Journal of Ship Research, Vol. 57, No. 1, March 2013, pp. 1–13

http://dx.doi.org/10.5957/JOSR.57.1.120007

Comparison of Measured Ship Squat with Numerical

and Empirical Methods

Michael J. Briggs,* Paul J. Kopp,† Vladimir K. Ankudinov,‡ and Andrew L. Silver§

*Research Hydraulic Engineer, Coastal and Hydraulics Laboratory, U.S. Army Engineer Research and Development Center, Vicksburg, Mississippi†Naval Architect, Naval Surface Warfare Center, Carderock Division, Seakeeping Department, West Bethesda, Maryland‡Formerly Group Director (now deceased), TRANSAS, Hydrodynamics and Research Department, Washington, DC§Engineer, Naval Surface Warfare Center, Carderock Division, Seakeeping Department, West Bethesda, Maryland

The Beck, Newman and Tuck (BNT) numerical predictions are used in the Coastaland Hydraulics Laboratory (CHL) Channel Analysis and Design Evaluation Tool(CADET) model for predicting underkeel clearance (UKC) resulting from ship motionsand squat. The Ankudinov empirical squat prediction formula has been used in theCHL ship simulator and was recently updated. The World Association for WaterborneTransport Infrastructure (formerly The Permanent International Association of Navi-gation Congresses, PIANC) has recommended several empirical and physics-basedformulas for the prediction of ship squat. Some of the most widely used formulasinclude those of Barrass, Eryuzlu, Huuska, ICORELS, Romisch, Tuck, andYoshimura. The purpose of this article is to compare BNT, Ankudinov, and PIANCpredictions with measured DGPS squat data from the Panama Canal for four ships.These comparisons demonstrate that the BNT, Ankudinov, and PIANC predictions fallwithin the range of squat measurements and can be used with confidence in deepdraft channel design.

Keywords: ship squat; numerical models; empirical formulas; ship measurements

1. Introduction

HISTORICALLY, THE World Association for Waterborne TransportInfrastructure (formerly The Permanent International Associationof Navigation Congresses, PIANC) has recommended severalempirical and physics-based formulas for the prediction of shipsquat. These include those of Tuck (1966), Guliev (1971), Hooft(1974), Huuska (1976), ICORELS (1980), Norrbin (1986),Yoshimura (1986), Romisch (1989), Millward (1992), Eryuzlu,Cao, and D’Agnolo (1994), and Barrass (2009). Most are func-tions of a limited number of ship and channel parameters in aneffort to minimize the number of free parameters and increase theease of use. Typical ship parameters include ship speed, Vk, blockcoefficient, CB, and ship dimensions of length between perpen-diculars, Lpp, beam, B, and draft, T. Channel parameters includewater depth, h, type of channel cross-section, Ac, side slope, n,

and bottom channel width, W. Channel types include idealizedcross-sections representing unrestricted (U) or open channels,restricted (R) or dredged with a trench, and canal (C) with sidesthat extend to the surface (PIANC 1997). However, many shipand channel parameters are not known with certainty. Channelcross-sections are usually not as simple as the three idealizedshapes and dimensions can vary considerably along the channellength. All parameters used in this article are listed in the Nomen-clature section.The PIANC formulas are usually considered the standard for

predicting ship squat if field or laboratory measurements are notavailable. Most of these PIANC formulas were developed over adecade ago from limited laboratory and field measurements butare used for the newer generation of containerships, tankers, andbulk carriers. Some such as Tuck and Romisch are more physics-based, whereas others like Barrass and Yoshimura are moreempirical in nature. Although many pilots and channel designershave their “favorites,” no one formula has demonstrated itself tobe universally better for all ship types and channel shapes. Briggs

Manuscript received at SNAME headquarters November 19, 2012; revised

manuscript received January 11, 2013.

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(2006) has recommended examining squat predictions with morethan one formula and comparing the results based on the type ofship, channel, and formula constraints. Because there is no oneformula that is universally accepted, an average, range, or maxi-mum value might be considered in channel design.The PIANC MarCom Working Group 49 (WG49) is updating

the 1997 PIANC guidance (1997) and expects to publish theirreport in 2013. The WG49 has reduced the number of squat pre-diction formulas to seven of the most popular formulas. Theseinclude the updated versions by Barrass, Eryuzlu, Huuska,ICORELS, Romisch, Tuck, and Yoshimura.The Channel Analysis and Design Evaluation Tool (CADET)

is a computer program originally developed by the US Navy todetermine the “optimum” dredge depth for entrance channels (2005)used by military vessels. It uses an external program, BNT (basedon the work of Beck, Newman & Tuck 1975), to predict shipsquat. BNT is a potential flow program that predicts sinkage andtrim from vertical force and pitching moment resulting from thedynamic pressure on the hull. Briggs et al. (2010a) found reason-able agreement between BNT and PIANC predictions for severalships including an aircraft carrier, two containerships, a tanker,and a bulk carrier for two different channels.The Ankudinov ship squat formula is an empirical formula that

predicts maximum squat resulting from midpoint sinkage andvessel trim (Ankudinov et al. 2000). It has been used in theCoastal and Hydraulics Laboratory (CHL) Ship/Tow Simulator toaccount for ship squat in the determination of instantaneous

underkeel clearance (UKC). Compared with other empirical for-mulas, it is somewhat more complicated because it includes addi-tional input parameters to account for the effects of the ship’spropeller, bulbous bow, stern transom, initial trim, and the chan-nel’s depth, blockage, and cross-section. Modifications haverecently been incorporated to improve its accuracy.Most mariners and pilots are more concerned with the static

UKC and not just ship squat. Static UKC is the safety margin or“what is left” after subtracting static draft and trim from thechannel depth. Ship squat and an allowance for vertical shipmotions (for ships exposed to waves) must be subtracted to getthe net UKC. A brief mention of the static UKC is provided foreach ship, but the main focus of the article is comparisons ofship squat.The purpose of this article is to compare the BNT, Ankudinov,

and PIANC predictions for four ships with measured ship squatdata from the Panama Canal. This article is an update of prelim-inary comparisons of Ankudinov and PIANC predictions (Briggs& Daggett 2009). These comparisons of almost 3000 squat mea-surements demonstrate that the BNT, Ankudinov, and PIANCpredictions fall within the range of the squat measurements andcan be used with confidence in deep draft channel design. Thesecond section in this article describes the ship and channelparameters in the Panama Canal study. Section 3 describes theCADET/BNT ship squat program. The Ankudinov ship squatformulas are presented in the next section. Section 5 describesthe five PIANC empirical squat formulas used in this study.

Nomenclature

Ac ¼ wetted cross-section area of canal, m2

As ¼ ship’s underwater amidships

cross-section, m2

B ¼ ship’s beam, m

BS ¼ bias or difference between predicted

and measured squat, m

BTr ¼ stern transom width, Ankudinov

CB ¼ ship’s block coefficient

CF ¼ correction factor for ship

shape, Romisch

CV ¼ correction factor for ship

speed, Romisch

Fnh ¼ Froude depth number

g ¼ gravitational acceleration (9.81 m/s�2)

h ¼ water depth, m

hm ¼ mean water depth, Romisch, m

hmT ¼ restricted channel water depth,

Romisch, m

hOut ¼ height outside underwater trench,

similar to hT, ft

hT ¼ height of dredged underwater trench, m

j ¼ location along Panama Canal

K ¼ channel coefficient, Barrass

Kb ¼ correction factor for channel

width, Eryuzlu

KTb ¼ bulbous bow factor, Ankudinov

KC ¼ canal channel correction factor,

Romisch

KSP ¼ propeller sinkage factor, Ankudinov

KTP ¼ propeller trim factor, Ankudinov

KR ¼ restricted channel correction factor,

Romisch

KS ¼ channel width correction factor, Huuska

KTr ¼ trim coefficient, Ankudinov

KTTr ¼ stern transom factor, Ankudinov

KTT1 ¼ initial trim effect factor, Ankudinov

KU ¼ unrestricted channel correction factor,

Romisch

KDT ¼ squat at critical speed, Romisch

Lpp ¼ ship length between perpendiculars, m

MAE ¼ mean absolute error between predicted

and measured squat, m

n ¼ inverse bank slope

nTr ¼ trim exponent, Ankudinov

PCh1 ¼ channel effect parameter, Ankudinov

PCh2 ¼ channel effect, trim correction

parameter, Ankudinov

PFnh ¼ ship forward speed parameter,

Ankudinov

PhT ¼ propeller effect in shallow water on trim

parameter, Ankudinov

Pþh/T ¼ water depth parameter, Ankudinov

PHu ¼ ship hull parameter for shallow water,

Ankudinov

R ¼ Pearson correlation coefficient

RMSE ¼ root mean square error, m

RS ¼ ratio between predicted and

measured squat

S ¼ blockage factor ¼ As/Ac

Sb ¼ bow squat, m

Sh ¼ channel depth factor for R and

C channels, Ankudinov

Sm ¼ midpoint ship sinkage, m/m�1,

Ankudinov and BNT

SMax ¼ maximum ship squat at bow or stern,

BNT, ft

Ss ¼ stern squat, m

SM,j ¼ measured ship squat at location j, mSP,j ¼ predicted ship squat at location j for

BNT, Ankudinov, or PIANC, m�SM ¼ average ship squat for measured

DGPS, m�SP ¼ average ship squat for predicted BNT,

Ankudinov, or PIANC, m

s1 ¼ corrected blockage factor, Huuska

T ¼ ship draft, m

Tap ¼ draft at aft perpendicular

Tfp ¼ draft at forward perpendicular

Tr ¼ ship trim, m/m�1, Ankudinov and BNT

Vcr ¼ critical ship speed, m/s–1

Ve ¼ enhanced ship speed, Yoshimura, m/s–1

Vk ¼ ship speed, knots

VS ¼ ship speed, m/s–1

W ¼ channel width, measured at bottom, m

WEff ¼ effective width of waterway, m

WTop ¼ channel width, measured at top, m

sM ¼ standard deviation of ship squat for

measured DGPS, m

sP ¼ standard deviation of ship squat

for predicted BNT, Ankudinov,

or PIANC, m

r ¼ ship volume of displacement, m3

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Results are presented and discussed in Section 6 for the fourships. Finally, the last section provides some conclusions.

2. Channel and ship parameters

This validation exercise involves field measurements (Daggett &Hewlett 1998a, 1998b) made in the Gaillard Cut section of thePanama Canal (Fig. 1) in December 1997 and April 1998 using theDifferential Global Positioning System (DGPS). The Gaillard Cut(Fig. 2) is a typical “canal” cross-section and stretches fromCulebra to Bas Obispo, a distance of approximately 9.1 km fromstation location 1670 to 1970 (in hundreds of feet). The channelwidth for all transits was 152 m. Ankudinov et al. (2000) reportedthat the minimum water depths in the center 91.4-m-section of thecanal were 13.6 to 15 m in the 1997 study and 12.4 to 13.7 m in the1998 study. Additional details on average depths and static UKCare presented for the individual ships. The DGPS measurementswere made using dual-frequency equipment mounted at three pointson each ship (bow and port and starboard bridge wings). The ver-tical accuracy levels were on the order of 1 to 5 cm. According toDr. Daggett (personal communication), a larger source of error oruncertainty is measurement of water levels and depth.Four of the ships from the 1997 and 1998 studies were selected

for comparison. Table 1 lists the parameters for these vessels thatincluded a Panamax tanker (Elbe), Panamax bulk carrier (GlobalChallenger), Panamax containership (Majestic Maersk), and con-tainership (OOCL Fair). These four ships represent 2978 individ-ual squat comparisons with measured DGPS data. The ships aregrouped in Table 1 in alphabetical order and by trim location atthe bow or stern.Figure 3 shows the ship speeds through the Gaillard Cut. All of

the ships were traveling northward from the Pacific to the AtlanticOcean or from right to left in this figure. When calculating shipsquat, one wants to avoid acceleration and deceleration. Thesetransits obviously have some periods with nonsteady ship speedsas a result of maneuvering concerns and bends in the channel(there are four bends in this section of the Panama Canal) but areincluded in the averages. The Elbe had the smallest ship speedsbecause it was somewhat overloaded for the drought conditions inApril 1998 and was required to go slower for the shallower depths

and UKC. The Majestic Maersk has the largest ship speeds and themost variation in speed.

3. Beck, Newman, and Tuck/Channel Analysis andDesign Evaluation Tool numerical model

The BNT potential flow program by Beck, Newman, and Tuck(1975) is the default option for predicting underway sinkage andtrim in CADET. Although included with and loosely coupled toCADET, BNT is completely independent and standalone. Becausechannel geometry can vary from reach to reach, CADET supportsthe ability to define multiple sets of sinkage and trim data sets forthe same ship and loading condition.The BNT sinkage and trim prediction program is based on early

work by Tuck (1966, 1967) investigating the dynamics of a slen-der ship in shallow water at various speeds for an infinitely widechannel and for a finite width channel such as a canal. Tuck’soriginal formula has been successfully used for tankers for manyyears. This work was expanded to include a typical dredgedchannel with a finite-width inner channel of a certain depth andan infinitely wide outside channel of shallower depth by Beck,Newman, and Tuck (1975).Figure 4 is a schematic of the simplified channel cross-section

used in BNT. In addition to the automatically specified insidechannel depth, h, the user has the option to include the channelwidth, W, and outside channel depth, hOut. The value of hOutremains the same for all h values. One restriction to BNT is thatthe sides of the channel are fixed as vertical.Fig. 1 Gaillard Cut in Panama Canal

Fig. 2 Ship transiting Gaillard Cut in Panama Canal

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In his early work, Tuck (1966) calculated the dynamic pressureof slender ships in finite-water depth and infinite and finite-waterwidth by modeling the underwater area of the hull. This under-water area was defined by the 21 equally spaced stations along theship’s length. Therefore, the ship’s geometry file, draft, speeds,and water depths are used in the BNT squat calculations. Withinthis analysis, the fluid is assumed to be inviscid and irrotationaland the hull long and slender. Input hull definition is provided interms of the waterline beam and sectional area at these 21 stations

between the forward and aft perpendiculars (Fig. 5). Typically,generic ship lines from a ship database are used and adapted fora particular ship because ship lines are proprietary and not readilyavailable for newer ships.The dynamic pressure is obtained for each depth Froude num-

ber, Fnh, by differentiating the velocity potential along the lengthof the hull. The Fnh is defined as

Fnh ¼ Vsffiffiffiffiffigh

p ð1Þ

where g is gravitational acceleration and VS is ship speed in m/s.The sinkage and trim predictions are obtained from the dynamicpressure by calculating the vertical force and pitching moment,which are translated to vertical sinkage and trim angle. Channeldepths should be the same order as the draft of the ship to satisfythe shallow-water approximations assumed in Tuck (1966).The BNT program numerically calculates midship sinkage, Sm,

and trim, Tr, as a function of Fnh. Because English units are usedin CADET, sinkage is measured in feet positive for downwardmovement. Trim in feet is the difference between sinkage at thebow and stern positive for bow down. The equivalent bow Sb andstern Ss squat are given by

Sb ¼ Sm þ 0:5 Trð ÞSs ¼ Sm � 0:5 Trð Þ ð2Þ

This is a simplistic representation of the squat at the bow andstern because it assumes they are equidistant, fore and aft, fromthe midpoint of the ship. An Excel spreadsheet was created fromthe BNT output to iterate between water depths and ship speeds ateach measurement location.

Fig. 3 Ship speeds during Panama Canal transits. All ships were north-

bound, sailing from right to left

Table 1 Ship parameters for Panama Canal study

Ship ID Type Location Date

Lpp B Tfp Tap

CB

Vk

(m) (m) (m) (m) (kt)

E T B 1998 222.0 32.2 11.3 11.3 0.84 5 to 7

G B B 1997 216.0 32.3 11.7 11.8 0.83 9 to 10

M C S 1997 284.7 32.2 11.8 11.8 0.63 7 to 12

O C S 1998 227.0 32.2 9.8 10.6 0.65 6 to 10

Ship ID: E ¼ Elbe, G ¼ Global Challenger, M ¼ Majestic Maersk, O ¼ OOCL Fair; Type: T ¼ tanker, B ¼ bulk carrier, C ¼ containership; Location: B ¼bow, S ¼ stern; Date: 1997 ¼ December, 1998 ¼ April.

Fig. 4 BNTchannel geometry variables

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4. Ankudinov model

Ankudinov and Jakobsen (1996) and Ankudinov et al. (1996,2000) proposed the MARSIM 2000 formula for maximum squatbased on Sm and Tr in shallow water. The Ankudinov method hasundergone considerable revision as new data were collected andcompared. The most recent modifications from a study of shipsquat in the St. Lawrence Seaway by Stocks, Daggett, and Page(2002) and correspondence between Ankudinov and Briggs inApril 2009 are contained in this study. These new revisionswere programmed and documented in a technical note by Briggs(2009). The Ankudinov formula has been used extensively in theCHL Ship Tow Simulator.The Ankudinov prediction is one of the most complicated for-

mulas for predicting ship squat because it includes many empiricalfactors to account for the effects of ship and channel. The restric-tion Fnh � 0.6 is applied. The maximum ship squat, SMax, is afunction of Sm and Tr given by

SMax ¼ SbSs

¼ Lpp Sm � 0:5Trð ÞLpp Sm þ 0:5Trð Þ

��ð3Þ

The SMax can be at the bow or stern depending on the value ofTr. The negative sign is used for bow squat, Sb, and the positivesign for stern squat, Ss.

4.1. Midpoint sinkage Sm

The Sm is defined as

Sm ¼ 1þ KSP

� �PHuPFnh

Pþh=TPCh1 ð4ÞThe ship, water depth, and channel parameters in this midpoint

sinkage equation are described subsequently. The propeller param-eter KS

P is defined as

KSP ¼ 0:15 single propeller

0:13 twin propellers

�ð5Þ

The ship hull parameter for shallow water, PHu, was recentlymodified by Ankudinov and Briggs (2009) as

PHu ¼ 1:7CBBT

L2pp

!þ 0:004C2

B ð6Þ

The ship forward speed parameter, PFnh, is given by

PFnh¼ F

1:8þ0:4Fnhð Þnh ð7Þ

which is a numerical approximation to the term “F2nh

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� F2

nh

p”

that is in many of the PIANC empirical squat formulas.The water depth effects parameter, Pþh/T, is defined as

Pþh=T ¼ 1:0þ 0:35

h=Tð Þ2 ð8Þ

The channel effects parameter, PCh1, for an R or C channel isgiven by

PCh1 ¼ 1:0þ 10Sh � 1:5 1:0þ Shð ÞffiffiffiffiffiSh

pð9Þ

where the channel depth factor, Sh, is defined by

Sh ¼ CBS

h=T

� �hTh

� �ð10Þ

and hT is the trench height measured from the bottom. The block-age factor S is a measure of the relative cross-sectional area of theship, As, to that of the channel, Ac, defined as

S ¼ As

Ac¼ 0:98BT

Whþ nh2ð11Þ

The “0.98” factor is the result of the radius on the corners of thehull. The Ac is a projection of the channel sides to the watersurface. However, for the Panama Canal comparisons, the mea-sured Ac was used.

4.2. Vessel trim

The Tr was also recently modified by Ankudinov and Briggs(2009) as

Tr ¼ �1:7PHuPFnhPh=TKTrPCh2 ð12Þ

In addition to the two parameters already described for themidpoint sinkage equation, the Tr also includes parameter Ph/T,coefficient KTr, and channel effect trim correction parameter PCh2

to account for the effects of the ship propellers, bulbous bow, sterntransom, and initial trim.The vessel trim parameter Ph/T accounts for the reduction in

trim resulting from the propeller in shallow water and is defined as

Ph=T ¼ 1� e2:5 1�h=Tð Þ

Fnh

h ið13Þ

The trim coefficient, KTr, is a function of many factors and isgiven by

KTr ¼ CnTrB � 0:15KS

P þ KTP

� �� KTB þ KT

Tr þ KTT1

� � ð14ÞThe first factor in this equation CnTr

B is the block coefficient CB

raised to the nTr power. The exponent is defined as

nTr ¼ 2:0þ 0:8PCh1

CBð15Þ

where Pch1 was previously defined in equation 9.

Fig. 5 CADETand BNTship lines hull geometry

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The next two factors define the propeller effect on the vesseltrim. The first factor KS

P is the same as the propeller parameter forthe midpoint sinkage and the second factor is the propeller trimparameter KT

P

KTP ¼ 0:15 single propeller

0:20 twin propellers

�ð16Þ

The last group of three factors define the effects of the bulbousbow KT

b, stern transom KTTr, and initial trim KT

T1 on the vessel trim.

The KTb is given by

KTb ¼ 0:1 bulbous bow

0:0 no bulbous bow

�ð17Þ

The KTTr is defined by

KTTr ¼ 0:1

BTr

B

¼ 0:04 stern transom

0:0 no stern transom

8<: ð18Þ

where BTr is the stern transom width and is typically equal to 0.4B,although values as high as 0.7B have sometimes been used.The KT

T1is given by

KTT1 ¼

Tap � Tfp� �Tap þ Tfp� � ð19Þ

where Tap is the static draft at the stern or aft perpendicular andTfp is the static draft at the bow or forward perpendicular.

Finally, the channel effect trim correction parameter PCh2 forboth R and C channels is defined as

PCh2 ¼ 1:0� 5Sh ð20Þ5. The World Association of Waterborne Transport

Infrastructure empirical formulas

In 1997, PIANC (1997) included 11 empirical squat formulas intheir design guidance for deep draft entrance channels. PIANCWG49 is updating this guidance to retain only seven formulas thatare the most appropriate and useful. Five of these squat formulasare evaluated in this article with the main emphasis on a canal (C)configuration. They include those of Huuska (1976), Yoshimura(1986), Romisch (1989), Eryuzlu, Cao, and D’Agnolo (1994), andBarrass (2009). The ICORELS formula was not used in this appli-cation because it was not intended for canal cross-sections and isalso very similar to the Huuska formula. The original Tuck for-mula was also not used because it is represented by the BNTnumerical formulation using potential flow theory.Briggs (2006) programmed these formulas in FORTRAN pro-

grams and Briggs et al. (2010b) summarized and illustrated themwith examples. Although some constraints and limitations forthese formulas are exceeded, they are included in the results forthis article because this seems to be the accepted practice withinthe deep-draft navigation community to relax these constraintswhere reasonable.All of these PIANC formulas give predictions of bow squat Sb,

but only the Romisch method gives predictions for stern squat, Ss,for all channel types. Barrass gives Ss for unrestricted channels andfor canals and restricted channels depending on the value of CB.According to Barrass (2009), the value of CB determines whetherthe maximum squat is at the bow or stern. Barrass assumes that full-

form ships with CB > 0.7 tend to squat by the bow and fine-formships with CB < 0.7 tend to squat by the stern. The CB ¼ 0.7 is an“even keel” situation with maximum squat the same at both bowand stern. Of course, for channel design, one is mainly interested inthe maximum squat and not necessarily whether it is at the bow orstern. With these constraints in mind, the average PIANC valuesconsisted of predictions for all five PIANC formulas for bow squatbut only Barrass and Romisch for stern squat.

5.1. Barrass

The Barrass ship squat formula has evolved and been revised atleast four times. The one in this article (Barrass 2002, 2009) isconsidered the third version for both Sb and Ss. It is a function ofCB, ship speed, Vk, in knots, and channel blockage coefficient, K,and is defined as

KCBV2k

100¼ Sb CB > 0:7

Ss CB � 0:7

�ð21Þ

If CB > 0.7, maximum squat occurs at the bow Sb. If CB � 0.7,it occurs at the stern and is equal to the stern squat, Ss. Barrass’channel coefficient, K, is based on analysis of over 600 laboratoryand prototype measurements for all three channel types. It isdefined as

K ¼ 5:74S0:76 1 � K � 2 ð22ÞThe limits on K are designed so that K ¼ 2 for C (also for R)

channels. The blockage factor S was previously defined. If S >0.25 for C (also for R) channels, the value of K is set to 2 to ensurethe limits required for K.

5.2. Eryuzlu

Eryuzlu, Cao, and D’Agnolo (1994) developed a formula for Sbbased on laboratory experiments. Although it is usually applied toonly unrestricted (U) and restricted (R) channels, it is included inthese comparisons because the Canadian Coast Guard (2001) usesit for ships in the St. Lawrence Seaway, a channel that is verysimilar to the Panama Canal. Therefore, it is included herealthough the CB constraint is technically exceeded. It is defined as

Sb ¼ 0:298h2

T

VsffiffiffiffiffiffigT

p� �2:289 h

T

� ��2:972

Kb ð23Þ

where the factor Kb is a correction factor accounting for relativechannel width according to the ratio of channel width W to shipbeam B.

Kb ¼3:1ffiffiffiffiffiffiffiffiffiffiW=B

p W

B< 9:61

1W

B� 9:61

8>><>>: ð24Þ

5.3. Huuska/Guliev

The next empirical squat formula was developed by Guliev(1971) and Huuska (1976) and is given by

Sb ¼ 2:4CBBT

Lpp

F2nhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� F2nh

p Ks ð25Þ

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The channel width correction factor Ks is defined as

Ks ¼ 7:45s1 þ 0:76 s1 > 0:031:0 s1 � 0:03

�ð26Þ

where the corrected blockage factor s1 ¼ S for C channels. Huuskadefined other values for s1 for the other two channel types, butthey are not presented here because we are only concerned withC channels in this article.

5.4. Romisch

Romisch (1989) developed formulas for both Sb and Ss fromphysical model experiments for a C channel. The Romisch squat isdefined as

Sb; Ss ¼ CVCFKDTT ð27Þ

The factors in this equation are correction factors for shipspeed, CV, ship shape,CF, and squat at critical speed,KDT, defined as

CV ¼ 8Vs

Vcr

� �2 Vs

Vcr� 0:5

� �4

þ 0:0625

" #ð28Þ

CF ¼10BCB

Lpp

� �2

Bow

1:0 Stern

8<: ð29Þ

KDT ¼ 0:155ffiffiffiffiffiffiffiffih=T

pð30Þ

Critical ship speed, Vcr, for a canal is a function of wave celer-ity, C, and a channel shape correction factor, KC, defined as

Vcr ¼ CKC ¼ffiffiffiffiffiffiffiffighm

p2 cos

p

3þ Arc cos 1� Sð Þ

3

� � 1:5ð31Þ

where the mean water depth, hm, is a function of the projectedwidth at the top of the channel, WTop, defined as

hm ¼ AC

WTop¼ AC

W þ 2nhð32Þ

However, because the hm was provided for the Panama Canaldata, that value was used in the Romisch formula.

5.5. Yoshimura

The last squat formula was developed by Yoshimura (1986) andincluded by the Overseas Coastal Area Development Institute ofJapan (2009) as part of Japan’s Design Standard for Fairways inJapan. It was enhanced by Ohtsu et al. (2006) to include predic-tions for C (also for R) channels. It is defined as

Sb ¼ 0:7þ 1:5T

h

� �BCB

Lpp

� �þ 15T

h

BCB

Lpp

� �3" #

Ve

g

2

ð33Þ

where the enhanced ship speed term, Ve, is a function of shipspeed, Vs, in m/s given by

Ve ¼ Vs

1� Sð Þ ð34Þ

6. Validation results

As mentioned previously, only five of the PIANC formulas wereincluded in this study. Although all five can predict bow squat, onlythe Barrass and Romisch formulas were appropriate for stern squatpredictions for canal channels. Therefore, only these two predic-tions were used to calculate the PIANC average for stern squat.Depending on the value of CB, only bow or stern squat predictionswere calculated as dictated by the constraints of each formulation.Again, the PIANC values were used to calculate an average bow(five averages) or stern (two averages) squat prediction at eachlocation for each ship to compare with the measured DGPS values.

6.1. Uncertainty analysis

Each of the parameters in the prediction of ship squat has inher-ent uncertainties. The channel depth, h, is assumed to have no biasor variability because it is a deterministic parameter. Uncertainty inthe static and dynamic drafts, T, comes from the estimation of thedraft at the pier, from the draft marks, and the sinkage and trimestimate, S. According to Kopp and Silver (2004), the error band inthe static draft is assumed to be known within a range of� 1%. Thecritical points of the bow, stern, and bilge are assumed to have anerror band within 4.5% of the actual value at the bow and stern and1.5% of the actual value at the bilge. The sinkage estimate is basedon an analytical or empirical method from experimental results.The uncertainty in the sinkage comes from the scatter of the datain model tests and how well the calculated results fit the model testresults. This gives a variability of the sinkage parameter of approx-imately 1% with no bias. Of course, the ship squat empirical esti-mates are based on many different parameters that are not includedin every formula. Even if included, they are “weighted” differentlyin each formula.

6.2. Error metrics

Statistical and error metrics used to quantify uncertainty in thepredictions are described in this section. The first two metrics arethe means and the standard deviations for the measured and pre-dicted squat data.Two simplistic “goodness-of-fit”measures to characterize the agree-

ment between the model squat predictions and measured DGPSdata are the ratio RS between predicted and measured data and thebias or difference between the two data sets. The RS is defined as

RS ¼ SP; jSM; j

ð35Þ

The bias BS is given by

Bs ¼ SP; j � SM; j ð36Þwhere SP,j is the predicted squat at the bow or stern at locationj and SM,j is the measured squat at location j.Three better error metrics include the root mean square error

(RMSE), Pearson correlation coefficient (R), and the mean abso-lute error (MAE). They are defined as

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(N

j¼1SP; j � SM; j

� �2N

vuuutð37Þ

MARCH 2013 JOURNAL OF SHIP RESEARCH 7

Journal of Ship Research

R ¼ CovðSP; SMÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVar SPð ÞVar SMð Þp

¼(N

j¼1SP; j ��

SP� �

SM; j ��SM

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(N

j¼1SP; j ��

SP� �2s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

(N

j¼1SM; j ��

SM� �2s ð38Þ

MAE ¼(N

j¼1j SP; j � SM; j j

Nð39Þ

where�SP is the mean of the predicted squat,

�SM is the mean of the

measured squat, and N is the number of measurement points.Values of RS > 1.0 indicate overprediction, RS < 1.0 under-

prediction, and RS ¼ 1.0 perfect match. Similarly, values of BS >0.0 indicate overprediction, BS < 0.0 underprediction, and BS ¼0.0 a perfect match. The RMSE and MAE measure the actual

differences between the two data sets. Although unbounded,smaller values indicate better agreement. The R error statisticprovides an indication of the correlation that may exist betweendata sets. It is bounded between –1 (perfect negative correlation)and 1 (perfect positive correlation). It is an indication of linearitybetween the data sets because an uncorrelated value of R ¼ 0 canindicate a nonlinear or random scatter relationship between datasets. Therefore, small values of R indicate that the data sets areuncorrelated, but this does not mean they are unrelated, just notlinearly related.

6.3. Squat comparisons

Figures 6 to 9 show time series comparisons of the BNT,Ankudinov, and PIANC squat predictions to the measured DGPSvalues for the four ships along the Gaillard Cut. The top plotshows the RS for these three predictors at each location. Thebottom plot shows the measured and predicted bow or stern squatfor each ship. A spline smoothing was applied to each of the threepredictors to illustrate the general trend of the data. The degree ofsmoothness was automatically selected using cross validation ofthe data. The entire data set was used in all the statistical andmetric calculations, however.In lieu of error bars on the plots, the statistical and error

metrics are listed in Tables 2 to 5 to improve readability of theplots. Table 2 compares the squat means and standard deviationsfor the measured DGPS and predicted BNT, Ankudinov, andPIANC data. Table 3 lists the minimum, average, and maximumvalues of RS for BNT, Ankudinov, and PIANC squat predictionsfor each ship. Table 4 lists minimum, average, and maximum BS

between measured and predicted bow and stern squat for eachship. Finally, Table 5 lists the RMSE, R, and MAE statistics foreach of the four ships.

6.3.1. Elbe tanker. Figure 6 shows the bow squat for the ElbePanamax tanker. The average water depth ranged between 13.0

Table 2 Mean and standard deviations for measured andpredicted squat data

Ship ID No. Statistic DGPS BNT Ankudinov PIANC

E 1015�S �0.42 �0.27 �0.54 �0.35

s 0.083 0.052 0.073 0.055

G 633�S �1.11 �0.76 �1.33 �1.00

s 0.132 0.045 0.057 0.048

M 649�S �0.84 �0.45 �0.74 �0.74

s 0.307 0.172 0.201 0.318

O 681�S �0.58 �0.45 �0.59 �0.55

s 0.099 0.077 0.058 0.053

Total 2978

�S ¼ Sample mean or average; s ¼ sample standard deviation.

Table 3 Ratios of predicted to measured squat

Ship

ID

No. of Squat

Measurements

BNT Ankudinov PIANC

Minimum

Ratio

Average

Ratio

Maximum

Ratio

Minimum

Ratio

Average

Ratio

Maximum

Ratio

Minimum

Ratio

Average

Ratio

Maximum

Ratio

E 1015 0.45 0.64 0.84 0.96 1.29 1.78 0.60 0.83 1.11

G 633 0.53 0.69 0.90 0.96 1.21 1.61 0.72 0.92 1.21

M 649 0.30 0.54 0.89 0.46 0.91 1.70 0.45 0.89 1.62

O 681 0.54 0.78 1.16 0.78 1.04 1.48 0.70 0.96 1.33

Ratio RS ¼ predicted squat divided by measured squat: 1.0 is perfect match; < 1.0 is underprediction; > 1.0 is overprediction.

Table 4 Bias between measured and predicted squat

Ship ID

No. of squat

measurements

BNT (m) Ankudinov (m) PIANC (m)

Minimum Average Maximum Minimum Average Maximum Minimum Average Maximum

E 1015 �0.28 �0.16 �0.04 �0.02 0.11 0.22 �0.21 �0.08 0.03

G 633 �0.65 �0.35 �0.08 �0.05 0.22 0.48 �0.38 �0.10 0.17

M 649 �1.62 �0.39 �0.05 �1.28 �0.1 0.29 �1.01 �0.10 0.41

O 681 �0.35 �0.13 0.07 �0.17 0.01 0.19 �0.21 �0.04 0.14

Bias BS ¼ predicted squat—measured squat: 0.0 is perfect match; < 0.0 is underprediction; > 0.0 is overprediction.

8 MARCH 2013 JOURNAL OF SHIP RESEARCH

Journal of Ship Research

and 13.2 m with an average static UKC of 1.8 m. Table 2 showsthe mean

�SM ¼ �0:42 m for the DGPS data with a standard

deviation sM ¼ 0.083 m. Comparing the BNT predictions, the�SP ¼ �0:27 m, an average underprediction of 15 cm, with asP ¼ 0.052 m. The RS ranged from 0.5 to 0.8 with an averageunderprediction of 0.6 times the measured bow squat. The BS

ranged from underpredictions of 28 to 4 cm with an averageunderprediction of 16 cm. The RMSEBNT ¼ 0.17 m and MAEBNT ¼0.16 m were relatively small, showing a strong correlation ofRBNT ¼ 0.75. For the Ankudinov predictions, the

�SP ¼ �0:54 m,

an average overprediction of 12 cm, with a sP ¼ 0.073 m. The RS

ranged from 1.0 to overpredictions up to 1.8 with an average of1.3 times the measured squat. The BS ranged from underpredictionof 2 cm to overpredictions of 22 cm with an average overpredic-tion of 11 cm. Again, the RMSEAnk¼ 0.13 m andMAEAnk¼ 0.12 mwere relatively small, showing a strong correlation of RAnk ¼ 0.77.Finally, for the PIANC predictions, the

�SP ¼ �0:35 m, an average

underprediction of 7 cm with a sP ¼ 0.055 m. The RS ranged from

0.6 to 1.1 with an average underprediction of 0.8. The BS rangedfrom an underprediction of 21 cm to an overprediction of 3 cmwith an average underprediction of 8 cm. The RMSEPIANC ¼0.09 m and MAEPIANC ¼ 0.08 m were very small, again showinga strong correlation of RPIANC ¼ 0.76.In general, the

�SP and the BS values tended to confirm each

other. The BNT predictions were smaller than measured valuesbut followed the same trends and were closer to the PIANCpredictions. The Ankudinov formula overpredicted by 30% and12 cm, whereas the PIANC underpredicted by 20% and 9 cm. ThePIANC had the best error metrics followed by Ankudinov andBNT. The Ankudinov and BNT predictions were approximately50% and 100% larger than the PIANC values, although relativelysmall maximum differences of 8 cm.

6.3.2. Global Challenger bulk carrier. Figure 7 shows the bowsquat for the Global Challenger Panamax bulk carrier. This shipwas trimmed 12 cm by the stern (i.e., deeper draft at the stern).The water depth ranged between 12.9 and 13.2 m with an averagestatic UKC of 1.4 m. Table 2 shows the mean

�SM ¼ �1:11 m for

the DGPS data (largest of the ships) with a standard deviationsM ¼ 0.132 m. For the BNT predictions, the

�SP ¼ �0:76 m, an

average underprediction of 35 cm with a sP ¼ 0.045 m. The RS

ranged from 0.5 to 0.9 with an average underprediction of 0.7times the measured bow squat. The BS ranged from anunderprediction of 65 cm to an overprediction of 8 cm with anaverage underprediction of 35 cm. The RMSEBNT ¼ 0.37 m andMAEBNT ¼ 0.35 m were twice as large as the Elbe values, show-ing a weak linear correlation of only RBNT ¼ 0.41. For theAnkudinov predictions, the

�SP ¼ �1:33 m, an average overpre-

diction of 22 cm with a sP ¼ 0.057 m. The RS varied from 1.0to 1.6 times the measured squat with an average overpredictionof 1.2. The BS ranged from underpredictions of 5 cm tooverpredictions of 48 cm with an average overprediction of22 cm. The RMSEAnk ¼ 0.25 m and MAEAnk ¼ 0.22 m weretwice as large as the Elbe values, showing the weakest correlationof only RAnk ¼ 0.37 for all ships. Finally, for the PIANC

Table 5 Error statistics between measured and predicted squat

Ship ID No. Statistic Units BNT Ankudinov PIANC

E 1015 RMSE m 0.17 0.13 0.09

R — 0.75 0.77 0.76

MAE m 0.16 0.12 0.08

G 633 RMSE m 0.37 0.25 0.16

R — 0.41 0.37 0.40

MAE m 0.35 0.22 0.13

M 649 RMSE m 0.44 0.21 0.21

R — 0.84 0.82 0.83

MAE m 0.39 0.15 0.16

O 681 RMSE m 0.15 0.07 0.08

R — 0.70 0.75 0.74

MAE m 0.13 0.06 0.06

Total 2978

RMSE ¼ root mean square error; R ¼ Pearson correlation coefficient;MAE ¼ mean absolute error.

Fig. 6 Panamax Elbe tanker values for (a) ratio RS and (b) measured and predicted bow squat Sb. Ship northbound, sailing from right to left.

DGPS ¼ black open circle; BNT ¼ blue solid line; Ankudinov ¼ red dash line; PIANC ¼ green dot–dash. Spline fit for predictors

MARCH 2013 JOURNAL OF SHIP RESEARCH 9

Journal of Ship Research

predictions, the�SP ¼ �1:00 m, an average underprediction of

11 cm with a sP ¼ 0.048 m. The RS were closer to the measuredbow squat, especially above the location at station 1850. Theyranged from 0.7 to 1.2 times the measured squat with an averageunderprediction of 0.9. The BS ranged from an underprediction of38 cm to an overprediction of 17 cm with an average under-prediction of 10 cm. The RMSEPIANC ¼ 0.16 m and MAEPIANC ¼0.13 m were relatively small, again showing a very weak correla-tion of only RPIANC ¼ 0.40.

Again, the SP and the BS values tended to confirm each other.The BNT underpredicted the measured squat but followed thesame trends as the two other predictors. In general, the Ankudinovformula overpredicted and BNT and PIANC underpredicted bowsquat. The PIANC formulas overpredicted the measured squat fora short section from location 1920 to 1970. Thus, the Ankudinovformula averaged 20% and 21 cm overprediction and the PIANC

10% and 9 cm underprediction. For design purposes, overpredic-tion is more conservative and potentially safer. The error metricsfor the Global Challenge were twice as large as the Elbe with theworst correlation of all the ships. However, the worst RMSEBNT ¼0.37 m for the BNT predictions was not a huge overprediction.

6.3.3. Majestic Maersk containership. As mentioned previouslyin Section 6, according to Barrass (2009), maximum squat willoccur at the bow for a ship with a CB > 0.7 and at the stern for onewith a CB < 0.7. Therefore, bow squat was reported for the firsttwo ships previously discussed. The next two ships will illustratesquat by the stern because they have a CB < 0.7. Thus, the averagePIANC value used in the plots is based on the average of only theBarrass and Romisch predictions.Figure 8 illustrates the stern squat for the Majestic Maersk

Panamax containership. The water depth ranged between 12.9

Fig. 7 Panamax Global Challenger bulk carrier values for (a) ratio RS and (b) measured and predicted bow squat Sb. Ship northbound, sailing from

right to left. DGPS ¼ black open circle; BNT ¼ blue solid line; Ankudinov ¼ red dash line; PIANC ¼ green dot–dash. Spline fit for predictors

Fig. 8 Panamax Majestic Maersk containership values for (a) ratio RS and (b) measured and predicted stern squat Ss. Ship northbound, sailing

from right to left. DGPS ¼ black open circle; BNT ¼ blue solid line; Ankudinov ¼ red dash line; PIANC ¼ green dot–dash. Spline fit for predictors

10 MARCH 2013 JOURNAL OF SHIP RESEARCH

Journal of Ship Research

and 13.2 m with an average static UKC of 1.3 m. Table 2 showsthat the mean

�SM ¼ �0:84 m for the DGPS data (largest of the

ships) with a standard deviation sM ¼ 0.307 m (largest of thefour ships). For the BNT predictions, the

�SP ¼ �0:45 m, an

average underprediction of 39 cm, with a sP ¼ 0.172 m. The RS

ranged from 0.3 to 0.9 with an average underprediction slightlylarger than half of the measured squat. The BS ranged from anunderprediction of 5 cm to 1.6 m (station 1940.73) with anaverage underprediction of 39 cm. The extreme underpredictionat station 1940.73 is probably an error in the measured databecause they all appear to look unusually large in this sectionof the canal from 1940 to 1950. The RMSEBNT ¼ 0.44 m andMAEBNT ¼ 0.39 m were largest of the four ships, but hadthe strongest correlation of RBNT ¼ 0.84. For the Ankudinovpredictions, the

�SP ¼ �0:74 m, an average overprediction of

10 cm, with a sP ¼ 0.201 m. The RS ranged from 0.5 to 1.7 timesthe measured stern squat with an average underprediction of0.9. The BS ranged from a worst underprediction of 1.3 m (sta-tion location 1940) to an overprediction of 29 cm with an aver-age underprediction of 10 cm. The RMSEAnk ¼ 0.21 m andMAEAnk ¼ 0.15 m were relatively small and second only tothe Elbe with the strongest correlation of RAnk ¼ 0.82. Finally,for the PIANC predictions, the

�SP ¼ �0:74 m, an average

underprediction of 10 cm, with a sP ¼ 0.318 m. The RS rangedfrom 0.4 to 1.6 with an average underprediction of 0.9 timesthe measured squat. The BS ranged from an underpredictionof 1.0 m to overprediction of 41 cm with an average under-prediction of 10 cm. The RMSEPIANC ¼ 0.21 m and MAEPIANC ¼0.16 m were similar to the Ankudinov predictions with the stron-gest correlation of RPIANC ¼ 0.83.The

�SP and the BS values confirmed one another because

they were identical. Although the BNT model underpredictedthe measured values, it showed the same trends as the mea-sured data and the other predictors. From station 1670 to 1850,both Ankudinov and PIANC tended to underpredict stern squatwith Ankudinov predictions slightly better. Around station1880 to 1960, the PIANC formula overpredicted stern squat.In general, both Ankudinov and PIANC tended to underpredict

stern squat by 10% with minimum and maximum predictionsabout the same.

6.3.4. OOCL Fair containership. Figure 9 shows the stern squatfor the OOCL Fair containership because CB < 0.7. This ship hadthe most trim with a value of 0.8 m by the stern. The water depthranged between 13.0 and 13.2 m with an average static UKC of2.9 m. Table 2 shows that the mean

�SM ¼ �0:58 m for the DGPS

data (largest of the ships) with a standard deviation sM ¼ 0.099 m.For the BNT predictions, the

�SP ¼ �0:45 m, an average

underprediction of 13 cm, with a sP ¼ 0.077 m. The RS rangedfrom 0.5 to 1.2 with an average underprediction of 0.8 timesthe measured squat. The BS ranged from underpredictions of35 to 7 cm with an average underprediction of 13 cm. TheRMSEBNT ¼ 0.15 m and MAEBNT ¼ 0.13 m were the smallest ofthe four ships with a strong linear correlation of RBNT ¼ 0.70. Forthe Ankudinov predictions, the

�SP ¼ �0:59 m, a nearly identical

value that overpredicts by 1 cm with a sP ¼ 0.058 m. The RS

ranged from 0.8 to 1.5 times the measured stern squat with anaverage ratio of 1.0 (near exact match with measured data). TheBS ranged from underprediction of 17 cm and overpredictionsof 19 cm with an average of 1 cm (near exact match). TheRMSEAnk ¼ 0.07 m and MAEAnk ¼ 0.06 m were the smallest ofthe four ships with a strong linear correlation of RAnk ¼ 0.75. TheAnkudinov predictions with the smallest RMSE and MAE errorswere the “best fit” of the three predictors for the OOCL Fair.Finally, for the PIANC predictions, the

�SP ¼ �0:55 m, an average

underprediction of 3 cm, with a sP ¼ 0.053 m. The RS varied fromunderpredictions of 0.7 to overpredictions of 1.3 with an averageof 1.0 (near exact match) times the measured stern squat. The BS

ranged from an underprediction of 21 cm to an overprediction of14 cm with an average underprediction of 4 cm. The RMSEPIANC ¼0.08 m and MAEPIANC ¼ 0.06 m were similar to the Ankudinovpredictions with a strong linear correlation of RPIANC ¼ 0.74.

Again, the�SP and the BS values were nearly identical. The BNT

underpredicted the measured data, but not by as much as some ofthe other ships. The Ankudinov predictions were about the sameas the PIANC predictions in this case and both excellent.

Fig. 9 OOCL Fair containership values for (a) ratio RS and (b) measured and predicted stern squat Ss. Ship northbound, sailing from right to left.

DGPS ¼ black open circle; BNT ¼ blue solid line; Ankudinov ¼ red dash line; PIANC ¼ green dot–dash. Spline fit for predictors

MARCH 2013 JOURNAL OF SHIP RESEARCH 11

Journal of Ship Research

6.4. Discussion

On average for the ships in this Panama Canal data set, theBNT underpredicts squat by 30% to 35% for the bow and 20%to 45% for stern squat. The Ankudinov formula overpredictssquat by 25% for the bow and underpredicts by 5% for thestern. The PIANC underpredicts squat by 15% for both bowand stern. All of the 2978 data points were used in the compar-isons although there are a lot of turns or bends in this section ofcanal. Ships experience acceleration, deceleration, and rollwhile turning, which affect squat but is not accounted for inthese squat predictors.Portions of the Majestic Maersk measurements look to be

inordinately large from station 1940 to 1950. According toDr. Daggett, who participated in field measurements of the Majes-tic Maersk, they conducted an acceleration test in this rangestarting from a “dead in the water” condition to “full ahead” tosee is if such an acceleration would cause extreme squat at thestern as the ship’s wheel dug into the water. They observed asteadily increasing heel to starboard, which affected both shipsquat and UKC. The ship heeled over three to five times and rolledup to a 3� heel. He also noted that there was quite a variation inship speed for this ship throughout the measurements. Theseunusual ship responses would explain some of the observed dis-crepancies between measurements and predictions.These comparisons indicate that the Ankudinov formulas are

conservative in most instances because they tend to overpredictship squat. The BNT predictions are generally lower than themeasured values. Possible reasons for the smaller BNT predic-tions might be that the actual ship lines were not used as a resultof proprietary issues. The use of generic ship lines, althoughappropriate, can misrepresent the water line beam and sectionalarea curves of the ships during transit. A sensitivity study by Kopp(2011) showed that there can be a 3% to 8% variation in predictedsquat as a result of 10% variations in fore and aft sectional areas.BNT has shown better agreement with US Navy projects whenactual ship lines and measured model-scale data are available forcomparison. As previously mentioned, Briggs et al. (2010a) foundgood agreement between BNT and PIANC predictions for a rangeof ship and channel types. Additionally, the BNT results havebeen found to be comparable to those produced by more expen-sive higher-order computational fluid dynamics predictions.Therefore, variations in the generic ship lines that are most repre-sentative of the ships in this study could have a significant effecton the predicted squat. It should be noted that CADET is notrestricted to using the BNT model results because the user canalways import other specific or model test squat data. The PIANCpredictions are based on averages of all five of the PIANC formu-las for bow squat but only two (Barrass and Romisch) for sternsquat. In some instances, one or more of the PIANC formulasmight match measured data much better than the averages.

7. Conclusion

This article has compared BNT, Ankudinov, and PIANC shipsquat predictions with DGPS measurements of four ships in theGaillard Cut section of the Panama Canal. These ships included aPanamax containership, Panamax bulk carrier, Panamax tanker,and a containership. In general, the BNT underpredicted bow and

stern squat by factors of 0.67 and 0.66, respectively. TheAnkudinov formulas overpredicted measured bow squat by afactor of 1.25 and underpredicted stern squat by a factor of0.98. PIANC underpredicted bow and stern squat by 0.93 and0.88, respectively. Thus, the BNT predictions were generallysmaller than the measurements but showed the same trends asthe other predictors. The Ankudinov predictions are slightlylarger than the PIANC predictions, although the Ankudinov pre-dictions match canal channel types like the Panama Canal betterthan PIANC. Thus, all three predictors appear to give reasonablepredictions of ship squat and can be used with confidence indeep draft channel design.

Acknowledgments

We acknowledge the Headquarters, US Army Corps of Engi-neers, Institute for Water Resources (IWR), the Naval SurfaceWarfare Center, Carderock Division, and TRANSAS for authoriz-ing publication of this article. It was prepared as part of theImproved Ship Simulation work unit in the Navigation SystemsResearch Program (CHL) and the IWR NETS program. This arti-cle is dedicated to Dr. Vladimir Ankudinov, who died March2012. We acknowledge the assistance of the CHL PrototypeMeasurements Branch and Larry Daggett, Waterway SimulationTechnology, for providing channel, ship, and DGPS data. We alsothank the reviewers for their useful suggestions.

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