ORIGINAL ARTICLE

Optimal ship tracking on a navigation route between two ports:a hydrodynamics approach

J. K. Panigrahi • C. P. Padhy • D. Sen •

J. Swain • O. Larsen

Received: 20 July 2010 / Accepted: 9 January 2011

� JASNAOE 2011

Abstract The optimal trajectory from Calcutta port to

Mumbai port is charted for a tanker transshipping from the

East coast to the West coast of India during rough weather.

Rough weather is simulated over Indian seas using the

state-of-the-art WAM numerical wave model (WAMDI

Group in J Phys Oceanogr 18:1775–1810, 1988), assimi-

lating satellite (IRS-P4) wind fields. These simulated wave

fields and two-dimensional (2D) directional wave spectrum

are an absolute representation of the irregular seaway.

Hence, the same for the monsoon month of August 2000

formed the input basis for this study. Loss of ship speed

due to the wave field (i.e., nonlinear motion of the tanker in

waves) and associated sea-keeping characteristics in the

seaway are estimated (Bhattacharya in Dynamics of marine

vehicles, Wiley, New York, 1978). The approach adopted

in this paper is unique in that it takes into account both

voluntary and involuntary speed reductions of the ship. It

helps in ship tracking by the optimum route using inverse

velocity as the weight function for the path in an efficient

way. Dijkstra’s algorithm [Numer Math 1(3):269–271,

1959] is applied in an iterative manner for determining the

optimum track. The optimum track information has broad

scope for use in modern shipping industry for obtaining

safe and least-time routing by avoiding schedule delays

with economic fuel consumption.

Keywords Ship behavior � Surface waves � Ship routing �WAM model � OTSR

1 Introduction

Optimum tracking of ship routes (OTSR) involves pro-

viding a vessel with a route recommendation prior to

sailing and thereafter closely monitoring the progress of the

vessel en route, and updating the master to ensure the

vessel achieves either the earliest possible safe arrival or

arrives safely at the required time. This service is also of

great use on coastal routes, by providing the master with

advance warning of heavy weather conditions which may

be encountered. Hence, ship routing is an essential pre-

requisite for all navigators for planning their voyage in any

part of the world ocean. The complexities involved in

OTSR demand multidisciplinary expertise such as wave

forecasting, ship behavior in the seaway, navigation, path

optimization, etc. In this study, the authors demonstrate the

minimal time path for a tanker in transit to Mumbai port

from Calcutta port sailing through the Bay of Bengal and

Arabian Sea. The state-of-the-art WAM wave model [1] is

implemented for establishing the wave climate over a

regional grid system of the Indian Ocean. Rough weather is

simulated, assimilating IRS-P4 [multifrequency scanning

microwave radiometer (MSMR)] analyzed wind fields into

the wave model for the monsoon month of August 2000.

The model predicted synoptic wave fields help in alerting

the ship in advance and simplify the navigator’s decision

regarding the optimum track to be navigated from depar-

ture to destination in a known wave field. This brings out

J. K. Panigrahi (&) � O. Larsen

DHI-NTU Research Centre, DHI Water and Environment,

200, Pandan Loop, Pantech-21, Singapore 128388, Singapore

e-mail: jkp@dhigroup.com

C. P. Padhy � D. Sen

Department of Ocean Engineering and Naval Architecture,

Indian Institute of Technology, Kharagpur, India

J. Swain

Naval Physical and Oceanographic Laboratory,

Kochi 682021, India

123

J Mar Sci Technol

DOI 10.1007/s00773-011-0116-3

the characteristics of the rough waves that the ship will

encounter en route. It is obvious that wind, wave, and cur-

rent impact on ship velocity in the open ocean. However,

waves have maximum impact on the ship due to their

periodic undulation, which significantly alters ship velocity

in an irregular seaway. While sailing, the cost of time spent

at sea by a ship has always been an important factor in the

overall cost of ship operation. Hence, the numerical model

under discussion is based on minimum travel time between a

specified origin and destination point. In the simplest case,

the behavior of the ship is represented by loss of speed due

to the wave field. A similar velocity reduction is estimated in

this study using sea-keeping characteristics as suggested by

Bhattacharya [2]. Furthermore, using the inverse velocity as

the weight function for a given path and optimization

algorithm [3], the minimal time path for the ship is obtained.

1.1 Historical perspective of OTSR

Many efforts [4] have been devoted to minimization of

time spent at sea, such as increasing the power for

achieving speed, aerodynamic and hydrodynamic shape of

the ship, etc. The state of the sea determines the upper

bound of the attainable speed for each ship. The advent of

extended range forecasting and the development of selec-

tive climatology, along with powerful computer modeling

techniques, have made ship routing systems possible. The

ability to effectively advise ships to take advantage of

favorable weather was hampered previously by forecast

limitations and the lack of an effective communication

system. In marine navigation, rough weather routing has

long been neglected due to the unavailability of systematic

wave observations and regular wave forecasts over the

Indian Ocean for the past several years [5]. Hence, it was

not possible to operationally forecast minimal time ship

routes based on sea state. In recent years, with the launch of

satellites and advancement of wave modeling, many

operational satellites and numerical models (WAVE-

WATCH III, WAM, SWAN, and OSW/SW-Mike21) can

give wind-wave parameters on a coarse and fine grid res-

olution over the world ocean. Hence, it has become feasible

to carry out routine wave forecasting and provide optimal

ship routes using these meteorological products.

1.2 Literature review

Originally, the concept of OTSR evolved in the US Navy

for minimal time routing of warships. Hence, most of the

literature in this area is of defense interest and classified.

Subsequently, the problem of obtaining an optimal ship

trajectory attracted the attention of many civilian

researchers. In this context, a few pertinent reports are cited

here to assess the evolution of research in this field.

In the past, Hanssen and James [6] demonstrated an

optimum ship route under stationary weather conditions.

Development of more realistic routing mechanisms using

variational methods assuming ship speed under maximum

power to be independent of time was attempted by Haltiner

et al. [7]. Models based on numerical methods to propose

an optimal trajectory were attempted by Faulkner [8].

Employing Pontryagin’s maximum principle, ship routing

problems were studied using a rectilinear, spherical coor-

dinate system, advocating solution of a system with three

nonlinear differential equations with appropriate boundary

conditions to provide an optimal ship trajectory. These

models, however, had constraints pertaining to undesired

ship motions and treatment of continental obstacles.

Zoppoli [9] formulated the minimal time algorithm as an

N-stage discrete process subjected to stochastic and

dynamic conditions. A deterministic dynamic program-

ming procedure was carried out on a grid system similar to

that used by Zoppoli [9] to investigate the total voyage

time. Mitchell and Papadimitriou [10] investigated the

shortest path through a weighted planar subdivision.

Optimum routing in the past was attempted based on long-

term weather conditions and sea-keeping criteria. Typi-

cally, routes were planned based on a set of generic

‘‘speed-reduction’’ curves to predict ship position and

anticipated weather conditions along the intended course

[11]. The weakness of this rather oversimplified method is

that it does not guarantee an optimal path, since combi-

nations of all possible paths may not be investigated. In

addition, factors related to involuntary speed reduction are

not taken into account. Several authors discussed the prob-

lem of ship weather routing in different ways to find a

minimal time solution, viz. variational calculus method,

broken extremal approach, isochrone method, etc. These

approaches, however, appear to have limitations in the sense

that they do not appear to handle well the part associated

with voluntary speed reduction. Hence, the path optimization

algorithm that we use here is Dijkstra’s algorithm. Routing

decisions of ships using advanced meteorological modeling

and satellite data employing Dijkstra’s algorithm [3] were

investigated by Anel [12] and the US Navy Meteorology and

Oceanography (METOC). Bekker and Schmid [13] investi-

gated the use of Dijkstra’s algorithm and a genetic algorithm

to achieve practical strategies and a method in which the two

optimization techniques interact to provide a safe route,

considering the risk of both sea mines and the environment,

making it applicable to sea mine avoidance.

2 Data

In the present study, wave model hindcast data are used by

assimilating IRS-P4 analyzed wind fields into the WAM

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123

wave model. IRS-P4 (Oceansat-I) was launched by India in

1999, with two payloads, namely MSMR and ocean color

monitor (OCM). The MSMR sensor is configured as an

eight-channel radiometer using four frequencies with dual

polarization. It has three resolutions, and the geophysical

products are also processed for three standard grid sizes

(150 9 150, 75 9 75, and 50 9 50 km2) covering the

whole globe. There are a total of eight geophysical prod-

ucts, including wind speed (accuracy ±2 m/s) without

direction [14]. Again, the exact repeat period of the satel-

lite is 2 days. However, both wind speed and direction are

essential for sea state prediction. Hence, the MSMR winds

over 150 9 150 km2 are blended with medium-range

global weather forecasts by the National Center for Med-

ium-Range Weather Forecast (NCMWF) along with the

various other available data, including data received

through Global Telemetric System (GTS) for the prepara-

tion of analyzed fields suitable for sea state nowcasting.

IRS-P4 gives scalar winds, which are converted to vector

winds by appropriate operators from observation space to

analysis space. The operational analysis and forecast sys-

tem at NCMWF is based on a T80L18 global spectral

model and spectral statistical interpolation (SSI) schemes

for data analysis. The analysis scheme is mainly based on

the Lorenc [15] concept of minimizing a cost function in

terms of the deviation of desired analysis from the first

guess field, which is taken as the 6-h forecast, and the

observations, weighted by the inverse of the forecast and

observation errors. The detailed methodology for comput-

ing vector winds is available in Kamineni et al. [16]. The

objective analysis scheme in SSI takes care of generating

six-hourly wind fields from 48 h satellite wind data gaps.

Atmospheric conditions over the Indian seas (Bay of

Bengal and Arabian Sea) are also unique, as the winds over

this basin reverse semiannually, blowing from the south-

west during the summer monsoon and from the northeast

during the winter monsoon. In the absence of any observed

data over the oceanic region, the 6-h model forecast

(serving as a background field as a guess) is retained in the

analysis. The assimilated products such as the zonal and

meridional components of winds at six-hourly intervals are

processed and supplied by NCMWF. This wind is known

as the IRS-P4 analyzed wind field, having u- and v-wind

components covering the south to north Indian Ocean,

being available for sea state forecasting. The same has been

assimilated into the third-generation wave model WAM for

simulating six-hourly wave parameters. Wind speed and

direction for the monsoon month of August are used here

for wave model hindcasting. The mean monthly wind fields

for the rough weather month of August 2000 are presented

in Fig. 1 as a contour map. The contours show the wind

speed in m/s, and arrows represent direction. These wind-

wave parameters are used for meteorological navigation.

3 Methodology

The basic principles involved in optimal ship routing are

the same as those for minimum flight time of an aircraft

flying at fixed level [17]. However, the two problems differ

primarily in the manner in which the environment impedes

the motion of the vehicle. The methodology adopted to

carry out this study is described in the following sections.

The first section describes the implementation of the wave

model for simulating an irregular seaway, while subsequent

sections describe the ship behavior in the seaway and route

optimization, respectively.

3.1 Wave modeling

The state-of-the-art third-generation WAM wave model

(Cycle-4) is used in this study, originally developed by the

WAMDI Group [1]. It integrates the basic transport equa-

tion without any prior assumptions on the shape of the

wave spectrum. WAM requires wind input on the pre-

scribed model grids and computes the evolution of two-

dimensional wave spectrums for the full set of degrees of

freedom [18]. It provides 25 frequencies and 12-directional

discretization for evolution of the wave spectrum by

solving 1.2 million equations for each grid. The source

terms and the propagation are computed with different

numerical methods and time steps. The model outputs are

significant wave height, peak and mean wave periods,

mean wind-wave directions, swell wave height, swell fre-

quency and direction, frictional wind velocity, wave-

induced stress, and the two-dimensional wave spectrum.

The model is being updated with new advances and con-

tinually validated with long-term measurements from

moored buoys and satellite data [19]. In the history of wave

modeling, in the early 1960s the first attempt was made to

develop wave models considering the source term as the

sum of atmospheric input and white-cap dissipation, known

30 40 50 60 70 80 90 100 110 120Longitude

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30

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itud

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Fig. 1 Mean monthly IRS-P4 analyzed wind fields for August 2000

J Mar Sci Technol

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as first-generation wave models. In the late 1970s, second-

generation wave models were developed, with increased

understanding of the physical processes responsible for

wind-wave generation, propagation, decay, and nonlinear

interaction (JONSWAP) [20]. The present model used in

this study is a spectral wave model that takes care of the

wind input term and associated spectral change, and the

complete spectral energy change due to nonlinear wave–

wave interaction and dissipation due to white-capping.

The sea state parameters are simulated using six-hourly

IRS-P4 analyzed wind fields. The mean monthly wave

climate for the monsoon month of August 2000 is estab-

lished using this model. The outputs plotted in Figs. 2 and

3 show the spatial distribution of significant wave height

and peak wave period with mean wave direction, respec-

tively, over the Indian Ocean. The time-series wave

parameters of the model have been validated against buoy

data by Panigrahi [21] and Panigrahi and Swain [22]. It is

reported [5, 23] that, over the Bay of Bengal, the significant

wave height shows root-mean-square deviation of 0.24 m.

The average 2D directional wave spectrum simulated at

central Bay of Bengal and Arabian Sea for the month of

August 2000 are presented in Figs. 4 and 5, respectively.

These are absolute representations of waves in an irregular

seaway. The multiple peaks of the spectrum show the

occurrence of various wave groups having different wave

periods. The spectral peaks show the maximum energy

levels associated with predominant wave directions.

3.2 Ship behavior in seaway

The theory of ship behavior in a seaway is referred to as the

sea-keeping characteristics of the vessel. Mostly, this

30 40 50 60 70 80 90 100 110 120Longitude

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0

10

20

30

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itud

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Fig. 2 Mean monthly WAM simulated significant wave height

(m) and mean wave direction (�) for August 2000

30 40 50 60 70 80 90 100 110 120Longitude

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-20

-10

0

10

20

30

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itud

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Fig. 3 Mean monthly WAM simulated peak wave period(s) and

mean wave direction for August 2000

Fig. 4 Two-dimensional directional wave spectrum at central Bay of

Bengal, August 2000

Fig. 5 Two-dimensional directional wave spectrum at central Ara-

bian Sea, August 2000

J Mar Sci Technol

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involves ship motion due to waves, which include several

factors, viz. (1) loss of speed in a seaway and added

resistance, (2) wave loads and structural design, (3) habi-

tation, comfort, and safety, (4) slamming, deck wetness,

and propeller emergence, and (5) the establishment of

operational limits for mission requirements.

All these aspects of sea-keeping or ship behavior in

waves result from the six fundamental modes of rigid-body

motion (Fig. 6) of the ship in waves, e.g., three linear

motions along longitudinal, transverse, and vertical axes

defined as surge (xb), sway (yb), and heave (zb), respec-

tively, and the three rotational motions around these axes

defined as roll (ø), pitch (h), and yaw (w). These funda-

mental motions further set for deriving other useful derived

responses. The various responses of the vessel in regular

monochromatic waves are represented by corresponding

transfer functions, generally termed response amplitude

operators (RAO). For an incident wave of amplitude fw and

absolute frequency x, the incident wave elevation f is

given by

fðx!; tÞ ¼ fxcosðk!

x!�xtÞ; ð1Þ

where k!

is the wavenumber. Any particular response r(t) of

the ship will be of the form

rðtÞ ¼ Rcosðxet þ erfÞ; ð2Þ

where R is the amplitude of the response, erf is the phase

angle between the response and the incident encountered

wave, and xe is the frequency of encounter, which

depends on the ship speed (V), the ship heading (b) with

respect to the waves, and the absolute wave frequency

(x):

xe ¼ x 1� xV

gcosb

� �: ð3Þ

The response can thus be written as

rðtÞ ¼ RAO � fWcosðxet þ erfÞ; ð4Þ

where

RAOðx;V; hÞ ¼ Rðx;V ; bÞ=fW: ð5Þ

It is noted that the amplitude of the response R depends

on the absolute frequency (x), the ship speed (V), and the

relative heading (b = ls - lw), and therefore RAO can be

determined over a range of these three variables. Here, ls

and lw represent the ship heading angle and wave

direction, respectively. In general, most of the ship

responses depend linearly on the incident wave amplitude

fw. Few response parameters such as added resistance

(RAW) that are a second-order function of incident wave

amplitude ðfWÞ are crucial. In this case, the appropriate

relation to determine the response spectrum can be

modified as

SAWR ðxeÞ ¼

RAW

f2W

SWðxeÞ; ð6Þ

where RAW is added resistance in monochromatic waves at

the given encountered frequency (i.e., at the given

combination of speed, heading, and absolute frequency),

and SRAW is the added resistance spectrum. The average

(mean) added resistance in an irregular wave field (i.e., in

an incident wave spectrum) is given by twice the area

under the spectrum,

�RAW ¼ 2ffiffiffiffiffiffim0

p; m0 ¼

Z1

0

SAWR ðxeÞdxe: ð7Þ

This differs from the other responses varying linearly

with wave amplitude. Statistical quantities such as the

average amplitude, significant amplitude, etc. are

proportional to the square root of the area under the

corresponding response spectra. Furthermore, a simplified

formulation for added resistance suggested by

Bhattacharya [2] can be plugged into the calculations,

RAW ¼x3

e

2gðbzz

2a þ bhh

2aÞ; ð8Þ

where za and ha are heave and pitch displacements for

corresponding heave ðbzÞ and pitch ðhaÞ damping,

respectively.

Several other simplified relations of varying levels of

complexity are available in literature, each producing

estimates of added resistance with progressively greater

degrees of accuracy. All formulae are based on hydrody-

namic characteristics of the hull in the vertical plane (heave

and pitch motions, total added mass and damping, sectional

added mass and damping, gradient of sectional added mass

along length, gradient of vertical relative motions along

length, etc.). As can be seen, determination of ship

behavior in an irregular wave field essentially requires

evaluation of the ship response in regular monochromatic

Fig. 6 Definition of ship motions

J Mar Sci Technol

123

waves (i.e., determination of the fundamental RAOs). This

is the most complicated and difficult part of evaluation of

sea-keeping characteristics. The RAOs represent charac-

teristics of a given hull, regardless of the prevailing wave

condition. For any given hull, a set of RAOs can be pre-

computed over a range of ship speeds, relative headings,

and wave frequencies. Using these as a database, the

response in any given sea condition (i.e., wave field: wave

height and wave direction) can be determined fairly by

interpolating the RAOs and carrying out spectral calcula-

tions. Furthermore, the mean added resistance, �RAW ¼�RAWðV ; b;HsÞ database can be prepared for equal intervals

of V, b, and Hs by adopting a similar interpolation scheme.

This database can help to arrive at the added resistance

(RAW) corresponding to a given sea state.

Adopting the above mathematical formulations, the

ship’s behavior in the seaway is determined for the present

study, and a database is generated. A vessel having overall

length (LOA) of 60.0 m and beam of 11.0 m is chosen

from Clarkson’s Registrar [24]. The trajectory of the ship,

from Calcutta to Mumbai (old name Bombay) and vice

versa, is assumed as the normal navigation track (Fig. 7)

for attending the call between the two ports. It is seen from

the figure that the commercial navigation route is via

Colombo port, covering a distance of 1244 nautical miles

from Calcutta to Colombo and 889 nautical miles from

Colombo to Mumbai [25]. The optimal back-and-forth

route is estimated following the algorithm described in the

subsequent section.

3.3 Ship route optimization

Route optimization of a ship involves optimization of

several factors, viz. (1) minimum transport time linked to

speed loss in the seaway, (2) minimum fuel cost linked

with added resistance in waves and total distance, (3)

minimum structural damage, (4) maintaining minimum

motions for specific operations (e.g., minimum relative

motion in vertical plane, roll motions, etc.), and (5) fixed

time of arrival. The various factors discussed above can

be related to: (a) involuntary speed reduction due to

increased resistance in the seaway and the decrease in

propeller efficiency in rough weather, and (b) voluntary

speed reduction, which is deliberate reduction in speed

Fig. 7 Commercial navigation route between the two ports (Calcutta–Mumbai)

J Mar Sci Technol

123

by the ship’s captain to ensure that ship behavior

remains within acceptable safety limits, since it is found

that in general a reduction in speed improves sea-keep-

ing characteristics. In our implementation, distance

between two nodal points in a spatial grid is expressed

as a weight function. These weights can be viewed as

the objective function, which are obtained by combining

weather information (WAM computed wave parameters)

along with vessel sea-keeping characteristics. The gen-

eral framework of Dijkstra’s algorithm [3] is described

here.

Considering a weight directed graph (Fig. 8) or digraph

(G), where G = (V, E), with a weight function w:

E ? R mapping edges to real-valued weights. The weight

of path P ¼ ðv0; v1; v2; . . .; vkÞ is the weighted sum of its

constituent edges and can be represented by Eq. 9

wðPÞ ¼Xk

i¼1

w vi�1; vi½ �; ð9Þ

where w is weight, vi is the ith vertex, and vi-1 is the (i - 1)th

vertex.

Edge weight dðu; vÞ can be defined as the shortest path

weight (w) from u to v, i.e., w(P) = d(u, v) and is given by

d u; vð Þ ¼ min wðPÞ : u! vf g if path exist from u to vð Þ¼ 1 otherwiseð Þ ð10Þ

Adapting this algorithm, a code has been developed. If

the start and destination nodes are defined for a given grid

with associated weights, the algorithm finds the optimal

path joining these two nodes through grid points. For a

distance L between two neighboring nodes with

corresponding sea-state parameters, the mean value of the

added resistance (RA) is determined. This computation

requires interpolation of the motion data (RAO) for

appropriate speed and ship heading relative to wave

direction. The weights can be represented as: w = L/V,

where V = P/(Rc ? RA), where ‘Rc’ is the calm water

resistance of the ship (which can be related to the ship’s

engine power through a set of parameters, all of which are

assumed to be known for a given ship). The weights are

thus the time taken by the ship to travel between the two

considered neighboring nodes in the wave environment. By

assigning these as weights wi,j over the network, the path

having minimum weight T is determined as the shortest

path [26].

T ¼X

ij

Lði;jÞ

.Vredði;jÞ

h ior; T ¼

Xij

Lði;jÞ

.VMði;jÞ

h i;

ð11Þ

where L (i,j) is the distance between two neighboring nodes

i and j, Vredði;jÞ is the (involuntarily) reduced speed of ship

between node i and node j due to its added resistance, and

VMði;jÞ is the maximum allowable speed between the path

connecting nodes (i, j) beyond which a certain response

(e.g., slam, acceleration, etc.) exceeds the prescribed limit

(RT).

4 Results and discussion

This paper aims at charting the optimal route from Calcutta

port to Mumbai port and vice versa using model simulated

wave data derived from remotely sensed winds. The mean

monthly wind speed and direction distribution (Fig. 1) for

August 2000 could clearly identify regions of highs and

lows and the mean wind directions. The satellite observed

wind in general shows an average wind velocity in the

range 4–10 m/s prevailing over the Indian seas for SW

monsoon month. In particular, closer to the Indian coast,

the wind varies between 4 and 6 m/s. Corresponding mean

monthly wave height ranges from 1.0 to 3.5 m over the

model domain (Fig. 2). The wave height contour of 1.5 m

runs closer to the east coast and the 2.0 m contour closer to

the west coast of India, whereas waves of order 2.0 and

2.5 m prevail over the central Bay of Bengal and Arabian

Sea, respectively. Similarly, the wave period ranges from 6

10

23

11

43

17

35

( , ) :12I Jw

I J K

L M N

( , ) : 87I Mw

( , ) :12L Iw

Fig. 8 Representation of the

directed weighted graph (G)

J Mar Sci Technol

123

to 10 s in the model domain (Fig. 3), revealing that the sea

state is composed of wind-seas and swells. Furthermore,

closer to the coast, the peak wave period varied between 8

and 10 s, which indicates that the sea state is dominated by

remotely generated swells as compared with locally gen-

erated wind-waves. The mean wave directions mainly

follow the mean wind pattern, i.e., predominantly prevail-

ing from the SW, excepting some local variations. The

predominant direction of wave approach at each grid point

with respect to the ship’s course determines the irregular

seaway, viz. head sea, astern sea or bow sea. Hence, a

sample mean monthly 2D directional wave spectrum is

presented for deeper section at central Bay of Bengal

(Fig. 4) and central Arabian Sea (Fig. 5). Both spectra

exhibit similar energy peaks of the order of 0.5–2.5 m2/Hz

in the frequency band 0.3–0.1 Hz. As the winds blow

consistently over longer fetch and duration due to the

active southwest monsoon in the open sea, the swell waves

were consistently present, excepting periods of very strong

winds, generating various wave groups ranging from 3 to

10 s period. Mainly, the mean monthly wind and wave

fields are quite promising, and they follow the standard

climatic wind and wave variability patterns for Indian seas

[27, 28]. Using this sea-state information, rough weather

routing is carried out from departure to destination. The

shortest path algorithm is utilized to obtain the optimal

route between the two national ports of Calcutta and

Mumbai (Fig. 7). The results of the optimal ship route

between these two locations are presented in Figs. 9 and

10, where the route circumnavigates seven Maritime

Provinces of West Bengal, Orissa, Andhra Pradesh, Tamil

Nadu, Kerala, Karnataka, and Maharashtra. The optimal

path line connecting these two ports passes around the

island country of Sri Lanka, covering a distance of 2155.25

nautical miles. The total time taken by the ship from

Calcutta to Mumbai and vice versa is 205.17 and 203.60 h,

respectively. In addition to sea-keeping characteristics,

slam response (=50/h) is considered in case of consider-

ation of voluntary speed reduction. The effect of added

resistance is to reduce the speed by the order of a few

percentage, and therefore large deviations in paths are not

usually obtained unless wave conditions drastically differ

between adjacent zones. This algorithm is promising and

extensively used in transportation planning.

5 Conclusions

The influence of sea-state dynamics on the safety and

economy of a ship’s route are of considerable concern to

the modern shipping industry. The most important envi-

ronmental factors relating to the safety and performance of

a ship on high seas are prevailing ocean surface winds and

waves. In this context, the WAM wave model predicts the

synoptic wave field using IRS-P4 analyzed winds with a

higher level of confidence. The simulated model outputs

are very useful for meteorological navigation and avoid-

ance of heavy weather damage. The formulations of

Bhattacharya [2] for estimating ship motion and sea-

keeping characteristics appear realistic. The results of this

study show that the chosen algorithm is capable of simu-

lating the optimal route between given departure and des-

tination nodes reasonably well. The numerical code

developed and implemented here differentiates continental

boundaries from water, and estimates the minimal routing

path, avoiding land barriers. In this paper, the problem of

minimal time ship routing is demonstrated using a static

mean monthly sea state for one of the rough weather

months (August) of the year in Indian seas. The same can

60 63 66 69 72 75 78 81 84 87 90 93

Longitude (degree East)

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0

3

6

9

12

15

18

21

24

Lat

itud

e ( d

egre

e N

orth

)

Calcutta

Mumbai

INDIA

Fig. 9 Optimum route from Calcutta to Mumbai

60 63 66 69 72 75 78 81 84 87 90 93

Longitude (degree East)

-3

0

3

6

9

12

15

18

21

24

Lat

itud

e (d

egre

e N

orth

)Calcutta

Mumbai

INDIA

Fig. 10 Optimum route from Mumbai to Calcutta

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be implemented more realistically by assimilating satellite

wind and wave data at higher spatiotemporal resolution.

However, this needs extensive validation with ship log data

before it can be made operational. Such a practical solution

to the problem of optimal ship routing is very important to

shipping communities. Furthermore, operational ship

routing services can also be provided in combination with

an onboard voyage planning system. This study should also

help navigators to track the ship via an optimum route,

taking into account safety, minimal time, and probably

minimum fuel consumption. It may be concluded that the

modern shipping industry will find this work very useful

and that it also has wider scope for use in warship routing.

Acknowledgments The authors are thankful to Head, Department

of Ocean Engineering and Naval Architecture, IIT, Kharagpur, India

and Director, DHI-NTU Water & Environment Research Centre and

Education Hub, Singapore, for extending support and encouragement

for this work. This study is a collaboration of Ph.D. works of Dr.

Chinmaya P. Padhy and Dr. Jitendra K. Panigrahi under the super-

vision of Prof. D. Sen and Dr. J. Swain, respectively.

References

1. WAMDI Group (1988) The WAM model—a third generation

ocean wave prediction model. J Phys Oceanogr 18:1775–1810

2. Bhattacharya R (1978) Dynamics of marine vehicles. Wiley, New

York, p 508

3. Dijkstra EW (1959) Note on two problems in connection with

graphs (spanning tree, shortest path). Numer Math 1(3):269–271

4. Marks W, Goodman TR, Pierson WJ, Tick LJ, Vassilopoulos LA

(1968) An automated system for optimum ship routing. Trans Soc

Nav Archit Mar Eng 76:22–55

5. Swain J (1997) Simulation of wave climate for Indian seas. Ph.D.

thesis, Cochin University of Science and Technology, India

6. Hanssen GL, James RW (1960) Optimum ship routing. J Inst

Navig 10(3):253–272

7. Haltiner GJ, Hamilton HD, Arnason G (1962) Minimal time ship

routing. J Appl Meteorol 1:1–7

8. Faulkner FD (1963) Numerical methods for determining opti-

mum ship routes. J Inst Navig 10(4):351–367

9. Zoppoli R (1972) Minimum-time routing as an N-stage decision

process. J Appl Meteorol 11:429–435

10. Mitchell JSB, Papadimitriou CH (1991) The weighted region

problem: finding shortest paths through a weighted planar sub-

division. J Assoc Comput Math 38(1):18–73

11. Panigrahi JK, Umesh PA (2008) Minimal time ship routing using

IRS-P4 (MSMR) analysed wind fields. Mar Geod 31:39–48

12. Anel AM (2005) Network shortest path application for optimum

track ship routing. M.S. thesis, Naval Postgraduate School,

Monterey, CA. Report No. A106534, p 93

13. Bekker JF, Schmid JP (2006) Planning the safe transit of a ship

through a mapped minefield. J Oper Res Soc S Afr 22(1):1–18

14. Bhatia RC (2001) Validation of water vapour and wind speed

products derived from MSMR payload onboard IRS-P4 Satellite

and meteorological applications of data products. In: Proceedings

of Megha-Tropiques 2nd scientific workshop. Directeur de la

Publication, Remy Roca, 2–6 July 2001, pp 1–6

15. Lorenc AC (1986) Analysis methods for numerical weather

prediction. Q J R Meteorol Soc 112:1177–1194

16. Kamineni R, Rizvi SRH, Kar SC, Mohanty UC, Paliwal RK

(2002) Assimilation of IRS-P4 (MSMR) meteorological data in

the NCMRWF global data assimilation system. In: Proceedings

of Indian Academy of Science (Earth and Planetary Science).

111(3):351–364

17. Arrow KJ (1949) On the use of winds in flight planning.

J Meteorol 6:150–159

18. Komen GJ, Cavaleri L, Donelan M, Hasselmann K, Hasselmann

S, Janssen PAEM (1994) Dynamics and modelling of ocean

waves. Cambridge University Press, UK

19. Swain J, Panigrahi JK, Vijayakumar D, Venkitachalam NRV

(2003) Performance of 3g-WAM using IRS-P4 winds for its

operational implementation in the Indian Ocean. In: Proceedings

of the symposium on microwave remote sensing applications.

CSRE IITB, India, p 1

20. Hasselmann K, Barnett TP, Bouws E, Carlson H, Cartwright DE,

Enke K, Ewing JA, Gienapp H, Hasselmann DE, Kruseman P,

Meerburg A, Muller P, Olbers DJ, Riehter K, Sell W, Walden H

(1973) Measurement of wind-wave growth and swell decay

during the joint North Sea Wave Project (JONSWAP). Deutsche

Hydrographische Zeitschrift A8(12)

21. Panigrahi JK (2007) Wind induced surface gravity waves in the

north Indian Ocean and their potential applications. Ph.D. thesis,

Berhampur University, India, p 91

22. Panigrahi JK, Swain J (2010) Numerical simulation and valida-

tion of deepwater spectral wind-waves. Mar Geod 33:39–52

23. Bhatt V, Sarkar A, Kumar R, Basu S, Agarwal VK (2004) Impact

of Oceansat-I MSMR data on analyzed oceanic winds and wave

predictions. Ocean Eng 31(17):2283–2294

24. Clarkson Registrar (2006) Bharat Book Bureau CD. Standard

edition

25. (1970) Admiralty charts and publications. Routing chart Indian

Ocean. UK Hydrographic Office, Taunton

26. Taha H (2002) Operations research: an introduction, 7th edn.

Prentice Hall, New Jersey, p 222

27. Hastenrath S, Lamb PJ (1979) Surface climate and atmospheric

circulation. Climatic atlas of the Indian Ocean. Part-I. The Uni-

versity of Wisconsin Press, USA

28. Young IR, Holland GJ (1996) Atlas of the oceans: wind and wave

climate. Pergamon/Elsevier Science, USA, p 1

J Mar Sci Technol

123