Li Bway Baek, ness eee ey q
U.S. Army
Coastal Engineering
Research Center

ie

eee
INTERACT HONS OF THE
BEACH - OCEAN -ATMOSPHERE
| SYSTEM

AT VIRGINIA BEACH, VIRGINIA

TECHNICAL MEMORANDUM NO. 7

DEPARTMENT OF THE ARMY
CORPS OF ENGINEERS

h Thbbeoo TOEO O

WNC CN

IOHM/181N

INTERACTIONS OF THE
BEACH -OCEAN-ATMOSPHERE
SYSTEM
AT VIRGINIA BEACH, VIRGINIA

by

W. Harrison
and
W.C. Krumbein

TECHNICAL MEMORANDUM NO.7
December 1964

Material contained herein is public property and not subject to copyright. Reprint or re-publication of any
of this material shall give appropriate credit to U.S.Army Coastal Engineering Research Center

LIMITED FREE DISTRIBUTION OF THIS PUBLICATION WITHIN THE UNITED STATES IS MADE BY THE U.S.ARMY
COASTAL ENGINEERING RESEARCH CENTER 5201 LITTLE FALLS ROAD, N. W., WASHINGTON D.C. 20016

FOREWORD

An understanding of the changes taking place in a beach environment requires
the sorting out and ranking in importance of the various processes acting upon each
of the environmental factors being analyzed. Also required is a knowledge of the
time lag between the inception of a group of "processes" and the moment of their
maximum effect on the "response" being analyzed.

The approach to these two problems followed in this study involves the use
of some 27 variables of the beach-ocean-atmosphere system at Virginia Beach, Virginia.
The sorting out and ranking of the variables in a given analysis is approached through
linear and quadratic multiregression analysis, as programmed for high-speed computers.
One of the variables in the system is selected as the dependent variable and studied
in relation to several controlling independent variables, by taking the latter one at
a time, two at a time, and so on until all of the independent variables are included
simultaneously. Thus the technique of sequential multiregression analysis is the
tool used in the investigation. In addition, the dependent variable at time to is
studied in its relation to the independent variables as measured at successive lag
perdodsmti. stowed backwards) dingtimer

This report was prepared by Wyman Harrison, formerly Associate Professor of
Marine Science, Virginia Institute of Marine Science, in pursuance of Contract
DA-49-055-CIV-ENG-64-5 with the Coastal Engineering Research Center, in collabora-
tion with W. C. Krumbein, Professor of Geology, Northwestern University, and
consultant to the Coastal Engineering Research Center.

This study was supported by the Coastal Engineering Research Center (formerly
the Beach Erosion Board of the Corps of Engineers), under the general supervision of
J. V. Hall, Jr., G. M. Watts, and N. E. Taney, of the Engineering Development Division.
The Computing Centers at Northwestern University and the College of William and Mary
extended every cooperation. Programs used in the analysis were largely developed with
supporting funds from the Geography Branch of the Office of Naval Research, under
general supervision of Evelyn Pruitt and R. A. Alexander. Betty Benson of the North-
western Computing Center was most helpful in developing the IBM 709 computer programs
used in the study, and R. Libutti and J. Curran of IBM aided in rewriting her programs
for use on the 1620 computer.

The Virginia Beach City Engineer, C. Kiley, and his assistant, A. Gregg, aided
by laying groundwork for the field studies. The following graduate students of the
Virginia Institute of Marine Science aided in sampling: D. R. Tuck, Jr., M. P. Lynch,
R. Morales-Alamo, R. B. Stone, and W. S. Wilson; C. Kyte of Antioch College, N.
Peterson of Hofstra College, and G. Williamson of Old Dominion College also aided in
field work and data reduction. Wave data were furnished by the Research Division of
the Coastal Engineering Research Center. The U. S. Army Transportation Corps, Fort
Story, Virginia, provided amphibious support for overwater sampling. Weather data
were supplied by the U. S. Weather Bureau Station at Fort Story and tide data were
furnished by the U. S. Coast and Geodetic Survey. D. R. Tuck, Jr. and R. Barnes,
Virginia Institute of Marine Science, were in charge of data reduction.

The authors are indebted to the following colleagues for critical review of
the manuscript: Douglas Inman of Scripps Institution of Oceanography: Per Bruun of
the University of Florida; and Arthur Brebner and Ian White of Queen's University
(Ontario). The freedom and encouragement given to Wyman Harrison in this study by
William J. Hargis, Jr., Director of the Virginia Institute of Marine Science, is
most gratefully acknowledged.

An addendum to this report represents a cooperative effort between the U. S.
Coast and Geodetic Survey, the U. S. Weather Bureau, and the U. S. Army Coastal
Engineering Research Center, in the general area of research on coastal vulnerability.
Time for the research was granted by the U. S. Coast and Geodetic Survey to Wyman
Harrison in his capacity as Chief of the Marine Geology Section and by the U. S.
Weather Bureau to Arthur Pore in his capacity as a member of the Storm Surge Unit of
the Office of Meteorological Research.

This report is published under authority of Public Law 166, 79th Congress,
approved July 31, 1945, as supplemented by Public Law 172, 88th Congress, approved
7 November 1963.

TABLE OF CONTENTS

Page No.
ABSTRACT - - - = - = = = = ie & a es a Mi 4
INTRODUCTION - - - - - = = = = = 2 es eS S 2
General Considerations - - - - - - = - = = 2
Area of Investigation - - - - = = = = = = 3
METHODS) - - - - = - = = = = = a = e ui 6
Measured Variables - - - - - = = - = © Z 6
Approach - - - mec - =e - = So 2 8
Method of Analysis used in this Report - - - - - - 9
The sum of Squares Criterion - - - - - - - - alal
Implications of Linear Regression Analysis - - - - - 16
LONGSHORE CURRENT VELOCITY - - - - - = = = = = 18
Results oF Amalyeais S| = SS te) 9s co a & = & 18
Discussion - See Mate hs ee a til eines sea 20
Summary - - - - - - - = = = = aR A = 27
THE PROBLEM OF NON-LINEARITY - - - ~ - = = = & Da
DEPOSITION AND EROSION ON THE LOWER FORESHORE - - - - - 30
Time Lag in Peak Interaction - - - - - - - - 30
Measurement of the Dependent Variable - - - - - - 32
Measurement of the Independent Variables - - - - - 36
Net Deposition (J¢) - - - - - - = = = = = 36
Net Erosion (K¢) - - - - = = = = = = = 39
BOTTOM SLOPE AND PARTICLE SIZE IN THE SHOALING WAVE ZONE - - - 44

Mean Slope in the Shoaling Wave Zone (Sa) - - - - - 44

Average Mean Grain Size in the Shoaling Zone (Mz) 5 - - - 48
UU RE SEU. eet anne Ue tien MERE RANI AMP eR AE it aS ee 51
REFERENCES) tate pene) cre Ee RS ina 1 clark IC DMO TMS Mar eNGE at 53
LIST OF FIGURES - - - - = - = = = = = = 2 56
APPENDIX A - (Description of variables and method of measurement )- SY
APPENDIX B - (Summary tables of the computer output) - - - - 65

ADDENDUM - "Alternative Multiregression Technique for Obtaining
Predictor Equations" by W. Harrison and N. A. Pore
(8 pp.) - = - - = - - = = = = = A-1

ry
An

r ult ta eu;

yar!

tee

INTERACTIONS OF THE BEACH-OCEAN-ATMOSPHERE SYSTEM
AT VIRGINIA BEACH, VIRGINIA

by

W. Harrison (1)
U. S. Coast and Geodetic Survey, Washington, D. C. and
Virginia Institute of Marine Science, Gloucester Point, Virginia

and

W. C. Krumbein
Northwestern University, Evanston, Illinois

ABSTRACT

A number of interactions among beach variables are investigated
by sequential linear multiregression analysis, as programmed for high-
speed computers. The study includes the influence of beach geometry,
wave characteristics, tidal effects, and local wind conditions on the
velocity of longshore currents, deposition and erosion on the lower
foreshore, and the response of grain size and beach slope to shore
processes,

Results show that if about six variables are segregated out of
any group of about a dozen, these six account for essentially all of
the variability that is explained by all twelve. Thus , the regression
method serves to condense relatively large data matrices to more com-
pact form,

The most-influential combinations of variables arbitrarily
designated as "process" variables are in general agreement with sig-
nificant variables of wave-tank experimentation, and substantiate
intuitive judgments regarding the relative importance of these vari-
ables on natural beaches. The results suggest that certain additional
variables, seldom examined under controlled conditions, be studied in
combination with variables normally examined in wave tanks.

The combination of six variables found to be most influential
in the determination of longshore-current velocity in the study area is
made up of wave period, wave height, lower-foreshore slope, wind velocity
onshore, wind velocity offshore, and angle of wave approach, in order of
decreasing importance. The significance of wind velocity on and off-
shore is believed to lie mainly in the ability of the wind to alter the
form of incoming swells. A special regression analysis for quadratic
effects reveals that water density is highly non-linear in its effect
on longshore-current velocity.

Bottom slope in the shoaling-wave zone, some 250 feet seaward of
the breakers, is found to be controlled primarily by average mean grain
size of the bottom materials, wave period, wave length, wave steepness,
water depth, and tidal-current velocity. This combination exerts its

(1) Present address, Study completed while at the Virginia Institute
of Marine Science.

maximum influence on the slope through a lag in time of between 4 and 8
hours, and apparently to a lesser extent between 16 and 20 hours. Mean
grain size of the beach slope in the shoaling-wave zone is found to de-
pend upon the combination: mean bottom slope, wave period, wave steepness,
wind velocity parallel to shore, angle of wave approach, and tidal-current
velocity, and this combination is most influential after an 8-12 hour lag
in time. When mean slope, the most-dominant independent variable, is
removed from the analysis, water density and tidal-current velocity appear
as the most influential variables on mean grain size. Wind velocity
parallel to shore is believed important because it will influence the
velocities of the tidal currents that flow parallel to the shore in the
study area. Angle of wave-front approach may at times significantly
augment or decrease tidal-current velocities near the bed and thereby the
sizes of particles moved. Wind velocity onshore and offshore at times
interlocks with water density, as density varies when stratified shelf
waters undergo turnover. Fluid drag velocities vary as water density
varies, and differing sizes of particles will be moved.

Net deposition on the lower foreshore during June and July is most
influenced by slope of the foreshore. Wave period, wave height, wind
velocity on-shore, angle of wave approach, and water density are variables
that form the most-influential combination when in conjunction with lower
foreshore slope. This combination expresses itself 8 to 12 and 20 to 24
hours prior to the low-tide time of measurement of net deposition; that
is, during times of rising tide. The regression analysis suggests that
net erosion on the foreshore is not nearly as much influenced by beach
slope angle as is net deposition, but that lower-foreshore slope is still
the most consistently important variable to net erosion through time.

The combination of five variables suggested as most influential to net
erosion during June and July includes lower-foreshore slope, wave period,
wind velocity offshore, angle of wave approach, and depth to the water
table at the top of the uprush. Of secondary importance, and manifest-
ing at times of falling tide, is the combination made up of lower fore-
shore slope, wave period, wave height, and angle of wave approach.

Problem areas reviewed in the study are related to redundant and
noisy data and to the linear model used. Descriptions of the regression
techniques for the linear and higher-order models are also given.

INTRODUCTION
General Considerations

Study of the beach-ocean-atmosphere system under natural field
conditions progresses from initial descriptive studies, consisting mainly
of masses of seemingly unrelated observations, to the analysis of "“cause-
and-effect" (process-and-response) relationships between oceanic and
atmospheric forces and their resulting products, the beaches. While in
the simplest sense it is possible to choose some intuitively rational
group of environmental processes that act as causal factors, and a like
group of environmental responses that act as effects, this simple picture

is complicated by numerous interrelationships among the various causal
factors as well as among the response elements.

This study is mainly a "search procedure" for identifying inter-
actions among subsets of the numerous variables that operate in the
natural beach environment. A number of "causes" and "effects" common to
the general environment are measured and evaluated, consistent with avail-
able time and resources. Least-squares techniques are employed in this
search for relationships between the various individual "effects" and their
attendant "causes."

When the beach-ocean-atmosphere complex is considered as a whole,
it is evident that no single functional equation can at present fully
express the complex interrelations that occur in nature. The tendency
has been to "fragment" the system into portions, either in controlled
wave-tank experiments or studies of a limited number of variables in the
field, that may help explain some facets of the many phenomena composing
the whole. It is one purpose of this paper to refrain from expressing
the data to be presented, either within the framework of process-response
models (Krumbein and Sloss, 1963, Chapter 7; Whitten, 1964), or as more
formal deterministic models. Rather, the intent is to "sort out" sets of
variables to see whether they may provide a basis on which more formal
models can be erected.

The method used here is a form of sequential multiple linear re-
gression to be described later. It is recognized that this is only one
of several search procedures that may be used, and it is anticipated that
opportunity may arise for extending the study in later publications by
using alternative methods, such as factor analysis and discriminant func-
tions, to see whether some optimum model for search can be identified.

Area of Investigation

This study was conducted at Virginia Beach, Virginia, an oceanic
beach situated near the entrance to Chesapeake Bay, at the center of the
mid-Atlantic Bight (fig. 1A). The study was concentrated along a strip
of shoreline between 6 1/4 and 8 miles south of Cape Henry (fig. 1B). A
relatively small, controlled inlet occurs near the center of the study
strip. (The strip is bounded by the northern and southern transects of
figure 1C). A study by Harrison, Krumbein, and Wilson (1964) indicates
that the influence of the inlet on the adjacent beach is confined to a
zone 500-900 feet to either side of its mouth and that it exerts no
measureable influence at the northern or southern transects where all of
the measurements for this study were made.

Surveys by the U. S. Army Corps of Engineers show that beach ‘slopes
in the foreshore area range from about 1:17 to 1:28, while beach slopes
between roughly the 6 and 20-foot contours range between 1:50 and 1:60
(We S. Congress, 1953, p. 13). Details of beach profile modifications
in the study area are presented by Harrison and Wagner (1964), along
with several maps of the nearshore bottom that show the presence of

NSXVL SJYSM SLNANSYNSVAW HOIHM LV SLOASNVYL GNV NOILVOILSSANI 3O VSYV ONIMOHS SdVW ‘| AYNOIS

NOILVOSILSSANI 40
VayV S3LON3G

EES 44 0008 ()

0009 OO0O0€E (0)
= [ = SS |

ABE
0091 008
L

SuaLaW

aay

dIHSLH9I1
130N!

330qnu
XY

LOSSNVYL 4
SJNGCINW

y1vmMayuvog \\ Se See 4
ON] Hinos—s-\ BS nod

JNVAdDVSIHD @

HOvad

MANTONEA

LOASNVYL
NYSHLYON

39V9 SAAVM~-

of
292

3AM =
ae

NOILVLS YSHLVAM AYNSH 3dVO

bar-trough rhythms. Because an artificial beach-nourishment project has
been in operation within the study area since 1956, the measurements for
this study were taken during times of little or no pumping of sand.

Physical and dynamic properties of 219 sand samples from the area
have been studied by Harrison and Morales-Alamo (1964), and the averages
of several properties for various dynamic zones of the beach may be sum-
marized as follows:

Zone Mean Size in mm Sorting Reynolds Number
Nominal |Sieve Coefficient (Under Average Sea-
Dia. Dia.* Water Conditions)
Shoaling-wave ©,25 0.22 O70 4.8
Breaking-wave 0.30 0.27 0.70 Soak
Swash 0.28 0.25 On55 6.2
Swash-berm 0.37 0.33 0.54 IDO

*Converted from nominal diameter (Int.-Agcy Comm, Wat. Res., 1957, fig. 5)

Sand samples with mean grain sizes approximating the above average values
exhibit about 10% (by number) of heavy minerals and 2% or less of rock

or shell fragments. Thus, the beach is composed largely of medium to fine
quartz sand. Small samples of the average sand of the beach exhibit rel-
atively low Reynolds numbers under average temperature and salinity con-
ditions of the seawater.

Tides at Virginia Beach are of the semi-diunal (equal) type and
have a mean range of 3.0 feet, as against a spring range of 36) FRAKES
Tidal currents in the study area are related to the ebb and flood through
the entrance to Chesapeake Bay. They are generally reversing in nature,
and are usually parallel to the shore. Harrison, Brehmer, and Stone
(1964, fig. 4) present evidence from which it may be calculated that peak
tidal-current velocities some 3,000 feet from shore and a few feet above
the bottom do not exceed roughly 0.5 knot (0.85 ft /sec) in the study area.
Peak velocities measured one meter above the bottom at the end of the 15th
Street fishing pier (fig. 1C, "northern transect") approximated 0.68 ft/
sec. This pier marks the northern terminus of the study strip, and it
is here that the tidal currents are at a maximum for the study area.

The wave climate for the study area may be estimated from an
analysis by Harrison of four years of wave records from a stepped-resis-
tance wave gage (fig. 1B) maintained by the U. S. Navy in 20 feet of water
off Cape Henry and from five years of wave observations at Chesapeake
Lightship (fig. 1A), in 60 feet of water. The brief summary that follows
indicates that roughly two-thirds of the waves that could be expected to
strike Virginia Beach come from only three directions and that mean heights

and periods are relatively low.

Significant Wave Analysis Visual Swell Observation

Wave Height

Mean ist Mode
(sec) (@Eo) (£b2 )

Wave Period Direction in Range 05° malolieg

(True N) From Which Swells Come

Direction

(sec)

Percent of Total
Swells

A breakdown of the 15 most-frequent associations of wave period, height,
and angle of approach at the Chesapeake Lightship is given in Harrison
and Wilson (1965), together with details of the refraction of these wave
fronts as they move into the study area.

Surf statistics at the Virginia Beach Life Boat Station, 1.6 miles
north of Rudee Inlet (fig. 1C), have been complied by Helle (1958). ‘Three
years' records, for observations every four hours, reVeal that surf is 4
feet or higher 10% of the year, 3 feet or higher 50% of the year, and 2
feet or higher 95% of the year. Surf tends to be highest in early fall
when the angle of wave approach tends to be from the east (or ENE). Surf
is also high in January, when it is largely out of the northeast. The
average period of the surf tends to be greatest April through July (around
6.0 seconds).

Wind data from the U. S. Weather Bureau's Cape Henry station (fig.
1B) fora 16-year period is summarized in the Virginia Beach, Va., Erosion
Control Study (U. S. Congress, 1953, plate 5). Analysis of these data
shows that whereas prevailing winds are from the southern quadrants, vel-
ocities and total wind movements are greater from the northern quadrants.
Northeast winds tend to be most common in September.

METHODS
Measured Variables

A list of the variables measured in the beach-ocean-atmosphere
system at Virginia Beach is presented in table 1, and descriptions of the
techniques used in their measurement are given in Appendix A. Of all of
the measurements used in this study, most were taken in the months of
February, June, and July. Some were taken in March and April. Thus, a
winter and a summer set of data were available for combination and study.
(Only summer data were used in the evaluation of the dependent variables
Jp and Kp).

Table 1.--Variables Used in Analysis of Interactions of Beach-Ocean-Atmosphere
System. (See Appendix A for Descriptions and Units Used. )

Symbol Dimensions Variable Description

Ba L Depth of water at wave breaking

C Tite Velocity of tidal current

Cy ur-t Velocity of tidal current flowing opposite to longshore
current

C. tr-t Velocity of tidal current flowing in same direction as
longshore current

D L Depth of water table at top of uprush

h L Still-water depth

Hy L Height of breaking wave

Ho L Height of wave (expressed as deep-water height)

H)/Lo 0) Wave steepness of wave (expressed as deep-water wave)

Jp L Net deposition at lower foreshore stations in 24. 5-hour
period

Kp L Net erosion at lower foreshore stations in 24.5-hour period

Lo Th Wave length (expressed as length in deep water)

(M,) 5 L Average mean nominal grain diameter over bottom in

shoaling-wave zone

R L Range of tide over one tidal cycle
8, O Mean slope of lower foreshore of beach
8, O Mean slope of beach over inner portion of shoaling-

wave zone

a8 ay Wave period

UW, ur? Mean wind velocity directed against the longshore current
Vor un Mean wind velocity in an offshore direction

ue tr Mean wind velocity in an onshore direction

Up Lt Mean wind velocity parallel to shore

Us ure Mean wind velocity in same direction as longshore current
Vv prt Mean velocity of longshore current

a O Angle of wave approach

Te po-t Rate of rise of still-water level

Ne Lr Rate of fall of still-water level

fe) Ma Water density

Wind, tide, and most wave measurements were made every two hours,
both day and night. Some of the other variables listed in table 1 (Ba,
D, Hp, @, and 0) were measured four times a day at 0700, 1100, 1500, and
1900 hours. (Values at 2300 and 0300 hours were arrived at by linear
interpolation.) The variables Jp, Kp, (M,),, Sp, and 5. were usually
measured oncé each day at low tide, while (>), and 5. were measured

Zz
every 4 hours during the February field period.

Approach

Designation of "cause" and "effect", or "stimulus" and "response",
is rather arbitrary in a system of complexly interlocked variables (like
those of table 1) 5 all of which may vary simultaneously. In a general
way it is possible to say that relationships between the variables in a
specific beach environment are conditioned by the natural ranges in
magnitude of the variables and by their natural frequencies of occurrence.
Ranges in the magnitude of wave height, tidal-current velocity, wind
velocity in various diections, and grain size distribution in the various
littoral zones are all characteristic for a given beach. Also character-
istic are the frequency of occurrence of waves of a given height, currents
of a given velocity, and so on.

Ideally then, to understand the interaction of a variable singled
out as an “effect" with a number of other variables designated as "causes",
one must measure both "causes" and "effects" over the range of values
that they assume in the study area. Understanding of erosion of the
foreshore of a beach as it is "caused" by winds, and currents, for example,
can come only when foreshore slope, wind velocity, wave height, longshore-
current velocity, etc., are measured through their expectable range of
values and over a long enough time period so that adequate representations
of natural frequencies are obtained.

It is to be noted that consideration must be given to the fact that
the moment of maximum interaction of an effect with its several causes may
be influenced by a time delay. The well-documented delay in the response
of laboratory beach slopes to unvarying wave trains in wave-tank experi-
mentation is a good example of the delay factor in this cause-and-effect
relationship.

It is also to be noted that a given variable designated as an
"effect" in the beach-ocean-atmosphere system may in fact be little in-
fluenced by some of the variables that are considered+as "causes". Wind
velocity parallel to the shore taken as an effect, for example, is in-
tuitively independent of water density, angle of wave approach, and mean
grain size on the foreshore taken as causes. For this reason it becomes
intuitively "unprofitable" to investigate wind velocity parallel to the
shore as a function of such improbable causes. Where it appears, however,
that one of the measured variables is significantly dependent upon most
of the other measured variables, it becomes desirable to investigate
that inferred dependency.

Method of Analysis Used in This Report

Among several methods available for analyzing observational data
that involve interrelationships among independent variables, the one
chosen here is that of sequential multiple linear regression. It in-
volves measures of the relationships of a given dependent variable
("effect") in terms of several controlling environmental "causes" (in-
dependent variables), by taking the latter one at a time, two at a time,
and so on, until all of the environmental processes are included simul-
taneously.

This sort of analysis is commonly called stepwise regression, and
it may be conducted with regression techniques or multiple and partial
correlation techniques. These methods also permit study of interrelations
among the independent variables themselves, and they are useful for eval-
uation of data redundancy.

Redundant variables, that is, variables that in large part restate
what some other variable has already measured, are common in early stages
of quantification in the observational sciences, especially when physical
models are not clearly discernible in the complex of observations. In
these cases a method for "sorting out" a set of independent variables in
terms of their importance or meaningfulness in controlling the response
of some dependent variable, Y, helps to reduce the number of variables
in the set to more manageable proportions.

We shall illustrate the method with a subset of data from a larger
example to be treated more fully in a later section of this paper. The
problem here is to examine the nearshore bottom slope just seaward of the
zone of breaking waves, as it responds to several shore process elements.

The full example includes a dozen independent variables, designated
as) Xi, K2, ..., X12. We sellect five of these, retaining the same number
designations that they have in the larger example, for ease of later com-
parsion. As set up, our introductory example includes the following var-
iables:

Bottom Slope In Shoaling-wave Zone M
Mean grain size xa
Wave period X2
Wave height x4
Angle of wave approach X9

Still-water depth at time of
slope measurement X10

In conventional step-wise regression the "strongest" single X-
variable is first obtained, and this is then "held constant," statisti-
cally, to identify the second strongest variable. Multiple and partial
correlation is commonly used, and some step-wise computer programs have
built-in provisions that fix the relative importance of a given X for
all subsequent stages of analysis. This sometimes leads to spurious
results, in that a variable that may be weak in combinations of two or
three Xs may rise in relative importance as combinations of four or five
Xs are considered. Hence, our analysis is based on a sequential regression
analysis of all possible combinations of Xs, so that every X has a chance
to enter into every possible combination. The reader is referred to
Kemeny, and others (1958, chap. 5), and to Rao (1952, chap. 1) for reviews
of the matrix algebra that follows.

The procedure adopted uses the computational form for the Y and
five Xs of our illustration, in terms of the general linear model:

Yi = 6, + BX1 + Bok2 + By X4 + BoX9 + By oX10 + e;
which is expressed computationally in matrix form as:
(1)
where g aS 2) (6) se cl vector-Y, S is a 6 x 6 matrix of squares and cross-pro-

ducts of the Xs, and 8 is a 6 x 1 vector of the estimated’ Bs “inedevanne
the matrix equation is:

{D>
I
loa

&

N Ba we bo) SG ff Bg |L | ase
2X1 mal mee ey, | abel | ey. || | mane
; (2)
2X5 Dod wie so wa || | sya) aon

The computer first inyerts the entire matrix and post-multiplies
the inverse by g, to obtain B. The proportion of the total sum of squares
of Y attributable to all five Xs is then computed and expressed as a per-
centage. This was found to be 78.7% in our example, which suggests that
the set of five Xs taken together yields a fairly satisfactory predictor
equation for nearshore bottom slope.

In extracting all possible combinations of Xs from the matrix in
equation (2), the computer program is so arranged that it always starts
with N in the upper left corner of 8, and always has ZY as the first
element of ¢ in equation (1). Thus, in the first computer loop the Xs
are taken one at a time, yielding the following as the first two sub-matri-
ees

N 2X1 Bo ZY N 2X2 Bo LY

and @ =
2X1 =x1° By 2ZX1LY 2X2 EXO= Bo 2ZXCY

When all such combinations are computed, the next computer loop takes the
3 xX 3 matrices having pairs of Xs, again starting with N as the upper left
element of S, and ZY as the first element of g. The first part of this
loop has the submatrix for Xl and X2, the second has X1 and x4, and so on
for all combinations of our example. The first submatrix in this stage
aS:

N IX1 IX2 Bo ZY
mal | eae ebe|O/E, = TX1Y
mS we wes | |. SX2Y

Of course, the individual Bs change in value with each stage, but
by continuing the looping process until all five Xs are used, it is pos-
sible to scan the computer output to identify the strongest individual X,
the strongest pair of Xs, and so on.

The data for this example are given in table A and the complete
output is given in table B. Before discussing the results, some remarks
on the method of computation are appropriate. The total number combina-
tions obtained by this procedure is 2*-l, where k is the total number of
Xs used with a given Y. In our example k = 5, and we obtain 31 elements
of output. This rises rapidly as k increases: for 12 Xs the output has
4095 elements and for 13 Xs it is 8191. Thus there are practical limi-
tations on the size of problem that can be economically handled, to say
nothing of the sheer bulk of output for large sets of Xs. For many
problems with k of the order of 10 or more, the sequence may be carried
as far as say six Xs at a time, which commonly yields most of the infor-
mation in the set of data. The computational procedure may be simplified
by using deviation matrices, but the present procedure is given here, to
agree with the computer program as described in Krumbein, Benson, and
Hempkins (1964).

The Sum of Squares Criterion
Once the Bs are estimated for each combination of Xs, the raw
sum of squares of the computed Y values is obtained by multiplying the
transpose of the B-vector by the vector g:
i Seo 8
The sum of squares of the observed values of Y is computed as:

es Neue 2a,
SSYopg = 2 (Yq - Y)" = 2Y,° - Y2Y,

Table A.- Subset of Data From Virginia Beach Study

Beach Mean Wave Wave Angle of Water

Sample Slope Grain Size Period Height Wave Approach Depth

Number (Sg) (M,) s (T) (Ho) (a) (h)
O1 0.68 0.79 7.80 1.82 30.00 12.0
02 0.85 0.65 8.00 8.84 25.00 11.40
03 0.66 0.81 9.03 512 35.00 10.70
ou 0.50 0.74 6.06 5.43 hO.00 aL 560)
05 1.86 0.22 5.90 AL 2) 30.00 Tal 30)
06 2.33 0.23 8.40 1.09 30.00 10.70
O7 2, Af 0.25 12.00 N15) 25.00 aa. NO)
08 1.83 0.26 4.80 8.53 25.00 12.80
09 1.68 0.41 10.80 So dal 10.00 13}, 30)
10 BeOS Os55 10.40 6 (EO 30.00 13}5:30
ala TGs} 0.47 10.80 1.04 30.00 14.10
1D 1.84 0.59 7.90 5 O2 35.00 13.40
13 ILS OH 0.47 4.30 Aga 30.00 13550
14 1.82 0.50 10.80 0.62 35.00 1334 3}0
15) 185 On52 3.80 1.69 30.00 14.40
16 LS 0.47 4.10 1622 20.00 14.10
ale( 15 Sul 0.42 4.50 QS} 30.00 15), 30
18 1.38 0.37 6.10 Ll 20.00 14.00

Table B.- All Combinations of Xs for Beach Data of Table A.

i

Independent Variable Combinations Percentage of Sum of
Squares of Y Accounted for:

X1 (Mean Grain Size) 63.1
X2 (Wave Period) Ao Al
x4 (Wave Height) OB. of
X9 (Angle of Wave Approach) 5.6
X10 (Water Depth) Dae
AS ae 65.5
i 4 bees
1 9 64,2
ale 10 66.4
2 ih. a4, 4

2 9 6.5

2 10 ©q a.

4 9 36.1

4 10 ah. y

one) 8.9

1 2 & 15-9
a ee 9 66.8
ee 2 10 HLS
1 4k 9 (leg Al
aL y 10 74.8
al S 10 68.6.
De Ih © 36.5

BD Mh 10 25.9

2 Q I 12.0

h © 1 36.2

Th ee ee) (do)
i. ak 10 Wok
Tg Ae iS) al 1503
iL « Meo, io 74.9
2 he ©) dO 36.5

1 2 hh 9 10 1801

where the last term on the right is the so-called "correction term", which
is then used to obtain the sum of squares of the computed Ys:

SsY = SSY
19

comp = Eas

aw
Thus, the proportion of the total sum of squares attributable to
linear regression, expressed as a percentage, can now be computed directly
as 100 SSY comp i; SSYops- For example, if SSY obs = 250, and SSY ay = W510).
then the linear relation "accounted for" 60% of the total sum of squares
of Yobs:
The quantity 100 SY om / SsYy is informally referred to as the
i t reduction in the stim of 88", h as in the ab i
percent reduction in the s squares , inasmuch as in the above example,
if the total sum of squares of Y is 250, the 150 units attributable to
Yeomp leave, in effect, only 40% of the original variability "unaccounted"
for. Hence, the reference to a percent reduction in SSY,p, means that the
regression relation has in effect "reduced" the initial variability by 60%.
The sum of squares criterion is generally most useful when the number of
samples (or times of observation of process elements) is several times as
great as the number of process elements measured. Thus, for six Xs the
minimum number of times of observation should be 15 to 18. As the number
of Xs measured approaches the number of observations, the sum of squares
reduction tends to be forced closer to 100%, and may give an impression
of explaining a greater part of the variability in Y than is correct.
(The unusually high percent reductions in the lower part ‘of table BuO,
described later, may be attributable to such effects).

The reduction of the sum of squares is a measure of the mathematical
association between variables, and is not necessarily the measure of a
physical relationship. Where the independent variable has physical mean-
ingfulness in the problem, however, it is not extreme to infer that the
strength of the mathematical relation is also a measure of the strength
of the physical relation.

We return to the output for this example, as given in table B. It
is interesting to note the wide range in the percentage of the sum of
squares of Y accounted for by the various combinations of Xs. In pre-
computer days, when only a few variables could be handled feasibly, it is
not surprising that different researchers, using different sets of variables,
might well have made different inferences, depending upon the particular
two or three variables that might have been used,

The material in table B can be conveniently arranged as in table
C, which lists only the three strongest combinations for each computer
loop. There seems to be no doubt in the present subset of data that the
effect of mean particle size is by far the strongest single variable in
slope response, even though mean grain size is itself influenced by the
process elements. For Xs taken two at a time, it is noteworthy that mean
grain size (X1) is consistently an element of each pair; and that the com-
bination of mean grain size and wave height (X4) is the strongest pair.

Table C. - The Three Strongest Combinations of Xs Taken 1, 2, 3, and MWaiti tal
Time From Table A

i

Independent Variable Combinations Percentage of Sum of Squares
of Y accounted for:

Xa, XB Wh KG) YALO

AL 63
h 2h One at a time
9 6
aL, 4 74
ll 10 66 Two at a time
ee 65
ae 4 16
iL 4 10 75 Three at a time
iL 4 9 74
cles 2 4 10 78
i 2 4 9 76 Four at a time
dL, } 9 10 ' ie
deere 4 9 10 79 Five at a time

NET CONTRIBUTIONS OF RANKED VARIABLES

xa 63.1 %
x4 11-0) %
x2 1.8 %
X10 2.2 %
x9 0.6%

78.7 %

This is borne out by the next stage, in which variables X1 and X4 are
consistently present, but now the added contributions by the other three
variables does not cover a very wide range. Thus, the three strongest
combinations differ by only about 1%. Similarly, combinations of four

Xs at a time show a limited range of contributions, with perhaps a sug-
gestion that variables X2 (wave period) and X10 (water depth) are slightly
stronger than angle of wave approach, X9.

In considering the implications of the analysis, we may estimate
the relative importance of the several variables in the following way:
select Xl first, with 63.1% of the total sum of squares of Y attributed
to it. Then, because the gap between the pair (X1, X4) and the next
competitor (Xl, X10) is about 10%, choose X4 as the second strongest
variable. Its contribution, in the presence of Xl, is (74.1 - 63.1) =
11.0%. From here on the choice is less clear, but if we tentatively
accept wave period (X2) as the third strongest, we obtain a contribution
of (75.9 - 74.1) = 1.8% from it. Similarly, if we tentatively accept
water depth (X10) as the fourth strongest, we obtain (78.1 - 75.9) =
2.2% from it. Lastly, the contribution of angle of wave approach (X9)
is found by the relation (Se % - [Sodl,) = 0.6%. The relatively small
contributions attributable to X4, X9, and X10 suggest that, except for
the two strongest variables, there is little to choose from among the
other three, which add on the average about 1.5% each, in contrast to
63% by mean grain size and 11% by wave height.

Examination of the weakest variable in the set is also illuminating.
As table C shows, wave period "accounts for" only 1.1% of the sum of
squares of Y. Yet on the strictly least-squares basis of choice used
here, this becomes of about equal rank with X9 and X10, which individually
contribute something more than 5% when taken alone.

The IBM 1620 and 709 programs used in this study compute the linear
coefficients and the sums of squares reduction for all combinations of Xs,
as stated, and the output lists the Xs involved and the corresponding per-
cent SS reduction. If intermediate output (such as the coefficients) is
desired, the program produces this by way of a control card. Details are
given in Krumbein, Benson, and Hempkins (1961).

Implications of Linear Regression Analysis

An important aspect of the present method of analysis is that
the general linear model, as used here, examines only the linear re-
lations among the variables, although the model itself can be extended
to some non-linear cases, providing that the Bs always remain linear.
It is sometimes instructive to examine the matrix of linear correlation
coefficients along with the multiple regression output, to see whether
additional light is shed on the regression by the linear relations
(interlock) among the various Xs used in the study.

Table D is the upper right half of the Pearson product-moment-

correlation-coefficient matrix for all pairs of variables in table A.
Inasmuch as rjj is the same as ry; the matrix is symmetrical and only
half of it need be examined. The diagonal in table D carries the value
1.0 for all entries, and merely means that any variable correlated with
itself always has r = 1. The top row of the table has the correlation
coefficients of Y with each X in turn, and the relation between the values
in table B and the corresponding r's in table D is that the 63.1% listed
opposite mean grain size in table B is the same as 100 ryx 1 = 100
(-.795)© = 63.2%, which agrees within rounding error.

Table D.-Correlation Matrix For All Pairs of Variables In Table A.

ij
Slope Mean Wave Wave Angle Still-
Grain Period Height of Wave Water
Size Approach Depth
1.00 -.795 . 104 aa NCH =5236 . 228 Slope
1.000 .062 -205 419 -.060 Mean Grain Size
1.000 - 049 =.035 26356 Wave Period
1.000 -.cl7 =-.305 Wave Height

1.000 =.222 Angle Wave App.

Note: Attention is called to negative

correlation between (Mz), 1.000 Still-Water Depth

and Ss which Is explained on p.45

The most interesting parts of table D are the linear relations
among the Xs in this subset of data. For example, ryjx9, between grain
size and angle of wave approach, is + 0.419, the largest r in any row
other than the first. This correlation with at least one of the process
elements is perhaps to be expected, considering that mean grain size is
itself in general dependent in part on shore process elements. For a set
of data having only 18 observations, any r less than about 0.40 is of
doubtful significance.

The lack of strong correlations among the variables (except for
mean grain size and slope) merely means that the linear relations among
the variables are weak, and neither the correlation table nor the re-
gression results gives any direct information on non-linear relations
that may occur. The problem of non-linearity is always present in linear
analysis, and it will be examined in the next section of this paper, where
an extension of the linear model is described that permits examination

17

of the second degree (quadratic) effects that may be present.
LONGSHORE-CURRENT VELOCITY

Longshore-current velocity typifies a variable measured in this
study that depends rather obviously upon most of the other measured
variables. We may expect from theoretical and laboratory studies (cf.
Brebner and Kamphuis, 1963, for example) that the following variables
will be significant ones in their influence on longshore-current velocity:
beach slope, wave period, wave height, and angle of wave approach. Other
measured variables that are not customarily considered in laboratory
studies may be expected to exert lesser, but possibly significant, effects
upon longshore-current velocity. Among these might be local wind veloci-
ties in onshore or offshore directions, which would affect the form of
the incoming swells, and water density, which would affect fluid drag and
the quantity of sediment transported. Tidal currents opposed to or in
the same direction as the longshore currents might also be of signifi-
cance.

Of the greatest importance in the evaluation of the influence of
the several variables considered as causes, on the one considered an
effect, is the assessment of the simultaneous influence of each of the
various possible combinations of the causal variables. The method used
in investigation of this simultaneous influence was described in the
previous section. Here we take longshore-current velocity as a dependent
variable.

Results of Analysis

The results of the first stage of the regression analysis are
shown in table 2. This table shows the extent to which 13 environmental
factors (independent variables, or Xs) reduce the sum of squares of
longshore-current velocity when the factors are taken one at a time. The
total q, SS reduction by all 13 Xs taken simultaneously is 69.97. (The
number of longshore-current measurements used was 53.) An initial in-
ference is that some 70% of the interaction of longshore-current velocity
with the environment is "explained" by these 13 measured environmental
variables. While it is possible that certain important variables may
have been omitted from the analysis, it is also likely that a relatively
high "noise-level" may be present, due in part to local fluctuations in
the phenomena studied, as well as to errors of measurement. Mean long-
shore-current velocity, V, for example, shows a great variation about
the true mean velocity because the point of actual observation relative
to the upstream and downstream rip-current boundaries was hever know.
Thus the value used for V actually ranged from zero velocity, up through
the true mean velocity, and beyond to the maximum expectable one.

Again, noise may have been introduced into the analysis owing to
the way that the independent variables were measured through time, relative
to the dependent variable. Wave height, for example, could have been
measured at the instant that the longshore-current measurement was taken
or up to 4 hours before its measurement. In view of these difficulties

Table 2.-Per Cent Reduction in Longshore Current Velocity Sum of Squares
Attributable to Thirteen Independent Variables, Taken Individually

Variable Symbol Position %, Reduction in SS

Mean Lognshore Current

Velocity Vv Me =
Mean Slope Angle, oH

Lower Foreshore Sr X1 D)o dll
Wave Period ay X2 21.98
Wave Length (deep water value) Lo X38 26.46
Wave Height (deep water value) ial xh 17.89
Wave Steepness (deep water value) Ho/Lo X5 47. 4e
Mean Wind Velocity Onshore esa X6 6.92
Mean Wind Velocity Offshore Use X/ 5.63
Mean Wind Velocity Directed Ml

with Longshore Current We x8 0.68
Mean Velocity Directed Us X9 0.00
against Longshore Current

Angle of Wave Approach a X10 1.19
Water Density fe XeLAL 0.65
Tidal Current Velocity

with Longshore Current C. x12 P25)
Tidal Current Velocity

against Current Cy X13 0523}

with the noise content of the data, it becomes more understandable that
the total % SS reduction by all 13 variables simultaneously was only
69.97%. In actual fact it would be slightly lower than the 69.97 figure
if variables X3 and X5, which exhibit considerable "data redundancy"
with variables X2 and X4, had been left out of the analysis. Data re-
dundaney is brought up again later.

Discussion

Table 2 shows that the variable X5, wave steepness, has the greatest
influence upon longshore-current velocity, followed by the wave factors of
Lo, T, and Ho. As noted earlier, the fundamental variables of T (X2) and
H(X4) could be expected from theoretical (Putman, Munk, and Traylor, 199)
and laboratory (Brebner and Kamphuis, 1963) studies to be among the most-
influential in determining longshore-current velocity. Also of expected
importance would be beach slope and angle of wave approach, but both of
these variables are outranked by wind velocity onshore and offshore when
the variables are taken one at a time.

As stated, when the independent variables are taken one at a time,
their relative rank cannot, in general, be determined directly from the
degree to which they reduce the sum of squares of the dependent variable,
Some of the variables may be redundant. That such redundancies are pre-
sent in the longshore-current example can be seen by adding the individual
reductions in the sum of squares of Xp in table 2. This sum is 135. 41%,
indicating that some variables, when taken one at a time, show stronger
relations than they show when taken in combination with other independent
variables. Thus, one may say that there must be, as a minimum, at least
35% of data redundancy and (or) noise in these observations.

Before going further it is well to note that variables X3 and X5
(table 1) will exhibit a certain amount of artifically introduced data
redundancy because they repeat information found in variables Xe and xh.
Thus, wave length L, being directly proportional to the square of the
wave period T, might be expected to exhibit a closely similar % SS re-
duction to T. Because X3 gives a 4.48% SS reduction (table 2) over that
for X2, however, the relation between V and T may be non-linear. More
will be said of non-linearity later. Variable X5, then, repeats informa-
tion found in the "fundamental" variables X2 and X4. It is stronger
than either, however, and by combining these two attributes wave steep-
ness be comes a useful variable to measure in studies involving prediction
of V. Its non-dimensionality may also enter, inasmuch as H,/Lo is free
of any scale factor.

The next step is to examine the computer output for the strongest
combinations of variables when they are taken in combinations of two,
three, four, and so on. Table 3 summarizes certain of the strongest
combinations for Xs two through nine at a time; it reveals that wave
steepness retains its dominant role in all of the strongest combinations.
The strongest doublet is not X5 and X3 (Ho/Lo and Lo), as might be ex-

20

Table 3.-The Strongest Per Cent Reductions in Longshore Current Velocity
Sum of Squares Attributable to Each of Several Combinations of
Thirteen Independent Variables. (Total Percent Reduction, All Xs =

69.97).

Percent

Independent Variable Combinations Reduction in SS
Two Xs 5 1 5h 2
ie @ wake al 5 53.91
5 10 50.77
3 5 50.09
5 6 49.86
Tiree Xs. 1 5 i 60.69
ateagtame: 1a: 3 5 Si ola
il 5 10 57.03
5 ie ila, 56.02
1 2 5 56.02
Four Xs al, 3 5 rie 62.23
aia came: 5 7 10 61.96
iL 5 a WL 61.85
1 2 5 if 61.50
il 5 ri 12 61.37
il 3 5 10 61.19
al Me 5 T 60.90
al 5 i 13} 60.72
Five Xs IN 2y/ 63 5 10 64.53
Bie @) waa IL 3 5 T 10 64.07
4 3 5 i TAL 63.52
12.3 5 ii 68.035
A 5 ih 10) SL 63.02
Six Xs ANG DY 3 5 i 10 66.39
io @, teams dL 3 5 i 16) 65.25
il eyes 5 10 13 65.02
ale Dames) 5 8 10 64.95
yy Be ely 10 64.92
Sevaa ke 1 2 3B 5 6 F 10 67.56
ae a wame lt 2 3 5 il 10 12 66.72
le yeas 5 "/ IO 1 66. 67
a By ln Gl 10 66. 60
AL Ge) aes 5 a 10 13 66.53

2

pected from table 1, but rather is X5 and X7 (H,/L, and Uj). The third
strongest doublet is X5 and X10 (a). Although these differ by only a
small percentage, they are being taken here as ranked by the least-squares
procedure.

The practice of inferring the relative rank of combinations of
variables from their rank when taken individually is generally not a sound
procedure. Slope angle of the lower foreshore (X1), for example, does
not appear (table 2) to contribute as much as fully six other variables
(Xs 2-7). The method of sequential multiregression analysis shows, however,
that Sp is present in the strongest triplet and remains prominent in several
of the stronger triplets. From an initial SS reduction of 5.11%, when
considered individually, it is seen that in the presence of variables X5
and X7 it contributes fully 6.48% (table 4), for an increase in its origi-
nal value of only Me SiGe Thus, the strongest individual variable may be
influenced by other variables. However, when the grounds for accepting
the strongest variable on a physical basis are sound, it is conventionally
taken at face value, and the effects of other independent variables grouped
with it are expressed in terms of the added reduction contributed by the
combined variable.

Continuing in table 3, we noted that the strongest triplet (Xl, X5,
X7) now includes beach slope (X1), another fundamental variable, which
ranked fully seventh in order of importance when the variable yere considered
individually (table 2). Table 4 indicates the change in the original %
SS reduction value of Xl when in the presence of X5 and X/. The value of
X1 has increased some 1.37%.

The strongest combinations of independent variables taken four at
a time is composed of Xl, X3, X5, and X/7. Interestingly, the original
SS-reduction value of X3 (26.46%, table 2) has been reduced by 24.92%
{table 4), in the presence of Xl, X5, and X7. This is largely because it
already occurs in variable X5.

At this point it is instructive to rewrite the empirical laboratory
expressions found by Brebner and Kamphuis (1963, p. 22) to represent the
mean longshore-current velocity as they computed it: a) using Airy wave
theory, b) Snell's law for wave refraction (assuming a gently sloping plane
beach), and c) by expressing the wave parameters in terms of deep-water
values.

V

1.9 (g Be)4/3 ne (Ho/Ly)'/© (sin 1.650, +0.1 sin 3.300,) Energy (1)
Approach

V

ll

4.0 (g co ae (fe Mesa 1.65, + 0.1 sin 3.300) Momentum (2)
pproac

Assuming for the moment that there is a reasonable relationship
between the major factors that determine longshore-current velocity in
the laboratory and those that influence it in nature, we may look for the
position of the combination of Xl, X3, X5, and X10 in the ranking of % SS

ae

‘ouTy @ Ye SX G JO suCTzeUTQUOD ayy UT (€ SeTqQe4) ySesucTys puodas ST UOTYeUTAWOO STUL yxy

9x ‘eX ‘OLX
Y9L°G fo aseeroeg MALT 95°19 Se CiTexGey Kar Gine c6°9 9X
AO
%99°6T JO eseeroeg Acar 6£°99 Sxacae Kae axa Ging 96°T? ox
OTX
%59°O gO eseetout 419 °T L0°t9 SKE KeIG Gineis OTXxx
%26*he FO eseeroaq 4S°T €3°c9 Cx ‘Tx 1x “Gx 91°92 eX
ZLE*T FO eseertouy ora) 69°09 ibaa Ka GxG TESS TC
ZOT*T gO eseerouT 6L°9 Te +S IEG €9°S 1x
ot’ lit GX on’ Lh GX
uoTANGTI7U0D X MON go uoTZeUTqUO) uOTAZeUTQUO)D (2 eTAeL 329) oTAeT Ie A
TeuUTSTIO 0 SATISTSY UoTIngT14U09 Akg ow ur SX ATTENPTATPUL 10,7
aseotoeq Jo sseatour pequnoooy gg % peyzunoosoy gs %

ATWuTor usyel,
SoTqetze, Vuepuedepurl 4seBsuor4g9 UsAeg 94 SuoUYy SeTqetze, Teuotatppy Aq suotangqtaquog - ‘th eTqeL

23

reductions for variables taken four at a time (table ane This combination
ranks sixth in importance, but is separated from the most-important combin-
ation by only 1.02%. The suggestion, then, is that the method of data
analysis reasonably duplicates the expectable influence of combinations

of important variables as they have been suggested by theoretical and
laboratory approaches, when studied by a straight forward least-squares
procedure. Then, an important corollary to this inference is that the

data set itself is a reasonably faithful representation of the environmental
interactions under study.

Exmination of the results (table 3) for five Xs at a time again
shows that the variables found to be of significance in the laboratory
exert the greatest % SS reduction on V. it is at this point that the
variable of angle of wave approach (X10 ) enters into the strongest com-
bination. X10 exhibits an increase in its original SS-reduction value of
0.65%, when in the presence (table 4) of X1, X3, X5, and X7. What effect
expressing @ and M (using Snell's law and Airy wave theory) would have
had in bringing the relative ranking of Xl, X3, and X5 in table 2 more in
line with the powers in expression (1) and (2) is unkown. It may be unim-
portant in this set of data, owing to complex wave-refraction patterns
that may be present.

The highest-ranking combinations of six variables at a time include
X/7, wind velocity offshore, which probably has a significant effect upon
wave steepness, X5. Thus the interlocking nature of the variables again
enters the picture. On several occasions during the measurement periods
wind shifts from onshore to offshore were noticed in connection with
passage of cold fronts. Waves were observed being "knocked down," as
spray blew from their crests in a seaward direction. Because of the apparent
"corroboration" of the method of analysis for the fundamental variables
mentioned above, it seems fair to attach significance to the additional
atmospheric variables that turn out to be significant in the least-squares
analysis when they combine with these fundamental variables. That is, the
indicated importance of variables X/ and possibly X11 (table 3) to long-
shore-current velocity may be believed to have true physical significance,
when in combination with the variables just mentioned. Once physical
significance is attached to these variables it is realistic to enquire into
the precise physical relationships involved. As mentioned, for example,
strong offshore winds will often rather effectively alter the wave form,
and water density enters into fluid drag in the longshore-current trough,
the amount of sediment entrained, and thereby the velocity of the longshore
current. Bruun (1963) and Inman and Bagnold (1963) give excellent reviews
of the problems in understanding the generation of longshore currents.
Any inferences advanced here are not capable of immediate verification
based upon field studies but may serve as a basis for the design of field
measurements.

Because of the strictly artificial introduction of data redundancy
in variables X3 and X5 (table 2), a new summary (table 5) was prepared
from the computer output. This new table lists the strongest. combinations
when variables X3 and X5 are removed from the analysis. As expected,

24

variables X2 and X4 are dominant throughout all of the combinations, and
variable Xl through all but one combination. Variable X6, wind velocity
onshore, takes the place occupied by X10 in the five Xs at a time combina-
tion of table 3. Angle of wave approach, X10, then appears in the com-
bination of six Xs at a time. Thus, wind velocity again appears important
in influencing lonshore-current velocity.

Table 5.-The Strongest Percent Reductions in Longshore Current Velocity
Sum of Squares Attributable to Combinations of Eleven Independent
Variables (x3 and X5 Removed From The Original Thirteen Xs).

Percent
Independent Variable Combinations Redceution in Ss
Two Xs 2 y h1.10
at a time
Three Xs 1 2 y U7 £23
at a time
owe Xs ah y 7 51.98
at a time
Page ots ik 2 6. 7 Sf oO
at a time
SaexaeXs lene? y © Ff 10 59.71
at a time
Seven Xs 1 2 4 6 7% 8. 10 62.04
at a time

Table 6 summarizes the combinations of independent variables that
least influence V. The table reveals certain variables that contribute to
both the weakest and to the strongest combinations. It is instructive to
refer to the weakest (table 6) and the strongest (table 2) combinations of
independent variables taken six at a time. Intuitively, we might expect
that none of the six variables that appear in the strongest combination
(Xs 1, 2, 3, 5, 7, 10) would appear in the weakest combination (Xs 8, 9,
TO, 1a, 12, 13). It is found, however, that X10 does appear in both com-
binations.

When the number of variables is fairly large, as ih this study, it
is not unusual to find that as the number of Xs per combination is increased,
a variable whose net contribution is of the order of lor 2 L, may "rise
to the top" when combined with a very strong combination of Xs, whereas
when combined with very weak Xs, its own slight contribution is not suffi-
cient to raise the total contribution by any significant amount.

25

Table 6.-The Five Weakest Per Cent Reductions in Longshore Current Velocity
Sum of Squares Attributable to Each of Several Combinations of
Thirteen Independent. Variables.

Percent

Independent Variable Combinations Reduction in SS
Two Xs 9 13 O28}
at a time 9 Lal 0.65
6 © 0.68
8 13 Ost
14 13 0.87
Three Xs 8 9 13 Oni
at a time 9 LAL 13 0.87
9 12 13 1.28
S} © TAL. ale
8 ileal 3} 1.50
Four Xs 8, 9 abit 13 50
at a time 9 dL 12 13 1.68
© iO) 13 1.88
8 9 Ae) gals} 2.00
©) 10) Tah 13 2.20
Five Xs 8 9 ii 12 18 245
at a time Sj ©) Sil) Tal 13 2.61
10 Ag 12 as Bo i3
3 © 10 1A 12 3.23
3 ©) NO Ti Je Bul
Six Xs eh ©) TO: al, ae 8 3.66
at a time (GOD mae Ove eniale 13 6.02
G8 © aa 2 8 6.16
Ge teh) lO) Ala. = a 6.44
8. 9) IO ey 3} 6.44
Seven Xs Ht}. S) MO) dad 13} 6.51
at a time 1 8 9, 10) th 1p. 13 9.23
1 7% & © I@ as ip 1 AS
IL Te (GP ao ala 13 11.88
aL "7 8 © IO 2) 113} i, C2
Eight Xs 1 “8 © IO da. 12 13 I 2A5O
at a time 1 6 (Gye). oss KO) ot abal, aay vals) 19.33
4 6 SS) A@- abl 12 18 25.76
i 4 6 8 9 i eats 26.95
aL 4 6 8) 6) © 1a ¢ alg 27-74

26

For example, the combination 1, 5, /, 10 in table 3 accounts for
61.96 %, an increase of 1.27 % over combination 1, 5, 7, In table 6, the
combination 8, 9, 13 accounts for 0.77 %, and the combination Sis Op. tO),

13 accounts for 1.88 %, an increase of 1.10%. Thus the position of X10
depends very largely on the strength of the combination in which it occurs,
and X10 seems to play a minor role in this set of data in that it first
shows up as weak or strong in combinations of 4 Xs at a time. It will be
recalled from table 2 that X10 accounts for 1.19 % of the SS of Y, a con-
tribution that seems to remain much the same throughout the analysis.

summary

Based on an analysis involving eleven independent variables, it is
possible to say that the combination of six factors (table 5) that is most
influential in the determination of longshore-current velocity in the study
area at Virginia Beach is the combination made up of wave period, wave
height, lower foreshore slope, wind velocity onshore, wind velocity offshore,
and angle of wave approach, in that order. (When the derived variables of
wave length and wave steepness are added_to the analysis, the six most-
important factors are Sf, T, Lo, Ho/Lo, Uon, and Upp). The wave factors
and the beach slope are variables agreed upon by workers in the field of
longshore-current generation to be of fundamental importance. The signi-
ficance of winds on and offshore is believed to lie in their ability to
alter the wave form prior to breaking.

The least-squares relations developed by this analysis appear to
adequately represent the combinations of variables of most significance in
nature. It is to be recalled that the framework of analysis is linear, but
it is not uncommon, when a large number of variables is involved, that a-
linear approximation yields reasonable results, even though some relations
may be known to be non-linear. Moreover, non-dimensional ratios among
some of the original variables, such as H [ils in this example, may rise to
greater relative importance than the original variables themselves, as
shown in table 2. This suggests that further analysis by use of non-dimen-
sional variables alone (perhaps as derived from use of Buckingham's Pi
theorem) may be useful and informative. We hope to extend the analysis
in this direction.

As a summary of the foregoing linear analysis, it seems fair to
say that the least squares techniques used here have helped "sort out"
the relative interplay of a group of variables as they affect longshore-
current velocity on a particular beach during a given time-span. Statis-
tically our model is "fixed", and extensions of our findings to general-
izations about other beaches is valid mainly in that the underlying variables
are perhaps the same, even though their relative rankings may vary from
beach to beach, or on the same beach from time to time. That non-linearity
is also a problem is discussed next.

THE PROBLEM OF NON-LINEARITY

The occurrence of non-linear relations among beach-ocean-atmosphere
variables was touched upon above, and it is discussed here in connection

27

with the study of longshore-current velocity. Non-linearity can be detected
in several ways, the simplest perhaps being the examination of scatter dia-
erams of each pair of variables in the entire set. Another search method

is to run the regression analysis first with the "raw" data and then with
all or some Xs transformed to logarithms. In this example we shall use an
extension of the linear model to seek for quadratic effects among the
variables.

When the linear regression of Y on some single X is studied, the
computational model reduces to the following form:

¥ = Bo + B,X

In terms of regression analysis, this involves the 2 x 2 matrix, vector-Y,
and B-vector discussed in connection with equation (2), and yields the "sum
of squares reductions" associated with one X at a time. However, this
model can be extended as follows to include higher powers of X:

seen “ “~ )
Y = Bo + BX + Box (3)
where the coefficient B, is now associated with the quadratic form of X.
The coefficients are still linear, and hence the same general technique
may be used, even, if desirable, to include such powers as 1S, Mv SUSo

The procedure used here is first to take each X by itself in terms
of Y, to obtain the values in table 2. Then, for each X, its square is
also included as in equation (3), to obtain a corresponding sum of squares
of Y attributable to the combined linear and quadratic effects of the X.
The difference between these two "sum of squares reductions" gives an
estimate of the second degree non-linearity associated with each X. This
was done with a computer program that computed the linear and quadratic
relations between Y and each of the 13 Xs, as well as all interlocks between
the Xs themselves. In this: latter analysis the order of entry of the Xs
is involved, in that the expression:

Nes ~ “~ D)

ele Bo + Bj Xe + Boxe
is not the same as the expression:

2@ = [i 2 eek > Bel

even though, as was mentioned earlier, ryjy5o is the same as r ST ELAUISE

the complete quadratic output for a problem involving 12 Xs is voluminous,
and we shall here emphasize mainly the analysis in equation (3) that contains
M% ChineCcwliyyrc

Table 2A shows the linear and quadratic counterpart of table 2. It
is apparent that in absolute terms, variables X5, X6, and X11, representing
wave steepness, wind velocity onshore, and water density, have the largest
quadratic components. In relative terms, the lower foreshore slope angle,

28

Table 2A.-Linear And Quadratic Relations Between Longshore Current Velocity
And The Independent Variables In Table 2.

Variable Positions %SS Reduc. %SS Reduc. 4% Added by
Attributable Attributable Quadratic
to Linear to Linear and

Quadratic

Mean Longshore Current

Velocity Me -= -= —=

Slope Angle Lower Foreshore X1 Fo dll AO), als} 5.02
Wave Period X2 21.98 2560 3.62
Wave Length (deep water) X3 26.46 27-43 0.97
Wave Height (deep water) x4 17.89 19.97 2.08
Wave Steepness (deep water) X5 Mire de) renal 25.30
Wind Velocity Onshore X6 6.92 25) OL 18.09
Wind Velocity Offshore Le 5.63 6.26 0.64
Wind Velocity Directed

with Longshore Current x8 0.68 ig als} 0.46
Wind Velocity Directed

against Longshore Current X9 0.00 0.00 0.00
Angle of Wave Approach X10 I.g19 2.38 1.19
Water Density X11 0.65 12.42 IL

Tidal Current Velocity
with Longshore Current X12 125 Ws Sk. 0.06

Tidal Current Velocity
against Longshore Current X13 Oo23 Ooss/ o.14

X1, has doubled, from 5.11% to 10.13%. Thus, these several variables may
enter the least-squares relations more effectively as logarithms or as
variables raised to some power. Of particluar interest is water density,
which in the linear sense is quite negligible (only 0.65%), but in the
quadratic has risen to nearly 12%. This is an illustration of one limita-
tion of a strictly linear analysis: some variables that have virtually no
linear effect may become quite strong in a model that explicity includes
non-linear effects.

DEPOSITION AND EROSION ON THE LOWER FORESHORE
Time Lag In Peak Interaction

In a sequential multiregression analysis of the interaction of the
Mission Beach, California, foreshore slope to the four independent variables.-
of wave height, wave period, angle of wave approach, and longshore-current
velocity, Krumbein (1961, p. 45) found that the maximum effect of these
combined independent variables occurred (in a least-squares sense at least)
sometime between 6 and 12 hours prior to the time of the measurement of
foreshore slope. In regard to wave period, the analysis indicated that
the greatest effect on foreshore slope was exerted some 30 hours prior to
the time of measurement of beach slope. Because of the laboratory and
field evidence for the delay in time in the peak interaction of certain
dependent variables like beach slope and the independent variables that
influence them significantly, the four interaction studies at Virginia
Beach that follow were investigated over five or six "lag periods."

In the case of modification of the segment of the lower foreshore
that is covered and uncovered by the tide, it is necessary to standardize
the times of slope measurement so that the measurement times are repre-
sentative of similar dynamic conditions on the beach. lLow-tide time is
a convenient reference time and was adopted here. Figure 2 shows the
scheme adopted at Virginia Beach for making foreshore measurements in
June and July of 1963. The ticks marks designated Pj, Po, P3,...-.-.---
Pjh9 represent the standard times during the 25-day period when the in-
dependent variables were measured or for which Sie ae values were
obtained. The tick marks designated Rj, Ro; R3 go00 0006 represent the
times of low tide when measurements of the eats ete of — lower fore-
shore stations were made. This measurement time (Ra) is seen (Gates 2)
to progress through the times of measurement of the other variables.
Because each lag period is 4 hours in length, the measurements for all
of the independent variables for lag period 1 will have been made O to
4 hours prior to the foreshore-altitude measurements, for lag period 2
they will have been made 4 to 8 hours prior, and so on. Because the
precise time of peak interaction between a dependent variable (Tp, Ke)
measured at low-tide time and one or more independent variables is un-
known, it is probably just as well that bias has not been introduced by
setting up lag times of fixed numbers of hours prior to measurement of
the dependent variable.

30

vooteetneees 2y ly) SINSW3TS 3SNOdS3Y ONY ( PSlqeee Od old)
SINJW373 S$S3D0Nd JO LNSW3YNSVI3W JO S3WIL ONIMOHS WVHOVIG ‘2 3yNdIS

SYNOH NI SWIL

006! ool! fele} x0) 0061 ooll lele}<fe) OOo6I ooll fole}=xe) 006! ooll tole} fo) 006! ooll fofe} xe) 006!
oo¢ge | oOos! 9020

Osly Sdly Obl, Sely ofl, Sely fely

izoal

eely Oely Silly Olly Sol, OOl, $6, £6

Sly
O¢ aune 9¢ eunt

2@qy 0s«COy S@q O%q Slq Olg Sy Ya fa 2a Ig

Vy

|] eune 01/9

3I

Measurement of the Dependent Variable

An attempt is made to understand the changes in the overall alti-
tude of the lower-foreshore surface along a 100-125 foot segment running
perpendicular to the shoreline (fig. 3). The dependent variable has the
dimensions of a length, the magnitude of which is believed sufficient to
adequately reflect the environmental processes producing the foreshore
changes, This length is the net thickness of material added to or sub-
tracted from the foreshore surface at five or six stations in a 24.5-hour
period; i.e., over two tidal cycles.

The section of the "lower foreshore'' referred to here includes the
zone of breaking waves somewhere in its lower half (fig. 4) at the time of
measurement of the altitude of the beach surface (time of low tide). The
remainder of the lower-foreshore surface studied here consists of the
region covered by the breaker and swash-backwash zones of the previous two
tidal cycles. Measurements of the altitude of the beach surface were made
at six stations at 15th Street (northern transect, fig. 1) and five stations
at the Camp Pendleton line (southern transect, fig. 1), at the low-tide
times designated by the letter "R' on figure 2. Thus, the measurements
cover 25 consecutive days in June and July.

Precision of the sounding-line or leveling-rod measurement is esti-
mated at +0.10 foot. Altitudinal variations at the stations were summed
over the five or six stations ahd the net erosion or deposition recorded.
Unfortunately, the large sampling distance involved (25 feet) undoubtedly
missed minor irregularities in the profiles that could have been signifi-
cant. Figure 4 shows the range of changes in lower-foreshore altitudes at
the two transects during the 25 consecutive days (fig. 2) of measurements.
Clearly, the beaches were not anywhere near an equilibrium condition, such
as might be expected (cf. Strahler, 1964) during the late summer. Daily
values for the increment of deposition, J¢, averaged about 0.7 foot at 15th
Street; for erosion, K¢, averaged about 0.5 foot. Respective average values
at the Camp Pendleton line were 0.5 foot (Kf) and 0.4 foot (Jf). The values
compare with daily net fluctuations of 0.2-0.3 foot on Cape Cod beaches
(Zeigler and Tuttle, 1961).

For this part of the study, the variations in the causal independent

variables were followed over six lag periods (two tidal cycles), as shown
below:

32

A|1JO1g abpsaay

aurq uo}a;pudq dwod
a}1jO1g aBboiaay
“‘paulwmsajap
SDM Y 9434M

489415 YdGl ad
$D UO!}1S0qg —

ew $224S YG! SNIT ALYsadOYd NOLATIGNSd dWvo GNV

Ysld ONIHSIS LAASYLS HSI
LV SLOSASNVYL
¢ 3YNdIS

(a4DW1xouddy)

S3NOZ SAVM-ONIIVOHS JNOZ YSsyNVSIYE \ Geis ]
\
Ny G
“XN
NN
ie ~~ — = =
auiq uojaipuad =A ety =
ABIg 499445 YIGI A
goik z SNOILVLNdWOD 3d07S 4YO4
Ss pun (Zw) aasn Hovaa 4O SIN3WO3S
alg 4aa44S ug! +y pud 4p oH
( aig 4224S ici 45
a i a ! !
00S OOvV (olelss 002 OO!

JNI1 3SV€ WOYS GYVMVSS L334 NI JONVISIC

SQN LILIV

1334 Ni

33

SNOILVLS SYOHSSYOS-YSMO1 LV SANIVA IWNIGNLILIV 40 SAdO1SAN3S % SYNDIS

JNI7 NOLAIGN3d dWVO

1334 0S Ge 0

AYOHSSYOS -—YSMO1 40
SSGNLILIV 40 SdOTSANS

(3d!i1 MO1 LV) G3SYNSVAW 3y¥3M

| #4 GNW 3f HOIHM YSAO ZONVLSIC |

| (Adil MO1 Lv) G3yNSvV3W 3yaM
I
| Jy anv +f HOIHM Y3AO JONVILSIC

Y3ld L3535NLS 4S! LV 30VSYNS peels

Lag Period Hours Prior To Tide Stage

Number Measurement Of
iL o-4 Falling (LW at O hrs)
2 4-8 Rising and falling about HW
(HW at 6.12 hrs)
3 Sal2 Rising
----------------------------------------- Low (IW at 12.25 hrs)
4 WAS Falling
5 16-20 Rising and falling about HW
(HW at 18.37 hrs)
6 20-24 Rising

----------------------------------------- Low (LW at 24.5 hrs)

The interval of two tidal cycles duration was chosen for convenience only,
as it is far easier to measure the dependent variable in daylight hours
than at night. It is true, however, that the single tidal cycle is the
more basic dynamic unit and is preferable to the two-cycle interval. Thus,
a small amount of erosion of the foreshore might occur during the first
tidal cycle under one set of independent variables only to be followed by
a greater amount of deposition during the second tidal cycle under a dif-
ferent set of independent variables. The dependent variable, however,
would be measured as an increment of deposition and it would be impossible
to realistically interpret the correlation between the set of independent
variables that resulted in erosion and the dependent variable that showed
deposition. Alternatively, a large amount of deposition might occur dur-
ing the first tidal cycle, and a lesser amount of erosion during the second
cycle. Misinterpretations could again arise.

The difficulties just mentioned are believed to be rather minimal
in the data set used for the regression analyses. The values for J¢ and
Kp usually occur in runs of two to seven successive days. The inference
is that Jp and Kp values for intermediate low-tide times would have shown
similar tendencies of net erosion and deposition to those at the actual
measurement times. For a number of reasons, then, the data sets used in
evaluating Jf and Kp are relatively noisy. This will generally be the
case when part of the work is done in the breaker zone and times of measure-
ment are not keyed to successive times of low tide.

35

Measurement of The Independent Variables

The variables now mentioned (and described in Appendix A) were
measured for the analyses of both the increment of net erosion (J) and
net deposition (Kp).

Slope of the lower foreshore was determined by first passing a smooth
curve through the plotted altitudes of the sample stations at which Jp and
Ke were determined (Cie 3)" A Sne of mean slope was then passed through
this curve by eye, and the line's angle with the horizontal was taken as
the mean slope angle (Sp) of the “lower foreshore." This angle was rela-
tively easy to measure. The two transects (fig. 4) provided a good range
of slope angles for the regression analysis, although the values were
somewhat clustered. Wind, wave, water density, and tide data were assumed
constant at both transects. Im addition to Sa, only the values for long-
shore-current velocity and depth to the water table at the top of the up-
rush varied between the two transects. Thus, the following relationship
could be postulated:

Jf or Ke = f (Se, T, Lo; Ho» lala) lic Un Use, Up, Qa, Ve Ps Nro Nee D) +46

where t = a "lag period," O-4 hours long, prior to measurement of the de-
pendent variables or prior to another lag period of equal duration.

Average mean size on the lower-foreshore slope was not included in
the analysis because of the considerable data redundancy that would have
been introduced. Also, the noise content of such data would have been
quite high, because grain-size variability is at a maximum when crossing
the breaker zone.

Net Deposition (Jp)

Results.--Table 7 shows the results of the first stage of the analy-
Sis, in which the independent variables are studied one at a time. Total
%-SS-reduction values are relatively high, considering the noise content
of the variables. The least-squares analysis shows that lag periods 1, 4,
and 5 have the most influence on net deposition on the lower foreshore.
If the total %-SS-reductions for lag periods for corresponding tide stages
are added together, lag periods 1 and 4, 2 and 5, and 3 and 6 have total
%-SS-reduction values of 173.38, 169.85, and 163.27, respectively. Thus,
the variables that act during falling tide levels appear to exert a some-
what greater influence on foreshore deposition than those acting during
the time of rising tide.

Slope angle of the lower foreshore (X1) emerges as the most con-
sistently important variable, reaching its maximum influence O-4 hours
before measurement of Jp. Other less-influential variables show relatively
less consistency by lag periods, with the possible exception of wave height
(x4), which increases in influence backward in time. The five strongest
7%, SS reductions by independent variables taken in combinations of two to

36

ysnaidgq jo

96° €2 16°2 646 99° ET 19°22 ot"). TX a doy ‘yydeq eTqey, 2072
TeAeT

00°0 TL°9 Okan 00°0 29°S gt°€ €TX Fu TIZVM-TTTIS “TTed JO 99ey
Taaey

96'2 oL’9 00°0O 80°0 88 °6 00°0O ox a) TOyemM=TITIG festy Fo e7ey
TO°O 6£°S 00°0 Qc °9T €€° Or €€°OL TIX fe) Aqtsueq 107eM
G9°0 6h H@°ST On'9 T0°0 gL°6 OTx A AyTooTEA WUetaND) sLoyssuoT
(oy nS TS°0 20°0 GS°0 Qt °O €Q°e 6X re) yoeorddy ovaem jo eTSsuy
aroug oF

er are, 02 °2 QS°S T6°2 €6°ST 92°t Qx dy TeTrereg AZTOOTSA PUTM
22°Q gla Olen TO'T GO"0 og'€ LX FO eroussIO AQTOOTSA PUTM
00°0O €9°T HL°9 GQ°T OL *TT OL*9T 9X ese} azoysug £4TOOTSA PUTM
QT’S ALAS LL°0 29°0 oH" Tete GX Ory /OH ssoudee4g ovem
ey °€2 88 °QT 9S°9T T2°€T 10° OT G6". +X Si JUSTSH oAeM
HE T G2°O 62°T? QT °6T ila 60°8T €X Ont yysueT oAeM
11°0 00°0 98°8T €8°9T Tm) 00°8T ox i potted aAeM
aLoyserog

mrss gle. ess Ormys my°GS- _- Gy2OE TX 3g JanoyT eTsuy edots
aLOUseto iT

i Ral JamMoT uotytsodeg 3eN

gl°2g 99°18 18°19 61° 08 6T'°2Q 1S°SQ uotyonpey gs % TedoL

potted G potted + potted € potted g potted T potted

spotzeg Sey Ag gg UT UOTJONpeY yuUsoTEg uoTITSOg Toquits aTqQetze pn

‘Q-T Spotseg Be] soz ‘ATTenpTATpuL ueyeL ‘seTqetsep quepuedepuyt uw9e4ummog
04 eTaeInqtsz14y Ssezenbg Jo umg suoT#e4g eLoYserZog, JamoT ye uoTZTSodeg 4eyl UT suOTJONpSY queg Jeg =-*), eTaeI,

37

six at a time are shown in appendix B (tables B1-B6), and the five weakest,
for lag periods 2, 3, 5, and 6, in tables B7, BS, B9 and BO. Frequency
tables of the computer output for lag periods 2, 3, 5, and 6 are show

in appendix tables Bll, Bl2, B13, and Bl}.

Discussion.--Considering the strongest combinations of Xs taken six
at a time (tables Bl-B6), it is noted that variable X1, lower foreshore
slope, always contributes to these strongest combinations. In an analysis
(not presented here) of the interactions of the lower foreshore slope with
twelve variables of the environment, it was noted that the strongest com-
binations of five Xs at a time, in the most-influential lag period, con-
sisted of H,, @, V, p, and R. It would seem reasonable to expect that
some or all of these variables would also be of significance in this analy-
sis for Jr. (The corresponding Xs in the analysis for Jp are XO KS), XO,
X11, and perhaps X12 and X13.) Tables B1-B6 show a tendency for most of
these variables to enter into the strongest combinations, but most also
enter into the weakest, and there is little consistency from one lag period
to the next.

Assuming physical validity for an analysis such as this--for changes
in a surface that crosses two differing dynamic zones and using a data set
that is noisy and redundant--the following few points should be noted.
Beach slope is a controlling factor in deposition on the lower foreshore,
as net deposition is measured at low tide, after two tidal cycles. (It is
understood that grain size of the slope material is also of great importance. )
The rate of fall of the still-water level is of some significance during
the two periods of falling tide(lag periods 1 and 4). The rate of rise of
still-water, however, has no significance during the two periods of rising
tide. Depth to the water table has the greatest effect on net deposition
around the time of high tide or during falling tide. The effect of the
longshore current enters into the strongest combinations of variables taken
six at a time during only one lag period. This is not surprising, inasmuch
as the longshore current serves largely to transport material parallel to
the shoreline. Wind velocity appears to have significance to net deposi-
tion during the falling tide immediately prior to the time of low-tide
measurements. Wind blowing offshore may set up a weak sea-surface current
that will aid in the removal of fine sand suspended in the breaker zone
but,at the same time, not cause an equal quantity of sand to be transported
onshore in the corresponding return flow on the bottom. As a result of
wind blowing onshore, large particles may be transported out of the breaker
zone by the seaward return flow of water (cf. King, 1959) 1p. 207-213) vom
the bottom. Local winds in onshore and offshore directions will also in-
fluence the form of the incoming swells, as discussed in the section on
the longshore currents.

Among the wave variables, H,, and occasionally Hoy ae are the most
important to net deposition. Statistically, water density plays a role
(tables B2 and B3) during the first part of the tidal cycle immediately
prior to measurement of J,. Presumably, 0 affects the rates of particle
movement by its effect on fluid drag velocities and turbulence at the bed.

38

Table 8 shows the results of the analysis with redundant variables
X3 and X5 removed. This procedure results in the disclosure that lag
periods 3 and 6, corresponding to periods of rising tide, are the most
influential on net deposition, and that the variables Sp, 2B SG) Isley elayol (OF
are the four most influential variables in combination.

Summary.--Based upon an analysis involving 12 independent variables,
it is possible to say that the 6 most-influential variables in determining
net deposition on the lower foreshore are S-¢, T, Hy, Usa a, and pe, and
that this combination of variables expresses itself 8 to 12 and 20 to 2h
hours prior to the low-tide time of measurement of net deposition; i.e.,
this combination of variables expresses itself during times of rising tide.

Net Erosion (Ke)

Results.--Table 9 shows the results of the first stage of the analy-
Siso IWowedl %-SS-reductions for the six lag periods are considerably lower
than those obtained for Jp (Gale Fo ~ Welesia individually, lag periods 1
and 2 exert the greatest influence on net erosion of the lower foreshore,
as net erosion is measured at low tide after two tidal cycles. If the
total %-SS-reduction values for lag periods for correlative tide stages
(p.35) are added together, however, lag periods 1 and 4, 2 and 5, and 3
and 6 have total %-SS-reduction values of 113.41, 104.26, and 72.60, re-
spectively. Thus, the variables acting during times of high tide or fall-
ing tides exert the greatest influence on foreshore erosion.

It is noted that variable X12, rate of rise of still-water level,
shows slight percent reductions in SS during lag periods 1 and 4, times
of falling tide level. This is due to the fact that the time of measure-
ment of K, was based upon the predicted time of low water. But because
the actual time of low water, used in computing 7, sometimes occurred
slightly before predicted low water, Kr was sometimes measured just after
the still-water level had begun to rise. The effect on the analysis in
lag periods 1 and h is believed negligible.

The influence (table 9) of lower foreshore slope, Xl, on net erosion
tends to be relatively uniform over the six lag periods, and amounts to
about one-third of its observed influence on net deposition (table 7).

The only variables exceeeding slope angle in influence in a given lag period
are Ny in lag period 3, and Usps Qa, and V in lag period 2.

The five strongest % SS reductions by combinations of variables
two to six at a time are shown in tables B15-B20, and the five weakest in
tables B21-B26. Frequency tables of the computer output are not presented
for Kr.

Discussion.--The strongest combinations of independent variables
taken six at a time, for the two most-influential lag periods (1 and 2).
simeludel (Si) lig,) He/ la, Usp, >) and D) (tablies Bild and Bl). y Because
variables X11 and xi contribute also to the weakest combinations (tables

39

Table 8.-The Five Strongest Per Cent Reductions in Net Deposition at Lower
Foreshore Stations Sum of Squares Attributable to Combinations of
Six Independent Variables (X3 and X5 of The Original 14 Xs Not
Used) for Each of Six Lag Periods.

Percent
Independent Variable Combinations Reductions in SS

hag al a] ae 6 9 10 13 73-31
iL. 2 6 9 aL 13 12.85

pete 6 9 13 ae DTT

AO y 6 9 13 (AoE

one y 6 9 10 12a 2B

Lag 2 1 6 1©, ii iI2 14 69.40
iL 6 ©. IQ ia. 14 68.04

alt 6 da, 12 13) ah 67-76

1 6 9 ni 12 1h 67.62

a, 4 6 id 12 14 66.95

Lag 3 Alt y 9 ia ID [Po eh
He fe y 9 al 14 70.96

lee 4 3 © ala! (OTS

a2 Th iL, IB 14 70.56

2 y 6 I 2 70.45

Lag 4 2 6 lal, 13 ah 61.26
al 6 9 LAL 13 Wh 61.02

ie iM alah 2) ah 60.80

aL 6 AMO aL, is) as 60.39

2 y 8 ala 14 59.46

Lag 5 1 2 4 6 @ i© 68.98
ree 4 6 10 13 68.80

2 4 6 10 12 68.75

aoe iH 6 8 10 68.56

age 4 8 TAL 13 68.54

Lag 6 Tee y 7 8g 76.14
ie) 4 6 8 9 75-64

W 4 h 109 ala 72.36

IL ye 6 7% S99 72.34

al 4 Tt 3 9) Ae 72.06

40

Or °@ €0°0 gE"T 0°0 6S°0 TX a ysnady go doy,
i y Z yydeq etaeL re7yeM

00°0 G9°0 ro) Sire! 00°O GS°T T6'T €TX Fu TeAsT TeVemM-TITIS
TTed FO eqey

96°E€ 8S°0 98°0 GL°OT T8°0O 99°T 2TX “ath TeAsT Toyem=-TTTIS
esty jo aqey

og" € 6E°h eH °9 S0°0 OMX) S0°O TIX d Aqtsueq r07eM
Ons 00°O Ont TLE 90°T?2 G2°0 OTX A queIIN) estoYssuoT
OT’O 6E°H 9T°S Go°) 0g °€T 64°S 6X 7) yoworddy evem go eTsuy
Z0°O T2°0 GEE Q2°T 05°98 €8°S QX dy eZoyg OF TeTTereg
AqTooTeA PUTM

tO QT °T OT’? €6°H 99 °¢2 99°L 1X Jon eLousssO AZTOOTAA PUTM
INS Cilane Hey HL°9 HEE gS°0 9X ei aroysug AZTOOTEA PUTM
qLt2 +0°0 EL°0 On'T 60°T 99°T GX Siva ssoudeeyg avey
1S°0O GO'T 62°0 9£°O GQ°e G9°T +X Set qZUSTOH oveM
ETE gS°0 €9°0O GT°O Ht BES €x Si yqsueT oveM
00°€ eh °O 22°0 00°0 OT °H €9°0O ox a potsed aseM
6E°€T 9S°T gL° OT 69°OT 16° OT OL: TX as] aLOYyse toy
JamMoT aTsuy edotg

A Jy aTOYse1o7g

JaMOT UOTSOTY J2N

GB°ge ge * et €9°HS GLE’ 88°19 gL°9S uotzonpsy Ss % TeqOL

potted G potsed + potted € potseg g potseg T potszeg

spotieg Sey Ag gg UT UOTIONpSY AusodtEeg UOTATSOg Toquvg STOUSPT TEN

"Q-T Spotseg Bey soy ‘ATTenpTATpUL ueyeL ‘seTqetsze, JUepusdeput ueeqmog
04 eTQeyngTtz44y Sserenbg jJo umg suoT e499 SLOYseto4 TamoT ye UOTSOAY JeN UT SUOTJONpeYy 4USD Jeg="°6 eTaeL

4

Bel and B22), it is difficult to assess their true influence. It appears,
however, that these variables will logically combine with wave steepness
(X5, tables B15 and B16) to erode the lower foreshore during falling tide.
Variable X11, water density, will affect the rate of sand-grain transport

by fluid drag. The variable D, X14, will be of importance in determination
of the magnitude of erosion or deposition during rising and falling tides

as it affects the infiltration of swashes and the ability of the backwash

to transport sand back toward the breaker zone; i.e., the ability to scour
the lower foreshore. The significance to net erosion of variable X/, wind
velocity offshore, is believed to lie in three areas: 1) in its interlock
with @ and V (table B9), noted earlier in the section on longshore-current
velocity, 2) in the postulated mechanism whereby offshore wind of suffi-
cient magnitude is capable of aiding in the offshore tranport of finer
particles thrown into suspension in the breaker zone at the time (lag period
2) that high tide covers the lower foreshore, and 3) in its interlock with
water density, wherein offshore winds cause denser water to move shoreward
in the lower layers, as mentioned in the section treating (M,),. Angle

of wave approach, X9, although interlocked to a degree with V, may influence
circulation over the lower foreshore at high tide in such a way that rip
eurrents are more effective in removing sand from the foreshore for certain
values of Q. The rate of rise of the still-water level (X12) is of impor-
tance to net foreshore erosion in lag periods 3 and 6, when the tide is
rising. The rate of rise and fall (X13, table B15) of the still water level
will determine the time over which any of the other independent variables
will be able to act at a given point.

Table 10 presents the results of the regression analysis for the
strongest variables taken six at a time after redundant variables X3 (L))
and X5 (Ho/L,) have been removed. If the maximum %SS-reduction values
for lag periods for correlative tide stages are added together, lag periods
1 and 4, 2 and 5, and 3 and 6 have values of 67.21, 84.82, and 58.09, re-
spectively. Thus, the influence of variables acting about the time of
high tide is highest on net erosion of the foreshore, as it is measured in
this study. And the four variables most influential at high tide will be
(table 10) Sp, Uor, @, and D.

Summary.--Based upon an analysis involving 12 independent variables,
it is possible to say that the combination of 5 variables that shows the
most influence on net erosion of the lower foreshore is composed of vari-
ables Sr, T, Vor; a, and D. This combination is most-influential about
the times of high tide. During times of falling tide, variables Sp, T, Ho
and Q@ compose the most-influential combination of variables. The rate of
rise of the still-water level, depth to the water table, and beach slope
are judged to be the most important variables during time of rising tide.

42

Table

Lag 1

Lag 2

Lag 3

Lag 4

Lag 5

Lag 6

10.-The Five Strongest Per Cent Reductions in Net Erosion at Lower
Foreshore Stations Sum of Squares Attributable to Combinations
of Six Independent Variables (X3 and X5 of The Original 14 Xs
Not Used) for Each of Six Lag Periods.

Independent Variable Combinations

BPRPPEPRP PRPRER

PRPPER

PRERPPRP PRRBPERB

PREPHPPR

@
2

NM Po

NM mw hw ~ P

iS iS

- Je Fr FHS

ON ON

ON ON ON ON ON

ODN ON ON ON OD

if

SJ 4qaqg

Od CO CO C

©0 CO OO

\O \O 10 \O \O \O \0 10 \0 \O \O \O \O 10 \O \O \O \O \0 \O \O 10 10 \0 \O

1O
10

ale)
10
1O

LO
LO
10
10
10

43

at,

ali
ala
ala
dal
ala

aa
aa
ala
aL
ala

dla
11

all,

12

1h

14

12 14

14

13) ah

12 1h
12
12
12
12
ae
12

14

14

14

14

14

12 14

12 14

12 1h

12 14

1A IS ale

Percent

Reductions in SS

BOTTOM SLOPE AND PARTICLE SIZE IN THE
SHOALING-WAVE ZONE
Mean Slope in the Shoaling-Wave Zone (S,)

Also studied was the interaction of 12 environmental variables with
a portion of the beach slope roughly 250 feet beyond the breaker zone. In
the case of measurement of the mean slope in the shoaling-wave zone (S,),
or average mean grain size on the slope £ ™2)s 7; the measurements of the
independent variables were nearly always made exactly ney, 2 US, ere BO
hours prior to measurement of the dependent variable. This exact lag-
time convention was considered more realistic than the 4-hour-long one
used for lower-foreshore measurements, because the measurement of the de-
pendent variable was not tied to a dynamic condition, such as obtained at
the time of low tide for lower-foreshore measurements.

Variables measured.--Recent laboratory work on the equilibrium char-
acteristics of sand beaches beyond the breaker zone (cf. Eagleson, Glenn,
and Dracup, 1963) commonly utilizes measurement or determination of the
following variables, where one is concerned with the mechanics of slope
alteration: bottom slope itself at the point in question, particle size
on the slope, wave period, wave length, wave height, water depth, specific
eravity of the sand particles and the fluid, and kinematic viscosity of
the fluid. The variables described in the paragraphs below were measured
in this study in an effort to approximate the variables found useful in
laboratory studies. The subset of data used earlier in the description of
the regression method was taken from the fourth lag period of this section
of the study.

The segment of the shoaling-wave-zone slope selected for study here
was one which was reasonably flat and yet which exhibited a relatively
large range of slope values during the periods of observation. As seen
in figure 3, the slope was determined from soundings at four stations at
the 15th Street pier, each of which was 25 feet apart. The slope was
determined by plotting the sounding values for these four, and for several
surrounding stations, on cross-section paper and then drawing an estimated
mean slope through the four stations by eye. Still-water depth (a) aie
the midpoint of this mean slope (fig. 3) was determined for each lag period
from precision tide data or direct measurement by sounding line. Mean grain
size, M,, determined as explained in appendix A, was obtained for each of
the four stations used in slope determination. The four values were aver-
aged to give (SE) values for the regression analyses. Wave parameters
were again estimated from wave-spectrum-analyzer records furnished by
the Coastal Fngineering Research Center. Specific gravity of the particles
was assumed constant (2.65, or that of common quartz). Rather than deter-
mine kinematic viscosity, we instead determined water density, from pre-
cision temperature and salinity measurements (see appendix A for details).

In addition to the above variables that are similar or identical to

44

those commonly measured in laboratory studies, we utilized measurements of
wind velocity (onshore, offshore, and parallel to shore), angle of wave-
front approach (which is not constantly parallel to the shoreline in nature,
as it usually is in the laboratory), and tidal-current velocity. Finally,
we inserted wave steepness as a variable in the least-squares analysis and
ran five lag periods.

Thus:
Sg = 2 [7 We)eo Bo Bop Hos Holos Uso Uses Up» @ hy p, Cle. L

Results.--Table 11 presents the results of the first stage of re-
gression analysis. The highest %-SS-reduction is found at lag period 5,
perhaps because of the dominance of x4 (Gels) and X/ (Gee) Ine, als} *ELLSO
noted in passing that lag periods 2 and 5, which show the highest %-SS-re-
ductions are lag periods that coincide with times of high tide for 6 of
the 18 slope observations used in the analysis.

Discussion.--It is seen in table 11 that when the variables are
considered individually, mean grain size and water density (Xisaa arid)
generally have the greatest effect on shoaling-wave zone slope, out-rank-
ing the wave parameters considerably. The significantly high effect of
(M,) s on ise could easily have been predicted from considerations of the
mechanics of slope formation and from the known relation between slope
and particle size, usually found on foreshore slopes (cf. Bascom, MEHL)
A difference exists in the mechanics of alternation of the lower foreshore
slope, however, where permeability of the sloping surface assumes con-
siderable importance in the process of transportation and deposition.
Fluid drag forces are of greater significance to grain movement on the
shoaling-wave slope.

Water density, a factor of unexpected importance, undoubtedly in-
fluences the shoaling-zone slope through its effect on threshold drag
velocities and turbulence at the bed. Wave tank experiments on beach slope
modification at the Coastal Engineering Research Center, using warm and
cold water, have revealed that under constant incoming wave energies the
slope modification was more rapid under cold-water conditions. (The final
slopes under both conditions were closely similar).

The strongest combinations of variables taken six at a time make
an interesting study. Referring again to lag periods 2 and 5 (tables B28
and B31), which show the greatest percent reductions in total SS, it is
seen that average grain size, wave period, wave steepness and tidal-current
velocity, appear in the two combinations of six variables for these lag
periods. Absent, but occurring in all of the other strongest combinations
of six (tables B27, B29, and B30) is wave height (x4), while wind velocity
offshore and parallel to shore appears in two of the other combinations.
This may be evidence that the slope in this region is shaped by tidal
currents in conjunction with wave characteristics at high tide (lag periods
2 and 5), but more by wave height and winds during half tides or low tide.
More observations for lag periods keyed to times of low tide will be need-
ed to settle this point.

45

UG? J, (ane) LIS 68°S 01° 0 otX 2) AgTooTaA yuEerM) TeptL

69° 1S GS°6S G9 °6S 98°1S 15° 0S TD fc) Aqq~sueg 107eM
9S°6 T2e°S ee GH*S Gee O—TX y yydeq reyemM-TTT99
2S°0 19°S ie peat, 2G°ch = 9° OT 6X fe) yoeorddy eAeM JO eTsuy
TO"0 6'2 66°) 95°), 09'2 x orn arog 04
TeTrerted AATOOTSA PUTM

GQ°OT = 99“ 66° L0°0 Oly 1x i) eLoussJO AYTOOTSA PUTM
9z°0 CLS 48°8 9S°0 T6°T gx Een ezoysug AYTOOTeA PUTM
SSS « @E°S 99°T SG°OS GO GX ony oH ssoudeeqg ovem
TG se GOAN, EI°OG ——-- GIEPE HX °H qUSTSH eAeM
org GT LL Ep°rr 1@°O ex Tip ygsueT oveM
G6°O JOE 99°T 9S°9 00°0 2X ah potted eveM
26°99 Grea) GWG) SPE) 02° €9 ie © (7i) su0Z eAeM-SuTTeoys
aZTS uUTeIy uesW

eon-- ----- ----- ----- wo--- i aC SuOZ
sASM=SUTTeoUS eTsuy edots

€T° 16 90°26 28 °6 26 °S6 GT°&6 uotyonpey ss % TetOL

G potted 4 potted € potted g potseg T potaeg

Spotdeg Seq Ag ag ut uoTIONpPeY queosZeg uoTITSOg Toquiks aTqetre

"G-T Spotted sey coz ‘ATTenpTATpuL ueyey, ‘seTqet ze,
qjUepusedepult SATEMT, 09 SeTQeIAnGT744y Serenbg Jo umg sdoTg suogy aAeM-SuTTeoug UT UOCTJONPaYy 4UEeDO Jeg="TT eTaeL

46

Removing the contribution of the dominant variable (X1) from the
analysis (tables B32-B36) reveals the new dominance of another geometric
variable, X10 (water depth), which may be considered as a mediator for the
various process elements in the environment.

Thus, ignoring X10, it is seen that lag periods 2 and 5 are still
the most-influential on shoaling-zone slope, when the variables are taken
six at a time. Because variables X3 and X5 exhibit data redundancy with
those of X2 and X4, as explained earlier, it is seen that wave period is
the important variable for measurement times weighted with those of high-
tide measurements (lag periods 2 and 5; tables B33 and B36), while wave
height (X4) is most important in altering the slope for measurement times
weighted with those of low-tide measurements (tables B32, B34, B35). This
result of the least-squares analysis, if truly representative of high and
low-tide conditions, would be in keeping with the knowledge that waves of
a given period and height will exert a greater effect at a fixed point on
the bottom when the tide is low than when it is high (cf. Inman and Nasu,
IHG, We 30). The wave-height effect might then be diminished at high
tide to the extent that the only effect that is felt is the effect of
wave length (wave period) as it influences drag over the bottom.

A study of the weakest combinations of the variables taken six at
a time shows (tables B37-B41) that wave steepness and tidal-current veloc-
ity, which are both present in four of the strongest combinations for the
five lag periods are also present in these weakest combinations. The
only variables that are present in the strongest combinations of six at
a time (tables B27-B31), and present in none of the weakest combinations
for the corresponding lag periods, are mean size (X1), wave length (X3),
wave height (X4), water depth (X10), and water density (X11).

Finally, reference to frequency tables B42-Bh46, for variables con-
sidered in combinations of six at a time, reveals that the distributions
of numbers of combinations by %-SS-reduction classes are polymodal for
lag periods 1, 3, 4, and 5, but essentially unimodal for lag period 2.

Lag periods 3 and 4 exhibit four modes, while 1 and 5 exhibit three each.
One inference is that sub-groups of variables that influence the slope

to dissimilar degrees are somewhat segregated during lag periods 3 and }.
These sub-groups, however, if they are in fact semi-discrete in a physical
sense, each seem to influence the shoaling-wave-zone slope to about the
same degree during lag period 2. If true, it might be said that the com-
(gama caloMN NG), Xo XO) KelOnmvandexale) (railte B33) is the most-significant
combination of variables to influence the foreshore slope and that this
influence is exerted between 4 and 8 hours prior to slope measurement.
Additional work with such frequency tables is planned. They are presented
in appendix B for the interested reader and for future reference.

Summary.--The bottom slope of the beach at 15th Street, some 250-
feet seaward of the breaker zone, may be thought of as being controlled
mostly by the following combination of six variables: average mean grain
size of the bottom materials, wave period, wave length, wave steepness,

47

water depth, and tidal-current velocity. This combination of variables
exerts its maximum influence on the slope through a lag in time of between
4 and 8 hours, and apparently to a lesser extent between 16 and 20 hours.
This delay may reflect the influence exerted on the slope during the time
of the previous two high tides. Wave height, angle of wave approach, and
water density become more influential through delays in time amounting to
O-4, 8-12, or 12-16 hours, and coinciding with or approaching times of
low tide.

Average Mean Grain Size in the
Shoaling-wave Zone (M,)

Variables measured.--The measurements used in this analysis were
the same ones as were used in the previous analysis for Sg, average mean
grain size being interchanged with S, as the dependent variable. Thus:

(M) nat (Ss, T, lo, Hos Ho/Lg, U on? Voges Up, a, hy Pp, Oe
=»)

The analysis was run for five periods, as was done for Sg.

Results.--Table 12 presents the results of the first stage of the
analysis where it is seen that the most-influential lag period is number
3, which occurs 8-12 hours prior to measurement of the independent variable.
Lag periods 2, 4, and 5 are of about equal influence, while lag period 1
is the least influential. Variable Xl is seen to be among the dominate
variables when they are taken individually, and this is to be expected in-
asmuch as a high interdependence between S, and (My) s was noted in the pre-
vious analysis. Because of the possible masking influence of this dimen-
sionless variable, Xl, two sets of tables have been prepared, both of which
show the strongest combination of independent variables influencing (M, )
One set (tables B7- B51) includes 411 12 Xs; the other set (tables B52-
B56) does not include Xl. Finally, a set of tables for the combination of
independent variables showing the weakest influence (tables B57-B61), and
a set of frequency tables (tables B62-B66) have been prepared.

s*

Discussion.--Turning to table B49, for lag period 3, one sees that
the most-influential combination of variables taken six at a time consists
of Se aE lile/dbas UO ” Q@, and C. Variables ay Ale tp and C also enter into
the weakest combination, when they combine with Lo, Ujp, and h (table B59).
Recognizing the importance of slope in its influence on particle-sizes
transported, we turn our attention to the other variables in this strongest
combination of six variables. The reaaer is reminded that T is redundant
with Telesis through the relationship Lo = 5.12Te, which relationship was
used in this study for obtaining L,. At any rate, the fluid forces at
the water-sediment interface that are induced by varying wave periods and
wave steepness will tend to move grains of various sizes over the shoaling-
zone slope. Superimposed upon this mass transport by the wave-drift cur-
rent will be the transport induced by tidal and wind-driven currents. For
the bottom slope in question (fig. 3), the major current will be the tidal

48

QS ° On Th 8d €o°S HE "tS Ted? dh oTX 0) Aq4TOOTEA JUetINO-TePTL

gE*HT Sore Goes «UOC C« "ULE TIX d Aytsueq 107eM
99°T E€"0 SOx OL°O 9g°T OLX Yy yydeq 107eM-TITIS
16°? HE°9 QL°ST cg° Ce MN POKE 6X 0 yoeorddy evaemM JO eTsuy
00°h 12°8 E}°O Toe COG gx “al aLoUg OF TEeTTered
AqpooTeA PUTM
wT L0°0 AO) GaSe GCL Ix ail aLoysssO AZTOOTSA PUTM
€0°0 90°T 09°ST C6 °t 6° 9x meh, ezoysug £yTOOTeA PUTM
ieee 10°0 He" € 9g°6T 95°S GX wef FR sgyoudeeyg oAeM
28 °9T QS°TT 2l°0 90°6 Wicheals 1X Hy qUusTOH oAeM
LE*O G80 Go°O TQ*H ASO) ex oa yysuey avem
ow 80'd 6cha 6h °? 6S°T ox i potdeg ahem
60°eh = 69°BH HL. BH €£°09 0Z* €9 aK aS) QUOT SeABM=SUTTEOUS
eTsuy edoTs
SESS eooon eocee= cene- ----- k = (2M) SUZ aAeM-SuTTeoys
aZzta uTery UesW
6g°€6 g9°h6 SL°g6 Ot°€6 = ST “98 uoTzonpey SS % TedOL
G potted + potted € potted g potsed T potsed

Spotdzeg Seq Ag ga UT UOTZONpeYy yused.TEg UOT TSOg Toquxts STAeTIe/\

sab SOE ke en eee Pes Ente as ee Ea ee ee ee ee ee ee eS eee
"G-[ Spotdeg Sey Jog “AT TeNPTATPUT ueyey, ‘seTqetsze, Juepuedspurt eATEM],
09 eTaegnqTz49y Sezenbg jo umg (eucZ eAeM=SUTTeOYS) ezZTS UTerH UeeW UT UOTJONpSY JUEeD 4ed-"*cT PTAPL

49

current and it will react a complex way with wind-generated bottom currénts
and wave-drift currents. Thus, it is possible to conceive of a situation
in which the angle of wave approach, the direction and velocity of the
wind, and the direction and velocity of the tidal current at a given time
will interact to reinforce or impede one another. A wind moving opposite
in direction to a tidal current, for example, will produce an increase in
the tidal-current velocity near the bottom (Reid, 1957) and, should the
wave fronts be traveling in the direction of flow of the tidal current, the
net current velocity could have a velocity up to about 16 percent (cf.
Collins, 1964) greater than the simple algebraic summation of the wave-
induced current and the reinforcing tidal current. As noted by Collins
(1964, p. 1051), "the effects of even very small currents on the mass
transport ios fine material / by waves could be very large." D. R. Tuck
(oral communication) has studied the interrelationships of wind velocity
and direction, angle of wave approach, and tidal-current velocity on the
average mean-grain-size values for the beach slope in question. In instances
where the wave-drift current was reinforced by the tidal current, water
velocities of 21-29 cm/sec were obtained near the bed. Mean particle sizes
actually observed at the bed under these circustances, and for the slopes
in effect at the time, were within the range of particle sizes predicted by
theory as being moved by the net current.

ey Thus, there is every reason to believe that a combination such as

Sg, I, Ho/Lo, Up, G, and C, mentioned at the outset as being the most-
influential Pemdmacson of six variables in the most-influential lag period,
is in fact a valid combination for this area of the beach at Virginia Beach.

Table B54 shows that the most-influential combination (Xl ignored
for 6 Xs at a time) includes: T, Lo, Up, a, h, and p. Variables X10 and
X11 take the place of variables X5 and X12 of the analysis in which X1 was
included. Water depth merely acts to mediate the various process elements,
while water density will affect fluid drag at the bed. As seen in all of
tables B52-B56, water density, X1l, is a dominant variable in the strong-

est combinations.

Of further and more general significance is the rather persistent
contribution of variables X6, X7, X8, and X11 and X12 in the strongest
combinations of Xs taken six at a time (tables B52-B56). Wind velocities
onshore (X6) and offshore (X7) have also been seen to be of importance in
net erosion on the lower foreshore. In that section of the study it was
postulated that when the offshore wind reaches a certain velocity it is
important in producing a weak surface current beyond the breaker zone that
is capable of transporting fine particles put into suspension in the break-
ers out into the shoaling-wave zone. The onshore wind, however, may pro-
duce a seaward return flow of water on the bottom that will transport
somewhat coarser particles out of the breaker zone and onto the slope in
the shoaling-wave zone. Winds parallel to shore will probably interact
significantly with tidal currents, which are generally parallel to the
shoreline in the area of investigation, and augment or decrease the current
velocities in the lower layers.

50

Also of considerable significance in consideration of the role of
onshore and offshore winds is their interlock with water density, X1l. If
the wind blows offshore long enough and strong enough, the wind stress at
the water surface produces a seaward movement of the upper layers of the
ocean and a shoreward return flow in the lower layers. (The procedure is
more or less reversed with an onshore wind.) This sort of "pseudo-upwell-
ing" results in the shoreward movement of colder, more-saline water in the
summer months and relatively warmer, more-saline water in the winter months.
This type of turnover of the continental-shelf waters has been undergoing
documentation by the Virginia Institute of Marine Science for over two
years, using sea-bed drifters and sea-surface drift bottles. In addition,
the temperature and salinity measurements made at the 15th Street pier
correlate well with observed offshore and onshore wind movements. Thus,
there is a clear interlock between Upor, Upon» afd P. And water density
will clearly influence fluid drag velocities and turbulence at the bed and,
thereby, the particle-size distribution in the shoaling-wave zone.

Summary.--Based upon an analysis involving 12 independent variables,
average mean size on the shoaling-zone slope at 15th Street, 250 feet
beyond the breakers, is most-influenced by the combination of variables
(taken arbitrarily in a combination of six) composed of mean slope, wave
period, wave steepness, wind velocity parallel to shore, angle of wave
approach, and tidal-current velocity. This combination of variables is
most influential on (Mz), after a lag in time of 8-12 hours.

Mean slope is the most-dominant independent variable at all times.
When removed from the analysis, it is in the main replaced by water density,
and this variable is found to be of considerable importance in all of the
strong combinations of variables.

The importance of wind velocity onshore and 6@fshore in the analyses
is found in their interlock with water density as it will change when shelf
water in the lower layers is moved to the shore during offshore winds and
out to sea under onshore winds. The shelf water in the upper and lower
layers is stratified as to density, especially during the summer months,
and the density affects the sizes of particles moved.

Wind velocity parallel to shore is significant in its ability to
reinforce or decrease tidal-current velocities near the bottom, because
tidal currents in the study area flow generally parallel to shore. The
angle of wave approach is probably of significance as it interlocks with
tidal-current velocity. Wave-drift and tidal currents interlock in their
effect on particle movement at the bed.

FUTURE STUDY

A major aim of any search for significant interactions in a natural
system is to provide a basis upon which more formal models may be erected.
The authors are continuing their efforts in this direction, using the pre-
sent data set. Additional techniques (for example, factor analysis and
discriminant functions) will be examined. The data set will also be

SI

augmented with additional field measurements in the study area. Develop-
ment of relatively informal "process-response" models is underway, while
construction of stochastic process (simulation) models will be attemped

in the near future. Expression and analysis of the present variables in
non-dimensional form will also, it is hoped, lead to better integration of
field and wave-tank studies of the beach-ocean-atmoshpere system.

52

REFERENCES

Bascom, W. N., 1951, The relationship between sand size and beach face
slope: Trans, Am. Geoph. Union, v. 32, p. 866-874.

Brebner, A. and J. W. Kamphuis, 1963, Model tests on the relationship
between deep-water wave characteristics and longshore currents:
Dept. Civ. Eng., Queen's Univ., Kingston, Ontario, Canada, Rpt.
No. 31, 25 pp.

Bruun, P., 1963, Longshore currents and longshore troughs: Jour. Geophys.
Res., v. 68, p. 1065-1078...

Collins, J. I., 1964, The effect of currents on the mass transport of
progressive water waves: Jour. Geophys. Res., v. 69, p. 1051-1056.

Eagleson, P. S., B. Glenne and J, A. Dracup, 1963, Equilibrium character-
istics of sand beaches: Jour. Hydraul, Div. Amer. Soc. Civ. Eng.
Proc., v. 89, p. 35-57

Harrison, W..,, M. L. Brehmer and R. B. Stone, 1964, Nearshore tidal and
non-tidal currents at Virginia Beach, Virginia, U. S. Army Coastal
Engineering Research Center Tech. Memo. No. 5, 20 pp.

Harrison, W. and W. Wilson, 1964, Development of a method for numerical
calculation of wave refraction: U. S. Army Coastal Engineering
Research Center Tech. Memo. No. 6, 64 pp.

Harrison, W., W. C. Krumbein and W. Wilson, 1964, Sedimentation at an inlet
entrance: Rudee Inlet, Virginia Beach, Virginia, U. S. Army Coastal
Engineering Research Center Tech. Memo (in press).

Harrison, W. and R. Morales-Alamo, 1964, Dynamic properties of immersed
sand at Virginia Beach, Virginia: U. S. Army Coastal Engineering
Research Center Tech. Memo. (in press).

Harrison, W. and K. A, Wagner, 1964, Beach changes at Virginia Beach,
Virginia: U. S. Army Coastal Engineering Research Center, Misc,
Paper (in press).

Helle, J. R., 1958, Surf statistics for the coasts of the United States:
U. S, Army, Corps of Engineers, Beach Erosion Board, Tech. Memo,
108, 22 pp.

Inman, D, L. and N. Nasu, 1956, Orbital velocity associated with wave

action near the breaker zone: U.S. Army, Corps of Engineers,
Beach Erosion Board, Tech. Memo. 79, 43 pp.

53

Inman, D. L. and R. A. Bagnold, 1963, "Littoral Processes" in Hill, M.N.,
The Sea: v. 3, pp. 529-553 (Interscience Publishers, New York).

Inter-Agency Comm, Water Res., 1957, Measurement and analysis of sediment
loads in streams: Subcomm. on Sedimentation, Rpt. 12, U. S. Gov't.
Printing Office, Washington, D. C., 55 pp.

Kemeny, J. G., J. L. Snell and G. L. Thompson, 1958, Introduction to finite
mathematics: Prentice-Hall, Englewood Cliffs, N.J., 372 pp.

King, C. A. M., 1959, Beaches and coasts; Edward Arnold, London, 403 pp.

Krumbein, W. C., 1961, The analysis of observational data from natural
beaches: U. S. Army Corps of Engineers, Beach Erosion Board, Tech,
Memo. 130, 59 pp.

Krumbein, W. C. and L. L. Sloss, 1953, Stratigraphy and sedimentation:
Freeman, San Francisco, 660 pp.

Krumbein, W. C., B. Benson and W. B. Hempkins, 1964, A computer program for
sequential linear multiple regression: ONR Tech. Rpt., Contract
Nonr-1228 (26), ONR Task No. 389-135, Northwestern University,
Dept. of Geology (in press).

Miller, R. G., 1958, The screening procedure, Studies in statistical
weather prediction: Final Rpt., Contract No. AF(604)-1590,
Hartford, Conn., Travelers Weather Research Center, p. 86-95.

Panofsky, H. A. and G. W. Brier, 1958, Some applications of statistics
to meteorology: Univ. Park, Pa., The Pennsylvania State University,
224 pp.

Putnam, J. A., W. H. Munk and M, A. Traylor, 1949, The prediction of long-
shore currents: Trans. Amer. Geophys. Union, v. 30, p. 337-345.

Rao, C. R., 1952, Advanced statistical methods in biometric research:
Wiley, New York, 390 pp.

Reid, R. O., 1957, Modification of the quadratic bottom-stress law for
turbulent channel flow in the presence of surface wind stress:
U. S. Army, Corps of Engineers, Beach Erosion Board, Tech. Memo.
93 33s pps

Stahler, A. N., 1964, Tidal cycle of changes in an equilibrium beach, Sandy
Hook, New Jersey: Tech. Rpt. 4, ONR Contract 266, (68), Dept. of
Geology, Coiumbia Univ., New York, 46 pp.

U. S. Congress, 1953, Virginia Beach, Va., beach erosion control study:
83rd Congress, 1st Session, House Doc. 186, 45 pp.

54

U. S. Navy Hydrographic Office, 1962, Tables for sea water. density:
H. O. Publ. No. 615, 265 pp.

Whitten, E. H. T., 1964, Process-response models in geology: Geol. Soc.
Amer. Bull., v.. 75, p. 455-464.

Zeigler, J. M. and Barbara Gill, 1959, Tables and graphs for the settling-
velocity of quartz in water, above the range of Stokes' Law: Woods
Hole Oceanographic Inst. Ref. No. 59-36, 13 pp.

Zeigler, J. M., G. G. Whitney and C. R. Hayes, 1960, Woods Hole rapid
sediment analyzer: Jour. Sed. Petrology, v. 30, p. 490-495.

Zeigler, J. M., and S. D. Tuttle, 1961, Beach changes based on daily
measurements of four Cape Cod beaches: Jour. Geol. v. 69,
p. 583-599,

55

LIST OF FIGURES

Maps showing area of investigation and transects at which measurements
were taken.

Diagram showing times of measurement of process elements (Py, Possess
Pi 5) and response elements (R), Ro,......-- - Ros)

Diagram showing average beach profiles at the 15th-Street pier (north-
ern transect) and Camp Pendleton property line (southern transect), and
segments of the beach for which slope and mean grain size were deter-
mined.

Diagram showing envelopes of altitudinal values at lower-foreshore

stations for 25 profiles each at the 15th-Street pier and Camp Pendleton
line during the months of June and July.

56

APPENDIX A

Descriptions of the variables used in this study and explanations
of how they were measured are given in this appendix, together with the
ranges in the values that were employed in the regression analyses. Addi-
tional details of the measurement techniques may be found in two other re-

ports in this series on Virginia Beach (Harrison and Wagner, 1964; Harrison
and Morales-Alamo, 1964).

57

Ba
Depth of Water at Wave Breaking

The average vertical distance, in feet and hundredths, from the
water surface immediately in front of a breaking wave to the bottom. This
is an average of ten measurements, recorded four times each day. Values
used ranged from 0.35 to 3.80 feet.

CS Con ©
Tidal-current Velocity, C3; Opposed to Longshore Current,
Co; And in Same Direction as Longshore Current, Cg.

The mean tidal-current velocity was measured one meter above the
bottom at a point 850 feet from shore. A Price current meter was used,
and measured or interpolated values were expressed to hundredths of feet
‘per second. The respective ranges of values used were: C=0.00 to 0.68 ft/sec,
Co=0.00 to 0.66 ft/sec, and C,=0.00 to 0.36 ft/sec. Some tidal-current-
velocity values were estimated from a relationship established between the
tidal range and the tidal-current velocity at the point.

D
Depth of Water Table at Top of Uprush
The depth to the water table at the top of the swash line was
measured to feet and tenths four times daily. Values used ranged from
0.01 to 1.83 feet.
h
Still-water Depth
The still-water depth was determined from tide-gage data. The values
used are for the depth from the still-water surface to the mid-point of the
slope (fig. 3) in the shoaling-wave zone that was investigated for changes
in inclination and sediment size. Values used in the analyses ranged from
10.6 to 16.1 feet.
Hy,
Height of Breaking Wave
The breaker height was measured directly with a graduated rod and
was taken as the average trough-to-crest distance of ten successive break-

ing waves. Breaker heights were measured four times daily. Values used
in the analyses ranged from 0.69 to 4.50 feet.

58

Ho

Deep-water Wave Height (Significant)

Wave-height walues were determined from wave-spectrum-analyzer re-
cords furnished by the Soastal Engineering Research Center, as they were
obtained from the relay-type wave gage in 20 feet of water at the end of
the 15th Street pier. The peak value on the linear-average curve for wave
heights of the dominant wave train was multiplied by 2.22. This was done
because J. M. Caldwell had found (written communication, 1963) the follow-
ing relationship in a study of 92 simultaneous wave recordings made by
both magnetic-tape and paper-tape methods:

Average height on analyzer record _ 45
Significant height on chart record

Height values were converted to deep-water ones by entering tables. Thus
conversion to deep-water wave height values did not involve consideration
of wave refraction or special shoaling effectS. Values for deep-water
significant wave heights ranged fran 0.56 to 12.97 feet. The absolute
validity of the wave-height values so determined is not of importance, as

all that was needed was a consistently objectively-determined measure of
this variable.

Ho/Lo
Deep-water Wave Steepness

Values of wave steepness, expressed in terms of the ratio for deep-
water waves Ho/Lo, ranged from 0.00100 to 0.07226.

Je
Net Deposition at Lower Foreshore Stations
in Previous 24.5-hour Period
Net deposition was determined from measured changes in altitude of
stations occupied on successive days at low tide. Net deposition was
measured over a distance of 125 feet, or at six stations (fig. 3, 4) at
15th Street and at five stations (Fig. 4) at the Camp Pendleton property
line. Values used ranged from 0.00 to 1.50 feet.
Ke
Net Erosion at Lower Foreshore Stations

in Previous 24.5-hour Period

The net erosion was determined in a like manner to that of the
increment of deposition, Jp. Values used ranged from 0.00 to 1.30 feet.

59

L
fe)

Wave Length in Deep Water

Values for wave length (expressed as a deep-water value) ranged
from 49.20 to 989.24 feet.

(Mz) 5
Average Mean Nominal Grain Diameter
Over Bottom in Shoaling-wave Zone

Mean-size values were averaged for four samples taken 25 feet apart
in the shoaling-wave zone (fig. 3) where MLW depths ranged between 10.1
and 14.1 feet. The sampling device, a pipe dredge, permitted taking an
integrated sample over a 15-foot distance parallel to shore, and so the
average value for the four samples is presumed to represent the average
mean size of a 15 x 100 foot area of the bottom. Individual (M,) 5 values
were determined using a Woods Hole Rapid Sand Analyzer (Zeigler, and others,
1960; Zeigler and Gill, ng5@))e and the procedure outlined in Harrison and
Morales-Alamo (1964). The statistics used to estimate the mean nominal
diameters were

12) ap 12 dp JP)
O O
Mo = e ? ee (for summer data)
5
12) ap Je) cr iP) ap dee Pp Ie
10 O O
NO = 2 2 (ec oe (for winter data)

D

Average (M,) values used in the analyses ranged form 0.234 to

0.843 mm.
R
Range of Tide Over One Tidal Cycle

The measured tidal range at 15th Street as taken from records furnish=
ed by the U. S. Coast and Geodetic Survey. Values used ranged between ~*
2.4 and 4.7 feet. This was the only independent variable whose values were
held constant over several lag periods, a practice which is to be discour-

aged.

Se
Mean Slope Over Lower Foreshore of Beach

The mean slope was determined for a 200-foot distance of the lower
foreshore (fig. 3) from measurements at nine stations that were made daily

60

at low tide. This measurement was for Sp taken_as a dependent variable.
Values ranged from 1.40 to 4.55 degrees. When Sp was taken as an independ-
ent variable, the slope was measured over the distances shown on figure }.

Bs

Mean Slope of Beach Over Inner
Portion of Shoaling-wave Zone

The mean slope was determined for a 100-foot length of beach in the
shoaling-wave zone (fig. 3) where MIW depths ranged between 10.1 and 14.1
feet. Measurements of the altitude of five stations were made daily at
low tide in the summer, and every four hours in the winter. Mean slope
values used in the analyses ranged between 0.50 and 2.22 degrees.

aly
Wave Period

Wave period was determined from wave-spectrum-analyzer records fur-
nished by the Coastal Engineering Research Center. The period used was
the one corresponding to the peak value on the linear-average curve of
wave heights for the dominant wave train. (Period values thus obtained
were nearly always larger than those obtained in a "significant-wave"
chart analysis.) Values used ranged from 3.10 to 13.90 seconds.

Une Us Upp U

U wy Us

Mean Wind Velocity Against the Direction of Longshore Current, U8

In An Offshore Direction, Vor; In An Onshore Direction, U,,3

Parallel to Shore Up: And in the Same Direction as
the Longshore Current, Ugo

Mean wind velocities and directions were recorded at 2-hour intervals
at the Cape Henry weather station (fig. 1B), some 7.5 miles north of the
study area. The deviations in wind directions permitted when describing
the direction as onshore, offshore, and so on, are given below:

(+ LO of wind vector directed opposite to longshore-
eurrent flow)

Uoe (+ 80° of wind vector perpendicular to shore and in
an offshore direction)

G 80° of wind vector perpendicular to shore and in
an onshore direction)

6

U (Ge 10° of wind vector directed in either direction
along trend of shoreline)

(+ iO of wind vector directed parallel to longshore-
current flow, and in the same direction)

values ranged from 0.00 to 16.00 M.P.H.
Uo¢ values ranged from 0.00 to 34.00 M.P.H.
values ranged from 0.00 to 26.75 M.P.H.

U. values ranged from 0.00 to 17.50 M.P.H.

Us values ranged from 0.00 to 18.00 M.P.H.

Vv
Mean Velocity of Longshore Current

The current velocity and direction was measured four times daily by
timing the movement of 2 or 3 fluorescene dye patches moving over (usually)
a 100-foot distance. Values used in the analyses are absolute values, a
zero-velocity value representing either a no-current condition or a rip-
current condition. ‘Values used ranged from 0.00 to 3.20 feet per second.

Qa
Angle of Wave Approach

The angle of wave-front approach of the dominant wave train as
measured with a pelorus in a zone 1000 to 1300 feet from shore in water
depths of 20-26 feet. Measurements were made 4 times daily. Values used
in this study were absolute values only and ranged between 2 and 75 degrees.

Tye
Rate of Rise of Still-water Level

The instantaneous rate of rise of the tide; in hundredths of feet
per hour, as determined from observed times and magnitudes of high and
low water. Values used ranged from 0.00 to 0.67 feet per hour, and were
computed for the mid-points of the 6 lag periods. Because times of low
tide in nature did not always coincide with anticipated times (CURE ries
2) che Nr and Ne computations sometimes resulted in values for rising
tide during a lag period when it should only have been falling, or vice
versa.

62

Te
Rate of Fall of Still-water Level

The instantaneous rate of fall of the tide,
JOSIP Inoehe,,

low water.

in hundredths of feet
as determined from observed times and magnitudes of high and

Values used ranged from 0.00 to 0.60 feet per hour, and were
computed for the mid-points of the 6 lag periods.

fe)

Water Density

The "sigmatee" values for the sea water as determined (U. S. Navy
Hydro. Office, 1962) from temperature (+ 0.05°C) and salinity (+ 0.02 °/oo)
data once daily at noon time in the surf zone, for the summer measurements,
and three times daily in the shoaling-wave zone, for the winter measure-
ments. Values used ranged from 1.0136 to 1.02500 gm/ cm A

63

NOTE: Summary Tables of Appendix B are started on

opposite page.

64

APPENDIX B
Summary tables of the computer output

are given in this appendix.

65

66°99
29°19
gL" 9
40°89
On *69

€6°29
60°€9
TL°€9
TL°€9
go°s9

gn" 6S
4S °6S
g9°6S
te" 09
SL°09

TL°TS
SG°eS
02 "SS
gt °9S
€6°9S

99°6£
gL°6€
Ze"ay
Leen
9° Li

SS UT uoTJONpSY

queg reg

HT
aT
uae
HI
qT
HT
HT

T

HI

€T

€T

as
as
ol
ot

as
as

ol
al

oT

ot

ot

oT

as

TT

OT
Ov

OT

OT

\0 190 \/0 \0 ‘0

\0 \0 \0 ‘0

SUOTYeUTQUOD SsTqeT re, JUepusdepuT

a) ta) tala) it) i dra litte taiGa! tal Gal oa!

ddd d4

Co fl Ga} Cal cal

owt, @ 4e
SX xTS

aut, eB 42
SX OAT

aut} & 4e
sx mo,y

aut}, e@ 42
SX 9eZUy,

outZ @ 4e
SX OM,

*Z potdeg Bey Joy ‘soTqetsre, yuepuedepuyt ueeqmoyg Jo suotyeutquog
Tedeneg Jo yoeq 0f aTqeqnqt144y serenbg jo umg suoTze99 aLoYsezog
FaMoT Ye uoTyTsodeg JeN UT suoTyJONpeYy queg Jeg ysoeSuor4g sAT A oUL-*cg_ aTaeL

69°hL
HL,
Te"
69°S2
2L92

oN,
SSaroy
diol
€G°eL,
6g°2l

68°99
16°99
Gs‘ l9
+S°OL
29°ol

92°65
€€°6S
80°09
€2°T9
o£ *T9

G6° Lh
48°8t
BT 6h
96°TS
GT°ES

SS UT uoTJONpeY

queoteg

qT
aT

€T

€T
€T

€T

€T
SE

9 G6 ~~ U
6 9 S¢ + TE
L G th a
gy 2 G i & T owrq 2 ye
@ 2 G4 @ i sX XTS
6 8 9 Ss TE
OT 6 9 GS T
6 9 GE
6 9 € T ourz @ 4e
6 V-G iT T SX aATa
6 9 € T
6 9 Gv
8 SG T
DyenG) 9 ¢ T ourg e@ 4e
ot 6 9 E sx mog
OT T
6 9 TE
S iE
9 S T oury 2 ye
OT 9 T sX sequy
€ ile
TL T
8
OT T owrq e 42
id iT SX OM],

SUOT}EUTQUOD STqeTreA JUepUedepuTy

*T potted Seq7

Joy ‘saTqetzep, yuepuedepuy useqmog Jo suotTyeuTquog Teteaeg jo
yoeg of aTqeqnqtaz4y4y Serenbs Jo umg suoTyzeyg etoYysatog IaNoT
qe uoTytsodeg yen UT suoTJONpeYy yueD Jeg 4yseBuor4g saTY eUL-*TA eTaeL

66

LT" 19
oe" 19
gL l9
TT"89
99°TL

Z6°€9
26° €9
Sh °19
THS9
9£°99

T6°9S
T6°9S
96°9S
99°09
16°29

G2°6h
gt TS
69°TS
ZO°HS
TS°9S

19°6E
19°Th
geet
09 2h
To*8t

$$ UT woTZONpeYy

qusozeg

OT G7 € et
Oo Gr & SE
J; Gi Set
6 G+ € 2 T oWTq B 48
fe} 6S. BE SX XTS
6 nv S$ @ E
L 7 & @ E
g i & @ E
OT u S& @G ET Sia Bae
G&S ot SX ATA
8 q a
L mn T
8) iy T
GS 7 T ourqy e 4e
i & GE sX mog
G € T
S @ iE
“ & G
+ oe T 3wrq @ 4e
a oS T sX 9e2ur
S U
€ T
+ 2
+ T o3wry 2 32
4 € SX OMT,

SUOTJEUTQUOD SaTqeTre, JYuepuedeput

‘tH POTdeg Bey toy ‘saTqetze, quepuedepuy useqmog jo suoTyeuTquiog
Tetaseg JO yoeg OF aTqeqngqtsz49y serenbg jo umg suoT{e19 ezoYyseto¥g
TemoT Ye uoTZTSOdeg JaN| UT suoTJONpeYy que Jeg 4yseSuorzg SAT Y oUuL-"*Ha STqeL

nec EL
TE*HL
0S HL
og*a
29 "tL,

Te°tL
LE*TL
49°Th
90°eL,
HEEL

Lg°€9
LT"49
at "99
en" l9
1T°oL

g9°1S
ASS
27°95
€S5°9S
99 09

HS°6E
TE*Oh
9° On
BETH
2S° lh

SS UT uoTZonpeYy

qusoreg

°€ potseg Sey toy ‘setqetzre, yuspuedepuy useqmmog so suotzeutquiog
Teteaag jo yoey 04 aTqeqnqtz44y serenbg jo umg suoT ze 99 ezoysezog

a

HI

I

ol
as
as
ot
as

ot

as

1

SUOTPEUTQUOD STqetzeA jJuepuedeput

a a ee a a at a at at st

ata

mm

MOAMM OY m4

tal eal ta} cal

dd do AAA

th ta tl tl

dddAaad

out, e@ 4e
SX XTS

awry e 42
SX eATH

owry B@ 42
sx moj

oury 2B 4e
SX 90rU],

out @ 42
SX OM,

ZanoT ye uotatsodeq Jan UT suoTyonpey queg Jeg yseSu0r49 SATA au-"Cq eTaeL

67

gn" 62
+S°61
16°62,
GL°6L
19°61

16°22,
eee
26 "tL
T6°LL
6€°6L

G69
gT°oL
Gc°oL
6S°0L
00°TL

99°9S
2° lS
TH? 1S
89°99
69°19

of" LE
2g" Le
T2°6€
OL*Ok
€2°€S

SS UT uoTJONpSY
queoteg

qT

qT
qT

qT

qT

uae
qT

cL

6 8 Gt ot
6 8 S So U
6 ¢@ 2 S Coal
TL 6 8 G co T o3wry @ 4e
6 8 S aq SX XTS
6 S € T
6 G cual
8 9 GS Caer
6 8 S € T owrq 2 4e
6 8 S @ € SX oATT
S BE
8 S eo
6 S eS
} S € T our 2 4e
L S GF sx mog
G €
6 t E
nm ils
S € T outa 8 ye
S 2 Tt SX secs
4 3]
q
S T our 2 48
+ T SX OM],

SuOTTeUTqUOD eTqetse, JUuepuedepuT

"9 potdeg Bey] soz ‘seTqetze, quepuedepuy useqymog jo suotzyeurquiop
Tedansg go yoey 04 aTgeqnqtsz44y setenbg jo umg suoTye4g aeTOYyserOg
ZaMOT 7e uoTATSOdag JeaN UT SuCTJONpeEY yueD Jeg yseBuo0r4g eaTty eUuL-"gg STAeL

95°92
02 °gL,
o€°gh
J9°gh
£e°el

0S°TL
qheol
og*eL
6TH
€erHh

00°S9
ST°99
TE"89
LE°g9
On * 89
Ee*HS
G6°hS
HE°9S
gor ls
+S°QS

€S°ge
00°6€
68°6€
gt eh
9S Lt

SS UT uOTJoNpeY

queoreg

*G potdeg Seq toy ‘setTqetre, quepuedepul usequmo¥g jo suoTyeuTquiog
Tetaneg Jo yoey of eTqQeqnqtz34y serenbg jo umg suoTze49 eLOYyseroy

€T
€T
€T
€T
€T
€T

€T

cL

as

an

as

cL

OT

Ov

SUOT}eUTQUOD eTaeTe\ Juspusdeput

LN LY LY LN LY

ey LN UN UN LN LN

LN LN LY LN LY LN LN LN

LN LN LN LN LN

at apa

MMMM O

mo

NNO

ANN OW

AdAAd GaGa) ral fea

ddd

awry @ ye
SX XTS

out e 4e
SX OATH

auty @ 4e
sx moj

aut, e@ 4e
SX d0TUZ

awry B 4e
SX OM]

JaMOT 7@ uoT4TSodaq yeaN UT suUOTJONpEeYy qusD Jeg yseBuomsg SATA suL-"Gq STqaeL

68

TT‘ OT eT 69.1 9 G

g6°6 ‘Sie GE OT 1 ©) &

26°6 €L er glogs

on°g €l et OT 6 L G auty &@ 12
€o°L €— eL 6 Fo nS sx xt9
6L°9 ei eck 6 L 9

66° €T 6 19S

88"t Ciseecie g 9 S

glee €l et I, & & aut, @ 42
02°2 €l eT 6 dh 4 SX eATA
06° ol 6 IL G

1I°? €T 6 J S

G9°T Ss ro 6 I,

c9°T SIE GE Jl G aut) 8 4728
Te*t SE GE 6 G sX moj
Te°T €T 6 G

6T'T qT 6 G

60°T €l er Jb

g9°0 StaecE G aut) & 48
£9°0 €l et 6 SX aerUI,
g9°0 2 G

€9°0 et 6

Z9°0 is G

SS°0 €T 6 aut, @ ye
go°o SE GE SX OM,

SS UT woTZONpeY
queso rag

SUOTPEUTGUIOD eTqeTre, JUepuedspuT

*€ potszeg Sey Toy ‘setqetzre, yuepuedepuy useqmog fo suotyeutquog
Tedansg jo yoeq 0 aTqeqnqt1z44y serenbg jo ung suoTyeqg ezLoYseto,g
ZanoyT ye uoTyTSodeg Yay] UT SUOTJONpeYy 4UaQ Jeg YSeyeem SATT SUT-*ga STAeL

TE*eT Se GE
zo°eT €— eT
90°2T GIL
€L° Il Se SE
+6°9 €T
1S°Q €T
g6°L €T
69°9 €T
GT°E

l2g*%

GL°T

GS°T

LE*T

ce°T

4S*0

+6°0

2S°O

gh°o

9t°0

+E°O

9 MAO
Same
[oomomeo ne)

S$ UT uoTZONpeY
queoreg

*g potsieg Se] toy ‘seTqetzepj quaepusdepuyt usee7ymog jo suotyeuTquog
Tetensg jo yoey of aTqeqnqtsz44y serenbg jo umg suotzeI9 ezroysezog

OT
OT
OT
OT
OT

OT
OT
OT
OT
OT

OT
OT
OT
OT

OT
OT
OT

TT

OT

OT

ON ON Ov OV ON OV OV OV OV OV ON

eRe RR RRR BRE

ce ee &

LN LN LN Tay UN LN LN LYN

UN LN LY LN UN LN LN

UN LN

mo

NUNN O

NNW

NAW

SUOTYSUTQUOD OTQeTIeA, JUapuedepuTy

aut, @ 48
SX “XTS

ouTy @ 428
SX eATT

aut? @ 4e
sX mog

out @ 42
SX derUy,

aut, @ 42
SX OM]

ZJanoT Ve uoTATSodeg 4aN UT suOTJONpeY qUeD Jeg ysoWeaM oaTA euT-*Jq eTaeL

69

G6") €l eT OT 8 9 z 8H ° OT qT 6 Qt ES
o6°L Cl = cl kt OF 9 € 0g°6 +T 6-9 th 9 €
€T°h Sk ae WE OE 9 Z oH°6 HT 6 19 SZ
+S°S €T THe OVE 8 9 € ouTy @ 48 ST°6 6 9 bo Cenc, auty @ 328
90°S €T GEOL 8 9 2 SX XTS 26°8 +1 6-9. J ©) ro sx xTS
eS°h €T TEE OE g 9 rh a or © @ F €
tot €T TE g 9 € €or) 41 6 @ 4-9
Gere €T Ee 8 9 2 gl°9 OE 6 @ J 2
Sige €L Tl OT 9 € aut, @ 42 TE°9 6 9 4 9 € ouTz @ 42
€9°T €T TEE. OL 9 2 SX OATT G2e°9 6g 4 9 ro SX ATA
€9°T LE OL 9 rf Olt qT 6 8 9
€9°T €T Oo 9 2 QT°t 6 9 2 €
ty €T TEE 9 € €6°€ + 6 ©) J, rd
6T°T €T GE OE @ aury 8 48 16°2 “6 3 9 € autz B 42
Gg°o €T Tt 9 Z sX moj Paras 6 8 9 2 SX Mog
S8°0 €T 9 ro TH’? 8 9 ro
G8°0 Tt 9 rf On°? 6 8 9
LL°0 €T ae ro €e°g 6 8 ro
G9°0 €T Tl OT ouTy @ 48 Gt°2 6 9 € awry 8 48
To°0 €T Tt 9 SX eerU], €9°T 6 9 Z SX 9024]
g9°0 cae OL 10°? 9 €
g9°0 €T OT €9°T 6 9
T0°0 €L TEU G9°T 9 ro
TO*O 1815 9 aut} @ 48 99°0 6 € ouTz B 48
00°0 €T 9 SX OML €S°0 6 Zz SX OMT
ss ut uoTYZONpsy suoTtyzeutquop eTqet rej quoepuedepuy SS UT WUOTJONPSY SUOTZeBUTqUON STQetse fp quopusedepul
queso ted qus0ceg
"9g potsed Bey Joy ‘saTqetze, yuepuedapuy useqsmog Jo suotyeutquo) *G potdeg Sey roy ‘satqetre, qyuepuedepuy usequmog fo suoTzeUTqUIOD
Teceaeg jo yoey 09 eTqeqnqtsz49y serenbs jo umg suoTqeqg etoYysetog Texzansg Jo youy of eTqeqnqtz44y sezenbg Jo umg suoTze4g sLOYysetog

ZaMOT 7 UOTZTSOdeg Jay UT SUOTJONpEY 4ueD Jag 4ySeHeaM SATY OUL-*OTE OTAVL JaMOT 18 uoTATSOdeg JeN UT SUOCTJONPEY qUEeD Jeg ySeyeom SATA SUT-"6a STAPL

70

o64°0Q = UCTZONDEY SS TeIOL

€00€ Zoo? TOOT H9E T6 tT 4 %60°2g = uoTJONPEY Ss TeIOL
Ds Se 6°4-0° OL E00E Zooe TOOT HOE T6 kT @
6yT zee 2 6°69-0°S9 we 6°69-0°S9
Sis Gor Ge © 6°49-0° 09 661 SE 2 6°t9-0° 09
Q6E got Oh 4 6°66-0°SS cee Gyr ce € 6°66-0°SS
GGh coe 8h On 6*4S-0° 0S Hee HET SE ft 6°HS-0°.0S
Heh e9e SQ TT T 6°6H-0°SH 266 ITE 16 OT T 6°64-0°SH
ges Thy 99T 6h € 6°tt{-0° OF 66h 9€€ THT TE 2 6 *th-0° OF
goe 2€2 9ST 94h 9 6°6E-0°SE Gs ose Ger Sy we F 6°6£-0°SE
eos Jie G5 Gy Of © 6*HE-O° OF 052 g6t 60t le € 6*H€-0° OF
sr om I) Gre 4 6°62-0°S2 ee whe 1m) a 6°62-0°S2
vat Whe oe wWS5 J 6° 2-0" 02 Ge wie se wy db T 6*t2-0° 02
a Gd cd G5 Cc & 6°61-0°ST 49 «46€T ERT T9 6 6°6T-0°ST
ae US «6hy US CSE 6*tT-0" OT Ge % 1% Ob Ge 6*HT-0° OT
yw OF Ge © 6°6-0°S @ tf Ge Jr oe @ 6°6-0°S
t ke Ge Ge 6*t-0°0 Z 6 Cr wet © 6°*t70°0
Gh STeeeGe Ome % SSeTO Os. (Se ee % sseto

soTqetzeA yuepuedepul fo LequNy{ uoTzyonpey Ss soTqette, Juopuedepuy jo czequmy uotTqjonpey sg

(€ potsed eT) SX qT SA etoysetog tanoT uo uotytsodeg 4eN-"cTd eTAeL (g potaed eT) SX _T SA eLoyseto,g JamoT uo uot4tsodeg JeN="TTa FTaeL

7|

%g1°eg = UoTZonpeYy Ss fedoL 499°)Q = uoTZONpeY gg TeqOZ,

EO00E Zoog TOOT HOE T6 HT g €O00E Zoos TOOT HOE T6 HT g
Ww 6°6L-0°S) (on 6°6L-0°SL
ote dh 6°*tL-0°0L 6TT €T 6*tL-0°0L
ae 0) Gr e 6°69-0°S9 JS i 6°69-0°S9
gos 98 6 6 "9-0" 09 H€h get ET 6*49-0° 09
Igy lye 49 9 6°6S-0°SS 96h 9S2 TS € 6°6S-0°SS
Ge Or gy G F 6745-0" 0S oe 9g 26 TT 6*4S-0°0S
gen noe G9 OL 6°6h-0°St €0€ Qc? 90L ST T 6°64-0°Sh
ty ae cxe JB % 6*hh-0° Oh EC mae G Mma CaeuT 6°tH-0° Oh
g6t Jze I6t QS 6 6°6£-0°SE ose 6l2 6hT 62 4 6°6£-0°SE
QST S8T €eT th 8 als 6°HE-0° OF Te GSe Cee 45 we t 6*HE€-0° 0€
WS Car Ge 44 Ge 6°62-0°S2 g6 66 +49 9f 9 6°62-0°S2
1 Oy @2 Ge 4b ze 6*Z-0° OZ Gon cing Cy 43 J 6 "2-0" 02
an 4S On we 6°6T-0°ST OG. Wier Go us ¢) ro 6°6T-0°ST
1 Wo Gb Ged 7 6*HT-0°0T Q@ yr ed 5 4G 6" T-0° OT
Te Qc Gb GS Cy & 6°6-0°S + Ww Gr Jk te & 6°6-0°S
9 Go gf 92 8 6°-0°0 G Ge ww g 6°hH-0°O
Q2= Get SS ee =r % SSeTO SoG a Soe ee % SSBTO

SoTqetre, JUapuedepuy jo Laqumyf uoTyonpey gg saTqetre, juepuedepuy jo Jaqumy uoTJoNpey gg

(9 POTted BVT) SX qT SA axoysetog TaMoT uo uoTqTsodeq JeN-"tHTA eTAeL (SG potaag 387) SX HT SA az0ysezog JamoT uo uoTATsodeg 4eN-"€TA eTaQeL

72

9S°2S +T ia 6 8 € E So°sh €T TELE 6 dL, S €
g2°€S HT TT 6 DE G1 TE°St TL 6 L SG € 2
6E°ES TL 6 L S € iE S8°st TEE 6 L Goitt 4
TIS qT TT JE S € T wry B ye G6°eh LE 6 I ©) © € out, 8 ye
oL°SS qT TI 6 dE € IE SX XTS 6L°TS TT 6 L S € T SX XTS
19 °8t €T 1 S € T So°Tth 6 8 9 S €
99°8h 6 dh G € I ET*ah TELL J, G € i
+0°6t OT L S € is GS*an 6 L G € T
61° 0S TT L S € T 9uty @ 7e LO et 6 6 7 & T suty B 78
64°0S qT LZ S € T SX eATA on Lh TT 6 “ds S € SX eATH
66° L H € E Z0°SE Tt 6 L €
go°en ot L S € 6G°SE L S € T
60° th qT L € 1 Te°9€ 6 L S €
29° th L S 2 T #3ewry e 48 05°9E Go § T owrq 2 42
Te* Lh ese aS € T sX mog cer LE Ee L S € SX mog
WE CS L 4 il TL 2 4 € T
By LE qT A, T G0°G2 6 4 €
go" Le L @ 1 65°92 Gy &
S9°6€ I S € aury @ 42 20°62 I, 4 € aur @ 4
6L°6E L € T sx eeruy QT 62 TI L € sX eeu,
62°62 on 2 90°ST € u
9E*62 L zZ 60°ST 6 it
OW ° OE OT 6 TS°ST 8 x
Og*Te db, € aut} @ 4e T9°ST J, T owrq e ye
2c" €€ L T SX OM], Ji QT JL, € SX OM],
SS UT wuoTJONpSYy SUOTJeUTqUOD SeTaeTre, yUepusedepuy SS UT UOTIONpSYy suOTZeUTqUON eTQeTre/ quepuedepul
qusotedg quooleg
ee ee SS SS ee
°g potsieg Bey] toy ‘saTqetze, yuepusdepuy useeqmoy Jo suotTyeutquog *T potseg Sey toy ‘setTqetxre, qyuepuedepul ueeqzmmoy Jo suotTyeutquop
Tezaneg jo yoeq 09 eTqeqnqt144y sezenbg jo umg suoTqeq49 sLOYysetog Teteaeg jo yoey OF eTaeqynqtTaz49y sexenbg jo umg suoTyeqg ertoysetog

FaMoT 7@ UOCTSOTY JeN UT suoTJONpeY 4ueD Jeg yseBuor4g eSATA SUL-"OTA eTAeL JAMOT Ye UOTSOAY Joey] UF SuOTJONPey qUeD Jeg yseBuor4g SATA euL-°STA STAeL

73

6T°9E
Zo"ge
90°R¢
ge"ge
€8° Oh

9g°t
To°€e
GT*EE
2g" ee
1g°€€

OL*6z
TI°62
€L°6%
€o°ze
Orc’

2°
TI°Sz
+S°S2
16°S2
GL lz

g6°HT
1&6°ST
62°9T
98° LT
Core?

S$ UT woTZONpSY
queozreg

oT
ae

6 9) GR oF
6 GS 7 & @ Tt
6 1 G a & 2g
L Ga & @ t
QQ G& 7 ES T
TEL 6 9 eG t
6 9 € T
6 9 a
6 9 “T&S E
6 9 + @
6 9 GS E
We 9 @
6 9 + T
6 9 € T
6 9 aU
9 SG
9 € i
9 a
EE 9 1
6 9 E
g it
HE 9
OT 6
6 T
9 T

SUOTJSUTQUOD eTQeTreA JUepuedepuTt

owt, @ 42
SX XTS

out, @ 4e
SX oATY

aur, @ 48
SX Mog

awry 8 48
SX 9eauy

out e 4e
SX OM,

oO —— ——— — — — ————— — ______

‘tH POTZed Bey Loy ‘saTqetxre, quepuedepuy ueey.mog jo suotyeutquiog
Tetasag jo yoeq oF aTqeqnqt2z94y sezenbg jo umg suot9e99 asoysetog
TaMOT 7B UOCTSOTY Jo UT suoTZONpeY JueD Jeg 4seBu0r4S SATA SUL-"QTa eTAeL

GL°Se
LL°Se
lt le
0S * de
9° TE

On tS
TS*h2
65°42
€9°HZ
glne

1g°€2
65° 2
26° €2
96°E2
66° €2

QS*T2
tQ°Td
GE°e2e
19°22
L0°€2

TO" et
G98°ST
00°8T
Ot * 02
Tg" 02

SS UE uot ONpSY

queoreg

cL

ol

or

ot
ol
an

ol
as
ot
ot
ot

as
as
as
as

oT

ot

6 G iS E
6 9S @ ©
G 7 © ¢ E
9 Cy €& St
6 Ga © @ TE
6 9 4 T
6 GS ot gt
TEL 6 G £
TEL 6 + E
6 Goh iL
6 8 E
6 9 iE
6 4 iL
? 6 d E
TI 6 ic
SG T
6 9 T
9 iT
dh TE
6 T
6
es ils
6 T
9 T
Tg

SUOT}SUTQUOD STQeTIeA JUepuedepuT

out, @ 42
SX XTS

aut} e 42
SX SAT

outy @ 42
SX Mog

out © 428
SX deruy,

auty @ 48
SX OAL

°€ potdeg Bey toy ‘seTqetze, quepusdepuy ueeqyamog fo suotyeutquog

Tetereg JO yoeq OF aTqeqnqts49y serenbg jo umg suoTyeyg aztoysezt0y

JaMOT 78 UOTSOAY JON UT suoTJONpey quan Jeg yseBuorgg aeaTty oug-")JTq etaez

74

Ly ee
GL°ee
6L°2€
06°2E
HT°€€

46°62
66°62
09° 0€
€9°0€
die Ze

19°S2
gt °92
GH°9e
HS°9e
68°62

Li*22
62°22
HE"ee
GE°ee
HE°SS

oL*9T
0° LT
+0° LT
4S° LT
96°6T

SS UT uoTJoNpeY

queoteg

EEE eee

ae. Ss GE
+ ra
+T ra
qT er
qT Gir
qT et
qT et
qT ot
+1 et
+T er
qT
qT ra
qT ral
qT et
+T et
qT rca
AIL
au
et
qT Al
+

OT 9 iT
Ot 9 + T
OT 8 9 U
TT OT 9 T
OIE 19 T
OT L AE
OT + il
Tt OT T
OL 6 t
OT 9 T
Ot 9 ils
€ T
@ U
9 it
Ot iT
on
G iT
9 T
ge 0
T
@
S T
9 T
Ts
ils

SUOTZEUTqUOD eTQeTTeA JUepuedepuyT

*9 POTteg Bey Tog ‘soTqetsze, quepuedepuy useymog Fo suoTyeuTquioD
Tetaaag Jo yoeq 0f eTQegnqTt144y Ssezenbs jo umg suoTZe4g etLOYserOg
JamMOT 4e UuCTSsOI Jan UT suUOTJONPEY yUeD Jeg 4seBu0r4g9 SATA SUL-"Ocd STAPL

aur} e@ 42
SX xTS

awty © 4e
SX OAT I

aut9 e 72
SX mog

out @ 42
SX serUy,

awry e@ 4e
SX OAD

cgr2e
glece
6L°zE
Gorge
ELeHe

€S°6¢
46 "62
GT" OE
OL’TE
TL°TE

Zo'l2
90°22
Lo: lz
€2°g2
66°92

€H° 02
0S°0¢
29°02
Ge"Te
tthe

2o°ST
65°ST
40° LT
€g° LT
+2°6T

S$ UE uoTJoNpeY

queoreg

uae

eeeh

co
UN UN LY LN LN
mom
Ce] Galiteal et Gal

fee)
Ne}

OT

OT

DAADAHN DAADAAN AAHAD
-
ddddae

AdAaAd

OT

=F
dddaed

No}
Al tl ee) tel

SUOTJEUTQUIOD STQeTTeA JUspuedepuy

°G potseg Sey Joz ‘sotqetze, yuepuedeput useqymoy jo suotzyeutquop
Teteaeg jo yoey of oTqeqnqtaz44y sezenhg jo-umg suoTye49 aztoysertog
JamMoT Ve uoTSOIG YON UT suUOTJONpEY queD Tag 4yseBu0r1g eATY oUL-“6TA aTAeL

duty e@ 48
sx XTS

eauty e 42
SX SATA

awry e@ 4e
sX mog

awry @ 4e
SX d0TU,

aut, @ 4e
SX OMT,

75

qT
qT
qT
qT
qT

qT
qT
qT
qT
qT

HT
I
qT
4T
a

tT
qT
qT
qT
qT

a] GAGINA fe
DoOAWWN

WY WSHGE
ee 8 ° °
AdMmMmM MmMatAatSt Atrerro

A+ +N
TaN neN A
°

Ohetretietict

oo0Odd

qT

a
HI

ODAM DMLUN
°

\O feo)
2 SS al Cah
e

SS UT uoTZONpEY
queoteg

cL
ol

aE G1 &
TL Gt é
TEs S & @
TELE 9 t out, @ ye
WE 9 S SX XTS
TT 9 S
9 SG
TT 9 GS
TEE S out, @ ye
EE 9 SX OATH
TEE S
TT 9
S
EE S out, @ 4e
Ts sX mo,
TL
S
TL S out, B ye
TEE SX doT4],
HE S
S
Tt
owty @ 472
TL SX OM],

SUOT}SUTQUOD STQeTZeA JUspuedepuT

*Z potdeg Sey Loy ‘soTqetzre, yuepusdepuy ueeqmog Jo suotyeuTquop
Teteveg JO yoey 0f STqeqnqtaz44y Serenbs jo umg suoTyeIg eroYysetOg
JaMOT 7B UCTSOLY JON! UT SUOTJONpPEY jUeD Teg ysoyeemM SATA OUL-*ccd OTAeL

€gre
gLre HT
ger’ 41
gerk +1.
ese qT
06°2 41
68°2
16°2 +T
GS°2 +T
TS*2 +1
Gore
to"?
€6°T
2Q°T +1
Dee +T
JAE +L
JeAE +T
ZO°T
gg°0 qT
Tg°0
0g°0 qT
TIPO)
99°0 qT
€9°0O
gc°0

SS UT woTJoNpeY
quooreg

€T
€T
€T
€T

WE OL 9 c
OT 9 zZ
EOL 3%
iG OL 9 out, e@ 4e
186 OL 9 A sx xXTg
BE OLE 9 é
Tt OT 9 ro
15 OE é
be O 9 outy @ 4e
1G CONE t ro SX dATA
EOF io)
OT 9 i
He OL 9
115 OLE fe) aut} B@ 4e
1515 (015 @ sx mog
TT ro
OT 3
it OF 9
TIE OE owty @ 4e
EOL ro) SX d0TU],
OT
TT 9
Gc
OT ro auty @ 4e
EOL SX OM],

SUOT}SUTQUOD eTqQeTre\ JUspusdeput

*T potieg Sey soy ‘fsoTqetre, quepuedspuy useqmmoy Jo suotyeutquog
Tetaaeg jo yoeq oF aTqeqnqtaz44y serenbg Jo umg suoTye99 atoYsatoy
JaMOT 78 UOTSOTY JaN UT suoTYONpEy quep Teg ysoHeomM OAT T 2UTL-"Ted STAeL

76

Gt°) +1 OT 1: S Sarc,
4°) qT ot OT J Se
+6°9 +1 et OT L Gay:
G9°9 +1 GE dr “ & @
€€°S HL et 1 S Sg
90°S HL J, S € @
Te*t 4 ot 1 + 3
glen HT ot I, aac
elt or d, S € 2
of *t HL CIE i & @
Elz qT ot 2 ro
lycz +1 oL L S

tte? HL rae G4

TE"? 4 al h €
g9°T 4 2. 4 Z
6S°T ral oL S

TH°T qT ot @
Gort +1 ot t

+2°T 4 q S
19°0 qT 4 2
gg°o + €
€L°0 +L S

64°0 4 2
LE*O + 4

Zf°0 €T 2

SS UT woTZONpsY
queodeg

SUOTPEUTQUIOD STqQeTre, JUepuedepuy

“ty POTTeg Bey toy ‘saTqetre, quepuedepuy ueeqmmog jo suotyeutquog
Texeveg JO yoey oF aTqeqnqtz44y serenbg Jo umg suotT9#e4g aaoYysetog

auty @ 42
SX XTS

aut, 8 42
SX eATY

aurty @ 428
SX mog

owt, eB 4e
SX der],

aur @ 42
SX OM],

°
mms at WN

ra

BRP eS

a
HI

HT
qT

qT

DW oO
Raa SA
3
daaaa

CHT HT

$3 UT uoTZONpeY
quaorag

ee Os
TE OL S

mma
UNUM

at at at —t
may
fal}

q
=
co ee) feokee) © 0 feoyice)

=
4d

tat at
Cat}

TL

mo

1
ic co

SUOTPEUTQUOD eTqeTre, JUspusdepuT

aut, @ 42
SX XTS

aut) @ 42
SX eATY

out @ 42
sx mog

aut} @ 42
SX deTU,

auty @ ye
SX OM],

———————— —————————————e—eeeeeeeeeeeeeSeSSSSSESSSESSSSMMesesese

°€ potdeg Bey soz ‘seTqetse, jyuepuedepuy ueeqmoyg Jo suotzeutquog

Tetensg go yoey 09 eTqeqnqT4z494y sertenbg go umg suoTqeIg aaoysetog

77

€6°E €T OT
Z6°€ €T on
6g°€ €T OT
HQrE €T

€9°2 €T OT
€9°2 OT
29°e €T OT
go°d €T Ot
06°T €T OT
HS°T €T

HS°T

On°T €T

92°T €T

00°T €T

0g°0 €T

0g°0

glo €T

+L°0 €T

T9°0O €T

TT°O €T

T9°0

1&S°0 €T

TT°0

OT°O €T

z0°Oo €T

§S§ UT woTJONpEY
queo reg

"9 pOTteg Bey toy ‘seTqetze, juaepuedepuy ueeqymmog jo suoTyeutquop
Tetaaeg jo yoey oF aTqeqnqtsz44y Sserenbg jo umg suoTye4g stoYsetoOg

OV ON ON ON OV O’V Ov OV OV ON ON

OV ON

foe) co feooe leo) ce) cc oO feo) 0 00 Cc ee) coccocada

8

RRR R BRR RR BEE E

LUNUN OLN

+t zt at + tt ata at at st tatoo

as

mM MM

SUOTIeUTQUOD eTQeTrTeA JUSepuedepuT

outy @ 4e
SX XTS

awty e 42
SX aATT

aurty @ 42
SX moj

our} @ 4e
SX aor,

aut, @ 42
SX OM],

TOMOT Je UOTSOTY JON UT sucTJONpey quan Jeg 4SeyeomM OATT OUL-*9cd STARL

99°0
2S°0
€2°0
T2"0
70°0

S$ UT uoTJONpsYy
queozeg

°G potdag Sey coz ‘soTqetsze, quepuedepuy useqmmog jo suotyeutquog
Tedeneg Jo yoey of atTqeqnqtsz4qy serenbg jo umg suoTyzeqyS eLoysetog

€T
€T
€T
€T
€I
€T

€T
€T

€T

€T

cL
as
ol
as
ot

ot
as
ol
ol
an

ol
ot
as
ol
ol

ot

OT
OT
OT
OT
OT

OT
OT
OT
OT
Ot

OT
OT
OV
OT

OT
OT

OT

OT

OT

OT
OT

suorqgeutquoy aTqet&eA Juepuedspuy

co 0c OO 000 OO

cc oc leocoooe) ee) amcocca

IN WN LN LN

+ at ata

mmm

NU

out, 2B 48
SX XTS

eut9 @ 48
SX eATY

awry @ 4e
sX mog

outa @ 48
SX derNZ

owt} @ 48
SX OM]

JaMOT 4@ UOTSOTY JaN UT SUOTJONpEY queD Teg ysSoeweomM SATA 9UL-"Geg eTAeL

78

Tg°18
Tg*1g
Sg° 18
46°88
66°98

on €8
c0°T8
0S "+8
Z€°Sg
og" 8

GS°gL
TL°9L
6L°6L
16°61
90°T8

TUS)
€T°S)
Lo°9L
Z2°9L
geo)

61°19
ZT °69
Jg*th
de°eL
6E°EL

$$ UT uoTJoONpeYy
queoteg

AL QI; =) + i
TL gy I + T

6 3 dS + T

yh oO G4 T

OT g 19 + ils

at Gt T
THE GH € T

J, Got @

L G1 T

8419 ut T

HE Gin Ts
Goi & Ts

THE 9 + iE

I; ©) + T

JE Goh E

OT Jh E

OT SG E

O- € I

rae + TE
TE + T

+ it

OT it

I ils

at il
alas rE

SUOTPEUTqUOD aTqQetre, Juepuedeput

aut}, @ 4e
SX. XTS

out} @ 328
SX eat

awty e@ 4e
sx mog

awry e@ 42
SX eer,

aut, e@ 42
SX OM],

JO suoTtzeuTquog

*T potdteg Seq Joy ‘sotqetze, yuepuedepuyt sATemy,
Tedeaaes Jo yoey 09 eTqe4nqT144y Serenbs jo umg

adoTg au0z avAemM=SutTeoyg ut suoTjZoNpeYy 4ueg Jeg yseBuor4g eATY our-*J2eq etaeL

79

€9°S6
€g°S6
Gg°S6
19°S6
69°S6

90°S6
gL°S6
gl°S6
€9°S6
G8°S6

al
ot
ol
as
ot

as
ot
oL

ot

$$ UP uoTZoNpeY
queoreg

Eee

"2 potseg Seq toy ‘setTqetzre, quepuedepul sATeaM],
jo suoTyeutTquog Tezeneg jo yoey 04 eTqeqynqT1z44y Sezenbg jo ums
adotg auoz aAemM-SutTecug UT suoTyonpaey queg Jog 4sesuor4g SATA SUL-*gcd FTAPL

WE
TT
EE

TL
TT

OT
OT
OT
OT
OT

OT
OT
OT
OT
OT

OX OV ON ON ON ON OV OD OD ON

RRR REE Y

‘oO oO

\o

iN LLIN

tats ao

tata

mma MmM~AMMY

UNAUNAN

NUNN

Gal teal Ga) Ga) Go

Gali taktal ta) tal

SUOT}SUTQUOD STQeTTe\ JUepuedeput

awry e@ 7e
SX uaz,

aut4 e 4e
SX 8UTN

66 "6
1a 46
HS “H6
0°S6
56

c9°€6
4qL°€6
Z0°H6
GO°H6
€f*H6

19°16
19°T6
T6°T6
€6°T6
th £6

S0°68
Bt 68
89° 06
416° 06
To‘ T6

To'9g
HT"98
Lt "98
64°98
oe" 1g

80°08
TH" 08
ge°28
S8°€8
68°48

2g°TL
ore)
gt’ EL
go's)
0S* LL

as
ot
cL
ot
ol

ol
al
ol
ol
oT
as
ot
ov
ol
as
as
ot

all

ot

ot

oT

SS UT uoTIONpsy

queoreg

HE OL

°
3 &
ANN BWA AQ
heh
UN LN LY LIN LN
Pep Sep
anauaa
ddd Adda

TE Of 6

°
4
psp
m mm MAMMAMNMY MAMMMNM

Added

EL 6
IT 6

Delve loe}
NOW
ded

re OE 6

°
q
Speers Sp Speapsp
au
ddd¢4

Tt 6

TL 6
OT

=)
ad
titted

1E
OT TE
T
TT
TT I

SuOTJeUTQUOD ATQeTIeA JUapuedepuTy

auTy e 4e
SX qUsTa

aut) @ 4B
SX uanAeg

out, eB 4e
SX XTS

aut} 8 42
SX OAT

our, @ 4e
sX moj

aurTy @ 4e
SX ery],

SS
*g potded Be] Joy ‘saTqetxze, yuepuedepuy aaTany

JO suoTzeuTquion TeteAsg Jo yoey 04 sTqeqnqtaz44y sexenbg zo umg
adoTg auoz sveM-SutTeoyg ut SuoTyonpey Jue Jag 4ySeSucrzg aaTy auL-"ged eTaeL

80

Z€*T6
GE°T6
gE°T6
4S°T6
Gg°T6

92°06
92°06
TH’ 06
95°06
+T°T6

12°68
62°68
G68
£9°68
G9" 68

or €g
g6"€8
05°
98°18
8°98

98°69
02° Ol
49° EL
TT'nh
et TS

al

oL

oT

oT
oT

SS UT uoTIONpeY

queoreg

W))
4

HE OL
TE Git
TE Ole
HE. OL 6
GE OL

fee) cc <O Oo
J Gl fl

EE 6
THE OE S
EE 6

TT 6 L

Te. OL 8

14

I cal tl

EE OE

TEE 8

11E L
TT 6

TEE

at at abet tattoos ata st at

Ga to Gal Gal col

Et 4

IT S T

It 6 T
ot tt T

It q

It

TT E

SUOTJEUTQUOD eTqeTIe\ JUepuedepuy

auty @ ye
SX xTS

aut, @ 4e
SX eat a

aut @ 42
sX, Mog

auty e ye
SX ddTU],

aura @ 42
SX OM],

‘tH POTZeg Sey coy ‘satqetsrea quepuedepuy oaTeny,
JO suUOTJeUTQUOD TeZaAag Jo Yyoey 04 aTqeqnqtz94y Serenbg Jo umg
adoTg eu0z sAeM-SuTTeoyg UT suoTJoNpeYy yuaD Jeg yseSuorzig eat aut-"ofG eTaeL

66°68
00°06
€0°06
80°06
€T°06

92°68
12°68
6£°6Q8
69°68
€g°68

66°98
16°98
TL*18
€6°Q8
TT'68

T6°H8
16°h8
9E°S8
S2°98
26°98

ThH°oL
gL eh
tO)
L1L°QL
€L°1g

QS UT UoTJonpsY
queo reg

ot

EE OL

TE ONE

at at at at
NU
Gal Galital tal ca]

Tt OT S
TE OL

Cot) it) Get) Goh tI

TE OL

q
4

St Spep Sper ep
a

tl
4d
uN
om
Gel Gal Om) ee) Go) AddAodd

WwW
AddAdaAe

TELE

SUOTJEUTQUOD STQeTreA JUepuedepuT

°C potazeg Sey rox ‘setqetze, quepuedepuy eATeny,
JO suOTJeUTQUOD TeZeAeg JO YOey OF 9TQeInqTz49y Sertenbg jo umg
adoTg saucy aAeM-SuTTecys UT suoTyJONpeY yquaD Teg ySeBu0rmqg oATA euT-*6cg eTaeL

out) B@ ye
SX xTS

auty B@ 48
SX AT

outy @ 42
sx moj

out, @ 4e
SX 9eTU],

aury @ 42
SX OM],

8l

61°96
08°96
06°96
26°96
20° 16

ot °96
81°96
0S °96
19°96
98°96

SS UT woTyonpeY
qusoteg

ol

ot

ot
ot

*G potseg Sey toy ‘saTqetrep, quepuedeput aaTemy,
JO SUOTJBUTQUOD TezZaAsg Jo YOey 01 eTqejngqtz44y Serenbs jo umg
adoTg auoz aAeM=SuUTTeoYg UT sUOTJONpeYy yueD Jeg ysoeBuorI4g SATA SUL-°TEA STAPL

TE KOE
EE OE
TE OIE
EE OL
TEE OE

Ee OE
Ts OE
TEE ONE
Tee OL

ON OD) OV ON ON OV ON ON ON OND

cc

BReRERR BREED

\0 10 10 \0 ‘0

9
9
9

LY LN LN UN LN LN UN LN LN

at at —t st Faas

mmm

€

NO

bal ta) Ga) Galital

ial a Gal cab

SuUOT}eUTqUIOD eTqetTIeA JUepusedepuT

aut @ 42
sx usz,

aut, @ 42
SX eUuTN

66°66
90°96
20°96
82°96
€£°96

95°56
Q9S°S6
65 °S6
69°S6
TL°S6

10°S6
10°S6
92°S6
HE°S6
6£°S6

€€°H6
6 °H6
+S °H6
9L*t6
G6 *t6

62°6
TE*26
96°26
T6°€6
Go"46

go: Lg
9T" Lg
1S°99
02 * 06
0S *T6

a EL
etHh
oL* LL
G6°TS
T9°€8

ol
ol

cL
ae

ol
ot
as
al

ot
oT
as
oL
as

ot
al

as
ol

as

oT

as
ral

oT

ot

SS UT uoTJONpeYy

queoteg

TL
TT

OT
OT
Oot

OT

OT
OT

OT
OT

ot

OV ON OV OV ON ON O’ OD OD ON OV ON OD OND ON O’ OD OD OD ON ON ON ON OD

\O \0 \O

UN LY LN

LN UY LN LN LN LN LN UY

mm mM ~m mmMmmM

SUOTJVUTQUOD STqeTIeA JUepuaedepuT

Gt falta ta) tr ial Celt) Gal Go) tal tal fed tl Adddd

Galt) ta) tal tl

Tre

aut, 2 48
SX JUSTE

aut} e 4e
SX uaAeg

owt e@ 98
SX XTS

aut, e 4e
SX aT

aut. e@ 42
SX mog

aut4 e@ 4e
SX s0muy,

aut} 8 48
SX OM],

——————
"G potteg Sey Joy ‘satTqetsze, yuepuedepuyt eaTamMy

JO SUCTICUTQUIOD TeaAeg FO YOey of eTqeqnqtaz4q9y serenbg jo umg
adoTg su0gz aSAeM-BuTTeoyg uT suotqyonpay quap Teg yseBuorg aaTa out--Teq etaes

82

Z0°T6
10°T6
To°T6
4e°T6
19°T6

62°98
10°68
G0°6g
8t°68
TO"T6

SHH
49 “418
98°48
TS°Sg
of" Lg

ST°9L
2o°9),
€9°9L
et 6L
Gg°eg

LL*€9
95°59
29°89
€6°99
go°sh

SS UT uoTZONpEY
queoteg

as
oL
as
ot
oT

as
oT
ol
oT
oT

oT
or
cL
oT
oT

oT
oT
as
as
oT

al

TT
TELE

OT
OT
Ov
OT
OT

OT

OT

OT

ON ON ON ON ON

ON ON ON OD ON ON ON OV ON

tats

tata

SUOTJSUTQUOD OTGQeTIeA JUepuedepuT

aut @ 42
SS SX XTS

outa 8 42
SX OAT

~m

aut @ 42
sx moj

aut @ 42
sX 90rU],

auty @ 42
SX OM,

*d potted

eT Joy (PesN FON TX SeTaetre,) seTqetze, yuepuedepuy ueseTy go
SUOTISUTQUOD TeteAes jo yoem 09 aTqeqnqt1z44y Serenbs jo umg odotgs
aUOZ SACM-BUTTEOYUS UT SUOTJONpeY quUeD Jeg yseBuor49 eATA euT-°C€a eTaeL

€o'Hg
€2°Q
6L°t8
gL°Sg
99°98

dH 61
€L°6L
25°08
GL 0g
66°28

6e°EL
On HL,
glihh
T6°SL
GEL,

Sg °6S
g0°T9
€€°T9
8°19
4S °S9

TS°€S
9S°€S
06°HS
T8°9S
6H °9S

SS UT uoTJoONpeY
quooteg

oT

as

BIE
TT

TT

OT
OT
OT

OT

ocaoadana cocacamaa

RRR RRe BEER

\0 \0 \O \O \O \0 \0 \0 \O \0 10 (0 10 \O

LN LN

LN LN

+t —t at —t i es an tata at tata

aut} @ ye
SX xTS

aut, e 42
SX eATy

auwty @ 42
SX mog

SZ out, @ 48
SX 907u,

aut @ 42
SX OM,

SUOTJSUTQUOD STqeTre, JUepuedepuT

"T potted

Bey Toy (Pas JON TX STqetse,) setTqetze, juepuedepuy ueseTy
gO SUCTZeUTQWUOD TeZeAeg JO YoRy 04 oTgQeyngqt449y Sezenbs jo umg
adoTg au0Z SABM=ZuTTeoug UT suOTJONpEY 4UaD Jeg 4SeBu0r49 aATA auy-"Zeq etaez

83

82°88
82°88
9E°88
09°88
06°88

0€°98
0£°98
3°98
at "98
12°88

gE°n8
ot t8
a8
59°18
10°98

TH
9S°hL
9e°Sh
65°SL
g6°€8

90°19
Ly °T9
et 19
98°69
49° EL

S§ UT uoTJONpSY
queoteg

or
ol
as
ot
as

ot
oT
ot
ot
cL

ol
oT
as
as
ot

as
oT
ot
ot

ol

1615

OT
OT
OT
OT
OT

OT

ot

OT

UN LN LN LN LN

UN LN LN LN

—t tatoo at ap at at tatoo

SUOT}SUTqUOD STqeTreA JoupuedepuyT

aut @ 4e
SX XTS

Zz auT4 8 428
SX eATY

owty @ 42
sx moj

out, e 42
SX 9erUZ

emty @ 328
SX omy,

*t POTded

Bey OJ (PesM ION TX eTaetzen) seTqetrea quepuedepul usaeTY jo
SUOT}eUTQUOD TeZeaAeg Jo yoey 09 STQeynqTz44y Serenbg jo umg sdots
auoZ oAeM-SuTTecyg UT suoTyonpey 4ueD Jeg yseSuor4g eATA euL-"Sfq eTaAeL

ger LL
ger LL
16°LL
99° LL
gre

6g°S)
96°S)
geo)
6S°9)
derh)

Heth
T9°Th
g0°eL
Teel
TS°SL

€1°S9
€2°99
SE°99
Gg°69
6T°OL

te" 09
LE°09
T2°19
29°29
ls°€9

SS UT uoTJONpeY
quso ced

oL

TL OL
EOL
TE OL
we OL
HE OE

EE. IE

TE OL

\O 10 10 10 \O \0 \0 10 \0 10

\O\O

UN UN LN LN UN UN LN LN LN LY

uN

tat ast att at at at att tata

at as a

q

ro auty 2 42
SX XTS

Z aut, @ 42
SX aATZ

Naw

out @ 4e
sx mo,

aura @ 48
SX de.TUI,

aut, eB 48
SX OM]

SUOT}SUTQUOD STqeTre, Juepusdepuyt

°€ potted

Se] OJ (Pesn FON TX eTqaeTseA) SeTqeTse, quepuedepuy ueAeTY Jo
SUOTJEUTQUOD TeTaAsg Jo yoRem 0F aeTqeqnqtz49y Sserenbg jo umg adots
auoZ SAVM—BUTTeOYS UT suUOCTJONpEY quUeD Jeg yseSuor4g aaTyT suT-"yCq STABL

84

€6°he ral
EES ot
GS°2e at
06°T2 Al
gh °9T ot
40°ST An
GL eq et
O€°€T et
9S°2T et
+t °oT ral
got ran
oy° OL et
€T°OT

9S°H et
G6°E ral
g6°E

gg°€ et
lg9°€ et
Th’? ot
6°T

€6°T

Z6°T

OT

T2°T Al
GLO As

SS UT UoTJONpEY
quedreg

Comeolice)
Linin
rat

Z out, & 9B
SX XTS

ee REEL
Nolo)
LN UN
”

OV, OV ON ON OV ON ON ON

\O 0 \O
ID IAIN iN
Nn

€ aut) @ 48
€ SX OATd

cee Oh
Ma

aut) @ 4e
sx mog

2 out, @ 48
cA sx s974]

LN LN UN TaN LN LN LN UN LN
nN

\O \O \o \0 \0 \O \O 10 \O

LN LN

€ aut, 8 42
Z SX OM,

SUOTJEUTQUOD eTQeTTeA JUSsPUedepuy

‘T potded Sey toy ‘setqetzep, yuepuedepuy
DATAM]T, JO SUOCTIeUTQMUOD TeZeAeg jo Yoey Of eTgeqnqtzz4y sertenbs
gO ung sdoTyg eu0Z aAeM-SuTTeous UT suoTyJONpey queD Jeg 4yseweom-"JET eTAeL

69 °T6
+L°T6
Te°T6
T6°T6
g0°26

12°16
lz*T6
92°16
ZE°T6
19°16

TH 06
05°06
65°06
96°06
QI 16

te°og
ZE°2g
GH°SQ
9T* 1g
0g°06

go°eg
66°29
gS °S9
69°89
G6°TR

SS Uy uoTYoONpeYy
quaoteg

TE OU
EE OIE
Tl OT
EOL
TC OT

EE OL
WE OE

HE OL
TE ONE

LOIS

Os

LN LN LN LN LN

SUOTZeUTQUOD STQeTTEh quepuedaput

at ap at st

aut} @ 42
SX XTS

Gn) GAY GA)
Ca iat}

aut, @ 42
SX eATY

out} e@ 4e
SX mMog

tat ast tat st at
N

aut, @ 42
+ SX eeTUy,

4H aut} © 42
SX OM]

°G potted

seq Loy (pesn FON TX aTQetzeA) SeTqetre, Juspusdepuy usaeTyY jo
SsuoTZeUTqUION TeteAseg jo yoey 09 eTqeqnqtz44y sezrenbg jo umg edotg
SUOZ SASM=BUTTeOUS UT suOTJONpSY queD Jeg yseSuoujg oaT yA eUL- "OCA POTABLE

85

41°68
€T 6
22° lg
26°98
29°98

zo'€g
TS*2Q
ge"eg
o£ °Tg
18°08

oe.
eo.

oL

oL
ol

SS UT UuoTZONpaYy

queoteg

TL 6 9-19 & R §€ &

LL Gg 1 9 Gy SS t

bE © 6 GY As & at

He of 6 GY Fo 6 ws ag

TE Of 6 @-1.9 & € ile

TT 6 Gg +9 S-h § g

It OL gj 1 @ © € T
Oo 6 9 4 9 G € 1E

He Oo 6 @ 1 ® & i &

EE OF © G LO GS i Z

SUOTVEUTQUOD STQeTIeA JUepuUedepuT

our} @ 4°
SX uaz

out, @ 42
SX ouUTN

‘Se potieg Sey soy ‘soTqetae, juepuedepuyt oaTemy jo
SUCTJBUTqUION Tetavsg Jo yoeq 04 aTqeqynqtsz44y seazenbg go umg adotg
au0Z SABA-FuTTBoyg UT SuoTJONpey quep Jog 4seyeoM OATY ouL-"gla eTaeL

ano.

€9°S),

1g°eL et
TL°eL et
on°99

69°49 et
00°H9 ot
OT*€9 ot
gL°T9 et
2l°TS ot
4€°0S

TL "6h eT
g8°sh ot
le°Sh a
Gorct

TO" LE

9H SE

T6°E€ oL
SUEOwWS ot
66°92 er
€0°S2

€6°02 ot
€L°9T

6E°ET

9S°eT

64°ST

16°6 cil
GT°6

6e°L

12°g ra
oth

16°9

+19 a
98°0

98 UT uoTZJONp|Yy
quaoteg

IT OT Lo & iT @
oogiig gS wwe
o 69 L9G i @
OT Q@to gS wy Ss oury @ 4@
OT 9 L9 Gx z SX JUSTH
OT 6 Log q zZ
Ob 9S G q
OT =O) an @
OT I OY -G ane & outy @ 48
Qh @ & iq rj SX UsAag
8 9G ip &
OT I, GY Ge ay Z
8 9 on g
ye) Gan & out @ 42
I, OG ap @ SX XTS
OT oO GS Ty &
9 & wy &
ko ax Z
OT I, 9) € ourTy @ 42
OT A; ©) z SX oAT A
LAO €
oT 8 9 tq
jh) z
OT 19 € auly 8 48
OT 8) rej sX Mog
OT I &)
OT 9 €
8 9
OT 9 Z out, @ 42
19 Z SX seru],
8
JL Z
9 Z
9 aut? @ 78
19 SX OMT

SUOTYBUTQUOD STQeTIeA JUSpuedepuT

*Z potmeg Bey wos ‘seTqetre, quapuedeput aaTemy, jo
SUOT}eUTQUIOD TetaAeg JO yoRy of aeTqeqnqtz49y sarenbs jo umg adots
UOT SAVM-ZuTTBoyg UT suoTyonpay yuep Jeg yseyeam OATT suL-"g@la STABL

86

16°ST

98°ST

€9°ST et
1g °+T et
9gE*HT et
ASME Al
GO°IT et
€L°6 et
11°6 or
9n°8 et
GL°9 2
Bn'9

412°9 ot
g's

To"t GAL
90°H cal
9g°€ et
9g°€ et
es er
Gere

66°2

46°32 et
€6°2

QS°2 et
We et

QS UT uoTZONpEY
queoteg

ot 6 8 9 G zZ
ot 6 19 S @
6 19 SG €
6 I; OE 2
6 8 9 S zg
Ot g 9 SG
6 9 SG Z
19 S €
8 9 SG re
I, Zz
8 9S
jG) -G €
9 Ss €
19 S ro
9 S ro
8 9
9 S
9 re
S 2
9 S 2
S g
8
9 Z
9
2

——_—_— ees

SuUOTJeUTQUIOD eTqetsrepA Juepuedeput

auty @ 7e
SX XTS

awry e@ 4
SX oATT

awry @ 42
sx moj

auty @ ye
SX set],

auty @ ye
SX OM],

ae eee — —

"ty POTred Be] 10s ‘satqetze, YUepuedeput
AaATAM], JO SuUOTJeUTQUOD TeTeAeg Jo Yoem OF STQeInqT4144y sezenbg

go umg adoqtg au0z anem-SuTTeoug UT suoT}JONpey 4UeD Taq ySeHeoM-" OHA STICL

HO"EE et
On°ee @IE
HE EE et
qL°Te ral
8c°Te GE
Ge LT ZT
0g°9T

92°ST

€9°€T Al
On eT Al
SESE ran
6 °IT

69°TI

EEE 2
T6°6 ral
gc°6 ral
ELS

LES

80°8 Ge
26° 2
6£°S

06° ral
€6°€

OT'€ 2T
Ger?

99 UT UuOoTJONpeY
queoreg

*€ potseg Sey roy ‘seTqetze, yuaepuedepuy jo
SUOTJSUTqUIOD TetaAag JO yoey oF eTqQeynqt444y Sezrenbs jo umg sdotg

OT
OT
OT
Ot

Ov
OT

ov

SUOTJEUTQUOD OTQeTIe, JUepuedepuT

c 00 Oo co coada

co cc

cco

el el So

LN UN UN LN LN LN LN LN LN

LN LN UN UN

+ —+—+t

mo

mm

(aUiat}

ANA AW

Lat}

i

aut @ 42
sX xTS

out, e@ 42
SX eAT I

aut e 42
sx mog

auty: e ye
SX derN,

auT4 e ye
SX OM],

SUZ SASM-SUTTeOUS UT suUOTJONPeY 4ueD Jeg YsoyeomM SATAY euT-“6Cq eTaeL

87

6£°S8
60°S8
H6°H8
46°€8
dye LL

6L°SL
9S°S)
TLheth
cert
00°S9

oT
oT

ol

oT

QS Up wot oNpeY

quaodeg

TT

TT
TT

TT

OT
OT
OT

OT

OT
Ot

OT
ot

ON OV ON ON ON ON ON ON ON

0 cc ~ leolcoKoowooyee)

REE R BREE

\0 \0 \O \0 \90 \0 \0 10

\o

UN LY LN LY LY LN LN LN LY LY

i i i a ata at at

MmMMAOAM mo 9 ~m

NUNN A

NUN

SUOTZBUTQUOD OTQeTTeA JUepUaedepuUTy

°G polteg Bey roy ‘“soTqetsze, quepuedeput eaTamy, Jo
SUOT}JEUTQUOD Teaeanag Jo yoey 09 eTqeqnqtaqyqy serenbg jo umg adots
aUuoZ SABM-BZuUTTeOYS UT SUOTJONpEY 4UED Teg JseyeoM ATT SUL-“THA eTABL

our, e@ 4e
SX uaz,

owty e@ 48
SX SUTN

89°29

19°19 OT
gE ES OT
er°TsS OT
96° Lh OT
60° Lh OT
TS*9h OT
€o°Sh OT
98° Or ot
88°gE OT
6H°QE ot
6E°HE OT
gL°TE OT
26° 0€ OT
oN ot

oL*€T OT
T9°ET OL
90°€T OT
29°eT O-
HT°TT ral

ZO*TL

GT°OT OT
Z6°H

20°d

Hen

96°T

GQ°T

GLiT

T9°O

€9°T

90°T

GS°0

GS°0

LE°0

Sg UT uOTJONpSY
quaorag

*¢ potszag Sey rox ‘satqetxe, quepusdapuy satemy, jo
SUOTYeUTQUOD TeaAsS JO Yyoey of aTqeqnqtz49y sexenbg go umg adots

ON OV Ov ON ON OD OD OF

OV OV OV OV ON ON ON ON OVO’ OD OD ON ON ON ON OV OV OV OD

pe) cc c< ccc OO ccc OO

feeyieo) oo)

foo)

ccc foe) foovoo}

8

See

i

eh heh

eh

\O \/0 (0 \0 10 \O\0 10 ‘0 10 \O 9 (0 \0 10 \0 \190 (0 \0 ‘10 \0 9 \0 10 \0

\O \0

LN LN

LN LN

UN LY LN LY LY

LN LY UN LY

tat at tas stat

at —t

mmm

mmo

mmm ~ (sa)

NAO

NW

SUOTVEUTQUOD STQeTIeA JUepUsdepuTy

ourTZ @ 4e
SX 4USTH

aura e 42
SX ueAeg

aur e 9e
SX XTS

ouwtZ e 7e
SX eATT

aura @ 3e
SX Mog

owTy e@ 4e
SX 9074]

aut, B 4e
SX OM],

auoZ SAVM-BUTTeOUG UT SUOTJONpeY queD Jog JSoyeoM OATT SUL-"THA eTABL

88

%26°S6 = UOTJONPEY SS TeIOL

99 Occ S6h 26h +26 c6L S6_h Oce 99 aT Kd
WE Y Z 6°66-6°S6
Gry se Whe oor we & 6°76-0°06
9 29 Tle 6c 961 © TT 6°68-0°S8
Al 66 SSe tle OST Oh S&S 6°H8-0° 08
Ot QOL g9¢ OS2 QTL ce 2 6°6L-0°S)
zg ws - 1 — Gee eg Se 6°4L-0° OL
7 wy OL ds Ge. 6 6°69-0°S9
7 @ Ww g 6°49-0°09
G me @ Se 6°65-0°SS
€ GE 4 — TE aR 6°4S-0° 0S
+ Ge aS- Gr Ts 6°64-0°St
6 = Ge GE 6*HH- 0° OF
2 6 cS 6°6€-0°SE
2 ee Je g u 6° E=0° OF
Z t 2 6°62-0°S2
T 9 + E 6°20" 02
i Z Z 6°6T-0°ST
cee More ee SE 6*HT-0° OT
€ dh + 6°6-0°S
E ro 6*+70°0
G9 Ue) 6 9 8 a % SseTo

soTqetzej juepuedepuy jo zrequmy uoT}ONpey Sg

(2 POTZed Bel) sX GT SA euoZ eAeM-DuTTeoys eTsuy edoTs-"€rq eTaeL

TL

ot

soTqetze, Juepuedepuy jo Tequmy

%ST°€6 = uoTJONpey Sg TeIoL

26) G6h O22 99 eT Zz
Z 6°68-0°S8
us € 6 "48-0708
gzezq 69 G 6°6L-0°S)
Cine (3 “Ge — § 6°4L-0°0L
Jy CSS pe «CS 6°69-0°S9
6 68 Ge G6 6*49-0°09
ig? 6 mW 2 6°65-0°SS
eS CE Ce 9g E 6°HS-0°0S
ns 1. ©) 6°64-0°Sh
Ole6 9 T 6° 70° Oh
Te 6 € 6°6E-0°SE
ew ar A 6°HE-0° OF
O§ 2. G zg 6°62-0°S2
ww We OC v 6°t2-0°02
SE -@& mw SG 6°6T-0°ST
q €2 96 Gt & 6*HT-0" OT
ES eh ee 6°6-0°S
T J, we A, 6*t-0°O
Got. MG ee — an 4 sseto

woTjonpey $s

(TI potaed 3eT) SX GT SA auoZ eAeM-SuTTeoys eTZuy edolTs-"cha eTaeL

89

926°G6 = woTZONpeY SS TeIOL %2Q"z6 = voTJONpEY gg TeqOL,

qT6 c6L S6h Oce 99 eT g HT6 c61 S6_ O22 99 aT gz
Gy 6*t6-0" 06 4 6°H6-0° 06
glT 68 @ + 6°68-0°S8 Sey Sse gs 6°68-0°Sg
te GL qe Gee 6"+8-0° 08 a 4 i Ge t 6*48-0° 08
€92 coe 99 SG 6°6L-0°S) We ioe WS Ge @ 6°6L-0°S)
i, hee Coe Gs ¢ 6*4L-0°0L WS th On Ce @ 6'tL-0°0L
62 «6€ Gs we - JL 6°69-0°S9 0) 6 Gd iw © T 6°69-0°S9
oe @. JE ce 9 —t 6*t9-0°09 Suk GEE 1) GE 9 6 "49-0" 09
ae 6 IE zZ T 6°65-0°SS oe Ge 6 1 + T 6°65-0°SS
ke org 6°S-0° 0S Th €D 6°HS-0°0S
gt eT Ce 6°6h-0°SH OS Une mW © 6°6H-0°SH
‘Ww Ge 9 ~¢ 6° -O° OF gs - NIG Je 6°HH-0° OH
cy Ot le 8 T 6°6E-0°SE Go. Gy. GS OF =F 6°6€-0°SE
re) as re 6*HE-0°0€ 4 MS 05> Ge 6*t€-0°0F
Oe ss Sr —€ 6°62-0°S2 GE GS. Gay 6°62-0°S2
6m 22 tt -_3 S T 6"t2-0" 02 ro L we Ce © il 6°HS70" 02
GE ats we J 6°6T-0°ST G Ge we 6°6T-0°ST
Oo @ tt a 47 6*HT-0" OT 3g Ge Ge 6 6*HT-0° OT
€ ke. US. Ge 6°6-0°S e @ -Ge y 6°6-0°S
Z 9 sie, 6°-0°0 T + S 6°4-0°0
OG. y= IS eae SE SSeTO Se Geet Ce chem % SSBTO

soTqetzep, JUuapuedeput jo sequmyy uotyonpsy Ss soTqetzea YUepuaedsepuy Taqumyy uoTyonpeY. Ss

(4 POTteg Be) sx ZT SA uz eAemM-BuTTeous aqtauy edotg -"Shd eTaeL (€ potzed BeT) sX ST SA auoZ aABM-SuTTeoyg eTsuy adoTs="hha STABL

90

HET LE = voTyonpay gg TeqoL

99 022 S64 c6L 426 c6) S6_h Occ 99 eT gz
GE GE GR. 1S 6°66-0°S6
61 WE Gone OS (fers ke) (SES 6°t6-0° 06
6 US 65 65 Ge sori iy 6°68-0°S8
Z ds et ie Siu Sor aS 1 - 6°t8-0°08
ie 8 Qe ws Gh 95. Cee iy a 6°6L-0°SL
2g SS. Jor Oe ire cy I G 6°hL-0°0L
iE & ob 68 6G) & Gt 9 6°69-0°S9
i aw GS € & Ge -& 6°49-0°09
g Go Ce EEE 4h Tg 6°65-0°SS
zZ 6 @3 6 at - 6°HS-0°0S
9 gw dS. C8 -OE TE 6°64-0°SH
¢ Gy GE we 6°th-0° On
S oe is he 6°6E-0°SE
It -CB- e— © 6*HE-0° OF
9 wa ) 6°62-0°S2
T € OF a oO. 6*42-0° 02
jie ate onc, 6°6T-0°ST
Ge os 6 TE 6*tHT-0° OT
6 S 2 6°670°S
Z Ge. 6 G 6°+-0°0O
Oo - 6 § FT 9 ¢ © 8 F % sseto
— seTqetzeA quepusdspuy seqimy | uoTyonpeYy Sg

(S potted BeT) sX ZI SA 9u0Z sAeM-FuTTeoug a TSsuy edoTs- gra eTqeL

9|

TT

A ABRB
BARRA

=

€6°2g ae TE
HO°ng ae ee

HE°TS at Ee
96°L9 Ge ee

99°2) et
"el TI

SS UT woTZONpSY
queoleg

OT

OT

OT

RRR RRR ERR BREE BREE

LUN LN UN LN LN

Tey LN tay UN LN LN LN

tats

at

tas

i.e) (oa) MmMOH

AddAde

da qd ul ood hit) tH

dodidd

da

iD

SUOTJEUTQUOD OTQeTIeA JUapUadepuUT

*g potseg Bey Joxy ‘saTqetre, yuepuedeputy
SATEM], JO SUCTJSUTQUIOD TeteAsg Jo yoey 0F aTqeqnqtsz49y sezrenbs
gO UMS S8ZTS UTeLH UeeW UT UOTJONpeY queD Jeg ysesuor4g sATY SUL-"QHaA STASL

outZ @ 4e
SX xTS

our, @ 42
SX eATH

aut, @ 4e
SX moj

owty e@ 72
SX seruy,

awty @ 42
SX OM]

6£°H8
THB
99°18
62°S8
€4°S8

89°28
29°2Q
cee
09" €8
TE*t8

€9° 08
86°08
Z0°TS
LI°Tg
8e°e8

ge"9l
TH'9h
96°9)
GT°6L
9L°6L

09 °S9
19°S9
18°99
T9°ol,
9T°Sh

SS UT UoTJonpey

queoteg

as
ol

ot

TE 6 yy; + T
IL 6 L a we
II 6 I, 9 T
TE 6 @ J E
TE Oe 6 Ih is
Te Oe 6 iis
TL 6 S aT
TL 6 9 iL
TE OE 6 L T
iE 6 A, T
THE + rE
IL 6 J E
IE L TE
THES 9 iE
Tae 6 ils
6 L E

4 TE

ia L U
6 iTS

Tt Tg
S T

6 I

9 il

L E

U

SUOTZSUTQUOD STQeTIeA JUepusedepuT

*T potaeg Bey tox ‘satTqet.re, yuepyedepuy
SATSMT, JO SUCTZeUTQUIOD TeIaAeg Jo YOR 09 aTQeqnqTz44y sazrenbs
gO UMg OZTS UTeIH UeaW UT UoTJONpeY quseg Jag qyseSuorzqgg aaTy out-")_ya eTAeL

awry 8 4e
SX XTS

awty e 42
SX oATT

awry e@ 428
SX mog

aut @ 3e
SX derTUL

owt & 48
SX OMT

92

00° €6
To0°€6
40° €6
He" 6
cE °€6

10°16
o£ °16
gn T6
11°26
69°26

65°68
61°06
02°06
92°06
gE° 06

99 “8
T8°H8
19°48
46°98
g£68

GT°€S
Zo°nS
+Q°SS
68°SL
ge°es

$$ UT uOTZONpEY
queorteg

eee ee SS —

ol
oT
er
al
ot

ot
ol
ov
ol
oL

ol
oT
ot
ol
ol

ot
Aas
as
oT
al

ol
ol

8 9 Ss € il
ot 8 9 € T
8 9 q eT
g 1 9 gE
@ 9) € E
19 € ic
) S et
L S € T
8 9 eT
8 9 € T
8 9 T
j q T
19 ie
Dh, a
L € rE
TI T
S i
+ I
9 t
L iL
€ 1!
BE T
Ot oT
Ta
il

——————————————————————

SuOT}eUTqUOD STqeTze, Juepuedepuy

* potted Bey toy ‘setqetze, quepuedepul
SATOMT, JO SUOTYeUTQUION TeZeAeg Jo YoeT oF eTGeInqTz44y serenbs
Jo mg ezTs uTedy weay] UT UOTJONpey qUeD Jeg yseBu0r4g SATA OUL-"OGd STISL

ouTy @ 42
SX xTS

aut}, @ 42
SX eATT

aut @ 4e
sX mog

aut e@ 4e
SX eeTy]

aut @ 4e
SX OAT,

TH 96
4°96
GL°96

g2°g6
92°86
+1°96

GT°96
82" 96
Ze*g6

6€°16
16°16
21°96

90°96
94°96
16°96

60°46
HE H6
69°46

25°26
64° £6
9S°€6

61°98
15°98
61° 06

0g 62
62°08
St" t8
6T"t9
61°69
99°eL

S$ UT uoTJONpeY

qusoreg

GE EE OF .6 9@ 2-9 § 7 Came
Gr oc CG Lg & 7S SE
a tme.@ 6 @ 1 Oo € oT
ot 6 G29 & ye So
GE WE 6-9 Lb 9 GS Sg
ot o 6 9 b D> G C-@ t
er 6-9 L9G @ AE
at 6 9 1.9 & S-@
GE or 6g bo & GU
ot 6 9 2 4 € Boe
er 6 @ 1 2 © € T
et 6 @-49 @
er 6g 2 Gh T
ot 6 g 2 S € T
CALE 6 g 2 S ao .
ot 6 9 S a
et 6 8 G € T
ot 6 8 G oS U
RAE 6 + eo
ral 6 S € ii
eL 6 S Cans
ot 4 eo t
ot 4 € if
ot 6 4 il
et eS U
ot € i
at G ils
OT is

Al T
G E

SUOTJEUTQUOD STqeTre, JUepuedepuT

*€ potdeg Seq tox ‘setqetre,j yuspuedeput

SATOEM]T JO SuUOTYEUTQMOD TetTeASg Jo Yyoey 09 sTqeynqt1z44y sezrenbs
gO UNg eZTS UTeIH UeaW UT SUOCTJONPEY queD Jeg yseSuorzg eeLU], eUL-"6ya TAP

out} & 42
SX usAeTE

aut, @ 42
sX uey,

aut, e@ 4e
SX eUuTN

out} @ 4e
SX USTa

awry B 4e
sx ueaeg

auty & 4e
SX XTS

aura e 42
SX OATY

aurty e 4e
sX mog

auty e ye
SX seTUI,

aura @ ye
SX OM],

93

16°62)
2g°6L
18°62.
SE*08
68°08

ante,
6o°hl,
Gheqd
eehh
+T°6),

To°tL
€T°th
Te°tL
THEL
49° EL

0S°69
29°89
86°89
£2°69
92°69

GT°LE
LL LE
eth
86°29
0£°S9

SS UT woTZoNpey
qusodeg

ot
ot
ot
as

oL

ol
cL
ol

ol

as
ol
oL

ot
as
cL
oL

ot

TE 6 9 t @

It 6 9 GS

Tl 6 19g t

Tt 6 8 9 4 out, e@ 48
ie OF 6 9 + SX XTS
TT 8 9 I

EE 6 19 4

TE 6 J; &)

TT 6 G4 aut, e@ ye
BE 6 9 t SX daATT
TEE J; 9)

HE OE © L

Tt 6 9

IT 9 4 aut} eB ye
ABLE 6 1 sx mog
TI €

TT 6

Tt dh

TEE 9 aut e@ ye
TT 6 L SX 9erq],
TEL g

Tt S

EE 9

TT L aut © 4e
TEE SX OM],

SUOTZEUTQUOD STqeTTeA JUepUsdepuUT

*T potted Be] Tog (pesn JON TX eTaetxe,) ‘seTqeTzea yuepuedeput
UsAeTY JO SUOTZeUTQUOD TeIaANg JO YOR 0fF oTQeqnqTz49y sezrenbg
gO uMg eZTS UTeIy UedW UT SUOTJONpeYy 4ueD Jeg 4yseBuor49 aeaty euzT-"*2Sq etaeL

TT°1g
€o°lg
29° lg
gT°8s
66°88

TE°eg
of "HB
ehmais)
91°58
TE*98

T6°6L
80°08
03° 0g
TE*TS
€s°€g

TH°OL
qe Lh
T6"LL
LL°gL
62°62

1S°9S
gt° LS
00°9S
6h°od,
09°Sh

S$ UT uoTJoNpeY

queoteg

Ce IE
Cie
cL
as
ot

ot, TL
ol
Cie Wee:
oT
Cy ING

ON ON OD ON ON ON ON ON O'N ON
ee)

et TL

et

Al 5 g
et

oc, EL 6

as
ol
a)
as
IE EE

or
ot
ot
Ch 195
al

Ree eh

heh

—t tapas a tatoo

at

SUOTPeUT QUOD aTqeTaeA juSpuS apul

ddoea

AddAd

T

*G potdeag Sey Joy ‘satqetre, yuepuaedeput
SATOMT JO SUOCTIeUTQUOD TeteAeg JO Yoem OF AaTQeqnqTaz4y9y Serenbs
gO umg 9ZT9 UTeIyH Wea UT SUOTJONPSY 4uaD Jag ysoesuor4g SAT eUD-*TGA eTABL

out, @ 3e
SX xTS

outy 8 4e
SX OATH

aut? @ 48
SX mog

auty @ 48
SX der],

auTy & 48
SX OMD

94

28°89
G2°69
ah 69
LL°69
T6°0OL

62°59
GE°S9
On °S9
18°99
€0°989

Z9°9S
gT°T9
64° T9
HT" 29
o£ °29

og" ot
19 "9h
68°9t
60° Lh
ge°0S

€L°62
61°62
€T°O€
9E°O€
00° €€

$$ UT woTJONpeY
qusodeg

as

ol

ol

ol

ol

TEE OL I=) BAy
He OE Oe Set
TEE OIE g 2 Coat
ee OL 9 nm & f outy @ 4e
TE Of © § S ¢ sX xTS
OT 9 “i & @
IT Oe Seti
Te OE i; Gok
THE. OE 8 ‘S- @ out, @ 42
TE OF 2 G SX aAT A
Tl Ot G4
1a G 4
HE. OE Ss ¢
ot 8 G @ auty e 42
OL 9 S sX mog
Qo 6K
THe Sac,
HE c
8 eg aut} @ 72
Ot Sg SX deTU],
It 9
9 ul
185 OL
TI 2 aut, @ 48
TL SX OMT

SUOTVEUTQUOD eTQeTIeA JUSspusdeput

4€ poftseg Bey Jog (pesn JON TX STaetxzeA) seTqetzre, quepusdepuT
uaAsTy JO suOTZeUTGUOD TeteAeg JO Yyoey 04 eTqeqnqt4z44y serenbs
go umg eZTg UTeIN UesW UT sUOTJONpeYy queD Jeg 4yseBuorzg SATA UL-*HSa PTAeL

Z€°98
9E°98
LE°98
19°98
gs" 1g

€2°ng
6S°h8
89°18
LE°Sg
qT"98

6L°T8
TE*2g
GE°e8
€6°28
7O°H8

ge°€
ra)
gt°oL
92°91,
HITS

go°ns
go°ss
02°SS
86°19
O6°cL

SS UT uoTJONpeY
queoteg

ZT

ot

as
as

ELE 6 dl S Z
TE d G4 3g
EE 6 9 2 S
TL 6 i, ©) & outy e@ 4e
TE 6 A, Gt SX XTS
Ee 6 I, €
It 6 y} J
EE 19 S
GE L Gh oury eB 4%
ara 6 L S SX eATA
Tl Q 1
Tl J 2g
TL 19
TI 6 L owt, @ 42
Ee 2), S sx mog
TE L 2
TT 19
Tl 6
TL L S our, @ ye
TL dD, SX seay],
Ee g

g G

6

ia oury @ ye
TE dh SX OMT

SUOTPEUTQUOD STQeTTe\ JUspusdepuy

*Z potted Bey 107g (pesn ON TX STaetrea) sotqetse, yuspusdepuy
uUasAeTY JO SUOTYeUTQUOD TeteAeS JO Yoem 0} STgQeynqTz99y Serenbs
jo umg ezTtg uTeIy UeOW UT sUOTJONpEY 4UeD Jeg ySeSu0rzg SAT ouT-°€Ga eTAeL

95

Tr‘ Jg
€3° 1g
29°18
8t°8e
66°88

LL°€8
Tg°€8
Ze Hg
9418
TE*98

+6)
gr 6L
49°61
gg°6L
€S°€Q

9g°eL
ST° eh
6g°eL
60°SL
62°62

0g" 8t
TO*0S
16°9S
00°8S
6h ol

SS UT uoTJONpEY
queso reg

ot
as
ot
as
oT

ot
oT
as
as
ot

cL
al
oT
as
as

ot
ol
oL
as
as

ot
ol
oT
ae
as

WE 6 L t o
TT 6 db m &
6-@ 2 nm &
6 de) tT io) outy @ ye
6 MN; &) H € SX XTS
TE 6 t co}
Ig 6 8 qt
6 L u &
EE 6 9 ar autTy @ 72
aa 6 L qT SX oATH
iE au rej
TEE tH €
6 L t
It } TT aut, e@ 42
SEE 6 t sX moj
We dh
EE 8
IE CHE
HE S suty & ye
TU t SX der]
OT
L
ut
S aut, & ye
TT SX OML

SuoT}euTquo) eTqetxzeA JUsepuedepuy

*G potseg Be] Jog (pesn JON TX eTQeTTeA) SeTaeTzeA quepusdepuy
uaAeTY JO SuOCTJeUTQUIOD TeteAeg Jo yoem 07 STQegnqtz49y Sezenbs
jo umg eztg uTezn ues UT SUOTJONPEeY 4UEeQ Jeg yseBuc0r4g SATA ouL-"QSGd eTAeL

95°98
65°98
09°98
96°98
Sq" 18

16°SQ
29°S9
10°98
11°98
oc°98

612g
29°28
Br ts
umes)
gS "ng

oL'9
4e°OL
Te'@l
gh gl
QT 2g

TS"TE
ga"€t
og HE
g6°9E
€S° eh

S§ UT uoTJonpeY
queoreg

TT 8 Ts @
EL OE jl + 2
TEE O16 ds, Gt
TT 6 L G4 outa eB 42
ELE 8 OG a SX XTS
Tt 8 nm &
1EIG ONE iy &
Tt 8 q 2 :
TEE OIE qT @ out, & ye
TT Ir Gt SX aAT qT
HE 6 4
TI ys; 4
TE: S G tf
Tt nm & out9 @ 4e
TL uf @ sx mod
ce L
TT 9
TT €
TL Z out @ 4e
II q SX deIU
9
6
OT
L out, B 4B
qT SX OMT

SUOT}eUTQUOD STQeTIeA JUepuedepuy

‘ty POTAed Be] Tog (PesM JON TX STAQeTzeA) seTqetTzep quapuedeput
usASeTY JO SUOTISUTQUOD TeteAag JO Yoey OF STQeqnqta4yq4y Serenbs
go umg eZTS UTeIN UBaW UT SUOTJONpSYy qUEeD Jeg 4SeBsuoI4S SATA suL-"SGd STaeL

96

+L°6t TT 6 9 uu & @

oL*6t OT 6 J, 8) & @

Ze" Qt et Or 19 € ¢@

lg lh OL 6 9 h € 2 ouTy @ 48
To°ge oT 19 y & @ SX XTS
GS°Ee OT J, -8) ES

90°€€ OT J,--©) t z

go°Te OT 9 G6 4 z

4°62 OT 9 qT &- 2 oury @ 42
6°92 OT 96 4 € SX eAT A
le*t2 OT Gc @

g9°€2 OT G aq &

16°22 OT J, ©) Z

€6°0g OT 9 & auty @ ye
TL°QT OT 9 + Zz sX mog
69° LT 9 + Z

Q°ST ot 9 t

1T°ST OT + z

T6°TT OT 9 € aura 8 3e
TS°8 OT 9 ro SX serUy
+€°6 OT +

on), 9 €

16°S OT @

19°S 9 2 auty @ ye
12°S OT 9 SX OM]

SS UT uoTZONpeY
queoteg

SuOTZeUTqMOD eTqeTreA JUepuedepuT

*Z potsteg Sey soy ‘setqetsze, yuepuedeput
AATAMT, JO SUOTYEUTQUIOD TeIeAeg JO YOR 04 aTqQeqnqt149y sezenbs
gO ung eZTg UTeIH UBaW UT UOTJONpeY 4ueD Jeg yseHeom SATA SUL-"QSd eTAeL

92°6g OT
18°8¢ OT
Ne OT
08° OT
HE "te et OT
+T°ST Al OT
Hg °qT ot

+0°€T Ot
00°) OT
T9°S Ot
TS°S OT
Lies OT
Las OT
RES OT
6E°H OT
€6°€ OT
SEE

TEES OT
Led

6€°2 OT
gz OT
88°T OT
88°T

19°T

6T°T

SS UT uoTJONpeY
queoreg

*T potdeg Bey toy ‘satqetsre, quepuedeput
SATOM], JO SUOCTPEUTQMUOD TeTaA|ag JO Yoey of aTqeqnqtz44y seztenbs

OV OV ON

cocacdan ccc oOo

00 <0 <=

Reet

te

tat ato ats is —+t

at at at

mo

mm

NUN

SUOTJEUTQUOD OTQeTTeA JUapuUedeput

auty e 4e
SX XTS

outa e 48
SX ATG

our} @ 42
sX mog

auty © ye
SX 90TU],

aurty e@ ye
SX OM,

gO umg aZzTS UTeIy Uee_ UT sUOTJONpeY 4UeD Jeg 4YSoyeoM oATA OUT-")G_ eTaeL

97

00° 16
€2°S6
19° €6
€6°26
cor Ll

€9°€)
T6°2L
1t-2h
Sg°tL
aT lipes

Sq" £9
T6°29
46°09
QT°TS
96° St

Toth
6£°Eh
Te°Th
G9 °6£
16-9

§S§ UT WoT onpsY

quso tog

GE EE OVE

Cie OL
Le ESO
I= OE

as OT
EOE
WE OLE

ce TE+ OL
iG:

Ge EOE

GE WE Oe
Ci GL OL
GE EE OVE

1s (OE
ol OT
CLES Ol
as OT

suoTzeutquog aTqe

ON OV ON OD OD ON ON ON ON ON ON ON ON ON ON OV OV OV

looKookoooolee) a 0 003 0 co cO 3 Cc co 0 0c CO

ee

= tal tall

aurty @ 4e
sy UdAOTH

UN LA LN LN
UNUNNY

2 auTy @ 32
2 sx uel,

(oa) Mmm mo mm

LN LN LN LN LN

UW
ata tata at at at a

2 aut} @ 4e
SX oUuTN

mmo m mm

awry @ ye
€ SX USTH

\0 \90 \0 \0 10 \0O\0 (10 10 10 \0 10 \0 ‘0 0 \O \0 10 10 \O

LN UN LN LN LN LN UN

d

ce, Juspusdepuy

“€ potzeg Seq toy ‘setqetae, quepuedeput
AATAMT, JO SuUOTYeUTQUOD TeaAeg Jo yoey 04 STqeqnqtaigy serenbg
gO uMg eZTg UTeIN UesW UT UOTJONpeYy quep Jog yseyeom OATT OUL-"66G OTAeL

Go°ok oL OT

Gg°Te ot

BH TE ot OT
c€°TE ot OT
1g°l2 eL OT
et €2 ot

68*27e ot

TEES ot OT
03 “12 Git

THOT ot OT
€T°9T GE OT
TO*ST et OT
+9°eT et

TWSCAE Bit

T9°8 ot

gg°h ot

Gt°) ,
95°9 :
ThH'S

88°¢

The

60°2

36°T

H9°T

0g°0

OV OV WO
OR Ot
ooood

SS UT uoTIONpsY
qusootedg

*€ potzeg Bey toy ‘satqetze, Juapuedepuyt
SATEMT JO suUCTYeUTQUIOD TeTeaAag FO yoRy 04 sTqQeqnqT1z44y serzenbg

ON ON ON OV OV OD OD ON

cc fee) 0c CO CO CO <O co cco CO c fo o}fe oo ooo ioe) 0 0c OO

a

ot

‘Oo ‘\O.O

LN LY

—t at a +t —t —t

t+

Gq) GA) Gay GaA)Go) GA)

mm

€

SUOT}EUTQUOD eTQeTIeA JUepuedepuT

out, @ 4e
SX ueAeg

outy @ 4e
Sx xTS

aut} e@ We
SX ATT

aut? 8 4e
SX moj

out B 42
SX deTY],

aut, @ 4e
SX OM],

go umg aeztg uTerH ues UT UOTJONpPSY quUeD Jog ysoeyeoM ATA OUT-*6Gg eTAeL

98

€T°e2 OT

8T°Te OT
GT* 0g OT
eL6T OT
6g° LT OT
ey" TT OT
OW TT OT
6€°TT OT
69°. OT
USO, OT
98°9

gS°S OT
Ges OT
6L°t OT
69°t OT
Q6°E

cians

6L°2

Ore OT
ELT OT
Q6°T

Gr OT
69°T OT
69°T

GS°0

SS UT uoTJoNpsY
qusotag

OV OV OV OV OV OV ON

ee) leowcekee) ee)

oO 0

\0 10 10 \0 ‘0 \0 19 \0 10 ‘0 \0 10 190 10 \O \0 \0 10 ‘0 10

UN WN LN UN LY

LN

mom (sa) mo

SUOT}JEUTQUOD STqeTIe, JUspuedepuy

awty @ 4e
SX XTS

aura @ 42
SX eATT

awry @ 42
SX Mog

aut, & 48
SX eerUy,

aury e@ 42
sX omy,

oO Cr

*G potdeg Bey toy ‘satqetaze, quepuedeput
SATEMT JO SUCTIJeUTQUOD TeteAag JO yoRey 04 sTqeqnqtsz44y sezenbs

Jo ung eZTS UTedp UeeW UT UOCTJONPaY que) Jed 4seyeeM SATA SUT--To eTaeL

lg? Lt
2S° LT
6€°9T
G2°€T
6T°TT

19°6
gL°6
12°6
6g°€
1T°2

gt°2
QT
GT°e
AAG
2n°T

OHT
O€°T
8e°T
9o°T
Hi" 0

Ot'T
96°0
TH°O
on*O
(oye)

SS UT UOT ONpeY
qusozedg

OT

OT
Ot
OT

Oot
Ov
ot
Ot
Ot

OT
OT
Ov
OT

ot

OT
ot
OT

Ov
Ov
OT

*1 POTted Sey Tog ‘suotzeutquog etqetze, qyuepuedepul
aATEM]T JO SUOTZeEUTQUOD TeLaAeg JO YoeT OF sTqeynqtayyy sezrenbg

OV ON OV OV

co 0

Kee M EEE Si

e eeere

—

\O \0 \O No} \0 \0 ‘0 \0 \0 10 (0 10 0 \0 \0 10 10

LN UN UN LN LYN LN LN UN UN LYN

LN LN LN LN

(99) mm

MMMM om

SUOT}EUTqUIOD STqetre, Juspusdeput

aury @ 32
SX XTS

auta @ 4e
SX oATY

outa @ 4728
SX Tog

aut @ 4e
SX d0TU,

aut @ 42
SX OM]

go umg eZTg UTeIH UeaW] UT UOTJONpeYy queD Jeg Jseyeom OATY OUT-*O9d OTAPL

99

Ht6 26L S6h Ode 99

Ge i
69 61 iE
Gs on Us
OSC icaate9)
5 CSE HOE
69 69 9€
PAS (6) th
es — Jhy 3
oth qh
S ge On
oT on
T € OT
q at
3g 8
ul
T
Ome Sant

qT
OT

i

= onic

(ny ep =r oe)

z

ot

T

soTqetze,\ JUepuedepuyt jo requmy

(Z potted Be]) sx ZT SA eztg weeW -*€9 q eTAeL

%OT°€6 = uotyonpey ss TeIOL

z
6°t6-0° 06
6°68-0°S8
648-0" 08
6°6L-0°S)
6°t1-0°0L
6°69-0°S9
6 "49-0" 09
6°65-0°SS
6°tS-0°0S
6°6h-0°SH
6*tH-0° OF
6°6E-0°SE
6*t€-0° OF
6°62-0°S2
6*t2-0" 02
6°6T-0°ST
6*HT-0°OT

6°6-0°S

6*4-0°0

% sseTo

woTjonpey $s

9

26.

GET

ooT

$

G6 022 99
6

Gh BE =o

NG Ge

66 6€ 9

Be ge «

T-

9t 2

qm

Cs Oe

Me AE. Ay

US Q €

Ws Se -&

Ge € 9

ee we. e

GT Oc &

OE Ss CL
E © 6

q oe 3g

ol

iN =

|

soTqetre, JUuepuedeput jo zJaqumN

(I potted B87) sX ZT SA ezTg uBaW -"zog FTABL

%ST°9Q = UoTJONpeY SS TeIOL

z
6°68-0°Sg
6°48-0°08
6°6L-0°SL
6°_L-0°0L
6°69-0°S9
6 "49-0" 09
6°66-0°SS
6°4S-0° 0S
6°64-0°Sh
6"tt-0° OF
6°6€-0°SE
6"HE-0° OE
6°62-0°S2
6°t2-0° 02
6°6T-0°ST
6*4T-0° OT
6°6-0°S
6*4-0°0

% sseTo

woTZONpSY ss

100

416 c6l S64 O22 99

Zot lz
gyT 66
gS 9
65 ot
19 Ot
ty oS
GS Ok
g2T got
Te) WH,
6€ ah
gE SH
ay aS
€€ 2S
€e — Ot
€t le
2 €2
BS ie
q
Z
Sees

in
is

6T

io)

as

iT

soTqetze, Juepuaedepuy jo zrequmN

(4 POFted Bel) sx 2T SA ezTS ueeW--Cog eTIeL

Hg9°h6 = voTZONpeEY Ss TeqOL

z
6°t6-0° 06
6°68-0°S8
6*tH8-0" 08
6°6L-0°SL
6°tL-0°OL
6°69-0°S9
69-0" 0
6°65-0°SS
6°4S-0°0S
6°64-0°SH
6*H- 0" Oh
6°6E-0°SE
6° E-0° OF
6°62-0°S2
6 "2-0" 02
6°6T-0°ST
6*HT-0° OT

6°6-0°S

6*4-0°0

% Sseto

uotyonpey $s

oge S6h 26)

St
99
Bt

TELE

(€ potted Sel) sX ZT SA ezTg wesw -"HOM eTAeL

8c 8
T2T 90T
get OT
th ee
TT 46
Sr
9S 1g
1S 2
GT 0S
G €€
g 62
S- Oe

@ fll

Q 2

+16

Lt
GTT
9eT

TEU

9

26h

TOT

$

G6h 02s 99
iT

6

or 2g

Gd &

6S 6ST (OT
6 rt &
oe 2 €
we
Gg €
us €
Goan

66 8

G5 Us §
Th 92 6
US RE ay
62 GE 9T
6 OE 4K
8 fis {3}
Teng) age aC
tee Cae ce,

oL

t

soTqetre, JUuaepuedepuy fo caqumyy

461°96 = uoTyonpey Sg TeqOL

iG
6°66-0°S6
6°t670° 06
6°68-0°S8
6*H8-0° 08
6°6L-0°SL
6°tL-0°0L
6°69-0°S9
69-0" 09
6°65-0°SS
6°4S-0° 0S
6°6h-0°SH
6"t-0" OF
6°6€-0°SE
6°HE-0° OF
6°62-0°S2
6"+2-0° 02
6°6T-0°ST
6°_T-0°OT

6°6-0°S

L-+70°0

% Sseto

wOTJoNpSY Ss

101

6g°€6 = uoTJOMpEY Sg TeIOL

+16 26 S6h O22 99 aT

Gere se- GS . Se

Sire We ny Ge

€or 78 4S 92

UN
+
=I
fo}
Coal
ty
q

\O
To]
=
a) |
nl
all

seTqetzre, JUuaspuedepuy jo requmN

(G potzad BeT) sX Zl SA aztg ueeW -"gg ETAL

woTyonpeYy ss

z
6°68-0°S8
6*8-0" 08
6°6L-0°SL
6°tL-0°0L
6°69-0°S9
6°49-0" 09
6°65-0°SS
6745-0" 0S
6°64 7-0°Sh
6*Hh-0° OF
6°6£-0°SE
6*HE-0° OF
6°62-0°S2
6 "42-0" 0g
6°6T-0°ST
6*HT-0° OT
6°6-0°S
6°4-0°0

% sseTo

102

ADDENDUM

ALTERNATIVE MULTIREGRESS ION TECHNIQUE FOR
OBTAINING PREDICTOR EQUATIONS

by

W. Harrison
U.S. Coast and Geodetic Survey, Washington, D.C,
and

N, A. Pore
U.S. Weather Bureau, Washington, D.C.

ABSTRACT

The data used in the study of interactions in the beach-
ocean-atmosphere system are subjected to a multiple-regression
screening procedure which is programmed so that all Xs, from
all time lags are subjected to a single screening, Examples of
predictor equations suggested by the analysis are given for the
following predictands: mean longshore current velocity, mean
bottom slope and mean grain size in the shoaling-wave zone, and
net deposition and net erosion on the lower foreshore during
June and July, The screening technique yields comparable or
somewhat stronger predictor equations than are obtained in the
sequential multiregression analysis, where combinations of Xs
are restricted to specific lag periods,

INTRODUCTION

The computer program for the least-squares search procedure used in
the first part of this report permitted identification of the strongest
combinations of 10-14 Xs (predictors) for each Y (predictand), when the
combinations of Xs were restricted to specific lag periods, This scheme
was adopted because of a program restriction that limited to 25 the number
of Xs that could be studied at any one time, and because of the great
numbers of combinations that would be generated if more than a few tens
of Xs were considered,

It is desirable, nevertheless, to have available an extension of the
method for searching large numbers of predictors so that each of many Xs
may be considered, regardless of its position in time, One such search
procedure - called here the screening procedure - identifies "optimum"
sets of predictors for given predictands, The data to which the screening
procedure will be applied will be identical in every way with those of the
first part of this report, to which the Krumbein, Benson, and Hempkins
(1964) computer program was applied,

The computer program for the screening procedure is similar to that
presented by R. G. Miller (1958). Analysis of the data will proceed
without the inclusion of variables X3 and X5 (ys Elavel hyo), alfel foewert
because of their redundancy with each other and with T. (The reader is
referred to the list of symbols in Table 1 for definitions of the various
notations used for the variables),

METHOD OF ANALYSIS

As in the main part of this report, the procedure adopted for select-
ing predictors involves expression of a predictand Y as a linear function
of a number of predictors X, (n=1,...,N).

Thus:

Y= Ag a AyXy + A5X5 oP BOOMS © 606 AyXn

where the coefficients A,(n=o, ...,N) are determined using the method of
least squares,

Because of the large numbers of predictors involved, the screening
procedure required the use of a high-speed, large-memory computer, The
IBM 7030 was used, Basically, the manner in which the predictors were
screened is shown below:

1) = VAC EE BEX:

1 V1
a). YC B A, + BAX, + CiX5
3) Ye AL + BX + Co-1%2 222 e NX

where Y is
By, B,, Cy> C

i)

predictand, the As are constants, the Xs are predictors, and
3? etc. are regression coefficients,

The procedure is to first select the best single predictor (X,) for
regression equation 1, The second regression equation contains thé first
predictor (X1) and the predictor (X2) that contributes most to reducing
the residual after the first predictor is considered, regardless of its
lag position, This means that the second regression equation contains the
best set of two predictors that includes the X selected at the first
screening step, The procedure will not necessarily select the best set
of r predictors out of the original set containing P predictors, In the
closely analogous field of meteorology, however, it has been shown that
the screening procedure can select a highly reliable set of predictors when
applied to problems that involve redundant, interrelated variables.

Generally, the significance of the improvement attained at each step
of the screening is tested and the screening discontinued when the amount
of improvement is found not to be significant. Near that point the addi-
tion of many more predictors usually lowers the predictive ability of the
system on independent data. As pointed out by Panofsky and Brier (1958),
however, objective standard significance tests may be misleading on data
such as those of this study, because the underlying assumptions may be
violated. The predictors used here are certainly interdependent in time
and space. Often the most practical and convincing test of significance
can be an application of the result to an independent set of data. This
will be the method used by the present investigators for "significance"
tests in future work with predictor equations for data of this sort.

RESULTS
Predictors Selected

As mentioned earlier, variables X3 and X5 (Lg and Ho/Lo) were not used
when screening for predictor equations, because of their redundance with
each other and with T (wave period). The results of the screening procedure
for 1 through 4 predictors are given in table Al, where the order of the
first four selected predictors is shown along with the lag and correlation
coefficients. For example, in run 1 the first predictor selected by screen-
ing was (T) with a lag of 0-4 hours and a correlation of 0.47. The second
predictor selected CHE, with lag of 0-4 hours) increases the correlation to
0.64. Four predictors bring the correlation to 0.72.

Predictor equations, each containing the four predictors of table Al,
are presented below for the five screening runs.

(1) VS Loi =O.AC)_ | > Oni) oe Oss GG)

eat ea 0.02 Use

(2) 8, 20.69 sls | i», | ag % O20 (zg > O03 EH), 7 0.64 (C)_,

(3) me 2 0.64 <O.17 (S_)_, = O40 (@)_, 0.02 (p)_., » 0.03 Cay,

(4) dg = 60,08; 20.28 G)_. 470 G4 = O59. CD, 2? ©.09 @)_,

(5) Ky, = Ola & 0,04 4 On (Gy = Ohl ES Le )_.

where p = (p-1) x 10° and the other variables and their units are as defined

and explained in Appendix A. The numbers subscripting the variable symbols
designate lag periods, each of which is of four hours duration.

Time Lag in Predictor-Predictand Relationships

As is amply demonstrated in the main part of this report, there is
a time lag in the peak interaction of the most-important combination of
independent variables, as they influence the dependent variables. The
program used for the screening procedure is written to print out all of
the simple correlation coefficients between predictors and predictands.
This permits an informative analysis of the time variance of specific
predictors selected by the screening procedure, Figure Al is a graphical
presentation of one such analysis, in which the simple correlation co-
efficient, r, for the first four predictors selected in run 5 (table Al)
is plotted versus time.

DISCUSSION

It is apparent that a great amount of useful information may be
derived from diagrams such as that of figure Al. The high positive
correlation of wind velocity offshore during the second lag period (4-8
hours prior to measurement of the predictand) indicates the lag in this
factor's influence on beach erosion. The alternating negative and positive
correlations of mean longshore-current velocity with beach erosion are to
a certain degree a mirror image of the curve for Waves which would be ex-
pected if one considers that onshore winds create waves that produce strong
longshore currents, The relatively constant value of the negative correla-
tion between slope of the lower foreshore and beach erosion suggests that
the slope itself, although having a real influence on beach erosion (table
Al, run 5), does not need to vary significantly for erosion to take place.
The correlation between the angle of wave approach is seen to be highest
in the immediate past and it does not show the clear lag tendency that is
shown by Wace As the data matrix from Virginia Beach is enhanced, a repeat
of the present screening procedure will permit the plotting of more meaning-
ful diagrams such as that of figure Al.

Turning to the predictor equations once again, it would be well to
remind the reader that these are best tested by their application to a set
of independent data. This will be done in the near future when additional
data become available, It is also noted that the number of cases available
for the screening procedure (table Al, "'N=..'') needs considerable augmenta-
tion.

A comparison between the best four-variable equations selected by the
two multiregression computer programs is presented in table A2. In the case
given for V as the predictand, both programs happened to select the same
combination of Xs and, because Xs from just one lag period were involved,
the "%-SS-reduction” values were also identical. The results for Se are
not quite the same. Although both computer programs selected the same kinds
of Xs for the best 4-variable predictor equations, the total variance reduc-
tion ("%-SS-reduction") is slightly greater for the four predictors selected

by the screening procedure, This is due to the fact that the screening
procedure is able to select variable X12 from lag period 2 where it is
stronger than it is in lag period 5, the lag period shown by the Krumbein,
Benson and Hempkins (1964) program to be the strongest combination for the
given lag period,

More pronounced differences exist in the kinds of variables chosen
by the two programs in the remaining runs, and the variance reduction is
perhaps significantly higher for the runs involving (Mz) and J¢, where
the screening procedure is used,

SUMMARY REMARKS

Although the alternative search procedure described in this addendum
will not necessarily select the best set of r predictors out of an original
set containing P predictors, the screening procedure does allow rapid con-
sideration of many more predictors than the Krumbein, Benson and Hempkins
(1964) procedure and, in terms of variance reductions, appears to yield
comparable predictor equations. Thus, the screening procedure appears to
be worthy of application in studies such as this in which large numbers of
interrelated predictors are to be considered, and in which the data extend
through time,

ACKNOWLE DGMENT

We wish to thank W. C. Krumbein for review of this addendum,

S9°O ¥z2-02 | 28° O 72-02 i || 26°@ 8p y 36°@ 8h 5)
ED"O ~H-O 5 || ZO 7-O i || G5"O 70 0 96°O O@-9T 20
: J : I : :
96$°O 9T-2T § | &2°O -0 || 8820. Ouse 9 160 O@-9T d
)

vy'0 8-pb 7m | 09°70 4-0 45 | 62°0 4-0 “s | 28°0 02-91 ~ (7N)
I Sey “IeA I ‘sey “reA I “Sey “IeA 3 °3e7T “IeA

(Ve=N) (OZ=N ) (ST=N) (ST=N)

¢ NOU py NOU € NAY @ Nou

Ss s
Fy AE Cw) s A + puejyITpartd

(sznoy ut passaidxa st [TeAzajuT 3e7)
SSHOOUd ONINAHYOS AM SYOLOIGHYd AO NOILOATAS

TV HTEaVL

Predictand

@)

Vv

TABLE A2

BEST FOUR-VARIABLE PREDICTOR EQUATIONS
(Variables X3 and X5 not used)

Sequential
Multiregression Method Screening Procedure
Predictors %-SS Predictors %-SS
(Xs) Reduction (Xs) Reduction
Oooo 7 52.0 1,2,4,7 52.0
Gigys. a) (Lag 1)
HG Peplielinelee 94.1 if @) cabal lz} 95.2
Caer) (Lags 5,5,5,2)
1 D732 90.3 I 20), HI 52) 95.0
(Lag 4) (Lags 1,2,1,4)
it. @, 9,0) ORO ib, 245, LO), ha US db
(Lag 1) (Lags 1,6,1,1)
1,2,7,14 42.5 it 7,9). 0) 43.0
(Lag 2) (lags 4,2,1,6)

=2

SIMPLE CORRELATION COelririCheN il i

as

falling HW rising ew falling yaw rising

! :

EAG ReRIOID

FIGURE Al. TIME VARIATION OF r FOR THE PREDICTAND K,

(BEAGH EROSION) AND THE FOUR PREDICTORS Use
(WIND VELOCITY OFFSHORE ), oC (ANGLE OF WAVE
APPROACH), V (MEAN LONGSHORE - CURRENT
VELOCITY), AND Ss (MEAN SLOPE OF THE LOWER
FORESHORE ). DESIGNATIONS ALONG THE
O-INTERCERT SRERER TO MDE WG ONDIiO NS:

A-8

“pa}esTSeaut Os]e ST ,,asuodsaz,, ay} uo 4azza
unuTxew Ireyz FO jUsuOW pue ,,sessaz0id,, Jo dnoiZ e zo uoTIdasuT uaaMjaq
Bel awry “syue}z aaem ur pautwexa AT[ewsou saTqetTien yzIM uoTJeUTqWOD UT
PaTpNzs aq “SUuOT}IpPUOD paT{or}UOI Tapun pauTueXxe wopyas ‘saTqeTieA [eUOTY
—Tppe urejJs> jy} 4Sassne sj[nsey “SayIeeq [ein}eu uo satqeTszeA asayi
JO 90ue}IOdwT aATyeTar Sutpiedax sjuewSpnf aatqrnjur azetiuesqns pue
“uoTjzejUuawtradxa yuej-ehem Jo satTqetsea ZUBITIJTUSTS YIM JUsWeceTSe Terauag
UT 91e SeTQeTIeA ,,ssa00r1d,, se pajeustsap A[Tie1jTqQIe SaTqerseA Jo suaTy
—euTquos TeTJUaNTFUT—-isSoW “sasssd01d azoys 0} adoTs yoeeq puke azts ures3
JO asuodseai pue ‘aioysa10z JamoT 343 uO UOTSOZa pue uotjitsodap ‘sjuarind
aioyssuo,T Fo Ajr20[aA uO suOT}IpUOD pUTM TeDI0T puke “syoezya Tept} ‘so14st
—JezIereYyD asAem ‘Ar}aQW0eS YIeaq FO aduUanTzuT sepn{out Apnjys “sszaynduos
paads-ysty roy pauwe1301d se stsAyeue uorsserSaz13 [nw AvaUTT Tet juenbss
Aq pazeSt}Saaur aze SaTqeties yeaq Suowe suotzIeIa{UT Jo Iaqunu y

a:
“y 'N ‘azog AT aad ISSVIONA Z N WRONYYOWSW TVOINHOSL

“Oo ‘mM ‘utaqunzy ITI “910q “y "N pue uostizey ~m Aq

“M “UOSTIZ®H II SNOILVNOT YOLOIGAYd ONINIVLEO YOI ANOINHOAL NOISSTNDTY

eT3FL I -ILIAW JALLYNUALTy ('dd g) wnpusppy pue ssotpuadde z

“yp “seTqei ZI ““SNTIt » Zutpnyour “dd zoT “p96T taquanon

weisoid 13j3ndwop “¢ utTaquniy “D°m pue uostszzey “mM Aq “yA ‘HOVSA VINIOUIA

erUrsItA LY WHISAS JYIHdSOWLY-NVSDO-HOVAE SHL JO SNOTLOVYALNI
“yotag STUTSITA “Zz

sassad0i1d a104yS “IT “D°d "HSWM

ae

3D “USINSO “SHY “OUONT IVISYOO AWUV “S7n

“payest}seaut osTe st ,,asuodsas,, ay, uo zoaezza
UnwExXeW ITIY4 FO JuewoW pue ,,sessed0id, Jo dnor# e Jo uotzdasut uaamjzaq
SET SWTL “syUue} AEM UT pauTwexa ATTeWIOU SOTQeTIeA YIM UOT JeUTQqUOD UT
P2eTpNys aq “SuUOTITpUOD paT{orjUOD Iapun pauTWexa woptas “saTqetiea [euorz
~TPPe UTeIII9 jeY }6ed3ns sjInsay “sayoeaq TeinjzeU uo satqetieA 9soyy
JO aouejsodwr satzeyar Sutpresa1 szuawSpnf aatztnjqur a}eT Ue ySqns pue
UOTE UsWTIadxs YUEZ-SAeM JO SaTqeTIeA jULITFTIUSTS YIM UaWaeISe Tersues
UF S18 SSTQUTIEA ,,S5920Id, se pajeustsap ATr1e1}IqGIe saTqerazeA jo suoty
~euTquiod TRTJUSNTJUT-ZsoW “Sassad0id aioys 0} adots yoeaq pue azts uteis
JO asuodsazr pue ‘asoysar0z amo, ay} uo uOTsosa pue uoT}rsodep ‘sjuazino
PIOYSSUOT JO AZTI0TSA uO suOT}Tpuod puTM TeDI0T pue “sz29yJ9 Tept} ‘soTyst
~29}IEIEYD aAeM ‘AIZaWOaS YOeaq JO adUaNTzZuI sapntsut Apnjys “siayndwo>
pesds-ysty 10zZ pawwesso1a se stsATeue UOTSSAI59IT [NW TeaUTT TeT}Uanbas
Aq paje8tjysaaut ase satqerieA yoeaq Suowe SUOT}IVID}UT JO Taqunu y

“y "N ‘az0g AT ddd ISSVIONA 2 “ON WNGNVYOWSW IVOINHOSL

“O “mM ‘utTaquniy Iii “910d “Vy "N pue uostizey *m Aq

“M ‘wosTz3eH II SNOILYNOT YOLOIGAYd ONINIVLEO WA ANDINHOAL NOISSaWay

eT3FL I -ILINW SALLYNYSLIV (‘dd g) wnpuappy pue sastpuadde z

“y “S9TQk} @T ‘°SNTTTt » Surpnpour “da zor “po6T saquaaon

weiso1d sayndwop -¢ urequiniy “D°M puke uostiiey “mM Aq “YA ‘HOVHA VINIDUIA

eLUTsITA LV WHLSAS dMaHdSOWLV-NVdOO-HOVEE FHL JO SNOILOVUAINI
“yoeeg erursiqa -z

sSassa003d a10qgS “I “D"q “HS¥M

HO ‘UHINSO “SFY “OYONT IVLSVOO AWYV “S"n

“pe eSTJS@AUT OSTe ST ,,asucdsaz,, a4} uo yoezzFa
unWTxeW ITIy} JO JUsWOW pue ,,Sassaz0rzd,, Jo dnoid e zo uorjydasut usamjzaq
ST awry “syue, anem ur paurwexea AT Tews0U SSTQeTIeEA YIM UOT eUTQUOD UT
PeTpnzs aq “suOoT}Tpuor paTTos}UuO> sapuN pauTwexa woptas “saTqetiea [euoT
Fppe UTeI99 34} YSesddns sjINsey ~saydeeq [eInNzeU UO SeTgetieA asayj
JO 220Ue}IOdwT aAT}eTaI Surpiesar1 Sjuswspnf{ aAtjInjur a,etjUe;Sqns pue
“uoT}ejuawt1adxa yue,-anem JO SOTQETIEA YULSTFIUSIS YIM JUaWeaISe Ter3uas
UF 918 SaTqeTseA ,,Ssa00Id,, Se pa}euStsap ATr1eI}TqQIe SatqeT3eA jo suory
~PUTqWOD TETIUaNTFUT-{SOW “Sassa2010 az0Yys 03 adoqTs y>e¥aq pue ezts uteis
JO esuodsaxr puke ‘szoysaroz 1amOy ay} uO UoTSosa pue uoTjISsedep ‘sjzuarino
sioyssuotT jo AzTI0T[aA uo SUOT}TpUOD pUTM TBIOT pue ‘s}IazzJe [TepT} ‘S2TYST
~FazIeTeYD BAEM * AT} eWOaS Yeaq JO aouanTzuT sepntour Apnyg ~“siza3zndwo>
peeqs-ysty OF pauwesS0zq se stskTeue UOTSS9IS59IT} [NW IeaUTT TetTjUuanbas
_4q pazestysaaur ase satqerien yoeaq Suowe SUOT}IEI3}UT FO Jsqunu y

“¥ °N ‘az0g AI GaIaISSVIONN 2 “ON WOGNVYOWSW TVDINHOAL

“O °M ‘Urequinzy TIT "910d “VY “N pue uostiiey “mM Aq

“M “UosTIIeH IT SNOILVADA HOLOIGI¥d ONINIVLEO YO ANOINHOAL NOISSauDay

eT3tL I -ILINW SAILVNUALIV (“dd g) umpuappy pue saztpusdde z

“pb “seTqe; ZT ““sn{rr » Sutpntour -dd zor -p96r saquaaon

wexso1d 1azndwod “¢ urequnzy “9 -m pue uostzzey “Mm Aq “VA “HOVIG VINISUIA

BIUTSITA LV WSLSAS 4YaHdSOWLY-NVIOO-HOVAG FHL JO SNOILOVYSINI
“ydeag erursimm -z

Sassa201d ar0ys “IT “D°d “HSWM ‘"3D ‘YaINaD “SHY “DYONA IVLSVOD AWUY “Sn

*pazesT}SaAUT OsTe st ,,asuodsar,, ay} uo Izzo
wWNWTxew IT3IYy} JO JUsWOW puke ,,sassaz0id,, Jo dnoi3 e Fo uot}daduT usaMjaq
Sel awry “syue, aAem ut pauTwexa AT[TeWIOU SeTGeTIBA YIM UOT eUTqWOD UT
Petpnys eq ‘suoT}IpuOD peT[OI}UO2 Iapun posuTwexa WoptTeas ‘SaTqeTieA [eUuOCT}
-Ipp® uUTe,199 jeu} 76a9Sns sj[nsey “saydeeq [eIn}eu uO SeTqeTIeA asay}
JO 20uej,IOdwI SAT eTer Surpiesax1 sjuewspnf 9AtT}INjuT |9}eTJUe,SQGnS pue
‘uOT}eUSMIIadxa yUe}-SAEM FO SATQGRTIEA JUBITFIUSTS YIM JUSWaaTZe TeTauas
uT 918 SatqetieA ,,ssad0id,, se pajyeustsap ATtIeI}TqQIe SaTqeTseA Jo suoT}
-eurquod TeTjUan][zJUuT-js0W “Sassaz0id aitoys 0} adots yoe¥aq pue ezrs utreid
jo asuodsai pue ‘as0ysatojz IaMOT ay} UO UOTSOIa pue UOoT}TSOdep ‘s}juazINnd
aioyssuoyT Jo A}TIOTSA uO SUOT}FpUOD PUTM TRIOT pue ‘s}Iazza Tept} ‘SOT4ST
-Iaq2e1ey? aaem *‘Ax}awoaS yIe|aq JO adUenTJUuT SapnyToIuT Apnjys ~siayndwos
peeds-ysty 10x powweirdso1d se stsAyeue uoTSSaIsait}[nw reauT,T Terjuaenbas
AQ pazeZrysSaaur aie BaTqeTIeA yIeaq Suowe suoT}IeIa}UT JO Jaqunu y

daIdISSVIONA Z “ON WAONVYOWSW TIVOINHOSL

“WN ‘920d AI
“OD “mM ‘utequnzy TIT "ai0q “VY "N pue uostzzey “mM Aq
“m ‘uostzzeH IT SNOILWNOA YOLOIGAYd ONINIVLEO YOd ANOINHOPL NOLSSTUOTa
PTITL I -ILINW FJALLVNUALIV (“dd g) wnpusppy pue saotpuadde z
b ‘SeTqe} 2T ““SNTIT p Surpnpour “dd 20T “p96T Tequaaon
wezsS30i1d sayndwop “¢ utTequniy “D°M pue uostizey “mM 4q “VA “HOVAd VINIDUIA
eIUTSITA LY WHLSAS SYIHdSOWLY-NVSOO-HOVAd SHL JO SNOILOVUFLNI

“yoeag BIUTSITA “Z

Sassa20id azoys “I “O°G “HSVM ‘dO “WHINSO “Sdu “SUONA IVLSVOO AWHV “SN

“pazestysaaut osTe st ,,asuodsas,, ay} uo yoazza
unuwtxew Itay} FO JUaWOW pue ,,sassaz0id,, Jo dnoi3 e jo uotzdasut uaamzaq
deT owty “syuez aAeM UT pouTWwexa ATTeWIOU SaTqeTIeA yZTM UOTZeUTQWOD UT
Petpnzys aq ‘SUOT}TPUOD paTTOr}UOD Iapun pauTWexa woptes ‘saTqeTIeA TeUuoT}
-Tppe® ute}1a9 3eY4} 3Sa3sns s}Tnsay “Sseydeaq [TeIn}eu uO SaTqeTIeA asay}
FO a0uezJodwt aAT}eTar Sutpiesgar sjuawSpnf aatztnjur azetzue,sqns pue
“uoTzeUsWTIadxa yUe-aAEM FO SATQETIVA jURITZJTUSTS YIM jUSWaeISe TeIauad
UT are SatqeTreA ,,ssad0id,, se pazeustsap ATTieIjTqQIe saTqeTIeA Fo suoT}
—euTquoD TerTjUuanTzFut-zsow “sassadox1d azroys 0} adoTs yoeaq pue azts uteid
JO asuodsair pue ‘atoysaroz IaMOT 9y} UO UOTSOIa puke UOoTZTSOdap ‘sjUazIn>
aTOYyssuOT FO A}TIOTSA UO SUOT}TPUOD pUTM TeRIOT puke ‘s}dazza TepT} ‘SoT4ST
-iozeIeYyD aAem ‘ArzaWOaS YyOeaq Jo aduany,Jur sapnyout Apnys ‘*siajyndwoo
peeds-ysty 10F pawwerSoid se stsATeue uoTSssaisa1ty[nNwW IeaUTT TeT}Uanbas
4q pazestzsaaut 31% saTqeTieA yoeaq Suowe suot}eIajzuT Fo equnu y

xe)
“V °N ‘azod AT daIdISSVIONN Z N WAGNVYOWSW TVOINHOEL

‘) ‘M ‘uTaquniy IIT “aI0q "VY "N pue uostaizey “mM Aq

"M ‘UOSTIZeH II SNOILVNOS YOLOIGHYd ONINIVLEO HOA ANMOINHOTL NOLSSTUOTY

eT3tTL I ~-ILINW AAILYNYALIV (‘dd g) wnpuappy pue saostpuadde z

“p ‘saTqe} 21 ‘*SMTIt p Surpntour ‘dd zor “p96T Jeqwaaon

ueiso1d rayndwod “¢ uraquniy *D°M pue uostazeH "M Aq “VA ‘HOVEE VINIDUIA

eTUTSITA LY WALSAS FYTHdSOWLV-NVSOO-HOVAA AHL JO SNOILOVUALNI
“yoeag eruTsdita ‘z

sassao00id a10ys “fT “D°ad “HSWM

HO ‘“UHINTD “SHY “OYONY IVLSVOD AWUV “S‘n

“pozeStysaaut osTe st ,,asuodsai,, ay} uo ydazza
unwExew ITey} FO JUusuOW pue ,,Sassad0zd,, Jo dnorZ e jo uotydaouT uaamzaq
Bey owt, “syuez adem ut pautwexa ATTewIOU saTqeTIeA YIM uoT}eUuTqWuOD UT
PaTpNzs aq ‘SuOTZTpUOD paT{Or}U0D Iapun pautTwexa wopras ‘satqeTIeA ypeuoty
~Fppe UTe}T99 ZeYy YSes5sns s}[nsay “saydeaq [eInjeu uO saTqeTIeA asayy
FO aouejztodwr sArzeyai Surpiezai syuawspn( aatztnqut 9}eT}JUeZSqGns pue
UOT JeJUsWTIAadx9 YUeL-aAREM JO SaTQUTIEA jURDTZTUSTS YIM JUaWaaISe Terauad
UT 91P SoTqetTieA ,,ssa0id,, se pajzeustsap ATT1e1}TqQIe satqeTIeA Fo suoT}
TeUTQqUOD TeTJUSENTFUT-ysoW “Sassad01d aroys 03 adoTs yoeaq pue azts utes
JO asuodsar puke ‘axr0ysa10z Jamo, ay} uO uoTSOIa pue uoTyrTSOdap ‘sjuaziIno
atoyssuoy Fo A}T20TaA UO suOT}TPUOD pUTM TeD0T pue “szo9gjJ9 Tept} ‘sotyst
~Tazoereyo aaem ‘ArjowWo0as yoeaq JO aduanTzur sapnqouT Apnjs ‘“s1aqndwos
peads-ysty 10Fz pawwerso1d se stsATeue uoTssazSait}[NW reauTT [eT Uanbas
Aq payestysaaut aie saTqetiea ydeaq Suowe suot} 2 eraquT JO taqunu y

“VY ‘N ‘az0q AT aadIdISSVIONN <2 “ON WONGNVYOWSW IVOINHOSL

“O °M ‘uraquniy TIT “910d "VY "N pue uostizey “mM Aq

"M ‘UOSTIIEH TI SNOILVNOA YOLOIGHYd ONINIVLEO WA SNOINHOSL NOTSSauDgTy

STFFL I -ILINW SAILVNYALTV (“dd g) wnpuappy pue sadtpuadde ¢

“bp ‘seTqey 21 ‘SMTTt p Surpntout “da zOT “p96T r1aquaaon

wersord rayndwoy *¢ uraquinzy "9°M pue uostizeH “Mm Aq “WA ‘HOVSE VINISUIA

erursiqA LY WHLSAS TUTHdSOWLY-NVAOO-HOVEA AHL JO SNOTLOVUSLNI
“yoeag erursita “Zz

Sesseo01d aroys “Tt “Od “HSVM ‘"dO ‘YHINHO “SHY “ODUONT IVLSVOO AWUY ‘S"A

“peyestjsaaut oste st ,,asuodsaz,, ay} uo poazza
unwExXeW ITIY4} FO JUeWoW puke ,,sassed0r1d,, Fo dnor3 e jo uotzdaout uaamzaq
Sey owrL “syuez oAem ur pautwexa ATT ewz0u SOTQETICA YIM UOT}LeUTqUOD UT
PeTpn}s aq ‘suoTzTpuod paT{or}u0d Japun pautwexa woptas “soTqerieA peuoty
~FPpe uTeyTI9 4eYy, YSadIns sj[nsay “sayoeaq TeInjeu uO saTqeTieA asayy
JO a0uezztodwr aatzetTax Surpresax szuawSpnf aatytnqut 9} eT UeYSqns pue
“uoT}e\UuoWTradxa yue,-aAeM Fo saTqeTseA JULITFTUSTS YIM JUsWaeTSe Terauad
UF ere SaTqeTieA ,,ssac0rd,, se pazeustsap A[r1e1}tqQIe SatqerieA JO suoty
~BUFQuUOS TeTIUANTFUT—-SOW “SaSsa00iIa azroys 0} adoyTs yoeaq pue ezts ureis
JO asuodsar puke ‘aroysaroz 1aMOT ay} uo uoTsora pue uot}tsodap ‘sjuarino
eroyssuoT Jo AjTOOTaA uO suOT}TpUOD puTM TedOT pue ‘s}59zJ29 Teptz ‘sotyst
~FazeIeYD aAeM ‘AI}aW0ag YOVIq JO adUaNTJUT sapntour Apnis “srajndwos
peeqs-ysty IoF pawwer3031q se stsAkTeue uOTSSaIsaIT} [NW IevauT,T [TeT}Uanbas
Aq pajzestysaaut are satqetzea yoeaq Suowe SUOT}I¥I9}UT Jo Taqunu y

“V CN ‘azog AT dadId ISSVIONO 4 “ON WAGNVYOWSW TVOINHOSL

“OD °M ‘utequnzy [IT "at0d “VY “N pue uostizey “mM Aq

“M “uOsTiieH ITI SNOILWNOA YOLOIGHYd ONINIVLEO YO’ INOINHOAL NOISSHuDau

eT#FL T[ -ILIAW SAILVNUSLIv (“dd g) wnpuappy pue saotpuadde 2

“yp “SaTqe} @T ‘‘sn{It » Sutpntour “dd got ‘poor azaquaaon

wergo31d rajndwod “¢ urequniy “D. "Mm pue uostizeH “Mm Aq “WA ‘HOVAE VINIOUIA

PTUTSITA LY WHLSAS SYxHdSOWLV-NVAOO-HOVEE FHL JO SNOILOVUSINI
“yoeaq Brursira “Zz

Sessed01d atoys “T “D"d “HSWM ‘"dO ‘URINED ‘sau “OUONT IVLSVOO AWYY “S*n

“pajzeSTj{SeAUT OsTe ST ,,asuodseai,, ay} UO IazzTa
wunutxew ItTay} FO Juawow pue ,,sassad0id,, Fo dnois e Fo uotydaout useMjoq
Set owtyL ‘syue, adem ut pauTwexa ATTewWIOU SaTqeTIeA YIM UOT}eUTqWOD UT
patpnzs aq ‘suorzrTpuod paT{OI}UOD Iapun pauTWexs WopTeas ‘SaTqerivA [euoTy
-Tppe uTe}19a9 jeu} 4Sadsns sjz[nsay ‘“saydeeq [eIn}eu uO SaTqeTIeA asayy
JO aouejrodwt aAtzepai Sutpresez sjuowdpn{ aAtjrnjur azetTj,Ueysqns pue
‘uoT}e}UawTIadxa yUe}-aAeM JO SATGETIPA jJUBITFTUSTS YIM JUaWadIde TeIo9Uad
UT a7 satqetieA ,,ssac0id,, se payeustsap ATtre1,1qQze SaTqetTIeA FO suOoTy
-euTqWO. TeTJUaNTJUT-}SOW ‘“Sassad0i1d ar0ys 0} adoTs yoeeq puke ezts uTeIs
Jo asuodsaz pue ‘ar0ysar1oy TaMOT ay} UO UOTSOIA puke UCT}TSOdap "eQuaneine
azoyssuo,T jo AZTIOTaA UO SUOT}TpUOD pUTM Ted0OT puke ‘s}d9zzZa Tepr} ‘SOTAST
-zayoereys aaem ‘Arjawoas yoeeq JO aduany,Jur SepntTouT Apnzys *siazndwos
peads-ysty 10x pawweiso1d se stsATeue uotTssaiseitj}[NW IBaUTT TetjUenbas
Aq pazestysaaut a8 satqetieA ydeaq Suowe suoT}IeIa}zUuT FO Jaqunu y

‘ dad IdISSVIONN Z “ON WNGNVYOWSW TVOINHOSL
“VY "N 940d AI

“) ‘m ‘utaqunay IIT *‘ar0q “Vv ‘N pue uostizey “mM Aq

‘mM ‘uOSTIz®H II SNOTLVNOA YOLOIGAYd ONINIVLEO YOs ANOINHOAL NOISSHUDTy

aTITL I -ILINW SALLVNUALIv (‘dd g) wnpuappy pue seotpuadde ¢

*p ‘satTqe} @T ‘"SNTIt p Sutpntour “dd 2OT “p96T TaquaAoNn

weiso0id raynduop ‘¢ utaquniy “O°M pue uostziey “mM Aq “YA “HOVAM VINIDUIA

eTUTSITA LV WALSAS HYIHdSOWLV-NVHOO-HOVA FHL JO SNOILOVYSINI
“yoeag BTUTSITA “2

sassaso0id azoys “T “Od “HSVM ‘“dO ‘UHINSO “SHU “OUONE IVLSVOO AWYV “S7n

bh
nea: ‘
Yili

Si
ai

*

y

*)

KK

mG ox

.