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“DAB 3: 3013'
WK ENDERY INC.

ABSTRACT

 

A STOCHASTIC MODEL FOR NEARSHORE COASTAL PROCESSES

By “tag
Louis Allen Orlowski

Quantification of the changes in nearshore topography

 

has proven difficult due to rapid. short-tern fluctuations
of bottom features. Structuring topographic transitions
into three states. (1) no significant deflections fro-
unifors slope. (2) positive deflections. and. (3) negative
deflections. and analysing state succession through tine
and space as a Markov chain allows for the description

of the evolution of nearshore coastal features. Data.

from the eastern shore of southern Lake Michigan (Davis

and Fox, 1971) indicate that such an approach is feasible.
Five clusters were defined, within which the topographic
response of the inner nearshore functions as a first order
larkov chain. Associated transition probabilities describe
process function through tine and space within the environ-
ments of the inner nearshore: bar. trough. subaqueous.\
terrace, and swash sons. Good correlation is obtained
between the larkov chain.nodel and eapirical interpretation
of nearshore topographic fluctuations. Stochastic process

Louis Allen Orlowski

aodels can serve as an accurate technique for the

description of coastal processes.

 

A STOCHASTIC MODEL FOR NEARSHORE COASTAL PROCESSES

BY
Louis Allen Orlowski

A THESIS

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

MASTER OF SCIENCE

Department of Geology

197“

 

TABLE OF

List of Tables. . . .

List of Figures . . .

IntrOdUCtiOne e e e e

Stochastic Processes.

Procedure . . . . . .

Discussion. . .
Cluster 1.
Cluster J.
Cluster K.
Cluster 0.
Cluster N.

Conclusions . .

Bibliography. .

ii

CONTENTS

Page
0 O O O O O O O O O O 111

O O O O O O O O O O C 1v

. . . . . . . . . . . h
. . . . . . . . . . . 8
. . . . . . . . . . . 27
. . . . . . . . . . . 30
. . . . . . . . . . . 30
. . . . . . . . . . . 32
. . . . . . . . . . . 33
. . . . . . . . . . . 33
. . . . . . . . . . . 34
. . . . . . . . . . . 36

 

Table 1.

Table 2 .

Table 3.

Table h.

LIST OF TABLES

Regression equations derived for
eaCh rmge I O O O O O O O O O O O O 0

10 by 3 contingency table for testing
of total independence along Range B .

2 by 3 contingency table for testing
of independence between stations
150.0 and 162.5. Range Be e e e e e e

Cluster characteristics for each
controlling station e e e e e e e e e

iii

Page

11

17

19

26

 

E]

 

Figure
Figure

Figure

Figure
Figure
Figure

Figure

1.
2.

3.

LIST OF FIGURES

Locality nap Of BtUdy areas e e e s e e

Tine-distance topographic map of
Range B, reproduced from Davis and
Fox (1971). O O O O O O O O O O 0 O O O

Lon shore profile along Range B on
7/2 70 with fitted regression line
and prediction limits . . . . . . . . .

Form of matrix used to develop tallies
and probabilities of state changes. . .

Station location map showing
independence of stations. . . . . . . .

Location map showing areas occupied
by first Order Clusters e e e e e e e e

Generalised nap of the inner and

outer nearshore zone of eastern
LakeMlchigan.............

iv

Page
3

9

12

20

22

28

 

INTRODUCTION

Fluctuations in bottom topography within the near-
shore sone are rapid and ephemeral. The types and sequences
of topographic transitions are known fron.observationa1
data. Quantification of such empirical data has proven
difficult due to the aforementioned variability and the
multiplicity of causative factors. These.factors. which
are both deterministic and probabilistic in nature. can
invoke sisilar responses in a natural system by any number
of process routes. This has limited the effectiveness of
deterministic modeling of the nearshore environment.)

A stochastic process model can describe these topog-
raphic variations on a probability basis. larkov chains.
a stochastic process. structure the response of the system
as a finite number of states and evaluate the probability
of state succession through tise. By identifying those
areas that have siailar probabilities of topographic
transition. areas as similar process response can be
delineated and investigated.

Such an analysis of the nearshore sone froa.eastern
Lake lichigan has been undertaken for this study. The
topographic data were obtained from a sequence of tine
distance maps ease-bled by Davis and Fox (1971) for the

J

 

 

2
Office of Naval Research. A thirty day tine series study
of the nearshore environment was conducted 2.2 ailes north
of the Black River at Holland. Michigan: Ottawa County.
ISN. R16W. Section 21 (Figure 1). At this locality nine
ranges 100 feet apart were surveyed along the north-south

 

trending shoreline. Along each range. bottom elevation
was aeasured at ten stations at 12.5 foot intervals fro:
50.0 feet. outward to 162.5 feet. Contouring was based
upon a 0.5 1,0.5 foot contour interval and reflected
depth of water or bottoa elevation with respect to stilled

 

water level.

Relating topographic variations in the nearshore
zone. at Holland. Michigan. to Markov chains has resulted
in the outlining of five clusters of similar process
response. Analyses of the probabilities of state occur-
rence. up. down. and unifbrn slope. and of the internal
chain structure within these clusters have indicated a
good correlation between the stochastic process model and
the known sequence of events occurring within the near-
shore sone. This study indicates that larkov chains are
an accurate model for the description and simulation of

nearshore processes.

Figure 1 - locality map of study area.

4 k” ___~’_——_————fi

3...: v

_ _ + L s
N _ o N: _

 

llll'llll'lllldoldgdHFQIIIIIIII

F)
was... xofim
ozqsnox

2<o=._o_s_ My?)

 

 

 

 

 

 

30E _N ._
20H owm .

 

 

 

 

 

h
STOCHASTIC PROCESSES

Natural phenomena can be approximated through model-
ing. Applicable mathematical models can be classed as
deterministic. random. and/or stochastic.

In a deterministic model the state of the system at
any point in time or space can be exactly predicted from
knowledge of the functional relations specified by appli-
cable differential equations. The variability and complexity
of geologic systems and the restriction to sampling of
available populations limits the accurate description of
the true geologic population by deterministic para-eters.

A random model is characterised by the independence
of the state of the system at any point in time or space
from any other point. Preceding events have no effect
upon the present state of the system. Occurrence of a
state is based upon fixsd probabilities determined by
the proportion of occurrence within the sample. That is.
the long term occurrence of a state in the model cannot
exceed that of the initial input data taken from the
natural system. Population estimators are biased and
predictions inaccurate.

Stochastic models. of which larkov chains are a type.
are intermediate between deterministic and random. lost
natural processes are stochastic. exhibiting a predictable
random behavior. That is. deterministic and probabilistic
elements of a process produce a response. the occurrence

of which is random. This random occurrence in stochastic

 

 

 

5

processes is controlled by probabilistic mechanisms which
allow statistically for the complete description of the
phenomena by its probability distribution. A stochastic
model specifies the complete Joint probability distribution
of the observations at each point of time. Viewed as a
continuous development in time. the natural process can
be termed stochastic.

Iathematically. a stochastic process can be described
in the following manner:

Xt. ts T (1)

where

It I a family of random variables.

t I the index time. and

T I the index set of the process. the population.
For each t in the population T. there exists a random
variable It. With the index t taken as time. It denotes
the state of the process at time t. When T is countable.
the process is a discrete-time process. All possible
values of It define the state space of the process. when
It is countable it is a discrete-state process. Thus. a
stochastic process consists of a family of random variables
that describes the evolution through time and space of
some natural process.

A discrete-time. discrete-state larkov chain is the
particular stochastic process investigated in this study.
A larkcv chain is a sequence of states. the order of

occurrence of which is determined by the transition

6
probabilities associated with the immediately preceding
state. A process so structured has a.memory relationship
termed first-order. Nathematically. it can be represented
as.

P(xt+1 I j/Xt I i) I P13 (2)
where

P I probability.

i: I the 11th cell in a matrix. Figure 4.

xt+1 I an observation at time t + 1.

It I an observation at time t.

i. J I states of the system. and

P13 I the probability associated with an 1: transition.
That is. the probability that the observation I at time
t + 1 will be equal to the state 1. given that at time t
the observation X is equal to the state i. is the Joint
probability P11. This defines the one-step transition
probabilities. A process satisfying Equation 2 is said
to possess the Markov property. This property indicates
that the order of state succession in the chain is indepen-
dent of the process route prior to the immediately preceding
state.

Observed transitions from a process lacking a first-
order dependence. the larkov property. are stochastically
independent random variables having a memory relationship
termed sero-order. It is represented as.

put“ - j/xt . 1) . 11,1 (3)

where

7

P1 I the probability associated with a J transition.
That is. the probability of a transition to the state 1.
at time t e 1. given that the process was in state i. at
time t. is equal to the probability P1. The immediately
preceding state of the system has no influence upon the
present state of the system. so the state transition is
independent. The probability P3 is determined by the

percentage of occurrence of the state 1 in the sample set.

8
PROCEDURE

Depths of water or bottom elevation at all station
locations were recorded daily by Davis and Fox (1971)
during the thirty day observation period. These observa-
tions. taken from the time-distance topographic maps
(Figure 2). are the source of all subsequent generated
data.

Subtle. small-scale. topographic variation within the
nearshore zone was objectively detected and classified by
a reference line defined by least squares linear regression
techniques (Draper and Smith. 1966). Use of linear regres-
sion replaces the subjective definition of states from
longshore profiles. Linear regression determines the
functional relationship between two variables. x and f.
in terms of a linear function. where a change in the value
of X is reflected in the response of the variable I. This
relationship allows regression to be used as a predictive
tool of the response of I. over the investigated range of
x. The regression equation is of the following form:

21 - b0 + blxl (‘0
where _

‘1 I the estimated value of ’1 for a given x1.

0 I the Y intercept.

b1 I the regression coefficient or slope factor. and
11 I the given value 11.
The parameters b0 and b1 are estimated by least squares

Figure 2 - Time-distance topographic map of Range B.
reproduced from Davis and Fox (1971).

 

 

 

 

 

 

 

 

 

Io<mm hum...

 

Ts..n.4_4_4ce__ssacc__s_r_4
hm nu MN_N m_tm.m_:m.\. on

10
which minimizes the sum of squares of deviation from regres-
sion. The regression calculations were performed using
the Stat System Version Three Least Squares Program
(Michigan State University. 1973). A measure of how
accurately the regression predicts the response of Y is
the coefficient of determination. R2. (Draper and Smith.
1966). which determines the proportion of the total vari-
ability in Y associated with the variability in X. With
depth of water the dependent variable I and distance out-
ward from the shore the independent variable x. the reference
line for each range was obtained. to which all other topog-
raphic variations are compared. Data for regression
analyses were chosen from a day during the study period
when the recorded elevations along a range approximated a
uniform slope and the longshore profile was featureless.
The regression equations obtained (Table l) from.these data
mathematically define the negative sloping longshore
profiles (Figure 3). Equations were chosen that give a
high coefficient of determination. a significant F test
in the analysis of variance. a low error mean square or
deviation from regression for the setting of prediction
limits. and a one per cent slope on the regression
coefficimt.

Prediction limits (Sokal and Rohlf. 1969) for a single
other predicted value of bottom elevation were computed to
the reference line. Observations that fall outside of these
limits are statistically significant topographic expressions

11
TABLE 1. Regression equations derived for each range.

 

Date of Data Used for

 

 

WW 31811511211
A r - -0.8‘+ + (-.01)x 7/1u/7o
100 Y = ~1.33 + (-.01)x 7/ 7/70
200 Y - -1.33 + (-.01)x 7/ 2/70
300 Y - -1.16 + (-.o1)x 7/ 2/70
s r . -0.87 + (-.01)x 7/ 2/70
500 Y - -1.12 4- (-.01)x 7/1u/7o
600 r . -1.21 + (-.01)x 7/ u/7o
700 r - -1.23 + (-.01)x 7/17/70
0 Y a -1.u6 + (-.01)x 6/30/70

 

Figure 3 - longshore profile along Range B on 7/2/70
3:111 fitted regression line and prediction
t'e

12

 

1L

m20_._.<._.m

0.00.

3..—23 20_._.0_Qmmn_

NZ... 20_mmwm0wm
N_x .xm ....mw>

0.0K.

 

xA .0.; 1. No.0- u>

0.00

 

Ihamo

1
e
p
t
d
. 0
lo
I e

 

13
in all stages of development and magnitude. The functional
relationship defined by X and Y in the regression equation
is employed to predict the outcome of other observations.
The standard error (Sokal and Rohlf. 1969). sy. for a
single predicted value of Y1 based on a given value of X1

is derived in the following manner:

2
A 2 -
SyI\Fy.x(l+%+L§l;zfl-) (5)

 

where
82
y o x I the error mean square obtained from the
regression.
Xi I the given values of X. i I 1. "’. n.
1' I the mean of the 1's. and
z x2 I the sum of the squares of the 1's.

With n-2 degrees of freedom. prediction limits (Sokal and
Rohlf. 1969) are computed by the following formula.

P(Y1 - tat s’ 3%,. :21 4- ta: 5),) I 1 -oc (6)
where

ii I the estimated value of T. for a given Xi.

ta: I the table value from the t distribution.
Vithci I .05 and n92 degrees of freedom.

S I the standard error of a predicted value of
y Yi' for a given x1. and

fl ’1 I the parametric value of Y1.
These limits delineate two hyperbole about the regression
line (Figure 3). The prediction lflmits form an envelope
of acceptance about the one per cent slope. Recorded
values of elevation. from the time-distance topographic
maps. that fall outside of the range of the prediction

14
limits represent statistically significant departures from
this slope.

All observations can now be categorized into three
states. (1) no significant deflection from slope (slope).
(2) positive deflections (up). and (3) negative deflections
(down) (Figure I) so that the response of the system to
any process is determined by bottom elevations. So
structured. the nearshore acne can be observed as a
discrete-time. discrete-state stochastic process. The
fsmily of random variables formed by the states and their
order of transition through time describes the evolution
of nearshore coastal features as a Markov chain.

Station locations based on a small interval of
separation were tested for independence of observations
based upon contingency table tests of observed and expected
frequency of state occurrence (Sokal and Rohlf. 1969).
Statistical independence exists if the probability of two
events occurring together is the product of their separate
probabilities. That is.

in - P(X - xi and Y - yj) . P(x - :1) - P(Y - yj) (7)

where
f11 I the obggrved frequency of occurrence in
the 1: cell of the tally matrix.
x. Y I distinct. random variables. and

x1. yJ I the corresponding states for X at i and
T at 3 for all 11.

Contingency tables. based upon this required property of
probabilities. determine if three properties occurring in

Figure I - Form of matrix used to develop tallies and
babilities of state changes. States are
l) slope. where there is no significant
deflection from a unifora 0.01 slope. (2)
up. where deflection is positive. as in a
bar. and (3) down. where deflection is
negative as in a trough.

SLOPE
STATE AT
UP
TTME t
DOWN

15

 

 

 

 

 

 

 

STATE AT TIME ++I
SLOPE UP DOWN
pn Ffiz W3
p2| p22 p23
p3! p32 933
l 2 3

 

16
two states are independent. Hypotheses of the forms

“0 a the row is independent of the column
classification (Figure h)

H1 3 H0 is not true
are tested from tables of the Chi-square distribution.
The computational formula (Sokal and Rohlf. 1969) is of
the following form:
x2 = 2 (0n - 31112 (8)
Eli

 

where

01: I thghobserved frequency of occurrence in the
ij cell. and

311 I th hexpected frequency of occurrence in the
i1 cell.

Bi: is computed as.
311-111011-1 (9)

nee
where

n11 I the number of observations in the 11th cell.

n1, I r I the column marginal totals.
2 nij
i- 1

no: I c I the row marginal totals. and
z nij
1.- l

n00 I re I the grand total
2 1101,10
13

Based upon (r-1)(c-1) degrees of freedom. this is a model
two test in which the n . j marginal totals are fixed.
Initially. 10 by 2 tables (Table 2) for an entire range were
computed to determine if total independence occurred. It

1?

 

 

 

 

 

TABLE 2. 10 by 3 contingency table for testing of total
independence along Range B.
mm IE 29!!!
WWW—.m— mi
50.0 25 21.51 0 3.53 5 4.9” 30
62.5 2“ 21.51 0 3.53 6 4.9# 30
75.0 27 21.51 0 3.53 3 b.9b 30
87.5 25 21.51 3 3.53 2 “.9“ 30_
100.0 26 21.51 3 3.53 l 4.94 30
112.5 18 21.51 12 3.53 O #.9# 30
125.0 18 21.51 12 3.53 0 n.94 30
137.5 25 21.51 4 3.53 l “.94 30
150.0 21 21.51 1 3.53 8 “.90 30
162.5 I 19.36 0 3.18 23 #.#5 27
"i. 213 35 “9 297 no-

 

 

 

 

 

e I observed frequency

e I expected frequency

2
x 005(9)(2)
Izusing Equation 8 I 172.2“57

I 28.869

Reject Ho

18
did not. Station location pairs. progressively outward
from the shore. were then compared in 2 by 3 tables
(Table 3). From these tables the areal distribution of
independence was determined (Figure 5). No discernible
patterns are apparent and dependence is not restricted to
any range of stations.

A random variable is a.mathematical entity occurring
from probabilistic mechanisms. just as a systematic
variable occurs from deterministic mechanisms (Ross. 1972).
A random variable is a real valued function defined on a A
probability space such that the probability of sets of
the following form can be defined:

(w/X(w)_<_x) (10)
where

to I an element of a random phenomena. and

x( ) I the set of points at which the value of X
“ does not exceed x.

For every x there is a distribution function of the random
variable x. such that.

”(x)-P(w/X(w):x) (11)

IP(X:X)

where

F is non-decreasing and right continuous.

0 §|PX(X ) 5_1. and

1im rx(x ) I 1.

x-Ho

Topographic transitions. observed as a process
response through time. were tested for the first order

19

TABLE 3. 2 by 3 contingency table for testing of
independence between stations 150.0 and
162.5. Range Be

 

 

 

 

.STA:E_ $1022. UP 2:00!n_____.
Statics....c__Ji__JL___._2__n____a___141__J1__sL__..nai.
150.0 21 13.15 1 0.52 8 16.31 30
162.5 a 11.84 0 o.u7 23 1n.63 27
n1. 25 l 31 57 no-

 

 

 

 

 

o I observed frequency

e I expected frequency
2
x .os(1)<2> ‘ 5'991

2 using Equation 8 I 19.7396 Reject Ho

Figure 5 - Station location map showing independence
of stations.

20

 

 

 

 

 

e . fl 1. _l
. 2 .1 u 0.8.
e .r H . ..I
H . . . . . I Ohm.
o . # a e e e M .I 0.00—
fi 1 e e e e e T 0.0“
H e e . H e e H ..l 0.0m
¢ _ — a _ T d a
00_ 00m 00m m 000 000 00x. 0

mw0z<m m20_._.<._.m

O

Hzmozwawog

elItllls ...zwozwawo

 

21
Markov chain property. as opposed to a sequence of
stochastically independent. identically distributed.
random variables. Testing is based upon the likelihood
ratio criterion (Anderson and Goodman. 1957) and takes
the following computational form:

-2 logel I 2 (n: r11.1 loge (PM) ) (12)
i3 P.1
where
.1 I the likelihood ratio criterion.

m I the number of states.

ni.1 I the number of observations in the 11th cell
of the probability transition matrix.

P13 I the probability in the iith cell. and

P.1 I the marginal probability for the 1th column.
The asymptotic distribution of -2 logell is based upon a
Chi-square distribution with (m-l)2 degrees of freedom.
A,map of the study area was obtained that outlines these
stations that possess the Markov property (Figure 6).
Several clusters are apparent. within which the associated
transition probabilities show a first order Markov
property. These clusters appear to delineate areas
subject to similar process. The resulting map is a fixed
distance analysis.

Tests for stationarity in time and space were
conducted on the transition probability matrices within
clusters to substantiate that the Markov properties of
the clustered stations justify grouping. In a stationary

Figure 6 - Location map showing areas occupied by
first order clusters. 1. J. K. 0. N
designate clusters.

22

 

 

 

 

 

 

1 0.09
r 0.3.
T 0.00_
T
.. 0.2.
T
T 0.0m
0m. 010m 0? m 00H 0mm 0%
802$ 020.25
a 20:30 405200 39000.. . 20.25

23

process the transition probabilities are approximately
equal and constant through time or space (Harbaugh and
Bonham-Carter. 1970). In a stationary first-order'Marhov
chain. P11 is the probability of transition from state i
at time t to j at time t + 1. In a nonstationary chain.
probabilities vary in time or space. and P13 (t) becomes
a function of time or space. The hypotheses tested are: 5‘7’
Ho 3 P13 (t) I P1: stationary

H1 1 P11 (t) I Pij nonstationary
Testing of these hypotheses is based upon the likelihood
ratio criterion and is similar in form to that used to
determine the order of the chain.(Equation 12). A station

is designated as the control station. to which all other

 

stations are compared for stationarity. by means of the

following equations T P (t)
I
-2 log. 1 - 2 (z 2 nn(t) 1.3. (gt-a ) (13)

t I 1 i:
where
l .. the likelihood ratio criterion.
t I l. ..., T I the number of subintervals.
m I the number of states.

n1j(t) .. the tally matrix in the 13‘” «11 of the
tth subinterval.

P1j(t) - the prob 111 matrix in the 11““ cell
of the t sub nterval. and

P1.1 I the probability matrix in the 13th cell
of the testing location or control station.

This is distributed as a Chi-square with m(mrl)(T-1)

2»

degrees of freedom. All stations within a cluster were
tested as the control station using Equation 13. All
clusters were found to be stationary with respect to at
least one station within it. This allows for the grouping
together of a number of station locations which possess the
Markov property of a first order chain. One station was
chosen from each cluster to characterise the cluster. This
station functions as the control station in comparison to
which the other stations in the cluster are stationary.

The one step transition probabilities. Pijl I P11.
was introduced in the discussion on.Markov chains. An
n-step transition probability (Ross. 1972). P1j(n). is
defined as the probability that a process in state i will
be in state 1 after h transitions. Mathematically. the
relation is represented as:

P13(n) , P13(t) x Pij(t+l) (1a)
where

P11(n) I the probability matrix after n transitions.

p11“) - the probability matrix at time t. and.

P11(t+1) I the probability matrix at time t+1
In essence the n-step transition.probability'matrix is
obtained by the matrix multiplication of the initial
matrix. Pij(1)' by itself n times. As n.+-I. P1J(n)
converges to a value which is identical for all is all
rows in the matrix are equal. This value. the limiting
probability. is the probability that after n transitions.
the process will be in state 1. independent of the initial

25
state. The limiting probability is also equal to the
proportion of the time that the process will occupy state
1. Limiting probability matrices were computed for the
control station for each cluster to obtain the limiting
probability matrix or equilibrium matrix characterising
the cluster. Table 0 lists the tally. probability. and
limiting probability matrices for each cluster.

State 3 communicates or is accessible from.state i
if P11”) > o. n z, o (Kemeny and Snell. 1960). The
communication is denoted by i‘r+j. Any two states that
communicate are in the same class. The concept of
communication divides the state space. E It. into
separate classes. A finite. irreduciblrfiarkov chain is
composed of one class. For any state i. let f1 denote
the probability that starting in i the process will ever
return to i. State 1 is recurrent if f1 I 1 and transient
if f1<z 1. If recurrent the process will return to state
i infinitely often. if transient there will be a positive
probability. 1 - f1. that the process will never return
to state i. Recurrence and transience are class properties.
If the process returns to state i in finite time. the
state i is positive recurrent. All recurrent states in
a finite state irreducible Markov chain are positive
recurrent. The communication of states and the internal
structure of the chain for the controlling station for
each cluster is evaluated in the separate discussions of
the clusters in the following section.

26

 

 

 

 

 

 

 

 

TABLE 4. Cluster characteristics for each controlling
station.
Cluster
(See Tally Probability Limiting Probability
....1uuuds11, M31213 IIIHJ£11_______.
s U 01'
I S 6 2 4 .50 .17 .33 .3757 .0852 .5391
U 3 1 0 .75 .25 .00 .3757 .0852 .5391
D 3 0 10 .23 .00 .77 .3757 .0852 .5391
J
Sub 1 6 0 2 .75 .00 .25 .3590 .0000 .6410
O 0 0 .00 .00 .00 .0000 .0000 .0000
3 O 18 .14 .00 .86 .3590 .0000 .6410
Sub 2 h 3 3 .40 .30 .30 .3212 .0350 .2309
2 7 0 .22 .78 .00 .3212 .4380 .2409
4 O 6 .40 .00 .60 .3212 .4380 .2409
K 11 1 2 .79 .07 .14 .4843 .0696 .4461
1 0 1 .50 .00 .50 .4843 .0696 .4461
2 1 10 .15 .08 .77 .4843 .0696 .4461
O 7 3 .70 .00 .30 .3478 .0000 .6522
0 0 0 .00 .00 .00 .0000 .0000 .0000
3 0 16 .16 .00 .84 .3478 .0000 .6522
N 3 0 .70 .30 .00 .3478 .6522 .0000
3 16 0 .16 .84 .00 .3478 .6522 .0000
0 0 .00 .00 .00 .0000 .0000 .0000

 

 

 

 

1. A11 3 by 3 matrices have the indices S. U. D. which
stand for the states slope. up. and down. See text
for explanation.

27
DISCUSSION

Davis and Fox (1971) have noted the presence of two
or more distinct bars at approximately 300 and 500 feet
outward from the shoreline. These true sand bars are
fairly continuous. are storm generated. and have an
associated trough on their shoreward side (Figure 7).

The boundary that separates the nearshore sone from the
nearest trough. the inner from the outer nearshore. is the
four foot depth contour line (Figures 2 and 7). and is
approximately 175 feet offshore. Within the inner near-

 

shore sone a shallow bar is usually present approximately
100 feet offshore. This wave generated ephemeral bar is
normally discontinuous. being interrupted by rip channels.
and is ocassionally eroded away completely by storm-
generated waves. There is usually an associated trough-
like structure in front of the ephemeral bar. The trough
sometimes grades off into a subaqueous terrace. The
ephemeral bar form does not migrate in the direction of
littoral drift but seems to oscillate within the 100.0 to
125.0 foot range.

The five defined clusters span the environments of
ephemeral bar and trough development along with the swash
some and the area dropping into the first true trough at
the four feet contour line. Analysis of the internal
chain structure of the controlling station for each cluster
quantifies and supports the observational data on the

Figure 7 - Generalised map of the inner and outer
nearshore some of eastern Lake Michigan.
Ladif'ifid from Davis and Fox (1971). Not
so e.

28

 

 

 

500' TRUE BAR

 

 

 

 

 

 

 

 

 

_1=_ __.\\\~_
OUTER TROUGH
NEARSHORE
\f’ "E—
300' TRUE BAR
MM
TROUGH

 

 

I75'jLTTTT-~\\\\\‘~‘~___fi__flafiflfl_ ...1
4 FOOT CONTOUR LINE
INNER

NEARSHORE

EPHEMERAL

IOO' BAR R”: EPHEMERAL BAR

CHANNEL

A

BEACH

 

29
behavior of the inner. ephemeral bar and trough structures
within the nearshore zone. It appears that there is a
tendency by the inner bar to develop the equilibrium
configuration that characterises the true. outer bars.
That is. the inner bar tends to develop a continuous bar
structure within the nearshore. However. large amounts
of water congregated into rip channels cut across the bar
form and prevent a continuous bar from forming. The
result is that the bar form oscillates back and forth
within its location range. The large central portion of
the study area (Figure 6) is probably one of these rip
channels. The continuous. rapid deposition and erosion
prevent the topographic fluctuations within this area
from having a significant effect upon each other. hence
they are stochastically independent. random.variab1es.
These areas that fall into the clusters (Figure 6) are
less likely to fluctuate than the central. rip channel
area. so they are more likely to show continuity of
process through time. An analyses of each cluster.
the environment the cluster spans. and the way in which
the sediment responds in topographic fluctuations. as
described by the chain structure. follows.

30

91:13:12.1

Cluster I is situated at the far end of ranges A and
100 (Figure 6) offshore of the area of ephemeral bar
activity and near the slope into the first_true trough.
The chain structure is that of a finite. positive recurrent.
irreducible Markov chain. Within the single class. all
states communicate (Table 4) and all states can succeed
themselves. The only single step transitions not possible
are up - down and the converse. The up/down states
communicate through the state slope. indicating a time
period of approximately one day as a.minimum. to proceed
through the transition. Storm induced topographic
activity is significant in this sons. but the response of
the system is slow. The limiting probability matrix
indicates approximately 54 per cent of process time is
spent in the down state. a reflection of the first true
trough. 38 per cent in slope. and 8 per cent in up. due
to the occasional development of large bars originating
from within the 100.0 to 125.0 foot range. Erosion

appears the dominant factor in this some.

911mm
Cluster J is located along the first three southern

ranges. A. 100. 200. and spans the area of ephemeral
trough and bar development. 0f the eleven included
stations five can function as the control station. with
the majority of these stations falling in the trough

area. Two subclusters which divide the environments. can

31

be recognised.

Snislusisz_1
In the trough sons. from 87.5 feet shoreward. the

structure is that of a finite. positive recurrent.
irreducible Markov chain with zero probability of the
occurrence of the state up (Table 4). All other states
communicate. either succeeding themselves or each other.
Erosion is the dominant factor with sediment accumulation
never surpassing uniform slope. Limiting probabilities
indicate approximately 64 per cent of overall time is
spent in the state down. reflecting vigorous trough
activity in front of the ephemeral bars. 36 per cent in
slope. A corollary to this is that bar development in

the adjacent area (Subcluster 2) occurs and the bar can
move lakeward but does not migrate shoreward. Transported
sediment from the crest of the bar into this area is
reflected in the slepe state. but the sediment is removed
by erosion before the bar form can migrate shoreward. This
substantiates and quantifies the empirical observation
that there is a net displacement down drift of the
sediment by littoral drift. but not of the bar form itself.

Suhslusta:_z
The chain structure within the bar development area.

100.0 to 125.0 feet offshore. is similar to that of Cluster
1 in structure. Single step transition between the states
up and down is impossible. with communication through

32
slope being required (Table 4). The major difference.
which is apparent in the limiting probability matrix. is
that within this area 44 per cent of process time is spent
in bar development and maintenance. Deposition is the
primary factor. The ephemeral nature of these bars is
indicated by the fact that within this area there is a
probability that the down state can be attained. reflecting
the complete removal of the bar form.

W
Cluster K is situated along Range 200 and is composed

of three stations located near the swash sons. The chain
structure (Table 4) is similar to that of the previous
clusters. In this area. which is characterised by ridge
and runnel systems. all states can succeed themselves
except up. When in the up state there is an equal
probability of transition to either the slope or down
states. This indicates that large amounts of sediment
introduced into this area are rapidly (approximately one
day maximum lag time) removed in either forward transport.
which results in welding to the beach proper. or transport
in the direction of littoral drift. or the sediment is
distributed evenly about the area. Direct transition
between the states up and down. which is possible in this
case. is also indicative of the rapid. erosive nature of
this some. Limiting probabilities show the long term
percentage of occurrence to be nearly equally distributed

between slope and down. 48 per cent and 45 per cent of

33
the time respectively. with an average of 7 per cent of the

time available for large sediment buildups.

mm
Cluster 0 is located along ranges 600 and 700 from the

near swash sone to the base of ephemeral bar development.
Its structure (Table 4) is identical to that of Subcluster
1 of Cluster J in that both clusters encompass similar
environments. Conclusions regarding this cluster are

similar to those of Subcluster l.

£13311I_fl

Cluster N covers four ranges. 500. 600. 700. and C.
in the outer. northern corner of the study area. The area
of the cluster is within the acne of ephemeral bar develop-
ment. The structure is that of a finite. positive recurrent.
irreducible Markov chain with mere probability of the
occurrence of the down state (Table 4). All states can
succeed themselves or one another. Deposition is the major
factor. as the limiting probabilities indicate 65 per cent
of process time is devoted to bar development. Comparison
with the station locations of Cluster J and 0. indicates
that the central area of bar activity is located approx-
imately 12 to 25 feet lakeward of that in the southern

part of the study area.

34
CONCLUSIONS

Identification of clusters that encompass several
station locations. all of which are stationary to more than
one station. allows for the objective selection of the
probability transition matrix that is used to characterise
that cluster. In.most clusters. the similarity in chain
structure among possible control stations allows for
selection of a representative station to characterise the
process response of the cluster. Where chain structures
vary. a control station can be chosen which best fits the
observed environment. but this is a subjective choice of
the controlling station. This is an inherent weakness in
a simple first order model based on a fixed distance
analyses. A more complex model which would limit the
number of possible control stations is desirable.

Within all stations that respond as a first order
Markov chain. a plethora of chain structures was observed.
The stationary chain structure defined in each cluster
was that of a finite. positive recurrent. irreducible
Markov chain. This structure represents a stable con-
figuration in which all states occupying a single class
have complete intercommunication. Topographic transitions
in the nearshore. when structured as a stochastic process.
operate in this equilibrium.configuration. The probability
of‘major fluctuations in topography in response to large

external factors. such as storms. is accounted for in the

35

stochastic model. However the long term probabilities of
occurrence. as defined in the stable configuration of the
chain structure. reflects the dynamic equilibrium in the
nearshore zone between land and water.

Fluctuations within the central portion of the study
area are independent. with the succession of events lacking
any dependence upon previous events. Hypothesising upon
the significance or stability of this area would require
extrapolation outside the range of the data. Coastal
process on Lake Michigan undoubtedly have seasonal
variations. The data for this study were collected during
the month of July. It is impossible to determine if the
derived probabilities are stationary throughout the year.
and the ephemeral or permanent nature of this central
portion of independence cannot be ascertained from the
available data.

Topographic response to external factors in the near-
shore sone. when structured as a stochastic process. can
be modeled on the basis of probability distributions. This
allows for the quantification of empirical data and the
computer simulation of the nearshore environment. Observa-
tional data of process evolution are substantiated and

mathematically defined by Markov chain analysis.

BIBLIOGRAPHY

BIBLIOGRAPHY

Anderson. T. W. and Goodman. L. A.. 1957. Statistical
Inference About Markov Chains: Annals of Mathematical
St8ti$ti°8g Va 28' P. 89-110e

Davis. R. A. and Fox. W. T.. 1971. Beach and Nearshore
Dynamics in Eastern Lake Michigan: Office of Naval
Research. Technical Report 4. NONR Contract 388-092.

Draper. N. R. and Smith. 11.. 1966. W
Analysis: John Wiley. New York.

Hubaugh. Jo we ma Bonham-Cartel‘. Ge. 1.970s mm
W: John Wiley. New York.

Kemeny. J. G. and Snell. J. L.. 1960. W:

D. Van Nostrand. New York.

Michigan State University. 1973. Stat System Version Three.
L. S. Program. Computer Laboratory. East Lansing.
Michigan.

Rosa. s. M” 1972. WWW:
Academic Press. New Ybrk.

Sokal. R. R. and Rohlf. F. J.. 1969. Big-gm: W. H.
Freeman. San Francisco. California.

36