ORIGINAL ARTICLE

Improved vessel squat modeling for hydrographic and navigationapplications using kinematic GNSS positioning

Alireza A. Ardalan • Mohammad-Hadi Rezvani

Received: 27 June 2012 / Accepted: 25 April 2013

� Springer-Verlag Berlin Heidelberg 2013

Abstract The squat phenomenon, that is, the sinkage of a

vessel due to its motion can affect the safety of navigation

and reduce the accuracy of hydrographic bathymetry.

Therefore, it is necessary to model and predict the squat of

vessels as a function of cruise speed. We present a Global

Navigation Satellite Systems–based squat modeling

method for both hydrographic and navigation applications.

For implementation of the proposed method, onboard GPS

antennae configurations are offered to model bow squat for

full-form ships such as supertankers or ore–bulk–oil car-

riers as well as stern squat for fine-form vessels such as

passenger liners or container ships. In the proposed meth-

odology, the onboard GPS observations are used to deter-

mine cruise ground speed, heave, attitude, and controlling

the quality of kinematic positioning via fixed baselines.

The vessel squat is computed from ellipsoidal height dif-

ferences of the onboard antennae with respect to a refer-

ence state, after removal of all disturbing effects due to

roll, pitch, heave, tide, vessel load, and geoidal height

variations. The final products of the proposed approach are

the analytical squat models usable for hydrographic and

navigation applications. As the case study, the method is

applied to a survey vessel in the offshore waters of Kish

harbor. Numerical results indicate that the experimental

precision of the derived analytical squat models is in the

range of 0.003–0.028 m. The computed navigation squat of

the test vessel at a speed of 12.64 knots is 30 % of the

vessel draft and about twice its hydrographic squat.

Although the field test was performed on a survey vessel,

the method can be applied to any ship at any waterway.

The proposed method can address the inevitable demand of

reliable squat models for delicate hydrographic projects

and high-speed marine traffic.

Keywords Ship squat modeling � GNSS � Hydrographic

surveying � Safety of navigation � Bathymetry � Marine

traffic

Introduction

The vessel squat, that is, the vertical drop caused by water

pressure fall beneath the keel of a moving vessel, can

increase the draft and the risk of grounding by reducing the

under keel clearance (UKC). The squat effect in hydro-

graphic surveying can also reduce the accuracy of

bathymetry if it is not considered and removed from the

observations. The sinkage of a vessel due to the squat can

be accompanied by changes in pitch and roll, which makes

the maximum squat to appear at either port or starboard

sides of bow or stern of the vessel depending on its

structural characteristics. Generally, full-form ships such as

supertankers or ore–bulk–oil carriers, which are technically

characterized by the block coefficient Cb [ 0.7, have bow

squat, while fine-form vessels such as passenger liners or

container ships (Cb \ 0.7) experience stern squat (Derrett

1999). Figure 1 schematically illustrates UKC, draft, stern

squat, and bow squat.

The squat in confined channels depends on the breadth

and depth of the channels, in addition to cruise speed

(Briggs 2006; Maynord and Briggs 2006; Delefortrie

A. A. Ardalan (&) � M.-H. Rezvani

Department of Surveying and Geomatics Engineering,

Center of Excellence in Geomatics Engineering and Disaster

Prevention, College of Engineering, University of Tehran,

P. O. Box 11155-4563, Tehran, Iran

e-mail: ardalan@ut.ac.ir

M.-H. Rezvani

e-mail: mhrezvani@ut.ac.ir

123

GPS Solut

DOI 10.1007/s10291-013-0326-8

2010). However, in unrestricted waterways or deep chan-

nels, the squat is mainly driven by cruise speed (Varyani

2006). According to Feng et al. (1996), squat is a signifi-

cant component of the UKC allowance. Therefore, the

modern demand for high-speed and safe marine transport,

especially in confined channels and restricted waterways,

requires prediction of the squat as a function of cruise

speed (Feng and O’Mahony 1999; EL-Kader et al. 2003).

Moreover, the development of reliable bathymetry for

delicate hydrographic operations, such as pipe laying,

drilling rig installation, and dredging, requires squat

correction. As Global Navigation Satellite Systems

(GNSS) like GPS became available, the geodetic commu-

nity began to explore GPS applications in squat modeling.

Generally, GNSS-based squat models rely on ellipsoidal

height differences of the onboard antenna with respect to a

reference motionless state while tidal changes are removed

(Gourlay and Cray 2009). In this respect, in addition to a

nearby base GPS antenna onshore, onboard GPS antennae

configuration at bow, port, and starboard of bridge wings is

used for bow squat determinations and UKC predictions of

full-form vessels (Maynord and Briggs 2006; Giorgi et al.

2010). The non-parallelism of geoid and reference ellip-

soidal surface as well as the pitch-derived sinkage must

also be considered for correct squat modeling (Zilkoski

et al. 1999; Harting et al. 2009).

We differentiate between squat modeling for hydro-

graphic and navigation applications. For the safety of

navigation, the modeled squat must encompass the UKC

variations resulting from roll, pitch, and the sinkage caused

by cruise speed. Therefore, the maximum squat, which

results in the minimum UKC, must be determined by

monitoring the squat at port and starboard sides of bow or

stern of a vessel. This in turn requires proper configuration

of the onboard GPS antennae. In contrast, for hydrographic

surveying, since bathymetry is usually corrected for the roll

and pitch effects, the computed squat must only include

vertical motion of the echo sounder transducer due to

cruise speed.

For squat modeling, we offer correction formulae which

consider the lever-arm coordinates of the onboard GPS

antennae with respect to the center of gravity (CG) of the

vessel, roll, pitch, heave, tide, vessel load, and geoidal

height variations. The roll, pitch, and heave information is

needed to remove the rotational and vertical disturbances

from ellipsoidal heights of the onboard GPS antennae

proportional to their lever-arms. The tidal information

during the field operations is required for the removal of

instantaneous sea-level variations from the ellipsoidal

heights. The vessel load variations, which are mainly

caused by the fuel consumption, must be observed in order

to remove the corresponding effects. Finally, the geoidal

height variations are needed to account for the effect of

non-parallelism of reference ellipsoidal surface and geoid.

The proposed method in this study can be explained as

follows: (1) Installation of three GPS antennae onboard,

and at least a base antenna at the wharf near to the area of

the field operations. The onboard GPS antennae must be

installed at the following locations depending on the type

of the ship: bow-port, bow-starboard, and stern-port/stern-

starboard for full-form ships, and stern-port, stern-star-

board, and bow-port/bow-starboard for fine-form vessels.

In the methodology, the onboard GPS observations provide

attitude, heave, and cruise ground speed information. In

(a)

(b)

UKC

UKC Squat

Sea surface

Sea surface

Seabed

Seabed

Draft

Squat

Sea surface

Seabed

Draft

Draft

UKC

Draft forward

Draft forward

Draft forward

Draft aft

Draft aft

Draft aft

UKC forward UKC aft

UKC aft

UKC aft

UKC forward

UKC forward

(c)

Fig. 1 The squat effect of a vessel on its UKC and draft. a Vessel at

rest, b in motion with stern squat for a fine-form vessel, and c in

motion with bow squat for a full-form ship

GPS Solut

123

addition, the fixed baselines between the onboard antennae

are used to control the quality of kinematic positioning. (2)

Starting GPS observations at the wharf before departure to

resolve the ambiguity and to develop the initial motionless

reference state of the squat modeling. (3) Departure from

the wharf and sailing with stepwise increasing cruise speed,

that is, keeping the speed constant for a while and then

increasing the speed in incremental steps while GPS

observations are continued. (4) Computation of the vessel

squat at 1 s interval with respect to the reference state

while considering the aforementioned corrections. (5)

Application of the least squares (LS) to the computed squat

values during the constant cruise speeds in order to take

advantage of the redundant values. (6) Developing the

analytical squat models as a function of cruise speed for

hydrographic and navigation applications by fitting suitable

functions to the corresponding LS-derived squat values

based on the robust M-estimations.

In the next section, the mathematical setup for squat

modeling based on the proposed method will be presented.

Then, the experimental setup for practical implementation

of the method will be offered in order to prepare for the

presentation of the results and discussions about the field

experiment in the subsequent sections. The last section is

devoted to the conclusions and final remarks.

Mathematical setup

Since the squat is the vertical displacement of a moving

vessel, it should be observable by kinematic GPS posi-

tioning with respect to the motionless reference state of the

vessel. According to Fig. 2, the difference of the ellipsoidal

height h(ti) of an onboard GPS antenna at epoch ti with

respect to the ellipsoidal height h(t0) at the reference epoch

t0 can provide us with the squat of the vessel, once all

disturbing effects have been removed. The disturbing

effects are due to attitude, heave, tide, vessel load, and

geoidal height variations between the two epochs. There-

fore, if S(ti) is the squat associated with the epoch ti when

the vessel is sailing at the cruise speed vðtiÞ, we have

SðtiÞ ¼ hðt0Þ � hðtiÞ þ dhAðtiÞ þ dhHðtiÞ þ dhTðtiÞþ dhLðtiÞ þ dhNðtiÞ ð1Þ

where dhAðtiÞ, dhHðtiÞ, dhTðtiÞ, dhLðtiÞ, and dhNðtiÞ are the

corrections due to attitude, heave, tide, vessel load, and

geoidal height variations between the two epochs. Note that

in (1), we have followed the convention of positive squat

‘‘down’’ (Briggs 2006).

According to Melachroinos et al. (2011), the effect of

attitude variations on the ellipsoidal height of the onboard

GPS antenna dhAðtiÞ can be computed by knowing the roll,

pitch, and the offset of the antenna from the CG of the

vessel (Fig. 3). Let the attitude parameters, which are the

rotation angles between the so-called body frame (b-frame)

and local-level frame (ll-frame), populate the rotation

Fig. 2 The ellipsoidal height difference of the onboard GPS antenna

between two epochs t0 and ti, which is equal to the squat at epoch tiplus the disturbing effects on the GPS antenna height at epoch ti with

respect to epoch t0

(a) (b)

Fig. 3 The ll- and b-frames, lever-arm coordinates {bkx , bk

y , bkz } of antenna k, and the proposed GPS antennae configurations for a fine-form

vessels, and b full-form ships

GPS Solut

123

matrix Cllb , then the transformation from the b-frame to the

ll-frame is given by

rll ¼ Cllbrb ð2Þ

where rll and rb are the position vectors of the onboard

GPS antenna in the respective frames.

As illustrated by Fig. 3, the b-frame is the coordinate

system fixed to the vessel with the center at the CG, x-axis

toward starboard, y-axis toward bow, and z-axis along the

vertical axis of the vessel that is positive upward. The ll-

frame is centered at the CG, with x-axis toward geodetic

east, y-axis toward geodetic north, and z-axis along the

ellipsoidal normal that is positive upward (Fig. 3). The

vector of attitude variations drllðtiÞ at epoch ti with respect

to the epoch t0 can be computed from (2) by

drllðtiÞ ¼ ðCllbðtiÞ � Cll

bðt0ÞÞrb ð3Þ

where the z component of drllðtiÞ, the so-called attitude

induced heave dhAðtiÞ, is as follows

dhAðtiÞ ¼ � sinðrðtiÞÞ cosðpðtiÞÞbx þ sinðpðtiÞÞby

�

þ cosðrðtiÞÞ cosðpðtiÞÞbzg � � sinðrðt0ÞÞ cosðpðt0ÞÞbxfþ sinðpðt0ÞÞby þ cosðrðt0ÞÞ cosðpðt0ÞÞbz

�

ð4Þ

where {rðt0Þ, pðt0Þ}, and {rðtiÞ, pðtiÞ} are the measured roll

and pitch angles at epochs t0 and ti, and the symbols bx, by,

and bz denote the lever-arm coordinates of the onboard

GPS antenna with respect to the b-frame.

For the roll and pitch determination by GPS, the fol-

lowing iterative procedure can be used. According to the

GPS antennae configurations shown in Fig. 3, for fine- and

full-form ships, the following relation holds

r ¼ �sin�1 dh13

l13

� �� r0 ð5Þ

where r is the roll angle, dh13 the ellipsoidal height

difference between antennae 1 and 3, l13 the spatial

distance between the two antennae, and r0 is the roll offset.

Next, after removing the effect of the roll angle from the

ellipsoidal height difference dh12 between antennae 1 and

2, the pitch angle p can be computed for both fine- and full-

form ships as

dh012 ¼ dh12 þ ðb1z � b2

z ÞðcosðrÞ � 1Þ � ðb1x � b2

xÞ sinðrÞ� �

ð6Þ

p ¼sin�1 dh0

12

l12

� �� p0 for fine-form vessels

�sin�1 dh012

l12

� �� p0 for full-form ships

8<

:ð7Þ

where l12 is the spatial distance between antennae 1 and 2,

dh012 the roll-removed height difference between the

antennae, and p0 is the pitch offset. Then, the effect of

the pitch angle on the ellipsoidal height difference of

antennae 1 and 3 can be removed by

dh013 ¼ dh13 þ ðb1z � b3

z ÞðcosðpÞ � 1Þ þ ðb1y � b3

yÞ sinðpÞn o

ð8Þ

Having derived the pitch-removed dh013, the procedure

must be started from (5) by substituting dh13 by dh013 until

the differences between the computed roll and pitch

angles in two consecutive steps become negligible. This

completes the iterative algorithm to determine the vessel

attitude.

The next correction dhHðtiÞ appearing in (1) is due to

heave at epoch ti with respect to epoch t0. Since GPS

antennae, like all other onboard sensors, is fixed to the

vessel, the GPS-derived heave is along the z-axis of the

b-frame (IHO 2005, p. 195). However, what we need is the

heave effect with respect to the ll-frame, therefore, dhHðtiÞcan be derived by using (2) as

dhHðtiÞ ¼ cosðrðt0ÞÞ cosðpðt0ÞÞHðt0Þf g� cosðrðtiÞÞ cosðpðtiÞÞHðtiÞf g

ð9Þ

where the heaves Hðt0Þ and HðtiÞ are determined via the

difference of the instantaneous ellipsoidal heights of the

onboard antenna from a time-averaged height in a moving

window with a certain time span, for example, 25 s

(Godhavn 2000).

Next, the tidal correction dhTðtiÞ at epoch ti with respect

to epoch t0 can be derived as

dhTðtiÞ ¼ TðtiÞ � Tðt0Þ ð10Þ

where Tðt0Þ and TðtiÞ are the tide gauge observations at

epochs t0 and ti at a nearby tidal station. Note that in (10),

we assume that the selected tidal station is representative of

the sea-level variations for the whole area of field opera-

tions. If this is not the case, then the co-tidal and co-range

corrections must be applied.

The next correction appearing in (1) is the vessel load

variations dhLðtiÞ at epoch ti with respect to epoch t0.

Therefore, by measuring water-level marks Lðt0Þ and LðtnÞat the beginning (t0) and the end (tn) of the field operations,

dhLðtiÞ can be derived using a linear interpolation as

dhLðtiÞ ¼ti � t0tn � t0

� �LðtnÞ � Lðt0Þ ð11Þ

The last correction dhNðtiÞ in (1) corresponds to the

variations of the geoidal height of the onboard GPS

antenna due to movements of the vessel. A global

geopotential model can be used to derive the geoidal

heights at epochs t0 and ti, that is, N0 and Ni, and from that

dhNðtiÞ can be computed by

dhNðtiÞ ¼ Ni � N0 ð12Þ

GPS Solut

123

So far there has been no differentiation between squat

determination for hydrographic and navigation applications.

Indeed, (1) corresponds to squat modeling for hydrographic

applications since the roll and pitch effects are removed.

However, for navigation applications, such effects must be

retained in order to arrive at the maximum squat which

results in the minimum UKC. Consequently, for navigation

applications, the squat computation formula for the cruise

speed vðtiÞ at the antenna location can be written as

SðtiÞ ¼ hðt0Þ � hðtiÞ þ dhHðtiÞ þ dhTðtiÞ þ dhLðtiÞþ dhNðtiÞ ð13Þ

where all the notations are the same as (1).

Since the field operations must be performed with a

stepwise increase in cruise speed, while the speed is kept

constant for a short time during each step, there will be

redundant squat values at each speed which allows the LS-

derived squat values. After that, the analytical squat models

for hydrographic and navigation applications can be

derived by fitting suitable functions to the corresponding

LS-derived squat values.

Experimental setup

We now present the details of practical implementation of

the proposed squat modeling using a test field in the off-

shore waters of Kish harbor and using the fine-form survey

vessel of National Geographic Organization (NGO) of Iran

with 27 m length, 6-m beam, and 1-m draft. To develop the

squat models for both navigation and hydrographic appli-

cations, three GPS antennae were installed onboard at the

bow-port, stern-port, and stern-starboard locations. In order

to increase the accuracy of GPS attitude determination, the

onboard antennae were mounted on outstretched arms from

both sides of the vessel as schematically shown in Fig. 3a.

In this way, the distances between the antennae along the

transverse and longitudinal axes of the vessel became

approximately 15 m.

The lever-arm coordinates of the onboard GPS antennae

and the echo sounder transducer were determined by sur-

veying measurements when the vessel was on a dry dock

for repair. Table 1 offers the measured lever-arm coordi-

nates of the aforementioned sensors with respect to the

b-frame.

Additionally, a base GPS antenna was established at the

wharf close to the test field for the relative kinematic

positioning. The field operations were performed in a calm

day to avoid excessive motions of the vessel and related

disturbances on the kinematic positioning. The GPS

observations were started about 20 min before departure

when the vessel was moored at the wharf and continued

throughout the sailing.

The sea-level information during the field operations

was taken from a nearby coastal tide gauge station. The

water-level marks were measured from four corners of the

deck at the beginning and the end of the field operations to

record the vessel load variations. Table 2 presents the main

features of the experimental setup.

Results and discussions

After the field operations, the Bernese GPS Software 5.0

(Dach et al. 2007) was used for precise point positioning as

well as carrier phase double-difference processing, in order

to obtain accurate kinematic coordinates of the onboard

GPS antennae. Among the processing settings, the 5-s

satellite clock corrections obtained from the Center of

Orbit Determination in Europe (CODE) (ftp://unibe.ch/

aiub/CODE/) and the 3� cutoff angle are worth mentioning.

The position information of the onboard GPS antennae was

used to determine the cruise ground speed, attitude, and

heave, which are needed for the squat modeling, as well as

controlling the quality of kinematic positioning through the

fixed baselines. Figure 4 shows the results of the quality

control with some outstanding disturbances. It is found that

those disturbances are related to the turns of the vessel,

which are accompanied by sudden speed reductions.

Table 1 The lever-arm coordinates of the onboard sensors

Sensors bx (m) by (m) bz (m)

Ant. 1 at stern-port -8.719 -3.985 4.650

Ant. 2 at bow-port -8.553 10.412 4.673

Ant. 3 at stern-starboard 6.325 -3.834 4.562

Echo sounder transducer 0.000 3.065 -0.965

Table 2 The main features of the experimental setup

Items Descriptions

Onboard GPS receivers Dual-frequency JAVAD

(TPS LEGACY)

Onboard GPS antennae Choke ring JAVAD

(JPSREGANT_DD_I)

Base GPS receiver LEICA GX1220

Base GPS antenna LEIAX1202

Range of the cruise ground speed 0.00–12.67 knots

Range of the water depth 22.08–44.89 m

Sampling rate of the GPS

observations

1 s

Sampling rate of the tide

gauge observations

10 min

Duration of the field operations 50 min

Maximum tracked satellites 11

GPS Solut

123

Moreover, we found that the biggest disturbance in Fig. 4

at about 5.3 9 104 s GPS time is connected to the U-turn

of the vessel. Further investigations revealed that such

disturbances could be resulted when wrong or no antenna

phase center variations (PCV) correction is applied, and/or

orientations of the onboard GPS antennae change (Kouba

J., personal communication, Geodetic Survey Division,

National Resources Canada). Both were the case in the

field operations.

Using the derived formulae, we computed the roll, pitch,

and heave of the vessel. Figures 5 and 6 show the results.

As Fig. 5 shows, the pitch angle increases as the vessel

speed increases, and its fluctuations are caused by the

sudden speed reductions during the vessel turns. Figure 5

also indicates that the U-turn has changed the trend of the

roll angle. According to Fig. 6, the range of the heave

variations are about 2 cm. Moreover, the turns of the vessel

have caused the heave jumps, and the maximum one is in

relation to the U-turn.

Next, the corrections due to the attitude, heave, tide, vessel

load, and geoidal height variations were computed. Due to

small extent of the test field, no corrections from the co-tidal

and co-range charts were applied to the coastal tide gauge

observations. The geoidal height variations within the test

field were computed from EGM2008 global geopotential

model (Pavlis et al. 2008). Table 3 offers a statistical sum-

mary of all applied corrections. According to this table, the

attitude and geoidal height variations were the dominant

disturbing effects. As the table shows, the biggest attitude

effects are related to the bow-port antenna, and the smallest

ones to the stern-port antenna, as the result of their lever-arms

given in Table 1. From Table 3, we can also conclude that

the load variations are the least among the disturbing effects.

For hydrographic applications, the squat values at the

echo sounder location were derived from the computed

squats at the onboard GPS antennae by knowing the lever-

arm coordinates. To determine the maximum squat for

navigation applications, the squats at the stern-port and

stern-starboard of the vessel were derived and compared

with each other. Figure 7 shows the derived squat values as

a function of GPS time at the transducer, stern-port, and

stern-starboard locations. According to the figure, the

maximum squat happened at the stern-starboard of the

vessel; therefore, it is the one that must be considered for

the navigation applications. We also plotted the GPS-

derived ground speed of the vessel as a function of GPS

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

−0.02

−0.01

0

0.01

0.02

GPS time (s)

Bas

elin

e di

ffere

nce

(m)

Mean = 0.0005 m, RMS = 0.0083 m

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

−0.02

−0.01

0

0.01

0.02

Bas

elin

e di

ffere

nce

(m)

GPS time (s)

Mean = −0.0049 m, RMS = 0.0194 m

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

−0.02

−0.01

0

0.01

0.02

GPS time (s)

Bas

elin

e di

ffere

nce

(m)

Mean = 0.0014 m, RMS = 0.0054 m(a)

(b)

(c)

Fig. 4 Comparison of the GPS and geodetic-derived distances

between the onboard antennae as a function of GPS time. a Baseline

2–3, b baseline 1–3, and c baseline 1–2

5.15 5.2 5.25 5.3 5.35 5.4x 104

−0.5

0

0.5

1

1.5

GPS time (s)

Atti

tude

ang

le (

deg)

Roll anglePitch angle

Fig. 5 The GPS-derived roll and pitch angles of the vessel as a

function of GPS time

GPS Solut

123

time in Fig. 8. The sudden speed changes during the turns

are evident in the figure. Comparison of the Figs. 5, 7, and

8 indicates the strong correlation of the squat with the pitch

and vessel speed, and its weak correlation with the roll.

The redundant squat values during the constant cruise

speed allowed LS estimation of the squats at different

speeds and their precisions by knowing the error budget of

the input data given in Table 4. Figure 9 shows the LS-

derived squat values as a function of cruise speed at the

transducer and stern-starboard locations. As can be seen in

the figure, the hydrographic squat values are considerably

less than the navigation ones, which supports the need of

having separate squat models for the two applications.

A summary of statistical information of the estimated

precisions of the LS-derived squat values at different speeds

for hydrographic and navigation applications is given in

Table 5. According to the table, the average precisions of

the LS-derived squat values for hydrographic and navigation

applications are 0.029 and 0.010 m, respectively.

According to Fig. 9, the LS-derived squat values are

contaminated with outliers. To suppress the effect of the

outliers in the squat models, the Tukey M-estimator with

bisquare weight function based on the iteratively

reweighted least squares (IRWLS) was used (Marrona et al.

2006) and a variety of fitting functions were tested. The

exponential functions were found the best based on the

RMS’s of fit among the tested ones. The parameters of

the exponential functions which present the analytical

squat models for hydrographic and navigation applications

are given in Table 6. Figures 10 and 11 show the analytical

squat models as the robust fit to the corresponding

LS-derived squat values. As can be seen from the figures,

the analytical squat models, while following the trend of

squat variations versus cruise speed, are free from the

effects of the outliers thanks to the robust M-estimation.

Having the estimated precisions of the parameters of the

analytical squat models and the GPS-derived cruise ground

speed, application of the error propagation law enabled us

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

GPS time (s)

Hea

ve (

m)

Fig. 6 The GPS-derived heave of the vessel as a function of GPS

time

Table 3 A statistical summary of the corrections for attitude dhAðtiÞ,heave dhHðtiÞ, tide dhT ðtiÞ, vessel load dhLðtiÞ, and geoidal height

dhNðtiÞ variations in the case study

Corrections Antennae Min.

(m)

Max.

(m)

Mean

(m)

STD

(m)

dhAðtiÞ Bow-port -0.013 0.373 0.116 0.085

Stern-port -0.125 0.107 -0.017 0.059

Stern-starboard -0.175 0.013 -0.050 0.043

dhHðtiÞ All -0.033 0.028 0.000 0.004

dhT ðtiÞ All 0.001 0.030 0.026 0.008

dhLðtiÞ All 0.001 0.010 0.005 0.003

dhNðtiÞ All -0.002 0.313 0.122 0.102

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

0

0.05

0.1

0.15

0.2

0.25

0.3

GPS time (s)

Squ

at (

m)

Stern−portStern−starboardTransducer

Fig. 7 The squat values as a function of GPS time at the transducer,

stern-port, and stern-starboard locations

5.15 5.2 5.25 5.3 5.35 5.4x 10

4

0

2

4

6

8

10

12

GPS time (s)

Cru

ise

grou

nd s

peed

(kn

ots)

Fig. 8 The GPS-derived ground speed of the vessel as a function of

GPS time

GPS Solut

123

to compute the precisions of the derived models, which are

summarized in Table 7. According to the table, the mean

precisions of the hydrographic and navigation squat models

are 0.009 and 0.012 m, respectively. Note that the

improved precision of the hydrographic squat model over

the LS-derived squat values, compare Tables 5 and 7, is

due to application of the robust M-estimation and reduction

in the outliers effect.

In order to see the significance of the squat modeling

for hydrographic applications, we refer to the IHO stan-

dards (IHO 2008) for bathymetry accuracy. By consid-

ering the average depth of the test field, that is, 33.5 m,

the required bathymetry accuracies based on the men-

tioned standards are 0.35, 0.66, 1.26, and 1.26 m for

special, 1st a, 1st b, and 2nd orders projects, respectively.

If we assume that the speed of the vessel during hydro-

graphic bathymetry is 12.5 knots, then in reference to

Fig. 10, the associated squat value at the transducer

location will be 0.14 m. Considering the other sources of

the bathymetry errors such as sound speed, heave, and the

orientation, it will be quite unlikely that the required

accuracy for the mentioned orders, especially the special

and 1st a orders, be obtained if the squat effect is not

considered.

For navigation applications, consider the vessel of the

case study which experienced at a speed of 12.64 knots a

squat of 30 cm, which is 30 % of its 1-m draft. This shows

how the squat can affect the draft and UKC of a vessel

0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

Cruise ground speed (knots)

Squ

at (

m)

Stern−starboardTransducer

Fig. 9 The LS-derived squat values as a function of cruise speed at

the transducer (for hydrographic applications) and the stern-starboard

(for navigation applications)

Table 5 A summary statistical information of the precisions of the

LS-derived squat values for hydrographic and navigation applications

Measures Precisions of LS-derived squats

Hydrographic Navigation

Min. (m) 0.005 0.003

Max. (m) 0.075 0.024

Mean (m) 0.029 0.010

Table 6 Parameters of the exponential functions SðvÞ ¼ a exp ðbvÞ þc as the analytical squat models versus the cruise speed v ðknots) for

hydrographic and navigation applications

Parameters Squat models

Hydrographic Navigation

a ðm) 0.012±0.001 0.025 ± 0.002

b ð1=knots) 0.202 ± 0.007 0.197 ± 0.005

c (m) -0.013 ± 0.002 -0.023 ± 0.003

RMS of fit (m) 0.006 0.008

Table 4 Error budget of the input data for the LS estimation of the

squats at different speeds

Input data Error budget

vðtiÞ 0.013 knots

hðt0Þ 0.001 m

hðtiÞ 0.007 m

dhNðtiÞ 0.011 m

Hðt0Þ, HðtiÞ 0.010 m

Tðt0Þ, TðtiÞ 0.010 m

Lðt0Þ, LðtnÞ 0.010 m

rðt0Þ, rðtiÞ 0.039�pðt0Þ, pðtiÞ 0.041�bx, by, bz 0.005 m

0 2 4 6 8 10 12

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Cruise ground speed (knots)

Squ

at (

m)

Fig. 10 The analytical squat model (dark blue line) for hydrographic

applications derived as the result of the robust M-estimation, and the

LS-derived squat values (light blue dots) at the transducer location

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123

which are among major concerns for the safety of navi-

gation, especially at high speeds in confined channels.

Conclusions and remarks

A GNSS-based squat modeling method is formulated and

presented for hydrographic and navigation applications.

Removal of the disturbing effects from ellipsoidal heights

of the onboard GPS antennae with respect to a reference

state provides the basis of the method. Suitable GPS

antennae configurations onboard full-form ships and fine-

form vessels are proposed to model bow and stern squats,

respectively. Performance of the method was tested by the

field operations. According to the numerical results, the

navigation squats were about twice the hydrographic ones,

which demonstrate the importance of differentiating

between hydrographic and navigation applications for

squat modeling. Though the method is implemented on a

survey vessel in open waters, it can be applied to any type

of ship at any waterway. A navigation squat model can be

integrated with the nautical charts for real-time UKC

determinations to assist the safety of marine traffic at high

speeds. Moreover, a hydrographic squat model can provide

real-time squat corrections for accurate bathymetry.

Acknowledgments The authors would like to thank National

Geographic Organization (NGO) of Iran for their cooperation and

support during the field operations. The help from Eng. N. Abdi for

data processing using the Bernese GPS software is gratefully

acknowledged. Last but not least, the authors would like to express

their gratitude to two anonymous reviewers for their constructive

comments and suggested corrections.

References

Briggs M (2006) Ship squat predictions for ship/tow simulator.

Coastal and hydraulics engineering technical note CHETN-I-72,

U.S. Army Engineer Research and Development Center, Vicks-

burg, MS

Dach R, Hugentobler U, Fridez P, Meindl M (2007) Bernese GPS

software. Version 5.0, Astronomical Institute, University of

Bern, Bern

Delefortrie G (2010) Squat prediction in muddy navigation areas.

Ocean Eng 37(16):1464–1476

Derrett DR (1999) Ship stability for masters and mates. Revised by

Barrass CB, 5th edn, Butterworth-Heinemann, ISBN: 0-7506-

4101-0

El-Kader FA, EL-Soud MSA, EL-Serafy K, Hassan EA (2003) An

integrated navigation system for Suez Canal (SCINS). J Navig

56(2):241–255

Feng Y, O’Mahony S (1999) Measuring ship squat, trim, and under-

keel clearance using on-the-fly kinematic GPS vertical solutions.

Navigation 46(2):109–117

Feng Y, Kubic K, O’Mahony S (1996) On-the-fly GPS kinematic

positioning for measuring squat and trim of large ships. In:

Proceedings of ION GPS, The Institute of Navigation, Kansas

City, MO, 9:367–373

Giorgi G, Gourlay TP, Teunissen PJG, Huisman L, Klaka K (2010)

Carrier phase ambiguity resolution for ship attitude determina-

tion and dynamic draught. In: XXIV FIG International Congress,

Sydney, Australia

Godhavn, J-M (2000) High quality heave measurements based on

GPS RTK and accelerometer technology. Ocean Conf Record

(IEEE) 1:309–314

Gourlay TP, Cray WG (2009) Ship under-keel clearance monitoring

using RTK GPS. In: Proceedings of Coasts and Ports,

Wellington

Harting A, Laupichler A, Reinking J (2009) Considerations on the

squat of unevenly trimmed ships. Ocean Eng 36(2):193–201

IHO (2005) Manual on hydrography, 1st edn. Publication M-13,

International Hydrographic Bureau, Monaco

IHO (2008) IHO standards for hydrographic surveys. 5th edn,

Special Publication No. 44, International Hydrographic Bureau,

Monaco

Marrona RA, Martin DR, Yohai VJ (2006) Robust statistics. Theory

and methods. Wiley, New York, ISBN: 0-470-01092-4

Maynord ST, Briggs MJ (2006) Ship squat measurements using GPS

at Charleston harbor. In: Proceedings of the international

conference on civil engineering in the oceans, Oceans VI:

449–463

Melachroinos SA, Tchalla M, Biancale R, Menard Y (2011)

Removing attitude-related variations in the line-of-sight for

kinematic GPS positioning. GPS Solut 15(3):275–285

Pavlis NK, Holmes SA, Kenyon SC, Factor JK (2008) An earth

gravitational model to degree 2160: EGM2008. In: The General

Assembly of the European Geosciences Union, Vienna, Austria

Varyani KS (2006) Squat effects on high speed craft in restricted

waterways. Ocean Eng 33(3–4):365–381

0 2 4 6 8 10 12

0

0.05

0.1

0.15

0.2

0.25

0.3

Cruise ground speed (knots)

Squ

at (

m)

Fig. 11 The analytical squat model (red line) for the navigation

applications derived as the result of the robust M-estimation, and the

LS-derived squat values (pink dots) at the stern-starboard location

Table 7 Precision of the analytical squat models for hydrographic

and navigation applications

Measures Model precisions

Hydrographic Navigation

Min. (m) 0.003 0.004

Max. (m) 0.022 0.028

Mean (m) 0.009 0.012

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Zilkoski DB, D’Onofrio JD, Fury RJ, Smith CL, Huff LC, Gallagher

BJ (1999) Centimeter-level positioning of a U.S. coast guard

buoy tender. GPS Solut 3(2):53–65

Author Biographies

Alireza A. Ardalan received a

M.Sc. degree in geodesy from

K.N. Toosi University Tehran,

Iran, and M.Sc. and Ph.D.

degrees from Stuttgart Univer-

sity Stuttgart, Germany. He is a

full professor of geodesy at the

Department of Surveying and

Geomatics Engineering, College

of Engineering, University of

Tehran. His research interests

include satellite positioning,

engineering geodesy, hydrogra-

phy, and gravity field modeling.

Mohammad-Hadi Rezvanireceived a M.Sc. degree in

hydrography from University of

Tehran, Iran. He is a research

fellow at the Department of Sur-

veying and Geomatics Engineer-

ing, College of Engineering,

University of Tehran. His

research interests include hydrog-

raphy, multibeam sonar calibra-

tion, and GNSS applications.

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