floor Spreading and Continental
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2. Rasmussen, L., "Calculations of a Velocity Distribution
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3. Burch, J., and Tokarski, J. "Production of Multiple Beam
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4. Archbold, J., Burch, J. and Ennos, A., "Recording of In-
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5. Cloud, 6., "Practical Speckle Interferometry for
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6. Asundi, A. and Chiang, F., "Theory and Applications of
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7. Asundi, A. and Chiang, F., "Perspective Effect in the
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8. Archbold, E. and Ennos, A., "Displacement Measurement
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9. Speckle Metrology, edited by R. Erf, Academic Press, New
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10. Thompson, B., "Coherent Optical Processing—A Tutorial
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11. Boone, P. and DeBacker, L., "Speckle Methods Using
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12. Burch, J. and Forno, C., "A High Sensitivity Moire Grid
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Optical Engineering, 14, 2, March-April, 1975.

 

 

I

l3. Khetan, R. and Chiang, F., "Strain Analysis by One—Beam
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Applied Optics, 15, 9, p2205-15, September, 1976.

14. Maddux, G., “A Programmable Data Retrieval System for
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Patterson AFB, Ohio, 1978.

15. Cloud, G. and others, "Feasibility of Non-Coherent
Speckle Photography for In-Vivo Measurement of Skin
Deformation," Proc. Optical Soc. of America Meeting on
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