873
1 INTRODUCTION
The article considers the relevance of the information
theory for the presentation of aleatory uncertainties of
observable random wind wave properties. The study
entrusts in the sequel on the information entropy
measure for environmental probabilistic uncertainty
assessment as well as on the Global Wave Statistics [8]
(GWS) (Hogben, Dacunha, and Olliver, 1986) and
online [3] BMT databases (2021). The article
demonstrates in the sequel how to derive the
information entropy from all tabulated annual and
seasonal GWS ocean areas. The wave data sets contain
observed frequencies of wave heights and periods for
all principal wave directions. At the end, the results
and examples are presented in the article by
mathematical expressions, tables, graphs, and charts.
The article at the beginning refers to the
information entropy concept that emerged earlier in
the information theory [7,18,13] (Hartley, 1928;
Wiener, 1948; Shannon & Weaver, 1949). The
information entropy was generalized later in the
probability theory as the probabilistic uncertainty
measure [9,12,1,16] (Khinchin, 1957; Renyi, 1970;
Aczel and Daroczy, 1975; Tsallis, 1988) and also
recognized as information based on uncertainty [10]
(Klir and Wierman, 1999). The usefulness of the
information entropy concept in the probability theory
for objective uncertainty assessment of various
systems of random events in engineering and
elsewhere was investigated earlier [20] Ziha (2000a).
The information entropy may also appropriately
express the redundancy and robustness of different
operational and failure system states [20] Ziha
(2000b). The entropy of marginal distributions was
earlier considered for the uncertainty assessment of
Review of Ocean Wave’s Uncertainties for Navigators
Z. Kalman
University of Zagreb, Zagreb, Croatia
ABSTRACT: The uncertainties of marine environments lastingly challenge navigation and safety of sea
transportation. Therefore, the article tackles the extraction, assessment, and analysis as well as the perceptive
presentations of probabilistic uncertainties of the random wind waves ocean-wide. The link of the probabilistic
uncertainties and statistical variabilities is accomplished in the article by using the reported Global Wave
Statistics of coherent and controlled wind wave data visually observed from ships in normal service. The
probabilistic uncertainty is defined in the information theory most coherently with the human experience of
randomness by the information entropy. The article reveals expressions, tables, graphs, and charts of
information entropy which objectively express the uncertainties of observed wind wave directions, heights, and
periods in all principal ocean areas. The combinations of areal entropy provide uncertainties of wider ocean
zones, sectors, and shipping routes for the assessment of all-around exposures of ships and other objects in
service at seas to random wind wave effects appropriately to sea-men’s experience of randomness.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 18
Number 4
December 2024
DOI: 10.12716/1001.18.04.13
874
ocean wind waves [21] (Ziha, 2007). The probabilistic
uncertainty can also be perceptively presented by
average numbers of events or outcomes closer to the
experience of gambling [23] (Ziha, 2013). More
recently the information theory was applied to
assessments of integral system safety in engineering
[24] (Ziha, 2022).
Additional information to mere statistics of ocean
wind waves provide a novel prospect for more
objective assessments of the effects of uncertainties of
maritime environments on ships and other ocean
objects, structures and vessels.
2 SETS OF OBSERVATIONS OF RANDOM
EVENTS
A discrete distribution PN of N probabilities pi, i=1,2,
...,N of outcomes of random events is presented as:
( )
12
, , ,=
NN
p p pP
(1)
Distributions of N probabilities PN (1) can also be
interpreted as systems SN of N random events Ei with
probabilities pi, i=1,2, ...,N.
The disjoint random events Ej configure a system
SN that can be written in a form of an N-element finite
scheme (Khinchin, 1957) as follows:
( ) ( )
( )
( )
12
12
=
jN
N
jN
E E E E
p E p E p E p E
S
(2)
The probability of N probabilities PN (1) of a
distribution or of a system of N events SN (2) is then:
1
( ) ( ) 1
=
=
N
N N i
i
p or p pPS
(3)
In (3), for complete distributions is
( ) 1=
N
p P
and
for complete systems is
( ) 1=
N
p S
.
3 VARIABILITY OF PROBABILITY
DISTRIBUTIONS
Variability of a single random outcome of a
probability distribution (1) can be experienced by an
equivalent number
i
of outcomes with
hypothetically equal probabilities pi out of N possible
in system SN as defined:
( ) ( ) / 1/
==
i N i i
p p p pP
(4)
The following terms related to variability apply
both for partial and complete distributions PN (1) (3).
The mean value of a probability distribution PN (1)
is by definition:
( ) (1/ ) ( ) 1/= = =
N N N
p p N p NPP
(5)
The statistical variability of N probabilities pi of a
distribution PN (1) is given by the average variance
VN(P):
2 2 2 2
11
( ) ( ) (1/ ) ( ) (1/ )
==
= = − = −
NN
N N i N i N
ii
V N p p N p pPP
(6)
The standard deviation of probabilities of
distribution of probabilities using (6) is then:
( ) ( )
=
NN
VPP
(7)
The reference value of variance can be calculated
(Ziha, 2013) as the limiting value of (6) under the
condition that one probability
()→
jN
ppP
is
dominant and all the others
0
ij
p
→
are
vanishing:
22
Ref
( ) ( ) ( 1)/ ( 1)/= − = −
NN
V p N N N NPP
(8)
The coefficient of variation of N probabilities of a
distribution PN (1) from (5 and 7) (Table 1) is:
( ) ( ) / ( )
= =
N N N N
CV p NP P P
(9)
The reference value of the coefficient of variation
follows from (9) as shown:
Ref
( ) ( ) ( 1) 1= − = −
N
CV p N NPP
(10)
4 INFORMATION ENTROPY OF SYSTEMS OF
EVENTS
The information entropy of a single event Ei in a
system SN (2) can be interpreted according to [18]
Wiener (1948) either as a measure of the information
yielded by the event or how unexpected the event was
and can be defined based on the equivalent number of
outcomes (4)
()
i
p
as follows:
2 2 2
( ) ( ) log ( ) log 1/ log
= = = = −
i i i i i
E H E p p p
(11)
The information based on the uncertainty of a
complete system SN (2) of N events is the weighted
sum of the unexpectednesses (11) expressed by the
Shannon’s entropy [13] (Shannon and Weaver, 1949)
(Table 1):
11
( ) log
==
= = −
NN
N j j j j
jj
H p p pS
(12)
Shannon's entropy (12) has properties of
uncertainty: continuity in its arguments, monotonic
increase with a number of equiprobable outcomes and
a composition rule [6] (Cover and Thomas, 2006). The
information based on the uncertainty of an incomplete
system SN (2) can be defined as the limiting case of
the Renyi’s entropy of order one (Renyi, 1970), using
the probability of a system of event (3) and definition
of the Shannon's entropy:
875
1
1
1
( ) log
()
=
=−
N
R
N j j
j
H p p
p
S
S
(13)
Tsallis (1988) extended the information entropy
formalism to the [12] Renyi’s entropy (1970) (13) of
order one.
Shannon's axiomatic derivation explains [13] (1949)
why the entropy is an intuitive measure of
uncertainty. The uniqueness theorem by [9] Khinchin
(1957) states that the information entropy is the only
function that measures the probabilistic uncertainty in
agreement with experience.
Let us consider system S of L subsystems of Li
elements
12
()=
i
i i i iL
s s sS
, i=1,2,…,Li (Ziha, 2000).
The probabilities of i
th
subsystems are
12
()= + +
i
i i i iL
p s s sS
and the conditional information
entropy of the system S concerning for the subsystems
Si is as follows:
1
( / ) log
( ) ( )
=
= −
i
L
kk
i
k
ii
ss
H
pp
SS
SS
(14)
The information entropy of the system of
subsystems
12
' ( )
L
= S S S S
can be presented
as shown:
1
( ') ( ) log ( )
=
= −
L
L i i
i
H p pS S S
(15)
The theorem of mixture of distributions following
[9] Khinchin (1957) and [12] Renyi (1970) provides the
conditional information entropy of system S
concerning for the system of subsystems S’ using (12-
15) (proof in APPENDIX B), as follows
1
11
1
( / ') ( ) ( / )
()
1
( ) ( ') ( ) ( ')
()
=
= =
= − = −
N
jj
j
RR
N L N L
H p H
p
H H H H
p
S S S S S
S
S S S S
S
(16)
The additional knowledge of subdivision of
system S to subsystems S’ reduces unconditional
information based on uncertainty (13) for an amount
of (15). The incompleteness
( ) 1p S
increases the
system uncertainty (16). Conditional information (16)
of system S may be viewed as the average entropy of
subsystems S’.
The information entropy HN(S) is equal to zero
when the state of the system S can be surely
predicted, i.e., no uncertainty exists at all. This occurs
when one of the probabilities of events pi, i=1,2,...,N is
equal to one, let us say pk=1 and all other probabilities
are equal to zero, pi=0, ik.
The information entropy is maximal for uniform
distribution of event probabilities when all the events
are equally probable with the probability equal to pi=
1/N, for i=1, 2, ..., N, and it amounts to HN(S)max=logN
that is the [7] Hartley's entropy (1928). Hartley’s
entropy relates to Renyi’s entropy of order 0.
The entropy increases as the number of events
increases. The entropy does not depend on the
sequence of events. The definition of the unit of
information based on uncertainty measure according
to [12] Renyi (1970) is not more and not less arbitrary
than the choice of the unit of some physical quantity.
If the logarithm applied in (12-16) is of base two,
the unit of information entropy is denoted as one "bit".
One bit is the uncertainty of two equally probable
events (a simple alternative) like flipping an ideal
coin. If the natural logarithm is applied, the unit is one
"nit". Relative measures for entropy hN(S)=HN(S)/logN
may be useful. Interpretations of statistical variability
vs. probabilistic uncertainty are given in Table 1.
Table 1. Statistical variability vs. probabilistic uncertainty
(information entropy)
________________________________________________
Variability (Statistics)
________________________________________________
Coefficient of variation of probabilities (9)
2
1
0 ( ) 1 1
=
=
= − −
iN
Ni
i
CV N p NP
Min: 0 – Fully invariable
(All N outcomes equally probable)
Max:
1−N
– Maximal variability
(One sure outcome all N-1 others impossible)
Unit: 1– (One sure, one impossible N=2)
1/pi equivalent number of equiprobable random event
________________________________________________
Probabilistic uncertainty (Information theory)
________________________________________________
The information entropy of system (12)
,
1
0 ( ) log log
=
=
= −
iN
N b i b i b
i
H p p NS
Max:
log
b
N
– Fully uncertain
(All N events equally probable)
Min: 0 – Maximal certainty
(One certain event, N-1 impossible)
Unit: 1 bit (2 equally probable events)
-log pi unexpectedness of a random event
________________________________________________
4.1 Average numbers of events
It was mentioned earlier that the average number of
equally probable events provides the same
information as the considered system of events [1]
(Aczel and Daroczy, 1975). The average number of
equally probable events FN(S) follows from the
condition of maximal information
2
( ) log ( )=
NN
HFSS
of
a system of N events with average probability
1/ ( )
N
F S
as another perceptive uncertainty indicator defined as
shown:
()
( ) 2=
N
H
N
F
S
S
(17)
Subsequently, uncertainties of random phenomena
with arbitrary numbers of outcomes can be expressed
by a number FN(SN) (17) of equally probable events or
outcomes. Recursively, the entropy of base B of the
average number FN(SN) (17) provides the entropy
value of logB[FN(SN)]=HN(SN)logBN.
Relative measures for the average number of
events fN(S)=FN(SN)/N may be useful. It is commonly
perceptive that flipping an ideal coin is as uncertain as
two events with the same probabilities equal to 1/2 of
a system S2=(1/2 12) with entropy H2(S2)max=log22=1
bit and tossing a perfect die as six events with
probabilities 1/6 of a system S6=(1/6 1/6 1/6 1/6
1/6 1/6) with entropy H6(S6)max=log26=2.58 bits,
which is equivalent to flipping of 2.58 coins. Since
entropy (12-16) are, in general, real numbers, so are
the average numbers of equally probable events FN(S)
(17), providing continuous scales for interpretations of
876
uncertainties (Ziha, 2013). More generally, the
information entropy n using a logarithm of base b
provides average numbers b
n
of equally probable
events (17) as gambling with n-sided ideal objects.
Note how the numbers of equiprobable outcomes or
events are generalized to not only integer values in a
continuous scale.
5 INFORMATION AND UNCERTAINTY OF THE
GWS
Visual observations of commercial ships have been
archived since 1861. From 1961 the collection is
systematized according to a resolution of the WMO.
The compilation of these observations for each of the
NA=104 Marsden’s squares (Appendix A, Fig. A1) is
the Global Wave Statistics (GWS) [8] (Hogben,
Dacunha and Olliver, 1986) that uses the past
experiences to eliminate biases.
An introductory example presents the information
based on uncertainty and statistical variability of
observed calm pcalm and wavy pwavy periods of sea
states and unexpectedness (11) (Table 2).
The GWS integrated the wind/wave climate
observations on the global level in scatter diagrams of
joint distributions of wave heights against wind speed
in terms of the Beaufort scale and separate sets of
normalized wind frequencies. However, the GWS
does not account for local climate conditions such as
the size, the topography within/surrounding the
region, the fetch, and ocean surface currents. Some of
the local effects are directly or indirectly assessable
from the attached GWS charts. Monthly frequency
tables of wave heights and wind forces against the
direction, together with information on ice conditions
and the occurrences of tropical cyclones were used to
decide upon seasonal subdivisions in the GWS. The
advantages of GWS [4] (Choi and Hirayama, 2000) are
the global approach, the duration of the collection
period, and its suitability to maritime applications.
The drawbacks are the lack of local wave/wind
climate information particularly outside the oceanic
areas, and the poor accuracy for periods, where
heights are well estimated and enhanced by
experienced observers. The accuracy of the Global
Wave Statistics Data is checked with specific
instruments [2] (Bitner-Gregersen and Cramer, 1994).
The basic World Wide Waves Statistics (WWWS)
consists of wave model time series for a great number
of positions calibrated against Topex, Jason, or other
satellite data such as the Atlas of the oceans: wind and
wave climate from the GEOSAT satellite [19] (Young
and Holland, 1996), The GWS often serves as a
reference guide for wave data reported by other
sources [11] (Nielsen and Ikonomakis, 2021). For long-
term predictions of ship responses in ocean a practical
method for comparison of GWS with other wave data
was proposed [14] (Shinkai and Wan, 1996).
5.1 Uncertainties of GWS areas
The GWS wave properties data sets A:s,d are available
for each of 104 ocean area A and for Ns=4+1 annual
and seasonal observations denoted s=(annual), March-
May, June-August, September-November, December-
February as well as for Nd=8+1 wave directions
denoted d=(all), NW, N, NE, W, E, SW, S, SE. Each data
set A:s,d,h,t present Nh=15 significant wave heights
from 0 meters to 14 meters and Nt=11 zero-crossing
wave periods from 4 seconds to 13 seconds, (for
example Fig. 1 and Table 3 for A25:s=annnual,d=all).
The GWS ocean wave data sets in tabular form
contain the jointly observed frequencies of wave
heights h and periods t per 1000 of observations as
:,
( , )
A s d
p h t
. The unconditional information of a data set
A:s,d,h,t as complete systems (2) is defined by the
Shannon’s entropy of all observations (12) (Table 3):
: , : ,
( : , , , ) ( , ) log ( , )= −
A s d A s d
all h allt
H A s d h t p h t p h t
(18)
The unconditional information entropy of the
marginal distributions [21] (Ziha, 2007) of all heights
h=hw
: , : ,
( ) ( , ) 1==
A s d w A s d w
allt
p h p h t
and for periods t=tz
: , : ,
( ) ( , ) 1==
A s d z A s d z
allh
p t p h t
of the data set A:s,d (Fig. 1,
Table 3) is:
: , : ,
( : , , ) ( ) log ( )= −
w
A s d w A s d w
allh
H A s d h p h p h
(19)
: , : ,
( : , , ) ( ) log ( )= −
z
A s d z A s d z
allt
H A s d t p t p t
(20)
In addition to the unconditional entropy of the
marginal distribution of heights (19) and periods (20),
the conditional entropy with respect to a selected
wave height hw and period tz is defined in (APPENDIX
C).
0
0.05
0.1
0.15
0.2
0.25
0.3
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Probabilty of heights
Significant wave heights in meters
a) Distribution of heights in A25 all seasons all directions
Uncertainty (h)
Entropy H(h)=2.50 bits(20)
Average distribution
F(h)=5.65 (17)
1/F(h)=0.18
Statistics (h)
E(h)=2.76 m (5)
Var(h)=2.17 m
2
(6)
(h)=1.47 m (7)
CV(h)=0.53 (8)
Range 0-10 m
0
0.05
0.1
0.15
0.2
0.25
0.3
4 5 6 7 8 9 10 11 12 13 14
Probabilty of height class
Classes of zero crossing periods in seconds
b) Distribution of periods in A25 all seasons all directions
Statistics (t)
E(t)=8.09 s (5)
Var(t)=2.15 s
2
(6)
(t)=1.48 s (7)
CV(t)=0.18 (9)
Range 4-14 m
Uncertainty (t)
Entropy H(t)=2.58 bits (19)
Average number
F(t)=6.00 (17)
1/F(t)=0.16
Figure 1. Distributions of wave heights (a) and periods (b) in
GWS area A25
877
Table 2. Variability and uncertainty (information) of calm and wavy periods of sea states
___________________________________________________________________________________________________
pcalm (pcalm) pwavy (pwavy) Condition CV(9) H(12) F(17) Variability Uncertainty
___________________________________________________________________________________________________
0 1 0 Always wavy 1 0 1 Maximal Certain
1/10 3.32 9/10 0.15 Prevailing wavy 0.82 0.47 1.38 High Low
1/4 2 3/4 0.41 1/4 time calm 3/4 time wavy 0.5 0.81 1.75 Moderate Moderate
1/2 1 1/2 1 1/2 time calm 1/2 time wavy 0 1 2 Invariable Uncertain
4/5 0.3 1/5 2.32 3/4 time calm 1/t time wavy 0.6 0.72 1.65 Moderate Moderate
19/20 0.07 1/20 4.32 Prevailing calm 0.90 0.29 1.22 High Low
1 0 0 Always calm 1 0 1 Maximal Certain
___________________________________________________________________________________________________
Table 3. GWS data set A25:s=annual, d=all of wave heights and zero-crossing periods (SUPPLEMENT 1)
___________________________________________________________________________________________________
H(t)(20) 2.583 pA:s,d(ht) 0.007 0.053 0.172 0.268 0.247 0.152 0.069 0.023 0.008 0.001 =1
___________________________________________________________________________________________________
>14 A25:s=annual, d=all tz
13-14 H(A25)=4.85 bits (18)
12-13 F(A25)=29 ane (17)
11-12 H(A25:h)=2.50 bits(19) pA:s,d(hw)
10-11 H(A25:t)=2.58 bits (20) =1
1 9-10 pA:s,d(h,t) 0.001 0.001
4 8-9 0.001 0.001 0.001 0.001 0.004
5 7-8 0.001 0.002 0.002 0.002 0.001 0.008
6 6-7 0.002 0.004 0.005 0.004 0.002 0.001 0.018
7 5-6 hw 0.002 0.006 0.012 0.012 0.008 0.003 0.001 0.044
9 4-5 0.001 0.006 0.019 0.029 0.025 0.014 0.005 0.002 0.001 0.103
8 3-4 0.002 0.018 0.048 0.059 0.041 0.019 0.006 0.002 0.195
8 2-3 0.008 0.046 0.089 0.082 0.044 0.016 0.004 0.001 0.290
8 1-2 0.002 0.022 0.073 0.088 0.053 0.020 0.005 0.001 0.264
6 0-1 0.005 0.020 0.027 0.015 0.005 0.001 0.073
Nt=10 h / t <4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 >13 2.497
Nht=62 Nh=10 2 5 6 8 9 10 8 8 5 1 H(h)(19)
___________________________________________________________________________________________________
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50 60 70 80 90 100
Wave directions
Information based on uncertainty and variability of wave directions in GWS areas
T
The northern GWS areas A1-A30
The eqatorial GWS areas A31-A80
T
T
T
M
T
Average number of wave directions
Entropy of wave directions in bits
COV of wave directions
The southertn GWS areas A81-A104
Figure 2. Information and variability of all wave directions on annual basis (21) (SUPPLEMENT 2)
5.2 Uncertainties of wind wave directions of all GWS
areas
The entropy of a GWS area A with respect to the
probabilities pA,d of eight wave directions can be
assessed as an incomplete system by the
unconditional [12] Renyi’s entropy (13) where pA=pA,d
1 (Fig.2), as follows:
1
,,
1
( : , ) log= −
R
A d A d
alld
A
H A s d p p a
p
(21)
Circular statistics [5] (e.g. Fisher, 1993) provides
the circular mean angle and coefficient of variation.
Note how the high entropy H (12) and low coefficient
of variation CV (9) indicate high uniformity of wave
directions in GWS area A86 (Fig. 2a) compared to A64
(Fig. 2b).
0
0.05
0.1
0.15
N
NE
E
SE
S
SW
W
NW
a) Wave directions
A86
H=3.03
F=8.16
CV=0.13
Circular mean
angle 240 degrees
0
0.2
0.4
0.6
N
NE
E
SE
S
SW
W
NW
b) Wave directions
A64
H=1.6
F=3.0
CV=1.6
Circular mean
angle 141 degrees
Figures 2a and 2b. Statistics of wind wave directions in A86
and A64 GWS areas
5.3 Uncertainties of combinations of GWS data sets
The combinations of GWS data require the
information of all GWS areas (18, 19, 20)
(SUPPLEMENT 3).
A combination D consists of k GWS data sets Ai:s,d,
i=1,2,…,k, of areas A, seasons s, and directions d.
878
The relative participations of all data sets
pA=pA1:s,d+pA2:s,d+…+pAk:s,d1 provide the unconditional
Renyi’s entropy (13) of the area A, (e.g. Table 4, for
A25:annaul,all, NW, N, NE, W, E, SW, S, SE), as
follows:
1
: , : ,
1
1
( ) log
=
= −
ii
k
R
A s d A s d
i
A
H A p p
p
(22)
The conditional information entropy of k combined
data sets of the combination D follows from the
theorem of mixtures of distributions (16) with respect
to aggregates A of all partaking areas Ai with
participations
:,A s d
p
that is the average information of
jointly observed heights and periods (18) of all
components, as shown:
1
:,
1
1
( / , ) ( : , , , )
=
=
i
k
R
A s d i
i
A
H D h t p H A s d h t
p
(23)
The average or conditional entropy of marginal
distributions of heights
( / )H D h
and of periods
( / )H D t
of the combination D of the conditional
entropy of heights
( : , , )H A s d h
(19) and periods
( : , , )H A s d t
(20) reported in GWS are as shown:
1
:,
1
1
( / ) ( : , , )
=
=
i
k
R
A s d
i
A
H D h p H A s d h
p
(24)
1
:,
1
1
( / ) ( : , , )
=
=
i
k
R
A s d
i
A
H D t p H A s d t
p
(25)
The overall or unconditional information of a
combination D of all jointly observed probabilities of
heights and periods
:,
( , )
Ai s d
p h t
in proportion to the
relative participations of components
:,Ai s d
p
, is equal to
the sum of the unconditional (22) and the conditional
information (23), as shown:
1
: , : , : , : ,
1
11
1
( ) ( , ) log ( , )
( / , ) ( )
=
= − =
=+
k
R
Ai s d Ai s d Ai s d A is d
i
A
RR
H Dht p p h t p p h t
p
H D h t H A
(26)
The overall or unconditional entropy of marginal
distributions of heights (24) and of periods (25)
implies the unconditional information
1
()
R
HA
(22)
according to the theorem of mixtures of distribution
(16) as shown:
1 1 1
( ) ( / ) ( )=+
R R R
H Dh H D h H A
(27)
1 1 1
( ) ( / ) ( )=+
R R R
H Dt H D t H A
(28)
Succinctly, the combinations D of selected GWS
data sets (26-28) comprise the average uncertainties of
all componential data sets (23-25) augmented by the
overall area A uncertainty H(A) (22).
Table 4. Information on combination of all wave directions for A25:s=annual,d= all (SUPPLEMENT 4)
___________________________________________________________________________________________________
d Nht pA25:d H(A:h,t)bits F(A:h,t) Nh H(A:h) bits F(A:h) Nt H(A:t) bits F(A:t)
GWS GWS GWS (18) 2
H
GWS GWS (19) 2
H
GWS GWS (20) 2
H
___________________________________________________________________________________________________
NW 66 0.1288 4.90 29.9 10 2.55 5.9 10 2.58 6.0
N 57 0.1770 4.67 25.5 9 2.35 5.1 10 2.53 5.8
NE 48 0.1887 4.39 20.9 8 2.19 4.6 8 2.38 5.2
W 66 0.1251 4.89 29.7 11 2.60 6.0 9 2.51 5.7
E 47 0.0914 4.32 19.9 8 2.16 4.5 8 2.33 5.0
SW 55 0.1046 4.53 23.0 10 2.36 5.1 8 2.35 5.1
S 50 0.1043 4.37 20.7 9 2.27 4.8 7 2.28 4.9
SE 44 0.0638 4.27 19.3 8 2.17 4.5 7 2.26 4.9
GWS all 433 pA=0.9838 4.86 28.9 73 2.50 5.6 67 2.58 6.0
Conditional 54 (23) 4.57 23.7 9 (24) 2.34 5.1 8 (25) 2.42 5.36
Unconditional (26) 7.52 183 (27) 5.29 39 (28) 5.37 41
___________________________________________________________________________________________________
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0 10 20 30 40 50 60 70 80 90 100
Wave heights in meters
Information based on uncertainty and variability of wave heights in GWS areas
T
The norhertn GWS areas A1-
The eqatorial GWS areas A31-
Maximal observed wave heights
Average number of wave heights
Variance of wave heights
Entropy of wave heights in bits
The southern GWS areas A81-A104
Mean value of wave heights
Mean value 4.9
Figure 3. Information and variability of all wave periods on annual basis (19)
879
0
1
2
3
4
5
6
7
8
9
10
11
12
0 10 20 30 40 50 60 70 80 90 100
Wave periods in seconds
Information based on uncertainty and variability of wave periods in GWS areas
T
The eqatorial GWS areas A31-A80
T
The norhertn GWS areas A1-A30
The southertn GWS areas A81-A104
Maximal observed wave periods
Average number of wave periods
Entropy of wave periods in bits
Variance of wave periods
Mean value of wave periods
Mean value 5.6
Figure 4. Information and variability of all wave periods on annual basis (20)
6 TRACING THE OCEAN-WIDE INFORMATION
BASED ON UNCERTAINTIES
The article also presents the statistical means (5),
variances (6) and maximal observed values as well as
the entropy in bits and the average numbers (17) of
wave heights (19) (Fig. 3) and periods (20) for all wave
directions on annual basis (Fig. 4) (SUPPLEMENTS
from A1-a-a to A104.a.a).
The study next presents the unconditional
information (22), the observed maximal values (GWS),
conditional and unconditional information of wave
heights/periods (23, 26) as well as of heights (24, 27)
and periods (25, 28) of the entire GWS and of zones of
combinations D of selected GWS areas (Table 5).
The study also presents the unconditional
information (22), the observed maximal values (GWS),
conditional and unconditional information of wave
heights/ (23, 26) as well as of heights (24, 27) and
periods (25, 28) of the oceanic GWS zones of
combinations D of selected GWS areas (Table 6).
The article also presents the charts of average
numbers (17) based on information entropy of wave
directions (21), jointly observed heights/periods (18),
wave heights (19), and wave periods (20) on annual
basis for wave directions in all 104 GWS areas
(Appendix A, charts). (SUPPLEMENT 6)
The annual numbers of wind wave directions (21)
are given in categories 3 to 8 (Appendix A, Fig. A2).
The minimal number of wind-wave directions in the
amount of 2.94 directions is encountered between the
Brazilian and African coast in the Mid-Atlantic areas
[A66-A68] where the E and SW wind-wave directions
prevail with about 90%. The maximal numbers of
wind-wave directions can even exceed the nominal
amount of 8 due to reported incompleteness of
observations (21) in areas where the wave directions
are almost uniformly distributed, for example in the
North Atlantic [A1, A4] and in the South Pacific areas
[A86, A93].
The annual numbers of wave heights (19) are
categorized from 2 to 7 (Appendix A, Fig. A3) in the
range from 2.9 in the Persian Gulf [A38] to 7.2 in the
south Indian Ocean [A99, A100] concerning the mean
value of 4.9 in the whole GWS (Table 5, Fig. 3). The
high uncertainty categories of 6-7 characterize the
North Atlantic areas [A3, A8, A9, A15, A16, A24].
Lower and moderate wave height uncertainty
categories are in gulfs and bays, for example, 2.9 in
the Persian Gulf [A38], 3.1 in the Gulf of Guinea
[A58], 3.5 in the Red Sea [A37], 3.9 in the Gulf of
Mexico [A32] and in the Bay of Bengal [A51]. In the
sea areas prevail the wave height category around 4,
like 3.8 in the Philippine Sea [A52], 4 in the Arabian
Sea [A39], 4-4.1 in the Mediterranean Sea [A26-A27],
4.2 in the Caribbean Sea [A47], 4.3 in the Sea of Japan
[A18] and in the Yellow Sea [A28]. The wave height
categories of 3.5-3.7 characterize the seas around the
islands of Micronesia [A63] and Melanesia [A71].
The annual numbers of wave periods (20) are
alienated in categories 4, 5, and 6 (Appendix A, Fig.
A4) in the range from 3.6 in the Persian Gulf [A38] to
maximally 6.2 in the mid-Pacific [A76-A77]
concerning the 5.75 mean value of whole GWS (Table
5, Fig. 4). Wave period categories below 4 characterize
only some gulfs and coastal seas, for example, 3.6 in
the Persian Gulf [A38] and 4.0 in the Sea of Japan
[A18].
Table 5. Information on all GWS areas and zones of selected ocean areas (SUPPLEMENT 5)
___________________________________________________________________________________________________
All GWS and ocean zones D % F(A) Max F(D/h,t) F(Dht) Max F(D/h) F(Dh) Max F(D/t) F(Dt)
GWS (22) GWS (23) (26) GWS (24) (27) GWS (25) (28)
___________________________________________________________________________________________________
All GWS areas [A1-A104] 100 91.4 34.2 24.2 2218 7.2 4.9 448 6.5 5.8 526
Northern areas [A1-A30] 21.7 25.1 34.0 27.0 677 6.9 5.6 141 6.1 5.6 140
Equatorial areas [A31-A80] 50.6 45.0 27.3 20.8 935 5.2 4.2 188 6.5 5.7 257
Southern areas [A81-A104] 27.7 22.1 34.2 29.7 656 7.2 6.0 131 6.2 7.0 132
___________________________________________________________________________________________________
Table 6. Information on zones of selected ocean areas
___________________________________________________________________________________________________
Ocean zones D (83% of the GWS) % F(A) Max F(D/h,t) F(Dht) Max F(D/h) F(Dh) Max F(D/t) F(Dt)
*some coastal areas not included GWS (22) GWS (23) (26) GWS (24) (27) GWS (25) (28)
___________________________________________________________________________________________________
North Atlantic Ocean (20 areas*) 15.5 19.0 34.2 24.8 473 6.9 5.0 96 6.5 5.7 109
South Atlantic Ocean (12 areas*) 12.8 8.8 33.1 25.6 289 6.75 5.0 57 2 6.0 67
North Pacific Ocean (17 areas*) 21.3 15.9 32.5 23.8 379 6.9 4.7 75 6.2 5.9 94
South Pacific Ocean (12 areas*) 13.2 10.7 32.8 25.8 276 6.8 5.0 53 6.2 6.0 65
Indian Ocean (16 areas*) 20.2 14.5 34.2 26.0 378 6.3 5.3 77 6.1 5.8 84
___________________________________________________________________________________________________
880
Table 7. Information of some northern hemisphere navigation routes (SUPPLEMENT 7)
___________________________________________________________________________________________________
R Route description GWS A:annual,all H(A) (22); F(D/ht)(23); F(D,h)(24); F(D,t)(25);
F(A) F(h,t)(26) F(h)(27) F(t) (28)
___________________________________________________________________________________________________
A English Channel-Gibraltar A11 A16 A17 1.48; 2.80 27.1; 75.9 5.84; 16.3 5.4; 15.1
B Gulf of Mexico-Gibraltar A24 A25 A32 A33 1.86; 3.62 25.6; 92.7 5.22; 14.6 5.6; 20.5
C Gibraltar - Port Said A26 A27 0.88; 1.84 17.2; 31.7 4.1; 11.3 4.7; 8.6
D Suez - Aden A37 000; 0.00 14.1; 14.1 3.5; 3.5 4.4; 4.4
E Aden - Arabian Gulf A39 A50 0.97; 1.96 18.5; 36.4 4.5; 12.6 4.6; 9.0
F Arabian Gulf-Colombo A39 A50 A60 1.52; 2.87 18.4; 52.8 4.4; 12.3 4.7; 13.5
G Colombo - Singapore A61 A62 0.97; 1.96 17.9; 35.1 3.9; 7.6 5.1; 10.0
H Singapore - Taiwan A40 A62 0.97; 1.96 19.9; 39.0 4.6; 12.9 4.9; 9.6
I Taiwan - Japan A29 A41 0.97; 1.96 23.6; 46.2 5.2; 14.6 5.2; 10.1
J North Sea A11 000; 0.00 23.0; 23.0 5.3; 5.3 5.0; 5.0
K North Atlantic Ocean A15 A16 0.97; 1.96 32.6; 63.8 6.6; 13.0 5.8; 11.5
L North Pacific Ocean A21 A22 A29 A30 1.72; 3.30 28.8; 95.0 5.7; 16.1 5.9; 19.4
___________________________________________________________________________________________________
Table 8. Uncertainties of some northern hemisphere combined navigation routes (SUPPLEMENT 6)
___________________________________________________________________________________________________
Compound route Routes R (Table 7) H(A) (22); F(D/ht)(23); F(D,h)(24); F(D,t)(25);
F(A) F(h,t)(26) F(h)(27) F(t) (28)
___________________________________________________________________________________________________
R1-Japan - Arabian Gulf F-G-H-I 2.00; 4.00 43; 171 12; 32 11; 43
R2-Germany-Arabian Gulf J-A-C-D-E 2.09; 4.25 29; 121 8; 23 7; 32
R3-Germany-Gulf of Mexico J-A-B 1.05; 2.07 78; 162 13; 28 17; 35
R4-Gibraltar - Suez – Aden C-D 0.99; 1.99 23; 45 7; 14 7; 13
R5-Colombo – Japan GF-H-I 1.55; 2.94 40; 117 11; 32 10; 30
___________________________________________________________________________________________________
6.1 Information based on uncertainties of navigation
routes
The effects of environmental uncertainties on
navigation routes, uncertainties in collision avoidance
maneuvering (Taylor, 1990), weather uncertainties in
ship route optimization are of lasting interest
andimportant for maritime safety [17] (Vettor,
Bergamini, and Guedes Soares, 2021).
Routs normally pass through more ocean areas
that may be jointly viewed as zones Z or a
combination of zones D (21-28) in proportion to the
length of a route or time spent in specific areas. The
article recalculates the annual uncertainties of all
wave directions, heights, and periods at some
northern hemisphere shipping routes (Table 7).
Overall uncertainties of northern hemisphere
combined navigation routes R are presented in Table
8.
Note that the entropy approach (21-28) is
applicable to uncertainty assessment of navigation
routes for different seasons, wave directions, heights,
and periods.
For those with gambling experience, the
navigation at the Mediterranean (Gibraltar - Suez –
Aden) (R4) is uncertain concerning jointly observed
heights and periods in all directions on an annual
basis (26) as flipping log2171=7.41 coins or dicing with
a log6171=2.88 six-sided die or drawing a card from a
stock of 171 cards.
The navigation through the North Sea and the
Atlantic Ocean from Germany to the Gulf of Mexico
(R3) is uncertain as flipping log2162=7.35 coins or
dicing with log6162=2.84 six sided dice or drawing
from a stock of 162 cards (Table 8).
7 CONCLUSION
The article demonstrated that the probabilistic
uncertainties expressed in terms of the information
theory perceptively present the random properties of
ocean wind wave climate data compiled in the Global
Wave Statistics. The information entropy objectively
defines the unexpectedness and uncertainty of ocean
wind waves regarding the environmental and
maritime experiences of randomness. Numerical
calculations provide summarizing uncertainty
formulae, tables, graphs, and charts of wind wave
directions, heights, and periods in all relevant oceanic
areas, zones, and routes. The ocean-wide distributions
of wind-wave information based on environmental
uncertainties are useful for maritime safety
management, navigation, shipping, and various
maritime activities as well as on exposures of ocean
structures and vessels in service.
NOMENCLATURE
A GWS areas (max 104)
CV coefficient of variation
D combined GWS data sets
d GWS directiones (max 8+1)
E directional exposure
H information entropy (bits)
h GWS wave heights (max 15)
F average numbers of events, directional wave effects
, unexpectedness, equivalent number of outcomes
NA, Ns, Nd Number of areas, seasons and directions
Nh, Nt, Number of wave heights and periods
p, P probability , probability distributions
R ocean navigation routes, robustness
S systems of events, directional exposability
s GWS seasons (max 4+1)
t GWS periods (max 11)
V variance
ACKNOWLEDGEMENT
This is curiosity-driven research of plausible application of
information theory to assessments of probabilistic
uncertainties of ocean-wide wind waves and their impact on
navigation, shipping as well as on ocean structures and
vessels in service.
881
APPENDIX A
9
94
102
56
86
81
82
95
54
73
72
7
1
104
103
96
21
31
44
97
22
14
43
7
13
30
20
12
45
49
46
48
47
55
,7
83
65
64
24
25
23
1
4
3
2
16
8
58
57
87
84
68
67
74
66
88
89
98
99
91
92
93
101
100
77
76
75
85
90
53
63
71
70
69
78
12
20
30
43
61
60
41
42
52
59
50
51
19
39
29
79
34
33
35
102
15
32
38
37
28
18
5
36
26
27
62
17
40
6
11
10
80
Figure A1. Marsden’s squares in GWS Hogben, Dacunha
and Olliver (1986)
7.2
7.8
3. 8
8.2
7.0
7.1
6.7
4
4.1
3.1
5.0
7
5
6.5
6.5
7.1
9
8.0
5.1
3.2
6.8
6.2
7
6.7
7.3
7.4
7.7
8.0
7.6
7.7
4.5
3.8
5
7.8
3.2
2
3,7
7.3
6
4.5
4.0
3.0
7.9
7.7
7.9
9
8.0
8.0
7.6
7.7
8.0
7.6
7.6
5,9
6.2
7.8
5.3
3.1
2.94
5.8
3
3.0
7.8
7.5
6.8
6.3
7.9
7.8
8.0
7.6
6.3
4.5
4.7
6,8
7
5.4
7.7
5
3.6
7.0
7.6
7.1
7.5
6.4
7.7
7.6
8.0
6.7
7.5
7.2
7.1
7.
02
7.6
2
6,2
6.1
7,3
7.1
6
7.
9
7.9
8.1
7.9
6.5
6.2
5.8
6.1
3
6.6
4.9
8
5.3
7.8
8.1
7.5
6.4
39
2.5-3.5
3.5-4.5
4.5-5.5
5.5-6.5
6.5-7.5
7.5-8.5
7.1
2
7.96
Max 8.16
Min 2.94
7.8
1
7.3
6.9
8.16
7.8
7.8
7.8
Figure A2. Average numbers of wave directions observed in
GWS on annual basis
6.9
5.8
3.6
5.6
5.0
4.6
6.3
3.7
4.1
4.0
3
5
5,7
6.9
5.4
5,7
4,6
6.7
4.4
5.6
5.0
6.2
6.1
6.4
6.6
6.7
4.1
3.9
3.8
4.0
000
00
4.2
3.3
4.5
3.6
3.5
6.1
5.6
4.9
5.3
5.8
6.1
4.5
6.8
5.3
6.9
6.7
3.1
3.4
5.1
4.6
3.7
3.7
4.1
3.4
5.9
6.0
6.3
7.1
6.3
6.1
5.5
6.2
7.2
5.1
5.0
5.1
5.6
6.3
4.4
3.7
3.5
4,1
4,0
5.0
6.7
6.6
6.4
5.0
3.8
4.0
4.3
5.2
5.0
4.5
3.8
4.9
3.9
5.3
5.2
4.7
4.5
4.6
4.2
4.5
4.2
6.2
5.6
6.4
3.9
4.1
2.5-3.5
3.5-4.5
4.5-5.5
5.5-6.5
6.5-7.5
Max 7.2
Min 2.9
6,1
2.9
3.5
5.6
5.5
4.0
4.0
4.3
4.3
3.7
5.9
4.3
Figure A3. Average numbers of wave heights in GWS on
annual basis for all wave directions
5.9
6.0
5.9
6.1
6.1
6.1
6.0
3
6.2
6.20
6.2
5
1
6.0
5.9
5.7
8
6.0
6.1
6.2
5.9
5.6
5.9
5.9
6.0
6.0
5.6
6.0
5.8
5.5
6.2
5.8
5
5.4
6.0
5.4
5.3
6.1
5.9
6.01
5.9
6.0
5.2
5.1
5.5
5.6
5.1
5.9
5.0
6.0
6.0
5.2
5.9
6,0
6.2
5.8
5.9
5.8
4
5.9
6.1
6.0
6.0
5.9
6.0
6.0
5.9
6.0
5.9
6.2
6.1
6.08
6.1
6.0
4
5.9
5.2
5.1
6.1
5.9
5.9
5.5
5.8
6.0
6.1
4.7
5.4
5.2
5.2
2
5.5
5
5.4
5.2
4.8
4.9
7
5.1
5.1
6
6.0
5.3
6.0
7
5.5
6.2
8
6.0
6.1
5.8
4.9
0
5.8
3.5-4.5
4.5-5.5
5.5-6.0
6.0-6.5
3.6
4.4
4.4
Min 3.6
Max 6.2
5.3
4.5
4.6
2
4.0
5.5
4.6
4.7
4.4
Figure A4. Average numbers of wave periods in GWS on
annual basis for all wave directions
APPENDIX B
Proof of the relation between the conditional and
unconditional entropies (16):
1 1 1
1 1 1 1
11
11
1
( / ') ( ) log
( ) ( ) ( )
1
log log ( )
()
1
log ( ) log ( )
()
1
( ) ( ') ( ) ( ')
()
= = =
= = = =
==
= − =
= − − =
= − + =
= − = −
i
ii
L
NL
kk
i
j i k
ii
LL
NL
k k i k
j i k k
NL
j j i i
ji
RR
N L N L
ss
Hp
p p p
s s p s
p
s s p p
p
H H H H
p
S S S
S S S
S
S
SS
S
S S S S
S
(B1)
q.e.d.
APPENDIX C
Instead of the unconditional entropy of the marginal
distribution of heights (19) and periods (20) the
conditional entropy with respect to a selected wave
height hw and period tz is:
: , : ,
: , : ,
( , ) ( , )
( : , / ) log
( ) ( )
=−
A s d w A s d w
w
allt
A s d w A s d w
p h t p h t
H A s d h
p h p h
(C1)
: , : ,
: , : ,
( , ) ( , )
( : , / ) log
( ) ( )
=−
A s d z A s d z
z
allt
A s d z A s d z
p h t p h t
H A s d t
p t p t
(C2)
Subsequently, the conditional entropy with respect to
all wave heights h and for periods t is:
,
( : , / ) ( ) ( : , / )=
A d w w
allh
H A s d h p h H A s d h
(C3)
,
( : , / ) ( ) ( : , / )=
A d w z
allh
H A s d t p h H A s d t
(C4)
The relation among the unconditional entropy of
joint distribution (18), unconditional entropy of of
marginal distributions (19, 20) and the conditional
entropy of heights(C3) and periods C4) holds:
( : , , , ) ( : , , ) ( : , / )
( : , , ) ( : , / )
= + =
=+
H A s d h t H A s d h H A s d h
H A s d t H A s d t
(C5)
The difference between the entropy of marginal
distributions of heights (19) and periods (20) and the
conditional entropy (C3) and (C4) are normally small.
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