Precise Multibeam Acoustic Bathymetry

This article was downloaded by: [Massachusetts Institute of Technology]On: 25 November 2014, At: 23:14Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

Marine GeodesyPublication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/umgd20

Precise Multibeam AcousticBathymetryXueyi Geng, Adam ZielinskiPublished online: 29 Oct 2010.

To cite this article: Xueyi Geng, Adam Zielinski (1999) Precise Multibeam AcousticBathymetry, Marine Geodesy, 22:3, 157-167, DOI: 10.1080/014904199273434

To link to this article: http://dx.doi.org/10.1080/014904199273434

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified withprimary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone is

http://www.tandfonline.com/loi/umgd20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/014904199273434

http://dx.doi.org/10.1080/014904199273434

expressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014

http://www.tandfonline.com/page/terms-and-conditions

Precise Multibeam Acoustic Bathymetry

XUEYI GENGADAM ZIELINSKI

Department of Electrical and Computer EngineeringUniversity of VictoriaVictoria, British Columbia, Canada

The maximum error in ocean depth measurement as speci� edby the International Hydro-graphic Organization is 1% for depth greater than 30 m. Current acoustic multibeambathymetric systems used for depth measurement are subject to errors from varioussources which may signi� cantly exceed this limit. The lack of sound speed pro� lesmay be one signi� cant source of error. Because of the limited ability of sound speedpro� le measurement, depth values are usually estimated using an assumed pro� le. Ifactual sound speed pro� les are known, depth estimate errors can be corrected usingray-tracing methods. For depth measurements, the calculation of the location at whicha sound pulse impinges on the sea bottom varies with the variation of the sound speedpro� le. We demonstrate that this location is almost unchanged for a family of soundspeed pro� les with the same surface value and the same area under them. Based on thisobservation, we can construct a simple constant-gradient equivalent sound speed pro� leto correct errors. Compared with ray-tracing methods, the equivalent sound speed pro-� le method is more ef� cient. If a vertical depth is known (or independently measured),then depth correction for a multibeam system can be accomplished without knowledgeof the actual sound speed pro� le. This leads to a new type of precise acoustic multibeambathymetric system.

Keywords bottom mapping, multibeam bathymetry, multibeam systems, precisebathymetry, swath sonar

Acoustic multibeam and side-scan bathymetric systems, as shown in Figure 1, are frequentlyused for ocean bottom depth measurements. These ef� cient systems have a wide swathcoverage, up to 150±. Usually, the depth z and horizontal range x are calculated using atriangulation method (Russel-Cargill, 1982), as shown in Figure 2 and given in Eqs. (1) and(2):

z = z0 +c0Tp

2cos q (1)

x = x0 +c0Tp

2sin q (2)

where (z0, x0) are the coordinates of the transducer array, c0 is the sound speed at theposition (z0, x0), and Tp is the sound pulse two-way propagation time in the direction ofinclination angle q . The accuracies of the depth and horizontal range measurements dependon the accuracies of Tp, q , and sound speed c0. In Eqs. (1) and (2), c0 is usually assumed tobe constant.

Received 5 March 1999; accepted 11 May 1999.This research work was supported by a grant from the Natural Sciences and Engineering Research Council

of Canada.Address correspondenc e to Dr. Adam Zielinski, Department of Electrical and Computer Engineering, Uni-

versity of Victoria, P.O. Box 3055 STN CSC, Victoria, B.C., Canada V8W 3P6. E-mail: [email protected]

Marine Geodesy, 22:157–167, 1999Copyright C° 1999 Taylor & Francis

0149-0419 /99 $12.00 + .00 157

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014

158 X. Geng and A. Zielinski

FIGURE 1 Multibeam acoustic bathymetry.

However, the sound speed in the ocean is not constant, but a function of space and time.As a result, the sound pulse propagation path is not a straight line. The triangulation methodas given in Eqs. (1) and (2) can result in signi� cant errors in the measurement results (Geng,1997).

Generally, the ray path and propagation time can be described using the followingequations:

T = *zB

z0

1

c(z)Ï 1 ¡ l 2c2(z)dz (3)

xB ¡ x0 = *zB

z0

l c(z)

Ï 1 ¡ l 2c2(z)dz (4)

l =cos a 0

c0(5)

where sound speed pro� le c(z) is assumed to be a function of depth z only, a 0 is the startinggrazing angle, c0 is the sound speed at the starting position (z0, x0), and (zB , xB ) is theposition at which the sound pulse impinges on the sea bottom.

No general solutions are known for Eqs. (3) and (4). If the sound speed pro� le c(z) isknown, the depth can be estimated using ray-tracing methods. In this article, we demonstratethat, for the same propagation time, the position at which a sound pulse impinges on thesea � oor is almost the same for a family of sound speed pro� les with the same surfacevalue c0 and the same area under them, which is de� ned as an integral of c(z) with respect

FIGURE 2 The triangulation depth measurement method.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014

Precise Multibeam Acoustic Bathymetry 159

to depth z. Based on this observation, a new method for depth estimate error correction isdeveloped. This new method allows correction of depth estimate error based on the relativearea difference under two sound speed pro� les. It is fast and easy to use compared with ray-tracing methods. Since a family of sound speed pro� les can result in almost the same depthestimate, each actual sound speed pro� le can be replaced by a simple, equivalent soundspeed pro� le which is one of its family members. The method, called the equivalent soundspeed pro� le method, is developed for depth estimates for acoustic bathymetric systems.We will later show that the equivalent sound speed pro� le can be a linear function of depthz and the actual sound speed pro� le measurement can be replaced by a reference depthmeasurement. Therefore, if a reference depth is given, precise depth measurements can beachieved without measuring the actual sound speed pro� le. This new method leads to a newtype of acoustic bathymetric system.

Depth Error Correction Algorithm

For depth measurements, the location (zB , xB ) at which a sound pulse impinges on theocean bottom is a function of sound speed pro� les. Assume the sound speed c(t , z) is afunction of ocean depth z and time t . Two sound speed pro� les c(t , z) and c(t0, z) whichare measured at time t and t0 with respect to the ocean depth z have the same surface valuec0 as shown in Figure 3. The area difference under the two sound speed pro� les is

D S = *dS = *zB

z0

[c(t0, z) ¡ c(t , z)] dz (6)

The integrand dS in Eq. (6) is a signed variable, that is, for all z,

if [c(t0, z) ¡ c(t , z)] > 0, dS > 0

and

if [c(t0, z) ¡ c(t , z)] < 0, dS < 0

The relative area difference is de� ned as

e S =D S

*z B

z0c(t0, z) dz

=D S

S(7)

FIGURE 3 Area difference under two sound speed pro� les.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


FIGURE 4 Sound speed pro� les, where the solid lines represent actual pro� les.

It was found that the depth estimate difference related to these two sound speed pro� lesis almost a function of only the relative area difference e S . The difference is zero whene S = 0. Therefore, a family of sound speed pro� les with the same sound speed c0 and thesame area under them, that is, where e S = 0, will result in almost the same depth estimatein an acoustic bathymetric system.

We will demonstrate this interesting property using some simpli� ed sound speed pro� letypes as shown in Figures 4a, 4b, and 4c. An actual sound speed pro� le in the ocean can beapproximately represented using one of these sound speed pro� le types.

In Figure 4a, the actual sound speed pro� le is represented by c0–cB with a constantgradient g 6= 0, and the assumed sound speed pro� le is represented by c0–cA, a constantspeed. We will show that the depth estimate difference between these two sound speedpro� les is a function of the relative area difference alone. With the help of Figure 4a, we have

S = c0(zB ¡ z0) = c0 D zB (8)

where D zB = zB ¡ z0. The area difference under these two sound speed pro� les is

D S =1

2(zB ¡ z0)(cB ¡ c0) =

1

2D zB (cB ¡ c0) (9)

and the relative area difference is

e S =D S

S=

cB ¡ c0

2c0(10)

If a sound pulse is transmitted from depth z0 and horizontal position x0 to the sea bottomwith starting grazing angle a 0, the measured one-way propagation time T is determined bythe actual sound speed pro� le c0–cB . However, if the assumed sound speed pro� le c0–cA isused to estimate the depth, it can be calculated using a triangulation method as

z 0B = z0 + T c0 sin a 0 (11)

x 0B = x0 + T c0 cos a 0 (12)

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


The relative errors, or error-correcting functions, are de� ned as

e z = fz( e S , a 0) =z 0

B ¡ zB

zB ¡ z0= (c0T

D zBsin a 0)¡ 1 (13)

e x = fx ( e S , a 0) =x 0

B ¡ xB

xB ¡ x0= (c0T

D xBcos a 0)¡ 1 (14)

where D xB = xB ¡ x0.Now, let us assume that the actual sound speed pro� le has a constant gradient in

Figure 4a,

g =cB ¡ c0

D zB(15)

The actual depth and horizontal range (zB , xB ) can be obtained in closed form (Urick, 1983)as

zB = z0 ¡c0

g cos a 0(cos a 0 ¡ cos a B ) (16)

xB = x0 ¡c0

g cos a 0(sin a B ¡ sin a 0) (17)

where a 0 is the starting grazing angle and a B is the grazing angle of the ray path impingingon the sea bottom. The one-way propagation time can be obtained in closed form fromEq. (3) (Geng, 1997) as

T =1

gln[cB(1 + sin a 0)

c0(1 + sin a B )] (18)

By Snell’s law (Urick, 1983),

cos a B

cB=

cos a 0

c0

thus

cos a B =cB

c0cos a 0 = (1 + 2 e S) cos a 0 (19)

sin a B = Ï 1 ¡ [(1 + 2 e S) cos a 0]2 (20)

Substituting Eqs. (16)–(20) into Eqs. (13) and (14), the error-correcting functions for thedepth and horizontal range can be obtained as

fz( e S , a 0) =sin a 0

2 e Sln[(1 + 2 e S)

(1 + sin a 0)

(1 + Ï 1 ¡ [(1 + 2 e S ) cos a 0]2)]¡ 1 (21)

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


fx ( e S , a 0) =(cos a 0 /2 e S) ln [(1+ 2 e S){(1+ sin a 0)/(1+ Ï 1 ¡ [(1+ 2 e S) cos a 0]2 )}]

[{Ï 1 ¡ [(1 + 2 e S) cos a 0]2 ¡ sin a 0 / cos a 0 ¡ (1 + 2e S) cos a 0}]¡ 1

(22)These errors are a function of e S alone for a given starting grazing angle a 0.

The error-correcting functions obtained using sound speedpro� les depicted in Figure 4bare slightly dependent on the parameter zm , where zm is the depth at which the actual soundspeed pro� le gradient changes. As shown in Figures 5a and 5b, when zm changes from itsminimum to its maximum. Typically, e S can be of the order of 1%. Thus the depth estimateerrors caused by this kind of dependence are much less than 1%. Numerical results shown

(a)

(b)

FIGURE 5 (a) Properties of depth error correction function fz(e S , a 0) for sound speedpro� le of Figure 4b. (b) Properties of horizontal range error correction function fx (e S , a 0)for sound speed pro� le of Figure 4b.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


(a)

(b)

FIGURE 6 (a) Comparison of depth estimate error correction functions between differentsound speed pro� le types. (b) Comparison of horizontal range estimate error correctionfunctions between different sound speed pro� le types.

in Figures 6a and 6b show pro� le models of Figures 4a and 4b give almost the same results(Geng, 1997).

For the sound speed pro� le type Figure 4c, numerical results show that when the twoarea differences D S1 and D S2 have the same value and different signs, the depth estimatedifference between using this sound speed pro� le and using the zero-gradient sound speedpro� le c0–cA is almost zero (Geng, 1997). Therefore, the error-correcting functions Eqs. (21)and (22) derived from the sound speed pro� le type Figure 4a can be used for other soundspeed pro� le types.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


For depth measurements using acoustic bathymetric systems, the depth estimated usingthe triangulation method can be corrected using the above error-correction functions whenthe sound speed pro� le is known and the relative area difference e S is calculated. Thecorrected depth and horizontal range estimates can be obtained as

zB = z0 +(z 0

B ¡ z0)

1 + fz( e S , a 0)(23)

xB = x0 +(x 0

B ¡ x0)

1 + fx ( e S , a 0)(24)

This depth error-correction method is based on the area difference under the two soundspeed pro� les. Compared with ray-tracing methods, this error-correcting method is easy touse and fast for bathymetry data processing and has similar accuracy for depth estimation.

Equivalent Sound Speed Pro� le Method

It follows from the above discussion that a certain family of sound speed pro� les results inalmost the same depth estimate. Therefore, for depth estimation an actual sound speed pro� lecan be replaced by an equivalent sound speed pro� le of its family member as illustratedin Figure 7. The actual sound speed pro� le may be a complicated curve; however, theequivalent sound speed pro� le can be a simple straight line segment. This equivalent soundspeed pro� le will simplify the depth estimate calculation and give almost the same resultsaccording to Eqs. (21) and (22) for the case of e S = 0.

If we use the zero-gradient sound speed pro� le c0–cA instead of the actual one, thedepth estimate error caused by this substitution is, for a ray with starting grazing angle a 0,

e z = fz( e S , a 0) (25)

where e S is the relative area difference between the actual sound speed pro� le and theassumed zero-gradient pro� le c0–cA.

Let us assume that a reference depth zB0 is known for a given beam angle (can bevertical) in an acoustic bathymetric system. This reference depth may be measured using anonacoustic method, such as a single laser beam. Then, we can calculate the depth as z 0

B0

FIGURE 7 Using an equivalent pro� le to represent an actual pro� le.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


using the triangulation method for the same given beam angle in the acoustic bathymetricsystem. Thus, the depth estimate error e z for this reference depth can be obtained as

e z =z 0

B0 ¡ z B0

zB0(26)

By substituting Eq. (26) into Eq. (25), the relative area difference between the actual soundspeed pro� le and the zero-gradient sound speed pro� le c0–cA can be obtained by solvingEq. (25). With the help of Figure 4a, the sound speed cB of the equivalent linear soundspeed pro� le on the bottom can be calculated as

cB = c0(1 + 2 e S) (27)

The constant gradient of this equivalent linear sound speed pro� le can be calculated as

geq =2 e Sc0

zB0 ¡ z0(28)

Therefore, this equivalent linear sound speed pro� le may be used to determine ocean depthin the acoustic bathymetric system, with closed-form solutions of Eqs. (3) and (4).

We assume that a sound pulse two-way propagation time is measured as k k for beamangle q k . The ray path starting grazing angle at this beam is a 0(k) = p /2 ¡ q k . Then, thedepth related to this beam angle can be estimated in closed form (Geng, 1997) as

z 0B(k)

= z0 + Req(cos[ a 0(k)] +2geq Req f 1 + sin[ a 0(k)] g c0 exp[( k k /2)geq]

(geq Req f 1 + sin[ a 0(k)] g )2 + f c0 exp[( k k / 2)geq] g 2)(29)

where

Req = ¡c0

geq cos[ a 0(k)]

is the radius of the corresponding equivalent ray path arc. A similar result can be obtainedfor the horizontal range estimate (Geng, 1997).

This new method for obtaining a depth estimate requires only the sound speed c0 atthe sea surface and one reference depth zB0 at the ocean bottom. It is not required that theactual sound speed pro� le be measured. It replaces the sound speed pro� le measurementwith a reference depth measurement. It can signi� cantly improve the depth measurementaccuracy for an acoustic bathymetric system. The relative depth estimate error can be muchless than 1%, as shown in Figure 9.

Application

We use the sound speed pro� les in Figure 8 as examples to estimate the depth for a multibeambathymetric system. It is assumed that the depth for the vertical beam is known, which maybe measured using a single-beam laser in the vertical direction in shallow water. Using this

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


FIGURE 8 Example of sound speed pro� les.

depth as a reference, an equivalent linear sound speed pro� le can be built using the abovemethod. Then, this equivalent sound speed pro� le is used to estimate the depths for otherbeam angles in an acoustic bathymetric system. The depth estimate errors caused by usingthe equivalent sound speed pro� le are shown in Figure 9, where the maximum relativeerror is less than 0.1% for the assumed sound speed pro� les. For comparison, the depthestimate errors caused by the triangulation method using a zero-gradient sound speed pro� le

FIGURE 9 Relative depth measurement errors caused by using the equivalent sound speedpro� le method. SSP1: ¤ ¤ ¤ ¤ , SSP2: + + + +, SSP3: o o o o, SSP4: ——, SSP5: £ £ £ £and SSP6: ¢ ¢ ¢ . . .

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014


FIGURE 10 Relative depth measurement errors caused by the triangulation method. SSP1:¤ ¤ ¤ ¤ , SSP2: + + + +, SSP3: o o o o, SSP4: ——, SSP5: £ £ £ £ and SSP6: ¢ ¢ ¢ . . .

assumption are shown in Figure 10. Notice that the maximum relative error can be as muchas 10%, even though the sound speed c0 is measured each time. Therefore, this equivalentsound speed pro� le method can signi� cantly improve depth measurement accuracy for abathymetric system without the necessity of sound speed pro� le measurements.

Conclusion

In this article, we demonstrate that for ocean depth measurement using acoustic bathymetricsystems, the position at which a sound pulse impinges on the sea � oor is almost the same for afamily of sound speed pro� les. Based on this property, two new methods for precise acousticbathymetry are developed. The equivalent sound speed pro� le depth estimate method usesone reference depth to build an equivalent pro� le instead of an actual sound speed pro� lemeasurement. Therefore, it provides a new method for acoustic bathymetric system designsuch that precise depth measurement results can be obtained without sound speed pro� lemeasurement.

References

Geng, X. 1997. Precise acoustic bathymetry. Master’s thesis, Department of Electrical and ComputerEngineering, University of Victoria, Victoria, B.C.

Russell-Cargill, W. G. A. 1982. Recent developments in side scan sonar techniques—with key articlesby B. W. Flemming, M. Klein, and P. N. Denbigh. Central Acoustics Laboratory, University ofCape Town, Cape Town, South Africa.

Urick, R. 1983. Principles of underwater sound for engineers. New York: McGraw-Hill.

Dow

nloa

ded

by [

Mas

sach

uset

ts I

nstit

ute

of T

echn

olog

y] a

t 23:

14 2

5 N

ovem

ber

2014

Documents

Precise Multibeam Acoustic Bathymetry